neurons ii ca6 – theoretical neuroscience romain brette ([email protected])
TRANSCRIPT
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Today1. The integrate-and-fire model2. Simulation of neural networks3. Dendrites
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Books Principles of Computational Modelling in Neuroscience,
by Steratt, Graham, Gillies, Willshaw, Cambridge University Press
Theoretical neuroscience, by Dayan & Abbott, MIT Press
Spiking neuron models, by Gerstner & Kistler, Cambridge Univerity Press
Dynamical systems in neuroscience, by Izhikevich, MIT Press
Biophysics of computation, by Koch, Oxford University Press
Introduction to Theoretical Neurobiology, by Tuckwell, Cambridge University Press
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The integrate-and-fire model
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The Hodgkin-Huxley model
Model of the squid giant axon
Nobel Prize 1963
nVndt
dnV
hVhdt
dhV
mVmdt
dmV
VEngVEhmgVEgdt
dVC
n
h
m
KKNall
)()(
)()(
)()(
)()()( 43
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The Hodgkin-Huxley model
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The spike threshold
spike threshold
action potential or « spike »
postsynaptic potential (PSP)
temporal integration
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The sodium current
mVmdt
dmVm )()(
)( VEmgI Na
max. conductance (= all channels open)
reversal potential(= 50 mV)
mVmdt
dmV
VEmgVEgdt
dVC
m
Nall
)()(
)()(
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Triggering of an action potential
mVmdt
dmV
VEmgVEgdt
dVC
m
Nall
)()(
)()(
The time constant of the sodium channel is very short (fraction of ms):we approximate )(Vmm
)())(()( VfVEVmgVEgdt
dVC Nall
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Triggering of an action potential
)())(()( VfVEVmgVEgdt
dVC Nall
V
f(V)
0V1 ≈ El V2 V3 ≈ ENa
fixed points
stable stable
unstable
What happens when the neuron receives a presynaptic spike? V -> V+w
Below V2, we go back to rest,above V2, the potential grows (to V3 ≈ ENa)
V2 is the threshold
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The integrate-and-fire model
spike threshold
action potential
PSP
« Integrate-and-fire »: RIVEdt
dVmL
m
If V = Vt (threshold)then: neuron spikes and V→Vr (reset)
(phenomenological description of action potentials)
« postsynaptic potential »
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Is this a sensible description of neurons?
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A phenomenological approach
Injected current
Recording
Model
(IF with adaptive threshold fitted with Brian + GPU)
Fitting spiking models to electrophysiological recordings
Rossant et al. (Front. Neuroinform. 2010)
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Results: regular spiking cell
Winner of INCF competition: 76%(variation of adaptive threshold IF)
Rossant et al. (2010). Automatic fitting of spiking neuron models to electrophysiological recordings(Frontiers in Neuroinformatics)
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Good news
Adaptive integrate-and-fire models are good phenomenological models!(response to somatic injection)
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The firing rateT = interspike interval (ISI)
Firing rateF= 10 spikes/ 100 ms = 100 Hz
nTF
1
(Tn = tn+1-tn)
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Constant current
Firing condition:
Time to threshold:
RIVEdt
dVmL
m
tL VRIE
R
EVI Lt
threshold Vt
reset Vr
tL
mL
VRIE
VRIET
)0(
log
tL
rL
VRIE
VRIET
log from reset
« Rheobase current »
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Current-frequency relationship1
log1
tL
rL
VRIE
VRIE
TF
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Refractory period
Δ = refractory period
tL
rL
VRIE
VRIET
log from reset
1
log1
tL
rL
VRIE
VRIE
TF max 1/Δ
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Simulating neural networks with Brian
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Who is Brian?
