Spike-Timing-Dependent-Plasticity (STDP) models or how to ......Introduction to plasticity...

Introduction to plasticityPhenomenological STDP models

Our modelSimulations and perspectives

Spike-Timing-Dependent-Plasticity (STDP) modelsor how to understand memory.

Pascal Helson

INRIA Sophia-Antipolis

- Advisor -Etienne TanréRomain Veltz

15 mai 2017

Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.1



Table of content

1 Introduction to plasticity

2 Phenomenological STDP models

3 Our model

4 Simulations and perspectives




Neuron network functioning




Biophysical ModelsBasic cellular processes :

Pathways mediating functional synapticplasticity :

Too complex⇒Phenomenological models




Main idea of STDP

Hebb’s law (1949) :

”Neurons that fire together , wire together .”[?]

Spiking-Time Dependent Plasticity (STDP)[Bi and Poo, 1998]




Table of content



3 Our model





Phenomenological STDP models

Simplify biophysical models

Neuron model

Plasticity model

Neuron network

Simulations




Plasticity rule

STDP learning rule

∆w = F+(w)G (∆t+)− F−(w)G (∆t−)

STDP experimental curve :

[Bi and Poo, 1998]

STDP deterministic curve (G) :

[Izhikevich and Desai, 2003]




Simulations

[Clopath et al., 2009]




Table of content



3 Our model





Objectives

Rich enough to reproduce biological phenomena

Simple enough to be analysed mathematically

Take into account time scales differences

Adapted to simulation of not too small networks (10 000neurons)

Observe global properties of the network




Individual neuron model

Neurons are either at rest(0) orspiking(1).

Wt ∈ RN × RN synaptic weightsmatrix.

Vt ∈ {0, 1}N neuron system state.

Inhomogeneous jump rates :

0α′i (Wt ,Vt )

β

1

S : R 7→ R+∗ bounded, positive and

nondecreasing, αm > 0 :

α′i (Wt ,Vt) = S

(N∑i=1

W ijt V j

t

)+ αm

System with 2 neurons




Plasticity rule

Jump of ∆w = cste

If the neuron i spikes at t−, ∀j 6= i :

if s jt < δ then wij + ∆w with probability p+(s jt ) = 1s jt<δA+e− s

jt

τ+ ;

if s jt < δ then wji −∆w with probability p−(s jt ) = 1s jt<δA−e− s

jt

τ− .

STDP curve :

[Izhikevich and Desai, 2003]




Markov Process (Wt , St ,Vt)t>0 with (W0, S0,V0) = (w0, s0, v0) and :

Wt ∈ RN × RN synaptic weight matrix ;

St = (S1t , ..., S

Nt ) ∈ RN inter-arrival time between spikes ;

Vt ∈ {0, 1}N neuron system state.

State space : E = (RN × RN)× RN × {0, 1}N




Gi =

w + ∆w

0 . . . 0...

0 . . . 0~γ i

0 . . . 0...

0 . . . 0

︸︷︷︸

N×N matrix

+

0 . . . 0 0 . . . 0...

......

...

...... ~ζ i

......

......

......

0 . . . 0 0 . . . 0

, (~γ i , ~ζ i ) ∈ F i

F i =

(~γ i , ~ζ i ), ~γ i = (γ i1, ..., γ

iN), ~ζ i =

ζi1...ζ iN

, γ ij , ζ

ij ∈ {0, 1}, γ i

i = ζ ii = 0

For (~γ i , ~ζ i ) 6= (0, 0) :

φ(s, ~γ i , ~ζ i ) =∏j 6=i

[γ ij p

+(sj) + (1− γ ij )(1− p+(sj))

] [(ζ ij p

−(sj) + (1− ζ ij )(1− p−(sj))]




Generator of the process (Wt , St ,Vt)t>0

∼Cf (w , s, v) =

∑i

δ1(v i )β[f (w , s, v − ei )− f (w , s, v)]︸︷︷︸B↓f (w,s,v)

+ φ(s,w ,w)∑i

αi (w , v)δ0(v i ) (f (w , s − siei , v + ei )− f (w , s, v))︸︷︷︸B↑f (w,s,v)

+N∑i=1

∂si f (w , s, v)︸︷︷︸Btr f (w,s,v)

+∑i

αi (w , v)δ0(v i )

∑∼w∈Gi ,

∼w 6=w

(f (∼w , s − siei , v + ei )− f (w , s, v))φ(s,

∼w ,w)

︸︷︷︸

∼Af (w,s,v)




Assumption on time scales∑(~γ i ,~ζ i )6=(0,0) φ(s, ~γ i , ~ζ i )� φ(s, 0, 0)

⇒ (St ,Vt)t>0 change fast compare to (Wt)t>0

First results

Wt fixed, (St ,Vt)t>0 has a unique invariant measure πWt

One can compute the Laplace transform of πWt

Slow fast analysis gives the limit model for the weights :

Cf (w) =

∫R2+×{0,1}N

Af (w , s, v)πw (ds, dv)




Table of content



3 Our model





Simulations2 neurons

Analytic

Numerical





Discontinuity of the density

��

��

��

��

��

��

��

��

��

��

�

��

��

��

��

��




SimulationsN neurons

STDP curve

-100 0 100-0.5

0.0

0.5

1.0

dt (ms)

dw/w

y1y2





Trajectories




Perspectives

Study the weights dynamics

Simulations to test with other plasticity rules

Neurons states from discrete to continuous.

Mean field approximations.




Bi, G.-q. and Poo, M.-m. (1998). Synaptic modifications in culturedhippocampal neurons : dependence on spike timing, synaptic strength, andpostsynaptic cell type. Journal of neuroscience, 18(24) :10464–10472.

Clopath, C., Büsing, L., Vasilaki, E., and Gerstner, W. (2009). Connectivityreflects coding : A model of voltage-based spike-timing-dependent-plasticitywith homeostasis. Nature.

Izhikevich, E. M. and Desai, N. S. (2003). Relating stdp to bcm. Neuralcomputation, 15(7) :1511–1523.

Morrison, A., Diesmann, M., and Gerstner, W. (2008). Phenomenologicalmodels of synaptic plasticity based on spike timing. Biological Cybernetics,98(6) :459–478.




Thank you for your attentionDo you have questions ?


Spike-Timing-Dependent-Plasticity (STDP) models or how to ......Introduction to plasticity...

Documents

Transcript of Spike-Timing-Dependent-Plasticity (STDP) models or how to ......Introduction to plasticity...