HodgkinHodgkin--Huxley ModelHuxley Modelandand

FitzHughFitzHugh--NagumoNagumo ModelModel

Nervous SystemNervous System

Signals are propagated from nerve cell to nerve Signals are propagated from nerve cell to nerve cell (cell (neuronneuron) via electro) via electro--chemical mechanismschemical mechanisms~100 billion neurons in a person~100 billion neurons in a personHodgkin and Huxley experimented on squids and Hodgkin and Huxley experimented on squids and discovered how the signal is produced within the discovered how the signal is produced within the neuronneuronH.H.--H. model was published in H. model was published in Jour. ofJour. of PhysiologyPhysiology(1952)(1952)H.H.--H. were awarded 1963 Nobel PrizeH. were awarded 1963 Nobel PrizeFitzHughFitzHugh--NagumoNagumo model is a simplificationmodel is a simplification

NeuronNeuron

C. George Boeree: www.ship.edu/~cgboeree/

Action PotentialAction Potential

10 msec

mV

_ 30

-70

V_ 0

Axon membrane potential difference

V = VV = Vii –– VVeeWhen the axon is excited, VV spikes because sodium Na+Na+ and potassium K+K+ions flow through the membrane.

Nernst Potential

VVNaNa , VVKK and VVrr

Ion flow due to electrical signal

Traveling wave

C. George Boeree: www.ship.edu/~cgboeree/

Circuit Model for Axon MembraneCircuit Model for Axon MembraneSince the membrane separates charge, it is modeled as a capacitor with capacitance CC. Ion channels are resistors.

1/R = g =1/R = g = conductance

Na+K+

ggNagK

C

inside

+

−

cell

NaVK rVV

outside

r

+

−axon

axon

inside

outside

membraneCl−

iiCC = C = C dV/dtdV/dt

iiNaNa = = ggNaNa (V (V –– VVNaNa))

iiKK= = ggKK (V (V –– VVKK))

iirr = = ggrr (V (V –– VVrr))

Circuit EquationsCircuit EquationsSince the sum of the currents is 0, it follows thatSince the sum of the currents is 0, it follows that

aprrNa

IVVgVVgVVgdtdVC KKNa +−−−−−−= )()()(

where IIapap is applied current. If ion conductances are constants then group constants to obtain 1st order, linear eq

apIVVgdtdVC +−−= *)(

Solving gives gIVtV ap /*)( +→

Variable ConductanceVariable Conductance

0

2

4

6

8

10

g_K

1 2 3 4 5

msec

0

2

4

6

8

10

12

14

g_Na

1 2 3 4 5

msec

g

Experiments showed that Experiments showed that ggNaNa and and ggKK varied with time and V. varied with time and V. After stimulus, Na responds much more rapidly than K . After stimulus, Na responds much more rapidly than K .

HodgkinHodgkin--Huxley SystemHuxley System

Four state variables are used:Four state variables are used:v(tv(t)=)=V(t)V(t)--VVeqeq is membrane potential,is membrane potential,m(tm(t) is Na activation,) is Na activation,n(tn(t) is K activation and ) is K activation and h(th(t) is Na inactivation.) is Na inactivation.

In terms of these variables ggKK=ggKKnn44 and ggNaNa==ggNaNamm33hh. The resting potential VVeqeq≈≈--70mV70mV. Voltage clamp experiments determined ggKK and nn as functions of tt and hence the parameter dependences on vv in the differential eq. for n(tn(t).). Likewise for m(tm(t)) and h(th(t).).

HodgkinHodgkin--Huxley SystemHuxley System

IVvgVvngVvhmgdtdvC aprrKNa

KNa+−−−−−−= )()()( 43

mvmvdtdm

mm )()1)(( βα −−=

nvnvdtdn

nn )()1)(( βα −−=

hvhvdtdh

hh )()1)(( βα −−=

110 mV

40msec

IIapap =8, =8, v(tv(t))

m(t)

n(t)

h(t)

10msec

1.2

IIapap=7, =7, v(tv(t))

FastFast--Slow DynamicsSlow Dynamics

m(t)

n(t)

h(t)

10msec

ρρmm(v(v) dm/) dm/dtdt = = mm∞∞(v(v) ) –– m.m.

