Spontaneous Dynamics and Information Transfer in Sensory Neurons

Spontaneous dynamics and information transfer in sensory neurons

A dissertation presented to

the faculty of the College of Arts and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree

Doctor of Philosophy

Hoai T. Nguyen

This dissertation titled Spontaneous dynamics and information transfer in sensory neurons

by HOAI T. NGUYEN

has been approved for the Department of Physics and Astronomy

and the College of Arts and Sciences by

Alexander B. Neiman

Associate Professor of Physics and Astronomy

Howard Dewald Dean, College of Arts and Sciences 3 Abstract

NGUYEN, HOAI T., Ph.D., August 2012, Physics and Astronomy Spontaneous dynamics and information transfer in sensory neurons (101 pp.)

Director of Dissertation: Alexander B. Neiman Peripheral sensory systems convey information about the outside world to the central nervous system and are organized in a feed-forward networks passing information through

a series of layers. Of high interest are basic problems of how sensory information is encoded in dynamical states of a network and how correlated activity progresses from one

network layer to another. Often peripheral receptors are characterized by spontaneous noisy oscillatory activity which introduces temporal and spatial correlations in neuronal

spike trains. This dissertation uses computational methods to elucidate the role of oscillations in shaping of spontaneous dynamics and information transfer in sensory receptors. In particular, we concentrate on hair cell - sensory neurons types of receptors

which mediate the senses of hearing, balance and electroreception. We develop a conductance-based Hodgkin-Huxley type model in which spontaneous noisy inputs from

sensory cells modulate stochastic synaptic transmission. The model is tuned to reproduce experimental data from electroreceptors of paddleﬁsh and from vestibular utricle sensory

neurons of a turtle. Using the model we study how the interplay of ionic currents and synaptic noise affects the statistics of spontaneous dynamics and response properties of the system. In particular, we isolate a region in the parameter space of the model which maximizes information transmission of the system. We contrasted linear and non-linear responses of the model and show that coherent oscillations from epithelial cells enhance significantly stimulus-induced synchronization of neural responses and information transfer. We develop a network model for peripheral electroreceptors to study how correlated noises common to several sensory neurons influence the overall dynamics of the network and transmission of sensory stimuli. In contrast to previous studies on noise 4 correlations which concentrated on weakly correlated white noise, we consider strongly correlated narrow-band fluctuations. We describe in detail transformation of input correlated signals through the network. Furthermore, we show that while coherent epithelial oscillations enhance information transmission for a single sensory neuron, the presence of spatially correlated noise introduces redundancy reducing stimulus coding efficiency and information rate on the network level. Interestingly, this information reduction can be minimized if the frequency of noisy oscillations is about one half of the

ﬁring rates of neurons in the network. Although the model is set to mimic electroreceptors in paddleﬁsh, it can be viewed in a more general context of a networks of oscillatory neurons processing correlated rhythmical signals.

Approved: Alexander B. Neiman

Associate Professor of Physics and Astronomy 5 Acknowledgements

Almost ﬁve years have passed since I began my PhD studies in Ohio University, and I could not have come this far without the guidance, support, help, advice, and encouragement of many people present along my journey. First and foremost, I would like to express my deepest gratitude to my PhD advisor Professor Alexander B. Neiman, for his excellent guidance, caring, patience, and providing me with an excellent atmosphere for doing research. In addition, I also appreciate his support and understanding in my life outside of physics. Without it, I could not have spent an incredible and beautiful time with my husband Anh Ngo and my son Chi Kien Ngo when we moved to Baton Rouge, Louisiana, one year ago and now in the lovely city Madison, Wisconsin. At the same time, I am thankful for all insightful discussion, support from my PhD degree committee: Professors Nancy Sandler, Michael H. Rowe, and David F. J. Tees. I would never have been able to ﬁnish my dissertation without the guidance of my committee members. I specially thank Professors M.H. Rowe and D.F. Russell for providing experimental data. Special thanks to Professor Sergio

Ulloa. I really appreciate all kindness and encouragement he gave to not only me but also my small family. I would also like to thank Department of Physics and Astronomy for the immense support I have had all along. Last but most certainly not least, I would like to call attention to my family: my Mom Pham Thi Than and my Dad Nguyen Van Khang, my husband Anh Ngo, my wonderful son Chi Kien Ngo. Thank you for your unconditional love and support. You are the world to me. I acknowledge ﬁnancial support from the National Institutes of Health under Grant

No. DC05063 and by the Biomimetic Nanoscience and Nanotechnology program of Ohio University. 6 Table of Contents

Page

Abstract ...... 3

Acknowledgements ...... 5

List of Figures ...... 8

1 Introduction ...... 13

2 Spontaneous dynamics and response properties of peripheral sensory receptors . . 19 2.1 Statistical measures of spontaneous activity ...... 19 2.2 Characterization of response dynamics of a single neuron ...... 23

3 Spontaneous dynamics of vestibular utricle and electroreceptor afferents ..... 30 3.1 Vestibular afferents ...... 30 3.2 Electroreceptors afferents ...... 34 3.3 Conclusion ...... 38

4 Single neuron models of stochastic spontaneous activity ...... 41 4.1 Conductance-based Hodgkin-Huxley type model ...... 41 4.2 Stochastic synaptic model ...... 43 4.3 Numerical simulations of the modiﬁed Hodgkin-Huxley model: comparison with experimental data ...... 47 4.4 Simpliﬁed neuronal model: theta neuron ...... 49 4.5 Conclusion ...... 52

5 Eﬀect of temporal correlations on spontaneous and response dynamics of sensory neurons ...... 54 5.1 Spontaneous dynamics of the Hodgkin-Huxley model of electroreceptors . 55 5.2 Linear response and stimulus encoding ...... 62 5.3 Nonlinear response ...... 64 5.4 Conclusion ...... 70

6 Eﬀect of spatial and temporal correlations on spontaneous dynamics and stimulus coding by small-scale networks of peripheral sensory neurons ...... 72 6.1 Network model ...... 73 6.2 Input – output correlation measures ...... 75 6.3 Stimulus encoding by a network ...... 78 6.4 Transformations of noise correlations by the network ...... 79 6.5 Stimulus encoding and information transmission by the network ...... 85 7

6.6 Conclusion ...... 90

7 Conclusion and Outlook ...... 92

References ...... 95 8 List of Figures

Figure Page

1.1 Schematic drawing of different classes of vestibular afferents. Shown are two types of sensory hair cells (I and II) and three classes of afferents: pure-calyx (C), dimorphic (D) and pure button (B). Mechanoelectrical transduction ion channels in stereocilia of hair cells are labeled as Met. Possible locations of spike initiation zones in afferents are labeled by asterisks. Grey unlabeled structures are efferents, which are neurons transferring feedback signals from the brain to hair cells. Modified from a courtesy figure obtained from Dr. Ruth Anne Eatock (see also [1])...... 14 1.2 The organization of an electroreceptor on the rostrum of paddlefish. Shown is a cluster of skin pores each leading to a canal. Sensory cells in epithelial layer at the bottom of canals innervated by a few afferent neurons. Modified from [2]. 15

2.1 Schematics of neuron perturbed by the stimulus s(t)...... 24

3.1 Spontaneous dynamics vestibular utricle afferents. (a) Raw extracellular recording (lower panel) and a sequence of identified spike times of a vestibular afferent. (b) Probability density function (PDF) of interspike intervals for three representative vestibular afferent discharges...... 31 3.2 Serial correlation coefficients (SCCs) (a) and power spectral density (PSD) for three representative utricle afferents. The data is the same as in Fig 3.1: black, red and blue colored lines show SCCs and PSD for the same afferents as in Fig 3.1. Grey lines in panel (a) show minimum and maximum SCCs values for corresponding renewal spike trains. In panel (b) grey lines show PSDs of renewal spike trains...... 32 3.3 Dynamical entropies for ISIs sequences of two VAs. Red line and symbols shows dynamical entropies calculated for the original sequence. Blue line correspond to corresponding renewal surrogate ISIs. Errorbars we estimated by calculating hn for 10 surrogate sequences...... 33 3.4 Spontaneous dynamics of paddlefish electroreceptor afferents. (a) Raw extracellular recording (lower panel) and a sequence of identified spike times ofanafferent. (b) Probability density function (PDF) of interspike intervals for three representative EAs...... 35 3.5 Power spectral density (PSD) (a) and serial correlation coefficients (SCCs) (b) and for three representative electroreceptor afferents. Black, red and blue colored lines show SCCs and PSD for the same afferents as in Fig 3.1(b). In panel (a) grey lines show PSDs of renewal spike trains. Grey lines in panel (b) show minimum and maximum SCCs values for corresponding renewal spike trains...... 36 9

3.6 ISIs correlation time calculated using Eq.(2.3) versus the frequency ratio of epithelial to afferent oscillators, w, for the sample of 66 paddlefish electroreceptor afferents...... 37 3.7 (a): Dynamical entropies for ISIs sequence of an electrorecepor afferent. Red line and symbols shows dynamical entropies calculated for the original sequence. Blue line correspond to corresponding renewal surrogate ISIs. Errorbars we estimated by calculating hn for 10 surrogate sequences. (b,c): Source entropies estimated for words of n = 6 ISIs for the sample of 66 EAs vs the correlation time (b) and the coefficient of variation (c). Red dashed lines show linear regression. Values of the correlation coefficient are indicated on the panels...... 39

4.1 Deterministic dynamics of the modified Hodgkin-Huxley model. (a) Periodic 2 spike train for Iext = 1 µA/cm . (b) Firing rate of the neuron vs Iext for the 2 indicated values of the AHP conductance (gahp is units of mS/cm ...... 43 4.2 Power spectrum density of synaptic conductance. The red and blue lines correspond to simulation and analytical result, respectively. The parameters are τs = 1 ms, λ = 1000 1/s, σ = 0, m = 0.4, B = 0.018, fe = 26 Hz, Q = 20, 2 2 2 gahp = 5 mS/cm , g0 = 7.378 10− mS/cm ...... 46 4.3 Comparison between the experiment× and simulations for a vestibular afferent. Three panels show probability density of interspike intervals (ISIs), the serial correlation coefficients (SCCs) and the power spectral density of corresponding spike train. Experimental data are shown by black circles. Results of numerical simulations of the model are shown by the red lines. The parameters of the 2 model were: gahp = 1 mS/cm , τs = 2 ms , λ = 700 1/s, σ = m = 0, 2 5 2 2 gs = 0.0032 mS/cm , var(gs) = 3.6 10− [mS/cm ] ...... 48 4.4 Comparison between the experiment× and simulations for an electroreceptor afferent. Three panels show probability density of interspike intervals (ISIs), the serial correlation coefficients (SCCs) and the power spectral density of corresponding spike train. Experimental data are shown by the black circles. Results of numerical simulations of the model are shown by the red lines. The 2 parameters of the model were: gahp = 3 mS/cm , τs = 2 ms, λ = 1000 1/s, 2 m = 0.45, B = 0.018, fe = 27.1 Hz, Q = 22, < gs >= 0.0032 mS/cm , 5 2 2 var(gs) = 10 10− [mS/cm ] ...... 49 4.5 Dynamics of× the theta neuron model on a circle. Modified from G.B. Ermentrout, Scholarpedia, 3(3):1398 (2007)...... 50 4.6 Statistical measures of spontaneous dynamics from the theta neuron model Eq.(4.20). (a) Probability density of ISIs. (b) Serial correlation coefficients (SCC). (c) Power spectral density (PSD). Red lines and symbols show the model with epithelial oscillations generated a non-renewal spike trains. Blue lines and symbols correspond to the model with oscillations turned off, but with other parameters tuned to match the probability density of ISIs of the non-renewal model. Modified from [3]...... 52 10

5.1 Probability densities of ISIs. The red lines and blue lines correspond to the original and renewal models,respectively. Parameters of the original model 2 2 2 3 were gAHP = 7 mS/cm , g0 = 8.68 10− mS/cm , B = 5.86 10− , × 1 2 × 2 m = 0.4, Q = 20, τs = 2 ms, λ = 1000 s− , gs = 9.27 10− mS/cm , 5 2 2 × var(gs) = 2.3 10− [mS/cm ] . Parameters of the renewal model were × 3 2 2 gAHP = 0, m = 0, g0 = 7.9 10− mS/cm , B = 1.62 10− , m = 0, 2 2 × 4 2×2 gs = 2.413 10− mS/cm , var(gs) = 1.316 10− [mS/cm ] ...... 56 5.2 The serial correlation× coefficients (SCC) of ISIs× for the original (red line) and renewal (blue line) models. The parameters are the same as in Fig. 5.1 ..... 57 5.3 Power spectrum densities (PSD) of spontaneous spike trains generated by the original and renewal models, fa,afferent oscillator peak, fe epithelial oscillator peak ...... 58 5.4 Fano factor for the original (red line) and renewal (blue line) models for the same set of parameters as in Fig. 5.1. Horizontal dashed line shows theoretical limit for the renewal process...... 59 5.5 Statistical properties of spike trains generated by the original model (with coherent epithelial oscillations) and by the no oscillation model whereby the narrow-band noise was substituted with a short-correlated Ornstein-Uhlenbeck noise. Probability density of ISIs (a), serial correlation coefficients (b) and power spectral density (c) of spontaneous stochastic dynamics. Solid red lines correspond to the original model with epithelial oscillations. Dotted green lines refer to the model with no epithelial oscillations. The parameters 2 4 2 are: gAHP = 6mS/cm , τs = 2ms, λ0 = 10 Hz, gs = 0.081mS/cm , 5 2 2 var(g ) = 3 10− [mS/cm ] , m = 0.5, Q = 20, f = 27.5Hz. These parameters s × e resulted in the mean firing rate fa = 55 Hz and CV = 0.185...... 60 5.6 First serial correlation coefficient, C(1) (a) and correlation time tcor (b) of spontaneous ISIs sequences versus the ratio of fundamental frequencies of EO to AO,w = fe/ f a for the indicated values of the quality factor Q of epithelial oscillations. Other parameters are the same as in Fig. 5.5...... 61 5.7 Stimulus - response coherence for Gaussian band-limited stimulus with the standard deviation σ = 0.5 and cutoff frequency fc = 200 Hz (a) and fc = 20 Hz (b). Red line corresponds to the original model with epithelial oscillations. Green line refer to the model with incoherent epithelial fluctuations (No oscil.). Both models had identical spontaneous mean firing rates and CVs. The parameters are the same as in Fig. 5.5...... 63 5.8 Lower bound estimate of the mutual information rate (a) and the information gain ∆I (b) versus the ratio of fundamental frequencies of EO to AO, w = fe/ fa for the indicated values of the quality factor of narrow-band epithelial oscillations. Stimulus parameters were σ = 0.5, fc = 20 Hz. Other parameters were the same as in Fig. 5.5...... 65 11

5.9 Cross-trial response variability. The original model and the no oscillation model were stimulated repeatedly with identical realization of Gaussian noise band limited to fc = 20 Hz. (a) Short segment of the stimulus. (b -d) Raster plots of models responses to 200 trial presentations of the stimulus. Vertical axis is the trial number; horizontal axis is the time with respect to the stimulus onset. Dots indicate appearance of a spike in a trial relative to the stimulus onset. Panel (b) is for the stimulus standard deviationσ = 0.5 (weak stimulus). Panels (c,d) is for the case of strong stimulus with σ = 2.0 applied to the original model with narrow-band noise (c) and to the no oscillation model with broad-band OU noise (d). Other parameters were the same as in Fig. 5.5. . . . . 67 5.10 Stimulus-response (SR) and response-response coherence for simulations shown in Fig. 5.9. (a) SR (dashed line) and RR (solid line) coherence functions for the original model with narrow-band noise. (b) RR coherence for the original model (oscil., solid line) and for the no oscillation model (no oscil., dotted line)...... 68 5.11 Lower and upper bounds of information rate (a) and information gain ∆I (b) versus the standard deviation σ of Gaussian stimulus. normalized to the standard deviation of epithelial oscillations. The stimulus cutoﬀ frequency is fc = 20 Hz. The parameters of the model are the same as in Fig. 5.5...... 69

