STOCHASTIC RESONANCE IN THALAMIC NEURONS AND RESONANT NEURON MODELS

by

STEFAN REINKER

Diplom-Mathematiker, Westf¨alische Wilhelms-Universit¨atM¨unster, 1997

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACULTY OF GRADUATE STUDIES

Department of Mathematics Institute of Applied Mathematics

We accept this thesis as conforming to the required standard

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THE UNIVERSITY OF BRITISH COLUMBIA

January 2004

c Stefan Reinker, 2004 Abstract

Neurons of the thalamus are major participants in gating sensory information for relay to the neocortex. Thalamic neurons are crucially involved in rhythmogenesis which determines the sleep/wake cycle. These roles require critical involvement of a T-type calcium current, conferring a frequency preference in response to subthreshold signals. We examine the interactions of this membrane resonance and noise using whole-cell patch clamp recordings in thalamocortical and reticular neurons of rat brain slices. We perform Monte-Carlo simulations and mathematical analysis using Hodgkin-Huxley-type and polynomial models of resonant neurons. We demonstrate stochastic resonance (SR) as maximal coherence between the input and stochastic output at intermediate noise levels. SR is measured by determining the signal-to-noise ratio under sine wave inputs, and from the reliability of detection measure under α-function inputs. In the experiments and neuron models with T-current, we demonstrate subthreshold resonance at 2-3 Hz, as well as noise dependent frequency dependence of SR for sine wave inputs. The simpler Hindmarsh-Rose model has a similar SR. This model also shows improved detection when the delay of consecutive EPSPs matches the preferred frequency. We show that the preferred frequency of the subthreshold and stochastic resonances depends on the time scale of the slow variable. The stochastic frequency preference arises from modulation of the firing probability of the fast subsystem. We develop a simple linear integrate-and-fire model with subthreshold resonance, which retains the main features of the more complicated models. An analytical solution of the stochastic equations shows that the eigenvalues determine frequency preferences in subthreshold resonance and stochastic resonance. SR can occur even with only noise. This autonomous SR depends on the resonance in our experiments and models. We demonstrate that preferred stochastic firing in the single neuron model translates into syn- chronized behaviour in a noisy network of resonant neuron models. With inhibitory synaptic coupling, noise can extend the parameter range of oscillations. With excitatory synaptic cou- pling, noise produces synchronized oscillations of the quiescent deterministic network. We speculate that combined subthreshold membrane resonance and stochastic resonance have physiological utility in coupling synaptic activity to preferred firing frequency, and in network synchronization under noise.

ii Table of Contents

Abstract ii

Table of Contents iii

List of Figures vi

Acknowledgements and Dedication ix

Chapter 1. Introduction 1 1.1 Stochastic resonance ...... 2 1.1.1 SR in experiments ...... 3 1.1.2 SR in models of neurons ...... 4 1.2 Physiology of thalamic neurons ...... 5 1.3 Noise and information processing in neurons ...... 7 1.4 Neuronal networks ...... 10 1.5 Aims, scope, and organization ...... 12

Chapter 2. Methods 15 2.1 Experimental methods ...... 15 2.1.1 Slice preparation ...... 15 2.1.2 Recording procedures ...... 15 2.1.3 Input signals ...... 17 2.2 Measures of stochastic resonance ...... 18 2.2.1 Signal-to-noise ratio with interspike interval histograms ...... 18 2.2.2 α-function stimulation ...... 21 2.3 Numerical methods ...... 23

Chapter 3. Experiments 25 3.1 Introduction ...... 25 3.2 Thalamocortical neurons ...... 26 3.2.1 ZAP function stimulation ...... 26 3.2.2 Sine wave stimulation with noise ...... 26 2+ 3.2.3 Blockade of IT with Ni ...... 32 3.2.4 α-function stimulation ...... 33 3.2.5 Noise stimulation with no signal ...... 35 3.3 Reticular neurons of the thalamus ...... 35 3.3.1 ZAP function stimulation in reticular neurons ...... 38 3.3.2 Noise stimulation in reticular neurons ...... 38 3.4 Discussion ...... 38

Chapter 4. Ionic Models of Thalamic Neurons 43 4.1 Introduction ...... 43 4.2 Impedance analysis ...... 44

iii Table of Contents

4.3 Stochastic resonance ...... 46 4.4 α-function stimulation ...... 49 4.5 Noise stimulation with no signal ...... 51 4.6 Reticular neurons ...... 52 4.7 Discussion ...... 55

Chapter 5. The Hindmarsh-Rose model 59 5.1 Introduction ...... 59 5.2 Bifurcation analysis ...... 60 5.3 Subthreshold resonance ...... 62 5.4 Noisy sine wave stimulation ...... 65 5.5 SR with α-function stimulation ...... 67 5.6 Noise stimulation with no signal ...... 69 5.7 Stochastic resonance in the fast subsystem ...... 70 5.8 Stochastic bifurcation analysis ...... 77 5.9 Discussion ...... 80

Chapter 6. The Resonant Integrate-and-Fire Model 82 6.1 Introduction ...... 82 6.2 Matching the model and parameter estimation ...... 83 6.2.1 Subthreshold properties ...... 83 6.2.2 Threshold properties ...... 86 6.3 Stochastic resonance in the RIF model ...... 88 6.4 Comparison with the non-resonant integrate-and-fire model ...... 89 6.5 Stochastic analysis of RIF ...... 93 6.6 First passage time ...... 101 6.7 Discussion ...... 103

Chapter 7. Network Synchronization 104 7.1 Introduction ...... 104 7.2 A network of Huguenard-McCormick neurons ...... 105 7.3 A network of nRT neurons with inhibitory coupling ...... 109 7.4 Physiological network of TC and reticular neurons ...... 113 7.5 Hindmarsh-Rose network ...... 116 7.6 Resonant integrate-and-fire neuron network ...... 120 7.7 Discussion ...... 124

Chapter 8. Conclusion 126 8.1 Summary of results ...... 126 8.2 Relevance and implications ...... 129 8.3 Problems and future research ...... 131

Glossary 134

Bibliography 135

iv Table of Contents

Appendix A. Model Parameters 149 A.1 Huguenard-McCormick model (HM) ...... 149 A.2 The model of reticular neurons of the thalamus (RET) ...... 150 A.3 Hindmarsh-Rose model (HR) ...... 151 A.4 Integrate-and-fire models ...... 151 A.5 Synaptic coupling ...... 152

v List of Figures

2.1 DIC-IR videomicroscopy in-slice image of a MGB neuron during whole-cell patch clamp recording...... 16 2.2 Illustration of the mechanism of stochastic resonance in a simple threshold model. .... 20 2.3 Typical SNR curve from a threshold model...... 21 2.4 Detection of α-function inputs by a threshold model with noise...... 22

3.1 Voltage responses and I-V plot of an MGB neuron to step current stimulation...... 27 3.2 Subthreshold voltage response and impedance plot of an MGB neuron with ZAP current input...... 28 3.3 Voltage response of an MGB neuron under subthreshold sine wave and noise stimulation...... 29 3.4 Interspike interval histogram distributions of an MGB neuron in dependence on input noise level and sine wave frequency...... 30 3.5 Signal to noise ratio of an MGB neuron computed from ISIHs...... 31 3.6 Sensitivity of frequency preference to Ni2+ application...... 33 3.7 α-function stimulation and noise in an MGB neuron...... 34 3.8 Response of an MGB neuron to noise stimulation with no input signal...... 36 3.9 Voltage response and I-V diagram of an nRT neuron under step function stimulation. 37 3.10 Subthreshold voltage response and resonance of an nRT neuron...... 39 3.11 Signal to noise ratio in an nRT neuron...... 40 3.12 Response of an nRT neuron to noise stimulation with no input signal...... 41

4.1 Voltage record of the HM model under subthreshold sine wave and noise stimulation. . 45 4.2 Impedance diagram of the HM model...... 46 4.3 Interspike interval histogram distributions in dependence on input noise level and sine wave frequency...... 47 4.4 Signal-to-noise ratio of the HM model with dependence on the noise level and signal frequency...... 48 4.5 Contour plot of the SNR dependence on noise level and input frequency for the HM model without IT -resonance...... 49 4.6 Voltage trace of the HM model under α-function stimulation with noise...... 50 4.7 Firing probabilities of the HM model under α-function stimulation...... 51 4.8 Three-dimensional interspike interval histogram of the HM model with dependence on the noise level σ without an input signal...... 53 4.9 Voltage trace of the RET model under subthreshold sine wave and noise stimulation. . 54 4.10 Impedance diagram of the RET model...... 55 4.11 Contour plot of SNR for the RET model depending on the noise level and input frequency...... 56 4.12 ISIH of the RET model with no input signal for different noise levels...... 57

5.1 Trace of the voltage-like variable x in the deterministic HR model...... 61 5.2 Bifurcation diagram of the deterministic HR model in dependence on I0...... 62 5.3 Impedance diagram of the HR model...... 64

vi List of Figures

5.4 Voltage trace of the HR model during sine wave stimulation with noise...... 66 5.5 Interspike interval histograms of the stochastic HR model from sine wave stimulation. 67 5.6 Signal-to-noise ratio dependence on the noise level and input period...... 68 5.7 Contour plot of the signal-to-noise ratio dependence on the noise level and input period with  = 0.0025...... 69 5.8 Voltage trace of the HR model under α-function stimulation with noise...... 70 5.9 Firing probabilities of the HR model under α-function stimulation with dependence on the noise level...... 71 5.10 Preferred frequency stochastic resonance with α-function stimulation and noise in the HR model...... 72 5.11 Three-dimensional interspike interval histogram of the HR model with dependence on the noise level without an input signal...... 73 5.12 Dynamics and bifurcation diagram of the reduced deterministic HR model...... 74 5.13 Stochastic resonance in the reduced HR model...... 76 5.14 Comparison of the optimal noise level for detection of periodic signals for different values of ...... 77 5.15 Probability distribution and stochastic bifurcation of the stochastic reduced Hindmarsh- Rose model...... 79

6.1 Result of the least square fit of the impedance magnitude of the HR and RIF models. 86 6.2 Trace of the HR model under step inputs, demonstrating threshold and reset...... 87 6.3 Trace of a realization of the stochastic RIF model...... 89 6.4 Interspike interval histogram of the RIF model...... 90 6.5 Signal-to-noise ratio dependence of the RIF model on the noise level and input period. 91 6.6 Preferred frequency stochastic resonance with α-function stimulation and noise in the RIF model...... 92 6.7 Stochastic resonance of the one-dimensional integrate-and-fire model...... 93 6.8 Plot of the moments of the probability distribution of the RIF model...... 100 6.9 Plot of the second moments of the RIF model...... 101 6.10 Plot of the probability distribution of the first passage time of the RIF model...... 102

7.1 Raster plots of simulations of a globally coupled network of 50 HM neurons in dependence on the noise level...... 109 7.2 Raster plots of simulations of a globally coupled network of 50 HM neurons in dependence on the coupling strength...... 110 7.3 Cumulative interspike interval histogram from a network of 50 HM neurons with global coupling...... 111 7.4 Raster plots of simulations of 50 HM neurons with nearest neighbour coupling...... 112 7.5 Raster plots of simulations of a network of 50 RET neurons with nearest neighbour coupling by GABAergic synapses...... 114 7.6 Cumulative interspike interval histogram from a network of 50 RET neurons with nearest neighbour coupling...... 115 7.7 Schematic of the network of RET and TC neurons connected by AMPA and GABAergic synapses...... 116 7.8 Raster plots of simulations of a network of 50 RET and 50 TC neurons...... 117 7.9 Raster plots of simulations of a network of a 50 HR neurons with global coupling. ... 119

vii List of Figures

7.10 Cumulative interspike interval histogram from a globally coupled network of 50 HR neurons...... 120 7.11 Raster plots of simulations of a globally coupled network of 50 HR neurons for  = 0.0025...... 120 7.12 Raster plots of simulations of 50 globally coupled HR neurons in dependence on the coupling strength...... 121 7.13 Raster plots of simulations of 50 HR neurons with nearest neighbour coupling...... 123 7.14 Raster plots of simulations of 50 RIF neurons with nearest neighbour coupling...... 124 7.15 Cumulative interspike interval histogram from a network of 50 RIF neurons...... 126

viii Acknowledgements and Dedication

I would like to acknowledge with gratitude the supervision of Robert M. Miura and Ernie Puil, as well as the support from the members of my supervisory committee: Yue-Xian Li, Wayne Nagata, and Don Ludwig. The development of the research presented in this thesis was a concerted effort between my supervisors and me, and would not have been possible as an isolated undertaking. Fruitful discussions with many other people helped the research along, most outstandingly with Yue-Xian Li, Leah Keshet and Rachel Kuske. This work was supported financially by research grants from the the Natural Sciences and EngineeringResearchCouncilofCanadaandtheCanadianInstitutesforHealthResearch,as well as Graduate Fellowships from the University of British Columbia. My gratitude for invaluable emotional, physical and intellectual support goes to my family andmanyfriendsandcolleagues.IwouldliketothankmylabmatesAmerGhavanini,Israeli Ran,StephanSchwarz,andXiangWanforprovidinghelpandinterestingdiscussions, sharingexperiences,andmakingandcleaningupofmesses.

ix Diese Arbeit ist meinen Eltern gewidmet, die mich immer treu unterst¨utzt haben.

This thesis is dedicated to my parents who faithfully supported me over the years.

x Chapter 1 Introduction

The main role of neurons in the brain is to integrate and process information in the form of electrochemical signals. In vivo, there is significant background activity due to randomly firing neurons, transmitter release, random ion channel openings, etc., which causes membrane poten- tial fluctuations. This constitutes noise, which may enhance detection of input signals through the stochastic resonance (SR) phenomenon. In SR, the noise can generate firing threshold crossings causing action potentials (APs). At an optimal noise level, the coherence between input signal and output spike train is maximal. Stochastic resonance occurs in a number of experimental systems and neuron models [61, 112, 124, 211] and may have importance as a mechanism that neurons use to amplify weak signals.

Many types of neurons also exhibit a deterministic membrane or subthreshold resonance, where a combination of electrical properties confers an increased voltage response in a preferred range of input current frequencies. This resonance occurs when the membrane potential is subthresh- old. Whereas the resonance may affect subthreshold signal processing, it does not directly influence the neuron’s action potential firing pattern. In thalamocortical (TC) and reticu-

2+ lar (RET) neurons, a low threshold Ca current, IT , interacts with the passive membrane electrical properties to produce this membrane resonance [84, 142]. Membrane resonance (or

“IT -resonance“) can selectively amplify subthreshold inputs at specific frequencies [87]. IT res- onance also is involved in the generation of oscillations in the thalamic network which can be in the same frequency range as the membrane resonance.

In this thesis, we propose that an interaction between the subthreshold and stochastic reso-

1 Chapter 1. Introduction nances confers a preferred frequency in the firing output which stems from amplification of small signals by resonance and stochastic threshold crossings. We will investigate this hypoth- esis in experiments on thalamocortical and reticular neurons and in simulations and analysis of a number of mathematical neuron models.

In addition to our single neuron model studies we will investigate noisy networks of neurons with resonance. We propose that the stochastic firing of neurons in a preferred frequency range that arises from subthreshold and stochastic resonances can play a role in the synchronization of neurons in a noisy network.

1.1 Stochastic resonance

Based on geophysical observations, Benzi et al. [13] suggested that small periodic variations of the orbit of the earth can induce ice ages through a noise-induced resonance, an effect they called stochastic resonance. Since then, SR has been found in a large number of physical and biological systems, and studied analytically in various dynamical system models. The current state of research has been reviewed by Moss et al. [124, 211], Gammaitoni et al. [61], and Petracchi [136].

A system is said to exhibit stochastic resonance if there exists an optimal amount of input noise to induce a maximal coherence between input and output signals. To investigate stochastic resonance experimentally, a deterministic signal is fed into the system and a measure of the response is used to quantify the input-output coherence. The appearance of stochastic resonance requires a barrier or threshold, which here can be explained tentatively as follows: The system will generate an output each time its internal state crosses the barrier. If the deterministic input is so small that no crossings occur in the absence of noise, weak noise will induce only rare, incoherent crossings. Strong noise, on the other hand, will induce frequent, but random transitions. At an intermediate noise intensity, though, the rate of the noise-induced crossings, and therefore, the firing pattern, will coincide with the input signal, yielding a coherent output signal. Thus, SR is an effect due to cooperativity between signal and noise.

2 Chapter 1. Introduction

There are various measures to quantify SR, including residence time distributions [61], power spectral density [13], signal-to-noise ratio (SNR) from the power spectrum or interspike interval histograms (ISIHs) [54, 108], or correlation to the input signal [30, 112]. In this work we employ SNR derived from ISIHs with periodic inputs, as well as the detection reliability of discrete synaptic inputs [152] to measure SR. SR occurs in a system if these a measures obtain a maximum at a particular noise amplitude.

1.1.1 SR in experiments

In biological systems, SR was first found experimentally in 1993 in crayfish tail fin sensory neurons [54]. In this system, SR in the mechanoreceptors may improve the detection of water currents, evoked by a predator in a turbulent or ”noisy” environment. Since then, SR phenom- ena have been reported in a number of neuronal systems, both in peripheral and central neurons as well as in networks of neurons. For example, in the crayfish mechanoreceptor/photoreceptor network, noise induces phase synchronization [5].

Patch-clamp experiments, similar to those presented in this thesis, have been used to feed noise into intact neurons in brain slices to demonstrate SR in neocortical and hippocampal neurons [154, 165, 166]. Stacey and Durand [167] showed that physiological noise from synaptic activity in a stochastic network can aid signal detection in hippocampal CA1 neurons. Membrane ion channels, which contribute to internal noise in neurons, can exhibit SR [14].

It also has been suggested that SR can act on the scale of the whole brain and influence animal and human behaviour [96, 208]. Freund et al. [57, 58] reported that young paddlefish can detect their plankton prey more readily in the presence of (electric) noise. In humans, a behavioral type of SR participates in the perception of ambivalent pictures [126]. SR also may influence affective disorders such as depression [81]. The effect of (internal, not acoustic) noise on hearing was investigated by Gebeshuber et al. [63]. Morse et al. [122, 123, 125] proposed a clinical application of SR for a cochlear implant for humans with impaired hearing. Jaramillo et al. [94] showed that noise present in the auditory system is close to the optimal level, suggesting that the system is tuned to take advantage of SR. Also, in the somatosensory system of live

3 Chapter 1. Introduction cats, SR improves detection of tactile stimuli [114].

Because of the experimental difficulty of recording from intact networks of neurons, there are not many studies of SR in networks of neurons. However, Gluckman et al. [65] found SR in extracellular recordings in hippocampal slices during stimulation with a noisy electrical field.

1.1.2 SR in models of neurons

In experimental systems, it is often difficult to maintain stationary conditions for a long enough time to collect sufficient data for statistical analysis of SR. Consequently, most studies of SR rely on modelling and numerical simulations. Longtin et al. [108] were the first to search for stochastic resonance in neuronal models. For an overview of SR in neuronal models, see [109].

SR has been studied analytically in simple neuron models [99], integrate-and-fire systems [22, 139, 140], and Wiener processes with threshold [25]. In general, the study of stochas- tic dynamical systems is difficult, and one cannot obtain analytical results for the probability distribution and threshold crossing times. However, even in nonlinear systems, a linear anal- ysis can be useful to approximate the effects of small noise and explain the occurrence of SR [41, 178]. In more complicated systems, such as physiological neuronal model, the study of SR is performed with numerical simulations.

In the Hodgkin-Huxley (HH) model, the standard physiological model of an excitable cell, SR is present and its dependence on the input signal shape was studied by Lee et al. [101]. HH-type models of pyramidal [154, 155] and hippocampal [166] neurons exhibit SR, implying a role of noise in the signal detection properties of these neurons. SR also was found in models of shark mechanoreceptors [17], auditory hair cells [93], and mammalian cold receptors [110].

SR also appears in simpler models of neuronal dynamics, such as the FitzHugh-Nagumo (FHN) model of tonic firing and the Hindmarsh-Rose (HR) model of burst firing neurons. The depen- dence on input signal and noise parameters was studied in the FHN model [69, 104, 105]. In the HR model, Wang et al. [206] also found that SR had increased selectivity for certain signal frequencies. The integrate-and-fire (IF) model, which constitutes the simplest ODE model of

4 Chapter 1. Introduction neuronal dynamics, exhibits SR as well as could be expected in a linear system with a threshold [107, 117].

Similar to noise, chaotic dynamics also can affect signal transduction in neuron models [27, 204]. In most physical and biological systems it is difficult to distinguish between noise and chaos, which have similar effects on firing dynamics in neuron models [24, 34]. Other forms of SR also can appear in neurons, including SR for nonperiodic inputs [35, 37] and single spikes [169]. So called coherence or autonomous SR is present as a preferred frequency in the stochastic spike train, even in the absence of an input signal [106, 111, 137].

1.2 Physiology of thalamic neurons

The thalamus is the first processing station of peripheral sensory inputs in the brain, which then are transmitted to the neocortex [1]. In addition to relaying information, the thalamus also is involved in the generation of large scale oscillations in the cortico-thalamic system, which appear in the electroencephalogram (EEG). By virtue of intrathalamic rhythmogenesis, the thalamus controls the sleep/wake cycle, and it also has a role in anesthesia and some forms of epilepsy [170].

The medial geniculate body (MGB) of the thalamus is the primary auditory nucleus. The MGB receives inputs from the sensory hair cells in the cochlea and has widespread cortical projections. The TC neurons in the MGB are part of a network that includes excitatory efferents from the auditory cortex and inhibitory inputs from the reticular thalamic nucleus [47]. We will investigate both experimentally and in model simulations, neurons of the nucleus reticularis thalami (nRT) that receive excitatory synapses from TC neurons. The nRT controls rhythms of TC networks by inhibitory GABAergic feedback. The interplay between nRT and TC neurons, such as those in the MGB, gives rise to spindle shaped oscillations, observed at the onset of sleep in the EEG as waxing and waning oscillations at about 10 Hz [44, 172].

A prominent feature of all TC, as well as nRT, neurons is their ability to fire bursts of action potentials in response to hyperpolarizing stimuli. These low threshold spikes (LTSs) arise from a

5 Chapter 1. Introduction

2+ low threshold Ca current, IT , which results from deinactivation at hyperpolarized potentials. In nRT neurons, which feature a similar set of ion currents found in TC neurons, the LTSs are longer than in TC neurons because the time constants of IT are larger [43]. The T-type calcium current has been described in the HH formulation by Coulter et al. [39] and by Huguenard and

McCormick [82, 83, 118], see also [84]. IT channels in TC and nRT neurons are concentrated mainly in the dendrites [43, 214], where they can influence synaptic integration.

In the subthreshold membrane potential range, IT interacts with the passive membrane electri- cal properties to produce a membrane resonance, that is, frequency preference for subthreshold stimuli, also called IT -resonance [142]. In dynamical systems, resonance is the process in which oscillations are produced, maintained, or enhanced by means of a periodic transfer of energy from an oscillating input, depending on the frequency. Resonance in different forms occurs in many physical and mathematical systems like the solar system, acoustics, and quantum interac- tions between particles. In dynamical systems, resonance is associated with complex eigenvalues in the linearization around a fixed point, see for example [91].

Many types of neurons are known to exhibit a subthreshold membrane resonance, which is manifest as a maximal deterministic voltage response to small periodic current inputs at the resonant frequency. This resonance can selectively amplify subthreshold periodic inputs and participate in rhythmogenesis [87]. We also refer to this maximal response to input signals as frequency preference.

Subthreshold resonances due to IT have been demonstrated in thalamocortical neurons [142, 184] and in their models [75, 76, 84] at a frequency around 3 Hz (22◦C). As a deterministic phenomenon, IT membrane resonance increases voltage responses with frequency content near the resonant frequency in thalamocortical neurons at subthreshold potentials. This amplifica- tion diminishes as the membrane potential approaches threshold. It has been observed that the

”window component” of IT , that is, the partial activation and deinactivation of the current at rest, is instrumental for resonance [84], as well as for bistable behavior [191]. In the thalam- ocortical network, IT and membrane resonance are believed to be involved in the generation

6 Chapter 1. Introduction of spindle oscillations [38, 49, 201], as well as in absence epilepsy [194]. Genesis of resonance is different in auditory thalamic neurons of the bird where a hyperpolarization-activated Na+ current (Ih) interacts with the passive membrane properties [176].

An Ih current, similar to IT also induces subthreshold resonance in neocortical neurons [85, 86,

150, 199]. The cortical nonspecific cation current Ih is activated by hyperpolarization, and it is mainly present in dendrites. The conventional HH model also has resonances at about 27 Hz [133] and 94 Hz [40], which are at a much higher frequency than the resonances studied in this work.

Resonance in neurons has a connection to burst firing [200], as demonstrated by simultaneous blockade of both in experiments [185, 186]. In synergy, membrane resonance and SR may boost inputs and evoke qualitative changes in the firing response of neurons. This occurs in shark electroreceptors where SR interacts with subthreshold membrane currents to influence the coding of input signals [17].

