hodgkin huxley 07

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Nervous SystemNervous System

Signals are propagated from nerve cell toSignals are propagated from nerve cell tonerve cell (nerve cell (neuronneuron) via electro-chemical) via electro-chemicalmechanismsmechanisms

~100 billion neurons in a person~100 billion neurons in a person Hodgkin and Huxley experimented onHodgkin and Huxley experimented on

squids and discovered how the signal issquids and discovered how the signal isproduced within the neuronproduced within the neuron

H.-H. model was published inH.-H. model was published inJour. ofJour. ofPhysiologyPhysiology(1952)(1952)

H.-H. were awarded 1963 Nobel PrizeH.-H. were awarded 1963 Nobel Prize FitzHugh-Nagumo model is a simplificationFitzHugh-Nagumo model is a simplification


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Action PotentialAction Potential

Axonmembranepotential

difference V = VV = Vii V Vee

When the axonis excited, VV

spikes becausesodium Na+Na+and potassiumK+K+ ions flowthrough the

membrane.

10 msec

mV

_ 30

-70

V

_ 0


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

NernstPotential

VVNaNa , VVKK and VVrr

Ion flow due to

electricalsignal

Travelingwave
http://www.ship.edu/~cgboeree/http://www.ship.edu/~cgboeree/http://www.ship.edu/~cgboeree/


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Circuit Model for AxonCircuit Model for Axon

MembraneMembraneSince the membrane separates charge, it ismodeled as a capacitor with capacitance CC. Ionchannels are resistors. 1/R = g =1/R = g = conductance

iiCC = C dV/dt= C dV/dt

iiNaNa = g= gNaNa (V (V

VVNaNa))

iiKK= g= gKK (V V(V VKK))

iirr = g= grr (V V(V Vrr))


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Circuit EquationsCircuit Equations

Since the sum of the currents is 0, it followsSince the sum of the currents is 0, it followsthatthat

aprrNa

IVVgVVgVVgdt

dVC KKNa += )()()(

where IIapap is applied current. If ion conductances are

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

apIVVgdtdVC += *)(

Solvinggives gIVtV ap /*)( +


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Variable ConductanceVariable Conductance

Experiments showed that gExperiments showed that gNaNa and gand gKK varied with timevaried with time

and V. After stimulus, Na responds much more rapidlyand V. After stimulus, Na responds much more rapidlythan K .than K .

g


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Hodgkin-Huxley SystemHodgkin-Huxley System

Four state variables are used:Four state variables are used:

v(t)=V(t)-Vv(t)=V(t)-Veqeq

is membraneis membrane

potential,potential,

m(t) is Na activation,m(t) is Na activation,

n(t) is K activation andn(t) is K activation and

h(t) is Na inactivation.h(t) is Na inactivation.

In terms of these variables ggKK=ggKKnn44 and

ggNaNa==ggNaNamm33hh.

The resting potential VVeqeq-70mV-70mV. Voltage clamp

experiments determined ggKK

and nn as functions of

tt and hence the parameter dependences on vv in


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Hodgkin-Huxley SystemHodgkin-Huxley System

IVvgVvngVvhmgdt

dvC aprrKNa

KNa+= )()()(

43

mvmvdt

dmmm )()1)(( =

nvnvdt

dnnn )()1)((

=

hvhv

dt

dhhh )()1)(( =


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

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

m(t)

n(t)

h(t)

10msec

1.2

110 mV

40msec


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Fast-Slow DynamicsFast-Slow Dynamics

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

slow.slow.

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

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

thanthannn(v) and(v) and hh(v). An(v). An

increase in v results inincrease in v results in

an increase in man increase in m(v)(v) andand

a large dm/dt. Hence Naa large dm/dt. Hence Naactivates more rapidlyactivates more rapidlythan K in response to athan K in response to a

change in v.change in v.

m(t)

n(t)

h(t)

10msec


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FitzHugh-Nagumo SystemFitzHugh-Nagumo System

Iwvfdt

dv+= )( wv

dt

dw5.0=and

II represents applied current, is small and f(v)f(v) is a cubicnonlinearity. Observe that in the (v,w)(v,w) phase plane

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

w=2vw=2v.

Iwvf

wv

dv

dw

+

= )(

)5.0(


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vv

I=0I=0 I=0.I=0.

22

Stable rest

state

Stable

oscillation

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

w w

vv


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FitzHugh-Nagumo OrbitsFitzHugh-Nagumo Orbits


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

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

3.3. L. Edelstein-Kesket, Mathematical Models in Biology,L. Edelstein-Kesket, Mathematical Models in Biology,Random House, New York, 1988.Random 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, 424-448,1952., 424-448,1952.

5.5. A.L. Hodgkin and A.F. Huxley,A.L. Hodgkin and A.F. Huxley,J. Physiol.J. Physiol.116,116, 449-566,449-566,1952.1952.

6.6. F.C. Hoppensteadt and C.S. Peskin,F.C. Hoppensteadt and C.S. Peskin, Modeling andModeling andSimulation in Medicine and the Life SciencesSimulation in Medicine and the Life Sciences, 2nd ed,, 2nd ed,Springer-Verlag, New York, 2002.Springer-Verlag, New York, 2002.

7.7. J. Keener and J. Sneyd,J. Keener and J. Sneyd, Mathematical PhysiologyMathematical Physiology,,Springer-Verlag, New York, 1998.Springer-Verlag, New York, 1998.

8.8. J. Rinzel,J. Rinzel, Bull. Math. BiologyBull. Math. Biology5252, 5-23, 1990., 5-23, 1990.

9.9.

E.K. Yeargers, R.W. Shonkwiler and J.V. Herod, AnE.K. Yeargers, R.W. Shonkwiler and J.V. Herod, AnIntroduction to the Mathematics of Biology: withIntroduction to the Mathematics of Biology: with

hodgkin huxley 07

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