6.1 Layered structure of the feed-forward sensory network in paddleﬁsh electroreceptors ...... 73 6.2 Coherence function between two inputs, Eq.(6.4). (a) Input coherence for τc = 0.1, fe = 26 Hz, c = 1, α = 5 and indicated values of the quality factor. (b) Input coherence for Q = 20 and indicated values of α. Other parameters are the same as in panel (a)...... 77 6.3 Transformations of noise correlations for two identical neurons driven by independent broad-band noises and a common narrow-band epithelial noise. (a) Input coherence (gray line) and output coherences for the indicated values of the frequency ratio w. (b) Contour lines of the coherence sensitivity χ( f ), Eq.(6.6), versus frequency and w. fe = 26 Hz indicates the location of epithelial oscillations frequency. Other parameters are A = 0.5, D = 0.2, λ = 0.02, β = 0.3, c = 1...... 80 6.4 Transformations of noise correlations for two identical neurons driven by independent broad-band noises and a common narrow-band epithelial noise for the indicated values of quality factor Q. (a) Output coherence. (b) Coherence sensitivity χ( f ). Other parameters are w = 0.5, A = 0.5, D = 0.2, λ = 0.02, β = 0.3, c = 1...... 82 6.5 Transformations of noise correlations for two identical neurons driven by independent broad-band noises and a common narrow-band epithelial noise for the indicated values of standard deviation D of spatially uncorrelated broad- band noise. (a) Output coherence. (b) Coherence sensitivity χ( f ). Other parameters are w = 0.5, A = 0.5, Q = 20, λ = 0.02, β = 0.3, c = 1...... 83 12

6.6 Noise correlations of a pair of non-identical neurons. Neurons are characterized by different firing rates, fa1 = fe/w1 and fa2 = fe/w2, where fe = 26 Hz and w1,2 are indicated in the figure. (a) Input and output coherences for w1 = 0.5 and indicated values of w2. (b) Low-frequency averaged output coherence coh as a function of neuronal firing rates parametrized by w1,2. Other parameters are A = 0.5, Q = 20, λ = 0.02, β = 0.3, c = 1...... 84 6.7 Information density for a network of M = 5 neurons stimulated by Gaussian band-limited stimulus with σ = 0.2 and cutoff frequency fc = 20 Hz, for the indicated values of the quality factor of epithelial oscillations Q (a) and the partial correlation parameter c (b). Firing rates of neurons were uniformly distributed from 42 to 62 Hz. Other parameters are A = 0.5, λ = 0.02, β = 0.3. . 86 6.8 Information density for a network of M = 5 neurons stimulated by Gaussian band-limited stimulus with σ = 0.3 and cutoff frequency fc = 20 Hz. (a) Network of non-identical neurons with firing rates uniformly distributed within interval [52 ∆ fa/2, 52 + ∆ fa/2]. Values of ∆ fa are indicated in the figure. Other parameters− are A = 0.5, Q = 20, D = 0.2, λ = 0.02, β = 0.3. (b) Network of identical neurons with the mean firing rates fa = 52 Hz, ∆ fa = 0. Values of independent broad-band noise intensity D are indicate in the figure. . 88 6.9 Lower bound of the mutual information rate versus the intensity of independent broad-band noise D for networks of identical neurons and indicate size M. Other parameters are w = 0.5, A = 0.5, Q = 20, λ = 0.02, β = 0.3...... 89 13 1 Introduction1

Oscillatory activity is observed in various regions of the central nervous system [5].

Rhythmical activity is well-known to mediate walking and breathing. However, the concept that many peripheral sensory systems undergo intrinsic oscillations instead of being just passive transducers of external stimuli has started to emerge only in recent decades. Self-sustained rhythms have been observed in several types of peripheral sensory receptors. For example, peripheral cold thermoreceptors produce periodic or chaotic bursts of spikes when chilled [6] which arise due to slow ion currents in terminals of sensory neurons [7, 8, 9]. Another example is oscillatory responses observed in olfactory

receptor neurons [10, 11]. This dissertation research focuses on 2-stage hair cell - primary aﬀerent receptors.

Such sensory receptors have stimulus-transducing hair cells or similar receptor cells synaptically coupled to sensory neurons (aﬀerents) that convey transduced stimuli to the central nervous system. 2-stage receptors are found in the auditory, vestibular, lateral line,

and electrosensory systems. Recordings of spontaneous firing from auditory afferent fibers in turtles, lizards and birds showed a series of similarly spaced peaks in interspike

interval histograms, indicating ”preferred intervals” [12, 13]. Some primary vestibular aﬀerents demonstrate extremely regular periodic ﬁring [14, 15]. Isolated hair cells of

auditory and some vestibular receptors of lower vertebrates undergo ringing (damped membrane potential oscillations) in response to step changes in membrane potential [16, 17, 18]. Such hair cells may undergo spontaneous oscillations of membrane potential

[19, 20, 21], or spontaneous periodic motions of ciliary hair bundles [22, 23, 24]. These spontaneous oscillations of hair cells are reﬂected in periodic sequences of post-synaptic

excitatory potentials and in periodic firing of primary afferents [21]. Intrinsic (spontaneous) oscillations have been found in both sensory cells and afferent neurons in

1Part of this section appeared in Nguyen and Neiman (2010) [4]. 14

electroreceptors (ERs) [25, 26, 27]. In particular, it was demonstrated that paddleﬁsh ERs have a 30- 70 Hz oscillator in each aﬀerent terminal, along with a population of 26 Hz oscillators in the sensory epithelia [28]. Interactions of these two types of oscillators cause

biperiodic afferent firing with two fundamental frequencies. The figures below sketch peripheral hair cell – afferent receptors. Fig. 1.1 shows

vestibular receptors and Fig. 1.2 represents ampullary electroreceptors in paddleﬁsh.

Met Met Met Met Met

I I I II II Kv7x

*Na K * * *

C D B

Figure 1.1: Schematic drawing of different classes of vestibular afferents. Shown are two types of sensory hair cells (I and II) and three classes of afferents: pure-calyx (C), dimorphic (D) and pure button (B). Mechanoelectrical transduction ion channels in stereocilia of hair cells are labeled as Met. Possible locations of spike initiation zones in afferents are labeled by asterisks. Grey unlabeled structures are efferents, which are neurons transferring feedback signals from the brain to hair cells. Modified from a courtesy figure obtained from Dr. Ruth Anne Eatock (see also [1]).

Both examples show sensory hair cells transducing external mechanical (in the case of vestibular receptors) or electrical (in the case of ERs) signals in voltage variations across their basal membranes. These membrane voltage variations are then transmitted via 15

Pore Canal Sensory wall Epithelia

Excitatory Canal hair cell- to-afferent synapses Afferent nerve (2−4 axons) Figure 1.2: The organization of an electroreceptor on the rostrum of paddlefish. Shown is a cluster of skin pores each leading to a canal. Sensory cells in epithelial layer at the bottom of canals innervated by a few afferent neurons. Modified from [2].

excitatory synapses to the aﬀerent neurons which encode external signals in sequences of

action potentials. Cellular oscillators, including sensory hair cells and sensory neurons are examples of non-equilibrium, nonlinear, open, complex systems and as such require approaches from

complex system theory, nonlinear dynamics, information theory and modern methods of signal processing. The main goal of this thesis is to gain better understanding of how hair

cell – aﬀerent receptors utilize their intrinsic dynamics to optimize responses and information encoding. In this study we use computational approaches to study the role of intrinsic oscillatory dynamics in the encoding of external signals.

What are possible functions of oscillations in peripheral receptors? 1) Frequency tuning. The existence of a natural frequency promotes frequency selectivity of a receptor. Examples include mechanical and electrical resonances in mechanosensory hair cells [16, 17, 18].

2) Nonlinear ampliﬁcation. Dynamical systems are most sensitive to external perturbations at bifurcations, where a system’s mode of operation changes abruptly as a control parameter exceeds some critical value. For example, one model for the high

sensitivity and selectivity of mammalian hearing proposes that auditory mechanisms are 16

normally poised near a Hopf bifurcation, such that a weak stimulus of appropriate frequency is greatly ampliﬁed [29, 22, 23]. 3) Precise coding. External analog stimulus information is encoded by an individual

peripheral sensory receptor into a time series of aﬀerent spikes, sent to the brain. Regularly ﬁring neurons may potentially provide precise sampling (coding) of external

stimuli [30, 15]. 4) Higher order correlations. Multi-modal oscillations may generate correlations in

sequences of action potentials and of receptor responses to external signals. These correlations provide a higher degree of ordering of neural responses and may enhance information transmission and detection performance of receptors. Serial correlations of

aﬀerent interspike intervals have received recently much interest (for review see [31, 32]) as the times between sequential spikes of some auditory aﬀerents and all electroreceptor

afferents tend not to be independent. Instead, a long interspike interval tends to follow a short interval, and vice versa. Such anticorrelations are a type of ordering, and result in reduced low-frequency noise in afferent firing. This ”noise shaping” yields higher

signal-to-noise ratios and increased discriminability and information transfer, at low frequencies. Paddlefish ER afferents show long-lasting serial correlations, that occur due to self-sustained epithelial oscillations [33, 28, 2]. Using simple theta neuron and integrate and fire models, it was shown that oscillatory internal noise, as found in paddlefish ERs,

can augment the information transfer through a sensory receptor [34, 3]. Here we extend these studies using a conductance-based, Hodgkin-Huxley type model with stochastic excitatory synaptic input.

Experimental validation of the functional role of oscillations in peripheral receptors involves several diﬃculties. An ultimate hypothetical test would involve abolishing oscillations and then analyzing how the sensitivity, encoding and detection characteristics of a sensor change as compared to normal conditions, i.e., when oscillations are present. 17

Abolishing of oscillations is achieved by pharmacological blockage of certain ion channels in the cell membrane. Such procedures may also change other properties of a sensor, such as mean ﬁring rate. Furthermore, pharmacological procedures are usually

done in in vitro preparation, which often leads to the question of whether a receptor system is still working normally. That is why modeling studies and methods of advanced

signal processing are crucial in understanding the functional signiﬁcance of coherent rhythmic activity in sensory systems. These methods allow the investigator to impose

clear constraints on a system while varying its parameters and studying its responses to various signals. The majority of previous theoretical and computational work studied the inﬂuence of oscillations and serial interspike interval correlation on responses to weak

stimuli, i.e. linear responses [35, 36, 3, 37]. The question of how internal oscillations and correlations aﬀect nonlinear neural responses has not been studied in full details. We note,

that strong stimuli may induce distinct neural responses in the form of bursts, as in paddleﬁsh ERs [38, 39] or as mode locking as in vestibular aﬀerents [40]. Peripheral receptors in many sensory systems are organized in limited scale,

feed-forward networks passing information through a series of network layers, and ultimately to the central nervous system (CNS). A classic example is the retina, a

feed-forward network serving as a preprocessor of visual stimuli [41, 42]. Often peripheral receptors are characterized by spontaneous, noisy oscillatory activity which

may introduce temporal and spatial correlations across elements of a neural network. Problems of how these correlations are processed by the network and how they inﬂuence information transmission is of high and continued interest

[42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. Examples include spontaneous stochastic oscillations in hair cells that are innervated by several primary sensory aﬀerents in auditory, vestibular and electro sensory receptors. Unlike the majority of theoretical works mentioned above that study weakly correlated, white inter-neuronal noise, we study 18

the inﬂuence of strongly correlated noise on spontaneous activity and information transmission in a model of a limited-scale network of electroreceptors. The structure of this work is as follows. Chapter 2 introduces measures of

spontaneous neuronal activity as well as stimulus-response measures, including estimates of the information rate. Chapter 3 is devoted to the analysis of spontaneous dynamics of

vestibular and electroreceptor aﬀerents. Chapter 4 introduces two classes of computational models used in this work. A modiﬁed Hodgkin-Huxley model is used in Chapter 5 to

study the eﬀect of epithelial oscillations on linear and nonlinear stimulus encoding and information transfer by a single sensory neuron. Finally, Chapter 6 is devoted to study of a small network of oscillatory neurons subjected to correlated internal noise. 19 2 Spontaneous dynamics and response properties of peripheral sensory receptors

2.1 Statistical measures of spontaneous activity

Experiments in sensory neurophysiology analyze the sequence of action potentials resulting from spontaneous activity of a neuron or the response of the neuron to external stimulation. Neuronal action potentials are believed to be the main carrier of information in the nervous system [56]. The generation and propagation of action potentials is

inherently noisy. Noise inevitably present in any dissipative physical system. In biological neurons the main sources of noise are: (i) ﬂuctuations of ionic currents due to

stochasticity of ion channels and pumps and (ii) synaptic noise arising from the inputs to a neuron from other cells. This inevitable noise results in two kinds of neuronal variability.

First, spontaneously firing neurons are characterized by fluctuating firing rates. Second, responses of a neuron to identical stimuli vary from trial to trial. Theory of stochastic point processes [57] is a powerful tool widely used in

computational neuroscience. In the framework of this theory a spike train is described by

a stochastic point process, i.e. discreet sequences of time events, t1, ..., tm, where tm is the time of arrival of the m-th spike. The intervals between neural ﬁring, called interspike

intervals (ISIs), are defined as I = t + t . The variability of neuronal firing can be m m 1 − m expressed in terms of the statistical properties of ISI distributions. The mean firing rate and the coefficient of variation are among most often used statistical measures of neuronal firing. Both these measures can be extracted from the probability density function (PDF) of ISIs, P(I). In neuroscience literature this measure is often called the interspike interval histogram (ISIH). Given a sequence of ISIs the mean firing rate, f¯ and the coefficient of

variation (CV) are deﬁned as

f¯ = 1/I¯, CV = std(I)/I¯, (2.1) 20

where I¯ is the mean ISI and std(I) is the standard deviation. From (2.1) it is clear that the CV is the standard deviation in units of the mean. It is instructive to consider a Poisson spike train as a reference model for a spike

train. The ISIs distribution for the Poisson spike train is exponential, P(I) = λ exp( λI), − where λ is the intensity (rate) of the Poisson process. The mean ISI is I¯ = 1/λ and the CV

of the Poisson spike train is 1, CV = 1. At the other extreme, a deterministic periodic ﬁring is characterized by zero variability, CV = 0.