1.3 Noise and information processing in neurons

Every physical system contains noise, often modeled as Gaussian white noise. In neurons, there are intrinsic and extrinsic sources of noise [64, 115, 196]. Intrinsic noise arises at the level of the neuronal dynamics when ions cross ions channels stochastically [168, 210]. Stochastic channel openings and closings also can contribute to membrane voltage fluctuations. This thermal noise in neurons can be well approximated by white noise, similar to noise in electronic circuits. Channel noise can influence the firing pattern of neurons [157, 209]

Extrinsic noise arises partly from spontaneous synaptic activity such as miniature potentials when a vesicle of neurotransmitter is released without an action potential arriving at the presy- naptic terminal. Because neurons fire spontaneously, there is also a large amount of action potential activity that does not carry information and gives rise to a random synaptic bom- bardment that contributes to the overall noise background in neurons. The irregularity of firing patterns of neurons is mainly attributable to the stochasticity of their synaptic inputs [51] and

7 Chapter 1. Introduction the probability of spike initiation in response to small input signals is influenced by synaptic reliability [213]. This is consistent with the observation that cortical neurons fire randomly when recorded in vivo, while generating regular spike sequences in slice preparations [174]. An estimate of noise that affects neocortical neurons in vivo has been obtained by Destexhe et al. [48], who found high frequency activity between 1 and 5 Hz at excitatory and inhibitory synapses. Another measurement was performed in the cat neocortex to compare noise in living animals and slice experiments. Noise levels as high as 8 mV standard deviation of the membrane potential were found in vivo under anesthesia compared to 0.18 mV in vitro in neocortical neu- rons [134]. This agrees with the finding by Stacey et al. [166], who found a noise level between 0.02 and 0.3 mV in hippocampal CA1 neurons in vitro. Another study by Steriade et al. [173] found an average firing rate of cortical neurons of 15.7 Hz in awake animals, resulting in high frequency continuous synaptic bombardment of neurons in vivo.

The influence of noise on the EEG has been studied by Steyn-Ross et al. [175], who also related noise changes to the transition between waking, anesthesia, and epilepsy. The stochastic background activity does not necessarily give rise to noise, but at very high activity, it also can increase the membrane conductance, leading to altered response properties of neurons [156]. This also has been studied in the IF neuron model where noise was generated by a network of IF neurons [117].

Due to the high background activity in the brain, only a small proportion of all inputs to a neuron will transmit relevant information. By definition, these inputs that carry no information are noise. Here, one should note that the mechanism of information coding in the brain has not been explained satisfactorily, and the distinction between noise and signal in vivo may be arbitrary. In our experiments, we use a clearly defined input signal and white noise to avoid this difficulty. The restriction to additive Gaussian white noise ignores autocorrelation as well as spectral properites of real noise. This approach simplifies the simulations and analysis because all different noise sources are treated as summed to a uniform distribution.

Godivier and Chapeau-Blonde [66] showed that the more realistic model of noise from multiple

8 Chapter 1. Introduction random synaptic inputs gives rise to stochastic resonance in neuronal models. When autocorre- lation times of the noise are long, it is likely that there will be interference with the membrane and resonance time scales of the model. However, when the noise autocorrelation time is com- paratively short, this interference would be negligible [71]. Another type of noise, 1/f noise, which is an important concept in psychophysics and many other areas, can evoke SR in neurons [130].

The importance of noise for the function of the brain was recognized recently, as reviewed by Traynelis and Jaramillo [192], and SR has been suggested as a possible explanation for the seemingly ”useless” stochastic firing activity in vivo. Hence, the brain may use the background noise for improved detection of input signals [180] and improved information processing (see the review by McCormick [119]). Information in neuronal output is carried by the timing of the spikes and various measures of the information content in spike trains have been proposed [15, 98], but the theory of information transfer between neurons is not complete. In general, it is believed that information is carried by the timing of spikes as well as the frequency. When the network is in a synchronized state, for example, in REM sleep states, neurons will experience periodic current inputs [171]. In this study, we only use periodic sine wave inputs and short sequences of α-function inputs that resemble excitatory postsynaptic potential (EPSPs), and the information is contained in the sine wave frequency and the timing of the inputs, respectively.

In the HH model, the precision of action potential generation has been studied in response to the shape of deterministic current inputs [55] as well as noise [33]. Noise helps in the detec- tion of subthreshold stimuli, but degrades the timing precision in response to suprathreshold stimuli [135]. When amplified by noise, subthreshold oscillations and dynamics also can have a significant impact on the coding of information in neurons [18, 19, 79, 80], and signals can be transmitted faithfully even by highly irregular spike trains [203, 205]. However, subthreshold dynamics such as oscillations also can hinder SR in neurons [68]. Burst firing, that is, the repetitive firing of a number of action potentials with quiescent periods between bursts, is a prominent feature of many neurons, such as LTSs in thalamic neurons. Bursts have stronger postsynaptic effects than single spikes and are more likely to generate a suprathreshold re-

9 Chapter 1. Introduction sponse. Hence, bursting is an amplification mechanism and is mostly caused by the activation of ion currents other than the action potential generating Na+ and K+ currents. For example,

IT can cause burst firing, but bursting also depends on the shape of the input and noise [7].

1.4 Neuronal networks

In a living organism, neurons are always part of a complex network and are connected by various types of synapses and gap junctions [73]. Synapses are activated by the arrival of a presynaptic action potential and hence constitute a type of pulse coupling. Synapses can be inhibitory or excitatory, that is, decrease or increase the probability of an action potential, depending on the type of post-synaptic receptor that receives the neurotransmitter released by the activated presynaptic terminal. The neurotransmitter bridges the gap between the membranes of the pre- and postsynaptic neurons. Gap junctions are pores in the membranes of adjacent neurons and couple the membrane potentials through diffusion of ions.

In the thalamus, TC neurons receive afferent projections from sensory neurons that carry, for example, visual or auditory information. In turn, TC neurons project axons with excitatory synapses to nRT and cortical neurons to relay this information. The neurons of the nRT form a GABAergic network that inhibits neurons of dorsal thalamic nuclei, such as the MGB. Hence, nRT neurons hyperpolarize TC neurons, evoking low threshold bursts. For a detailed description of the physiology of the thalamus and properties of thalamic neurons, see [171].

Thalamocortical neurons at rest display tonic firing in response to excitatory inputs and relay the information to the neocortex. However, when TC neurons are hyperpolarized by inhibitory inputs, an excitatory input of sufficient strength may produce an all-or-nothing burst response, which carries less information than frequency coded tonic firing [52]. These two modes of thalamic behaviour are related to wake and sleep activity, respectively. In the hyperpolarized sleep mode, the thalamus displays large scale oscillations, such as spindling (waxing and waning oscillations of 7-15 Hz). Isolated TC neurons in many in vitro preparations do not oscillate spontaneously, and the thalamic oscillations are a consequence of the structure of the network

10 Chapter 1. Introduction of thalamocortical, and reticular nuclei [47]. The full cortico-thalamocortical loop, including TC, nRT, pyramidal neurons of the neocortex and interneurons of the thalamus was modelled by Destexhe et al. [50] in order to explain the generation and synchronization of the spindle oscillations.

The types of neurons in the thalamus along with their synaptic connections, as well as their role in states of vigilance are reviewed by Steriade [172]. Steriade also has suggested that the spontaneous activity in the thalamic network not only produces noise, but also determines the receptivity of the brain to external inputs. When thalamic oscillations display pathological hy- persynchronization, information transfer is disrupted and an individual becomes unresponsive, leading to absence epilepsy [60].

The genesis of synchronized oscillations in neural networks depends on a number of factors. A theory of synchronization was developed to predict conditions necessary for synchronization or entrainment in weakly coupled deterministic oscillators [20, 78]. In general, both inhibitory and excitatory synapses can synchronize deterministic neuronal networks. Synchronization of spiking times can occur [158], and inhibitory coupling can promote synchronzed bursting [163]. There is no general theory for noisy networks, but studies show that simple oscillators can synchronize under noise [128], and in stochastic HH neurons, noise-induced synchronization of tonic firing was observed by Wang et al. [207].

Noise can give rise to stochastic resonance and synchronization in systems of coupled oscillators [72, 143]. In networks of coupled neurons, coherent firing [144] and improved detection of input signals compared to single neurons was found [202]. This is called array-enhanced stochastic resonance [88, 89, 95, 103]. Cooperation of neurons in a stochastic network can lead to a broadening of SR [31, 36], and hence the tuning of the noise level for optimal detection is less critical. SR even is present in the output of an ensemble of uncoupled neurons [161]. Synchronization of simple oscillators by noise has been observed and analyzed by Neiman et al. [129]. SR and stochastic synchronization also can occur in the crayfish mechano- and photoreceptor system [4, 5]. In a network of pancreatic β-cell models, a type of excitable cell

11 Chapter 1. Introduction similar to neurons, noise induces a transition between tonic and burst firing [53]. In the same cells, network synchronization by noise also has been described under gap junction coupling [159]. When the signal is subthreshold and periodic, stochastic phase locking to the signal can occur, and rich dynamic behaviour such as stochastic phase locking has been observed [113, 182]. In a stochastic network, it is difficult to quantify synchronization and most studies were performed in spontaneously oscillating systems where a phase can be defined. Shuai and Durand [162] were able to define a phase in the deterministic HR model, and they showed that two coupled neurons can synchronize their chaotic oscillation.

1.5 Aims, scope, and organization

In this thesis, we investigate the effects of noise on the signal processing capabilities of thalamic neurons and their models by means of stochastic resonance. Specifically, we study the coding of periodic stimuli into spike trains in the presence of white noise. We focus on subthreshold stimuli which would not elicit spikes in the absence of noise. Our main objective is to describe the interaction between stochastic resonance and subthreshold resonance in thalamic neurons and their models. We hypothesize that the subthreshold resonance can influence the firing behaviour through the noise and thus confer a frequency preference in the suprathreshold firing.

The work presented here demonstrates a new aspect of SR which could be an important mech- anism of neuronal information processing under in vivo noisy conditions and rhythmogenesis in neuronal networks. The interaction between SR and IT resonance also suggests a physiological role for the influence of subthreshold resonance on action potential patterns.

This thesis is organized as follows: Experimental methods, as well as the models and measures of SR are described in Chapter 2. The details and parameter values of the models can be found in the Appendix. Chapter 3 presents the experimental results of this thesis. We confirm earlier studies of membrane resonance near 2 Hz in thalamocortical neurons and show, for the

first time, IT resonance at 1.5 Hz in nRT neurons. We also determine the influence of this subthreshold resonance on stochastic resonance. We investigate the interactions of membrane

12 Chapter 1. Introduction resonance and SR by performing noise and sine wave stimulation in whole-cell patch clamp recordings of rat thalamocortical and reticular neurons.

In vitro slice experiments are not conducive to stable recordings over long time periods, limiting the collection of statistical data for SR studies. A comparison of data obtained from different neurons also is difficult because of differing membrane properties. Consequently, we complement the experimental studies by computer simulations and analysis of HH-type models of TC and nRT neurons in Chapter 4. Our computer simulations give a more detailed picture of the dependence between SR optimal noise and frequency. In the models, there is a preferred frequency of SR at 1.75 Hz (TC) and 1.5 Hz (nRT), i.e., for a certain noise level, this frequency is detected best. With α-function stimulation and the detection reliability measure, there also is SR, but no strong indication of a preferred frequency.

In Chapter 5, we study the interactions between the nonlinearities and input signals with noise in the Hindmarsh-Rose (HR) model of bursting neurons. This simple model is more accessible to mathematical analysis than the more complicated HH-type models. We analyze the sub- threshold membrane resonance and study its effect on the transduction of excitatory inputs into a train of action potentials. Employing both computer simulations and analysis methods on the Hindmarsh-Rose system, we demonstrate the occurrence of a subthreshold resonance as well as SR under sine-wave stimulation. The model also shows a preferred frequency. The depen- dence of this stochastic preferred frequency on the noise level is approximately an exponential function.

Additionally, we study the influence of noise on more realistic forms of input signals. With the probability of detection measure, we can demonstrate both SR and frequency dependent SR when α-functions are used as inputs. The preferred frequency in the spike train also manifests itself in the absence of any input signals, with a similar dependence between preferred frequency and noise. We show that this autonomous stochastic resonance-like phenomenon results from the subthreshold resonance in the HR model.

A resonant integrate-and-fire (RIF) is derived from the HR model in Chapter 6, which then is

13 Chapter 1. Introduction studied analytically. In this simple linear model, the solution of the stochastic equations can be obtained in closed form. We show that the preferred frequency SR is directly related to the subthreshold resonant frequency in the moment equations.

We extend our results to networks of coupled neurons in Chapter 7. A main observation is that in neurons with a subthreshold resonance, noise can induce network synchronization at approximately the resonant frequency. We demonstrate this stochastic synchronization in networks of HR and RIF neurons, and we present additional results on a model of the thalamic network including TC and nRT neurons.

In Chapter 8, we summarize our results, discuss the significance of the findings, and indicate directions for future research.

14 Chapter 2 Methods

2.1 Experimental methods

All experiments followed protocols approved by the Committee on Animal Care of the University of British Columbia.

2.1.1 Slice preparation

We prepared thalamic slices as previously described in detail [186]. Sprague-Dawley-rats (13-15 days old), were deeply anesthetized with isoflurane and decapitated. The cranium was opened and the brain rapidly removed and submerged in cold (4◦C) artifical cerebrospinal fluid (ACSF).

The ACSF had the following composition in (mM): 124 NaCl, 4 KCl, 1.25 KH2PO4, 2 CaCl2,

2 MgCl2, 26 NaHCO3, and 10 D-glucose. The pH of the ACSF, continuously saturated with

95% O2:5% CO2, was 7.4. The cortex was removed and slices ca. 400 µm thick were sectioned with a Vibroslicer (Campden Instruments, London, UK). For at least 2 h prior to recording, the slices were maintained in a holding chamber containing oxygenated ACSF at room temperature (∼ 22◦C).

2.1.2 Recording procedures

The recording electrodes were prepared from borosilicate glass (1.5 mm OD, WPI Instruments, Tokyo, Japan), using a micropipette puller (Model PP-81, Narishige Instruments, Tokyo, Japan). The pipettes were filled with a solution containing (in mM): 140 K-gluconate, 10 ethylene-glycol-bis-(b-aminoethyl ether)-N,N,N’,N’ - tetraacetic acid (EGTA), 5 KCl, 4 NaCl,

3 MgCl2, 10 N-[2-hydroxyethyl] piperazine-N’-[2-ethansulfonic acid] (HEPES, free acid), 1

15 Chapter 2. Methods

Figure 2.1: DIC-IR videomicroscopy in-slice image of a MGB neuron during whole-cell patch clamp recording. The width of the picture is approximately 20 µm.

CaCl2, 3 Na2AT P , and 0.3 NaGT P . Equilibration of this solution with 10% gluconic acid resulted in a pH of 7.3. We used electrodes with resistances between 6 and 10 MΩ.

Before recording, a slice was transferred to a submersion-type chamber (∼ 1 ml) and perfused with oxygenated ACSF at room temperature (23 − 25◦C) at a flow rate of 1-2 ml/min. For blockade of the T-type calcium current, 1.0 mM NiCl was applied to the slices in the ACSF [142, 177].

In the chamber, we identified the medial geniculate body (MGB) by differential interference contrast (DIC) microscopy and infrared (IR) illumination (Axioskop, Zeiss Instruments, Jena, Germany) with an IR-sensitive video camera (Hamamatsu, Tokyo Japan). Whole-cell measure- ments were conducted on visually selected neurons in the ventral portion of the MGB as shown in Figure 2.1.

Access resistance of the electrode was compensated using capacitance compensation and bridge

16 Chapter 2. Methods balancing. Whole-cell patch-clamp recordings were performed using an Axoclamp 2A amplifier in the current-clamp mode and pClamp software (version 8.1, Axon Instruments Inc., Foster City, CA, USA). Voltage and current measurements were sampled at 20 kHz with an anti- aliasing filter (time constant, 0.2 ms), and an output bandwidth of 30 kHz. The experimental data were analyzed offline using Clampfit 8 software.

2.1.3 Input signals

All experimental inputs such as sine waves, swept sine waves, Gaussian white noise, and α- functions (see below) were generated in a computer (programmed with Microsoft Visual C++ 5.0) and injected into the cell as currents using a Reactive Current Clamp (RCC) device [85].

Gaussian white noise was generated using a Box-Mueller [141] algorithm and injected into the neuron at a rate of 8 kHz. A potential of 11 mV was subtracted from all voltage measurements to compensate for the junction between the ACSF and electrode solution [186].

We measured the subthreshold frequency dependent voltage response of neurons on injection

b of a swept sine wave signal (ZAP, [142]). That is, Isignal(t) = sin(ωt ), and the impedance at frequency ω and resting voltage V0 can be computed from the ratio of Fourier transforms of voltage response and current input

FT (V ) imped(ω, V ) = 0 FT (I) where V and I are the voltage and current traces, respectively, and FT denotes the complex Fourier transform. Then, the absolute value of the impedance, |imped|, describes the magnitude of the voltage response relative to small sine wave inputs and a preferred or resonant frequency appears as maximum impedance. We do not investigate the phase behaviour of the impedance. In our experiments, we used ω = 0.299, and b = 3.32 for a total ZAP duration of 10 s, which results in a frequency sweep from 0 to 10 Hz. Fourier transforms were computed with the built-in functions of Matlab 6.1 (The MathWorks, Inc.).

In order to study stochastic resonance, we also employed sine wave inputs Isignal(t) = Asin(ωt),

17 Chapter 2. Methods with amplitude A and wavelength λ = 2π/ω, as well as series of α-function inputs that simulate the shape of excitatory postsynaptic potentials (EPSPs) with fixed arrival times ti:

X Isignal(t) = αi(t), i where

t − ti − t−ti α (t) = α H(t − t ) e τ . i max i τ

Here, αmax is a size constant, t denotes the decay time constant of the postsynaptic potentials

(PSPs), H is the Heaviside function, and ti is the arrival time of the ith PSP. Sine wave and α-function inputs were combined with Gaussian white noise.

Membrane voltage was continuously monitored with the RCC and occurrence of action poten- tials was defined by a Schmidt trigger (or Poincar´emap) as depolarization of V through 0 mV. Spike times were saved for offline analysis with Matlab software. Interspike interval histograms (ISIHs), signal-to-noise ratio (SNR) plots, and probability of detection were generated with Matlab.

2.2 Measures of stochastic resonance

2.2.1 Signal-to-noise ratio with interspike interval histograms

Classical studies of stochastic resonance in dynamical systems have employed statistical analysis of the responses to stimulation with sine waves. When presented with a subthreshold signal and noise, a neuron model can respond with firing of action potentials. These usually occur near the peaks of the signal where, on average, the system is closest to firing threshold and can be pushed over this threshold by noise. Using different noise amplitudes (σ) and sine wave inputs of different periods (λ), we simulate the equations for long time (1000 times the period of the input). The times of action potential occurrence are identified by a Poincar´emap and plotted in ISIHs.

In the absence of periodic stimulation, the ISIH of a model exhibiting neuronal tonic firing, such as the FitzHugh-Nagumo model or the Hodgkin-Huxley model, decays approximately

18 Chapter 2. Methods exponentially [196]. This implies that the neuron fires randomly with no correlations between subsequent spikes. Typically, a multi-peak structure for the ISIH is found on application of a periodic input, e.g., a sine wave, shown in Figure 2.2. The first peak near zero interspike interval times in the ISIH represents action potentials that form multiple spikes near the maximum of a sine wave (burst). The next ISIH peak occurs at an interspike interval time that is roughly equal to the period of the input signal. At integer multiples of this time, the harmonic peaks correspond to skipping of maxima of the sine wave signal. The locations of the peaks in the ISIH generally do not change with increasing noise levels, in contrast to their shapes and relative heights. The area under the ISIH peak at the stimulus period is a measure of how much of the input sine wave frequency is present in the spike train. Hence, varying the noise level changes the area under the histogram peak at the period of the stimulus, and this can be used to define the SNR for the period and noise level,

] of spikes near input period SNR = 10 log , 10 total ] of spikes as a measure of the correlation between the input signal and the output spike train. This approach, similar to that of Longtin et al. [108] and Douglass et al. [54], ignores the contribution of the higher order peaks and isolates the response at the input frequency.

Stochastic resonance is demonstrated by the ISIH peak at the input period and hence SNR going through a maximum at an intermediate noise level where the spike train has a strong component at the stimulus frequency. For low noise levels (Figure 2.2A), the probability of crossing the spike threshold and the number of action potentials are too small to correlate well with the signal. Because not every signal peak evokes an action potential, leading to skipping of one signal period, a large proportion of the interspike intervals is longer than the input wavelength, leading to decreased SNR. For high noise levels (Figure 2.2C), extraneous action potentials appear in the spike train, implying that the neuron fires due to the noise level and not to the periodic sine wave stimulation. Again, the SNR decreases. This gives rise to the typical stochastic resonance shape of SNR in Figure2.3 with a maximum at an intermediate (optimal) noise level.

19 Chapter 2. Methods

A low noise

0.1

0.08

0.06

0.04

0.02

0 2 4 6 8 10 12 B medium noise 0.1

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0 0 1 2 3 4 5 6

C high noise ISIH distribution 0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0 0 0.5 1 1.5 2 2.5 3 Input signal

time

Figure 2.2: Illustration of the mechanism of stochastic resonance in a simple threshold model. A: At low noise levels, crossings (denoted by black bars) of the threshold (dotted line) most likely occur on the peaks of the sine wave, but there are not many crossings overall. Frequently, signal peaks do not evoke a threshold crossing which leads to skipping. Thus, the ISIH has a peak at the input period, as well as peaks at multiples of the input period due to skipping. There is also a peak near 0 interspike time. B: At intermediate noise levels, threshold crossings most likely occur near the peaks of the input signal when the system is close to threshold. Consequently, the ISIH peak at the input period is large, while the harmonic peaks are reduced compared to A. C: At high noise level, threshold crossings appear at any phase of the input signal and the ISIH peak at the input period is reduced and obscured by the background activity.

20

Chapter 2. Methods SNR

Noise level

Figure 2.3: Typical SNR curve from a threshold model. The maximum at an intermediate noise level signifies the optimal noise level for transduction of the input frequency into the output spike train.

2.2.2 α-function stimulation

In many cases, living neurons receive discrete inputs such as excitatory postsynaptic potentials (EPSPs) that result from membrane depolarizing actions of synaptic neurotransmitter. We used α-function inputs to mimic EPSPs. In vivo, neurons often generate single EPSPs or short sequences of EPSPs as carriers of information. In the context of EPSP inputs, stochastic reso- nance can be determined by a measure of reliability of detection. If the EPSP is subthreshold, additional noise can evoke a threshold crossing in Figure 2.4A. Figure 2.4B shows that with enough noise, threshold crossings can occur even when no EPSP is present. We can define the probability of making a detection error for a single EPSP as

PE = qPF + (1 − q)PM (2.1) where q is the probability of an EPSP, PF is the probability of a spike without an EPSP, and

PM is the probability of no spike with an EPSP [152]. Therefore, PE describes the probability of making an error, including false firings in response to no EPSP or not detecting an EPSP.

Then, PE should be minimized for optimal signal detection.

In order to compute PF , we perform 10,000 simulations of the model equations with noise but

21 Chapter 2. Methods

A optimal noise C

PM

PF

B high noise D

Probability 1-PE

Noise

Input signal time

Figure 2.4: Detection of α-function inputs by a threshold model with noise. A: At optimal noise, an action potential is unlikely at rest, but likely to occur during the EPSP input, when the system is closest to threshold. B: At high noise, there is a high probability of threshold crossings from rest any time. C: Dependence of PM and PF under noise. D: Dependence of 1 − PE on noise with one maximum at the noise level where detection reliablility is optimal.

no input signal, where each run is 100 time steps long. Then, PF for this noise level σ is the fraction of trials where an action potential was evoked without an EPSP. Similarly PM can be obtained from trials with an EPSP. Then PM is the fraction of trials where no action potential is evoked in response to the EPSP within 100 time steps after the onset of the EPSP. We use

10,000 simulations each with and without an EPSP and hence q = 0.5. All probabilities PM ,

PF , and PE are dependent on the noise level, σ, because an increased noise amplitude produces more spike discharges. Thus, an increase in the noise amplitude results in increased PF and decreased PM (Figure 2.4C). Consequently, 1 − PE goes through a maximum for intermediate noise levels, demonstrating the occurrence of stochastic resonance in Figure 2.4D.

22 Chapter 2. Methods

2.3 Numerical methods

The stochastic model equations are simulated using a Runge-Kutta method [141] modified to incorporate the stochastic term [97]. A system of deterministic ordinary differential equations given by dy = f(t, y) dt with initial value y(0) = y0 can be solved numerically on the grid 0, t1 = δt, . . . , tn, tn+1 = tn + δt, . . . by a variety of numerical integration schemes. When y(tn) = yn is already known, the Runge-Kutta method approximates yn+1 = y(tn+1) using f values at tn, tn + δt/2 and tn + δt. Define

k1 = δtf(tn, yn),

k2 = δtf(tn + δt/2, yn + k1/2),

k3 = δtf(tn + δt/2, yn + k2/2),

k4 = δtf(tn + δt, yn + k3).

