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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 • • Source: http://en.wikipedia.org/wiki/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. Synapsesspecialized structures where neurotransmitter chemicals are released in order to communicate with target 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 dV Cm gNa m3h(V VNa ) gK n 4 (V VK ) gL (V VL ) dt H-H variables: dm m (V ) m (V ) m dt dh h (V ) h (V ) h dt dn n (V ) n (V ) n dt V-potential difference m-sodium activation variable h-sodium inactivation variable n-potassium activation variable Cm-membrane capacitance 3 gNa= gNa m h sodium conductance gK= gK n 4 potassium conductance gL = 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 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: dv dw f (v,w), g(v,w) dt dt 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: dv f (v,w) v(v a)(1 v) w dt dw g(v,w) v bw dt Fitzhugh-Nagumo model t • Defining a short time scale by T and defining V(T) = v(t), W(T) = w(t), we obtain: • dV f (V,W ) V (V a)(1 V ) W dT dW g(V,W ) (V bW ) dT • 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 h1 h2 h3 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. 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 w* 1 t2 dw w0 G3 (w) 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: w* 1 1 tp ( )dw G1 (w) w* G3 (w) Fitzhugh-Nagumo model The solution have a unique unstable steady state at (0.1,0), surrounded by a stable periodic relaxation oscillation. 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. Fitzhugh-Nagumo model Reference • Britton N.F. Essential Mathematical Biology, Springer U.S. (2003)