Computing with spikes Romain Brette Ecole Normale
Suprieure
Slide 2
Spiking neuron models
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Input = N spike trains Output = 1 spike train What
transformation ?
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Synaptic integration spike threshold action potential or spike
postsynaptic potential (PSP) temporal integration Integrate and
fire model: spikes are produced when the membrane potential exceeds
a threshold
Slide 5
The membrane Lipid bilayer (= capacitance) with pores (channels
= proteins) outside Na + Cl - inside K + 2 nm specific capacitance
1 F/cm total specific capacitance = specific capacitance *
area
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The resting potential At rest, the neuron is polarized: V m -70
mV The membrane is semi-permeable Mostly permeable to K+ A
potential difference appears and opposes diffusion V Membrane
potential V m =V in -V out K+ diffusion
Slide 7
Electrodiffusion Membrane permeable to K+ only Diffusion
creates an electrical field Electrical field opposes diffusion
Equilibrium potential or Nernst potential : or reversal potential F
= 96 000 C.mol -1 (Faraday constant) R = 8.314 J.K -1.mol -1 (gas
constant) z = charge of ion extra intra
Slide 8
The equivalent electrical circuit I Linear approximation of
leak current: I = g L (V m -E L ) leak or resting potential leak
conductance = 1/Rmembrane resistance = capacitance E L -70 mV : the
membrane is polarized (V in < V out )
Slide 9
The membrane equation I inj outside inside =1/R VmVm membrane
time constant (typically 3-100 ms)
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Synaptic currents synaptic current postsynaptic neuron synapse
I s (t)
Idealized synapse Total charge Opens for a short duration I s
(t)=Q (t) Dirac function ELEL Spike-based notation: at t=0 w=RQ/ is
the synaptic weight
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Temporal and spatial integration Response to a set of spikes {t
i j } ? Linearity: i = synapse j = spike number Superposition
principle
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Synaptic integration and spike threshold spike threshold action
potential PSP Integrate-and-fire : If V = V t (threshold) then:
neuron spikes and V V r (reset)
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The integrate-and-fire model Differential formulation spike at
synapse i Integral formulation If V = V t (threshold) then: neuron
spikes and V V r (reset)
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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 excellent
phenomenological models! (response to somatic injection)
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Advanced concepts Synaptic channels are also described by
electrodiffusion Neurons are not isopotential ionic channel
conductance synaptic reversal potential g s (t) presynaptic spike
open closed The cable equation
Slide 21
Simulating spiking models with PiPe Ce Ci P from brian import *
eqs=''' dv/dt = (ge+gi-(v+49*mV))/(20*ms) : volt dge/dt =
-ge/(5*ms) : volt dgi/dt = -gi/(10*ms) : volt ''
P=NeuronGroup(4000,model=eqs, threshold=v>-50*mV,reset=v=-60*mV)
P.v=-60*mV+10*mV*rand(len(P)) Pe=P.subgroup(3200)
Pi=P.subgroup(800)
Ce=Connection(Pe,P,'ge',weight=1.62*mV,sparseness=0.02)
Ci=Connection(Pi,P,'gi',weight=-9*mV,sparseness=0.02)
M=SpikeMonitor(P) run(1*second) raster_plot(M) show()
briansimulator.org Goodman, D. and R. Brette (2009). The Brian
simulator. Front Neurosci doi:10.3389/neuro.01.026.2009.
Slide 22
Computing with spikes I: rate-based theories
Slide 23
Statistics of spike trains Spike train: A sequence of spike
times (t k ) A signal Rate: Number of spikes / time (= lim k/t k )
Average of S: t1t1 t2t2 t3t3 t n+1 t n = interspike interval (ISI)
(Up to a few hundred Hz)
Slide 24
Rate-based descriptions I s (t) I s (t) = total synaptic
current F F1F1 F2F2 F3F3 F4F4 Rate F Can we express F as a function
of F 1, F 2,..., F n ? Inputs with rates F 1, F 2,..., F n
Slide 25
Approach #1: the perfect integrator Neglect the leak current:
More precise: replace V m by
Slide 26
The perfect integrator Normalization V t =1, V r =0 et si V=V t
: V V r = change of variable for V: V* = (V-V r )/(V t -V r ) Same
for I
Slide 27
The perfect integrator Integration: Firing rate: if otherwise
Hz Brette, R. (2004). Dynamics of one-dimensional spiking neuron
models. J Math Biol
Slide 28
The perfect integrator with synaptic inputs J k = postsynaptic
current (superposition principle) timing of spike at synapse k (= 0
for t