Then these approximations give δt y = y + (2k + k + k + 2k ) + O(δt5). n+1 n 6 1 2 3 4 This fourth-order Runge-Kutta method often is used for stiff systems, such as neuron models, as a good compromise between computational efficiency and accuracy.

In this work, we study stochastic systems where Gaussian white noise is introduced as an additive term in the first (voltage-like) equations of our models. This models current noise as discussed in the Introduction. Then, a solution of the stochastic differential equation is a stochastic process, or time dependent probability distribution.

With an additive stochastic term in the equation, we write instead

dy = f(t, y)dt + σdW, and a realization of the stochastic differential equation can be obtained by a similar Runge- Kutta method. Here, dW is a vector of independent Gaussian white noise random variables

23 Chapter 2. Methods with mean 0 and variance 1, and σ is the associated noise strength vector. In our studies, there is noise only in the first, voltage-like equation and consequently only σ1 6= 0. We use an approximation of dW by an Euler step:

δt √ y = y + (2k + k + k + 2k ) + δt σ∆W, n+1 n 6 1 2 3 4 where the kis are the same as in the deterministic method above. In general, this method is only accurate of order 1, and does not offer any advantage over simpler one-step methods like the Euler method [21, 97]. However, in our equations the noise is only present in the first equation and dynamic behaviour, such as action potentials and burst firing, is mainly governed by the deterministic terms, which are approximated by the Runge-Kutta scheme. The noise term √ δt σdW , in contrast, is continuous and nowhere differentiable, validating the use of a one-step scheme [70]. Breuer et al. [21] have given an empirical justification for the use of this method in similar systems. The stochastic increments ∆W are independent Gaussian white noise random variables with mean 0 and unit variance, generated with the Box-Mueller algorithm.

24 Chapter 3 Experiments

Part of the results in this and the following chapter have been accepted for publication by the Journal of Computational Neuroscience [146].

3.1 Introduction

2+ In thalamic neurons, low threshold Ca currents, denoted IT , interact with the passive mem- brane electrical properties to produce a membrane resonance [84, 142]. This membrane res- onance (or IT -resonance), demonstrated in thalamocortical neurons, can selectively amplify subthreshold periodic inputs and participate in rhythmogenesis [87]. However, the membrane resonance is dependent on the membrane potential and disappears when approaching threshold, raising the question whether it can influence the output firing frequencies of neurons.

Another influence on the firing behaviour in vivo is noise from various sources. Noise can increase the conversion of subthreshold inputs into action potential trains at specific frequencies. Such SR phenomena have been demonstrated in many peripheral neurons as well as in patch- clamp experiments in neocortical slices and in models of neocortical and hippocampal neurons [150, 154, 165]. Here, we investigated SR in thalamocortical neurons in the MGB, as well as in neurons of the nucleus reticularis thalami neurons, and determined if an interaction between the subthreshold resonance and SR could influence the firing behavior and give rise to a suprathreshold frequency preference.

We measured input resistance in all neurons by applying positive and negative step surrent in- puts as shown in Figure 3.1A. When depolarizing current steps were sufficiently large to exceed

25 Chapter 3. Experiments threshold, thalamocortical neurons responded with tonic firing. After applying a hyperpolar- izing step of sufficient magnitude, deinactivation of IT produced a low threshold spike (LTS) which sometimes could exceed firing threshold and produce an inhibitory rebound burst. A current-voltage (I-V) relationship was plotted (Figure 3.1B) from the voltage responses to the current steps. The I-V diagram gives information about the input and slope resistance as well as activation of membrane ion currents. From our experiments, we obtained an average slope resistance of 267 ± 48 MΩ (mean ± SEM, n = 32) at rest.

3.2 Thalamocortical neurons

3.2.1 ZAP function stimulation

We used the ZAP function to test the frequency responses of MGB neurons. Figure 3.2A shows that the amplitude of the subthreshold voltage response varied with frequency in a neuron held at -70 mV. Figure 3.2B shows the corresponding impedance magnitude curve with one maximum at ∼ 2 Hz. Hence, neurons at subthreshold potentials exhibited resonance as a frequency preference for inputs near 2 Hz. Depolarization of neurons from rest (-70 mV) to near threshold diminished the resonance. This confirmed the observations of subthreshold resonance in MGB neurons by Tennigkeit et al. [186].

3.2.2 Sine wave stimulation with noise

The application of a subthreshold sinusoidal current input with white noise to a neuron near -70 mV evoked action potentials, most often near the peak of the voltage response (Figure 3.3). However, not every sine wave peak elicited an action potential, and the output of the neuron exhibited a stochastic firing pattern that depended on the sine wave input frequency. Using different noise amplitudes (σ) and sine wave inputs at different frequencies (f), we recorded from a neuron for 40-80 s in each run in order to obtain a sufficient number of action potentials for statistical analysis of the firing pattern.

The ISIH resulting from the noisy sine wave input had a multi-peaked distribution. Figure 3.4 shows ISIHs for different values of σ and f. The first peak of each ISIH was near zero and

26 Chapter 3. Experiments

Figure 3.1: Voltage responses and I-V plot of an MGB neuron to step current stimulation. A: Neuron was held with DC at -70 mV and stimulated by current steps of 500 ms duration. Depolarizing steps gave rise to tonic firing when membrane potential exceeded threshold. After a sufficiently large hyperpolarizing step, a low-threshold spike (LTS) occurred at the offset of the current pulse, often with action potentials. B: I-V plot obtained from current steps, as in A. The neuron’s resistance can be calculated from the slope by fitting the curve with a straight line. Here, the slope resistance was 203 MΩ.

27 Chapter 3. Experiments

A

-70 mV -

0nA

10 mV

500 pA 10 s B 300

250 -70 mV 200 -55 mV 150

100

50 0 2 4 6 8 10 Frequency [Hz]

Figure 3.2: Subthreshold voltage response and impedance plot of an MGB neuron with ZAP current input. A: Swept sine wave stimulation with constant amplitude current (0.1-10 Hz) input evoked a voltage response of non-uniform amplitude. B: Averaged (from 3 consecutive ZAP runs) frequency dependence of impedance showed maximum impedance near 2 Hz, demon- strating a frequency selectivity for inputs. The neurons were held at -70 mV (upper record) and at -55 mV (lower record), showing diminished resonance on depolarization.

28 Chapter 3. Experiments

Figure 3.3: Voltage response of an MGB neuron under subthreshold sine wave (1 Hz) and noise stimulation. The combined current input led to stochastic firing, most likely on the peaks of the sine wave signal. Multiple action potentials occurred on a single peak with occasional skipping of signal peaks. resulted from multiple action potentials at the top of one sine wave. The second ISIH peak occurred at an interspike interval that was roughly equal to the period of the input sine wave and hence corresponded to action potentials on consecutive maxima of the input. More peaks in the ISIH appeared at integer multiples of the period, corresponding to one or more intervening sine wave maxima without action potentials (cf. Figures 3.3, 3.4B).

The interspike interval distribution depended on the noise level and sine wave frequency of the combined input. With low noise levels inputs, action potentials frequently did not occur on sine wave maxima (Figure 3.4A). Consequently, the ISIH peaks were large at multiples of the sine wave period whereas the peak at the input period (1000 ms) was relatively small. More action potentials occurred at higher, intermediate noise levels (Figure 3.4B), resulting in relatively small peaks at higher multiples of the input period. The ISIH peak at 1000 ms increased because inputs with higher noise amplitudes produced action potentials on almost every sine

29 Chapter 3. Experiments

Figure 3.4: Interspike interval histogram distributions of an MGB neuron in dependence on input noise level (σ) and sine wave frequency (f). The ISIHs were computed from membrane potential recordings of long duration (A, 67 s; B, 76 s; C, 55 s; and D, 75 s). wave peak. A greater number of action potentials occurred at high noise levels (Figure 3.4C), even when the input sine wave signal was not near its peak. Often, bursts of two or more action potentials occurred on a peak, resulting in interspike interval events at less than the period of the sine wave signal. The peak in the ISIH at the input period moved to lower interspike times due to discharges that did not coincide with the peaks of the input signal. A sine wave current input at a different frequency shifted the ISIH peaks at multiples of the input period (Figure 3.4D).

The evolution of the ISIH showed the correspondence between the input signal and the output spike train as a function of noise level (Figure 3.4A-C). The ISIH peak at the input period

30 Chapter 3. Experiments

Figure 3.5: Signal to noise ratio (SNR) computed of an MGB neuron computed from interspike interval histograms obtained from the experiments in Figure 3.4. A: SNR dependence on noise strength for an input frequency of 1 Hz. The curve shows one maximum at an intermediate noise level and declines for higher and lower noise levels. This shape is typical for stochastic resonance. B: SNR dependence on input frequency for fixed noise input with an amplitude of 0.6 nA. The SNR curve shows a broad maximum between 1 and 3 Hz. This frequency preference of stochastic resonance is similar to the subthreshold resonance in Figure 3.2, but it appears in the action potential pattern. was the largest relative to the other peaks when the noise was at an intermediate level. The SNR, shown in Figure 3.5A, also went through a maximum at these noise levels. Since the probability of crossing threshold was low at low noise levels, the number of action potentials was too small for a good correlation with the input signal, leading to a low SNR. At high noise levels, the firing probability was too high, resulting in extraneous action potentials. This implies that the neuron discharged in response to the noise level, and not to the sine wave

31 Chapter 3. Experiments stimulation, resulting in decreased SNR at high noise levels. The typical SNR shape shown in Figure 3.5A demonstrates that noise can increase signal detection over a certain frequency range in thalamocortical neurons. The occurrence of an optimal noise level with a maximal presence of the input frequency in the output train is the hallmark of SR. SR was found in 21 out of 21 cells that we investigated.

The SNR depended on input frequency, as well as noise level. This is evident in Figure 3.5B, which shows that the SNR is high between 1 and 3 Hz and is lower for higher and lower frequencies. This represents a maximal coherence between the sinusoidal current input and the output spike train at the input frequency. The SR was in the same frequency range, but was less pronounced than the subthreshold resonance. A preferred SR frequency near 2 Hz was found in 12 out of 15 cells that we investigated for this phenomenon.

2+ 3.2.3 Blockade of IT with Ni

In order to demonstrate the connection between the T-type calcium current, subthreshold resonance and frequency dependence of SR, we performed experiments with 1 mM Ni2+, a known blocker of IT [142, 177]. Blockade of IT typically occurred after 6-10 min of perfusion with Ni2+ in ACSF. Figure 3.6 shows the results of ZAP and noisy sine wave stimulation before and during IT blockade. Figure 3.6A demonstrates that the subthreshold resonance near 2 Hz was reduced by Ni2+, confirming the result of Tennigkeit [187] that the subthreshold resonance

2+ arises from IT . In Figure 3.6B, we illustrate the effect of Ni on the frequency dependence of SNR. The maximum of SNR at 2 Hz was reduced by application of Ni2+, while the SNR did not change substantially at higher frequency inputs. At lower frequencies, SNR also was reduced, consistent with the reduction of low frequency impedance by Ni2+ (cf. Figure 3.6A). The subthreshold resonance was eliminated in all 12 neurons that were investigated, while the SNR was reduced in 7 out of 7 neurons. Partial recovery from Ni2+ application was observed for both subthreshold resonance and frequency dependence of SNR.

32 Chapter 3. Experiments A

300

200 Control

100 Ni2+

0 0 2 4 6 8 10 Frequency [Hz]

B 0.6 Control

0.4

ISIH SNR 0.2 Ni2+

0 1 2 3 4 Input frequency [Hz]

Figure 3.6: Sensitivity of frequency preference to Ni2+ application. A: Impedance calculated from ZAP (cf. Figure 3.2). Nickel (1000 µM)blocked the subthreshold resonance at -70 mV. B: SNR dependence on input frequency also was reduced by Ni2+.

3.2.4 α-function stimulation

Because neurons in vivo mostly receive inputs in the form of postsynaptic potentials, we per- formed experiments with α-function stimulation and noise, and investigated stochastic reso- nance with the reliability of detection measure (see Methods). In response to an α-function input under noise, the neuron could either fire an action potential or just have a subthreshold voltage deflection, see Figure 3.7A. PM , PF , and PE for a specific noise level σ were estimated from a large number of such stimulations. Stochastic resonance in this context is demonstrated

33 Chapter 3. Experiments

A

20 mV 1nA 200m s B 1.0

0.8

E 0.6 1-P 0.3

0.2

0 0.5 1.0 1.5 2.0 Noise level [nA] C 1.0

0.8

E 0.6 1-P 0.3

0.2

0 500 1000 1500 2000

Figure 3.7: α-function stimulation and noise in an MGB neuron. A: Voltage response to stimulation with a pair of α-function inputs with a time difference of ∆t=1000 ms and noise of strength σ=0.6 nA. B: SNR dependence on noise strength for an input frequency of 1 Hz. The curve shows one maximum at an intermediate noise level and declines for higher and lower noise levels. This shape is typical for stochastic resonance. C: SNR dependence on input frequency for fixed noise input with an amplitude of 0.6 nA.

34 Chapter 3. Experiments

by 1 − PE going through a maximum at an intermediate noise level in Figure 3.7B.

Under stimulation with pairs of α-function inputs, we investigated if the detection of the second

EPSP was dependent on the delay ∆t after the first EPSP. Figure 3.7C shows a plot of 1 − PE for detection of the second EPSP. There was no apparent variation of detection reliability with the delay, and hence this measure of SR did not reveal preferred frequency.

3.2.5 Noise stimulation with no signal

When performing experiments with noise but no input signal, the neurons fired single action potentials and bursts in a random manner (Figure 3.8A). The interspike-interval histogram plot in Figure 3.8B reveals an intrinsic frequency in the output spike train. The majority of interspike events occurred at short times, arising from multiple spikes on a burst. There was also increased interspike activity between 0.5 and 1 s interspike time, indicating a preferred frequency at which the neuron fired without any external forcing. This was blocked under Ni2+, demonstrating that the preferred frequency was dependent on the T-type calcium current.

This demonstrates that the preferred frequency stochastic resonance is not simply caused by the input signal but is a manifestation of the internal dynamics of the system. This is in analogy to the autonomous or coherence stochastic resonance [111] found in bursting neuron models, where an optimal noise level maximizes firing at the burst frequency.

3.3 Reticular neurons of the thalamus

We performed the same series of experiments as in the MGB on neurons in the nRT. These neurons have similar ion currents and passive properties as TC cells but with some significant differences. In particular, the reticular IT current is slower than its counterpart in TC neurons [42, 43, 83], and consequently bursts in reticular neurons have an increased duration, visible in Figure 3.9A.

As in TC neurons, input resistance was measured as I-V slope resistance from responses to current steps and was found to be 178 ± 46 MΩ (mean ± SEM, n = 13). The voltage response

35 Chapter 3. Experiments

A

20 mV 1nA1000m s

B

0.35

0.3

0.25

0.2 Ni2*

0.15

relative # of events 0.1 Control 0.05

0 0.5 1 1.5 2 2.5 Interspike time [s]

Figure 3.8: Response of an MGB neuron to noise stimulation with no input signal. A: Voltage response to noise stimulation of strength 1.0 nA. B: Interspike interval histogram from noise stimulation with no input signal. There was a preferred firing interval between 0.5 and 1 s. After application of Ni2+ (1 mM), the preferred frequency disappeared and the ISIH approximately has the shape of an exponential function.

36 Chapter 3. Experiments

Figure 3.9: Voltage response and I-V diagram of an nRT neuron. A: Voltage traces under step function stimulation. Depolarizing steps from rest at V = -75 mV gave rise to an LTS and tonic firing when the membrane potential exceeds firing threshold. B: I-V plot obtained from current steps like in A. The slope resistance of the shown cell was 137 MΩ.

37 Chapter 3. Experiments to current steps with slow bursting, and the I-V relationship are shown in Figure 3.9B.

3.3.1 ZAP function stimulation in reticular neurons

The results of ZAP stimulation in an nRT neuron are shown in Figure 3.10A. The corresponding impedance magnitude in Figure 3.10B was a curve with a flat maximum near 3 Hz. Hence, neurons at subthreshold potentials exhibited subthreshold resonance as a frequency preference for inputs near 3 Hz. Depolarization of neurons from rest ( -70 mV) to near threshold diminished the resonance. A subthreshold membrane resonance was found in all 13 reticular neurons we investigated.

3.3.2 Noise stimulation in reticular neurons

A noisy current fed into nRT neurons revealed stochastic resonance. As in TC neurons, noise in combination with sine wave signal gave an SNR curve with a maximum at an intermediate noise level, shown in Figure 3.11A. This stochastic resonance was present in all 13 reticular neurons we investigated.

The SNR also was dependent on the frequency of the input, as shown in Figure 3.11B. SNR is highest between 1 and 3 Hz, demonstrating a preferred frequency. This phenomenon was reversibly blocked by 1 mM Ni2+ (Figure 3.11C). Hence, reticular neurons exhibited stochastic resonance and frequency dependent stochastic resonance, similar to thalamocortical neurons.

Even with no signal and only noise, an IT dependent autonomous stochastic resonance appeared in the spike train, shown in Figure 3.12. Hence, reticular neurons also have an IT dependent intrinsic frequency that can influence stochastic firing.

3.4 Discussion

Our experiments demonstrate the occurrence of stochastic resonance in thalamocortical and nRT neurons as a maximum in SNR at a resonant frequency. This was evident from interspike interval histograms, showing the effects of combined noise and subthreshold periodic signal inputs on the spike train output. SR also depended on the frequency of the periodic subthreshold

38 Chapter 3. Experiments A

-70 mV

10 mV 0nA

500 pA 10 s

B 300

250

200 -70 mV

150 -55 mV 100

50 0 2 4 6 8 10 Frequency [Hz] C 300

250

200 Control

150 Ni2+

100

50 0 2 4 6 8 10 Frequency [Hz]

Figure 3.10: Subthreshold voltage response and resonance of an nRT neuron. A: ZAP stimula- tion of constant amplitude current swept from 0.1 to 10 Hz shows input evoked voltage response of non-uniform amplitude. B: Averaged (from 3 traces) frequency dependence of impedance showed maximum impedance near 3 Hz, demonstrating a frequency selectivity for inputs. The neurons were held with DC at -70 mV and -55 mV, showing no resonance on depolarization. 2+ C: Blockade of IT by 1 mM Ni also abolished the membrane resonance. 39 Chapter 3. Experiments

Figure 3.11: Signal to noise ratio in an nRT neuron. A: SNR dependence on noise strength for an input frequency of 1 Hz. The typical stochastic resonance curve with one maximum at an intermediate optimal noise level appears. B: SNR dependence on input frequency for fixed noise input with an amplitude of 1.0 nA. SNR is increased between 1 and 3 Hz. C: As in TC neurons, the preferred SR frequency at 2 Hz was reduced by Ni2+.

40 Chapter 3. Experiments

0.3

0.25

0.2

0.15 Ni2+

0.1 relative # of events Control 0.05

0 0.5 1 1.5 2 2.5 Interspike time [s]

Figure 3.12: Response of an nRT neuron to noise stimulation with no input signal, cf. Figure 3.8 A: Voltage trace under noise stimulation of strength 1.4 nA. B: Interspike interval histogram from noise stimulation with no input signal. Similar to TC neurons, there was increased activity at interspike times between 0.5 and 2 s which was abolished by 1 mM Ni2+. current input signal. Although SR occurs in sensory cells [17, 54, 93, 110] and in neocortical neurons [154], our study is the first report of frequency preference of SR in neurons.

In TC neurons, the SR was frequency dependent with maximal SNR in the range between 1 and

2+ 3 Hz. Using the IT -blocker Ni , we demonstrated that the frequency dependence of SR results from subthreshold IT -resonance which diminished on depolarization to near threshold. The frequency selectivity of SR can be explained by an amplification of input signals at preferred frequencies by the IT -resonance which is boosted above this threshold by noise. Because of this boosting, the highest firing probability occurs near the peaks of the sine wave signal. Compared to the cell bodies, the dendrites generate more low threshold calcium current which is instrumental for producing subthreshold resonance [46]. The dendrites also receive the majority of excitatory postsynaptic potential (EPSP) inputs. The juxtaposition of IT -resonance and SR between dendritic and axon hillock sites may have physiological utility in EPSP-action potential coupling.

41 Chapter 3. Experiments

A similar stochastic resonance occurred in MGB neurons under α-function stimulation with the probability of detection measure. This demonstrates that noise at an optimal amplitude also can aid in the detection of single EPSPs. The data for detection of pairs of α-inputs did not give evidence for a frequency preference with the detection measure. Compared to the frequency dependence of SNR with sine wave inputs, this suggests that long signal inputs at the resonant frequency are necessary for the subthreshold resonance to have a significant effect on the firing response.

There also was an IT dependent autonomous stochastic resonance in TC cells. This appeared as a preferred firing frequency under noise stimulation with no input signal. The autonomous stochastic resonance was caused by the T-current, as demonstrated by blockade with Ni2+.

Additional experiments in reticular neurons of the thalamus demonstrated a similar stochastic resonance under sine wave stimulation and noise. Because of a lower resistance of these neurons, a higher noise level was necessary for optimal detection of the input than in the MGB. We also found a frequency preference to input signals with frequency content near 2 Hz in nRT neurons, as well as an autonomous stochastic resonance. In general, the nRT neurons show similar reactions to sine wave and noise inputs as observed in MGB neurons. However, the subthreshold resonance appears at a higher frequency and is less pronounced than in TC neurons. Because of a lower input resistance, higher noise were needed for stochastic resonance.

In summary, we have demonstrated stochastic resonance in association with the preferred firing frequency in thalamocortical and nRT neurons. The preferred frequency of SR results from an

IT dependent subthreshold membrane resonance. This resonance enables the neuron to select and amplify inputs at the resonant frequencies. Then, the addition of white noise current can induce neuronal firing with a preferred frequency that stems from the IT membrane resonance in the stochastic output spike train.

42 Chapter 4 Ionic Models of Thalamic Neurons

4.1 Introduction

Following the introduction of the Hodgkin-Huxley [77] model of the squid giant axon in 1952, similar differential equations were used to model other types of neurons and excitable cells. These models of membrane dynamics are derived from experimental measurements, and the variables and parameters reflect physiological properties.

We investigate the effects of noisy current inputs on the firing dynamics in detailed models of thalamic neurons. Huguenard and McCormick [82, 118] developed a Hodgkin-Huxley type ionic model that captures the essential features of thalamocortical neurons (HM model). Some mistakes in the parameter values were identified and corrected by Hutcheon et al. [84]. The membrane potential is determined by

dV C = −(I + I + I + I ) + I + I (t)). (4.1) m dt l Na K T 0 signal

Here, the ionic currents included in the typical HH form are a fast voltage-dependent sodium current (INa), a voltage-dependent potassium current (IK ), a voltage-dependent low threshold calcium current (IT ), and a voltage-independent leak current (Il), as outlined in the Appendix. The model is capable of producing tonic and rebound burst firing and has a subthreshold resonance similar to thalamocortical neurons. The magnitude and frequency content of Isignal determine the evoked firing pattern [84].

In a similar manner, Huguenard and Prince [83] and Destexhe et al. [42, 43] obtained a math- ematical model of reticular thalamic neurons, where the membrane potential is determined by

43 Chapter 4. Ionic Models of Thalamic Neurons the same equation (4.1) as HM. The currents have similar forms as well, but with different parameter values.

These models have been established as accurate descriptions of the dynamics of thalamocortical and reticular neurons and were used in studies of channel densities [43], subthreshold and firing dynamics [191, 201], and synchronization of the thalamocortical network [47]. Like TC and nRT neurons, the models exhibit tonic firing, rebound bursting, and subthreshold membrane resonance, as well as stochastic firing in response to noisy sine wave inputs as illustrated in Figure 4.1.

In stochastic resonance studies, models have proven especially useful because it is possible to perform long simulations for different parameters values. This is a major advantage of models over experiments where conditions are often non-stationary and it is difficult to obtain sufficient data for statistical evaluation.

4.2 Impedance analysis

In the HM model, the T-type calcium current, in conjunction with the passive properties, such as the membrane capacitance and leak current, gives rise to a subthreshold membrane resonance

[84, 142], whereas in the absence of IT , that is gT = 0, the model exhibits no subthreshold resonance. Hutcheon et al. [84] obtained an analytical expression for IT -resonance in the model subject to oscillatory inputs, yielding the small signal impedance dependent on the voltage and input frequency:

" 0 2 mT ∞(V ) 1 impedHM (ω, V ) = gleak + iCmω + gT mT ∞(V ) hT ∞(V ) 2 + mT ∞(V ) 1 + iωτmT (V ) −1 0 ! # hT ∞(V ) 1 1 + (V − VCa) . (4.2) hT ∞(V ) 1 + iωτhT (V ) V − VCa

The analytical impedance curve shows the frequency response and resonance of the model simi- lar to the impedance curves obtained from ZAP stimulation in experiments, see Chapter 3. The three-dimensional graph of the impedance magnitude |impedHM | in Figure 4.2 demonstrates the subthreshold resonance of the HM model as a peak impedance. The maximum is attained

44 Chapter 4. Ionic Models of Thalamic Neurons

Figure 4.1: Voltage record, V , of the HM model under subthreshold sine wave (1 Hz) and noise (0.4 nA) stimulation. Resting membrane potential is -71 mV. at 3 Hz and -70 mV. The impedance is smaller for higher and lower voltages, as well as for higher and lower frequencies. This demonstrates the presence of a frequency preference to small signals with components near 3 Hz near rest, similar to the subthreshold resonance at 2 Hz of TC neurons in Figure 3.2.

