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: http://en.wikipedia.org/wiki/Neurons

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

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Cm

dV

dt= −g Nam3h(V −VNa ) − g K n4 (V −VK ) − gL (V −VL )

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τm (V )dm

dt= m∞(V ) −m

τ h (V )dh

dt= h∞(V ) − h

τ n (V )dn

dt= n∞(V ) − n

H-H variables:

V-potential difference

m-sodium activation variable

h-sodium inactivation variable

n-potassium activation variable

Cm-membrane capacitance

gNa= sodium conductance

gK= potassium conductance

gL = leakage conductance€

g Nam3h

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g K n4

Suppose V is kept constant. Then m tends exponentially to m(V) with time constant τm(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 gNa and gK 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:

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ε dv

dt= f (v,w),

dw

dt= g(v,w)

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:

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ε dv

dt= f (v,w) = v(v − a)(1− v) −w

dw

dt= g(v,w) = v −bw

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).

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T =t

ε

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dV

dT= f (V ,W ) =V (V − a)(1−V ) −W

dW

dT= εg(V ,W ) = ε(V −bW )

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 ε=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.

h1 h2 h3

Fitzhugh-Nagumo model

• There are 3 solutions of f(v,w) = 0 for w*≤w≤w* given by v =h1(w), v=h2(w) and v=h3(w) with h1(w)≤ h2(w)≤

h3(w). • Time taken for excited phase:

– We have f(v,w) = 0 by continuity v=h3(w), and w satisfies w’ = g(h3(w),w) = G3(w). Hence w increases until it reaches w*, beyond which h3(w) ceases to exist. The time taken is

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t2 =1

G3(w)w0

w*

∫ dw

Fitzhugh-Nagumo model

Fitzhugh-Nagumo model

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:

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tp = (1

G3(w)−

1

G1(w)w*

w*

∫ )dw

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)