hodgkin-huxley model

7/27/2019 Hodgkin-Huxley model

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HodgkinHodgkin--Huxley ModelHuxley Model

andand

FitzHughFitzHugh--NagumoNagumo ModelModel


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Nervous SystemNervous System

Signals are propagated from nerve cell to nerveSignals 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 person

Hodgkin and Huxley experimented on squids andHodgkin and Huxley experimented on squids and

discovered how the signal is produced within thediscovered how the signal is produced within theneuronneuron

H.H.--H. model was published inH. model was published in Jour. ofJour. ofPhysiologyPhysiology

(1952)(1952) H.H.--H. were awarded 1963 Nobel PrizeH. were awarded 1963 Nobel Prize

FitzHughFitzHugh--NagumoNagumo model is a simplificationmodel is a simplification


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NeuronNeuron

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


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Action PotentialAction Potential

10 msec

mV

_ 30

-70

V

_ 0

Axon membrane

potentialdifference

V = VV = ViiVVee

When the axon isexcited,VV spikesbecause sodium

Na+Na+ andpotassium K+K+ions flow throughthe membrane.


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Nernst Potential

VVNaNa ,VVKKandVVrr

Ion flow due toelectrical signal

Traveling wave

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


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Circuit Model for Axon MembraneCircuit Model for Axon Membrane

Since the membrane separates charge, it is modeled asa capacitor with capacitance CC. Ion channels areresistors.

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(VVVNaNa))

iiKK== ggKK (V(VVVKK))

iirr == ggrr (V(VVVrr))


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Circuit EquationsCircuit Equations

Since 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 areconstants then group constants to obtain 1st order, linear eq

apIVVgdt

dVC += *)(

Solving gives gIVtV ap /*)( +


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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 thatExperiments showed that ggNaNa andand 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 .


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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==ggNaNamm

33hh.The resting potentialVVeqeq--70mV70mV. Voltage clampexperiments determined ggKKand nn as functions oftt and

hence the parameter dependences on vv in thedifferential eq. for n(tn(t).). Likewise for m(tm(t)) and h(th(t).).


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HodgkinHodgkin--Huxley SystemHuxley System

IVvgVvngVvhmgdt

dvC aprrKNa

KNa

+= )()()( 43

mvmvdt

dmmm )()1)(( =

nvnvdt

dnnn )()1)(( =

hvhvdt

dhhh )()1)(( =


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FastFast--Slow DynamicsSlow Dynamics

m(t)

n(t)

h(t)

10msec

mm(v(v) dm/) dm/dtdt == mm(v(v)) m.m.

mm(v(v) is much smaller than) is much smaller than

nn(v(v) and) and hh(v(v). An increase). An increase

in v results in an increase inin v results in an increase in

mm(v(v)) and a large dm/and a large dm/dtdt..Hence Na activates moreHence Na activates more

rapidly than K in responserapidly 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.


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FitzHughFitzHugh--NagumoNagumo SystemSystem

Iwvfdt

dv+=

)( wvdt

dw

5.0=and

IIrepresents applied current, 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 slowmanifoldis the cubic w=w=f(v)+f(v)+IIwhichwhich is the nullclineof the fast variable

vv. And ww is the slow variable with nullclinew=2vw=2v.

Iwvf

wv

dv

dw

+

=

)(

)5.0(


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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

R fR f


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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 andpropagation 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, RandomHouse, 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. Physiology116116, 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 inModeling and Simulation in

Medicine and the Life SciencesMedicine and the Life Sciences, 2nd ed, Springer, 2nd ed, Springer--VerlagVerlag, New, NewYork, 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. BiologyBull. Math. Biology5252, 5, 5--23, 1990.23, 1990.

9.9. E.K.E.K.YeargersYeargers, R.W., R.W. ShonkwilerShonkwiler and J.V. Herod, Anand J.V. Herod, An IntroductionIntroductionto the Mathematics of Biology: with Computer Algebra Modelsto the Mathematics of Biology: with Computer Algebra Models,,BirkhauserBirkhauser, Boston, 1996., Boston, 1996.

hodgkin-huxley model

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