When the membrane potential approaches the threshold for an action potential, resonance diminishes, resulting in a flat frequency response. The disappearance of resonance due to inactivation of IT on approaching threshold prompts the question whether the resonance would influence the firing behavior of thalamocortical neurons. Impedance is defined for small inputs in the linear regime, whereas nonlinear dynamics, such as the activation of channels, determine the behaviour for larger inputs. Hence, a subthreshold resonance, even near the threshold for action potentials, may not translate into a preferred firing frequency.

45 Chapter 4. Ionic Models of Thalamic Neurons

300

200

100

0 2 4 6 -50 f [Hz] 8 -70 10 -90 V [mV]

Figure 4.2: Impedance diagram of the Huguenard-McCormick model. 3-D plot of impedance magnitude as a function of resting membrane potential, V , and signal frequency, f, obtained from equation (4.2). The peak at V = -70 mV, f = 3 Hz signifies a maximum response to inputs at this frequency. As V approaches the firing threshold (at ∼ −50mV ), the membrane resonance is reduced. The Na+ and K+ currents do not play a role in this frequency range.

4.3 Stochastic resonance

In analogy to our SR experiments, we perform long (1000 signal cycles) simulations of the stochastic HM equations with sine wave inputs and noise (Figure 4.1) and analyze the spike trains. The resulting ISIHs (Figure 4.3) exhibit the expected multipeak structure with peaks near zero interspike time as well as at multiples of the input period. As in the experiments, multiple spikes on a sine wave peak correspond to a large ISIH peak near zero interspike time. The first peak away from zero interspike times is associated with the period of the sine wave. There are also peaks at multiples of the input period, corresponding to skipping events when a sine wave peak does not evoke an action potential. The basic shape of the ISIH is preserved despite changing noise and input frequency. Peak heights change with noise (Figures 4.3A-C) and peak positions change with input frequency (Figure 4.3D).

46 Chapter 4. Ionic Models of Thalamic Neurons

0.1 0.1

0.08 0.08

0.06 0.06

0.04 0.04

0.02 0.02

0 123450 12345 D 0.2 0.1

0.08 0.15 CELLD36 relative number of events 0.06 0.1 0.04 0.05 0.02

0 123450 12345 Interspike time [s]

Figure 4.3: Interspike interval histogram distributions in dependence on input noise level (σ) and sine wave frequency (f). The ISIHs were computed from simulations with sine wave inputs and noise.

In order to investigate SR in the model, we calculate SNR from the ISIHs in the same way as for experimental data. Figure 4.4A shows a three-dimensional graph of SNR and its dependence on noise level and input frequency. Each point in the graph is calculated from an ISIH for a particular combination of noise strength and input frequency. At fixed frequency (parallel to the σ-axis) and varying noise level, the curve has the typical SR shape with one SNR maximum at an intermediate noise level. Thus, the model exhibits SR similar to that observed in the experiments. At each fixed noise level (parallel to the f-axis), the SNR also has a maximum which is visible in the contour plot of SNR (Figure 4.4B). The absolute maximum of SNR is obtained at σ = 0.1 nA and f = 1.75 Hz. The optimal noise level is lower at this frequency

47 Chapter 4. Ionic Models of Thalamic Neurons

A SNR of model 0.2 *

0.1 SNR

0 1 5 0.6 4 3 0.2 2 0 0 1 f [Hz]

B Contour plot of SNR 1

0.8

0.6

0.4

0.2

1 2 3 4 5 f [Hz] C Contour plot of SNR without I T 1 Figure 4.4: Signal-to-noise ratio of the HM model in dependence on the noise level and signal frequency. A: 3-D plot of SNR. For each fixed frequency, the SNR has the typical SR shape with one maximum. The0.8 location of maximum SNR depends both on input frequency and noise level, with the absolute maximum (marked by an *) attained at 1.75 Hz and 0.1 nA. B: Contour plot, derived from0.6 A, of the SNR dependence on noise level and input frequency. For each frequency, the typical stochastic resonance curve has one maximum along the σ-axis. The thick line denotes the frequency of maximum SNR for each noise level used in the simulations. In general, a higher input0.4 frequency needs higher noise level for optimal detection. Only at f = 1.75 Hz a lower noise level is optimal than the required noise for higher and lower frequencies. 0.2

48 1 2 3 4 5 f [Hz] A SNR of model 0.2 *

0.1 SNR

0 1 5 0.6 4 3 0.2 2 0 0 1 f [Hz]

B Contour plot of SNR 1

0.8

0.6

0.4

0.2

1 2 3 4 5 f [Hz] Chapter 4. IonicC ModelsContour of Thalamic plot Neurons of SNR without I T 1

0.8

0.6

0.4

0.2

1 2 3 4 5 f [Hz]

Figure 4.5: Contour plot of the SNR dependence on noise level and input frequency for the model without IT -resonance, that is gT = 0. There is stochastic resonance for every input frequency. The optimal noise level (denoted by a solid black line) increases monotonically with the frequency. than at higher or lower frequencies. This constitutes a preferred SR frequency of the model. The noise level required for optimal detection (denoted by the thick curve in Figure 4.4B) increases for higher frequencies. This frequency preference is absent when gT = 0, that is, in a

2+ model without IT and resonance (Figure 4.5) in analogy to block by Ni . In this non-resonant model, the optimal noise level increases monotonically with input frequency. When compared to Figures 4.4A and B, this shows that the appearance of the preferred frequency is dependent on IT . In summary, a frequency- and noise-dependent SR is present in the model under sine wave stimulation.

4.4 α-function stimulation

In order to investigate SR in HM with α-function inputs (Figure 4.6), we perform 10,000 simulations each with and without an EPSP. Figure 4.7A shows that 1 − PE for detecting one

49 Chapter 4. Ionic Models of Thalamic Neurons

V

Isignal

20 mV 200 pA 100 ms

Figure 4.6: Voltage (Vm) trace of the HM model under α-function (t1 = 100, t2 = 600) stimu- lation with noise (σ = 0.3 nA). Action potentials may or may not appear on top of an EPSP.

EPSP goes through a maximum for intermediate noise levels, exhibiting another occurrence of stochastic resonance.

In response to pairs of EPSPs and noise, a subthreshold resonance may confer an increased detection reliability at the preferred frequency. In order to test this, we perform 10000 simula- tions each for varying time delays ∆t = t2 − t1 and noise levels σ, giving PE for the detection of the second EPSP, as shown in Figure 4.7B. There is no well-defined maximum in 1 − PE, and hence, no preferred frequency appears. Only at high noise levels is there a slightly increased detection probability around 400 ms delay. The increased detection probability at 100 ms delay is caused by the well-known temporal summation phenomenon, where the second EPSP rides on top of the tail of the first.

50 Chapter 4. Ionic Models of Thalamic Neurons

A 0.54

0.53

0.52 E

1-P 0.51

0.5

0.49 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

B 0.54

0.3 0.52 0.2 0.5

0.4 E

0.48 0.5 1-P

0.46 0.6 0.44

0.42 100 200 300 400 500 600 700 800

Figure 4.7: Firing probabilities of the HR model under α-function stimulation. A: 1 − PE for detecting a single EPSP in dependence on σ. The typical stochastic resonance shape with a maximum at intermediate noise levels appears. B: Detection probability (1 − PE) of the second EPSP in a pair depends on the time delay (∆t) after the first. The noise levels for the curves are marked in units of nA.

4.5 Noise stimulation with no signal

Even in the presence of only noise with no input signal, we showed experimentally that stochastic resonance can occur in the spike output at an intrinsic frequency. In order to investigate whether

51 Chapter 4. Ionic Models of Thalamic Neurons

this stochastic resonance is present in the HM model, we simulate the equations with Isignal = 0 and noise level σ for 107 ms and compute the interspike-interval histogram for each noise level. Figure 4.8A shows the three-dimensional plot of the ISIH against σ. Most of the spikes occur close together at small interspike intervals. However, there is also activity at intermediate times for each noise level above 0.2 nA. This signifies that longer interspike periods also are present in the output spike train. In the figure, there is no single preferred firing period, but rather a range of periods. The interspike time of the maximal activity decreases with increasing noise, similar to Figure 4.4 where it increases with the input frequency.

The occurrence of a preferred frequency with just noise input demonstrates that the preferred frequency stochastic resonance is not simply caused by the input signal but is a manifestation of the internal dynamics of the system. This is in analogy to autonomous or coherence stochastic resonance [111] found in bursting neuron models, where an optimal noise level maximizes firing at the burst frequency.

Without IT , there is no activity at intermediate interspike times (Figure 4.8B). Consequently, the preferred firing must be due to the IT resonance.

4.6 Reticular neurons

Similar to our biological experiments, we additionally perform a series of SR experiments on a model of reticular neurons of the thalamus. The main difference between the HM model and the reticular thalamic model (called RET in the following) lies in the T-current [83], which generates longer bursts (Figure 4.9) because of a slower inactivation time constant.

The impedance of the RET model again is given by the formula (4.2) with the reticular IT activation and inactivation variables in the Appendix. A subthreshold resonance appears in the impedance plot in Figure 4.10. Compared to the resonance in the HM model, the voltage range where IT amplifies the responses to periodic inputs is narrower and near -75 mV. The peak resonance is at 1.5 Hz, which is a lower frequency than in the HM model.

In response to sine wave and noise stimulations, the reticular model also exhibits stochastic

52 Chapter 4. Ionic Models of Thalamic Neurons

A rel. # of events

Interspike time [ms]

B rel. # of events

Interspike time [ms]

Figure 4.8: Three-dimensional interspike interval histogram of the HM model with dependence on the noise level σ without an input signal. Each line is an ISIH for fixed σ. A: In the HM model there is a noise level dependent local maximum in the ISIH distribution. B: The model without subthreshold resonance has only one ISIH peak near the zero interspike time. resonance. In a contour plot of SNR of the reticular model in Figure 4.11 SR again appears as SNR going through a maximum at an intermediate noise level, depending on the frequency. The optimal noise level increases with frequency, similar to Figures 4.4 and 4.5. The SNR maximum occurs at f = 1.5 Hz and σ = 2.3 nA, but the preferred frequency is not as pronounced as in HM.

53 Chapter 4. Ionic Models of Thalamic Neurons

Figure 4.9: Voltage trace of the RET model under subthreshold sine wave (1 Hz) and noise (1.5 nA) stimulation. Resting membrane potential is -85 mV. Noise can evoke burst and single spike firing.

The interspike-interval histogram of the reticular model held at -85 mV with noise only stim- ulation in Figure 4.12A shows no preferred frequency and hence no autonomous SR. Most of the spiking activity occurs at small interspike intervals, with only small activity at longer times for small noise levels. However, in a model neuron held at -75 mV, the potential where the subthreshold resonance is maximal (Figure 4.10), a maximum at intermediate interspike times appears in Figure 4.12B, demonstrating that the IT -resonance can influence firing in response to noise with no signal. As in the HM model (Figure 4.8) the interspike time of maximal activity decreases with increasing noise.

54 Chapter 4. Ionic Models of Thalamic Neurons

300

200

100

0 2 -60 -70 4 -80 f [Hz] -90 6 -100 V [mV]

Figure 4.10: Impedance diagram of the RET model. The peak is located at V = -75 mV, f = 1.5 Hz. The resonance is restricted to a narrower voltage range compared to the HM model, cf. Figure 4.2.

4.7 Discussion

Our simulations of the Huguenard-McCormick and reticular models show that SR in the models is consistent with the experimental findings. SR appears at input frequencies between 0.5 and 5 Hz, demonstrated by the typical SNR shape with one maximum for changing noise level. The optimal noise level depends on signal frequency, with high frequencies requiring high noise levels, as observed in a double-well system [12] and other neuron models [8, 145].

Similar to our experimental results, we found that in the HM model, SR is frequency dependent, with maximal SNR at a preferred frequency of 1.75 Hz. The actual SNR levels are higher in experiments than in the model, implying that the model neurons fire less reliably. A possible explanation for the increased reliability in real neurons is that the injected noise is not nec- essarily close to the site for action potential generation. Between the microelectrode tip and axon hillock, high-frequency filtering by the membrane may have altered the spectral content

55 Chapter 4. Ionic Models of Thalamic Neurons

6

5

4

3

0.12 0.1 0.08 0.06 0.04 0.02 2 0.16 0.14

1

0 1 2 3 4 5 f [Hz]

Figure 4.11: Contour plot of SNR for the RET model depending on noise level and input frequency. For all frequencies, there is an optimal noise level with maximal SNR. The optimal noise level increases with the input frequency, similar to the HM model in Figure 4.4. There is no outstanding resonant frequency, but the absolute maximum of SNR is attained at 1.5 Hz and 2.3 nA. The optimal noise level (thick black line) again increases with frequency. The model was held at -75 mV where the resonance is strongest, see Figure 4.10. of the noise. Electrode RC filtering was negligible in the experiments because we used capaci- tance compensation and the electrode resistance was small compared to the input resistance of neurons. Because of the membrane filtering, the effective noise level at the axon hillock would be lower than at the point of noise injection into a neuron. In contrast, the equations model a point neuron without any spatial attenuation.

The frequency dependence of SR results from subthreshold IT -resonance which diminishes on depolarization to near threshold. The IT -resonance amplifies input signals at the preferred frequencies which noise then boosts above threshold. Because of this boosting, the highest firing probability occurs near the peaks of the sine wave signal. The preferred frequency of

56 Chapter 4. Ionic Models of Thalamic Neurons

A

0.1 0.08 0.06 0.04 6 0.02 5 0 4 0

rel. # of events 3 500 2 1000 1 1500 Interspike time [ms] 2000 0

B

0.14 0.12 0.1 0.08 0.06 0.04 2.5 0.02 2 1.5

rel. # of events 0 500 1 1000 0.5 1500 Interspike time [ms] 2000 0

Figure 4.12: ISIH of the RET model with no input signal for different noise levels. A: In neurons held at rest (-85 mV), there is activity for low noise at intermediate interspike times, but no clear maximum appears like in the HM model, cf. Figure 4.8. B: When held at -75 mV, where the subthreshold resonance is strongest, there is increased activity at intermediate noise levels.

SR (maximal at 1.75 Hz) depends on the noise level but is close to the subthreshold resonance frequency (3 Hz).

We also found SR with α-function stimulation, suggesting that noise can aid in the detection of single EPSPs. The detection probability of pairs of EPSPs shows no marked dependence on

57 Chapter 4. Ionic Models of Thalamic Neurons

EPSP timing, thus exhibiting no preferred frequency for input detection. Similar to our exper- imental results, this demonstrates that a long, periodic input is necessary for the subthreshold resonance to influence the stochastic resonance. Under noise stimulation without input sig- nals, the preferred frequency appears as activity in the ISIH, similar to autonomous stochastic resonance.

In the RET model, the narrow IT resonance at 2 Hz also can influence SR, and maximal SNR is attained at 1.5 Hz. However, the preferred frequency is not as pronounced as in HM. Also similar to the HM model, the optimal noise level for detection of periodic inputs increases with signal frequency. Autonomous SR only appears in response to noise only when the neuron is held at a depolarized potential of -75 mV, where the the subthreshold resonance is strongest.

In summary, we have demonstrated stochastic resonance in the Huguenard-McCormick and RET neuron models. The preferred frequency of SR results from a subthreshold membrane resonance. This IT -resonance enables the neuron to select and amplify inputs at the resonant frequencies. Then, the addition of white noise current can induce neuronal firing with a pre- ferred frequency that stems from the membrane resonance in the stochastic output spike train. The frequency dependence of the optimal noise level may have significance for synaptic inte- gration and rhythmogenesis in a noisy environment. Thus, noise intensity may act as a control parameter for information processing reliability in the thalamus.

58 Chapter 5 The Hindmarsh-Rose model

Part of the results in this chapter have been published in the Bulletin of Mathematical Biology [145].

5.1 Introduction

Physiological models of neurons such as HH and HM were developed directly from experimen- tal measurements, and the variables and parameters reflect physiological properties. However, in order to gain a deeper understanding of the principles underlying the neuronal dynamics by mathematical analysis, simplified models are more accessible. The well-known FitzHugh- Nagumo equations [56, 127] constitute a polynomial model of tonically firing neurons, derived from the Hodgkin-Huxley equations. Using a similar approach, Hindmarsh and Rose [74] de- veloped a polynomial model of a bursting neuron from a thalamocortical neuron model with detailed ionic currents. The Hindmarsh-Rose (HR) model has the form

dx = y − ax3 + bx2 − z + I + I (t), (5.1) dt 0 signal dy = c − fx2 − y, (5.2) dt dz  1  =  x − (z − g) , (5.3) dt 4 where x is a voltage-like variable, y controls the voltage recovery after an action potential, and z describes the slow dynamics of an adapting current.

The polynomial form of the HR model simplifies both the mathematical analysis and numerical simulation in comparison to more physiological HH-type models. The deterministic HR equa-

59 Chapter 5. The Hindmarsh-Rose model tions are capable of burst firing [200] as well as chaotic dynamics [162, 204]. Stochastic versions of the HR model have been used before to investigate SR, see [8, 109, 112, 131, 132, 212]. Sim- ilar to our studies in vitro and in the HM model, we set out to analyze subthreshold resonance, stochastic resonance, and the frequency dependence of stochastic resonance in this model.

The first two equations of the deterministic HR system (5.1, 5.2) are similar to the FitzHugh- Nagumo model of tonic firing, while the z equation (5.3) has a slower time scale because  is a small parameter. Then the slow variable, z, controls the transition between bursting and quiescent periods (Figure 5.1A). The dynamics of burst firing of action potentials can be illustrated in a three-dimensional phase space plot, shown in Figure 5.1B. During a burst, the x and y variables go through an oscillation. Because of the high x values, the RHS of (5.3) is positive and z increases steadily during a burst. When z becomes too large, the RHS of (5.1) becomes negative, x decreases and the burst is terminated. Then x and y are near their rest values while z slowly recovers to a negative value. Eventually dx/dt becomes positive again and the next burst starts.

5.2 Bifurcation analysis

In deterministic dynamical systems, classical bifurcation theory describes qualitative changes of behaviour. In order to understand the dynamics of the deterministic HR system, we plot the bifurcation diagram and its dependence on inputs I0 in Figure 5.2. For values of I0 < 0.03, the system has one stable steady state. A change from the steady state to burst firing occurs when the input current, I0, is increased. At I0 = −0.03, a subcritical Hopf bifurcation takes place and the system jumps to a burst firing cycle that arises from a supracritical Hopf bifucation at

I0 = 23.96. Thus the stable fixed point loses stability at x0 = −1.33. In the following, we use the value I0 = −0.5 for our simulations, which corresponds to a resting level of x0 = −1.44.

Additional signal current inputs, Isignal(t), as well as noise, can evoke firing from this rest state.

The dynamics also include chaotic regimes (see [188, 189]) not shown in the bifurcation diagram. In the stochastic case that we study, however, the chaotic dynamics can be ignored because

60 Chapter 5. The Hindmarsh-Rose model

A 1.5

1

0.5

x 0

-0.5

-1

-1.5

0 50 100 150 200 250 300 350 400 450 500 t l

B

0.15

z

-0.2 -1.5 0

x y

1.5 -12

Figure 5.1: Trace of the voltage-like variable x in the deterministic HR model. A: For I0 = 0, there is periodic burst firing of a number of action potentials with long quiescent periods between the bursts. B: Three-dimensional trace of a burst like in A. During burst firing, z increases until dx/dt becomes negative and a quiescent period starts. Then, z decreases again until the next burst.

61 Chapter 5. The Hindmarsh-Rose model (a)

2

1

x 0

-1

-2 0 5 10 I 15 20 25 (b) 0

Figure 5.2: Bifurcation diagram of the deterministic HR model in dependence on I . As I 0 0 0 increases, the model goes through a subcritical Hopf bifurcation¸ at I0 = −0.03 that gives rise to burst firing. ¸ the noise will remove-5 any dependency on initial conditions. There are also two additional Hopf

bifurcations at I0 = 4.01 and 4.90 that do not contribute to the dynamics of tonic or burst y firing. -10

5.3 Subthreshold¸ resonance -15 Like many neurons, the deterministic HR equations exhibit a classical frequency preference or -1.5 -1 -0.5 0 x 0.5 1 1.5 2 resonance(c) for small, subthreshold inputs. As in the Huguenard-McCormick model, the response to small periodic variations can be obtained in analytical form. Assuming that Isignal(t) =

iωt 2 δe , the system (5.1-5.3) can be linearized around its I0-dependent steady state (x0, y0, z0) by

1 x

0

-1 62

-2

-2 0 2 4 6 8 10 12 I0 Chapter 5. The Hindmarsh-Rose model

introducing the perturbations X = x − x0, Y = y − y0, Z = z − z0:

dX = Y − 3x2X + 2bx X − Z + δeiωt, dt 0 0 dY = −2fx X − Y, dt 0 dZ  = X − Z. dt 4

Letting X = Aeiωt, Y = Beiωt, Z = Ceiωt, and δ = 1 and then dividing by eiωt, the system becomes

2 Aiω = B − 3ax0A + 2bx0A − C + 1,

Biω = −2fx0A − B,  Ciω = A − C. 4

Solving for A in the first equation after eliminating B and C yields the result:

 2fx 4 −1 imped (ω, x ) = A = iω + 0 + + 3ax2 − 2bx . (5.4) HR 0 1 + iω  + 4iω 0 0

This expression corresponds to the complex impedance for a biophysical neuron model and describes the magnitude and phase of the voltage responses to current inputs Isignal(t). Figure

5.3A shows the dependence of the impedance magnitude |impedHR| on x0 and period λ = 2π/ω. For short periods (high frequencies), the x response to a periodic signal is small whereas longer periods give a nearly constant response. There is a singularity at λ ∼ 336 and x0 ∼ −1.33 (for this z-time scale  = 0.005, which we also use in the following), as seen by setting the denominator of A (the quantity in brackets) to 0. The infinite impedance of the linearized system does not correspond to an infinitely increasing solution of the sine wave stimulated system, because away from equilibrium, the nonlinear terms dominate the dynamics. The impedance is large for a wide frequency range around λ = 336 for x0 just below −1.33, signifying an amplification of the impedance over this range of frequencies. Away from this x0 value, the impedance magnitude quickly diminishes. There also exist three linearly independent solutions of the homogeneous linear equations. However, as shown in Figure 5.3B, these solutions are unstable for x0 > −1.33 because at least one eigenvalue associated with these solutions has a positive real part. At x0 = −1.33, the real part of the pair of complex conjugate eigenvalues

63 Chapter 5. The Hindmarsh-Rose model

A

6

4

2

0

-2

-4 0 -0.5 1000 800 -1 600 -1.5 400 200 B -2 0

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2

Figure 5.3: Impedance diagram of the Hindmarsh-Rose model. A: Log10 (|imped|) vs. x0 and stimulus period, λ. The peak near x0 = -1.33, λ = 336 signifies a maximum response to inputs of this wavelength. B: Eigenvalues of the linearized matrix of the HR model with dependence on x0. At x0 = -1.33, a pair of complex-conjugate eigenvalues crosses the x0-axis for increasing x0, resulting in a loss of stability of the solution (5.4). The third eigenvalue is negative and does not influence the stability.

64 Chapter 5. The Hindmarsh-Rose model crosses 0, and the corresponding homogeneous solution becomes dominant. This is the point of the Hopf bifurcation, which gives rise to burst firing behavior (see above), the solution Aeiwt (5.4) loses stability, and the nonlinear terms in (5.1-5.3) dominate the dynamics with action potential firing.

The subthreshold resonance is dependent on the time scale of the z variable, which can be seen by changing . In the two-dimensional model with  = 0, the impedance plot is flat without a resonant frequency (not shown). In general, the location of the resonance maximum depends on .

This analysis demonstrates that the HR system has one single subthreshold preferred frequency at λ = 336 that is dependent on the z variable.

5.4 Noisy sine wave stimulation

When a noise term is added to equation (5.1), the stochastic version of the HR model,

3 2 dx = (y − ax + bx − z + I0 + Isignal(t))dt + σdW, (5.5)

dy = (c − fx2 − y)dt, (5.6)   1  dz =  x − (z − g) dt, (5.7) 4 exhibits stochastic action potential firing in response to input signals. Again, W is the standard Wiener process and σ is a constant that determines the noise strength.

Simulations of the stochastic HR equations with sine wave input for long periods of time (1000 times the period of the sine wave input as shown in Figure 5.4) gave enough action potential data for ISIH plots. The typical multipeak structure appears in the histograms (Figure 5.5), and the histogram peak at the input period goes through a maximum for an intermediate noise level indicating stochastic resonance.

The surface plot of the SNR of HR in Figure 5.6A shows typical stochastic resonance curves for fixed input period λ with one maximum as the noise level σ varies. The plot also shows that for fixed noise level σ, the SNR is dependent on the input period λ of the sine wave.

