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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This note was uploaded on 04/04/2010 for the course BME 402 taught by Professor Mel during the Spring '06 term at USC.
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