The spiking activity of a neuron can be described as either a renewal or a non-renewal point process. The renewal process is one in which all memory is erased after each action potential so that consecutive ISIs are statistically independent. The Poisson

process is one example of a renewal process that is often used to describe the ﬁring of cortical neurons [58]. However, many types of neurons in a variety of sensory and motor

systems display non-renewal spiking statistics with extended correlations between ISIs [31, 32]. These neurons display a memory in their spiking activity that persists over multiple ISIs. This memory can be quantified in terms of the normalized autocorrelation function of ISI, and by serial correlation coefficients (SCCs) C(k), The serial correlation coefficients (SCCs) are given by

2 I I + I¯ C(k) = m m k− , (2.2) var(I)

where represents averaging over ISIs and var is the variance of ISIs I2 I¯2. The serial m− correlation coeﬃcient measures the statistical dependence of pairs of ISIs separated by the intervals (lag). For a renewal process C(k) = 0 for values of k. Positive serial correlations (C(k) > 0) on average, a long ISI is followed by even longer ISI. Negative serial correlations (C(k) < 0) mean that ISI tend to be followed by a shorter ISIs and vice versa. The temporal extent of serial correlations is characterized by the correlation time [4],

∞ t = I¯ C(k) , (2.3) cor | | k=1 21

where the sum is taking over lags with SCCs signiﬁcantly diﬀerent from 0. A fundamental problem of neuroscience is to understand how neurons encode, decode and transmit information [30, 56]. Perhaps the simplest model for neural coding is

the so-called rate code where a stimulus is encoded in spike counts in a succession of non overlapping time windows, each of the length T. These spike counts n(T) can be also used to characterize the variability neuronal dynamics. The variability in the spike count distribution is most conveniently characterized by the Fano factor deﬁned as [59],

F(T) = var[n(T)]/ n(T) , (2.4) where var[n(T)] and n(T) are the variance and the mean of the spike count, respectively. The Fano factor is 1 for Possion process, i.e., F(T) = 1 at any T. At short time scales, i.e. for T I¯ any process approaches to this Poissonian limit. Consequently F(T) 1 as ≪ → T 0. For a renewal process F(T) CV2 as T . We note also that the Fano factor → → → ∞ can be used to characterize the discrimination capacity of sensory neurons: smaller values of the Fano factor (i.e. smaller variability) indicate better discrimination between spike count distributions with and without stimulus [34, 59]. The power spectral density (PSD) is used to study variability of neuronal ﬁring in the

frequency domain. If a particular shape of the action potential does not matter, then the spike train can be represented as a sequence of delta functions centered at the spike times with subtracted mean ﬁring rate as,

M x(t) = δ(t t ) f¯. (2.5) − m − m=1

The PSD, Gxx( f ), of a stationary stochastic process x(t) is deﬁned as [60],

G ( f ) = X( f )X∗( f ) , (2.6) xx

where X( f ) is the Fourier transform of a realization of x(t), asterisks indicates complex conjugate and the averaging is performed over the ensemble of realizations of x(t). In 22

practice a single long realization of x(t) is partitioned into overlapping windows to mimic an ensemble of realizations. Fast Fourier transforms (FFT) are performed for each window and then the squared amplitude of the Fourier transforms are averaged to obtain

an estimate of the PSD [60]. For the spike train given by (2.5) the PSD has units of (spikes/sec)2/Hz or simply Hz. A Poisson spike train with the rate λ is characterized by a

uniform PSD, Gxx( f ) = λ. As we noted above, any point process approach Poissonian at short time scales. Thus, the PSD of any spike train should approach to the mean ﬁring rate

f¯ at high frequencies, f f¯. A purely periodically ﬁring neurons, by contrast,is ≫ characterized by a delta-peak in its PSD centered at the mean ﬁring rate, δ( f f¯). − To characterize nonlinear correlations in sequences of ISIs, we employ the so-called

dynamic entropies. These measures characterize correlations not just between pairs of ISIs, but also among various ISIs patterns [61, 62]. In this work we used these high-order

correlation measure to inquire whether they can be used as additional classiﬁers of variability for utricle and electroreceptor sensory neurons. Given an ISI sequence I , we { m}

introduce a binary alphabet as follows: S m = 0 if Im < I¯ and S m = 1, otherwise. Then for

the resulting binary sequence we calculate probabilities of words containing n letters, Pn, and then the n-block Shannon entropies,

Hn = Pn log2 Pn, (2.7) −

where the summation is carried out over all n-words with positive probabilities, Pn > 0.

The dynamical entropies hn are deﬁned as

h = H + H , h H , (2.8) n n 1 − n 0 ≡ 1 and indicate the average information necessary to predict the n-th +1 symbol in a sequence, given knowledge about preceding n symbols. hn decreases with increase of n and saturates to the so-called source entropy in the limit of large n. For a renewal process, hn rapidly saturates to a positive number determined by the symmetry of ISI distribution. 23

In particular for the Bernoulli string, hn = 1 for all n, i.e. the Bernoulli symbolic sequence

is completely unpredictable. For a periodic ISI sequence hn rapidly drops to zero, so that only a ﬁnite number of symbols must be observed to predict with certainty the next

symbol. Correlation and spectral measures of a non-renewal stochastic process can be

compared with those of an equivalent renewal process.For such a comparison we need to generate a sequence of events which will retain certain properties of the original process,

but otherwise will lose serial correlations between ISIs. In this work we used a shuffling of ISIs to generate such renewal surrogates as follows. A given series of ISIs, I , { m} (obtained either from experimental data or from numerical simulations) was randomly shuffled, i.e. the positions of I were randomly exchanged, to obtain a new sequence I˜ . m { m} Although the probability distribution, and thus the mean and the coefficient of variation of

ISIs, are retained by this procedure, all serial correlations are removed. The spike times of

the shuﬄed ISIs are obtained as t˜m+1 = t˜m + I˜m. The statistical measures introduced above were used to analyze data from numerical

simulations and experimental data from utricle and electroreceptor aﬀerents. For experimental data, all computations were performed in Matlab with a custom made

package using Matlab’s signal processing toolbox. A similar package was developed in Fortran to process spike trains obtained in numerical simulations.

2.2 Characterization of response dynamics of a single neuron

Understanding of how a sensory stimulus is represented in neural response and mechanisms by which sensory neurons encode information remains important problems in

neuroscience. In this section we introduce and discuss various measures of response dynamics which will be used further on in the dissertation. Figure 2.1 shows 24 spike train stimulus neuron stimulus estimate

Figure 2.1: Schematics of neuron perturbed by the stimulus s(t).

schematically a procedure often used in the studies of response properties of a neuron. To probe the response dynamics a time varying stimulus, s(t), is administered to a neuron. In response the neuron generates or modulates a spike train. Often a broad band Gaussian

stimulus s(t) is used. The response is then measured as a sequence of neuron spike times t and the response spike train x(t) is represented by a sequence of delta functions { n} centered at t , Eq.(2.5). In the case of weak stimulus, one can then use linear response { n} theory to estimate the sensitivity of the neuron. The transfer function (also called the sensitivity) is calculated as [60]: G ( f ) H( f ) = | xs |, (2.9) Gss( f )

where Gxs( f ) is the cross-spectral density of the spike train and the stimulus, and Gss( f ) is the PSD of the stimulus. The units of the transfer function depend on the units of the

stimulus. For example, the stimulus for an electroreceptor afferent is time-varying external electric field with units of [V/m] and the transfer function has units of [(spk/s)/(V/m)]. For a mechanosensory neuron, e.g. vestibular afferent, the stimulus is displacement [m], so that the transfer function has units [(spk/s)/m]. Thus the transfer function measures by how much the firing rate changes per unit of external stimulation. The advantage of using broad-band Gaussian noise as a stimulus compared to ,e.g., a sinusoidal stimulus is that Eq.(2.9) allows estimation of the frequency response of a system within the whole

frequency band of the stimulus at once instead of at the single frequency of the sinusoidal stimulus. 25

The stimulus-response (SR) coherence is another linear input-output measure defined as [60] 2 Gxs( f ) CSR( f ) = | | . (2.10) Gss( f )Gxx( f ) The SR coherence is essentially the normalized cross-correlation coefficient in the frequency domain, i.e. it measures linear correlations between Fourier components of stimulus and response. As such the SR coherence is a dimensionless quantity varying from 0 (no stimulus-response correlations; stimulus and response are incoherent) to 1 (perfect correlations; stimulus and response are completely coherent). For an ideal linear system with no internal noise, the SR coherence is 1. Since a real physical system is contaminated by noise actual SR coherence values are typically < 1. Any non-linearity in a system will also result in coherence values < 1. Thus, SR coherence can be used to assess both the internal noise and the nonlinearities of a system. More importantly, SR coherence measures the linear encoding properties of a neuron. That is, high SR coherence values in a frequency band indicate faithful encoding of stimulus in the firing rate of a neuron as we discuss further on.

Response-response (RR) coherence is a measure that does not require the assumption of response linearity [63, 30, 64, 38]. A neuron is stimulated by the same realization of a

stimulus repeatedly and then the variability of responses xk(t) is characterized using a

coherence function between pairs of responses xk(t) and x j(t):

2 K 2 K(K 1) k=1 j

characterizes response variability that cannot be accounted for by the stimulus. The RR coherence changes from 0 (no synchrony between sets of responses) to 1 (perfect 26

stereotypical responses). It sets the upper bound for the SR coherence, C ( f ) C ( f ). For a linear response the values of SR coherence and the square root SR ≤ RR of RR coherence are close, C √C . Deviations of the square root of RR coherence SR ≈ RR from the SR coherence thus indicate nonlinearities at certain frequencies. Both the SR and RR coherence functions can be used for estimation of the amount of information which

carried by the spike train about the stimulus. Information theory is the rigorous way to quantify how much information a neural

response carries about a stimulus [30, 56]. Ideally, a sensory neuron should respond to wide variety of stimuli, possessing a high information capacity. At the same time, for a given stimulus,the ideal neuron responds stereotypically to sequential presentation of that

stimulus. However, internal and external noise will result in variability of neuronal responses to identical stimuli. The lower the response variability the higher information

transmitted by a neuron. Mutual information is a measure that characterizes the amount of information transmitted by a neuron about a stimulus,

I(S, R) = H(R) H(R S ), (2.12) − | where H(R) is the entropy of the response, and H(R S ) is the conditional entropy of the | response given the stimulus. For the ideal neuron with no response variability, H(R S ) is 0. | A direct method of estimation of the mutual information [65] involves calculating the

average information transmitted by the neuron. In this method a sequence of spikes is mapped to a symbolic binary alphabet with just two letters, 0 and 1 corresponding to ”no

spike” or ”spike” in a time window. Then the probabilities of words of various length pn and conditional probabilities qn of observing words of length n given the stimulus are estimated, and the entropies are calculated as

H(R) = p log p , H(R S ) = q log q , − n 2 n | − n 2 n n n { } { } 27 where the summation is taken over all possible words of length n. This method is conceptually simple, but it does not reveal what aspect of stimulus are being encoded and also requires enormous amounts of data.

In this work we will use lower and upper bound estimates of the information rate [30] which are considerably less computationally expensive and involve cross-correlation measures in frequency domain, allowing for physical interpretation of encoding mechanisms. Let us go back to Fig. 2.1 and assume that the stimulus is a Gaussian process. If the neuron is represented as a Gaussian information channel, then the amount of information it transmits per unit of time, i.e. the information rate, is given by the celebrated Shannon equation (2.13),

B S ( f ) I = log2 1 + d f, (2.13) 0 N( f ) where S ( f ) and N( f ) are PSDs of the signal and noise, respectively, B is the bandwidth of the channel in Hz, noise and the signal are assumed to be statistically independent. The units of the information rate in Eq.(2.13) are bits/s. We can represent the neuron as a linear ﬁlter with the response function h(t) and reconstruct the stimulus from the spike train using the convolution,

sest(t) = h(t t′)x(t′)dt′. (2.14) −

This linear reconstruction implies that the stimulus is encoded in the ﬁring rate of the neuron. The quality of such reconstruction is characterized by the ”reconstruction noise”, n(t) = s(t) s (t). We then can choose the linear ﬁlter that minimizes the variance of this − est noise, a well-known procedure in the signal processing. The result is the optimal

Wiener-Kolmogorov ﬁlter [60, 56]. It is convenient to work in the frequency domain. The

Fourier transform of the estimated stimulus is given by S est( f ) = H( f )X( f ), where H( f ) is the transfer function of the ﬁlter (i.e. Fourier transform of h(t)) and X( f ) is the Fourier 28

transform of the spike train. The transfer function of the optimal ﬁlter which minimizes the variance of the reconstruction noise is given by [56]

G ( f ) H( f ) = ∗sx , (2.15) Gxx( f ) where Gsx( f ) is the cross-spectral density between stimulus and response (i.e. spike train)

and Gxx( f ) is the response PSD. The PSD of reconstruction noise is

2 G ( f ) = G ( f ) + G ( f ) H( f ) G ( f )H∗( f ) G∗ ( f )H( f ), and using (2.15) we obtain nn ss xx | | − sy − sy [66], 2 Gsx( f ) Gnn( f ) = Gss( f ) | | = Gss( f ) 1 CSR( f ) . (2.16) − Gxx( f ) − Thus, the PSD of the reconstruction noise is expressed in terms of the SR coherence. For

perfect coherence, CSR = 1, i.e. a linear noiseless system, the reconstruction noise

vanishes. Deﬁning the signal-to-noise ratio as SNR( f ) = Gss( f )/Gnn( f ) we obtain the equation for the lower bound of the information rate [30, 66, 59]

fc ILB = log 1 CSR( f ) d f, (2.17) − 2 − 0

where fc is the cutoﬀ frequency of the Gaussian stimulus. Thus, the lower bound of the information rate can be calculated directly from the SR coherence (2.10), which makes the

last quantity extremely useful in the judging linear encoding properties of a neuron. The information rate estimate (2.17) sets the lower bound on the true information rate, because of the explicit assumption of linear encoding. The upper bound estimate of the information rate uses the RR coherence. In this, the ”signal” is calculated by averaging

over the ensemble of responses xk(t) in response to repeated presentation of a single realization of Gaussian stimulus, while the ”noise” is the deviation from that mean. This gives the following equation for the upper bound of the information rate [64],

f c I = log [1 C ( f )]d f, (2.18) UP − 2 − RR 0 29

where the RR coherence, CRR is given by Eq.(2.11). This estimate of the information rate does not make any assumption about particular encoding mechanisms, but since the stimulus is Gaussian (i.e. the stimulus entropy is maximal for a given variance) it sets the

upper bound for the true information rate. 30 3 Spontaneous dynamics of vestibular utricle and electroreceptor afferents

In this section we apply the statistical measures discussed in Sec.2.1 to experimental

data from two distinct types of sensory neurons: vestibular utricle afferents of the turtle and electroceptor afferents of the paddlefish. Data sets were sequences of afferent spike times collected by Dr. Rowe (vestibular afferents) and Dr. Russell (electroreceptor

afferents). Details of experimental procedures can be found in [40] (vestibular afferents) and in [28] (electroreceptor afferents). In the following we use the abbreviations VA for

vestibular utricle aﬀerents and EA for electroreceptor aﬀerents. We have analyzed n = 15 VAs and n = 66 EAs.

3.1 Vestibular aﬀerents

Vestibular afferents are known for their diverse variability and response properties [14]. Figure 3.1(a) shows an example of a raw extracellular recording of a spontaneously firing VA. Simple visual inspection indicates a rather noisy spikes sequence with no preferred interspike period. This is further illustrated in Fig. 3.1(b) in terms of probability distributions of for three afferents. The mean firing rate of VAs ranges from 3.906 to

30.206 Hz. The coeﬃcient of variation of the sample of VAs is 0.713 0.251, (range 0.381 ± – 1.237) indicating a rather high variability of VAs. In fact some of VAs with low ﬁring

rates showed CVs larger than 1, which can be interpreted as being more irregular than a corresponding Poisson spike train with the same ﬁring rate, e.g, the VA showed by the red line in Fig. 3.1(b).

Nine out of ﬁfteen VAs in the sample showed weak, but signiﬁcant serial

correlations. The absolute values of SCCs were smaller than 0.2 for all aﬀerents in the sample. Fig. 3.2(a1-a3) shows representative examples of VAs with no signiﬁcant SCCs 31

(a) (b)

0.5 s

Figure 3.1: Spontaneous dynamics vestibular utricle afferents. (a) Raw extracellular recording (lower panel) and a sequence of identified spike times of a vestibular afferent. (b) Probability density function (PDF) of interspike intervals for three representative vestibular afferent discharges.

(a1) with positive SCCs (a2) and negative SCCs (a3). While the negative correlations extended only to the ﬁrst ISI lag, positive correlations extended to several ISI lags. For example, positive correlations in Fig. 3.2(a2) extended to 10 ISIs. This leads to the

correlation time of 53.05 ms, which is smaller than the mean ISI interval (135.26 ms for that aﬀerent). These results indicate that the majority of utricle aﬀerents in our sample are

characterized by non-renewal stochastic dynamics.