65 Chapter 5. The Hindmarsh-Rose model

(a)

x

t (b) 1.5 Figure 5.4: Voltage (x-variable) trace of the HR model during sine wave (λ = 325, smooth curve) stimulation1 with noise (I0 = −0.5, σ = 0.5).

For high noise0.5 levels, the SNR goes through a maximum for an input period of λ between

250 and 400, near0 the resonant period of the deterministic system for subthreshold signals.

x At lower noise levels, however, the SNR peaks at periods up to λ = 1200 and reaches an -0.5 absolute maximum at λ = 750 for σ = 1.0. The relationship between optimal noise level and input period is-1 an exponential-like function shown in Figure 5.6B. This frequency dependent stochastic resonance is a maximal firing response (SNR) to small periodic inputs at a resonant -1.5 frequency. 0 100 200 300 400 500 600 700 t The frequency dependence of SR is determined by the value of the small parameter , as shown in Figure 5.7. The plot shows a contour plot of SNR for  = 0.0025, cf. Figure 5.6B. Qualitatively, the dependence of SNR on σ and λ is the same as with  = 0.005, but the noise level for optimal detection is higher for smaller .

66 Chapter 5. The Hindmarsh-Rose model

A B 7 7

6 6

5 5

4 4

3 3

2 2

1 1

0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 500 1000 1500 2000 2500 3000 3500 4000 4500

C D

8 7 log(# of events)

7 6

6 5

5 4

4

3 3

2 2

1 1

0 0 0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 Interspike time

Figure 5.5: Interspike interval histograms from sine wave stimulation as in Figure 5.4 (I0 = −0.5). Spikes during a burst produce a large peak near zero interspike times, and the first peak away from zero interspike times is associated with the period of the sine wave. The multipeak (harmonic) structure occurs because one or more subsequent peaks of the sine wave may fail to produce action potentials. A: At low noise levels, spikes are most likely to occur on the peak of the sine wave. Thus, the ISIH has peaks at the input period (here λ = 1000), at multiples of λ (skipping peaks), and near 0 (multiple spikes). B: At intermediate noise levels, skipping of consecutive peaks of the signal is less likely, and the peak at λ is large. C: At high noise levels, the system can cross the firing threshold during all phases of the input signal, and hence, there is not a distinct peak at the input period. D: For different sine wave frequencies, the resonant peaks are at the input period (here λ = 400) and its multiples. The basic shape of the ISIH is preserved.

5.5 SR with α-function stimulation

In order to investigate SR in HR with α-function inputs, we perform 10,000 simulations each with and without an EPSP and hence q = 0.5. An increase in noise amplitude results in

67 Chapter 5. The Hindmarsh-Rose model

A

2.5

2

1.5

SNR 1

0.5

0 0 0 0.5 500 1

1.5 1000 2 2.5 1500 B

0.09 0.12 0.03 2.2

2 0.18 0.06 1.8

1.6

1.4

1.2 0.15 0.21 1

0.8

0.6 0.09 0.12 0.06 0.4 0.03

0.2

100 200 300 400 500 600 700 800 900 1000 1100

Figure 5.6: Signal-to-noise ratio dependence on σ and λ. A: Stochastic resonance is demon- strated by the maximum in SNR for varying σ. Stochastic resonance is also frequency depen- dent. B: Contour plot of SNR. The dependence of the optimal noise level (maximum in A) for each input wavelength is denoted by the solid black line.

increased PF and decreased PM (Figure 5.9A), and 1 − PE goes through a maximum for in- termediate noise levels (Figure 5.9B), exhibiting another occurrence of stochastic resonance.

68 Chapter 5. The Hindmarsh-Rose model

2.5

0. 02 2 0 0. .0 04 8 0 .06

0 1.5 0 .1 .12 0 .14

0 1 0 .16 .0 6 0. 0. 08 14 0.1 0.5 0 0. .02 04

200 400 600 800 1000 1200 1400 1600 1800 2000

Figure 5.7: Contour plot of the signal-to-noise ratio dependence on σ and λ with  = 0.0025. For this , corresponding to a subthreshold resonance period of 471, the dependence of the optimal noise level on the input wavelength (solid black line) also is approximately exponential.

As with sine wave inputs, the simulations exhibit a preferred frequency; the detection probability of an EPSP depends on the time since a previous EPSP. Again, 10,000 simulations, each for varying time delays ∆t = t2 −t1, give PE for the detection of the second EPSP. Figure 5.9 shows

1−PE for the second EPSP as a function of ∆t. The error is minimized (1−PE maximized) for delays between 400 and 700 time steps, depending on the noise level. These delays are in the same range as the time scale of the preferred stochastic frequency, observed with periodic sine wave stimulation and noise. The preferred frequency again shows a dependence on the noise level where an increase in noise decreases the preferred frequency.

5.6 Noise stimulation with no signal

Similar to our investigations of autonomous SR in the HM model, we simulate the HR equations with no input signal for 107 time steps and compute the interspike-interval histogram for each

69 (a)

x

Chapter 5. The Hindmarsh-Rose model t (b) 1.5

1

0.5

0

x

-0.5

-1

-1.5

0 100 200 300 400 500 600 700 t

Figure 5.8: Voltage (x-variable) trace of the HR model under α-function (t1 = 50, t2 = 475) stimulation with noise (I0 = −0.7, σ = 0.5). noise level. The three-dimensional plot of the ISIH against σ in Figure 5.11 shows that again most activity occurs at short interspike intervals. There is also a local maximum at intermediate times for each noise level. This is a preferred intrinsic frequency of the system because it fires at this rate without any external forcing. In the figure, there is no single preferred firing period, but rather a range of periods between approximately 100 and 600. The dependence of the preferred frequency on the noise level again appears to be an exponential-like function (cf. Figures 5.6, 4.4).

This demonstrates autonomous SR in the HR model similar to our studies of HM, and other research on autonomous or coherence stochastic resonance [111].

5.7 Stochastic resonance in the fast subsystem

The different time scales of the noise, the z and the x, y dynamics, and the complicated bifur- cation dynamics of the full HR model motivate us to simplify the system of equations. In order

70 Chapter 5. The Hindmarsh-Rose model

A 1

0.9 PM 0.8

0.7

0.6 E

0.5 1-P 0.4

0.3

0.2 PF 0.1

0 0.5 1 1.5 2 2.5

0.65 B

0.6

0.55 E

1-P 0.5

0.45

0.4 0.5 1 1.5 2 2.5

Figure 5.9: Firing probabilities of the HR model under α-function stimulation with dependence on σ. A: PF , PM for detecting a single EPSP. B: 1 − PE. The typical stochastic resonance shape with maximum signal-to-noise ratio at intermediate noise levels appears. to separate the different time scales, we observe that equation (5.3), which governs z, contains the small parameter . Therefore, z is a slow variable, and we can formally simplify the model

71 Chapter 5. The Hindmarsh-Rose model

1.5

1.25

1.0

E 0.75 1-P

∆t

Figure 5.10: Preferred frequency stochastic resonance with α-function stimulation and noise in the HR model. The probability of reliably detecting the second EPSP (1 − PE) depends on the delay (∆t) following the first one. Traces shown are for σ = 0.75, 1.0, 1.25, and 1.5, with higher noise level corresponding to higher probabilities. For all noise levels, the curve goes through a maximum for ∆t between 400 and 700. Similar to sine wave stimulation (cf. Figure 5.6), the resonance period decreases as the level of noise increases.

by setting  = 0 and z to its resting value z0. Thus, we obtain the reduced two-dimensional stochastic HR model

3 2 dx = (y − ax + bx + I + Isignal(t))dt + σdW, (5.8)

dy = (c − fx2 − y)dt, (5.9) so that I = I0 − z0. Then, z0 plays the role of an additional current bias. The reduced model is capable of firing action potentials, but for the given parameter values it does not exhibit bursting and is similar to the FitzHugh-Nagumo model of tonic firing neurons. We study stochastic resonance in this model under sine wave, α-function, and no signal stimulation with noise to understand the effect of the slow variable.

The dynamics of a two-dimensional dynamical system such as the reduced HR model can be visualized in a phase plane plot (Figure 5.12A). The nullclines and the fixed point are plotted

72 Chapter 5. The Hindmarsh-Rose model

0.05 0.04 0.03 0.02 3 0.01 2.5 0 2

rel. # of events of # rel. 0 1.5 500 1 1000 0.5 1500 Interspike time 20000

Figure 5.11: Three-dimensional interspike interval histogram of the HR model with dependence on the noise level σ without any input signal. Each line is an ISIH for fixed σ. There is a noise level dependent local maximum in the ISIH distribution. Similar to Figure 5.6, this indicates a preferred firing frequency even when no signal is present. together with an exemplary trajectory of the noisy reduced HR model. Most of the time, the trajectory stays in a neighborhood of the stable fixed point, (x0, y0) = (−1.72, −14.1), driven by the stochastic term. Only when the firing threshold is exceeded will there be a large excursion around an unstable fixed point, resulting in the action potential. With a sufficiently large positive input current, I, the cubic nullcline is shifted upwards so that the stable fixed point collides with the unstable fixed point in a saddle node bifurcation, eliminating both steady states. Then, solutions can move towards a periodic limit cycle solution around the remaining fixed point, resulting in continuous action potential firing, called tonic firing. Figure 5.12B summarizes the bifurcation dynamics of the reduced HR model.

With the presence of three fixed points for every value of I between −0.92 and 0.23, correspond- ing to the resting level, threshold, and center of the oscillation, respectively, small changes of total input current I can switch the model between the fixed point and tonic firing. Such small changes can arise from noise or changes in the z variable of the full system. Under sine wave stimulation with noise (Figure 5.13A), the typical stochastic resonance curve with one maxi-

73 (a)

2

1

x 0

-1

-2 Chapter 5. The Hindmarsh-Rose model 0 5 10 15 20 25 A

0 ¸

¸

-5 y

-10

¸ -15 -1.5 -1 -0.5 0 x 0.5 1 1.5 2 B

2

1 x

0

-1

-2

-2 0 2 4 6 8 10 12 I0

Figure 5.12: Dynamics and bifurcation diagram of the reduced deterministic HR model A: Phase plane of the reduced deterministic HR model for I0 = −0.5. The broken lines denote the nullclines and the diamonds are the three fixed points. Overlaid is the trace of a trajectory of the noisy reduced system model. Most of the time, the trajectory stays near the stable fixed point, (x, y) = (-1.72,-14.1). However, when threshold is crossed, a large excursion takes place that encircles the right-most fixed point and corresponds to an action potential. B: Bifurcation diagram of the reduced deterministic HR model. As I0 increases, the stable fixed point becomes unstable through a saddle-node bifurcation at the right lower knee (I0 = 0.23). A homoclinic bifurcation gives rise to tonic action potential firing at I0 = −0.92. Thus, there is hysteresis for I0 in (−0.92, 0.23) because there are both a stable fixed point and a stable cycle present. Another Hopf bifurcation near the left knee does not contribute to firing of action potentials. mum for varying noise levels appears as expected. However, in the reduced model, SNR decays for increased signal periods and there is no frequency that is detected optimally. A comparison

74 Chapter 5. The Hindmarsh-Rose model with Figure 5.6A shows that the z dynamics boost the stochastic resonance for longer periods and a preferred frequency arises from the interplay between the firing and the slow z variable.

A preferred firing frequency also is absent under α-function stimulation with the probability of detection measure (Figure 5.13B). Only for a short delay between two EPSPs is 1−PE increased when the second EPSP catches the tail of the first one for an increased x-deflection; this is the well-known temporal summation phenomenon. There is no maximum for intermediate delays like in the full model (cf. Figure 5.10). In the reduced model with  = 0, the optimal noise level also is dependent on the wavelength of the input signal. Figure 5.14 plots the dependence of the optimal noise level on the λ for different  values, compare to Figures 5.6, 5.7, 5.13A. For smaller  = 0.0025, more noise is needed for optimal detection of all frequencies than for  = 0.005. Another way of looking at the figure is that at for a fixed noise level σ, a longer period signal is detected optimally for smaller . This is as expected because the smaller time constant corresponds to a longer subthreshold resonance period of 471. The reduced HR model, that is  = 0 (from Fig. 5.13A), also exhibits frequency dependence of the optimal noise level. However, in this non-resonant model, the frequency dependence is not as pronounced and varies less with λ than in the system with  6= 0. The figure demonstrates that the subthreshold resonance produced by the slow z dynamics is the major cause of the frequency preference in stochastic resonance in the HR model.

Similarly, the ISIH distribution of the fast subsystem without an input signal (Figure 5.13C, cf. 5.11) is decaying for each σ and no inherent frequency of the model is revealed by the noise input.

In summary, all three forms of inputs that we investigated in the reduced model show no preferred firing frequency. This demonstrates that the z dynamics give rise to the preferred frequency stochastic resonance.

75 Chapter 5. The Hindmarsh-Rose model

A SNR

0.75

1.5 B 0.7

0.65 1.25

E 0.6 1.0

0.55 1-P

0.5

0.45 0.75

0.4 100 200 300 400 500 600 700 800 900 D t C

0.05 0.04 0.03 0.02 0.01 3

frequency 2.5 0 2 0 1.5 500 1 1000 1500 0.5 Interspike time 2000 0

Figure 5.13: Stochastic resonance in the reduced HR model. A: SNR dependence on σ and λ under sine wave stimulation (cf. Figure 5.6A). Again, stochastic resonance is apparent in the maximum for each frequency input. There is no frequency dependence of the maximum for varying noise levels σ. B: Detection probability of the EPSP following an earlier EPSP (cf. Figure 5.10). For all noise levels, the detection probability decays or is approximately constant. C: ISIH distribution of the fast subsystem (cf. Figure 5.11). There is no preferred frequency, and the ISIH distribution is decaying for each noise level σ.

76 Chapter 5. The Hindmarsh-Rose model

2.5

2

1.5

1

0.5

200 400 600 800 1000 1200 1400 1600 1800 2000

Figure 5.14: Comparison of the optimal noise level for detection of signals with period λ for the HR system with  = 0.005 (solid curve),  = 0.0025 (dotted curve), and  = 0.0 (broken curve).

5.8 Stochastic bifurcation analysis

In stochastic systems, bifurcation analysis is replaced by the study of invariant measures that describe the probability distribution of the variables. These can be obtained from the Fokker- Planck equation associated with the model or by computing the probability distribution of the variables from long simulations of the equations [62]. However, stochastic bifurcations have been mainly studied in two-dimensional systems where it is possible to visualize the evolution of the probability distribution function in dependence on the bifurcation parameter.

In the reduced HR model (5.8, 5.9), the stochastic term σdW perturbs the trajectories away from the stable deterministic trajectories and the solution of the model can be represented as R a probability distribution p(x, y), representing a stochastic process. Then D pdxdy over a set 2 D ⊂ R denotes the probability of finding a realization of the stochastic differential equation in the set D. Figure 5.15 shows the development of the probability distribution for increasing I when there is noise with σ = 0.5 and no additional input signal. The plots are two-dimensional histograms of residence probability for x and y obtained from long (106 time steps) simulations

77 Chapter 5. The Hindmarsh-Rose model of (5.8, 5.9).

When I0 is far from the bifurcation point (I0 = −1.5, corresponding to I = 0.13; cf. Figures 5.13A and 5.13B), the real parts of the eigenvalues for the fixed point are large and negative (not shown). Thus, the noise perturbs the trajectories in a small neighborhood of the fixed point (Figure 5.15A) and almost never evokes an action potential such that the probability distribution approximates a Gaussian centered at the fixed point. For I closer to the bifur- cation point, the noise can boost the system above threshold, producing an action potential. During such an action potential, the trajectory roughly follows the nullclines, giving rise to a positive probability near the nullclines (compare Figure 5.15B with 5.12A). Action potentials are more likely to occur close to the bifurcation point where the probability distribution has a bimodal shape with the probability maxima corresponding to the fixed point and the lower limit of the oscillation. This change from a unimodal to a bimodal distribution during stochastic bifurcations is called a phenomenological or P-bifurcation, as described by Arnold [2].

Beyond the bifurcation, tonic firing occurs, and the noise only influences the timing of the action potentials. Hence, the probability distribution is concentrated around the trajectory of the deterministic action potential. Residence probability is positive all along the firing orbit and highest near the minimum value of the action potential (x-variable) because that is where the dynamics are slowest. For details, see Meunier and Verga [120], who performed the first investigations of stochastic bifurcations. Arnold [2] gives an overview of modern stochastic bifurcation theory, and Jansons and Lythe [92] provide an analytical treatment of stochastic bifurcations.

The stochastic bifurcation in the reduced HR system corresponds to the saddle-node and homo- clinic bifurcations of the deterministic two-dimensional system. The appearance of the stochas- tic homoclinic bifurcation is similar to the stochastic Hopf bifurcations in the Fitzhugh-Nagumo and HH models described by Tanabe and Pakdaman [179, 181] The difference of the stochastic homoclinic bifurcation in our model is that it immediately shows a large excursion from the rest state, unlike a Hopf bifurcation. The destruction of an unstable and a stable fixed point

78 Chapter 5. The Hindmarsh-Rose model

I =-0.7 A 0 I =-1.5 B 0

0.07 0.02 0.06 0.018 0.016 0.05 0.014 0.04 0.012 p p 0.01 0.03 0.008 0.02 4 0.006 4 2 0.004 2 0.01 0 0 -2 0.002 -2 0 -4 0 -4 -2 -6 -2 -6 -1.5 -8 y -1.5 -1 -8 y -1 -0.5 -10 -0.5 -10 0 -12 0 -12 0.5 -14 0.5 -14 x 1 x 1 1.5 -16 1.5 2 -16 2

I =+0.1 C 0 I =-0.5 D 0

0.014 0.016 0.012 0.014 0.01 0.012 0.008 0.01 p p 0.008 0.006 0.006 0.004 4 4 2 0.004 2 0.002 0 0.002 0 -2 -2 0 -4 0 -4 -2 -6 -2 -6 -1.5 -1 -8 y -1.5 -8 y -0.5 -10 -1 -0.5 -10 0 -12 0 -12 0.5 -14 0.5 -14 x 1 1.5 -16 x 1 2 1.5 2 -16

Figure 5.15: Probability distribution and stochastic bifurcation of the reduced stochastic Hindmarsh-Rose model for σ = 0.5 and varying I0. A: I0 = -1.5, the distribution is close to a delta function. B: I0= -0.7, decreasing the distance from threshold broadens the distribution, and occasional action potential firing gives positive residence probability near the nullclines, cf. Fig 5.12A. C: I0 = -0.5, the double peaked distribution indicates that a stochastic bifurcation has taken place. D: I0 = 0.1, after the bifurcation, there is tonic firing with positive residence probability along the nullclines, cf. 5.12A. Because action potentials are narrow in time, the residence probability is highest near the baseline of the action potential where the orbit spends most of its time. through a saddle-node bifurcation manifests itself in the stochastic case as the disappearance of the lower maximum in Figure 5.12A.

This stochastic bifurcation is the mechanism through which action potential firing occurs under noise. When an excitatory (Isignal > 0) subthreshold signal arrives, the probability distribution goes through the bifurcation in Figure 5.15 and an action potential is likely to occur. Because the z dynamics are slower, the bifurcation of the fast subsystem is a good approximation of the firing in the full model, where z acts as a modulating parameter. Hence, a resonance from the slow dynamics can influence firing through the z term in (5.3).

79 Chapter 5. The Hindmarsh-Rose model

5.9 Discussion

The HR model, which is a simple polynomial ODE model of bursting thalamic neurons, exhibits subthreshold resonance, stochastic resonance, and a preferred frequency stochastic resonance similar to the experiments and the HM model.

We observed that in the HR model, the noise level required for most faithful transduction of the sine wave inputs to the spike train, i.e., the point of stochastic resonance, decreases in an exponential-like manner with the input period (Figures 5.6B, 5.10). The range of the preferred frequency stochastic resonance period depends on the noise level between λ = 250 and 1200, compared to subthreshold resonance at λ = 336.

We also investigated stochastic resonance in a more realistic context of α-function stimulation and reliability of detection. The stochastic resonance curves are similar to those of the sine wave case, including a preferred firing frequency that depends on the noise level.

Thus, the stochastic and subthreshold resonance phenomena may interact to improve the de- tection of pairs of EPSPs. Even in the absence of an input signal, a preferred firing frequency appears in the ISIH. Our observations of analogous results with ISIH and probability of detec- tion measures imply that the dependence of the optimal noise level on the input frequency is a feature of many systems under noise stimulation.

In order to separate stochastic and subthreshold resonance for a mathematical analysis of the stochastic firing dynamics, the three-dimensional stochastic differential equation system can be split into slow and fast subsystems. The time scale of the slow subsystem determines the subthreshold resonance while the two-dimensional fast subsystem produces action potentials. Addition of noise to the fast variables results in a stochastic bifurcation that gives rise to firing. The evolution of the underlying probability density from a unimodal to a bimodal shape describes this stochastic bifurcation.

Our simulations of the fast subsystem of the HR equations reveal no preferred stochastic firing frequency. This illustrates that subthreshold resonance, introduced by the slow subsystem,

80 Chapter 5. The Hindmarsh-Rose model is necessary for preferred frequency stochastic resonance. The underlying slow dynamics and input signals push the system through the bifurcation to stochastic firing. Hence, subthreshold resonance may influence firing behaviour through this mechanism. From our analysis of the fast and slow subsystems, one might expect that the stochastic resonance peaks near the frequency of subthreshold resonance. However, the interplay between subthreshold resonance, firing, input signal, and noise is complicated and produces the exponential-like dependence of the preferred frequency on the optimal noise level.

Massanes et al. [116] explained such a frequency dependence of SR on the basis that high frequency inputs are less likely, during a signal period, to evoke an action potential on a signal peak because of their short period and hence, shorter residence time near threshold. Since one action potential on top of each signal peak is needed for optimal detection, i.e., a maximal SNR, more noise is necessary at higher frequencies for the voltage to cross threshold in the briefer time. Consequently, the change in preferred frequency of SR arises from the interaction of the signal period with the neuron’s firing probability during a signal period.

However, we found that in the reduced model without resonance the optimal noise level changes less with frequency, implying a contribution of the subthreshold resonance. Also, our findings with α-function stimulation and probability of detection measure with pairs of EPSPs cannot be explained as simply a feature of any SR system. The shape and duration of the EPSPs are constant and the noise needed to evoke a spike should be independent of the timing of the EPSPs. The maximum probability of reliably detecting an EPSP at 400 to 700 time steps after the first EPSP arises from the subthreshold resonance of the system. The frequency dependence in this case is more subtle and an additional mechanism might be involved.

In summary, we have demonstrated stochastic resonance with a noise-dependent preferred fre- quency (or equivalently frequency dependent optimal noise level) in the Hindmarsh-Rose model of the thalamic neuron. Classic and stochastic bifurcation analysis revealed that an interac- tion of the subthreshold resonance with the threshold dynamics gives rise to the frequency preference.

81 Chapter 6 The Resonant Integrate-and-Fire Model

6.1 Introduction

Even the simple polynomial Hindmarsh-Rose system is too complicated for mathematical anal- ysis in the stochastic case. In order to study the stochastic firing behaviour analytically, we require a model that captures the special features such as subthreshold resonance and SR, but is even simpler than the HR system. The simplest differential equation model for neuronal dynamics is the integrate-and-fire model [9], which has been used extensively for the study of firing and network behaviour [100, 160, 161]. Consequently, we set out to create a linear integrate-and-fire model that includes resonance and matches the subthreshold features of the Hindmarsh-Rose model. This is in analogy to conductance based IF models developed from HH models of neuronal dynamics [45]. Modified integrate-and-fire neurons that incorporate a subthreshold resonance or bursting have been introduced recently by Izhikevich [91]. Richard- son et al. [23, 150] created and analyzed an integrate-and-fire model with Ih resonance and demonstrated that noise in conjunction with this resonance can affect the firing response rate. A similar, but more complicated nonlinear integrate-and-fire model was used by Smith et al. [164] to model thalamocortical neurons and burst firing.

The features we want to capture from HR are resting level, resistance, and subthreshold res- onance, as well as its firing dynamics, such as threshold and post-action potential recovery reset. In the following, we use this model to investigate stochastic resonance and frequency dependence of stochastic firing in connection with subthreshold resonance.

82 Chapter 6. The Resonant Integrate-and-Fire Model

6.2 Matching the model and parameter estimation

6.2.1 Subthreshold properties

As the simplest ODE model of neuronal firing, we create a resonant integrate-and-fire model (RIF) where the first two equations of the HR system are replaced by one linear equation, namely

dV = AV + BZ + I, (6.1) dt dZ = CV + DZ + E, (6.2) dt where V is a voltage-like variable and Z describes the slow dynamics of an adapting current, similar to the linear third equation of HR (5.2). The parameters A, B, C, D, E, and I are necessary for matching the properties of the HR system. The constant term in (6.1) is a current-like input and consequently is named I. As in the one-dimensional integrate-and-fire model [195], when V reaches the threshold Vthres, V is reset to Vreset, and an action potential is recorded. For the Z variable, an action potential in the HR model corresponds to an increase by a certain amount and, in RIF, we reset Z to Z + Zreset.

