Post on 16-Feb-2018
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
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.