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Computing with spikes. Romain Brette Ecole Normale Supérieure. White Nights of Computational Neuroscience 2012. Spiking neuron models. Spiking neuron models. Output = 1 spike train. Input = N spike trains. What transformation ?. Synaptic integration.

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computing with spikes

Computing with spikes

Romain Brette

Ecole Normale Supérieure

White Nights of Computational Neuroscience 2012

spiking neuron models1
Spikingneuronmodels

Output =

1 spike train

Input =

N spike trains

What transformation ?

synaptic integration
Synapticintegration

action potential or « spike »

postsynapticpotential (PSP)

spikethreshold

temporal integration

« Integrate and fire » model: spikes are producedwhen the membrane potentialexceeds a threshold

the membrane
The membrane
  • Lipidbilayer (= capacitance) with pores (channels = proteins)

outside

Na+ Cl-

2 nm

inside

K+

specific capacitance ≈ 1 μF/cm²

total specific capacitance = specific capacitance * area

the resting potential
The restingpotential
  • Atrest, the neuronispolarized: Vm≈ -70 mV
  • The membrane issemi-permeable
  • Mostlypermeable to K+
  • A potentialdifferenceappearsand opposes diffusion

Membrane potential

Vm=Vin-Vout

V

diffusion

K+

electrodiffusion
Electrodiffusion
  • Membrane permeable to K+ only
  • Diffusion creates an electricalfield
  • Electricalfield opposes diffusion
  • Equilibriumpotential or « Nernst potential »:

or reversal potential

extra

intra

F = 96 000 C.mol-1 (Faraday constant)

R = 8.314 J.K-1.mol-1 (gas constant)

z = charge of ion

the equivalent electrical circuit
The equivalentelectrical circuit

I

= capacitance

leak or restingpotential

Linear approximation of leakcurrent: I = gL(Vm-EL)

leak conductance = 1/R

membrane resistance

EL-70 mV : the membrane is « polarized » (Vin < Vout)

the membrane equation
The membrane equation

Iinj

outside

=1/R

Iinj

Vm

inside

membrane time constant

(typically 3-100 ms)

synaptic currents
Synapticcurrents

synapse

synapticcurrent

postsynapticneuron

Is(t)

postsynaptic potentials psps
Postsynapticpotentials (PSPs)

Presynapticneuron (extracellularelectrode)

Postsynapticneuron (intracellularelectrode)

(cat motoneuron)

idealized synapse
Idealized synapse
  • Total charge
  • Opens for a short duration
  • Is(t)=Qδ(t)

Dirac function

EL

Spike-based notation:

w=RQ/τis the « synapticweight »

at t=0

temporal and spatial integration
Temporal and spatial integration
  • Response to a set of spikes {tij} ?
  • Linearity:

i = synapse

j = spikenumber

Superposition principle

synaptic integration and spike threshold
Synapticintegration and spikethreshold

action potential

PSP

spikethreshold

« Integrate-and-fire »:

If V = Vt (threshold)

then: neuronspikes and V→Vr (reset)

the integrate and fire model
The integrate-and-fire model

Differential formulation

Integral formulation

spikeat synapse i

If V = Vt (threshold)

then: neuronspikes and V→Vr (reset)

a phenomenological approach
A phenomenologicalapproach

Fittingspikingmodels to electrophysiologicalrecordings

Injectedcurrent

Recording

Model

Rossant et al. (Front. Neuroinform. 2010)

(IF with adaptive thresholdfittedwith Brian + GPU)

results regular spiking cell
Results: regularspikingcell

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)

good news
Good news

Adaptive integrate-and-firemodels are excellent phenomenologicalmodels!