We attempt to choose the parameters such that the resting level, input resistance, threshold, variable reset, and impedance to periodic inputs are the same in the two models in order to make the current and noise levels and the input frequencies comparable between the RIF and HR models.

The equilibrium state of HR can be obtained by setting the LHS of the equations (5.1-5.3) to 0, and elimination of y and z yields

3 2 0 = −ax + (b − f)x − 4x + g + I0 + c. (6.3)

This cubic equation can be solved for our parameter values to give one stable real steady state x0 = −1.44027, corresponding to z0 = −0.66109.

Furthermore, we want to match the resistance of HR, that is, the deflection of x resulting from

83 Chapter 6. The Resonant Integrate-and-Fire Model

change in I0. Taking the derivative of (6.3) with respect to I0 gives

dx dx dx 0 = −3ax2 + 2(b − f)x − 4 + 1 dI0 dI0 dI0 dx 1 ⇒ = 2 . dI0 3ax − 2(b − f)x + 4

The resistance for our parameter values and the steady state x0 is dx/dI0 = 0.224112. Addi- tionally, the z-deflection in response to I0 has to have equal magnitude. The derivative of the z-equation of HR (5.3) set equal to 0 gives

dz dx = 4 , (6.4) dI0 dI0 another condition for the parameter matching.

For the rest state of RIF to match HR, we solve (6.2) for Z and eliminate it from (6.1), which results in the equation B 0 = AV − (DV + E) + I. D Taking the derivative with respect to I gives the condition

dV BC dV 0 = A − + 1. dI D dI

In the RIF model, the resting value for Z and the deflection dZ/dt can be obtained from (6.2):

0 = CV + DZ + E, dV dZ 0 = C + D . dI dI

These equations have to be satisfied under V = v0, Z = z0, and dV/dI = dx/dI0 while matching the parameters. The condition (6.4) translates into C = −1/4D, reducing the number of parameters by one.

In the HR system, we also found a subthreshold resonance to input periods of λ = 336 from linear analysis under sine wave stimulation. The impedance is given by

 2fx 4 −1 imped (ω, x ) = iω + 0 + + 3ax2 − 2bx . HR 0 1 + iω  + 4iω 0 0

84 Chapter 6. The Resonant Integrate-and-Fire Model

In the linear RIF model, we want a similar subthreshold resonance. The resonance structure can be computed using a periodic input signal I(t) = δeiwt, and the system can be linearized around its equilibrium state (V0,Z0) = (−1.44027, −0.66109),

dV = AV + BZ + δeiwt, dt dZ = CV + DZ. dt

Letting V = V¯ eiwt, Z = Ze¯ iwt, and δ = 1 in these equations, and then dividing by eiwt, the system becomes

V¯ iω = AV¯ + BZ¯ + 1,

Ziω¯ = CV¯ + DZ.¯

Solving for Z¯ in the second equation and eliminating Z¯ in the first yields the result:

 BC −1 imped = V¯ = iω − A − . RIF iω − D This expression corresponds to the complex impedance and describes the magnitude and phase of the voltage responses to current inputs. Note that impedRIF is independent of I, E, Vthresh, and Vreset.

Now we can try to obtain values for A, B, C, D, E, and I such that resting value, resistance, and the resonance of the RIF model is similar to those of the HR model. In order to do this matching, we implement a stochastic minimization algorithm to minimize the mean square difference between |impedRIF | and |impedHR|, restricted by the equilibrium and resistance conditions. However, numerical experiments did not lead to a satisfactory match, and we had to introduce a size factor K for the impedance, so that the mean square difference of

|impedRIF |/K and |impedHR| is minimized. Figure 6.1 shows a satisfactory match with the parameter values A = −0.0320, B = −1.3258, C = 0.00024997, D = −9.9988 · 10−4, and K = 14.8543. The factor K means that all inputs (noise and signal) into RIF have to be divided by K in order to match them to similar results from the HR model. This does not pose

85 Chapter 6. The Resonant Integrate-and-Fire Model

2.5

2 HR

RIF

1.5 | imped 1

0.5

0 0 100 200 300 400 500 600 700 800 900 1000 wavelength

Figure 6.1: Result of the least square fit of the impedance magnitude of the HR and RIF models with dependence on ω. The figure shows the dependence of |imped| of the HR and the RIF models on the period λ = 2π/ω. Here, it is important to note that the resonance curve of the HR system is dependent on the rest level, and we fit RIF to HR at the steady state x0 = V0 = −1.44027. a problem for our stochastic resonance simulations. The values of I and E are determined from the required steady state, resulting in I = −0.9226 and E = −3.0099 · 10−4.

6.2.2 Threshold properties

The passive properties of the two models are determined from matching, and it remains to determine Vthresh, Vreset, and Zreset so that the firing dynamics are similar. These parameters can be determined by applying current steps until the model starts firing. Figure 6.2 shows a trace of the HR equations under square wave steps. The firing starts approximately at x = −1.1, which we consequently choose as Vthres. The minimum value of x after an action potential during the burst is above the firing threshold of -1.1. For the integrate-and-fire model, we have to choose Vreset < Vthres, and here we take Vreset = −1.11. Obtaining Zreset is more difficult

86 Chapter 6. The Resonant Integrate-and-Fire Model

2

1

x 0

-1

-2 0 200 400 600 800 1000

-0.3

-0.2

-0.4

z -0.5

-0.6

-0.7 0 200 400 600 800 1000 t

Figure 6.2: Trace of the HR model under step currents of 500 time steps duration with Iinj(t) = 0.0742 (no action potentials evoked) and 0.0743 (burst of two action potentials evoked), demonstrating threshold and reset. because the amount by which Z is increased by an action potential varies in a burst. The figure shows that during an action potential evoked by a step current, z is increased approximately by 0.1, and we take this value for Zreset. This turns out to produce burst firing behaviour similar to HR. In general, there is more confidence in the subthreshold parameters than in the firing parameters, and we cannot expect the same firing patterns exactly. Weaknesses of the integrate-and-fire model include the fact that an action potential is instantaneous and not followed by an absolute refractory period and the lack of nonlinear terms.

87 Chapter 6. The Resonant Integrate-and-Fire Model

6.3 Stochastic resonance in the RIF model

When Gaussian white noise is added to the first equation (6.1), the RIF model becomes a two-dimensional Ornstein-Uhlenbeck process until V reaches threshold

dV = (AV + BZ + I + Isignal(t))dt + σdW, (6.5)

dZ = (CV + DZ + E)dt. (6.6)

The noise, in combination with an input signal Isignal(t), makes it possible for V to reach the firing threshold and evoke an action potential.

The RIF model is capable of firing bursts and single action potentials in response to only noise inputs (Figure 6.3), much like the HR model. An interspike interval histogram of the output with no input, plotted in Figure 6.4, reveals that the frequency of the subthreshold resonance also appears in the random spike train. The location of the resonant peak does not change with increasing noise, unlike in the HR model.

There is also classical stochastic resonance in response to subthreshold sine wave inputs and noise. A plot of SNR against the noise level in Figure 6.8 shows that the optimal noise level depends on the input period with higher noise level required for optimal detection of short period inputs. Near the resonance period, λ ≈ 336, the SNR drops off less sharply for increased noise than at other periods.

Under α-function stimulation, there is also SR resonance as measured by detection reliability.

Again, 1−PE goes through a maximum at intermediate noise levels (Figure 6.6A). The preferred firing frequency of the model also appears in response to α-function inputs that mimic EPSPs. As in the HR model, the detection probability of an EPSP depends on the time since a previous

EPSP. Figure 6.6B shows 1 − PE for the second EPSP as a function of ∆t. A second EPSP is detected most reliably (that is 1 − PE is maximal) 300 time steps after the first one.

88 Chapter 6. The Resonant Integrate-and-Fire Model

0 bursts Threshold

-2

-4

x

-6

-8

-10 0 100 200 300 400 500 600 700 800 900 1000

-0.3

-0.4

z -0.5

-0.6

-0.7 0 100 200 300 400 500 600 700 800 900 1000 t

Figure 6.3: Trace of a realization of the RIF model with σ = 1.8. The noise acts mainly in the V direction because it is only present in the first equation, while Z dynamics are smoother. When V reaches Vthres = −1.1, the system is reset and a new trace starts. A number of consecutive threshold crossings results in an increase of Z that is large enough to make the LHS of equation (6.6) negative which causes a decrease of V away from threshold. Thus, after a burst, V makes an excursion away from threshold during which Z can recover towards the Z equilibrium, and another burst can occur.

6.4 Comparison with the non-resonant integrate-and-fire model

The one-dimensional integrate-and-fire model has been studied extensively as a simple model for neuronal dynamics (see [9] for references). It consists of a linear differential equation

dV = GV + I + I (t) (6.7) dt signal where G signifies the conductance and I is a current bias that determines the rest level. We use this model to mimic the dynamics of the reduced Hindmarsh-Rose model, and hence we want to match the resistance and steady state values. Resistance of the reduced HR model can be

89 Chapter 6. The Resonant Integrate-and-Fire Model

0.9 0.8 0.7 0.6 0.5 0.4 0.3 2.8 0.2 2.6

#0.1 of events 2.4 0 2.2 0 2 1.8 500 1.6 1000 1.4 1500 1.2 1 2000

Figure 6.4: Interspike interval histogram of the RIF model with Vreset = −1.11, Zreset = 0.1, showing its dependence on σ. The first peak near 0 interspike interval time arises from tonic- like firing with small interspike times. For higher noise levels, a second peak appears in the histogram that reflects firing at a preferred frequency, here near λ = 336. calculated in the same way as for the full HR system, yielding

dx = (4x + 3x2)−1. dI 0 0

At the resting level x0 = −1.44027, this gives a value of 2.1640. The one-dimensional integrate- and-fire model has a resistance of −1/G, and consequently we choose G = −0.4621, leading to I = −0.6656. The threshold and reset parameters of RIF are the same as for IF.

The impedance of the integrate-and-fire model can be calculated easily as

−1 impedIR = (iω − G) which does not have a maximum (not shown), and hence the IF model has no resonance.

Consequently, the stochastic IF model does not show a preferred frequency of the stochastic resonance in Figure 6.7A. In response to sine wave stimulation, the optimal noise level is nearly constant and only for short wavelengths is significantly more noise needed to evoke stochastic firing that is coherent with the input. Also, autonomous SR is absent and the ISIH, in response to noise stimulation, shows only one peak near 0 interspike time (Figure 6.7B). These results are in analogy to the reduced HR model (5.8, 5.9) which does not exhibit the frequency preference

90 Chapter 6. The Resonant Integrate-and-Fire Model

A 0.4

0.3

0.2 SNR 0.1

0 0 0 0.5 500 1 1000 1.5 1500 2 2000

B 1.8

1.6

1.4

1.2

1 0.05 0.15 0.25 0.1 0.8 0.2

0.6

0.4

0.2

200 400 600 800 1000 1200 1400 1600 1800

Figure 6.5: Signal-to-noise ratio dependence of the RIF model on σ and λ (cf. Figure 5.6). A: SNR plotted against σ and λ. B: Contour plot of SNR, with the optimal noise level denoted by the thick black line.

91 Chapter 6. The Resonant Integrate-and-Fire Model

Figure 6.6: Preferred frequency stochastic resonance with α-function stimulation and noise in the RIF model. A: 1 − PE for detecting one EPSP goes through a maximum at intermediate noise. B: The probability of reliably detecting the second EPSP (1 − PE) depends on the delay (∆t) following the first one. Traces shown are for σ = 0.75, 1.0, 1.25, and 1.5 as marked. There is a maximum in detection reliablity for ∆t near 300. Compare this to the corresponding data from HR in Figure 5.10, where the preferred frequency depends on the noise level.

92 Chapter 6. The Resonant Integrate-and-Fire Model

A 0.2

0.15

0.1 SNR 0.05

0 0 0 0.1 0.2 200 0.3 400 0.4 0.5 600

B

0.1 0.08 0.06 0.04 1

frequency 0.02 0.8 0 0 0.6 200 400 0.4 600 800 0.2 1000

Figure 6.7: Stochastic resonance of the one-dimensional integrate-and-fire model. A: Plot of SNR dependence on σ and λ under sine wave stimulation (cf. Figure 6.5). The dependence of the optimal noise level on the input period is not as strong as in RIF. B: ISIH distribution with no noise input (cf. Figure 6.4). There is no peak at an intermediate interspike time, and hence no preferred frequency. of the full HR model.

6.5 Stochastic analysis of RIF

Because of the simple linear form, a mathematical analysis of the RIF equations can give a solution in closed form. A solution of a stochastic process, such as (V,Z) described by (6.5,

93 Chapter 6. The Resonant Integrate-and-Fire Model

6.6), takes the form of a probability distribution P that is determined by the Fokker-Planck (FP) equation associated with the system. The FP equation is a convection-diffusion type parabolic equation that determines the evolution of P in time. In simple systems, the FP equation can be solved or approximated analytically [153, 197]. In our system, following Risken ([151], pp. 54ff), the drift coefficients for FP are

DV (V, Z, t) = AV + BZ + I,

DZ (V, Z, t) = CV + DZ + E, and there is only one diffusion coefficient (due to the presence of noise only in the first equation)

2 D11(V, Z, t) = σ /2, and D12 = D22 = 0. Here, it is useful to shift the variables V and Z in order to bring the rest state (V0,Z0) to (0, 0) and eliminate I and E. That is v = V − V0, and z = Z − Z0, resulting in D1 = Av + Bz, and D2 = Cv + Dz. The Fokker-Planck operator for this system is

X ∂ X ∂2 L P = − D P + D P. FP ∂x i ∂x ∂x ij i i i,j i j ∂ ∂ σ2 ∂2P = − (Av + Bz)P − (Cv + Dz)P + . ∂v ∂z 2 ∂v2

Then, P satisfies the deterministic partial differential equation

∂P = L P (6.8) ∂t FP with the initial condition given by a delta distribution at a deterministic starting point (v1, z1),

P (v, z, 0) = δ(v − v1)δ(z − z1).

For example, after an action potential, we would have v1 = Vreset +V0 and z1 = Z +Zreset +Z0.

Note that here z1 depends on Z, the state of the system before the action potential, which is be important for calculating the ISIH. Here, we treat z1 as a constant.

Because the RIF equations, and hence its Fokker-Planck operator, are linear, it is possible to solve the PDE (6.8) analytically. We can take advantage of the linearity of LFP because starting with an initial delta distribution, the solution P = P (v, z, t) must be a Gaussian distribution for all times t > 0 [151].

94 Chapter 6. The Resonant Integrate-and-Fire Model

Then, the equation can be solved by expressing P through its Fourier transform P˜ with respect to (v, z)

Z −2 P (v, z, t) = (2π) exp(i(k1v + k2z))P˜(k1, k2, t)d(k1, k2).

P˜ is also a Gaussian and fulfills a PDE that is derived from the Fokker-Planck equation for P by transforming ∂/∂v = ik1, ∂/∂z = ik2, and, by integration by parts, v = i∂/∂k1, z = i∂/∂k2, resulting in the first order PDE

˜ ˜ ˜ 2 ∂P ∂P ∂P σ 2 ˜ = (Ak1 + Bk2) + (Ck1 + Dk2) − k1P. (6.9) ∂t ∂k1 ∂k2 2

The initial condition is transformed into

P˜(k1, k2, 0) = exp(−i(k1v1 + k2z1)). (6.10)

Because P˜ is a Gaussian, we can use the Ansatz:

 1 1  P˜(k , k , t) = exp −i(k M + k M ) − k2N − k k N − k2N (6.11) 1 2 1 1 2 2 2 1 11 1 2 12 2 2 22 where Mi, the moments, and Nij, i, j = 1, 2, the variances of P˜, are functions of t. The moments and variances can be calculated by inserting (6.11) into the PDE equation for P˜ (6.9). Then

˜ ˜ ˜ 2 dP ∂P ∂P σ 2 ˜ = (Ak1 + Bk2) − (Ck1 + Dk2) − k1P dt ∂k1 ∂k2 2 ! dM dM 1 dN dN 1 dN ⇔ − ik 1 − ik 2 − k2 11 − k k 12 − k2 22 P˜ 1 dt 2 dt 2 1 dt 1 2 dt 2 2 dt

= Ak1(−iM1 − k1N11 − k2N12) + Bk1(−iM2 − k1N12 − k2N22) ! σ2 + (Ck (−iM − k N − k N ) + Dk (−iM − k N − k N ) − k2 P.˜ 2 1 1 11 2 12 2 2 1 12 2 22 2 1

95 Chapter 6. The Resonant Integrate-and-Fire Model

Collecting terms of the same powers in k1,2 shows that solution of this equation requires dM 1 = AM + BM , dt 1 2 dM 2 = CM + DM , dt 1 2 dN 11 = 2AN + 2BN + σ2, dt 11 12 dN 12 = CN + (A + D)N + BN , dt 11 12 22 dN 22 = 2CN + 2DN . dt 12 22

Initial conditions for Mi(t) and Nij(t) are

M1(0) = v1,M2(0) = z1,Nij(0) = 0, i, j = 1, 2, so that P˜ fulfills (6.10). This system of ODEs can be solved using the Green’s function of (6.1, 6.2), G(t) = exp(γt) where γ is the matrix   AB γ =   . CD G can be used to calculate the moments     M1 v1   (t) = G(t)   . (6.12) M2 z1 Similarly, the ijth variance satisfies [151] Z t σ2 Nij(t) = Gi1(τ)Gj1(τ) δτ. (6.13) 0 2 This can be seen by taking the second derivative of this equation, d2N σ2  d d  ij = G G + G G . (6.14) dt2 2 j1 dt i1 i1 dt j1 Then, because G is a Green’s function and dG = γG, that is dt dG 11 = AG + BG , dt 11 12 dG 22 = CG + DG , dt 12 22 dG 12 = AG + BG , dt 12 22

= CG11 + DG12,

96 Chapter 6. The Resonant Integrate-and-Fire Model we can replace the derivatives on the RHS of (6.14) to get d2N σ2 11 = ((AG + BG )G ) + G (AG + BG )) dt2 2 11 12 11 11 11 12 d = (2AN + 2BN ), dt 11 12 d2N σ2 22 = ((CG + DG )G + G (CG + DG )) dt2 2 11 12 21 21 11 12 d = (2CN + 2DN ), dt 12 22 d2N σ2 12 = ((AG + BG )G + G (CG + DG )) dt2 2 11 12 21 11 11 12 d = (AN + BN + CN + DN ). dt 12 22 11 12

2 Integrating this with the conditions Nij(0) = 0, dN11/dt(0) = σ /2, dN12/dt(0) = dN22/dt(0) = 0 gives the solutions for the variances.

In order to perform the integration, it is necessary to split γ into its spectral decomposition. This is possible if the determinant det(γ) = AB − CD is not equal to zero and the matrix has full rank. This is the case for our parameter values. A complete biorthogonal set of γ is given by the eigenvalues 1 p λ = (A + D) ± (A + D)2 − 4BC, 1,2 2 associated eigenvectors  λ1,2 − D  C u1,2 =   , 1 and adjoint eigenvectors 1 v1,2 = ± (CD − λ2,1) . λ1 − λ2 Then   v1 (u1 u2)   = I, v2 and γ has the spectral decomposition

t t t t γ = λ1u1v1 + λ2u2v2.

We can use this to compute G,

λ1t t t λ2t t t G(t) = exp(γt) = e u1v1 + e u2v2,

97 Chapter 6. The Resonant Integrate-and-Fire Model and

1   λ1t λ2t G11(t) = e (λ1 − D) − e (λ2 − D) , λ1 − λ2 B   λ1t λ2t G12(t) = e − e , λ1 − λ2 C   λ1t λ2t G21(t) = e − e , λ1 − λ2 1   λ1t λ2t G22(t) = −e (λ2 − D) + e (λ1 − D) . λ1 − λ2

When this form of G is inserted into (6.13), the variances can be computed as

σ2 (λ − D)2 2BC 1 2λ1t (λ1+λ2)t N11(t) = 2 (e − 1) + (e − 1) 2(λ1 − λ2) 2λ1 λ1 + λ2 (λ − D)2  + 2 (e2λ2t − 1) , 2λ2 σ2B λ − D A − D 1 2λ1t (λ1+λ2)t N12(t) = 2 (e − 1) − (e − 1) 2(λ1 − λ2) 2λ1 λ1 + λ2 λ − D  + 2 (e2λ2t − 1) , 2λ2 σ2B2  1 2 2λ1t (λ1+λ2)t N22(t) = 2 (e − 1) − (e − 1) 2(λ1 − λ2) 2λ1 λ1 + λ2 1  + (e2λ2t − 1) . 2λ2

We are now in a position to perform the inverse Fourier transform and compute P ,

Z −2  P (v, z, t) = (2π) exp ik1(v − M1) + ik2(z − M2)) 1 1  − k2N − k k N − k2N d(k , k ). 2 1 11 1 2 12 2 2 22 1 2

The covariance matrix   N11 N12 N =   N12 N22 is positive definite for t > 0, and hence its inverse N−1, as well as the root N1/2 and its inverse N−1/2, exist. The inverse can be written explicitly as   1 N22 −N12 N−1 = . det(N)   −N12 N11

98 Chapter 6. The Resonant Integrate-and-Fire Model

Then we can define new integration variables

1/2 1/2 −1/2 a1 =[N(t) ]11k1 + [N(t) ]12k2 + i([N(t) ]11(v − M1(t))

−1/2 + [N(t) ]12(z − M2(t))),

1/2 1/2 −1/2 a2 =[N(t) ]12k1 + [N(t) ]22k2 + i([N(t) ]12(v − M1(t))

−1/2 + [N(t) ]22(z − M2(t))), which changes the exponent to

1 1 ik (v − M ) + ik (z − M )) − k2N − k k N − k2N = 1 1 2 2 2 1 11 1 2 12 2 2 22 1 1 − (a2 + a2) − [N(t)−1] (x − M (t))2 2 1 2 2 11 1 1 −[N(t)−1] (x − M (t))(z − M (t)) − [N(t)−1] (z − M (t))2. 12 1 2 2 22 2

The Jacobian of this change of variables is det(N−1/2) and

Z Z ∞ 2 1 2 2 2 −1 exp(− (a1 + a2))d(a1, a2) = exp(−a /2))da = (2π) . 2 −∞

We obtain an expression for P as long as v is below threshold:

 1 P (v, z, t) = (2π)−1det(N)−1/2exp − [N(t)−1] (v − M (t))2 (6.15) 2 11 1 1  − [N(t)−1] (v − M (t))(z − M (t)) − [N(t)−1] (z − M (t))2 , or 12 1 2 2 22 2 " 1 1 (v − M )2 P (v, z, t) = exp − 1 (6.16) p N 2 2π det(N) 2(1 − 12 ) N11 N11N22 #! 2N (v − M )(z − M ) (z − M )2 − 12 1 2 + 2 . N11N22 N22

The second expression shows that P is a bivariate Gaussian distribution centred around (M1,M2)

2 with variances N11 and N12 and correlation coefficient N12/N11N22.

A plot of the moments M1(t), M2(t) in Figure 6.8 shows the time course of the two moments of RIF starting from Vreset just below threshold and z incremented by Zreset = 0.1 from its rest value. M1, the v moment, decreases steeply at the beginning and undershoots 0 before returning to rest after about 300 time steps. There is a small overshoot at t ∼ 400. This behaviour can

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Figure 6.8: Plot of the moments M1(t) (above) and M2(t) (below) of the probability distribution of the RIF model, starting from v = 0.331 and z = 0.1, corresponding to V1 = −1.11 and Zreset = 0.1. M1, the first moment of v, performs a large excursion into the negative regime before approaching its resting value 0. explain the peak near 300 interspike time in Figure 6.4, because after a reset, the average v value moves away from threshold making crossing the stochastic threshold unlikely. M2, the z moment, decreases initially because of the high v values, turns around near t = 300, before it approaches rest after about 400 time steps. The behaviour of the first moments is independent

t t of the noise level σ, since (M1M2) (t) = (v1z1) exp(γt). The return time is approximately the resonance period of 336. This can be explained by the form of the expression (6.12) which shows that the moment equation has the same eigenvalues as the deterministic, linear part of RIF.

The time evolution of the variances Nij is shown in Figure 6.9, and both N11 and N22 increase

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Figure 6.9: Plot of the second moments N11(t) (above), N12(t) (middle) , and N22(t) (below) of the RIF model with the same starting values as in Figure 6.8, and σ = 1.0. from an initial value of 0 towards a steady state. This indicates the growing spread of the probability distribution function over time towards a steady state distribution. The covariance

N12 initially decreases from 0 before it turns around and approaches a negative steady state. The plots are for σ = 1.0, but in the expressions for the variances, the noise level appears as a multiplicative factor and different noise levels will change the variance magnitudes but not the shapes. This explains why the ISIH peak in Figure 6.4 does not change location with noise.

