CNS 221 Spring 2006

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CNS 221 - Spring 2006 Lecture 6
(2006-Apr-13) Wolfgang Einhäuser Treyer
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Propagating action potentials
Rirad/4
Ri
d/4Ri (d2V/dx2) C dV/dt GK,max n4 (V- EK)
GNa,max m3h (V- ENa) Gleak (V-
Vrest) Diffusion equation of the form dV/dt
D (d2V/dx2) F(V), with Dgt0 Wave solution of
the form V(x,t)V(x-ut) (from experimental fact
that AP shape does not change) Arrive at
(d/(4Riu2C)) (d2V/dt2)(dV/dt)(INa IK
Ileak)/C (d/(4Riu2C)) must be independent of
diameter gt usqrt(d)
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Myelinated axons
Modeling the myelin as thin, concentric sheets of
membrane (capacitors in series) yields
(whiteboard) ud as compared to usqrt(d) in
unmyelinated fibers.
assume the myelinsheet to have infinite
resistance
Dr
a
b
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Phase plane analysis
Assume we have a 2-dimensional model, given by
differential equations which at time t is at
its starting point and moves in time Dt to the
state then for sufficiently small Dt i.e.
we can represent the system by its flow-field
(phase portrait) Problem HH-model is
4-dimensional gt reduction to 2D
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Phase plane analysis
v(t)
u(t)
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Reduction of HH-model
Hodgkin-Huxley model is 4 dimensional
C dV/dt GK,max n4 (EK--V) GNa,max m3h (ENa-V)
Gleak (Vrest-V) Iext dm/dt
am(1-m)-bmm dn/dt an(1-n)-bnn dh/dt
ah(1-h)-bhh
or equivalent
C dV/dt GK,max n4 (EK--V) GNa,max m3h (ENa-V)
Gleak (Vrest-V) Iext dm/dt
-(m-minf)/tm dn/dt -(n-ninf)/tn dh/dt
-(h-hinf)/th
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Reduction of HH-model
First observation m is faster than h, n

• Replace m(t) by minf V(t) Quasi-steady state
  approximation
• m(t) -gt minf(V)

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Reduction of HH-model
Second observation ninf is similar to (1-hinf)
and tn is similar to th at any voltage

• Replace n and (1-h) by a single variable W
• or a bit more general (b-h) an W with
  constants a and b

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Reduction of HH-model
Replace the 4-dimensional HH-system
C dV/dt GK,max n4 (EK--V) GNa,max m3h (ENa-V)
Gleak (Vrest-V) Iext dm/dt -(m-minf)/tm
dn/dt -(n-ninf)/tn dh/dt -(h-hinf)/th
By a 2-dimensional model
C dV/dt GK,max (W/a)4 (EK--V) GNa,max minf3
(b-W)(ENa-V) Gleak (Vrest-V) Iext dw/dt
G(V,W)/tW
or more compact
dV/dt (F(V,W)RIext)/t dW/dt G(V,W)/tW
with tRC and and some functions F and G
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Reduction of HH-model
dV/dt (F(V,W)RIext)/t dW/dt G(V,W)/tW
The choice of F(V,W) and G(V,W) now determines
our model
FitzHugh-Nagumo model F(V,W) V - V3/3 -
W G(V,W) a bV - W
Morris-Lecar-model F(V,W) -RIionic(V,W) G(V,W)
Winf - W Iionic(V,W) GCa,max minf(V-ECa)
GK,max W(V-EK) Gleak (Vrest-V) minf(V,W)
0.5(1tanh((V1mV)/15mV))) motivation see
whiteboard
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Notation for FHN-model
To simplify notation, in the following slides, I
set t 1 and summarize the constants into one
FitzHugh-Nagumo model F(V,W) V - V3/3 -
W G(V,W) (a bV - W)c c is the ratio of t and
tW
dV/dt F(V,W)I dW/dt G(V,W)
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Phase plane analysis
W
V
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Phase plane analysis Nullclines
dW/dt0
W
dV/dt0
V
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Phase plane analysis Nullclines
small I 1 fixpoint
dW/dt0
W
dV/dt0
V
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Phase plane analysis
large I (and blt1) potentially 3 fixpoints
dW/dt0
W
dV/dt0
V
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Phase plane analysis
bgt1 1 fixpoint (no matter I)
W
V
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Phase plane analysis
bgt1 1 fixpoint (no matter I)
W
V
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Phase plane analysis
bgt1 1 fixpoint (no matter I)
W
V
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Phase plane analysis Stability of fixed points
For a fixed point to be stable, the real part of
both eigenvalues of the matrix must be
negative, For the FitzHugh-Nagumo model, we
have
FitzHugh-Nagumo model F(V,W) V - V3/3 -
W G(V,W) (a bV - W)/c
dV/dt F(V,W)I dW/dt G(V,W)