briansimulator.org
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Example: current-frequency curvefrom brian import *
N = 1000tau = 10 * mseqs = '''dv/dt=(v0-v)/tau : voltv0 : volt'''group = NeuronGroup(N, model=eqs, threshold=10 * mV, reset=0 * mV, refractory=5 * ms)group.v = 0 * mVgroup.v0 = linspace(0 * mV, 20 * mV, N)
counter = SpikeCounter(group)
duration = 5 * secondrun(duration)plot(group.v0 / mV, counter.count / duration)show()
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Simulating neural networks with Brian
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Synaptic currents
synaptic current
postsynaptic neuron
synapse
Is(t)
smLm RIVE
dt
dV
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Idealized synapse Total charge Opens for a short duration Is(t)=Qδ(t)
sIQ
Dirac function
)(tRQVEdt
dVmL
m EL
t
Lm eRQ
EtV
)(
RQVV
VEdt
dV
mm
mLm
Spike-based notation:
at t=0
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Example: a fully connected networkfrom brian import *
tau = 10 * msv0 = 11 * mVN = 20w = .1 * mV
group = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)
W = Connection(group, group, 'v', weight=w)
group.v = rand(N) * 10 * mV
S = SpikeMonitor(group)
run(300 * ms)
raster_plot(S)show()
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Example: a fully connected networkfrom brian import *
tau = 10 * msv0 = 11 * mVN = 20w = .1 * mV
group = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)
W = Connection(group, group, 'v', weight=w)
group.v = rand(N) * 10 * mV
S = SpikeMonitor(group)
run(300 * ms)
raster_plot(S)show()
R.E. Mirollo and S.H. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM Journal on Applied Mathematics 50, 1645-1662 (1990).
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Example 2: a ring of IF neuronsfrom brian import *
tau = 10 * msv0 = 11 * mVN = 20w = 1 * mV
ring = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)
W = Connection(ring, ring, 'v')for i in range(N): W[i, (i + 1) % N] = w
ring.v = rand(N) * 10 * mV
S = SpikeMonitor(ring)
run(300 * ms)
raster_plot(S)show()
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Example 2: a ring of IF neuronsfrom brian import *
tau = 10 * msv0 = 11 * mVN = 20w = 1 * mV
ring = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)
W = Connection(ring, ring, 'v')for i in range(N): W[i, (i + 1) % N] = w
ring.v = rand(N) * 10 * mV
S = SpikeMonitor(ring)
run(300 * ms)
raster_plot(S)show()
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Example 3: a noisy ringfrom brian import *
tau = 10 * mssigma = .5N = 100mu = 2
eqs = """dv/dt=mu/tau+sigma/tau**.5*xi : 1"""
group = NeuronGroup(N, model=eqs, threshold=1, reset=0)
C = Connection(group, group, 'v')for i in range(N): C[i, (i + 1) % N] = -1
S = SpikeMonitor(group)trace = StateMonitor(group, 'v', record=True)
run(500 * ms)subplot(211)raster_plot(S)subplot(212)plot(trace.times / ms, trace[0])show()
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Example 3: a noisy ringfrom brian import *
tau = 10 * mssigma = .5N = 100mu = 2
eqs = """dv/dt=mu/tau+sigma/tau**.5*xi : 1"""
group = NeuronGroup(N, model=eqs, threshold=1, reset=0)
C = Connection(group, group, 'v')for i in range(N): C[i, (i + 1) % N] = -1
S = SpikeMonitor(group)trace = StateMonitor(group, 'v', record=True)
run(500 * ms)subplot(211)raster_plot(S)subplot(212)plot(trace.times / ms, trace[0])show()
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Example 4: topographic connectionsfrom brian import *
tau = 10 * msN = 100v0 = 5 * mVsigma = 4 * mVgroup = NeuronGroup(N, model='dv/dt=(v0-v)/tau + sigma*xi/tau**.5 : volt', threshold=10 * mV, reset=0 * mV)C = Connection(group, group, 'v', weight=lambda i, j:.4 * mV * cos(2. * pi * (i - j) * 1. / N))S = SpikeMonitor(group)R = PopulationRateMonitor(group)group.v = rand(N) * 10 * mV