6.6 First passage time

We are interested in analyzing the First-Passage-Time (FPT) problem for the stochastic reso- nant integrate-and-fire model [148, 149, 198]. Our aim is to study the probability distribution

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Figure 6.10: Plot of the probability distribution of the first passage time of the RIF model obtained from Monte-Carlo simulations. The starting value for each trial is V1 = 0.331 and Z1 = 0.1, the same as in Figures 6.8, 6.9. Compare this to Figure 6.8. of the timing of an action potential,

Tthres = inf {t ≥ 0; V (t) ≥ Vthres | (V,Z)(0) = (Vreset,Zreset)} (6.17) the time of the first threshold crossing after a reset. However, the FPT problem for Ornstein- Uhlenbeck processes only can be solved analytically in some special cases [102]. Consequently, we employ Monte-Carlo simulations of the RIF model to obtain an approximation of the first- passage times. Note that first passage time plots are not identical to the ISI histograms, because after an action potential Z is incremented by Zreset, while the FPT Monte-Carlo simulations always start at the same Z value. Figure 6.10 shows a plot of the FPT distribution of the RIF model calculated from 10,000 realizations starting with the same initial conditions. In many trials a threshold crossing occurs almost immediately when v is still near threshold, and

102 Chapter 6. The Resonant Integrate-and-Fire Model

firing probability is high at short times. If no such crossing occurs, it is unlikely for v to reach threshold for about 150 time steps. The threshold crossing distribution reaches a maximum between 350 and 400 time steps.

6.7 Discussion

We have constructed a resonant integrate-and-fire model that resembles subthreshold and firing properties of the HR model. Simulations of this stochastic model show stochastic firing prop- erties and SR similar to HR, including a dependence of the optimal noise level for detection of sine wave inputs on the frequency. As in our other models, the ISIH without an input signal, as well as EPSP detection reliability, shows a maximum near the resonance frequency. However, in contrast to HR, the autonomous SR maximum does not shift with increasing noise. A math- ematical analysis of the equations reveals that the moments of the RIF model are independent of the noise level and the probability of a stochastic threshold crossing is low for about 300 time steps after reset. The eigenvalues of the moments are the same as the eigenvalues of the deter- ministic part of the RIF model, and hence the preferred frequency in the output spike train, visible in the ISIH, is the same as the subthreshold resonance frequency. The SNR dependence on noise level and frequency of sine wave inputs is an exponential-like function, similar to our HR simulations. In the RIF model, this exponential dependence can be entirely explained by the mechanism suggested by Massanes and Vicente [116], see the discussion in Chapter 5. The comparison of RIF with the simple IF model without subthreshold resonance shows that the frequency preference of SR is a consequence of the subthreshold resonance. The RIF model is linear, and hence nonlinear dynamics are not necessary for the frequency dependence of SR and the autonomous SR.

These results show the value of a very simple model for a mathematical analysis of the features of the more complicated models. The RIF model also is an obvious choice for simulations of large networks of resonant neurons because of its computational simplicity.

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7.1 Introduction

In animals and humans, neurons are always part of a network and receive synaptic inputs from other neurons. A prominent feature of many neural networks is large scale synchronization, such as appears in an EEG. Synchronized activity is especially important in the thalamus where is involved in the generation of the sleep/wake cycle. Some of these oscillations occur in the frequency range of the subthreshold resonance of thalamocortical neurons.

Membrane resonance may play a role in the genesis of these oscillations and synchronization in neural networks. With our models of resonant neuronal dynamics, we are in a position to investigate the role of resonance in a noisy network. In the following, we investigate the influence of noise on networks of the resonant neuron models from the previous chapters. We want to answer the question whether subthreshold resonance can give rise to synchronized network activity at or near the resonant frequency. Because of this inherent frequency preference, we expect the appearance of synchronized burst oscillations due to noise at a frequency close to the subthreshold resonance frequency. This is an extension of earlier work by Wang et al. [207] who found synchronization of tonic firing by noise in an HH network.

The thalamic network of TC and nRT neurons is connected by both inhibitory and excitatory synapses [50]. Thalamic oscillations are controlled by GABAergic inhibitory activity of nRT neurons [47]. This leads to the question if both inhibitory and excitatory coupling can lead to stochastic synchronization, as possible in a deterministic inhibitory network [163].

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For constructing a neural network of our model neurons, we have to describe the synaptic connections that form connections between thalamic neurons.

After the presynaptic neuron fires an action potential, the resulting depolarization of the presy- naptic terminal causes secretion of neurotransmitter which diffuses across the synaptic cleft between the neurons. Interaction of the neurotransmitter with receptors in the postsynaptic membrane causes either an inhibitory or excitatory postsynaptic potential response. In pulse coupling, which models synaptic coupling, the individual neuron models are independent as long as no action potentials occur [90]. We add a synaptic current

Isyn = gsyns(V − Vsyn)

to the voltage equation of our physiological models where the maximal conductance gsyn, the gating variable s which describes activation after an action potential, and the equilibrium potential Vsyn depend on the type of synapse and neurotransmitter involved. Gating obeys

ds s (V ) − s = ∞ presyn dt τs(Vpresyn) where s∞ is a sigmoidal function that describes the activation of the synapse when the presynap- tic neuron Vpresyn fires an AP; the details are given in the Appendix. We investigate networks with global (all gsyn = g = const. for all synapses) and nearest neighbour coupling (gsyn = g 6= 0 when | i − j |= 1 for neuron numbers i and j). The neurons form a ring without a boundary, that is, neuron 1 also is connected to the last neuron.

7.2 A network of Huguenard-McCormick neurons

Thalamocortical neurons normally are not connected to each other but only project to other nuclei. However, we still investigate a network of HM neurons coupled by AMPA synapses in order to see if excitatory coupling can interact with resonance to create synchronized oscillatory states.

Figure 7.1 shows the firing behaviour in a network of 50 globally coupled HM neuron in a raster plot where a black dot in row i at time t denotes that this neuron fired an action

105 Chapter 7. Network Synchronization potential. For low noise, the spiking activity in A is scattered and appears to be random. The coupling strength g is low so that one AP cannot recruit other neurons to fire in response to a synaptic input. For increased noise level in Figure 7.1B, more APs appear and are grouped in vertical bands. This signifies synchronized activity where all or most neurons tend to fire bursts of APs at approximately the same time. Time between the bursts is approximately 660 ms, corresponding to an oscillation frequency of 1.5 Hz. For even higher noise, shown in Figure 7.1C, synchronized activity persists but the interburst time is reduced, the oscillation frequency is higher, and because of more stochastic spiking, the bands are less sharply defined. In Figure 7.1D, there is almost continuous stochastic activity in response to high noise and the synchronized bands have disappeared.

The dependence of the firing pattern on the synaptic coupling strength for a fixed noise strength is shown in Figure 7.2. For low values of g, coupling does not have a significant effect and the firing is mostly random. Stochastic synchronization occurs when g is large enough in B. For larger g in panel C, the bursts are prolonged and the bursting frequency is decreased. Very strong coupling at physiologically unrealistic levels increases the duration of synchronized bursts until there is continuous firing (not shown).

The dependence of the frequency of the synchronized bursting oscillation can be visualized in a cumulative ISIH in Figure 7.3 from all neurons in the network. In addition to a peak near 0 interspike times, which stems from fast firing during a burst, there is a second peak at a few hundred milliseconds which denotes the time between bursts as in Figure 7.1. In Figure 7.3A, the location of this second peak depends on the noise level and decreases in a nonlinear manner from 600 ms for low noise to 300 ms for high noise. For varying coupling strengths (Figure 7.3B), the location of the second peak does not change, but stays constant at 400 ms (for this constant noise level σ = 0.4 nA). For high coupling strengths, the peak and bursting oscillations disappear. Note that the interspike time does not directly correspond to the frequency of the oscillations because the duration of bursting activity is not constant and varies with the noise level. Also, the ISIH plot does not demonstrate synchronization in itself but only shows the preferred interspike time, which is the time between consecutive bursts.

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Figure 7.1: Raster plots of simulations of a globally coupled network of 50 HM neurons in dependence on the noise level for a coupling strength gsyn = 0.002 µS. For low noise in A, only a small number of action potentials are evoked and there is no synchronized activity. Stochastic synchronization occurs at intermediate noise levels in panels B and C as vertical bands of black dots, denoting synchronized action potentials. For high noise a large number of action potentials occurs and the synchronized activity is destroyed.

A similar progression of synchronized activity appears with nearest neighbour coupling as demonstrated in Figure 7.4. Similar to SR, stochastic synchronized activity occurs at inter- mediate noise levels and diminshes under high noise, A-C. The stripes of synchronized firing are not vertical anymore, compare to the synchronized bands in Figure 7.1, but spread through the network with a finite speed. For the dependence on g, D-F, stripes appear in the raster plot when one neuron has a large enough impact on a neighbouring neuron to recruit it to fire a burst, see Figure 7.2. Note that with nearest neighbour coupling the synaptic strength g has to be higher than with global coupling, because each neuron only receives inputs from two other neurons, instead of from the whole network.

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Figure 7.2: Raster plots of simulations of a globally coupled network of 50 HM neurons in dependence on the coupling strength for a constant noise level σ=0.4 nA. Stochastic synchro- nization occurs when the coupling strength gsyn is increased. This is visible in B when action potential firing, denoted by the black dots, occurs approximately at the same time for many neurons. The period of the synchronized activity bands without any input signal is about 300 ms in B and 1000 ms in C. 108 Chapter 7. Network Synchronization

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Figure 7.3: Cumulative interspike interval histogram from a network of 50 HM neurons with global coupling and its dependence on the noise level σ (A) and coupling strength g (B). The peak for short interspike times arises from spikes during a burst. For each noise level, the second peak near 500 ms shows the period of the antiphasic oscillations that appeared in Figure 7.1B. This second peak broadens with increasing σ and the location of the maximum decreases. In contrast, the interspike time between bursts does not change with g.

7.3 A network of nRT neurons with inhibitory coupling

In contrast to TC neurons, which are not coupled to each other in vivo, neurons in the nRT form a network with inhibitory coupling by GABAA synapses, and we show results of similar simulations of the RET model as with the HM network. We did not observe synchronized activity in a globally coupled network even for a wide range of noise and coupling values tested.

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Figure 7.4: Raster plots of simulations of 50 HM neurons with nearest neighbour coupling. For the dependence on the noise level (left column A-C), synchronized activity occurs at interme- diate noise level, similar to global coupling, compare to Figure 7.1. Dependent on the coupling strength g (right column D-F), synchronized activity appears when g is large enough, cf. Figure 7.2.

110 Chapter 7. Network Synchronization

With nearest neighbour coupling, there is also no immediately obvious synchronization in Fig- ure 7.5A. However, the firing pattern seems to have a checkerboard pattern, suggesting that neighbouring neurons fire bursts at the same frequency but with opposite phase. This can be elucidated by plotting the odd and even numbered neurons separately. Figure 7.5B shows that the even and odd form two synchronized clusters of neurons discharging at a frequency of approximately 1.8 Hz in antiphase to each other. As shown in Figure 7.5C, higher noise disturbs the synchronized pattern, but does not destroy it. These simulations were performed for a coupling strength of g = 0.2 µS, which was obtained by Destexhe et al. [43] by fitting ex- perimental data. However, the value of the coupling strength does not have a strong influence on the synchronization pattern, as shown in Figure 7.5D.

This inhibitory network is quiescent when no noise is present, however, once the synchronized pattern is established, noise is not necessary to maintain this. Figure 7.5E shows that even after the noise is switched off, synchronous oscillations of two clusters are maintained and are more regular than with noise. Hence, the antiphasic synchronization is not a stochastic phenomenon but a stable state of the deterministic network which persists for long periods of time (not shown). This deterministic oscillatory state is dependent on the coupling strength, illustrated in Figure 7.5F. For small g, noise evokes firing with small synchronized clusters, but the whole network does not show distinct synchronized oscillations. When the noise is switched off, all firing activity eventually tapers off. Thus the noise stabilized the synchronization even when the deterministic system cannot sustain this activity.

The presence of antiphasic activity in the deterministic network was found by Destexhe et al. [42], who developed the model of reticular thalamic neurons and networks. Our stochastic simulations are similar to findings by Tiesinga and Jose [190], who studied clustering in a similar network with inhibitory coupling depending on the network size and the noise level. De Vries and Sherman [53] also found that noise can increase the parameter range of synchronized activity in gap-junction coupled excitable cells.

A cumulative ISIH of all neurons in the network, in Figure 7.6, similar to Figure 7.3, shows the

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Figure 7.5: Raster plots of simulations of a network of 50 RET neurons with nearest neighbour coupling by GABAergic synapses. A: No obvious synchronized states appear, but there seems to be a checkerboard pattern. B: Plot of the same data as in A with even and odd neurons shown separately. There are two clusters which are synchronized, with neighbouring neurons bursting in antiphase to each other. C: The antiphasic pattern persists even for higher noise, but the firing is less regular. D: With increased coupling strength (4 times as high as in A,B) the antiphasic activity persists. E: The appearance of synchronized clusters is not due to the noise, because they persist when the noise is switched off at t = 5000 ms. F: For low coupling strength, noise evokes firing and some synchronized activity, which is not maintained without noise.

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Figure 7.6: Cumulative interspike interval histogram from a network of 50 RET neurons with nearest neighbour coupling, cf. Figure 7.3. The peak for short interspike times arises from spikes during a burst. For each noise level, the second peak near 500 ms shows the period of the antiphasic oscillations that are visible in Figure 7.5B. This second peak broadens with increasing σ and the location of the maximum decreases. different nature of the synchronized activity under inhibitory coupling. The peak at 500 ms interspike time which corresponds to the interburst time is largest and most distinct for small noise levels. At this low coupling strength (g = 0.2 µS), noise is not needed to sustain the oscillation. Also, the location of this peak does not shift much to shorter interspike times for higher noise, in contrast to the excitatory network. With increasing noise level, the peak flattens out, reflecting the more stochastic firing that can occur before and after a burst, resulting in shorter interspike times.

7.4 Physiological network of TC and reticular neurons

Oscillations in the thalamus stem from a combination of TC and nRT neurons, and we combine our models of the inhibitory RET network and excitatory HM network to study the influence of noise on synchronization in a physiological model of the thalamus. The network connection parameters are taken from Destexhe et al. [44, 50], who developed a deterministic network of the thalamus. A similar network was studied by Bazhenov et al. [10, 11].

113 Chapter 7. Network Synchronization

Figure 7.7: Schematic of the network of RET and TC neurons connected by AMPA and GABAergic synapses. The first and the last neuron in the network are connected to form a ring without a boundary. The figure is adapted from [44].

In addition to the GABAA mediated synapses between nRT neurons, the nRT projects to TC neurons in other thalamic nuclei. These connections are formed by GABAA as well as slower

GABAB synapses. The TC neurons in turn project AMP A synapses back to the nRT. Hence, in our network consisting of HM and RET model neurons, the HM neurons are not connected to other HM neurons but only project excitatory AMP A synapses to the RET neurons, see Figure 7.7. Here, we let the ith TC neuron project to three RET neurons i − 1, i, i + 1. The same connectivity applies to the inhibitory connections from RET to TC neurons. As in our other network models, the last and first neurons are connected to avoid boundary effects.

The network with neuron and coupling parameters from [44] does not exhibit apparent syn- chronized activity, shown in Figure 7.8A. There are, however, occasional bands of nearly si- multaneous bursts in the TC neurons while the population of RET neurons does not show synchronization or synchronized clusters. For higher noise (Figure 7.8B), the pattern does not change significantly; however, when the noise is switched off at t = 5000 ms, a stable pattern of oscillation both in TC and RET neurons appears, which persists for long periods of time. The noise is necessary for the initiation of the pattern because the deterministic network is quiescent, similar to the RET network discussed above.

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Figure 7.8: Raster plots of simulations of a network of 50 RET and 50 TC neurons. A: Even for low noise, there is no synchronized firing of the whole network, and only in the TC network do bands of synchronization appear. B: The TC synchronized activity still appears for higher noise. When the noise is switched off at t = 5000 ms synchronized oscillations of both the TC and RET network appear. C: Increased GABAA conductance yields synchronized oscillations of the whole network. D: Voltage traces from a TC and a RET neuron from C.

Under noise, RET neurons are not recruited to oscillate with the TC neurons for the parame- ter values from [44]. In contrast, for increased inhibitory coupling strength gGABAA =0.06 nA (tripled from the fitted value), there are stochastic oscillations that involve both TC and RET neurons as demonstrated in Figure 7.8C. The same effect can be achieved by spreading the con- nections between the RET and TC neurons to 5 neurons of the other nucleus, which effectively also is an increase in coupling strength. Figure 7.8D shows voltage traces of a TC neuron and a RET neuron from Figure 7.8C. The TC bursts at a frequency of about 2.8 Hz are rebound responses to IPSPs from RET neurons. RET bursts are evoked by EPSPs from TC neurons, and the burst frequency of the RET neurons of 1.4 Hz is approximately one half of the TC

115 Chapter 7. Network Synchronization neurons. The RET neurons split into two clusters which fire at the same frequency but with 180 degree phase difference.

7.5 Hindmarsh-Rose network

In the Hindmarsh-Rose model, we model synaptic coupling in the standard way with an addi- tional variable that describes synaptic activation. By analogy with the physiological synapses that we used in the thalamic network, a current-like term, Isyn = gijs(x − Vsyn), is added to the first x-equation of HR (5.1).

We investigate an HR network with global and nearest neighbour coupling. In this model, we were not able to find synchronized activity under inhibitory coupling (Vsyn = −2), even in trials with different values of coupling strength, noise level, and synaptic time constant. This lack of synchronization can be explained by the fact that the HR model does not exhibit rebound bursting. That is, an IPSP cannot evoke action potentials and activity does not spread through the network but instead suppresses firing. In contrast, excitatory coupling (Vsyn = 1) by definition increases the likelihood of an action potential in a postsynaptic neuron, and we investigate if stochastic synchronization of burst firing is possible in this context.

Figure 7.9 shows the noise dependent behaviour of a network of 50 globally coupled HR neurons in a series of raster plots. Coupling strength is small so that one synaptic potential does not exceed threshold, and for low noise there is little activity and no synchronization. For higher noise (Figure 7.9B), scattered bands appear in the raster plot when the synchronized bursting activity spreads over the whole network. However, the synchronization is not sustained and disappears after one or two synchronized bursts. For even higher noise, the synchronization is stable and sustained (Figure 7.9C). In the plot, there is synchronized bursting at a period of approximately 1000 time steps, which is stable even for long periods of time (not shown here). When the noise is very high (Figure 7.9D), many action potentials are evoked which destroys the synchronization.

The frequency of the synchronized bands is dependent on the noise level which can be demon-

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Figure 7.9: Raster plots of simulations of a network of a 50 HR neurons with global coupling. Stochastic synchronization occurs at intermediate noise levels. The bands of synchronized activity are nearly vertical, showing a rapid recruitment of the whole network. At high noise, some synchronized oscillations remain but additional spikes appear. strated by a plot of the cumulative ISIH from all cells in Figure 7.10. There is a maximum at interspike times between 400 and 1000 time steps. At higher noise levels, the time between consecutive bursts is shorter. This is in analogy to the ISIH of single HR neurons, plotted in Figure 5.11.

The appearance of bands of synchronized activity is dependent on the resonance of the neuron models, as demonstrated by changing the value of the slow time scale . When  = 0.0025, half of the value in Figure 7.9, the frequency of burst firing, plotted in Figure 7.11, is about half, as expected. If the same experiment is performed with  = 0, that is, in a model without resonance, no synchronized activity is observed.

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A progression of network activity from random firing to synchronization also occurs for constant noise and changing coupling strength. At low coupling strength, the noise evoked activity in the network is unsynchronized, as shown in Figure 7.12A. For larger values of g, bands of

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119 Chapter 7. Network Synchronization synchronized activity appear when the coupling is strong enough for recruitment of the whole network, but the oscillations do not stabilize into a consistent pattern (Figure 7.12B). Figure 7.12C shows stable stochastic synchronized activity at high coupling strength, but the burst are prolonged and the oscillation frequency is lower.

In a network with nearest-neighbour coupling, network synchronization also occurs. Figure 7.13 shows raster plots of such a network and its dependence on noise level (left column) and coupling strength (right column). In general, higher values of g are necessary to observe synchronization because each neuron only receives inputs from two other neurons, not the whole network, cf. Figure 7.9. The bands are not vertical but spread throughout the net with a delay between the burst of neighbouring cells. With the dependence on noise (Figures 7.13A-C), stochastic synchronization occurs for intermediate noise levels. However, the coupling strength has to be large enough so that bands of synchronized activity can spread even at low noise, in order to observe synchronization in Figure 7.13A. For fixed noise and variable coupling strength (Figures 7.13D-F), again bands of synchronization appear for sufficiently large g, but the pattern is not as regular as with global coupling, cf. Figure 7.9C.

7.6 Resonant integrate-and-fire neuron network

In our RIF model, we model synaptic coupling by simply reseting Vj to Vj + Vsyn,ij when a threshold crossing occurs in neuron i . Typically, one excitatory synaptic input is not enough to kick a resting neuron over threshold from rest, and consequently Vsyn should be chosen smaller than Vthres − V0. Then

X dVi = (AVi + BZi + I + Vsyn,ijδ(t − tj))dt + σdWi, (7.1) j

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where the reset amount, Vsyn,ij, denotes the coupling strength between neurons i and j, and tj denotes the times neuron j fires an action potential.

Figure 7.14 demonstrates that such a network of pulse coupled RIF neurons is able to produce

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Figure 7.13: Raster plots of simulations of 50 HR neurons with nearest neighbour coupling. Synchronized activity occurs at intermediate noise level (left column A-C), similar to the pattern with global coupling in Figure 7.9. With the dependence on the coupling strength g (right column D-F), synchronized activity appears when g is large enough.

121 Chapter 7. Network Synchronization

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Figure 7.14: Raster plots of simulations of 50 RIF neurons with nearest neighbour coupling. With the dependence on the noise level (left column A-C), synchronized bands appear at inter- mediate noise levels. When the coupling strength g is varied (right column D-F), synchroniza- tion occurs when g is large enough. In this model, the coupling is instantaneous and the bands of synchronized activity are vertical, in contrast to the HR network shown in Figure 7.13.

122 Chapter 7. Network Synchronization

synchronized activity under excitatory coupling, that is Vsyn > 0. The raster plot shows vertical bands of stochastic synchronization for an intermediate noise level. In contrast for a low noise level, there is not much overall activity and no sustained synchronization activity appears. For σ = 1.0, there is a sustained synchronization, demonstrated by vertical bands of synchronized action potentials. As in the other models, the bands are made up of bursts of multiple action potentials because V stays near the threshold after a threshold crossing. There are, however, less APs in a burst than in HR, cf. Figure 7.9. The stochastic oscillations are destroyed at higher noise levels. The intervals between the synchronized bands are approximately 300 time steps long which is close to the subthreshold resonance (336 time steps). In contrast to the other model networks we investigated, which had synaptic coupling modelled by a dynamic variable, the bands are nearly vertical even with nearest neighbour coupling. This can be understood by the resetting mechanism in the RIF network where synaptic spread is instantaneous. With the dependence on the resetting value, Vsyn, which corresponds to the coupling strength, syn- chronized bands of threshold crossings occur when the reset is large enough (Figure 7.14D-F).

The frequency of the network oscillations does not change with increasing Vsyn. In this model, global coupling gives rise to similar synchronized bursting as with nearest neighbour coupling because the synaptic reset is immediate.

The dependence of the frequency of the synchronized bands on the noise level is shown in Figure 7.15. The ISIH has a maximum at interspike times between 400 and 1000 time steps, and the time between consecutive bursts is shorter at higher noise levels.

In contrast to the HR model, cf. Figure 7.10, the cumulative ISIH in Figure 7.15 shows that the frequency of the synchronized oscillation does not change as strongly with increasing noise. At low noise levels, the firing is more irregular than in the HR model, and for intermediate noise levels, synchronized activity gives rise to a peak near 300 time steps. For increasing noise, the location of this peak shifts towards shorter interspike times. This is analogous to the autonomous stochastic resonance of a single RIF neuron, which occurs at a fixed frequency independent of the noise level.

123 Chapter 7. Network Synchronization

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Figure 7.15: Cumulative interspike interval histogram from a network of 50 RIF neurons, dependent on the noise level. When the noise is strong enough, spiking activity splits into bursts with short interspike times and intraburst firing which stems from the synchronized activity of the network. This second interspike peak is located near 300 time steps, and does not change strongly with increasing noise level σ. Here, Vsyn = 0.07.

The synchronization period is close to the resonant period of the deterministic RIF model because the eigenvalues of the stochastic first moments of the probability distribution function are the same as the eigenvalues of the matrix that describes the single deterministic neuron model. This can be seen by performing a moment analysis of the network similar to the one presented in Chapter 6 for a single neuron. The matrix describing the network is a tridiagonal matrix with blocks of the RIF coefficient matrix on the diagonal, because the single neurons are independent of each other as long as no threshold crossings occur. Hence, the eigenvalues of the network moment matrix are the same as of the single neurons with multiplicity of the size of the network.

7.7 Discussion

In our studies of networks of neuron models with a subthreshold resonance, we found syn- chronized oscillations that are evoked or sustained by noise. In the HM, HR, and RIF models with excitatory coupling, noise generates oscillations that involve the whole network, which is

124 Chapter 7. Network Synchronization quiescent without noise. These oscillations take the form of bursts of action potentials with a bursting frequency close to the subthreshold resonance frequency of the single neuron mod- els. In all of these networks, the oscillation frequency increases in a nonlinear manner with increasing noise.