(response to somatic injection)

advanced concepts
Advanced concepts
  • Synapticchannels are alsodescribed by electrodiffusion
  • Neurons are not isopotential

gs(t)

ionicchannel conductance

open

closed

synaptic reversal potential

presynapticspike

The « cableequation »

simulating spiking models with
Simulatingspikingmodelswith

Goodman, D. and R. Brette (2009). The Brian simulator. Front Neurosci doi:10.3389/neuro.01.026.2009.

frombrianimport *

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()

Ci

P

Pi

Pe

Ce

briansimulator.org

statistics of spike trains
Statistics of spike trains
  • Spike train:
    • A sequence of spike times (tk)
    • A signal
  • Rate:
    • Number of spikes / time (= lim k/tk)
    • Average of S:

tn+1 – tn = « interspikeinterval » (ISI)

t1

t2

t3

(Up to a few hundred Hz)

rate based descriptions
Rate-based descriptions

Inputs with rates F1, F2, ..., Fn

F1

Is(t) = total synapticcurrent

F2

F3

F4

Is(t)

F

Rate F

Can we express F as a function of F1, F2, ..., Fn?

approach 1 the perfect integrator
Approach #1: the « perfectintegrator »
  • Neglect the leakcurrent:
  • More precise: replace Vm by
the perfect integrator
The perfectintegrator
  • Normalization
    • Vt=1, Vr=0

et si V=Vt: V → Vr

= change of variable for V:

V* = (V-Vr)/(Vt-Vr)

Same for I

the perfect integrator1
The perfectintegrator
  • Integration:
  • Firing rate:

if

otherwise

Hz

Brette, R. (2004). Dynamics of one-dimensional spiking neuron models. J Math Biol

the perfect integrator with synaptic inputs
The perfectintegratorwithsynaptic inputs

(superposition principle)

timing of spikeat synapse k

constant (from change of variables)

Jk = postsynapticcurrent

(= 0 for t<0)

the perfect integrator with synaptic inputs1
The perfectintegratorwithsynaptic inputs
  • Input firing rates F1, F2, ..., Fn.
  • Let the « synapticweight » of synapse k
  • Output firing rate is

([x]+ = max(x,0))

Proof: first prove

where F is the rate of events (tj)

approach 2 mean field
Approach #2: meanfield
  • Meanfield approximation:
    • replace I(t) by itsmean
    • use the current-frequencyfunction: F=f(I)

where

mean field vs perfect integrator
Meanfield vs. perfectintegrator

meanfield:

neglect variations of I(t)

perfectintegrator:

neglect the leak

approach 3 poisson inputs
Approach #3: Poisson inputs
  • What if ? (balancedregime)
  • Assumption: input spike trains are independent Poisson processeswith rates Fi

F1

F2

Fn

...

w

w

identical synapses

w

If Ti = Poisson with rate Fi, thenUFi = Poisson with rate Fi

Conclusion: Fout = f(Fi)

summary
Summary:
  • Neglectleak:
  • Meanfield
  • Independent inputs

(+Poisson

+1 synapse type)

integral of synapticcurrent

(perfectintegrator)

current-frequencyfunction

(meanfield)

transmission probability

undeterminedfunction

Variation: diffusion

Otherwisetheremight not be a univocal input-output rate relationship

the precision of spike timing
The precision of spike timing

The same variable currentisinjected 25 times.

Spike timing is reproductible evenafter 1 s.

IF model:

Mainen & Sejnowski (1995)

(cortical neuron in vitro, somatic injection)

the neural code
The neural « code »

Time

Rank

Count

  • Code:
  • spike count
  • (rate code)
  • spike timing
  • (temporal code)
  • spikerank
  • (rankordercode)

Thorpe et al (2001), Spike-based strategies for rapid processing. Neural networks.

decoding rank order
Decodingrankorder
  • How to distinguishbetween AB and BA?
  • Solution: excitation and inhibition

A

B

Excitatory PSP

Inhibitory PSP

-

-

+

+

prey localization by the sand scorpion
Prey localization by the sand scorpion

Inhibition of opposite neuron

→ more spikesnear the source

(polar representation of firing rates)

