CNS 221 Spring 2006

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CNS 221 - Spring 2006 Lecture 7
(2006-Apr-18) Wolfgang Einhäuser Treyer
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Stability of fixed points
For a fixed point to be stable, the real part of
both eigenvalues of the matrix must be
negative. Hence (whiteboard)
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Linear example
dV/dt aV-W dW/dt c(bV-W) (Note that c
regulates the relative time-course of V and W
without affecting nullclines) b and c are
positive constants, a is a constant
around fixed point
a lt 0
W
dW/dtlt0
dW/dtgt0
dV/dtlt0
dV/dtgt0
V
stable as dF/dVdG/dW a-c lt 0
and (dF/dV)(dG/dW)-(dF/dW)(dG/dV) c(b-a)gt0
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Linear example
dV/dt aV-W dW/dt c(bV-W) (Note that c
regulates the relative time-course of V and W
without affecting nullclines) b and c are
positive constants, a is a constant
around fixed point
a gt 0
W
dW/dtlt0
dW/dtgt0
dV/dtlt0
dV/dtgt0
V
dF/dVdG/dW a-c (dF/dV)(dG/dW)-(dF/dW)(dG/dV)
c(b-a) gt unstable if a gtc or agtb
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Compare to FitzHugh-Nagumo
small I
dV/dt F(V,W)I dW/dt G(V,W)
dW/dtlt0
dW/dtgt0
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/dtlt0
W
dV/dtgt0
V
stable
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Compare to FitzHugh-Nagumo
larger I
dV/dt F(V,W)I dW/dt G(V,W)
dW/dtlt0
dW/dtgt0
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/dtlt0
W
dV/dtgt0
unstable if slope of V-nullcline gt c
V
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Limit-cycle
larger I
dV/dt F(V,W)I dW/dt G(V,W)
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
W

• flow at boundary is
• inwards,
• fixpoint is unstable
• limit cycle

V
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Transition from stable fixed point to limit cycle
where real part of Eigenvalues changes from
positive to negative, we have Re(l)0 and
hence oscillatory solution with
frequency sqrt((dF/dV)(dG/dW)-(dF/dW)(dG/dV))
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FitzHugh-Nagumo model
b gt 1
b lt 1
W
W
V
V
one or three fixed points (depending on I)
one fixed point
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FitzHugh-Nagumo model
b gt 1
b lt 1
W
W
V
V
one or three fixed points (depending on I)
one fixed point
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FitzHugh-Nagumo model
b lt 1
b gt 1
small I
1 fixpoint, stable
1 fixpoint, stable
large I
3 fixpoints, 1 unstable, 2 stable (bistable
system)
1 fixpoint, unstable (limit cycle)
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FitzHugh-Nagumo model

• Transition from stable fixed point to
  limit-cycle
• onset of oscillation at finite frequency
  (homework set 2!)
• type II neuron

