Hodgkin-Huxley Model and FitzHugh-Nagumo Model

Nervous System Signals are propagated from nerve cell to nerve cell (neuron) via electro-chemical mechanisms Signals are propagated from nerve cell to nerve cell (neuron) via electro-chemical mechanisms ~100 billion neurons in a person ~100 billion neurons in a person Hodgkin and Huxley experimented on squids and discovered how the signal is produced within the neuron Hodgkin and Huxley experimented on squids and discovered how the signal is produced within the neuron H.-H. model was published in Jour. of Physiology (1952) H.-H. model was published in Jour. of Physiology (1952) H.-H. were awarded 1963 Nobel Prize H.-H. were awarded 1963 Nobel Prize FitzHugh-Nagumo model is a simplification FitzHugh-Nagumo model is a simplification

Neuron

Action Potential Axon membrane potential difference V = V i – V e V Na+ K+ When the axon is excited, V spikes because sodium Na+ and potassium K+ ions flow through the membrane. 10 msec mV _ V _ 0

Nernst Potential V Na V K V r V Na, V K and V r Ion flow due to electrical signal Traveling wave

Circuit Model for Axon Membrane C Since the membrane separates charge, it is modeled as a capacitor with capacitance C. Ion channels are resistors. 1/R = g = 1/R = g = conductance i C = C dV/dt i Na = g Na (V – V Na ) i K = g K (V – V K ) i r = g r (V – V r )

Circuit Equations Since the sum of the currents is 0, it follows that I ap where I ap is applied current. If ion conductances are constants then group constants to obtain 1 st order, linear eq Solving gives

Variable Conductance Experiments showed that g Na and g K varied with time and V. After stimulus, Na responds much more rapidly than K. g

Hodgkin-Huxley System Four state variables are used: v(t)=V(t)-V eq is membrane potential, m(t) is Na activation, n(t) is K activation and h(t) is Na inactivation. g K g K n 4 g Na =g Na m 3 h In terms of these variables g K =g K n 4 and g Na =g Na m 3 h. V eq ≈-70mV g K nt v n(t).m(t)h(t). The resting potential V eq ≈-70mV. Voltage clamp experiments determined g K and n as functions of t and hence the parameter dependences on v in the differential eq. for n(t). Likewise for m(t) and h(t).

Hodgkin-Huxley System

I ap =8, v(t) I ap =7, v(t) m(t) n(t) h(t) 10msec mV 40msec

Fast-Slow Dynamics v, m are on a fast time scale and n, h are slow. ρ m (v) dm/dt = m ∞ (v) – m. ρ m (v) is much smaller than ρ n (v) and ρ h (v). An increase in v results in an increase in m ∞ (v)and a large dm/dt. Hence Na activates more rapidly than K in response to a change in v. ρ n (v) and ρ h (v). An increase in v results in an increase in m ∞ (v) and a large dm/dt. Hence Na activates more rapidly than K in response to a change in v. m(t) n(t) h(t) 10msec

FitzHugh-Nagumo System and Iεf(v) (v,w) I represents applied current, ε is small and f(v) is a cubic nonlinearity. Observe that in the (v,w) phase plane f(v)-w+I=0 w=f(v)+I which v w w=2v which is small unless the solution is near f(v)-w+I=0. Thus the slow manifold is the cubic w=f(v)+I which is the nullcline of the fast variable v. And w is the slow variable with nullcline w=2v.

v I=0I=0.2Stable rest stateStable oscillation f(v)=v(1-v)(v-a) Take f(v)=v(1-v)(v-a). w w v

References 1. C.G. Boeree, The Neuron, 1. C.G. Boeree, The Neuron, 2. R. FitzHugh, Mathematical models of excitation and propagation in nerve, In: Biological Engineering, Ed: H.P. Schwan, McGraw- Hill, New York, L. Edelstein-Kesket, Mathematical Models in Biology, Random House, New York, A.L. Hodgkin, A.F. Huxley and B. Katz, J. Physiology 116, , A.L. Hodgkin and A.F. Huxley, J. Physiol. 116, , F.C. Hoppensteadt and C.S. Peskin, Modeling and Simulation in Medicine and the Life Sciences, 2nd ed, Springer-Verlag, New York, J. Keener and J. Sneyd, Mathematical Physiology, Springer- Verlag, New York, J. Rinzel, Bull. Math. Biology 52, 5-23, E.K. Yeargers, R.W. Shonkwiler and J.V. Herod, An Introduction to the Mathematics of Biology: with Computer Algebra Models, Birkhauser, Boston, 1996.