Department of Mathematical Sciences B12412: Computational and Neuroinformatics

Why use models? Successes in Mathematical Neuroscience.

1950’s

The 1952 work of & Andrew Huxley model led to the award of the 1963 Nobel Prize in Physiology or Medicine (together with Sir John Eccles) for their work on understanding nerve action potentials in the squid giant . The experimental measurements on which the pair based their theory represent one of the earliest applications of a technique of clamp. The second critical element of their research was the so-called giant axon of the Atlantic squid (Loligo pealei). The large diameter of the axon (up to 1 mm in diameter; typically around 0.5 mm) provided a great experimental advantage for Hodgkin and Huxley as it allowed them to insert voltage clamp electrodes inside the lumen of the axon. The pair published their theory in 1952. In the paper, they describe one of the earliest computational models in biochemistry, that is the basis of most of the models used in Neurobiology during the following four decades. In the course of their research, Huxley and Hodgkin were surprised

The giant axon of the squid, not the axon of the giant squid! to learn that, contrary to earlier hypotheses, the outer of a nerve fiber is not equally permeable to all ions (charged particles). While a resting cell has low sodium- and high potassium- permeability, Huxley and Hodgkin found that, during excitation, sodium ions flood into the axon, which instantaneously changes from a negative to a positive charge. It is this sudden change that constitutes a nerve impulse. The sodium ions then continue to flow through the membrane until the axon is so highly charged that the sodium becomes electrically repelled. The stream of sodium then stops, which causes the membrane to become permeable once again to potassium ions. Action potentials

80 0 m/s

The Hodgkin-Huxley model of an axon is now widely recognised as a model of excitable media -

1 Basic Components of Hodgkin-Huxley type Models. Hodgkin-Huxley type models represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a (Cm). Voltage-gated and leak ion channels are represented by nonlinear (gn) and linear (gL) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (Ip). at every point along an axon a patch of membrane is described by a set of four nonlinear ordinary differential equations. Activity spreads in a regenerative way along the axon via diffusive coupling between neighbouring patches of excitable tissue. The form of the action potential is predicted extremely well by the model. For this, integration of the differential equations was required. Huxley devised an algorithm for use on a mechanical hand calculator. Each run of the algorithm producing a 5 millisecond theoretical voltage trace took about 8 hours of effort – Quote from Hodgkin: The propagated action potential took about three weeks to complete and must have been an enormous labour for Andrew [Huxley].

A ”Millionaire” mechanical calculator.

For a wonderful review of the Hodgkin-Huxley framework see the paper by John Rinzel1 For the past forty years our understanding and our methods of studying the biophysics of excitable membranes have been significantly influenced by the landmark work of Hodgkin and Huxley. Their

11990, Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and an update, Bulletin of Mathematical Biology, Volume 52, 3-23.

2 work has had far reaching impact on many different life science subdisciplines where concepts of cell biology have come to be important. These include not only neurophysiology but also endocrinology, muscle and cardiac physiology, and developmental biology. The ionic currents and electrical signals generated by neuronal membranes are of obvious importance in the nervous system. But ionic fluxes also play important roles in affecting cellular functions such as secretion, contraction, migration, etc. Voltage-dependent conductances like those which Hodgkin and Huxley discovered, and the ionic channels which they imagined (and which are now known to exist), are ubiquitous in the cells of animals and plants. A number of java applets showing the dynamics of excitable media (in both neural and cardiac) contexts can be found online.

See for example the model of an action potential at http://lcn.epfl.ch/tutorial/english/hodhux/html/axon.html

Numerous computations and analytic studies have now been performed with Hodgkin-Huxley like equations to account for the rich repertoire of nonlinear phenomena in excitable membranes, as well as to uncover new behaviors in the models, and to suggest further experiments. In example, the figure below shows the result of a mathematical analysis of travelling periodic wave speed (multiple spikes) in the Hodgkin-Huxley model.

1960’s

In the 1960s Rall developed the cable model of the dendritic tree. are strikingly exquisite and unique structures. They are the largest component in both surface area and volume of the brain and their specific morphology is used to classify into classes: pyramidal, Purkinje, amacrine, stellate, etc. Cable theory describes how spreads along the dendritic branches in response to a local conductance change (synaptic input) In the last thirty years, cable theory for dendrites, complemented by the compartmental modeling approach has played an essential role in the estimation of dendritic parameters, in designing and interpreting experiments and in providing insights into the computational function of dendrites.