ρρmm(v(v) is much smaller than ) is much smaller than

ρρnn(v(v) and ) and ρρhh(v(v). An increase ). An increase in v results in an increase in in v results in an increase in mm∞∞(v(v)) and a large dm/and a large dm/dtdt. . Hence Na activates more Hence Na activates more rapidly than K in response rapidly than K in response to a change in v.to a change in v.

v, m are on a fast time scale and n, h are slow.v, m are on a fast time scale and n, h are slow.

FitzHughFitzHugh--NagumoNagumo SystemSystem

Iwvfdtdv

+−= )(ε wvdtdw 5.0−=and

II represents applied current, εε is small and f(vf(v)) is a cubic nonlinearity. Observe that in the ((v,wv,w)) phase plane

which is small unless the solution is near f(v)f(v)--w+w+II=0=0. Thus the slowmanifold is the cubic w=w=f(v)+f(v)+II whichwhich is the nullcline of the fast variable vv. And ww is the slow variable with nullcline w=2vw=2v.

Iwvfwv

dvdw

+−−

=)(

)5.0(ε

Take f(vf(v)=v(1)=v(1--v)(vv)(v--a)a) .

Stable rest state

vv

w

I=0I=0 Stable oscillation I=0.2I=0.2

w

vv

FitzHughFitzHugh--NagumoNagumo OrbitsOrbits

ReferencesReferences1.1. C.G. C.G. BoereeBoeree, , The NeuronThe Neuron, , www.ship.edu/~cgboeree/.2.2. R. R. FitzHughFitzHugh, Mathematical models of excitation and , Mathematical models of excitation and

propagation in nerve, In: Biological Engineering, Ed: H.P. propagation in nerve, In: Biological Engineering, Ed: H.P. SchwanSchwan, McGraw, McGraw--Hill, New York, 1969.Hill, New York, 1969.

3.3. L. EdelsteinL. Edelstein--KesketKesket, Mathematical Models in Biology, Random , Mathematical Models in Biology, Random House, New York, 1988.House, New York, 1988.

4.4. A.L. Hodgkin, A.F. Huxley and B. Katz, A.L. Hodgkin, A.F. Huxley and B. Katz, J. PhysiologyJ. Physiology 116116, 424, 424--448,1952.448,1952.

5.5. A.L. Hodgkin and A.F. Huxley, A.L. Hodgkin and A.F. Huxley, J. J. PhysiolPhysiol.. 116, 116, 449449--566, 1952.566, 1952.6.6. F.C. F.C. HoppensteadtHoppensteadt and C.S. and C.S. PeskinPeskin, , Modeling and Simulation in Modeling and Simulation in

Medicine and the Life SciencesMedicine and the Life Sciences, 2nd ed, Springer, 2nd ed, Springer--VerlagVerlag, New , New York, 2002.York, 2002.

7.7. J. Keener and J. J. Keener and J. SneydSneyd, , Mathematical PhysiologyMathematical Physiology, Springer, Springer--VerlagVerlag, New York, 1998., New York, 1998.

8.8. J. J. RinzelRinzel, , Bull. Math. Biology Bull. Math. Biology 5252, 5, 5--23, 1990.23, 1990.9.9. E.K. E.K. YeargersYeargers, R.W. , R.W. ShonkwilerShonkwiler and J.V. Herod, An and J.V. Herod, An Introduction Introduction

to the Mathematics of Biology: with Computer Algebra Modelsto the Mathematics of Biology: with Computer Algebra Models, , BirkhauserBirkhauser, Boston, 1996., Boston, 1996.