Conversion rankorder code → rate code

Stürtzl et al. (2000). Theory of arachnidpreylocalization. PRL

predictive coding
Predictivecoding

input

output

linearread-out

Each output neuronspikes to reducesomeerrordefined on the read-out

References:

Boerlin & Denève (2011). Spike-Based Population Coding and Working Memory. PLoS Comp Biol

S Denève (2008). Bayesian spiking neurons I: inference. Neural Computation

S Denève (2008). Bayesian spiking neurons II: learning. Neural Computation

reliability of spike timing
Reliability of spike timing

In spiking model:

Z. Mainen, T. Sejnowski, Science (1995)

Spike timing isreproducible in vitro for time-varying inputs

Brette, R. and E. Guigon (2003). Reliability of spike timing is a general property of spiking model neurons. Neural Comput

Brette (2012). Computing with neural synchrony. PLoS Comp Biol

selective synchronization
Selectivesynchronization

Consequence:

Similarneuronssynchronize to similar time-varying inputs

?

What impact on postsynapticneurons?

coincidence detection principle
Coincidencedetection: principle

ThresholdVt

d

Same rate F

Delay d

threshold

threshold

spike

no spike

Spike if

coincidence detection in noisy neurons
Coincidencedetection in noisyneurons

How about in realistic situations?

Balancedregime

= VmDpeaksbelowthreshold

Rossant C, Leijon S, Magnusson AK, Brette R (2011). Sensitivity of noisyneurons to coincident inputs. J Neurosci

synchrony based computation
Synchrony-based computation

A

B

C

no response

D

signalssimilaritiesbetween A and C

synchrony receptive fields
Synchronyreceptivefields

A

B

no response

« Synchronyreceptivefield » = {S | NA(S) = NB(S)}

Whatdoesitrepresent?

synchrony receptive fields examples
Synchronyreceptivefields: examples

Synchronywhen:

S(t-dR-δR)=S(t-dL-δL)

dR-dL = δR +δL

Independent of source signal

synchrony receptive fields examples1
Synchronyreceptivefields: examples

Synchronywhen:

S(t-δA)=S(t-δB)

PeriodicsoundwithperiodδA-δB

Synchronyreceptivefields signal someregularity or « structure » in the sensorysignals

structure in stimuli
Structure in stimuli

A structured stimulus:

M

The transformation M introduces structure in sensorysignals

S=M(X)

X

Sensory signal

Object in environment

Examples:

Binaural hearing

M

source

sound S

M is location-specific

(FL*S,FR*S)

stimulus

Pitch

M

M is pitch-specific

sound in phase space

structure and synchrony
Structure and synchrony

dR-dL = δR +δL

Brette (2012). Computing with neural synchrony. PLoS Comp Biol

non trivial example binaural sound localization
Non-trivial example:binaural soundlocalization

FR,FL = location-dependentacousticalfilters

(HRTFs/HRIRs)

Delay:

highfrequency

lowfrequency

Intensity:

Sound propagation is more complexthan pure delays!

binaural synchrony receptive fields
Binaural synchronyreceptivefields

FR,FL = HRTFs/HRIRs (location-dependent)

NA, NB = neural filters

(e.g. basilar membrane filtering)

input to neuron A: NA*FR*S (convolution)

input to neuron B: NB*FL*S

Synchronywhen: NA*FR = NB*FL

Independent of source signal S

SRF(A,B) = set of filter pairs (FL,FR)

= set of source locations

= spatial receptivefield

Goodman DF and R Brette (2010). Spike-timing-based computation in sound localization. PLoS Comp Biol

decoding synchrony structure
Decodingsynchrony structure

basilar membrane

MSO

cochlear nucleus

Each source location isrepresented by a specificassembly of binaural neurons

= neuronswhose inputs contain the location in their SRF

proof of principle
Proof of principle

Sounds: noise, musical instruments, voice (VCV)