Cable theory

Dendrites are thin tubes wrapped with a membrane that is a relatively good electrical insulator compared to the resistance provided by the intracellular core or the extracellular fluid. Because

3 Left: Computed dispersion relations, conduction speed vs. firing frequency, for periodic traveling wave solutions to the Hodgkin-Huxley equations for three temperatures. Speeds of the fast and slow solitary pulses (shown dashed) represent the limiting speeds of the periodic waves as the frequency approaches zero. Right: The relation between conduction speed and wavelength for traveling wave solutions to the Hodgkin-Huxley equations. From R N Miller and J Rinzel Biophys J. 1981 May; 34(2): 227259. The dependence of impulse propagation speed on firing frequency, dispersion, for the Hodgkin-Huxley model. of this difference in membrane vs. axial resistivity, for a short length of , the electrical current inside the core conductor tends to flow parallel to the cylinder axis (along the x-axis). Cable theory considers dendritic segments to have just one spatial dimension (x) The other fundamental assumptions in cable theory are:

1. The membrane is passive (voltage-independent) and uniform.

2. The core conductor has constant cross section and the intracellular fluid can be represented as an ohmic resistance.

3. The extracellular resistivity is negligible (implying extracellular isopotentiality).

Importantly the cable equation can be solved analytically for arbitrarily complicated passive den- dritic trees. The cable equation is a second order partial differentiation equation ∂V V 1 ∂2V C = − + , 2 ∂t R ra ∂x where R is the membrane resistivity, ra is the cytoplasm resistivity and C the membrane capaci- tance for a unit length of cable with fixed diameter.

An equivalent cylinder

Rall argued that when a cylinder with diameter dp bifurcates into two daughter branches with diameters d1 and d2 , the branch point behaves as a continuous cable for current that flows from the parent to daughters, if 3/2 3/2 3/2 dp = d1 + d2 A branch point obeying Rall’s law is electrically equivalent to a uniform cylinder (looking from the parent into the daughters). Rall extended this concept (impdeance matching) to trees and showed that (from the soma viewpoint out to the dendrites) there is a subclass of trees that are electrically equivalent to a single cylinder whose diameter is that of the stem (near the soma) dendrite. Dendrites of many types (e.g., some α-motoneuron) obey, to a first approximation, the 3/2 rule. However, the dendrites of several major types of neurons (e.g.,

4 A Purkinje cell. cortical and hippocampal pyramidal cells) do not obey this rule. Still, the equivalent cylinder model for dendritic trees allows for a simple analytical solution and, indeed, it has provided the main insights regarding the spread of electrical signals in passive dendritic trees.

Compartmental modelling

The compartmental modeling approach complements cable theory by overcoming the assumption that the membrane is passive. Mathematically, the compartmental approach is a finite-difference (discrete) approximation to the (nonlinear) cable equation. It replaces the continuous cable equation by a set, or a matrix, of ordinary differential equations and, typically, numerical methods are employed to solve this system (which can include thousands of compartments and thus thousands of equations) for each time step. Conceptually, in the compartmental model dendritic segments that are electrically short are assumed to be isopotential and are lumped into a single RC (either passive or active) membrane compartment. Compartments are connected to each other via a longitudinal resistivity according to the topology of the tree. Hence, differences in physical properties (e.g., diameter, membrane properties, etc.) and differences in potential occur between compartments rather than within them. It can be shown that when the dendritic tree is divided into sufficiently small segments (compartments) the solution of the compartmental model converges to that of the continuous cable model. A compartment can represent a patch of membrane with a variety of voltage-gated (excitable) and synaptic (time-varying) channels.

1970’s

The 1970s saw the development of neural field equations2. These are tissue level models that describe the spatio-temporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons.

2For more info see http://www.scholarpedia.org/article/Neural Fields

5 From Idan Segev: http://www.genesis-sim.org/GENESIS/iBoG/iBoGpdf/chapt5.pdf Dendrites (A) are modeled either as a set of cylindrical membrane cables (B) or as a set of discrete isopotential RC compartments (C). In the cable representation (B), the voltage can be computed at any point in the tree by using the continuous cable equation and the appropriate boundary conditions imposed by the tree. An analytical solution can be obtained for any current input in passive trees of arbitrary complexity with known dimensions and known specific membrane resistance and capacitance and specific cytoplasm (axial) resistance . In the compartmental representation, the tree is discretized into a set of interconnected RC compartments. Each is a lumped representation of the membrane properties of a sufficiently small dendritic segment. Compartments are connected via axial cytoplasmic resistances. In this approach, the voltage can be computed at each compartment for any (nonlinear) input and for voltage and time-dependent membrane properties.

