h-h_model - Hodgkin­Huxley Model Hodgkin­Huxley Model and...

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Unformatted text preview: Hodgkin­Huxley Model Hodgkin­Huxley Model and FitzHugh­Nagumo Model Nervous System Nervous System Signals are propagated from nerve cell to nerve cell (neuron) via electro­chemical mechanisms ~100 billion neurons in a person 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. were awarded 1963 Nobel Prize FitzHugh­Nagumo model is a simplification Neuron Neuron C. George Boeree: www.ship.edu/~cgboeree/ Action Potential Action Potential mV _ 30 _ 0 V ­70 10 msec Axon membrane potential difference V = Vi – Ve When the axon is excited, V spikes because sodium Na+ and potassium K+ ions flow through the membrane. Nernst Potential VNa , VK and Vr Ion flow due to electrical signal Traveling wave C. George Boeree: www.ship.edu/~cgboeree/ Circuit Model for Axon Membrane Circuit Model for Axon Membrane Since the membrane separates charge, it is modeled as a capacitor with capacitance C. Ion channels are resistors. 1/R = g = conductance iC = C dV/dt iNa = gNa (V – VNa) iK= gK (V – VK) ir = gr (V – Vr) Circuit Equations Circuit Equations Since the sum of the currents is 0, it follows that dV C = − g Na (V − V Na ) − g K (V − V K ) − gr(V − Vr ) + Iap dt where Iap is applied current. If ion conductances are constants then group constants to obtain 1st order, linear eq dV C = − g (V − V *) + Iap dt Solving gives V (t ) → V * + Iap / g Variable Conductance Variable Conductance g Experiments showed that gNa and gK varied with time and V. After stimulus, Na responds much more rapidly than K . Hodgkin­Huxley System Hodgkin­Huxley System Four state variables are used: v(t)=V(t)­Veq is membrane potential, m(t) is Na activation, n(t) is K activation and h(t) is Na inactivation. In terms of these variables gK=gKn4 and gNa=gNam3h. The resting potential Veq≈­70mV. Voltage clamp experiments determined gK 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). Hodgkin­Huxley System Hodgkin­Huxley System dv 3 4 C = − g Na m h(v − VNa ) − g K n (v − VK ) − gr (v − Vr ) + I ap dt dm = αm( v )(1 − m ) − βm( v )m dt dn = αn ( v )(1 − n ) − βn ( v )n dt dh = αh ( v )(1 − h ) − βh ( v )h dt 110 mV Iap =8, v(t) 1.2 m(t) n(t) 40msec h(t) 10msec Iap=7, v(t) Fast­Slow Dynamics Fast­Slow Dynamics m(t) ρm(v) dm/dt = m∞(v) – m. ρm(v) is much smaller than n(t) h(t) 10msec ρn(v) and ρh(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. v, m are on a fast time scale and n, h are slow. FitzHugh­Nagumo System FitzHugh­Nagumo System dv ε = f (v ) − w + I dt and dw = v − 0 .5 w dt dw ε ( v − 0.5w) = dv f (v ) − w + I I represents applied current, ε is small and f(v) is a cubic nonlinearity. Observe that in the (v,w) phase plane 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. Take f(v)=v(1­v)(v­a) . Stable rest state w I=0 w Stable oscillation I=0.2 v v References References 1. 2. 3. 1. 2. 3. 4. 5. 6. C.G. Boeree, The Neuron, www.ship.edu/~cgboeree/. R. FitzHugh, Mathematical models of excitation and propagation in nerve, In: Biological Engineering, Ed: H.P. Schwan, McGraw­Hill, New York, 1969. L. Edelstein­Kesket, Mathematical Models in Biology, Random House, New York, 1988. A.L. Hodgkin, A.F. Huxley and B. Katz, J. Physiology 116, 424­ 448,1952. A.L. Hodgkin and A.F. Huxley, J. Physiol. 116, 449­566, 1952. F.C. Hoppensteadt and C.S. Peskin, Modeling and Simulation in Medicine and the Life Sciences, 2nd ed, Springer­Verlag, New York, 2002. J. Keener and J. Sneyd, Mathematical Physiology, Springer­ Verlag, New York, 1998. J. Rinzel, Bull. Math. Biology 52, 5­23, 1990. E.K. Yeargers, R.W. Shonkwiler and J.V. Herod, An Introduction to the Mathematics of Biology: with Computer Algebra Models, Birkhauser, Boston, 1996. ...
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