Gammatonefilterbank

Spiking: noisy IF models

Coincidencedetection: noisy IF models

Acousticalfiltering: measuredhumanHRTFs

Additionaltransformations: all HRTFs

band-passfilteredHRTFs:

Location = assembly of coincidence detectors (1/channel)

synchrony RF of inputs contain location

Goodman DF and R Brette (2010). Spike-timing-based computation in sound localization. PLoS Comp Biol 6(11): e1000993. doi:10.1371/journal.pcbi.1000993.

results
Results

Activation of all assemblies as a function of preferred location.

Spatial receptive fields

categorization

azimuth

elevation

Estimation error

hebb s rule
Hebb’srule

When an axon of cell A is near enough to excite cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A's efficiency, as one of the cells firing B, is increased. (1949)

A

B

D. Hebb

Neuron A and neuron B are active: wABincreases

Physiologically: « synapticplasticity »

PSP size isincreased

PSP

(or: transmission probabilityisincreased)

synaptic plasticity at spike level
Synapticplasticityatspikelevel

(STDP = Spike-Timing-DependentPlasticity)

potentiel d’action présynaptique

pre post: potentiation

post pre: depression

Dan & Poo (2006)

potentiel d’action postsynaptique

  • causal rule
  • favorssynchronous inputs
phenomenological model
Phenomenological model

Presynapticspike:

Postsynapticspike:

learning a synchrony code example a synchrony based code for duration
Learning a synchrony codeExample: a synchrony-based code for duration

A and B fire in synchrony for duration 500 ms

Neuronswithreboundspiking

duration

inhibition

synchronyreceptivefield

Spike latencydepends on duration

A postsynapticneuronfirespreferentiallyatduration 500 ms

Brette (2012). Computing with neural synchrony. PLoS Comp Biol

homeostasis
Homeostasis
  • A coincidence detector must onlyfire to coincidences, i.e., rarely
  • Homeostaticmechanism: enforce a targetfiring rate F
  • Example, synapticscaling:

w→(1-a)w

when the neuronspikes

dw/dt = b.w

otherwise

Weight change = -a.w.F.dt +b.w.dt

Equilibrium: F=b/a

Brette (2012). Computing with neural synchrony. PLoS Comp Biol

learning the synchrony code for duration
Learning the synchrony code for duration

STDP

+ synapticscaling

(onlypotentiation)

Potentiation of coincident inputs

pre

pre

pre

post

learning the synchrony code for duration1
Learning the synchrony code for duration

STDP

+ synapticscaling

(onlypotentiation)

Durationtuning in postsynapticneurons

Duration

learning the synchrony code for duration2
Learning the synchrony code for duration

Brette (2012). Computing with neural synchrony. PLoS Comp Biol

slide65

Publications on synchrony-basedcomputing

  • Reliability of spike timing in models: Brette, R. and E. Guigon (2003). Reliability of spike timing is a general property of spiking model neurons. Neural Comput 15(2): 279-308.
  • Coincidence detection: Rossant C, Leijon S, Magnusson AK, Brette R (2011). Sensitivity of noisy neurons to coincident inputs. J Neurosci 31(47):17193-17206.
  • Computing with synchrony: Brette R (2012). Computing with neural synchrony. PLoS Comp Biol
  • Sound localization with binaural synchrony: Goodman DF and R Brette (2010). Spike-timing-based computation in sound localization. PLoS Comp Biol 6(11): e1000993. doi:10.1371/journal.pcbi.1000993.
  • Simulation: Goodman, D. and R. Brette (2009). The Brian simulator. Front Neurosci doi:10.3389/neuro.01.026.2009.
  • Invariant structure in perception(psychology): James J Gibson (1979), The ecological approach to visual perception. Boston: Houghton Mifflin.

romain.brette@ens.fr

http://audition.ens.fr/brette/