f
I
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Now formal models
So far Conductance based models, realistic
models of spiking Hodgkin Huxley model 4
dimensional FitzHugh-Nagumo-model 2
dimensional Now Formal models - AP
stereotyped, fully given by spike time - fire
whenever membrane potential crosses threshold
(from below)
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Integrate Fire models
I(t) V(t)/RC(dV/dt) t(dV/dt) -V(t)RI(t)
I(t)
Reset
VmgtVt? gt spike and reset
Soma
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Integrate Fire models
AP from axon of pre-synaptic neuron at time t0
delta pulse d(t-t0)
I(t)
low pass filtered current pulse a(t-t0)
Soma
synaptic input
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Integrate Fire models constant input current
leaky IF model I(t) V(t)/RC(dV/dt) I(t)I0
V(t0)0 reset to 0 V(t-gttspiketgttspike)0
(limit from t larger tspike) whiteboard Current
threshold V(t-gtinf spiking off) gt Vthresh gt
Ithresh Vthresh/R Firing rate f for I0gt
Ithresh 1/f- t ln(RI0 /(RI0 -Vthresh)) If we
add refractory period DT 1/f- t ln(RI0 /(RI0
-Vthresh)) DT
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Integrate Fire models time-dependent input
current
leaky IF model I(t) V(t)/RC(dV/dt) I(t)
reset to Vreset last spike at t0
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Non-linear Integrate Fire models
instead of linear IF model I(t)
V(t)/RC(dV/dt) more general t(dV/dt)F(V)G(V)I
For example quadratic IF model t(dV/dt)a(V
-Vrest) (V-Vcritical)RI (with agt0
VcriticalgtVrest) For I 0 V-gtVrest if V(t0)
lt Vcritical V-gtinf (i.e. -gt Vthresh) if V(t0)
gt Vcritical
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IF-neurons Synaptic input
modeling synapses by their currents current at
postsynaptic neuron i Here wij is the
synaptic weight, and a(t-tj,spike) the
time-course of the postsynaptic current in
response to a spike of the presynaptic neuron j
at time tspike,j. (which is often modelled as a
function) As we saw some lectures ago, this
formulation only holds for small conductance
changes (i.e. e.g. single synaptic input),
otherwise (saturation, etc.)
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Spike response models
IF-models voltage dependent parameters SRM
dependence on time since last spike IF-models
differential equations SRM integral over
(past) time Definition SRM if no spike for
sufficiently long time V-gtVrest incoming spike
perturbs system for finite time e action
potential shape given by function h input
resistance k and Vthresh may depend on time
since last spike
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Spike response models
Definition SRM if no spike for sufficiently
long time V-gtVrest incoming spike perturbs
system for finite time e action potential
shape given by function h input resistance k
and Vthresh may depend on time since last
spike For neuron i, which has fired last at
time ti we then have Note that all terms
depend on t-ti i.e. on the time elapsed since the
neuron last spiked, i.e. in the notation here
timaxn(ti(n))
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Spike response models
Kernel describing the shape of the AP
Synaptic weight
Postsynaptic potential Impulse response
(dependent on time since last postsynaptic spike!)
shape of AP by h kernel
FHN-Model
responses to small external current pulses as
given by k-kernel (longer for larger
t-ti) (Figures modified from GK, 4.2)
V
ti
t
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Spike response models lt-gt IF models
IF model
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Spike response model lt-gt HH model
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
last week reduction to 2-dimensions
(-gtFHN-Model) today reduction to 1 dimension
(rescale such that Vrest 0 t0 time of last
spike) we use that the AP is stereotypical,
and the system (for small current) is linear
(gtGreens function) we need h, k and Vthresh as
function of t-t0
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Spike response model lt-gt HH model
Define h as the shape of the AP (apply supra
threshold current pulse to HH model,measure
V(t) relative to Vrest, define t0 as the time of
threshold crossing and set h(t-t0)V(t) Q(t-t0)
(using only the part of V(t) after crossing the
threshold) Define k as the response to a weak
current pulse (depending on t-t0), i.e. measure
V(t) (relative to rest) and subtract it from the
V(t) you get without the pulse (as we did before
for the FHN model) Simplify by taking Vthresh
as free parameter (see difficulty of definition
two lectures ago)
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Coding
Latency of first spike Assume last spike was
long ago, no external input, suddenly at ts you
generate spikes at a fraction p of your N
presynaptic neurons (assume all w equal for
simplicity) As e has the form of an alpha
function (or anything first rising with t), the
larger p, the earlier you cross the threshold. gt
The stronger the stimulus, the earlier you fire.
Compare e.g. to spike-latency code Visual
system is extremely fast in recognizing complex
objects. Suggestion (Thorpe and others)
information encoded in spike latency.
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Coding

• Synchrony
• Assume your N presynaptic neurons (all w1) fire
  at frequency f asynchronosly. Then (on average,
  for large N), there is a spike every
• 1/(fN).
• as on average every there is a spike at 1/(fN),
  2/(fN), 3/(fN), ....
• or in continuous time

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Coding

• Asynchronous firing
• Synchronous firing (at t0)
• with

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Coding
Typical t in the order of 10ms, so for typical
frequencies f Vasyn will be substanitally smaller
than Vsyn gt synchronous input is more likely to
cross threshold and to trigger a response (note
absence of noise, large number of input neurons,
etc)