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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Romain Brette
Ecole Normale Supérieure
White Nights of Computational Neuroscience 2012
action potential or « spike »
postsynapticpotential (PSP)
spikethreshold
temporal integration
« Integrate and fire » model: spikes are producedwhen the membrane potentialexceeds a threshold
outside
Na+ Cl
2 nm
inside
K+
specific capacitance ≈ 1 μF/cm²
total specific capacitance = specific capacitance * area
Membrane potential
Vm=VinVout
V
diffusion
K+
or reversal potential
extra
intra
F = 96 000 C.mol1 (Faraday constant)
R = 8.314 J.K1.mol1 (gas constant)
z = charge of ion
I
= capacitance
leak or restingpotential
Linear approximation of leakcurrent: I = gL(VmEL)
leak conductance = 1/R
membrane resistance
EL70 mV : the membrane is « polarized » (Vin < Vout)
Presynapticneuron (extracellularelectrode)
Postsynapticneuron (intracellularelectrode)
(cat motoneuron)
Dirac function
EL
Spikebased notation:
w=RQ/τis the « synapticweight »
at t=0
i = synapse
j = spikenumber
Superposition principle
action potential
PSP
spikethreshold
« Integrateandfire »:
If V = Vt (threshold)
then: neuronspikes and V→Vr (reset)
Differential formulation
Integral formulation
spikeat synapse i
If V = Vt (threshold)
then: neuronspikes and V→Vr (reset)
Fittingspikingmodels to electrophysiologicalrecordings
Injectedcurrent
Recording
Model
Rossant et al. (Front. Neuroinform. 2010)
(IF with adaptive thresholdfittedwith Brian + GPU)
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)
Adaptive integrateandfiremodels are excellent phenomenologicalmodels!
(response to somatic injection)
gs(t)
ionicchannel conductance
open
closed
synaptic reversal potential
presynapticspike
The « cableequation »
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,
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
tn+1 – tn = « interspikeinterval » (ISI)
t1
t2
t3
(Up to a few hundred Hz)
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?
et si V=Vt: V → Vr
= change of variable for V:
V* = (VVr)/(VtVr)
Same for I
if
otherwise
Hz
Brette, R. (2004). Dynamics of onedimensional spiking neuron models. J Math Biol
(superposition principle)
timing of spikeat synapse k
constant (from change of variables)
Jk = postsynapticcurrent
(= 0 for t<0)
([x]+ = max(x,0))
Proof: first prove
where F is the rate of events (tj)
where
meanfield:
neglect variations of I(t)
perfectintegrator:
neglect the leak
F1
F2
Fn
...
w
w
identical synapses
w
If Ti = Poisson with rate Fi, thenUFi = Poisson with rate Fi
Conclusion: Fout = f(Fi)
(+Poisson
+1 synapse type)
integral of synapticcurrent
(perfectintegrator)
currentfrequencyfunction
(meanfield)
transmission probability
undeterminedfunction
Variation: diffusion
Otherwisetheremight not be a univocal inputoutput rate relationship
The same variable currentisinjected 25 times.
Spike timing is reproductible evenafter 1 s.
IF model:
Mainen & Sejnowski (1995)
(cortical neuron in vitro, somatic injection)
Time
Rank
Count
Thorpe et al (2001), Spikebased strategies for rapid processing. Neural networks.
A
B
Excitatory PSP
Inhibitory PSP


+
+
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
input
output
linearreadout
Each output neuronspikes to reducesomeerrordefined on the readout
References:
Boerlin & Denève (2011). SpikeBased 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
In spiking model:
Z. Mainen, T. Sejnowski, Science (1995)
Spike timing isreproducible in vitro for timevarying 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
Consequence:
Similarneuronssynchronize to similar timevarying inputs
?
What impact on postsynapticneurons?
ThresholdVt
d
Same rate F
Delay d
threshold
threshold
spike
no spike
Spike if
How about in realistic situations?
Balancedregime
= VmDpeaksbelowthreshold
Rossant C, Leijon S, Magnusson AK, Brette R (2011). Sensitivity of noisyneurons to coincident inputs. J Neurosci
A
B
no response
« Synchronyreceptivefield » = {S  NA(S) = NB(S)}
Whatdoesitrepresent?
Synchronywhen:
S(tdRδR)=S(tdLδL)
dRdL = δR +δL
Independent of source signal
Synchronywhen:
S(tδA)=S(tδB)
PeriodicsoundwithperiodδAδB
Synchronyreceptivefields signal someregularity or « structure » in the sensorysignals
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 locationspecific
(FL*S,FR*S)
stimulus
Pitch
M
M is pitchspecific
sound in phase space
FR,FL = locationdependentacousticalfilters
(HRTFs/HRIRs)
Delay:
highfrequency
lowfrequency
Intensity:
Sound propagation is more complexthan pure delays!
FR,FL = HRTFs/HRIRs (locationdependent)
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). Spiketimingbased computation in sound localization. PLoS Comp Biol
basilar membrane
MSO
cochlear nucleus
Each source location isrepresented by a specificassembly of binaural neurons
= neuronswhose inputs contain the location in their SRF
Sounds: noise, musical instruments, voice (VCV)
Gammatonefilterbank
Spiking: noisy IF models
Coincidencedetection: noisy IF models
Acousticalfiltering: measuredhumanHRTFs
Additionaltransformations: all HRTFs
bandpassfilteredHRTFs:
Location = assembly of coincidence detectors (1/channel)
synchrony RF of inputs contain location
Goodman DF and R Brette (2010). Spiketimingbased computation in sound localization. PLoS Comp Biol 6(11): e1000993. doi:10.1371/journal.pcbi.1000993.
Activation of all assemblies as a function of preferred location.
Spatial receptive fields
categorization
azimuth
elevation
Estimation error
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)
(STDP = SpikeTimingDependentPlasticity)
potentiel d’action présynaptique
pre post: potentiation
post pre: depression
Dan & Poo (2006)
potentiel d’action postsynaptique
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
w→(1a)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
STDP
+ synapticscaling
(onlypotentiation)
Potentiation of coincident inputs
pre
pre
pre
post
STDP
+ synapticscaling
(onlypotentiation)
Durationtuning in postsynapticneurons
Duration
Brette (2012). Computing with neural synchrony. PLoS Comp Biol
Publications on synchronybasedcomputing
romain.brette@ens.fr
http://audition.ens.fr/brette/