The PSDs of the three representative VAs are shown in Fig. 3.2(b). The eﬀect of

positive serial ISI correlations is seen as enhanced power at low frequencies (red line in Fig. 3.2(b2)) compared to the PSD of a surrogate renewal spike train. This reﬂects a known eﬀect of positive serial correlations in enhancing low-frequency variability [67]. On other hand, negative serial correlations suppress low-frequency variability [35, 68, 64],

as seen in the somewhat lower power of the original spike train in the low frequency domain compared to a corresponding renewal spike train (Fig. 3.2(b3)). At high frequencies the PSD saturates to the mean ﬁring rate. 32

(a1) (b1)

cv =0.47

(a2) (b2)

cv =0.68

(a3) (b3)

cv =0.38

Figure 3.2: Serial correlation coefficients (SCCs) (a) and power spectral density (PSD) for three representative utricle afferents. The data is the same as in Fig 3.1: black, red and blue colored lines show SCCs and PSD for the same afferents as in Fig 3.1. Grey lines in panel (a) show minimum and maximum SCCs values for corresponding renewal spike trains. In panel (b) grey lines show PSDs of renewal spike trains.

The high-order dynamical entropies Eq.(2.8) were similar for the original and renewal spike trains. Figure 3.3 shows typical examples. Because of ﬁnite length of the ISI sequences used estimation of dynamical entropies for word lengths longer than 6 ISIs 33

(a) n h

(b) n h

Figure 3.3: Dynamical entropies for ISIs sequences of two VAs. Red line and symbols shows dynamical entropies calculated for the original sequence. Blue line correspond to corresponding renewal surrogate ISIs. Errorbars we estimated by calculating hn for 10 surrogate sequences.

was problematic. This is indicated by a steep decrease of h for n 6 for the renewal n ≥ processes (blue lines in Fig. 3.3) and the growth of the errorbars for n 6. We thus used ≥ 34

h6 as an estimate of the source entropy for ISI sequences [62]. The closeness of hn for the original and renewal data reflects the fact that serial correlations, although present, do not significantly affect spontaneous dynamics. There were no significant correlations between

the dynamical entropies and the coeﬃcient of variation.

3.2 Electroreceptors aﬀerents

The spontaneous dynamics of electroreceptors in paddleﬁsh was studied in [2, 28]. In

this section we review statistical properties of EAs and apply two new measures, correlation time and dynamical entropies of ISIs sequences. Figure 3.4(a) shows a typical

firing pattern of an EA. In contrast to the random firing of VA (Fig. 3.1), EAs clearly demonstrate periodic firing. ISI distributions show clear peaks at the preferred ISI corresponding to the inverse of the mean firing rate,(Fig. 3.4). The mean coefficient of variation for the sample of 66 EAs was 0.188 0.047, range 0.11 – 0.310. ±

An electroreceptor aﬀerent is characterized by a biperiodic ﬁring pattern that is a result of the unidirectional coupling of two distinct oscillators [28]. One oscillator resides in the population of epithelial cells, the so-called epithelial oscillator (EO). The other

oscillator is associated with the afferent neuron, (afferent oscillator, AO). The epithelial oscillations modulate the membrane potential of the receptor cells which results in periodic modulation of synaptic transmission and consequently of afferent firing. Power spectra of afferent spike sequences clearly show peaks associated with these two oscillators (Fig. 3.5(a1)). The peak at fa = 53.2 Hz corresponds to the natural frequency of the AO and matches the mean firing rate of this afferent. The EO is represented by a peak at f = 22.0 Hz. The sideband peaks at f f occur due to nonlinear mixing of the e a ± e

two natural frequencies, fa and fe. While the EO frequency is almost the same for different electroreceptor afferents and different paddlefish, f = 25.5 1.7 Hz [28], the e ± 35 (a)

0.1 s

(b)

Figure 3.4: Spontaneous dynamics of paddlefish electroreceptor afferents. (a) Raw extracellular recording (lower panel) and a sequence of identified spike times of an afferent. (b) Probability density function (PDF) of interspike intervals for three representative EAs.

mean ﬁring rate and the natural frequency of the AO varies over a wide range, 30 – 70 Hz.

The frequencies of the two oscillators within the electroreceptor were characterized by the frequency ratio w = fe/ fa. For the entire sample of 66 EAs, the average frequency ratio is w = 0.490 0.07, (range 0.4 – 0.7), indicating that for all EAs in the sample f > f . The ± a e majority of EAs are characterized by values of w close to 0.5, i.e. on average there are two spikes per cycle of epithelial oscillations. Figure 3.5(a1-a3) shows representative examples of EAs power spectra for w < 0.5 (a1), w 0.5 and w > 0.5 (a3). ≈ 36

fa (a1) (b1) fa + fe fe

fa - fe w=0.4

f a (a2) (b2) fa + fe fe

w=0.5

fa (a3) (b3) fa + fe fe

fa - fe

w=0.6

Figure 3.5: Power spectral density (PSD) (a) and serial correlation coefficients (SCCs) (b) and for three representative electroreceptor afferents. Black, red and blue colored lines show SCCs and PSD for the same afferents as in Fig 3.1(b). In panel (a) grey lines show PSDs of renewal spike trains. Grey lines in panel (b) show minimum and maximum SCCs values for corresponding renewal spike trains.

The unidirectional coupling, EO AO, results in non-renewal dynamics of aﬀerent → ﬁring [28, 2]. The SCCs shown in Fig. 3.5(b1-b3) indicate long-range ISI correlations

lasting for more than 50 ISIs in some EAs. The ﬁrst SCC is always negative, indicating 37

that a longer ISIs always follows a shorter ISIs and vice versa. The structure of SCCs depends crucially on the frequency ratio w and presumably on the magnitude of epithelial oscillations. Serial correlations are longest for aﬀerents with values of w close to 0.5 as in

Fig.3.5(b2). Figure 3.6 displays this tendency by plotting ISI correlation time, tcor, versus the frequency ratio for the sample of EAs. Note that ranking of EAs by the correlation time is complicated by the diversity of cells with respect to other parameters, such as magnitude of epithelial oscillations, quality factor of epithelial oscillations and the amount of inevitable ion channel noise, to name a few.

Figure 3.6: ISIs correlation time calculated using Eq.(2.3) versus the frequency ratio of epithelial to afferent oscillators, w, for the sample of 66 paddlefish electroreceptor afferents.

The dramatic interaction between serial correlations and oscillations is best seen by

comparing the power spectra of the original spike train with the PSD of its renewal surrogate (grey lines in Fig.3.5(b)). Three main eﬀects of serial correlations on the PSD spike train can be observed: (i) Spectral peaks associated with epithelial oscillations 38

disappear in the renewal spike train; (ii) The main peak corresponding to the mean ﬁring rate is signiﬁcantly wider for the renewal process; (iii) The power at low frequencies

( f < fe) is signiﬁcantly lower for the original spike train. Finally, the dynamic entropies calculated from the original ISI sequences and their renewal surrogate counterparts show a strong eﬀect of ISI correlations on high-order

variability. Figure 3.7(a) shows that the dynamical entropies of the original ISI sequences are signiﬁcantly lower than those for the renewal surrogates, indicating that serial

correlations lead to regularization of ISI sequences. Furthermore, the estimate of ISI

source entropy h6 is negatively correlated with ISI correlation time for the sample of EAs, as Fig. 3.7(b) shows. That is, longer ISI correlations lead to more regular ISI sequences

with lower values of h6. At the same time, the dependence of h6 on the coeﬃcient of variation is signiﬁcantly weaker (Fig. 3.7(c)). Note, that the CV does not take into account any serial correlation, i.e., the CVs of the original ISI sequences and their surrogates are indeed the same.

3.3 Conclusion

From the analysis of experimental data, non-renewal statistics are seen in some vestibular utricle afferents and in all electroreceptor afferents. Vestibular afferents are

known to possess considerable variability, characterized by the coeﬃcient of variation (CV) of ISI distributions [14]. In turtles, vestibular aﬀerents are rather noisy with CVs

larger than 0.15 [40]. Since vestibular sensory hair cells are spontaneously quiescent, the main source of ISI variability is ﬂuctuations due to stochastic synaptic transmission [69, 70, 71, 40]. Short-range, weak negative serial correlations thus may result from

spike-frequency adaptation due to speciﬁc ionic currents. In contrast, in paddleﬁsh electroreceptors the main source of ISIs variability is noisy oscillations of the epithelial 39

(b)

R = 0.62 (a) 1

0.8 n h

0.6 (c)

0.4 0 2 4 6 8 10 Word length R = 0.35

Figure 3.7: (a): Dynamical entropies for ISIs sequence of an electrorecepor afferent. Red line and symbols shows dynamical entropies calculated for the original sequence. Blue line correspond to corresponding renewal surrogate ISIs. Errorbars we estimated by calculating hn for 10 surrogate sequences. (b,c): Source entropies estimated for words of n = 6 ISIs for the sample of 66 EAs vs the correlation time (b) and the coefficient of variation (c). Red dashed lines show linear regression. Values of the correlation coefficient are indicated on the panels.

cells modulating stochastic synaptic transmission [28, 4]. Combined with spike-frequency adaptation, epithelial oscillations result in extended serial ISI correlations, reﬂecting a much higher degree of order in corresponding spike trains. Dynamical entropies further characterized higher order ISIs correlations. In particular, we showed that the epithelial oscillations of electroreceptors result in higher-order correlations and in extended memory. This is demonstrated in Fig. 3.7(a), red line and symbols: the entropy decreases progressively with the increase of the word 40

length, indicating that more information can be inferred by taking into account various spike patterns. In other words, a spike sequence can be predicted with a smaller uncertainty if longer prior sequences of ISIs are considered. In contrast, a renewal

surrogate, blue line and symbols in Fig. 3.7(a), is indeed characterized by a ﬁrst order Markov process with no memory propagated beyond one interspike interval. Vestibular

afferents (Fig. 3.3) shows significantly less higher order correlations, indicating that a first-order Markov model is appropriate for most vestibular afferents in our sample.

A spike train with negative ISI correlations displays less variability than a spike train with the same mean ﬁring rate but with independent ISIs. This has an important role for detection of weak, low-frequency signals [72, 34]. In contrast to negative ISI correlations,

positive ISI correlations tend to increase spike train variability and compromise signal detection [73, 74]. 41 4 Single neuron models of stochastic spontaneous activity

This chapter describes several models used to study eﬀects of correlations on spontaneous and response dynamics of sensory neurons. We used two classes of models: a

conductance based Hodgkin-Huxley type system and a phase model for oscillatory neurons. We demonstrate that the parameters of the models can be tuned to reproduce spontaneous dynamics of vestibular and electroreceptor aﬀerents.

4.1 Conductance-based Hodgkin-Huxley type model 2

We use a variant of Hodgkin-Huxley model proposed in [75, 76] for studying the so-called spike frequency adaptation which is observed in sensory neurons [67, 1, 14]. For example, the firing rate of electroreceptor afferents adapts almost completely to a constant stimulus [28]. Spike frequency adaptation opposes variations of firing rate due to either

internal ﬂuctuations or external stimuli. Although several biophysical mechanisms of spike frequency adaptation exists, they all rely on an eﬀective negative feedback which tends to bring the neuron system back to a stable steady state [67]. Here we have chosen

the adaptation mechanism based on the so-called afterhyperpolarization due to Ca-gated potassium ionic currents. Such ionic currents have been documented in several types of

sensory neurons, including vestibular aﬀerents [1, 14].

In addition to fast Na and slow outward K currents (INa, IK) the model includes Ca and adaptation currents, (ICa, Iahp). The dynamics of the membrane potential is given by the currents balance equation, dV C = I + I , (4.1) dt − ion ext where Iion is current through membrane ion channels and Iext is an external current. The model combines 5 ionic currents: fast sodium (Na), slow outward potassium (K), leak (L),

2Stochastic version of this model was developed and published in Nguyen and Neiman (2010) [4]. 42

fast calcium (Ca), and calcium gated potassium afterhyperpolarization (ahp) currents,

Iion = INa + IK + IL + ICa + Iahp. (4.2)

The sodium, potassium, leak, and calcium currents are given by the Hodgkin-Huxley

formalism,

3 4 INa = g¯ Nahm (V ENa); IK = g¯ Kn (V EK); IL = gL(V EL); ICa = g¯Cam (V ECa). − − − ∞ − (4.3) where the parameters were the same as in [75, 4]: E = 50, E = 67, E = 100, Na L K − 2 ECa = 120 mV;g ¯ Na = 100, gL = 0.1,g ¯K = 80,g ¯Ca = 1 mS/cm ;

(V+25)/5 2 m (V) = 1/(1 + e− ); C = 1 µF/cm . ∞ The gating variables h, m, n obey the following rate equations,

x˙ = α (V)[1 x] β (V)x, x = m, h, n; (4.4) x − − x 0.32(54 + V) 0.28(V + 27) α = , β = m(V) (V+54)/4 m(V) (V+27)/5 ; (1 e− ) (e 1) − 4 − α = . ( (50+V)/18), β = h(V) 0 128e − h(V) (V+27)/5 ; (1 + e− ) 0.032(V + 52) (57+V)/40 α (V) = , β (V) = 0.5e− . n 1 e (V+52)/5 n − − The AHP current is given by

[Ca2+] I = g¯ (V E ), (4.5) AHP AHP 30 + [Ca2+] − K

and the slow calcium dynamics by

d[Ca2+] = 0.002I 0.0125[Ca2+]. (4.6) dt − Ca −

In the absence of any external current, Iext=0, the model is in an excitable regime. That is, a weak external current pulse produces a weak perturbation, while a large enough

pulse results in generation of an action potential [77, 78]. With an increase of Iext the 43

system bifurcates to periodic limit cycle oscillations [75]. The model belongs to the so-called type-I excitability according to the classiﬁcation adopted in computational neuroscience [77, 78]. The hallmark of this type of excitability is a smooth increase of the

mean ﬁring rate (natural frequency) versus the injected current Iext. The alternative type II excitability is characterized by the Andronov-Hopf bifurcation in which stable limit cycle

of a non-zero frequency is born. Figure 4.1 shows a typical periodic spike train and the

dependence of the mean firing rate on Iext, the so-called f-I curves, for different values of the AHP conductance. The firing rate increases progressively starting from 0 at a bifurcation point (Fig. 4.1(b)). Furthermore, the AHP current results in a postponement of the bifurcation point and a lower firing rate for a given value of the injected current.

(µA/cm2 )

Figure 4.1: Deterministic dynamics of the modiﬁed Hodgkin-Huxley model. (a) Periodic 2 spike train for Iext = 1 µA/cm . (b) Firing rate of the neuron vs Iext for the indicated values 2 of the AHP conductance (gahp is units of mS/cm .