run(5000 * ms)subplot(211)raster_plot(S)subplot(223)imshow(C.W.todense(), interpolation='nearest')title('Synaptic connections')subplot(224)plot(R.times / ms, R.smooth_rate(2 * ms, filter='flat'))title('Firing rate')show()
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Example 4: topographic connectionsfrom brian import *
tau = 10 * msN = 100v0 = 5 * mVsigma = 4 * mVgroup = NeuronGroup(N, model='dv/dt=(v0-v)/tau + sigma*xi/tau**.5 : volt', threshold=10 * mV, reset=0 * mV)C = Connection(group, group, 'v', weight=lambda i, j:.4 * mV * cos(2. * pi * (i - j) * 1. / N))S = SpikeMonitor(group)R = PopulationRateMonitor(group)group.v = rand(N) * 10 * mV
run(5000 * ms)subplot(211)raster_plot(S)subplot(223)imshow(C.W.todense(), interpolation='nearest')title('Synaptic connections')subplot(224)plot(R.times / ms, R.smooth_rate(2 * ms, filter='flat'))title('Firing rate')show()
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A more realistic synapse model Electrodiffusion: )( msss VEgI
ionic channel conductance
synaptic reversal potential
gs(t)
presynaptic spike
open closed
))(( mssmLm VEtRgVE
dt
dV
« conductance-based integrate-and-fire model »
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Example: random networkfrom brian import *
taum = 20 * msecondtaue = 5 * msecondtaui = 10 * msecondEe = 60 * mvoltEi = -80 * mvolt
eqs = '''dv/dt = (-v+ge*(Ee-v)+gi*(Ei-v))*(1./taum) : voltdge/dt = -ge/taue : 1dgi/dt = -gi/taui : 1'''
P = NeuronGroup(4000, model=eqs, threshold=10 * mvolt, reset=0 * mvolt, refractory=5 * msecond)Pe = P.subgroup(3200)Pi = P.subgroup(800)we = 0.6wi = 6.7Ce = Connection(Pe, P, 'ge', weight=we, sparseness=0.02)Ci = Connection(Pi, P, 'gi', weight=wi, sparseness=0.02)P.v = (randn(len(P)) * 5 - 5) * mvoltP.ge = randn(len(P)) * 1.5 + 4P.gi = randn(len(P)) * 12 + 20
S = SpikeMonitor(P)
run(1 * second)raster_plot(S)show()
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Example: random networkfrom brian import *
taum = 20 * msecondtaue = 5 * msecondtaui = 10 * msecondEe = 60 * mvoltEi = -80 * mvolt
eqs = '''dv/dt = (-v+ge*(Ee-v)+gi*(Ei-v))*(1./taum) : voltdge/dt = -ge/taue : 1dgi/dt = -gi/taui : 1'''
P = NeuronGroup(4000, model=eqs, threshold=10 * mvolt, reset=0 * mvolt, refractory=5 * msecond)Pe = P.subgroup(3200)Pi = P.subgroup(800)we = 0.6wi = 6.7Ce = Connection(Pe, P, 'ge', weight=we, sparseness=0.02)Ci = Connection(Pi, P, 'gi', weight=wi, sparseness=0.02)P.v = (randn(len(P)) * 5 - 5) * mvoltP.ge = randn(len(P)) * 1.5 + 4P.gi = randn(len(P)) * 12 + 20
S = SpikeMonitor(P)
run(1 * second)raster_plot(S)show()
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Temporal coding
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Reliability of spike timing
Mainen & Sejnowski (1995)
The same constant current is injected 25 times.
The timing of the first spike is reproducible
The timing of the 10th spike is not
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Reliability of spike timing Time of spike n:
Constant current:
Similar constant current: I+δ
nnn Ttt 1
1
11
n
iin Ttt
)()1()1( 11 IfntTnttn
))()()(1()()( IfIfnItIt nn the distance between spikes grows linearly
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Reliability of spike timing Constant current + noise:
Variance:
1
11
n
iin Ttt noise smallTTi
)noise smallvar()1()var( ntnthe distance between spikes grows as n
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But
Mainen & Sejnowski (1995)
The same variable current is injected 25 times
The timing of spikes is reproductible even after 1 s.