The number of neurons and synapses in even a small piece of cortex is immense. Because of this a popular modelling approach has been to take a continuum limit and study neural networks in which space is continuous and macroscopic state variables are mean firing rates. Perhaps the first attempt at developing a continuum approximation of neural activity can be attributed to Beurle in the 1950’s and later by Griffith in the 1960’s. By focusing on the proportion of neurons becoming activated per unit time in a given volume of model brain tissue, Beurle was able to analyse the triggering and propagation of large scale brain activity. However, this work only dealt with networks of excitatory neurons with no refractory or recovery variable. It was Wilson and Cowan in the 1970’s who extended Beurles work to include both inhibitory and excitatory neurons as well as refractoriness. Further work, particularly on pattern formation, in continuum models of neural activity was pursued by Amari under natural assumptions on the connectivity and firing rate function. Amari considered local excitation and distal inhibition which is an effective model for a mixed population of interacting inhibitory and excitatory neurons with typical cortical connections (commonly referred to as Mexican-hat connectivity). Since these seminal contributions to dynamic neural field theory similar models have been used to investigate EEG rhythms, visual hallucinations, mechanisms for short term memory and motion perception.

6 A caricature of a neural field. Such models can include not only the effects of axonal delays, but the filtering of firing rate signals by both synapses and dendrites.

Geometric visual hallucinations

Geometric visual hallucinations are seen in many situations, for example: after being exposed to flickering lights, after the administration of certain anaesthetics, on waking up or falling asleep, following deep binocular pressure on ones eyeballs, and shortly after the ingesting of drugs such as LSD and Marihuana.

Phosphene hallucination produced by deep binocular pressure on the eyeballs. Honeycomb and lattice-tunnel hallucination generated by Marihuana.

To understand geometric visual hallucinations we first need to calculate what visual hallucinations look like, not in the standard polar coordinates of the visual field, but in the coordinates of V1 (layer 1 of visual cortex). It is well established that there is a topographic map of the visual field in V1, the retinotopic representation, and that the central region of the visual field has a much bigger representation in V1 than it does in the visual field. The reason for this is partly that there is a nonuniform distribution of retinal ganglion cells, each of which connects to V1 via the lateral geniculate nucleus (LGN). The transformation of various geometric shapes (except very close to the fovea) is summarised in the next picture. Hence, if there are cortical mechanisms that give rise to doubly periodic cellular patterns of squares and hexagons or rolls along some constant direction then, by the inverse retino-cortical map, this could underly many common forms of visual hallucination. Such a mechanism exists based upon

7 Spiral, spiral tunnel and tunnel hallucinations generated by LSD. the variation of a bifurcation parameter that leads to a Turing instability 3 of a homogeneous steady state. For the right choice of parameters (drugs!) a uniform state in a neural field with Mexican-hat connectivity can go unstable in favour of a periodic pattern. Basically a neuron must be under a delicate balance between excitation and inhibition to prevent it from being constantly active, or from being completely silent. Drugs such as LSD upset this balance. One of the effects of these drugs is to, in effect, reduce the threshold for the neurons to become active. As a result, the visual cortex produces spontaneous patterns of activity, unrelated to the outside world.

3after the famous British mathematician Alan Turing – of code-breaking and computing fame.

8 Corresponding patterns under the retino-cortical map.

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One and two-dimensional Mexican-hat functions.

1980’s

The dawn of Artificial Neural Networks — tracing back to the 1940’s .

1980 Adaptive Resonance Theory (Grossberg), self-organisation & competitive learning.

1981 Kohonen self-organising maps

1982 Hopfield recurrent networks, energy landscapes, memory metaphors, Hebbian learning rules (correlation, Widrow-Hoff, Pseudo-inverse, Gardner) and the use of powerful techniques from statistical physics.

1985 Boltzmann machine and the use of simulated annealing to solve hard computational prob- lems.

1986 Back propagation algorithm for multi-layer .

9 Density plot of doubly periodic square and hexagonal functions.

Action of the inverse retinocortical map on square planforms (left) and hexagonal planforms (right).

1988 Universal approximation theorem.

1988 Neural learning algorithms for PCA.

1988 Radial basis functions

1990’s to date

Functional magnetic resonance imaging (fMRI) is the use of MRI to measure the hemodynamic response related to neural activity in the brain. In 2003 Nobel prize in Physiology or Medicine went to Sir Peter Mansfield @ Nottingham. Mansfield is credited with showing how the radio signals from fMRI can be mathematically analysed, making interpretation of the signals into a useful image a possibility. The advent of computer simulation tools such as XPP and NEURON has also had an impact on the experimental neuroscience community, and these tools will be explored in future lectures.

10 For your entertainment why not colour the picture to mimic one or more geometric visual hallucinations!

S Coombes 2009

11 Left: An MRI scan of the lecturers brain performed in the Nottingham Brain & Body Center. Right: fMRI data (yellow) overlaid on an average of the brain anatomies of several humans (gray).

Left: Output from XPP – http://www.math.pitt.edu/∼bard/xpp/xpp.html Right: Output from NEURON – http://www.neuron.yale.edu/neuron/

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