4.2 Stochastic synaptic model

Quantal synaptic transmission is modeled as an inhomogeneous Poisson process with a rate modulated by the stimulus transmitted by hair cells and by spontaneous ﬂuctuations 44

of epithelial cells. The synaptic current in the model is

I = g (t)(V V ) (4.7) syn syn − syn

and corresponds to the ionic currents of the Hodgkin-Huxley model. Here gsyn(t) is the

time-dependent synaptic conductance, and Vsyn is the reversal potential for the synapse. In

our study we consider only excitatory synapses, i.e. Vsyn is more positive than the resting potential of the cell so that the synaptic current tends to depolarize the cell. The synaptic conductance follows the equation:

1 g˙ syn = (gsyn g0) + B δ(t ti). (4.8) −τs − −

The sum in the equation (4.8) represents random pre-synaptic events, τs is the synaptic

time constant, g0 is a constant component of the conductance and B is the size of elementary event. The sequence of synaptic release times t is modeled by a non-uniform { i} Poission process with time-dependent rate, λ(t), modulated by spontaneous activity of epithelial cells h(t) and/or by a stimulus s(t):

λ(t) = λ0(1 + x(t))Θ(1 + x), (4.9)

x(t) = mh(t) + σs(t), (4.10)

where λ0 is a constant rate, m and σ are the amplitudes of modulation for epithelia and stimulus respectively, h(t) is epithelial ﬂuctuations, s(t) is time-dependent stimulus and Θ(x) is the Heaviside step function,

1, if x > 0 Θ(x) =  (4.11)   0, if x 0  ≤  The Heaviside step function is introduced here to make sure that the rate of Poission process is always a positive value. 45

In the absence of rate modulation, m = 0, σ = 0, the mean and the variance of synaptic conductance are

g = g + Bτ λ , (4.12) s 0 s 0 1 var(g ) = g2 g 2 = λ τ B2. (4.13) s s − s 2 0 s

The power spectral density (PSD) of the synaptic conductance, Ggg( f ) can be calculated analytically for the case of weak modulation, m, s 1 [4]. In particular, for ≪ electroreceptors epithelial oscillations may be considered as source of narrow-band internal noise which makes a major contribution to the variability of spontaneous

dynamics [2, 28]. To this end we used Gaussian narrow-band noise h(t) to mimic epithelial oscillations. The PSD of this process is

C 4Q(1 + 4Q2) f 3 G = , C = e , (4.14) hh 16Q4( f 2 f 2)2 + 8(Q f )2( f 2 + f 2) + f 4 π − e e e e characterized by the peak frequency, fe and the quality factor, Q, describing the sharpness of the spectral peak. The PSD given by Eq.(4.14) is parameterized such that the variance

of h(t) is 1, i.e. var(h) = 1. The PSD of the synaptic conductance, Ggg( f ) can be calculated by noticing that Eq.(4.8) represents a low pass ﬁltering of the presynaptic events, so that Ggg( f ) is the product of the ﬁlter’s squared transfer function and the PSD of

the presynaptic events, Gξξ( f ), (Bτ )2 = s . Ggg( f ) 2 Gξξ( f ) (4.15) 1 + (2π f τs) . For weak modulation, m, σ 1, the PSD of Σδ(t t ), G ( f ) is given by ≪ − i ξξ

2 2 2 Gξξ( f ) = λ0 + λ0 m Ghh( f ) + σ Gss( f ) , (4.16)

where Ghh( f ) and Gss( f ) are the PSDs of epithelial and stimulus signals, respectively. For spontaneous dynamics, σ = 0, the variance of synapses conductance is

∞ var(gs) = Ggg( f )d f = (4.17) −∞ 46

λ τ B2 Q2 + 2Qτ (π f + m2Qλ ) + π f τ2[π f (1 + 4Q2) + 4m2Qλ ] = 0 s s e 0 e s e 0 . 2 2 2 2 Q + 2π feτs + (π feτs) (1 + 4Q )

Figure 4.2 shows excellent correspondence between numerical simulations and analytical calculations of the synaptic PSD. / Hz / 2 ) 2 PS (mS/cm PS

Frequency (Hz)

Figure 4.2: Power spectrum density of synaptic conductance. The red and blue lines correspond to simulation and analytical result, respectively. The parameters are τs = 1 ms, 2 λ = 1000 1/s, σ = 0, m = 0.4, B = 0.018, fe = 26 Hz, Q = 20, gahp = 5 mS/cm , 2 2 g = 7.378 10− mS/cm . 0 × 47

4.3 Numerical simulations of the modiﬁed Hodgkin-Huxley model: comparison with experimental data

The modified Hodgkin-Huxley model described above can be tuned to fit experimental data from vestibular utricle afferents. Since utricle sensory hair cells normally do not exhibit spontaneous oscillations we set the epithelial narrow-band noise to 0, h(t) = 0 and chosen parameters for the synaptic model from published experimental

data [70, 71]. Spontaneous ﬁring of some of the utricle aﬀerents is characterized by negative serial correlations, see e.g. Fig.3.2(a3,b3). This can be accounted for in the

model by the effect of the spike-frequency adaptation due to the AHP current. This is illustrated in Figure 4.3 showing a representative example of a spontaneously firing utricle afferent matched with numerical simulations of the model. Quantitative agreement of

experimental and simulation data is clearly seen in three distinct statistical measures: the probability density of interspike intervals (ISIs), the serial correlation coeﬃcients (SCCs)

and the PSD. Weak negative serial correlations (Fig.4.3b) are due to the AHP current. Positive serial correlations, see e.g. Fig.3.2(a2,b2), can be modeled by introducing

additional exponentially correlated noise into synaptic input, which can be achieved by choosing small values of the quality factor, Q < 1, for the narrow-band noise h(t).

The modified Hodgkin-Huxley model can be also tuned to fit the spontaneous dynamics of electroreceptor afferents. Although parameters of synaptic transmissions

were not measured, previous experimental and theoretical studies [33, 28, 2] have shown that the structure of the SCCs and aﬀerent’s PSD is determined by the ratio of epithelial to aﬀerent oscillator frequencies. Figure 4.4 shows a spontaneous statistics of a

representative electroreceptor aﬀerent matched with numerical simulation of the modiﬁed Hodgkin-Huxley model. All three statistical measures shown are reproduced exremely well by the model, indicating that the model captures the essence of dynamical processes 48

Figure 4.3: Comparison between the experiment and simulations for a vestibular aﬀerent. Three panels show probability density of interspike intervals (ISIs), the serial correlation coeﬃcients (SCCs) and the power spectral density of corresponding spike train. Experimental data are shown by black circles. Results of numerical simulations of the 2 model are shown by the red lines. The parameters of the model were: gahp = 1 mS/cm , 2 5 2 2 τ = 2 ms , λ = 700 1/s, σ = m = 0, g = 0.0032 mS/cm , var(g ) = 3.6 10− [mS/cm ] . s s s ×

in the system. We notice extended serial correlations (right upper panel in Fig. 4.4) due to

epithelial oscillations. As was discussed in Chapter 3, the existence of two oscillators in the electroreceptor system results in speciﬁc structure of the PSD (lower left panel in

Fig. 4.4) showing two independent peaks at the fundamental frequencies of epithelial ( fe) and aﬀerent ( f ) oscillators and the sidebands ( f f ). a a ± e 49

Figure 4.4: Comparison between the experiment and simulations for an electroreceptor aﬀerent. Three panels show probability density of interspike intervals (ISIs), the serial correlation coeﬃcients (SCCs) and the power spectral density of corresponding spike train. Experimental data are shown by the black circles. Results of numerical simulations of the 2 model are shown by the red lines. The parameters of the model were: gahp = 3 mS/cm , τs = 2 ms, λ = 1000 1/s, m = 0.45, B = 0.018, fe = 27.1 Hz, Q = 22, < gs >= 2 5 2 2 0.0032 mS/cm , var(g ) = 10 10− [mS/cm ] . s ×

4.4 Simpliﬁed neuronal model: theta neuron

There are many simpliﬁcations of the Hodgkin-Huxley system which have been proposed for modeling neuronal spiking activity [78, 77]. In this work we used the so-called theta neuron model, also known as the Ermentrout-Kopell model [79] as an element of feed-forward networks. The theta neuron model is the normal form of the saddle-node on a limit cycle bifurcation. It describes the spiking of a type-I excitable 50

neuron. The original theta model is one-dimensional dynamical system in the form:

θ˙ = 1 cos θ + (1 + cos θ)R, (4.18) − where the parameter R represents the input current to the neuron. The variable θ (hence the name of the model) is deﬁned on the circle, (0 2π) illustrated in Fig. 4.5. A “spike”

R<0 R=0 R>0

Figure 4.5: Dynamics of the theta neuron model on a circle. Modiﬁed from G.B. Ermentrout, Scholarpedia, 3(3):1398 (2007).

occurs when θ = π. For R < 0 the model is excitable and possesses two states, one stable (blue circle in Fig. 4.5) and one unstable saddle (red circle in Fig. 4.5). At R = 0 the these states converge to a saddle-node which disappears for R > 0. Thus, for R > 0 the model describes a non-uniform rotation on a circle , i.e., periodic spiking with a period π/ √R. The theta model is closely related to an integrate and fire neuron with quadratic nonlinearity [77, 78]. A modified theta neuron has been used in several studies to model spontaneous and response dynamics of single paddlefish electroreceptor [3, 2]. These modifications are the following. First, the parameter R was modulated by the narrow-band noise h(t), a broad-band exponentially correlated Ornstein-Uhlenbeck noise ξ(t) and by a stimulus s(t).

The narrow-band noise models the input from stochastic epithelial oscillations, while the Ornstein-Uhlenbeck noise mimics background synaptic ﬂuctuations. Second, an 51

adaptation variable u was introduced to account for the spike-frequency. The modiﬁed theta neuron model reads,

θ˙ = 1 cos θ + (1 + cos θ)[R + mh(t) + Dξ(t) u + s(t)] (4.19) − 0 − u˙ = λu + βδ(θ π), − −

where R0 stands for the constant component of the input current which sets the fundamental frequency of afferent oscillator, m is the strength of the narrow-band noise (epithelial oscillations), D is the standard deviation of the Ornstein-Uhlenbeck noise, λ is the rate of spike-frequency adaptation, and β is the strength of adaptation. The autocorrelation function of the Ornstein-Uhlenbeck process ξ(t) is < ξ(t)ξ(t + τ) >= exp( τ /τ ), where the correlation time τ was chosen to be much −| | c c smaller than the mean firing rate of the model, so that noise ξ(t) was effectively white. The

N(t) spike train generated by the model is represented as x(t) = n 1 δ [θ(tn) π], where tn are − − threshold crossing times (i.e. when θ(t) = π) and N(t) is the spike count, e.g. the number of spikes in the interval [0 t]. The theta model equations (4.20) are dimensionless. To

represent spike times sequence in seconds we took advantage of the fact that the fundamental frequency of the epithelial oscillations is almost the same across paddleﬁsh

electroreceptors. Thus, time in seconds t(s) can be introduced by t(s) = t/(2π fe), where fe is the frequency of epithelial oscillations in Hz [2]. The modiﬁed theta neuron model was shown to reproduce very well the spontaneous and response dynamics of electroreceptors.

Figure. 4.6 shows results of numerical simulation of the model. In particular, this ﬁgure contrasts the statistics of a non-renewal spike train generated by the model with epithelial oscillations with those of a renewal spike train with parameters tuned to preserve the

probability density of interspike intervals. While the probability densities of both are identical, the PSDs of the renewal model shows signiﬁcant enhancement of low-frequency

power, indicating a role of oscillations in suppressing the low-frequency variability [3]. 52

(a) (b)

(c)

Figure 4.6: Statistical measures of spontaneous dynamics from the theta neuron model Eq.(4.20). (a) Probability density of ISIs. (b) Serial correlation coefficients (SCC). (c) Power spectral density (PSD). Red lines and symbols show the model with epithelial oscillations generated a non-renewal spike trains. Blue lines and symbols correspond to the model with oscillations turned off, but with other parameters tuned to match the probability density of ISIs of the non-renewal model. Modified from [3].

4.5 Conclusion

This chapter described two classes of models used in this work. A conductance-based

Hodgkin-Huxley type model was used for studying spontaneous and response dynamics of single sensors. Such a model allows inclusion of biophysical details of various ionic conductances and synapses as they become available from future experimental studies. A modiﬁed theta neuron model will be used as an element of peripheral sensory feed-forward network, as it is much less computationally expensive. Both models allowed 53 imposition of clear constraints on a system while varying its parameters and studying responses to various stimuli, which is a situation hard and often impossible to achieve in neurophysiological experiments. 54 5 Effect of temporal correlations on spontaneous and

response dynamics of sensory neurons 3

Biological neurons are always noisy. Spiking of a neuron can be described by two

major classes of stochastic point processes, renewal and non-renewal [57]. A renewal spike train is characterized by statistical independence of ISIs, i.e. the occurrence of a spike is independent of the past history of the spike train. For a renewal process, the ISI

serial correlation coeﬃcient C(k) = 0 for all k , 0. As such, the value of a given ISI is independent of the value of the previous ISIs. Renewal models are often used to describe

the ﬁring of cortical neurons [80, 58]. However, many sensory neurons and neurons in various brain areas exhibit non-renewal spiking activity (for review see [31, 32]). For

example, the serial correlation coefficients persist over hundreds of ISIs in electroreceptors afferents of paddlefish [28, 2], or over a few lags in electrorecepotrs of weakly electric fish [72, 61], somewhat similar to vestibular afferents as we showed in the previous chapter.

Neuronal serial ISI correlations may appear result form to two factors. The ﬁrst, is the internal dynamics of a neuron, e.g. spike-frequency adaptation ionic currents observed in many sensory neurons [67]. the second, is correlated ﬂuctuations of synaptic current [4, 73]. In contrast to many other studies, we consider combination of both these factors.

Furthermore, we study how ISI correlations inﬂuence linear and nonlinear responses of a neuron to time-varying stimuli. The eﬀects of serial correlations on nonlinear response is a neglected topic in the current literature.

5.1 Spontaneous dynamics of the Hodgkin-Huxley model of electroreceptors

We use the modiﬁed Hodgkin-Huxley-type system described in Ch.4 to model oscillatory electroreceptors. In order to investigate the role of epithelial oscillations (EO)

3Material from this chapter was published in Nguyen and Neiman (2010) [4]. 55 in shaping of spontaneous dynamics and information transmission in a neuron system, we compared three kinds of models. First, the original model with spike-frequency adaptation due to the afterhyperpolarization (AHP) current and with EO. Second, a renewal model in which the AHP current and the EO were turned off (gAHP = 0 and m = 0 in the model of Ch.4.1-2). Third, the so-called “no oscillation” model where the coherent epithelial oscillations were replaced with incoherent broad-band noise. We impose constraints on the probability density of ISIs, so that for a given set of parameters the mean firing rates and the coefficients of variation (CVs) are the same for all three models. For the renewal, model this was achieved by tuning the parameters of synaptic transmission, g0 and var(gs), such that the mean firing rate and CV match those of the original model. For the no-oscillation model, we replaced the epithelial narrow-band noise with a Gaussian broad-band noise with a short correlation time ( 0.5 ms) and then tuned the variance of that noise to match the mean firing rate and the CV of the non-oscillation model to those of the original model. Thus, all three models had identical firing rates and CVs, but distinct correlation structure of generated spike trains. When stimulus is applied, the coherence functions and information rates are calculated for the original model and the model with no oscillations, allowing us to infer a role of coherent epithelial oscillations.

First, we compare the original and renewal models. The probability densities of ISIs shown in Figure 5.1, demonstrate that the synaptic parameters of the renewal model can be tuned to match the ISIs distribution of the original model. The second-order statistics of the original model are characterized by extended serial correlations and sequences of characteristic peaks in the power spectrum. Figure 5.2 shows extended serial correlations for the original model with epithelial oscillations. Serial correlations are absent for the renewal model even though it possesses identical distribution of ISIs. Finally, Figure 5.3 shows the PSDs of spontaneous spike trains generated by the original and renewal models. 56

250

200

150

100 Probability density

10 15 20 25 30

ISI(ms)

Figure 5.1: Probability densities of ISIs. The red lines and blue lines correspond to the original and renewal models,respectively. Parameters of the original model were 2 2 2 3 gAHP = 7 mS/cm , g0 = 8.68 10− mS/cm , B = 5.86 10− , m = 0.4, Q = 20, τs = 2 ms, 1 × 2 2 × 5 2 2 λ = 1000 s− , gs = 9.27 10− mS/cm , var(gs) = 2.3 10− [mS/cm ] . Parameters of × 3 × 2 2 the renewal model were gAHP = 0, m = 0, g0 = 7.9 10− mS/cm , B = 1.62 10− , m = 0, 2 2 ×4 2 2 × g = 2.413 10− mS/cm , var(g ) = 1.316 10− [mS/cm ] . s × s ×

The power spectrum captures the frequency content of a stochastic process. For the original model, we have a contribution of two oscillations: the epithelial (EO) and the afferent (AO) oscillations. Peaks at fe and fa represent the fundamental frequencies of these oscillations, while the sidebands at f f and higher harmonics are due to a ± e nonlinearity of the neuron. In contrast, the PSD of the renewal model has a single broad fundamental peak (and its higher harmonics) corresponding to the mean firing rate of the neuron. More importantly,the power at low frequencies of the original model is an order of magnitude lower than that of the renewal model, indicating significant decrease of low-frequency variability in the original model due to coherent epithelial oscillations. 57

0.6

0.4

0.2

0.0

-0.2 SCC

-0.4

-0.6

-0.8

1 10 100

ISI lag

Figure 5.2: The serial correlation coeﬃcients (SCC) of ISIs for the original (red line) and renewal (blue line) models. The parameters are the same as in Fig. 5.1

This oscillation-induced suppression of variability can be illustrated further in Fig. 5.4 with the aid of the Fano factor Eq.(2.4) which characterizes spike count variability in a

time window T. For short time scales, as T 0, both, the original and renewal models → can be approximated by a Poisson process. Consequently, the Fano factor for both models is close to 1. The renewal model has no signiﬁcant serial correlations amongst ISIs.