Under inhibitory coupling, we observed synchronized activity only in the RET model, which is capable of rebound burst firing in response to hyperpolarization. Under nearest neighbour cou- pling, the deterministic model splits into two clusters which are synchronized with antiphasic firing between the clusters. The addition of noise disturbs this pattern, but a large amount of noise is necessary to destroy it. The deterministic oscillations are dependent on the strength of the synaptic coupling. We found that for coupling strength that are too small for deterministic oscillations, noise could stabilize synchronized cluster oscillations. Similar clustering and sta- bilization of burst oscillations have been observed before in a number of networks of excitable systems [53, 67, 190]. In contrast, we showed that networks of HR and RIF neurons are not able to produce global or cluster oscillations when coupled by inhibitory synapses. A physiological network that incorporates TC and RET neurons with excitatory coupling and inhibitory feed- back also produces synchronized activity with TC neurons firing two bursts for each RET burst. TC neurons are all synchronized, while the RET neurons split into two antiphasic clusters.

Our results show that subthreshold resonance can be involved in synchronization; this has been suggested before by Richardson et al. [150] but not investigated in a neuronal network. There are some studies of noise effects in networks of tonically firing neurons without subthreshold res- onance and bursting, and synchronization of fast spiking was observed [207] at an intermediate noise level. Other studies have dealt with spontaneously oscillating or tonically firing neurons where it is easier to define a phase and quantify synchronized activity [5, 59]. In our systems, the periodicity of spontaneous network oscillations is not exactly the same as the subthresh- old resonance period, which otherwise could serve as a natural phase measure. These findings suggest a possible mechanism for the generation of brain oscillation and a role for noise in the brain.

125 Chapter 8 Conclusion

8.1 Summary of results

In this thesis, we have investigated the influence of noise on the firing behaviour of thalamic neurons and their models. Our preeminent finding is that the subthreshold resonance can interact with the noise to confer a preferred frequency in the stochastic spike train. This preferred frequency is manifest as a maximum in the signal-to-noise ratio in response to the combination of periodic signal and noise inputs. In neuronal networks with excitatory coupling, noise generates synchronized burst firing through its interaction with subthreshold resonance.

Our whole-cell patch clamp experiments on nRT and MGB thalamocortical neurons in rat brain slices demonstrate the occurrence of stochastic resonance. SR depends on the frequency of the periodic subthreshold current input and exhibits a frequency preference for periodic in- puts between 1 and 3 Hz. Although SR occurs in sensory cells [17, 54, 93, 110], neocortical neurons [154], and hippocampal neurons [166], this is the first experimental report of an inter- action between subthreshold resonance and stochastic resonance in excitable cells. Both the subthreshold resonance and the frequency dependence of SR arise from the T-type calcium cur- rent, as demonstrated by blockade of both phenomena by application of Ni2+, a known blocker of IT in thalamic neurons.

Simulations of the model equations of nRT and TC neurons show SR, consistent with the experimental findings. The preferred frequency in SR depends on the optimal noise level in a nonlinear manner, with higher noise favouring higher frequencies. That is, the noise level

126 Chapter 8. Conclusion required for most faithful transduction of the input signal frequencies, the point of stochastic resonance, increases with the input frequency. Subthreshold resonance in these neurons occurs at 2 Hz and the preferred stochastic firing frequency varies between 0.5 and 6 Hz. In the polynomial Hindmarsh-Rose model, the preferred frequency of SR is between λ = 250 and 1200 input period, compared to subthreshold resonance at λ = 336. Thus, an interaction between input signal, noise, and subthreshold resonance selects a preferred frequency of stochastic firing in the models.

Our simulations of the fast subsystems of the HR, HM, and RET equations revealed no pre- ferred stochastic firing frequency. This illustrates that subthreshold resonance, introduced by the slow subsystem, is necessary for a preferred frequency in stochastic resonance. The un- derlying slow dynamics and input signals move the system through a stochastic bifurcation to stochastic firing. Hence, subthreshold resonance may influence firing behaviour through this mechanism. From our analyses of the fast and slow subsystems, one might expect that the stochastic resonance peaks near the frequency of subthreshold resonance. However, the inter- play between subthreshold resonance, firing, input signal, and noise is complex in producing the exponential-like dependence of the preferred frequency on the optimal noise level.

Even with only noise (with no other input signal), stochastic resonance occurs in the experiments and model simulations. Interspike interval histograms obtained from recordings from RET and TC neurons and simulations of the HM, RET, and HR models show a maximum at interspike times corresponding to timing between bursts evoked by noise only. This is a preferred frequency of the system because it fires at this rate without external forcing. The location of the maximum depends on the noise level in a nonlinear manner, similar to the frequency dependence of maximal SNR with sine wave inputs. The preferred frequency in stochastic autonomous firing, or autonomous SR, is in the same frequency range as that of subthreshold resonance. This stochastic frequency preference is dependent on the subthreshold resonance, as demonstrated by blockade of IT . This also demonstrates that the preferred frequency in stochastic resonance is not simply caused by the input signal but is a manifestation of the internal dynamics of the system.

127 Chapter 8. Conclusion

In the experiments and models, we also investigated stochastic resonance in a more realistic context of α-function stimulation using a measure based on reliability of detection. The stochas- tic resonance curves are similar to those of the sine wave case, demonstrating that noise can improve the detection of single, or pairs of, EPSPs. In the HM and HR models there also are preferred frequencies with α-function stimulation that are dependent on the time lag between consecutive EPSPs.

The simple resonant integrate-and-fire model that we constructed from HR retains most of the essential features of the more complicated models, such as subthreshold resonance, burst firing, stochastic resonance, and a preferred stochastic firing frequency. With sine wave input, the SNR in RIF displays a frequency dependence similar to the other models.

The explanation for the frequency dependence of the optimal noise level given by Massanes and Vicente [116] (discussed in Chapter 5) applies to the exponential dependence that we found in the sine wave case in all models. However, this explanation does not apply to our findings with α-function stimulation and probability of detection measure with pairs of EPSPs. The shape and duration of the EPSPs were constant and the noise needed to evoke a spike should be independent of the timing of the EPSPs. The maximum probability of reliably detecting an EPSP following a previous EPSP arises from the subthreshold resonance of the system. The frequency dependence in this case is more subtle and might involve an additional mechanism. In the linear RIF model, the preferred frequencies of α-function detection reliability and autonomous SR do not change significantly with the noise level. Hence, this phenomenon is not a consequence of the interaction between subthreshold resonance and noise, but must be caused by the nonlinear dynamics in the more complex models.

In the linear RIF model, it was possible to obtain an analytical description of the solution as a time dependent probability distribution in the form of an evolving Gaussian distribution. This solution reveals an explanation for the fixed preferred frequency because the eigenvalues of the moments are independent of the noise level and are the same as the deterministic eigenvalues which give rise to the subthreshold resonance. Consequently, the preferred frequency of SR is

128 Chapter 8. Conclusion the same as the subthreshold resonance frequency. With sine wave stimulation, this preferred frequency is modified by the mechanism discussed above, and SNR has a nonlinear dependence on the noise level.

Our network studies show that noise can promote synchronization of a population of neurons by interaction with the subthreshold resonance. Noise evokes firing in a network of quiescent neurons, and under excitatory coupling, this stochastic activity can synchronize to an oscillatory state that involves the whole network. The frequency of this network oscillation varies with the noise level, similar to the frequency of maximal SNR.

With inhibitory coupling, a different behaviour arises and synchronization only appears in the RET model, which exhibits LTS burst firing in response to hyperpolarization. The other models do not show rebound behaviour and consequently, inhibitory potentials simply decrease the firing activity in the network. The deterministic RET model, on the other hand, has two different stable states, one quiescent and one oscillatory with two synchronized clusters that fire in antiphase. Noise destabilizes the quiescent state and forces the network to oscillate in a stochastic cluster pattern that is stable even under higher noise. When the coupling strength is low, the deterministic oscillations are unstable. However, noise can induce random firing which shows some limited synchronized activity. Hence, noise can extend the parameter range of synchronized activity, as observed in other networks of coupled oscillators [53]. A physiological network of the thalamus that incorporates HM and RET neurons with excitatory coupling and inhibitory feedback also produces synchronized activity. We observed synchronization in the TC neurons and synchronized clusters of the RET neurons, which fire at half the frequency of the thalamocortical bursts.

8.2 Relevance and implications

The research presented in this thesis illustrates a novel aspect of stochastic resonance in neu- rons and suggests a physiological role for membrane resonance. The subthreshold membrane resonance confers a preferred frequency for detection of sine wave and EPSP inputs and au-

129 Chapter 8. Conclusion tonomous stochastic resonance. In the context of neurons receiving inputs, which take the form of periodic waves or single PSPs [172], we can interpret the interdependence between noise and preferred frequency as a neuron modulating its output frequency through changes in noise. That is, the preferred output frequency changes when the noise level in a neuron changes because of altered membrane noise or an increase in the overall firing activity in the brain as observed experimentally in neocortical neurons in vivo [48].

In thalamic neurons, as in other resonant neurons, the preferred frequency would amplify post- synaptic responses to input spike trains that arrive at the matching frequency, selecting specific inputs out of the synaptic bombardment experienced by neurons. Thus, noise intensity may act as a control parameter for information processing reliability. Our studies show that subthresh- old resonance generates the preferred frequency of stochastic resonance. Thus, the detection of periodic signals, as well as EPSPs, improves when the input frequency matches the reso- nant frequency. This suggests a role for subthreshold membrane resonance, a feature of many different types of neurons. In this context, it is interesting to note that the ionic current, IT , which is responsible for subthreshold resonance, is present mostly in the dendrites of thalamic neurons, the location of EPSP generation. In contrast, the cell bodies have less resonance [46] and IPSPs, which arrive mostly at the soma, would interact less with the resonance. The frequency dependence of the optimal noise level has significance for synaptic integration and rhythmogenesis in a noisy environment. In the cortico-thalamocortical network that generates oscillations [170], the preferred frequencies for SR would amplify input action potential trains that arrive at the matching frequency.

Our studies of noisy neuronal networks suggest a role for subthreshold resonance in the gener- ation and control of thalamic oscillations. Synchronization of the thalamus as we observed in our model studies is critical for brain function, including conscious behaviour and sleep states. A combination of noise and resonance properties can control the frequency and stability of these oscillations. Increased synchronization based on combined membrane resonance and SR may facilitate generalized absence seizures, a condition when the cortico-thalamocortical system displays increased synchronization and disrupted brain function [6].

130 Chapter 8. Conclusion

A connection is plausible between the SR frequency preference and the strong 5 Hz frequency component in brain activity. Because our experiments and the model simulations were per- formed at room temperature, the time constants of membrane ion currents would be shorter in vivo, resulting in a higher resonance frequency at normal body temperature of mammals.

8.3 Problems and future research

A limitation of our studies is the use of Gaussian white noise that ignores autocorrelation in the noise. Godivier and Chapeau-Blonde [29] showed that noise from multiple random synaptic inputs also can evoke SR in neuron models. Long noise autocorrelation times on the order of one-tenth of a second can interfere with the membrane time scale of the model and alter SR. When the noise autocorrelation time is much shorter than the resonance time scale, however, the noise autocorrelation has negligible influence on resonance [71]. In our studies, the period of the resonant frequencies ranged in tenths of seconds, justifying the use of uncorrelated white noise. However, the distinction between signal and noise is at best difficult in the brain, and information content of an input may depend on the type and state of a neuron.

For our simulations, we only considered point neurons, ignoring geometry and heterogeneous ion channel distributions of the cells. These factors can have an effect on the integration and firing behaviour of neurons. However, our goal was to investigate the basic mechanism of interaction between noise and membrane resonance. The effects that we observed can be expected not to be significantly different in more detailed neuron models. Our experimental and modelling studies could be extended to other types of neurons and models, other types of excitable cells, noise, resonances, and oscillatory systems with different coupling mechanisms, and in in vivo studies of resonance and synchronization. Recently, Richardson et al. [150] published research on the interaction of noise and Ih resonance in models of hippocampal neurons and drew conclusions similar to ours concerning the significance of noise in resonant neurons.

Studies of neuronal networks in live animals are beyond the scope of this thesis, but experiments on an intact thalamus in unanesthetized animals could confirm if our suggested synchronization

131 Chapter 8. Conclusion by noise occurs in vivo. Many types of drugs, including anesthetics, could be used to influence the noise level by blockade of overall brain activity, or blockade of IT for abolishing membrane resonance. Technical difficulties of drug application and noise input would have to be overcome first.

On the mathematical side, the mechanism of interaction of noise and neuronal dynamics is not well understood, mainly because the tools available for the analysis of stochastic differential equations are limited to very simple systems. In the small noise case, a multiscale analysis of the model equations and even networks might provide approximations of dynamical behaviour and firing statistics. This could be performed most easily in the RIF and HR models. The first passage time problem has only been solved for one-dimensional systems [138, 147] because of a singularity in the corresponding integral equation. New approximation methods would be necessary to solve the problem for multi-dimensional systems with a subthreshold resonance, as in the RIF and HR equations. In the RIF model, the preferred SR frequency is dependent on noise level for sine wave inputs but does not change in α-function detection reliability and autonomous SR. This linear system is probably most accessible to an analytical study. However, the threshold introduces a discontinuity.

A theory of coupled stochastic oscillators, similar to the studies of weak coupling in determin- istic coupled oscillators [16, 78] is needed to explain our network results. The deterministic theory can predict synchrony or entrainment depending on the type of oscillators and coupling parameters. In a stochastic network, synchronization is a probabilistic phenomenon of the random spike trains of individual neurons. A better measure of stochastic synchronization of neuronal firing is needed to quantify the effects of noise, coupling strength, and other parame- ters on the generation and synchronization of networks. So far, stochastic synchronization has only been studied analytically under periodic inputs with a natural definition of phase [26, 215], and in discrete oscillators [183]. In our studies, however, there is no input signal and the noise gives rise to spontaneous synchronized activity at a frequency depending on the noise strength. Hence, no natural definitions of phase and synchronization measures are apparent.

132 Chapter 8. Conclusion

In conclusion, we have demonstrated stochastic resonance with a noise-dependent preferred frequency in thalamocortical and reticular thalamic neurons and their models. The preferred frequency of SR results from a subthreshold membrane resonance. This IT -resonance enables the neuron to select and amplify inputs at the resonant frequencies. Then, the addition of white noise current can induce neuronal firing with a preferred frequency that stems from the membrane resonance in the stochastic output spike train. This suggests a physiological role for subthreshold resonance and noise in information processing by these neurons. The frequency dependence of the optimal noise level has implications for the integration of synaptic inputs and generation of oscillations in the noisy environment that neurons experience in vivo. Based on our network studies with coupled resonant neurons we propose that noise can produce synchronized oscillations, linking them to the brain rhythms produced by the thalamus and possibly in absence epilepsy.

133 Glossary

ACSF Artificial cerebrospinal fluid AMPA Amino-3-hydroxy-5-methylisoxazoleproprionic acid AP Action potential CNS Central nervous system DC Direct current DIC Differential interference contrast microscopy EEG Electroencephalogram EPSP Excitatory postsynaptic potential FP Fokker-Planck equation FPT First-passage time FT Fourier transform GABA γ-amino butric acid HH Hodgkin-Huxley HM Huguenard-McCormick HR Hindmarsh-Rose model I-V Current-Voltage relationship IF Integrate-and-fire IPSP Inhibitory postsynaptic potential IR Infrared ISI Interspike interval ISIH Interspike interval histogram LTS Low-threshold spike MGB Medial geniculate body ODE Ordinary differential equations nRT Nucleus reticularis thalami PSP Postsynaptic potential RCC Reactive current clamp RET Reticular neurons of the thalamus RIF Resonant Integrate-and-fire RK Runge-Kutta method SNR Signal-to-noise ratio SR Stochastic resonance TC Thalamocortical ZAP Swept sine wave

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148 Appendix A Model Parameters

A.1 Huguenard-McCormick model (HM)

The Huguenard-McCormick model of thalamocortical neurons has the form dV C = −(I + I + I + I ) + I + I (t)) (A.1) m dt l Na K T 0 signal with a fast voltage-dependent sodium current (INa), a voltage-dependent potassium current (IK ), a voltage-dependent low threshold calcium current (IT ), and a voltage-independent leak current (Il), [82, 118]. The ionic currents are in the typical Hodgkin-Huxley form with

3 INa = gNamNahNa(V − VNa), (A.2) IK = gK hK (V − VK ), (A.3) 2 IT = gT mT hT (V − VCa), (A.4) Il = gl(V − Vl). (A.5)

The activation (mNa) and inactivation variables (hNa, hK ) for INa and IK were taken from a model of Traub and Miles [193], as used in models of thalamc neurons by Destexhe et al. [46, 47] and obey the differential equations dm = α (V )(1 − m) − β (V )m, dt m m dh = α (V )(1 − h) − β (V )h, dt h h with V + 50 α = 0.32 , mNa 1 − exp(−(V + 50)/4) V − 23 β = −0.28 , mNa 1 − exp(−(V − 23)/5)

αhNa = 0.128 exp((−(V + 46)/18), 4 β = , hNa 1 + exp((−(V + 23)/5)

149 Appendix A. Model Parameters

V + 48 α = 0.032 , hK 1 − exp((v + 48)/5)

βhK = 0.5 exp(−(V + 53)/40).

IT is determined by activation and inactivation variables that obey dm m (V ) − m T = T ∞ T , dt τmT (V ) dh h (V ) − h T = T ∞ T dt τhT (V ) where mT and hT are the activation and inactivation variables that relax to their voltage dependent equilibrium values mT ∞ and hT ∞(V ), respectively, with time constants τm,hT (V ), and −1 mT,∞(V ) = (1 + exp(−(V + 62)/6.2)) , −1 τm,T (V ) = (exp(−(V + 132)/16.7) + exp((V + 16.8)/18.2)) + 0.612, −1 hT,∞(V ) = (1 + exp((V + 84)/4)) ( exp((V + 467)/66.6), if V < −80 mV, τh,T (V ) = exp((V + 22)/(−10.5)) + 28.0, if V ≥ −80 mV.

The parameter values are VNa = 50.0, VK = −100.0, Vl = −63.0, (in mV), gNa = 100, gK = 80, 2 2 gl = 0.04 (in mS/cm ). The membrane capacitance is Cm = 1.0µF/cm with a total membrane area of 40,000 µm2. All time constants were scaled for a temperature of 34◦C.

The conductance of IT is dependent on the membrane potential V and has the form [Ca2+] − [Ca2+] exp(−V ξ) F g (V ) = 2P F V ξ i o (V − V )), ξ = . T T 1 − exp(−V ξ) Ca RT Here F = 96500 C mol−1 is the Faraday constant and R = 8.3145 J K−1mol−1 is the universal −6 3 2+ gas constant. We take the maximal T-conductance PT = 0.005×10 cm /s, [Ca ]o = 2 mM, −1 ξ = 1/13 mV , and VCa = 120.0 mV. We assume that the internal free calcium concentration 2+ −8 [Ca ]i = 5 × 10 M does not change significantly under T-current activation, and hence, the driving force for calcium, V − VCa, is linear in V .

A.2 The model of reticular neurons of the thalamus (RET)

The model of reticular thalamic neurons consists of the same form of differential equations as the HM model (A.1- A.5), but with different parameter values. The form of activation and inactivation dynamics of the reticular T-type calcium current IT res were taken from [43, 44]. −1 mT,∞ = (1 + exp(−(v + 50)/7.4)) , −1 hT,∞ = (1 + exp((v + 78)/5.0)) , −1 τm,T = 1 + 0.33(exp((v + 25)/10) + exp(−(v + 100)/15))) , −1 τh,T = 28.5 + 0.33(exp((v + 46)/4) + exp(−(v + 405)/50))) .

150 Appendix A. Model Parameters

2 Parameter values for the RET model are gNa = 100, gK = 80, gT = 3, gl = 0.05 (in mS/cm ) and reversal potentials VNa = 50, VK = −90, VCa = 120, Vl = −90 (in mV) in a cell with a 2 2 membrane area of 14,300 µm and membrane capacitance Cm = 1.0 µF/cm .

A.3 Hindmarsh-Rose model (HR)

The Hindmarsh-Rose model was derived from an ionic Hodgkin-Huxley-type model of bursting neurons [74]. It comprises of a three-dimensional system of ordinary differential equation with polynomial RHS, which the simplifies analysis and simulation, dx = y − ax3 + bx2 − z + I + I (t), (A.6) dt 0 signal dy = c − fx2 − y, (A.7) dt dz  1  =  x − (z − g) . (A.8) dt 4

The parameter values for our model are a = 1, b = 3, I0 = −0.5, c = 1, f = 5, e = 0.005, g = 5.1,  = 0.005. With these values, the model stays at a steady state of (x0, y0, z0) = (−1.44027, −9.37193, −0.66109), and it can exhibit subthreshold resonance and burst firing when stimulated with appropriate inputs Isignal(t). When  = 0, the z variable is constant, and the model is still capable of firing action potentials but does not exhibit burst firing or resonance.

A.4 Integrate-and-fire models

The one-dimensional integrate-and-fire model is the simplest possible ODE model for neuronal dynamics. It consists of a linear differential equation dV = GV + I + I (t) (A.9) dt signal where G signifies the conductance and I is a current bias that determines the rest level. When V reaches a threshold, Vthres, an action potential is recorded and V is reset to Vreset. We use this model to mimic the dynamics of the reduced Hindmarsh-Rose model with parameter values G = −0.4621, I = 0.6656, Vthres = −1.0, and Vreset = −1.01. In order to introduce a resonance in an integrate-and-fire model, we add a second, linear equa- tion to (A.9), similar to the slow z equation in the Hindmarsh-Rose model (A.8)

dV = AV + BZ + I + I (t), (A.10) dt signal dZ = CV + DZ + E. (A.11) dt By matching the rest state, input resistance, and resonance through a stochastic optimization algorithm to the HR model, we obtained the parameter values A = −0.0320, B = −1.3258,

151 Appendix A. Model Parameters

C = 0.00025, D = −0.001, I = −0.9226, E = −3.0099 · 10−4. Additionally, a scaling factor for all inputs Isignal(t) and noise, K = 14.8543, has to be introduced to match the impedance of RIF to HR. Then the equilibrium of RIF is (x0, z0) = (−1.44027, −0.66109). Threshold parameters are Vthres = −1.1 Vreset = −1.11, and Z also is reset after an action potential to Z + Zreset where Zreset = 0.1. In their stochastic form, both integrate-and-fire models follow Ornstein-Uhlenbeck dynamics until V reaches threshold.

A.5 Synaptic coupling

We couple our model neurons by synapses that mimic the dynamics of biological synapses. In the RIF model, the membrane potential is simply reset by a value Vsyn, in the spirit of creating the simplest network that includes coupling and resonance. In the Hindmarsh-Rose equation, we use the standard method of describing a synapse by one activation variable s that is described by an ODE ds = αk(x)v(1 − s) − s/τ , (A.12) dt syn where the activation of the synapse is described by a sigmoidal function dependent on the voltage-like variable x, k(x) = (1 + exp(−x/0.02))−1, and the decay of the postsynaptic potential is governed by the time constant τsyn = 5. This synapse is activated when the presynaptic HR neuron fires an action potential, that is when x exceeds 0, and the postsynaptic neuron experiences an input current Isyn = gsyni,js(x − Vsyn). Here gsyni,j is the coupling strength between cells i and j, and Vsyn is the equilibrium value for the synaptic potential. For excitatory synapses, we used Vsyn = 1, and for inhibitory synapses Vsyn = −2.

For the network of thalamocortical and reticular neurons, we need to model GABAA, GABAB, and AMP A synapses which form synaptic connections in the thalamic network. For GABAA and AMP A synapses, a simple one-step reaction can model the synaptic dynamics [44],

α C + T β O, where C is the fraction of closed synaptic channels, T is the neurotransmitter concentration, and O is the fraction of open channels. This leads to a law of mass action equation that describes synaptic activation dr = α[T ](1 − r) − βr, (A.13) dt and the synaptic current is described by

Isyn = gsynr(V − Vsyn), (A.14) for syn = AMP A, GABAA, respectively. Parameter values, following Destexhe et al. [44] are α = 0.94 mM−1ms−1, β = 0.18 ms−1 for AMPA, and α = 20 mM−1ms−1, β = 0.16 ms−1 for

152 Appendix A. Model Parameters

GABAA with reversal potential VAMP A = 0 mV, VGABA = −85 mV. In both types of synapses, neurotransmitter release is assumed to be a pulse of 0.3 ms duration and 0.5 mM amplitude when the presynaptic neuron fires an action potential.

GABAB synapses have more complicated dynamics that involve activation of ion channels by a G protein step, and consequently, the opening dynamics have to be modeled with an intermediate step: dr = K [T ](1 − r) − K r, dt 1 2 ds = K r − K s, so that dt 3 4 s4 IGABAB = gGABAB 4 (V − VK ). s + Kd

4 −1 −1 −1 Parameter values are Kd = 100 µM , K1 = 0.5 mM ms , K2 = 0.0012 ms , K3 = −1 −1 0.18 ms , K4 = 0.034 ms .

153