(cortical neuron in vitro, injection at soma)
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In the integrate-and-fire model
)(tRIVEdt
dVmL
m and if Vm=Vt: Vm → Vr
in silico in vitro
Spikes are precise if the injected current is variable
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Dendrites
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The dendritic tree
So far we assumed the potential is the same everywhere on the neuron (« point model » or « isopotential model »)
Not true!
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Electrical model of a dendrite
outside
inside
Assumptions:• Extracellular milieu is conductor (= isopotential)• Intracellular potential varies mostly along the dendrite (not across)
dx
Let V(x) = Vintra(x) - Vextra
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Electrical model of a dendrite
V(x) V(x+dx))(xI i
x
V
RdxR
dxxVxVxI
aai
1)()()(
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Electrical model of a dendrite
dx
Capacitance: dxdCdxC m
d
specific membrane capacitanceMembrane resistance:
dxd
R
dx
R m
specific membrane resistance
Axial resistance:
2
4
d
dxRdxR i
a
intracellular resistivity
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The cable equation
LEVt
V
x
V
2
22
Kirchhoff’s law at position x:
mmCR
i
m
R
dR
4
membrane time constant
space constant or« electrotonic constant »
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SynapsesIs(t) = synaptic current at position x
Discontinuity of longitudinal current at position x:
)(),(),( tItxItxI sii
)(),(1
),(1
tItxx
V
Rtx
x
V
R saa
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Stationary response
LEVt
V
x
V
2
22
LEVdx
Vd
2
22
Let EL=0 (reference potential)
Vdx
Vd
2
22
Solutions: //)( xx beaexV
We need boundary conditions to determine a and b
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Stationary response: infinite cableI = (constant) current at x=0
V(x) is bounded
Itx
V
Rt
x
V
R aa
),0(1
),0(1
Exponential solution on each half-cylinder:
/
2)( xa e
IRxV
d
RR m
a
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Green functionI(t) = current at position x=0
bounded V(x)
)(),0(1
),0(1
tItx
V
Rt
x
V
R aa
Let G(x,t) the solution of the cable equation with condition LV=δ(t).Then:
0
)(),(),( dsstIsxGtxV
G = Green function
(convolution)
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Green functionI(t) = current at position x=0
bounded V(x)
)(),0(1
),0(1
ttx
V
Rt
x
V
R aa
)()4
exp(4
1),(
2
2
tHt
xt
tCtxG
(Fourier transform)
(Gaussian for x)
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Dendritic delay
t
V
time to maximum = t*(x)
0*),(
txt
V
0*),()(log
tx
t
V
simpler (and equivalent:)
)/1(82
)(* xoxxt
(Pseudo) propagation speed:2
v proportional to d
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The dendritic tree
not really a cylinder
but almost: connected cylinders
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Cable equation on the dendritic tree On each cylinder k:
At each branching point:
kkk
k Vt
V
x
V
2
22
i
mkk R
Rd
4
0
2
1
Continuity of potential:
Kirchhoff’s law
)0()0()( 210 VVLV
L),0(
1),0(
1),(
1 2
2
1
1
0
0
tx
V
Rt
x
V
RtL
x
V
R
2
4
k
ik d
RR
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Cable equation on the dendritic tree At endings:
At the soma:
Kirchhoff’s law:
No current across:
0),(
tLx
Vk
0isopotential
),0(),0(),0(1
000
0
tVgtt
VCt
x
V
R L
transmembrane current
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Cable equation on the dendritic tree Synapses: Is(t) = synaptic current at position x
)(),(1
),(1
tItxx
V
Rtx
x
V
R sk
k
k
k
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Cable equation on the dendritic tree Summary:
kkk
k Vt
V
x
V
2
22
Linear homogeneous PDEs
)0(
)0(
)(
2
1
00
k
k
kk
V
V
LV
),0(1
),0(1
),(1 2
2
1
10
0
0
tx
V
Rt
x
V
RtL
x
V
Rk
k
k
kk
k
k
Synapses:
0),(
tLx
Vk
),0(),0(),0(1
000
0
tVgtt
VCt
x
V
R L
Linear homogeneous conditions:
Lj(V)=0
linear operator
)(),(1
),(1
tItxx
V
Rtx
x
V
R sk
k
k
k
)()( tIVS s
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Superposition principle Postsynaptic potential PSPj(t):
solution at soma of
Solution for multiple spikes at all synapses:
kkk
k Vt
V
x
V
2
22
)()( tIVS jj
Lj(V)=0
synaptic current at synapse jfor a spike at time 0
PSPj(t)=V(0,t)
cable equation
homogeneous conditions(branchings, etc)
kj
kjttPSPtV
,
)(),0( kth spike at synapse j
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Dendrites: summary Membrane equation ⟶ partial
differential equation « the cable equation »
Linearity ⇒ superposition principle
Different story with non-linear conductances!