2 Hence, for large T the Fano factor tends toward its theoretical limit, limT F(T) = CV →∞ [57]. Negative serial ISI correlations decrease the Fano factor beyond this value, indicating a less variable spike train. We note that the Fano factor determines discrimination capacity of a sensor: smaller values of the Fano factor indicates better discrimination between spike count distributions with and without stimulus [72, 34]. 58

f +f

a e

1000 /Hz)

f -f 2/ 100 a e

10 PSD([spike/s]

0.1

0 20 40 60 80 100 120

Frequency(Hz)

Figure 5.3: Power spectrum densities (PSD) of spontaneous spike trains generated by the original and renewal models, fa,aﬀerent oscillator peak, fe epithelial oscillator peak

In the original model, the coherent epithelial oscillations, h(t), is a narrow-band Gaussian noise. By replacing the coherent oscillations with incoherent broad band noise

h(˜t), we obtain the so-called no oscillation model, used to reveal the eﬀect of coherent epithelial oscillations on response properties and information transfer. In particular, we used an exponentially correlated Ornstein-Uhlenbeck (OU) process, h(˜t), with the correlation time of 0.5 ms. The variance of h(˜t) was tuned numerically to match spontaneous mean ﬁring rate and CV of the original model with coherent oscillations.

Figure 5.5 compares statistical properties of the original (with oscillation) and no

oscillation models. The constraint on the CV and the ﬁring rate imposed on the models resulted in almost identical ISIs probability densities, (Fig. 5.5a). Nevertheless the SCCs of the models diﬀer dramatically. We picked the parameters such that the original model 59

) 2 CV T

( F

T (ms) Figure 5.4: Fano factor for the original (red line) and renewal (blue line) models for the same set of parameters as in Fig. 5.1. Horizontal dashed line shows theoretical limit for the renewal process.

had two spikes per cycle of epithelial oscillations, e.g. the frequency ratio w = fe/ fa was exactly 0.5, resulting in extremely long ISI correlations (red line in Fig. 5.5(b)). We note that both models are non-renewal, so that short-range negative ISI correlations appear due to the self-inhibitory AHP current [76] even in the model with no epithelial oscillations (dotted green line in Fig. 5.5(b)). The power spectrum of the model with no oscillations showed a peak at fa = 55Hz corresponding to the mean firing rate of the neuron. Because of negative serial correlations, the power spectrum decreased towards low frequencies demonstrating the effect of noise shaping discussed in [35, 68]. Because of the larger magnitudes of SCCs, the effect of noise shaping is more pronounced for the original model with coherent epithelial oscillations, leading to larger reduction of low-frequency power as compared to the model with no oscillations (Fig. 5.5(c)). 60

1.0 (b) 160 (a) 0.5 120 0.0 80 SCC

40 -0.5

Probability desity 0 -1.0 10 15 20 25 01020304050

ISI (ms) ISI lag

10 3 (c)

10 2 /s) 2 10 1

10 0 PSD (spk 10 -1

050 100 150 200 Frequency (Hz) Figure 5.5: Statistical properties of spike trains generated by the original model (with coherent epithelial oscillations) and by the no oscillation model whereby the narrow- band noise was substituted with a short-correlated Ornstein-Uhlenbeck noise. Probability density of ISIs (a), serial correlation coeﬃcients (b) and power spectral density (c) of spontaneous stochastic dynamics. Solid red lines correspond to the original model with epithelial oscillations. Dotted green lines refer to the model with no epithelial oscillations. 2 4 2 The parameters are: gAHP = 6mS/cm , τs = 2ms, λ0 = 10 Hz, gs = 0.081mS/cm , 5 2 2 var(g ) = 3 10− [mS/cm ] , m = 0.5, Q = 20, f = 27.5Hz. These parameters resulted in s × e the mean ﬁring rate fa = 55 Hz and CV = 0.185.

The shape of SCCs is determined by the ratio of two fundamental frequencies of

epithelial and aﬀerent oscillators, w = fe/ fa, [2] as demonstrated for experimental data in Fig. 3.5. The experimentally measured value of w for paddleﬁsh electroreceptors is w = 0.49 0.08 [28], i.e. two oscillators embedded in the electroreceptor system operates ± close to a 1:2 mode locking regime. Modeling allows us to study how the extent and the 61 (a)

(b)

Figure 5.6: First serial correlation coeﬃcient, C(1) (a) and correlation time tcor (b) of spontaneous ISIs sequences versus the ratio of fundamental frequencies of EO to AO,w = fe/ f a for the indicated values of the quality factor Q of epithelial oscillations. Other parameters are the same as in Fig. 5.5.

magnitude of serial correlations depend on the frequency ratio w, while all other parameters are fixed. Figure 5.6 shows the results. The magnitude of the first serial correlation coefficient peaks at w = 0.5 (Fig. 5.6(a)) indicating that the SCCs are stronger 62

when the frequencies of the two oscillators are in a ratio of 1:2. At the same time the correlation time is sharply peaked at w = 0.5 as Fig. 5.6(b) shows. This is consistent with the previous results obtained with a toy model of the circle map [2]. Furthermore, the

enhancement and extension of serial correlations is stronger for higher values the quality factor of epithelial oscillations.

5.2 Linear response and stimulus encoding

The response properties of the model to weak stimuli and linear stimulus encoding were characterized with stimulus-response (SR) coherence explained in Chapter 2 and

given by Eq.(2.10). We remind the reader that for a noiseless linear system the SR coherence equals its maximum value of 1. Internal noise, measurement noise and nonlinearities all result in values of SR coherence less than one. We used a Gaussian

2 stimulus with a cutoﬀ frequency fc. The stimulus PSD is given by Gss( f ) = σ /(2 fc) for f f and G ( f ) = 0 for f > f , where σ is the stimulus standard deviation (SD). Notice, ≤ c ss c that the stimulus variance was invariant with respect to variations of the band fc. In the following we compared two variants of the Hodgkin-Huxley model: the original model with adaptation and narrow-band epithelial oscillations and the no oscillation model with adaptation, but with incoherent epithelial noise. The parameters of the no oscillation model were tuned to match the mean ﬁring rate and the CV of the original model with coherent epithelial oscillations, as explained in the previous section.

Figure 5.7 shows SR coherence for two values of the stimulus cutoﬀ frequency, fc.

For the wide-band stimulus with fc = 200 Hz, the SR coherence of the original model showed a wide peak at 7 Hz and a notch at the peak frequency of narrow-band epithelial ≃ oscillations, fe, followed by a smaller maximum. Similar coherence structure was reported recently in experiments with paddleﬁsh electroreceptors [38]. The existence of the notch 63 (a)

Original (b) No oscil.

Frequency (Hz) Figure 5.7: Stimulus - response coherence for Gaussian band-limited stimulus with the standard deviation σ = 0.5 and cutoff frequency fc = 200 Hz (a) and fc = 20 Hz (b). Red line corresponds to the original model with epithelial oscillations. Green line refer to the model with incoherent epithelial fluctuations (No oscil.). Both models had identical spontaneous mean firing rates and CVs. The parameters are the same as in Fig. 5.5.

at fe is due to the independence of the frequency of the epithelial oscillations from the stimulus. Thus, epithelial oscillations represented in the model by narrow-band Gaussian noise act to reduce the coherence at fe. The no oscillation model demonstrated a uniform 64

coherence curve with signiﬁcantly smaller values. As shown in the previous section, extended serial correlations induced by epithelial oscillations suppress spectral power at low frequencies, as compared to the model with incoherent epithelial ﬂuctuations, Fig.

( 5.5c). This reduction of noise in the form of background spiking results in the increased coherence for the original model with epithelial oscillations. Stimuli with smaller

frequency band, fc = 20 Hz, matching the response band of paddleﬁsh ERs, (Fig. 5.7b) shows similar behavior of the SR coherence for the original and the no oscillation model.

We used the SR coherence for calculation of the lower bound estimate of the mutual

information rate, ILB using Eq.(2.17). The dependence of ILB on the ratio of frequencies of

EO to AO, w = fe/ fa is shown in Fig. 5.8(a) for the original model. The information rate peaks at w 0.5 corresponding to the maximal effect of epithelial oscillations on serial ≃ ISI correlations (Fig. 5.6). To compare information rates of the original model with coherent epithelial oscillations to those of the model with incoherent epithelial fluctuations, we calculated the difference between information rates of these models,

∆I = I I , i.e. the information gain, shown in Fig. 5.8(b). This information gain osc − nonosc shows signiﬁcant enhancement of information transfer due to coherent epithelial

oscillations which is maximum at w 0.5. The enhancement of information rate increases ≃ with the increase of the quality factor of epithelial oscillations. These results conﬁrms the

previous ﬁnding made with a toy theta neuron model [3].

5.3 Nonlinear response

The SR coherence used in the previous section characterizes linear response properties of a system, so that the lower bound estimates of the information rate based o

SR coherence may signiﬁcantly underestimate the true information rate if a stimulus gets stronger. A formal reason is that even for a noiseless system the value of SR coherence 65 (a) (bit/spk) I

w (b) (bit/spk) I ∆

w Figure 5.8: Lower bound estimate of the mutual information rate (a) and the information gain ∆I (b) versus the ratio of fundamental frequencies of EO to AO, w = fe/ fa for the indicated values of the quality factor of narrow-band epithelial oscillations. Stimulus parameters were σ = 0.5, fc = 20 Hz. Other parameters were the same as in Fig. 5.5.

can be smaller than 1 because of nonlinearities. In a sensory peripheral system strong stimuli are equally important as weak stimuli. For example, in paddlefish electroreceptors strong stimuli result in nonlinear responses in the form of stimulus-induced bursts [39], 66 while strong periodic stimulation of vestibular afferents result in mode locking of spikes to periodic signals [40]. To characterize nonlinear response dynamics, we used the so-called response-response (RR) coherence which characterizes response variability of a neuron that cannot be accounted for by a given stimulus. As seen from the definition of RR coherence given in Eq. (2.11), the RR coherence does not involve stimulus power spectra. Instead, it characterize the degree of synchrony among the ensemble of multiple neural responses to an identical stimulus. Like SR coherence, RR coherence is a normalized quantity ranging from 0 (no correlations across responses) to 1 (perfectly synchronized responses) and satisfies the following inequality, C ( f ) C ( f ). S R ≤ RR We applied identical realizations of a Gaussian band-limited stimulus repeatedly to the original and no oscillation models to generate an ensemble of responses. This method, known as “frozen noise”, allows characterization of response variability [30]. Fig 5.9 (b,c) shows raster plots of responses of the original model with coherent epithelial oscillation to repeated presentations of a weak, Fig 5.9(b), and strong, Fig 5.9(c), Gaussian stimulus. The strong stimulus results in a bursting responses similar to those observed experimentally [38, 39]. Vertical stripes in Fig 5.9(c) clearly indicate that the strong stimulus synchronized the afferent bursting responses across the stimulus trials.

Such reliable and precise response dynamics of the model with coherent epithelial oscillations is contrasted to the no oscillation model with incoherent epithelial ﬂuctuations in Fig. 5.9(d), which shows a signiﬁcantly lower degree of cross-trial synchrony.

The RR coherence is a quantitative measure of cross-trail synchrony in the frequency domain. For the linear response, i.e for the case of weak stimuli, the SR and RR coherence function match. For a strong stimulus resulting in a bursting response, Fig 5.9(d), SR and RR coherence functions diﬀer signiﬁcantly. Fig 5.10(a) shows that while the SR 67 (a)

0.1 s 200 (b)

0 200 (c)

0 200 (d)

0 Figure 5.9: Cross-trial response variability. The original model and the no oscillation model were stimulated repeatedly with identical realization of Gaussian noise band limited to fc = 20 Hz. (a) Short segment of the stimulus. (b -d) Raster plots of models responses to 200 trial presentations of the stimulus. Vertical axis is the trial number; horizontal axis is the time with respect to the stimulus onset. Dots indicate appearance of a spike in a trial relative to the stimulus onset. Panel (b) is for the stimulus standard deviationσ = 0.5 (weak stimulus). Panels (c,d) is for the case of strong stimulus with σ = 2.0 applied to the original model with narrow-band noise (c) and to the no oscillation model with broad-band OU noise (d). Other parameters were the same as in Fig. 5.5.

coherence is zero beyond the stimulus cutoﬀ frequency ( fc = 20 Hz), non-zero values of RR coherence extend well beyond the stimulus band. Large values of RR coherence for high frequencies ( f > 50 Hz) reﬂects synchronization of spikes and bursts across the ensemble of neural responses. Such behavior of RR coherence is indicative of nonlinear 68

(a) 1.0 SR Coher 0.8 sqrt(RR coher) 0.6

0.4

Coherence 0.2

0.0 050 100 150 200 (b) 1.0 0.8 Oscil. No Oscil. 0.6

0.4

Coherence 0.2

0.0 050 100 150 200 f (Hz) Figure 5.10: Stimulus-response (SR) and response-response coherence for simulations shown in Fig. 5.9. (a) SR (dashed line) and RR (solid line) coherence functions for the original model with narrow-band noise. (b) RR coherence for the original model (oscil., solid line) and for the no oscillation model (no oscil., dotted line).

neural responses. The role of narrow band epithelial oscillations in enhancing of cross-trial synchrony is clearly seen in Fig 5.10(b) which compares RR coherence for the original model with narrow-band noise to RR coherence for the no oscillation model with incoherence epithelial ﬂuctuations. In a wide frequency range of 22-120 Hz, the RR coherence is about 3 times larger for the model with non-zero epithelial oscillations.

For a nonlinear response such as shown in Fig. 5.9(c,d) the upper bound of information rate can be calculated from the RR coherence of Eq.(2.18) and compared with the lower bound estimate. Figure 5.11(a) compares these estimates for the original model 69

(a) (bit/spk) I

(b) (bit/spk) I ∆

Figure 5.11: Lower and upper bounds of information rate (a) and information gain ∆I (b) versus the standard deviation σ of Gaussian stimulus. normalized to the standard deviation of epithelial oscillations. The stimulus cutoﬀ frequency is fc = 20 Hz. The parameters of the model are the same as in Fig. 5.5.

and diﬀerent values of stimulus strength, σ. For a Gaussian information channel, it is indeed expected that the true information rate increases monotonically with increases of a signal. However, the lower bound estimate of information rate shows diﬀerent behavior.

With the increase of σ, the lower bound ILB ﬁrst increases, reaches a maximum, and then decreases. Such behavior is a manifestation of nonlinearity of the system resulting in a 70

decrease of SR coherence and consequently in decrease of ILB. The calculation of RR coherence does not involve the stimulus directly. Furthermore, it quantiﬁes stimulus-induced synchronization. Consequently, the upper bound of information rate, IUP increases monotonically with the increase of stimulus strength, which is the expected behavior. Both estimates coincides for weak stimuli, indicating a range of stimulus strengths in which a linear model adequately describes stimulus encoding. Figure 5.11(b) compares information transmission by the original model with coherent epithelial

oscillations to that of the no oscillation model with incoherent ﬂuctuations using lower and upper bounds of information rate. In the range of linear response both estimates coincides and predict enhancement of stimulus coding due to coherent oscillations, as the

information gain ∆I grows with σ. However, for the nonlinear response the information gain based on the lower bound estimate passes through a maximum and then decreases,

while ∆I based on the upper bound keeps growing.

