Neural Modeling Suparat Chuechote

Introduction Nervous system - the main means by which humans and animals coordinate short-term responses to stimuli. It consists of : - receptors (e.g. eyes, receiving signals from outside world) - effectors (e.g. muscles, responding to these signals by producing an effect) - nerve cells or neurons (communicate between cells)

Neurons Neuron consist of a cell body (the soma) and cytoplasmic extension ( the axon and many dendrites) through which they connect (via synapse) to a network of other neurons. Synapses- specialized structures where neurotransmitter chemicals are released in order to communicate with target neurons Source:

Neurons Cells that have the ability to transmit action potentials are called ‘excitable cells’. The action potentials are initiated by inputs from the dendrites arriving at the axon hillock, where the axon meets the soma. Then they travel down the axon to terminal branches which have synapses to the next cells. Action potential is electrical, produced by flow of ion into and out of the cell through ion channels in the membrane. These channels are open and closed and open in response to voltage changes and each is specific to a particular ion.

Hodgkin-Huxley model They worked on a nerve cell with the largest axon known the squid giant axon. They manipulated ionic concentrations outside the axon and discovered that sodium and potassium currents were controlled separately. They used a technique called a voltage clamp to control the membrane potential and deduce how ion conductances would change with time and fixed voltages, and used a space clamp to remove the spatial variation inherent in the travelling action potential.

Hodgkin-Huxley model H-H variables: V-potential difference m-sodium activation variable h-sodium inactivation variable n-potassium activation variable C m -membrane capacitance g Na = sodium conductance g K = potassium conductance g L = leakage conductance Suppose V is kept constant. Then m tends exponentially to m  (V) with time constant  m (V), and similar interpretation holds for h and n. The function m  and n  increase with V since they are activation variable, while h  decreases.

Hodgkin-Huxley model Running on matlab

Hodgkin-Huxley model Experiments showed that g Na and g K varied with time and V. After stimulus, Na responds much more rapidly than K.

Fitzhugh-Nagumo model Fitzhugh reduced the Hodgkin-Huxley models to two variables, and Nagumo built an electrical circuit that mimics the behavior of Fitzhugh’s model. It involves 2 variables, v and w. V - the excitation variable represents the fast variables and may be thought of as potential difference. W - the recovery variable represents the slow variables and may be thought of as potassium conductance. Generalized Fitzhugh-Nagumo equation:

Fitzhugh-Nagumo model The traditional form for g and f - g is a straight line g(v,w) = v-c-bw - f is a cubic f(v,w) = v(v-a)(1-v) -w, or f is a piecewise linear function f(v,w) =H(v-a)-v-w, where H is a heaviside function Consider the numerical solution when f is a cubic:

Fitzhugh-Nagumo model Defining a short time scale by and defining V(T) = v(t), W(T) = w(t), we obtain: The two systems of ODE will be used in different phases of the solution (phase 1 and 3 use short time scale, phase 2 and 4 use long time scale).

Fitzhugh-Nagumo model There are 4 phases of the solutions -phase 1: upstroke phase - sodium channels open, triggered by partial depolarization and positively charged Na+ flood into the cell and hence leads to further increasing the depolarization (the excitation variable v is changing very quickly to attain f = 0). -phase 2: excited phase - on the slow time scale, potasium channel open, and K+ flood out of the cell. However, Na+ still flood in and just about keep pace, and the potential difference falls slowly (v,w are at the highest range). -phase 3: downstroke phase-outward potassium current overwhelms the inward sodium current, making the cell more negatively charged. The cell becomes hyperpolarized (v changes very rapidly as the solution jumps from the right-hand to the left-hand branch of the nullcline f=0). -phase 4: recovery phase-most of the Na+ channels are inactive and need time to recover before they can open again (v,w recovers from below zero to the initial v, w at 0).

Fitzhugh-Nagumo model Numerical solution for f(v,w) = v(v-a)(1-v) -w and g(v,w) = v-bw with  =0.01, a =0.1, b =0.5. The equations have a unique globally stable steady state at the origin. If v is perturbed slightly from the stead state, the system returns there immediately, but if it is perturbed beyond v = h2(0) = 0.1, then there is a large excursion and return to the origin. h1h2h3

Fitzhugh-Nagumo model There are 3 solutions of f(v,w) = 0 for w * ≤w≤w * given by v =h 1 (w), v=h 2 (w) and v=h 3 (w) with h 1 (w)≤ h 2 (w)≤ h 3 (w). Time taken for excited phase: –We have f(v,w) = 0 by continuity v=h 3 (w), and w satisfies w’ = g(h 3 (w),w) = G 3 (w). Hence w increases until it reaches w *, beyond which h 3 (w) ceases to exist. The time taken is

Fitzhugh-Nagumo model

When g is shifted to the left: g(v,w) = v -c -bw The results have different behavior. In recovery phase, w would drop until it reached w *, and we would then have a jump to the right-hand branch of f =0. This repeats indefinitely and have a period of oscilation equal to:

Fitzhugh-Nagumo model A numerical solution of the oscillatory FitzHugh-Nagumo with f(v,w) = v(v-a)(1-v) -w and g(v,w) = v-c-bw. The solution have a unique unstable steady state at (0.1,0), surrounded by a stable periodic relaxation oscillation.

Fitzhugh-Nagumo model

Reference Britton N.F. Essential Mathematical Biology, Springer U.S. (2003)