ji
jiim ttPPStV
,
)()(
jiii VEgV
x
V
t
V
,2
22 )(
See: Biophysics of computation, by Christof Koch, Oxford University Press
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Variations around the integrate-and-fire model
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The « perfect integrator » Neglect the leak current:
Or more precisely: replace Vm by
)(tRIVEdt
dVmL
m
2rt VV
)(2
tRIVV
Edt
dV rtL
m
2rt VV
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The perfect integrator
Normalization Vt=1, Vr=0
)(tIdt
dV and if V=Vt: V → Vr
= change of variable for V:V* = (V-Vr)/(Vt-Vr)idem for I
)(2/)()(* tRIVVEtI rtL
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The perfect integrator Constant current
Integration: Time to threshold:
Firing rate:
Idt
dV
ItVtV )0()(
IT
1
IT
F 1
if I>0
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The perfect integrator Variable threshold
Integration:
Firing rate:
)(tIdt
dV
t
dssIVtV0
)()0()(
IF
)2sin(2
1)( fttI
0Iif
0F otherwiseHz
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Comparison with integrate-and-fire: constant current
R
Good approximation far from threshold(proof: Taylor expansion)
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Spike timing: the perfect integrator is not reliable
2 different initial conditions:
2 slightly different inputs (ex. noise):
)(1 tIdt
dV )(2 tI
dt
dV
constant difference modulo 1
)(1 tIdt
dV )()(2 ttI
dt
dV
the difference grows ast
dss0
)( tO
(modulo 1)if ε(t) = noise
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The quadratic IF model Comes from bifurcation analysis of HH model,
i.e., for constant near-threshold input.
More realistic current-frequency curve (contant input)
But not realistic in fluctuation-driven regimes (noisy input)
RIVVVVadt
dVt ))(( 0
Resting potential Threshold
Spikes when V>Vmax (possibly infinity)
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The exponential integrate-and-fire model
Boltzmann functionVa=-30 mV, ka=6 mV(typical values)
(Fourcaud-Trocmé et al., 2003)
The exponential integrate-and-fire model
Spike initiation is due to the sodium (Na) channel
)()( 3 VEhmgVEgdt
dVC Nall
INa
)exp(~a
TNa k
VVI
(spike when V diverges to infinity)
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The exponential integrate-and-fire model
(Fourcaud-Trocmé et al., 2003)
Na+ current (fast activation)
Brette & Gerstner (J Neurophysiol 2005)Gerstner & Brette (Scholarpedia 2009)
With adaptation, it predicts output spike trains of Hodgkin-Huxley models
Badel et al. (J Neurophysiol 2008)
V
The exponential I-V curve fits in vitro recordings
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Izhikevich model
Fig. from E. M. Izhikevich, IEEE Transactions on Neural Networks 14, 1569 (2003).
Many neuron types by changing a few parameters
Comes from bifurcation analysis of HH equations with constant inputs near threshold→ not correct for fluctuation-driven regimes
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Adaptive exponential model Same as Izhikevich model, but with exponential initiation
Spike:V → Vr
w → w + bwEVadt
dw
IwVV
gEVgdt
dVC
lw
T
TTLLL
)(
)exp()(
More realistic for fluctuation-driven regimes
Brette, R. and W. Gerstner (2005). J Neurophysiol 94: 3637-3642.