5.4 Conclusion

In this chapter we have studied spontaneous and stimulus-induced response dynamics

of a Hodgkin-Huxley type system. The parameters were tuned to reproduce statistical properties of peripheral electroreceptors in paddleﬁsh, which are characterized by the biperiodic background activity of their primary aﬀerents. To elucidate the role of

epithelial oscillations observed in paddleﬁsh ERs, we contrasted two models: the original, with narrow-band noise served to mimic epithelial oscillations, and the no oscillation

model, where harmonic noise was replaced with short-correlated Ornstein-Uhlenbeck noise. The parameters of this second no oscillation model were tuned to match the mean

firing rate and the coefficient of variation of the original model. Our main results are: 1. Epithelial oscillations modulating the excitatory synaptic transmission of an afferent induce long lasting serial correlations of afferent interspike intervals, leading to 71

suppression of low-frequency power in spontaneous spike train. The strength and

extension of the serial correlation is maximal when the frequencies of epithelial fe and afferent fa oscillations are in rational relation, fe/ fa = 0.5. 2. Suppression of background power at low frequencies lead to significant enhancement of information transmission by the neuron, estimated with a linear stimulus reconstruction method. In other words, coherent noise allows for dramatically more efficient stimulus encoding by a neuron as compared to a neuron with incoherent internal noise. Information rate and the enhancement of information due to oscillations are maximal when the frequency ratio of epithelial to afferent oscillators, fe/ fa is close to 0.5. 3. Strong stimuli result in a nonlinear response in which neuronal spikes are grouped in bursts. Furthermore, strong stimuli synchronize neuronal responses, so that neuronal responses become stereotypical across multiple presentations of a stimulus.

4. The degree of such stimulus-induced synchrony is higher for the original model with coherent epithelial oscillations as compared to the model with no oscillatory internal noise.

5. Consequently, nonlinear stimulus coding estimated with response-response coherence and the upper bound of information rate is enhanced by coherent ﬂuctuations.

To conclude, our results show that epithelial oscillations enhance information transmission in electroreceptors for both weak (linear response) and strong (non-linear response) stimuli. More generally, we showed that oscillations may shape information transmission by a sensor, and that coding eﬃciency of the sensor can be enhanced signiﬁcantly by tuning fundamental frequencies of oscillations. 72 6 Effect of spatial and temporal correlations on

spontaneous dynamics and stimulus coding by small-scale networks of peripheral sensory neurons

Hair cell – aﬀerent receptors can be viewed as a layered cell networks passing information about external stimuli to the central nervous system. The primary layer in such network is formed by sensory hair cells which are innervated by sensory neurons

(afferents) forming a second layer. In the vestibular sensory system, an afferent may innervate multiple hair cells [1, 21] (see also Fig. 1.1). Higher afferent innervation is

observed in peripheral electroreceptors in paddlefish whereby a few (3 – 7) primary afferents innervate 3 – 30 sensory canals (Fig. 1.2). Thus, primary afferents may receive common inputs from multiple sensory cells. Furthermore, because of the stochastic nature of synaptic transmission these common inputs are stochastic. Such input noise is correlated not only in the time domain (temporal correlations), but also across neurons, that is, inputs to afferents are correlated. The general term of noise correlations is used in the neuroscience literature for random, but correlated inputs to a network of neurons.

General questions of how these correlations propagate through a network, and how they aﬀect encoding of sensory information is of major interest [43, 44, 45, 81, 46, 50, 54, 82].

In this chapter we use a general model of a small-scale parallel network of oscillatory electroreceptors to study the eﬀect of temporal, inter-neuronal and serial ISI correlation on spontaneous and response dynamics. Previous theoretical and computational studies used

small-amplitude – white noise to approximate input noise [82, 46, 50, 53]. These approximations certainly fail for hair-cell – aﬀerent receptors and for electroreceptors, in

particular: stochastic inputs from spontaneously oscillating hair cells possess signiﬁcant temporal correlations and represent the major source of variability in aﬀerent neurons (i.e.

inputs can not be considered as weak). For colored non-weak noise, analytical treatment 73

of a network is hard to implement, and thus we will rely on extensive numerical simulations. Previous studies mentioned above used a simple cross-correlation coeﬃcient between spike counts of network elements. In contrast, we employ a more advanced

coherence analysis which provides a full picture of cross-correlations in the frequency domain.

6.1 Network model

Figure 6.1 sketches a feed-forward network of ampullary electroreceptors. This network represents a cluster of canals innervated by few sensory aﬀerent neurons. Each sensory epithelium contains about 1000 receptor cells in each canals, converging onto 3 –7 aﬀerents which converge onto secondary neurons embedded in the central nervous system.

1 2 3 ... afferents

1 2 3 4 ... 30 canals

... recep. cells ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ EXTERNAL STIMULUS (aquatic voltage gradient, µV/cm) Figure 6.1: Layered structure of the feed-forward sensory network in paddleﬁsh electroreceptors

Because of the stochastic spontaneous activity of epithelial cells and external stimuli, afferent neurons receive correlated input. What is the effect of coherent epithelial oscillations on the dynamics of such a network? How does correlated stochastic activity propagate from layer to layer and how is the information transmitted by the network 74 affected by these correlations? We will address these general questions using a network which will combine individual neuronal units described in Chapter 4. We consider M uncoupled neurons (M =1 – 10) each modeled by the modified theta neuron system Eq.(4.20):

θ˙ = 1 cos θ + (1 + cos θ )[R + η (t) u + s(t)] (6.1) k − k k 0k k − k u˙ = λu + βδ(θ π), k − k k − where subscript k indicates k-th neuron. For simplicity we assume identical adaptation dynamics, so that the adaptation parameters λ and β are the same across the network. Each neuron receives a common stimulus s(t) and noise ηk(t) and generate a spike train xk(t).

Two stochastic processes contribute to the noise term ηk(t): (i) temporally exponentially correlated broad-band Ornstein-Uhlenbeck process, ξk(t), modeling stochastic synaptic release; (ii) A correlated Gaussian process modeling the spontaneous activity of epithelial cells. The Ornstein-Uhlebeck process is uncorrelated across the network,

ξ (t)ξ (t + τ) = δ exp ( t /τ ). The epithelial process is partially correlated, so that k m k,m −| | c the noise term ηk(t) is:

ηk(t) = D ξk(t) + A √c h0(t) + √1 c hk(t) , k = 1, ..., M, (6.2) − where D and A are the standard deviations of the Ornstein-Uhlenbeck and epithelial noise, respectively; h0(t) is a component of epithelial noise common to all neurons; hk(t) is independent component of epithelial noise. Both h0 and hk have identical temporal auto-correlation functions (and hence the power spectral densities), but are not cross-correlated: h (t)h (t + τ) = 0. In other words, each aﬀerent receives a combination 0 k of common and independent signals from spontaneously active epithelial cells. In Eq.(6.2) c [0 1] is a parameter which sets the degree of inter-neuronal correlations for epithelial ∈ noise: c = 1 correspond to perfectly correlated epithelial noise across the network; c = 0 75

refers to uncorrelated epithelial inputs. The correlation properties of the input noise are given by the correlation matrix with the elements η (t)η (t + τ) , k m

t 2 | | 2 2 ηk(t)ηm(t + τ) = δk,m D e− τc + A c h(t)h(t + τ) + A (1 c) h(t)h(t + τ) ,(6.3) − k, m = 1, ..., M,

where h(t)h(t + τ) is the auto-correlation function of epithelial signal and δ is the k,m Kronecker’s delta.

In the following, the parameters of the model Eqs.(6.2) were chosen to reproduce the spontaneous activity of individual electroreceptor aﬀerents as explained in Ch. 4.4.

Epithelial noise h0, hk(t) was modeled as narrow-band Gaussian noise with the PSD given

by Eq.(4.14) and the peak frequency fe=26 Hz.The parameters R0k, λ, β where chosen such that output spike train generated by the elements of the network had ﬁring rates in the

range 40 – 70 Hz and showed extended serial ISI correlations (see Fig. 4.6 for an example of spontaneous activity of the theta neuron). To consider an inhomogeneous network we

introduce a diversity parameter ∆ fa such that the mean ﬁring rate of the network elements. f is uniformly distributed in the range [ f ∆ f /2, f ∆ f /2], where f =52 Hz. ak a − a a − a a We note that although the model is intended to describe peripheral electrorecepors, it can also be applied to study the dynamics of oscillatory neurons in the central nervous system receiving common inputs from collective rhythms, e.g. beta rhythm [5].

6.2 Input – output correlation measures

We used coherence functions to characterize how noise correlations are transferred from the input to the output of our network model. In the absence of a stimulus, s(t) = 0 in Eqs.(6.2), the inputs are given by Eqs.(6.2). The coherence function between k-th and

m-th inputs is deﬁned as 2 (in) Gkm( f ) Ckm = | | , Gkk( f )Gmm( f ) 76

where Gkm( f ) is the cross-spectral density between inputs, ηk(t) and ηm(t), and Gmm is the

PSD of ηm(t). For the input noise Eqs.(6.2) this coherence function can be calculated analytically, 2 (in) c α Ghh( f ) Ckm ( f ) = , (6.4) Gξξ( f ) + α Ghh( f ) where α = (A/D)2, i.e. α is the ratio of intensities of narrow-band and broad-band noises,

Ghh( f ) is the PSD of narrow-band noise given by Eq.(4.14), and Gξξ( f ) is the PSD of

2 broad-band Ornstein-Uhlebeck noise, Gξξ( f ) = 2τc/(1 + 2π f τc) . Indeed the input coherence vanishes for c = 0, e.g., when there is no correlated component at the input. For c = 1 the input coherence tends to 1 for α , i.e when the intensity of broad band noise → ∞ vanishes, D 0. Figure 6.2(a) shows the input coherence for various values of the →

quality factor Q. For coherent epithelial oscillations the input coherence has a sharp peak at the frequency of the oscillations, f = fe and small values of coherence at low frequencies. The coherence peak becomes wider for small Q, and larger values of coherence are observed over wider range of frequencies, Fig. 6.2(a). Input correlations are also shaped by the parameter α as shown in Fig. 6.2(b). For small values of α the

coherence exists in a narrow frequency range centered at fe. With the increase of α, i.e. with a decrease of incoherent broad band noise, the coherence peak becomes wider, so

that the coherence between inputs occur in over a wider frequency range. The output network correlations, i.e. cross-correlations between neural spike trains will be characterized by the coherence function between pairs of aﬀerents’ spike trains.

The deﬁnition of this output coherence is the same as for the input coherence,

2 (out) Gkm( f ) Ckm = | | , (6.5) Gkk( f )Gmm( f )

where Gkm is the cross-spectrum of k-th and m-th spike trains, and Gkk, Gmm are the PSDs of k-th and m-th spike trains, respectively. To characterize how noise correlations are transferred by the network we introduce the output-input ratio of coherences, or coherence 77

Figure 6.2: Coherence function between two inputs, Eq.(6.4). (a) Input coherence for τc = 0.1, fe = 26 Hz, c = 1, α = 5 and indicated values of the quality factor. (b) Input coherence for Q = 20 and indicated values of α. Other parameters are the same as in panel (a).

sensitivity as C(out)( f ) = km χ( f ) (in) . (6.6) Ckm ( f ) 78

The values of χ( f ) > 1 indicate that noise correlations are enhanced by the network. Oppositely, values of χ( f ) less than one indicate suppression of noise correlations by the network.

6.3 Stimulus encoding by a network

We discussed in previous chapters how to assess information transmission by a single neuron using a lower bound estimate of the mutual information rate. In this section we focus on how stimulus transfers through from layer to layer of small scale neuronal network model. For a single neuron, the lower bound of the mutual information rate is given by Eq.(2.17). It can be written as

fc ILB = log2[SNR( f )]d f, − 0

1 where SNR( f ) = [1 C ( f )]− and C ( f ) is the stimulus-response coherence. A similar − S R S R stimulus reconstruction technique can be developed for neural networks [83]. We consider a network of M neurons stimulated by a common Gaussian stimulus s(t).

In the frequency domain a linear stimulus estimate from M neural spike trains is given by

S est( f ) = Hm( f )Ym( f ), (6.7) m=1 where S ( f ) is the Fourier transform of the stimulus estimate, Hm( f ) is the transfer function of the optimal ﬁlter for the m-th neuron and Ym( f ) is the Fourier transform of its response (spike train). The transfer functions of the optimal ﬁlters, Hm( f ), are the solution of the following linear matrix equation,

G11( f ) G12( f ) ... G1M( f ) H1( f ) G∗s,1( f )              G21( f ) G22( f ) ... G2M( f )   H2( f )   G∗s,2( f )      =   . (6.8)        ......   ...   ...                     GM1( f ) GM2( f ) ... GMM( f )   HM( f )   G∗s,M( f )              79 where Gm,n( f ) is the cross-spectrum between m-th and n-th spike trains and Gs,m( f ) is the cross-spectrum between stimulus and m-th spike train. The optimal filters Hm( f ) minimizes the variance of effective reconstruction noise, n(t) = s(t) s (t). In the − est frequency domain, N( f ) = S ( f ) S ( f ) = S ( f ) M H ( f )Y ( f ), which allows − est − m=1 m m calculating the PSD of the reconstruction noise, G ( f ) = N( f )N∗( f ) and thus the nn signal-to-noise ratio, SNR( f ) = Gss( f )/Gnn( f ). Finally, the lower bound ILB is calculated as fc ILB = log2[SNR( f )]d f, (6.9) 0 in units of bits per second. Dividing it by the mean firing rate across the network provides the lower bound of the mutual information rate in units of bits per spike. We will also use the quantity ILB( f ) = log2[SNR( f )] which is the information density (units are bit per second per Hertz) as an alternative for the stimulus-response coherence function of a single neuron. The information density allows quantification of stimulus encoding as a function of stimulus frequency content.

6.4 Transformations of noise correlations by the network

In this section we study the eﬀect of coherent stochastic oscillations on dynamics of the network model and in particular how the input correlations aﬀect statistics of neural spike trains generated by the network. We will consider only paired correlations, so that a comparison of only pair of inputs and outputs is necessary.

We start with a network of two identical neurons driven by independent broad-band noises and by a common narrow-band noise. Figure 6.3(a) shows the input coherence

(gray line) and output coherence between aﬀerent spike trains for various values of the frequency ratio of epithelial to aﬀerent oscillations, w = fe/ fa. The output coherence shows a peak at the frequency of epithelial oscillations fe and its higher harmonics, indicating a non-linear correlation transfer. Furthermore, noise correlations are enhanced 80

at low frequencies. We are particularly interested in the low frequency band, because stimuli to our network, simulate those experienced by electroreceptors, have signiﬁcant low-frequency content. The enhancement of noise correlations at low frequencies can be

χ(f )

Figure 6.3: Transformations of noise correlations for two identical neurons driven by independent broad-band noises and a common narrow-band epithelial noise. (a) Input coherence (gray line) and output coherences for the indicated values of the frequency ratio w. (b) Contour lines of the coherence sensitivity χ( f ), Eq.(6.6), versus frequency and w. fe = 26 Hz indicates the location of epithelial oscillations frequency. Other parameters are A = 0.5, D = 0.2, λ = 0.02, β = 0.3, c = 1. explained by a nonlinear eﬀect of extraction of a slowly ﬂuctuating envelope of narrow-band noise. Interestingly, the transformation of noise correlations depends 81

crucially on the fundamental frequencies of network oscillations. The effect of low-frequency correlation enhancement is minimized for afferents with the mean firing rates about twice as high as the epithelial oscillations, i.e. for w close to 0.5. Fig. 6.3(b)

summarizes these results in a contour plots of the coherence sensitivity χ( f ) versus frequency for diﬀerent values of the frequency ratio w. We can see clearly that noise

correlations are enhanced for aﬀerents with w > 0.56 and w < 0.51. However, noise correlations are suppressed for 0.51 < w < 0.56. We note that the values of w close to 0.5

correspond to the longest serial ISI correlations (see Fig. 3.5) leading to a maximal subtraction of the low-frequency power in spontaneous ﬁring. This minimization of low-frequency variability is then reﬂected in suppression of low-frequency noise

correlation at the network output. Temporal correlations have a signiﬁcant eﬀects on

noise correlation transfer by the network, as shown in Fig. 6.4. For the narrow-band epithelial collective input, the quality factor Q is the parameter which sets the temporal correlations. Small values of Q results in no peak in either input or output coherence

(black lines in Fig. 6.2(a) and Fig. 6.4(a)). The output coherence attains large values across a wide range of frequencies. Large values of Q correspond to a narrow peak in the

PSD of epithelial oscillations and refer to large correlation times of h(t). In fact, for large

Q the correlation time of the narrow-band noise is π fe/Q and sets up a slow time scale for amplitude fluctuations. These slow amplitude fluctuations are seen as a low-frequency peak in the output coherence, red line in Fig. 6.4(a). Fig. 6.4(b) shows that temporally coherent input fluctuations lead to significant enhancement of noise correlations by the

network. On the contrary, correlations of incoherent epithelial noise (Q 1) are ≤ suppressed by the network.

