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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/7/27/2019 Hodgkin-Huxley model
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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/7/27/2019 Hodgkin-Huxley model
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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.