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Computing with spikes Romain Brette Ecole Normale Suprieure

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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

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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

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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

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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 )

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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)

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Postsynaptic potentials (PSPs) Presynaptic neuron (extracellular electrode) Postsynaptic neuron (intracellular electrode) (cat motoneuron)

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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

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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.

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Computing with spikes I: rate-based theories

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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)

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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

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Approach #1: the perfect integrator Neglect the leak current: More precise: replace V m by

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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

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The perfect integrator Integration: Firing rate: if otherwise Hz Brette, R. (2004). Dynamics of one-dimensional spiking neuron models. J Math Biol

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The perfect integrator with synaptic inputs J k = postsynaptic current (superposition principle) timing of spike at synapse k (= 0 for t