We next study how the relative strength of noise correlations at the input aﬀects the network correlations. The relative strength of input correlations is characterized by the 82

Figure 6.4: Transformations of noise correlations for two identical neurons driven by independent broad-band noises and a common narrow-band epithelial noise for the indicated values of quality factor Q. (a) Output coherence. (b) Coherence sensitivity χ( f ). Other parameters are w = 0.5, A = 0.5, D = 0.2, λ = 0.02, β = 0.3, c = 1.

parameter α = (A/D)2, i.e. the ratio of variances of coherent narrow-band noise to incoherent broad-band Ornstein-Uhlenbeck noise, Eq.(6.4). Indeed, both input and output

coherence increases with the decrease of uncorrelated noise as Fig. 6.5(a) shows. For large values of D the input and output coherences are signiﬁcant only around the frequency of epithelial oscillations fe. However, the ampliﬁcation of noise correlations is 83

, α = 0.25 , α = 25 , α = 2500

Figure 6.5: Transformations of noise correlations for two identical neurons driven by independent broad-band noises and a common narrow-band epithelial noise for the indicated values of standard deviation D of spatially uncorrelated broad-band noise. (a) Output coherence. (b) Coherence sensitivity χ( f ). Other parameters are w = 0.5, A = 0.5, Q = 20, λ = 0.02, β = 0.3, c = 1.

larger for less correlated inputs, as Fig. 6.5(b) shows. In other words, the network is more

sensitive to weak correlation and less sensitive to strongly correlated noise. This nonlinear eﬀect is similar to the phenomenon of compressive nonlinearity observed in auditory and 84 vestibular hair cells [84, 85], whereby a sensor exhibits high sensitivity to weak stimuli and low sensitivity to strong stimuli.

Figure 6.6: Noise correlations of a pair of non-identical neurons. Neurons are characterized by different firing rates, fa1 = fe/w1 and fa2 = fe/w2, where fe = 26 Hz and w1,2 are indicated in the figure. (a) Input and output coherences for w1 = 0.5 and indicated values of w . (b) Low-frequency averaged output coherence coh as a function of neuronal firing 2 rates parametrized by w1,2. Other parameters are A = 0.5, Q = 20, λ = 0.02, β = 0.3, c = 1. 85

In padddlefish electrorecepors afferents innervating a cluster of sensory epithelia are characterized by diverse firing rates ranging from 40 – 70 Hz [28]. In our network model

this ﬁring rates diversity is parameterized by the frequency ratio, wk = fe/ fak, where

fe = 26 Hz is fixed for all neurons and fak is the firing rate of k-th afferent in the network. Figure 6.6(a) shows input and output coherences for two afferents with distinct firing rates

characterized by w1 = 0.5 and various values of w2. As expected, noise correlations are suppressed in cases of non-identical neurons. Furthermore, noise de-correlation is

maximal for the frequency ratios w which are in the range w [0.48 0.58]. This is 1,2 1,2 ∈ demonstrated in Fig. 6.6(b) showing values of output coherence averaged in the low

frequency range [0 3] Hz as a function of w1 and w2. Thus, for inhomogeneous network noise, correlations are minimal for aﬀerents ﬁring rates close to twice the frequency of common oscillatory input. For those values of w, the low-frequency power in neuronal spike trains is suppressed by extended serial ISIs correlations, resulting in minimal cross-correlations (noise correlations) between neuronal spike trains.

6.5 Stimulus encoding and information transmission by the network

In this section we add a band-limited Gaussian stimulus s(t) common to all elements in the network and study how temporal and spatial noise correlations aﬀect stimulus transmission by the network. The stimulus PSD is given by G ( f ) = σ2/(2 f ) for f f ss c ≤ c 2 and Gss( f ) = 0, f > fc, where σ is the stimulus variance and fc is the stimulus cutoﬀ frequency. In the following, we set fc = 20 Hz. Stimulus encoding by the network was characterized using the technique described in the section 6.3. In particular, we calculated lower bounds of the information density ILB( f ) = log2[SNR( f )] and the lower bound of the total mutual information rate, Eq.(6.9). 86

(a)

(b)

Figure 6.7: Information density for a network of M = 5 neurons stimulated by Gaussian band-limited stimulus with σ = 0.2 and cutoﬀ frequency fc = 20 Hz, for the indicated values of the quality factor of epithelial oscillations Q (a) and the partial correlation parameter c (b). Firing rates of neurons were uniformly distributed from 42 to 62 Hz. Other parameters are A = 0.5, λ = 0.02, β = 0.3.

We start with a network of M = 5aﬀerents those ﬁring rates are uniformly distributed

from 42 to 62 Hz. Figure 6.7 shows the information density for indicated values of the quality factor Q of epithelial oscillations and the parameter of partial correlations c.

Stimulus encoding becomes more eﬃcient for more coherent epithelial signals, as 87

indicated by larger values of the information density over the whole frequency range, Fig. 6.7(a). We remind the reader that the information rates of individual aﬀerents grow with the increases of Q (see, e.g. Fig. 5.8), because of suppression of low-frequency noise

power. This explains similar behavior of the information density for the network in Fig. 6.7(a).

The parameter c sets the level of partial input correlations for the narrow-band epithelial oscillations. Epithelial oscillations are completely independent across the network for c = 0. The opposite limit, c = 1, refers to perfectly correlated epithelial oscillations, i.e. all neurons in the network receive common epithelial noise. Fig. 6.7(b) shows that the information density of the network is at maximum for completely independent noise and decreases progressively as the parameter c increases. For our model, noise in network elements is stimulus independent. Consequently, input noise

correlations enhanced by neuronal nonlinearities degrade information capacity of the network.

Information capacity of a network can be increased if network elements eﬀectively decorrelate a common stimulus [44, 45]. For example, if each neuron in the network

encodes different aspects of the stimulus, then overall network encoding is larger than the mere sum of contributions from all network elements. In our model neurons possess similar response properties which are set up by their dynamics and so encode similar stimulus features. Nevertheless a diversity can be introduced by spreading the mean firing rates of the network elements or by an increase of broad-band noise which is uncorrelated across the network. Figure 6.8 shows the effect of diversity parameters on stimulus encoding for a network of M = 5 neurons perturbed by correlated epithelial noise (c = 1),

uncorrelated broad-band noise and stimulated by the same stimulus as in Fig. 6.7. First we consider the eﬀect of distribution of the mean ﬁring rates set by the diversity parameter 88

Figure 6.8: Information density for a network of M = 5 neurons stimulated by Gaussian band-limited stimulus with σ = 0.3 and cutoff frequency fc = 20 Hz. (a) Network of non- identical neurons with firing rates uniformly distributed within interval [52 ∆ f /2, 52 + − a ∆ fa/2]. Values of ∆ fa are indicated in the figure. Other parameters are A = 0.5, Q = 20, D = 0.2, λ = 0.02, β = 0.3. (b) Network of identical neurons with the mean firing rates fa = 52 Hz, ∆ fa = 0. Values of independent broad-band noise intensity D are indicate in the figure.

∆ fa, Fig. 6.8(a). As expected, the case of a network of identical neurons ∆ fa = 0 is characterized by the smallest values of information rate. The information rate increases

with the increase of inhomogeneity of the network. This eﬀect is explained by the decrease of noise correlations for non-identical neurons illustrated in Fig. 6.6. 89

Second, we consider identical neurons, ∆ fa = 0, and vary broad-band noise intensity D, Fig. 6.8(b). Because broad-band noise is uncorrelated across the network, the increase of D decreases noise correlations enhancing stimulus information transfer. On the other

hand, however, increase of noise retards stimulus encoding. As a result there is an optimal noise intensity at which the information rate is at maximum. This eﬀect of network

stochastic resonance [86, 87] is further illustrated in Fig. 6.9 which shows the information rate versus independent noise level for networks of diﬀerent sizes. For a single neuron,

Figure 6.9: Lower bound of the mutual information rate versus the intensity of independent broad-band noise D for networks of identical neurons and indicate size M. Other parameters are w = 0.5, A = 0.5, Q = 20, λ = 0.02, β = 0.3.

M = 1 the information rate monotonically decreases with the increase of noise. Stochastic resonance, i.e. the existence of an optimal noise level maximizing the information rate, appears ﬁrst for pairs of neurons, M = 2, and is more pronounced for larger networks. 90

6.6 Conclusion

In this Chapter we have investigated spontaneous and response dynamics of small-scale feed-forward networks. Elements of the network were subjected to temporally and spatially correlated noise. Temporal correlations came from stochastic oscillations and resulted in extended serial correlations of interspike intervals of individual neurons. Spatial correlations were due to component of noise common to all elements in the network. In contrast to previous studies which considered only weakly spatially correlated white noise, we considered strongly correlated narrow-band noise as a plausible model for electrorecepors and more generally for neural networks processing correlated rhythmical signals. Our main results are the following. 1. Noise correlations are transformed nonlinearly by the network. Coherence of pairs of neuronal spike trains showed peaks at frequencies higher than that at the input. More importantly, the network enhances low-frequency correlations which came from slow amplitude fluctuation of common narrow-band noise, as shown in Fig. 6.3. 2. The network sensitivity to noise correlations is higher for weak correlations and lower for strong correlations (Fig. 6.5). Thus, our oscillatory network exhibited the effect of compressive nonlinearity often observed in peripheral sensory receptors. 3. Transformation of noise correlations depends non-monotonously on the firing rate of network elements. In particular, the network sensitivity to noise correlations is minimal for a network those firing rates are in rational ratio 2 : 1 with the frequency of coherent noisy oscillations as shown in Fig. 6.6(b). Thus, a network can be tuned to minimize or maximize noise correlations.

4. Information transmission by the network is enhanced by oscillatory noise, Fig. 6.7(a). In the network model of electroreceptors narrow-band (oscillatory) noise simulates spontaneous epithelial oscillations which induce long-range serial ISI correlations. As we showed in Chapter 5, coherent epithelial oscillations result in 91

significant enhancement of information transmission by a single electrorecepor afferent. This effect becomes more pronounced on the network level. 5. Inter-neuronal noise correlations degrade information transmission by the network model in which noise correlations are stimulus-independent, Fig. 6.7(b). Noise correlations can be suppressed for inhomogeneous networks, resulting in higher

information transmission, Fig. 6.8(a). 6. For a network of identical elements, the information transmission can be

maximized via the phenomenon of stochastic resonance, in which the intensity of spatially uncorrelated noise is tuned to maximize the mutual information rate, Fig. 6.9. This ﬁnding underlines a possible positive role of internal noise in the collective dynamics of the

network. 92 7 Conclusion and Outlook

This dissertation presented research that contributes to a better understanding of how sensory receptors utilize their intrinsic dynamics to attain optimal responses and information encoding. We have focused on the role of intrinsic oscillatory dynamics in encoding of external signals by peripheral electroreceptors. However, models and methods developed in this study are applicable to other sensory systems. In particular, the modiﬁed Hodgkin-Huxley model with excitatory synaptic input can be adapted to study vestibular sensory neurons. Likewise, the network model and conclusions drawn from its study can be applied to a general setup of a parallel neural array subjected to correlated intrinsic noise. The main results of this dissertation can be formulated as follows. 1. Coherent noisy oscillations modulating excitatory synaptic inputs to a neuron induce long lasting serial correlations of ISIs generated by the neuron. The strength of the serial correlations is maximal when the frequencies of presynaptic oscillations and the

ﬁring rate of the neuron are in rational relation, i.e. two neuronal spikes per cycle of presynaptic oscillations.

2. Serial ISI correlations induced by coherent presynaptic oscillations decrease the variability of spike counts at low frequencies. This suppression of background power at low frequencies leads to signiﬁcant enhancement of information transmission by the neuron, as estimated with a linear stimulus reconstruction method. Thus, correlated noise allows for more eﬃcient neural stimulus encoding, compared to a neuron with uncorrelated internal noise. Information rate and the enhancement of information due to oscillations are maximal when the frequency ratio of presynaptic (epithelial) to neuronal oscillators is close to 0.5.

3. The enhancement of information encoding due to coherent noise is observed also for non-linear regimes, i.e. for strong stimuli. In such regimes, presynaptic oscillations 93 enhance the stimulus-induced synchronization of neuronal responses, leading to precise coding of a stimulus with bursting patterns of spikes. 4. For a network of oscillatory neurons with correlated input noise, noise correlations are transformed nonlinearly by the network. In particular, the network sensitivity to noise correlations is higher for weak input correlations and lower for strong input correlations, demonstrating the effect of compressive nonlinearity. 5. Transformation of noise correlations depends non-monotonously on the firing rate of network elements. In particular, the network sensitivity to noise correlations is minimal for a network those firing rates are in rational ratio 2 : 1 with the frequency of coherent noisy oscillations. Thus, a network can be tuned to minimize or maximize noise correlations. 6. Information transmission by the network is enhanced by temporal oscillatory noise correlations. On the other hand, inter-neuronal noise correlations degrade information transmission by the network model in which noise correlations are stimulus-independent. Inter-neuronal noise correlations can be suppressed for inhomogeneous networks, resulting in higher information transmission. 7. For a network of identical elements, the information transmission can be maximized via the phenomenon of stochastic resonance, in which the intensity of uncorrelated noise is tuned to maximize the mutual information rate.

The results 3 – 7 generate predictions which can be verified in neurophysiological experiments. In particular, the in vivo preparation of paddlefish ERs is a good candidate, because it allows simultaneous recording of spike trains from several afferents innervating a cluster of sensory epithelia, and thus receiving a correlated synaptic drive. It will be interesting to measure the distribution of afferent firing rates innervating clusters of canals 94

and then to compare it with our prediction of optimal frequency ratios which maximize information transfer. We note that in this dissertation, internal presynaptic noise due to epithelial cells was

considered stimulus independent. While it is true for electroreceptors, it is probably not for auditory receptors. Also, for a neuron embedded in a network in the CNS, the

presynaptic noisy input from other neurons is often stimulus-dependent. Thus, another direction for a future study is to consider stimulus-mediated correlated noise for a single

neuron and for a network, e.g. when common narrow-band noise is modulated by a stimulus. In this study we considered just a two layer network: sensory cells + afferent neurons. It is indeed of interest to consider a third layer of secondary neurons integrating output from multiple afferents. In particular, a coincidence detector model for the secondary neurons is promising, whereby the secondary neurons fire a spike when several spikes of primary afferents coincide within a small time window. For such a model inter-neuronal correlated noise may be beneficial, as it increases the probability that secondary neuron will fire and thus carry information about a stimulus. 95 References

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