spike-timing-dependent-plasticity (stdp) models or how to ......introduction to plasticity...
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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
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Table of content
1 Introduction to plasticity
2 Phenomenological STDP models
3 Our model
4 Simulations and perspectives
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.2
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Neuron network functioning
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.3
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Biophysical ModelsBasic cellular processes :
Pathways mediating functional synapticplasticity :
Too complex⇒Phenomenological models
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.4
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Main idea of STDP
Hebb’s law (1949) :
”Neurons that fire together , wire together .”[?]
Spiking-Time Dependent Plasticity (STDP)[Bi and Poo, 1998]
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.5
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Table of content
1 Introduction to plasticity
2 Phenomenological STDP models
3 Our model
4 Simulations and perspectives
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.6
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Phenomenological STDP models
Simplify biophysical models
Neuron model
Plasticity model
Neuron network
Simulations
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.7
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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]
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.8
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Simulations
[Clopath et al., 2009]
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.9
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Table of content
1 Introduction to plasticity
2 Phenomenological STDP models
3 Our model
4 Simulations and perspectives
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.10
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.11
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.12
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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]
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.13
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.14
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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))]
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.15
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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)
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.16
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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)
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.17
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Table of content
1 Introduction to plasticity
2 Phenomenological STDP models
3 Our model
4 Simulations and perspectives
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.18
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Simulations2 neurons
Analytic
Numerical
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.19
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Simulations2 neurons
Discontinuity of the density
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Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.20
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
SimulationsN neurons
STDP curve
-100 0 100-0.5
0.0
0.5
1.0
dt (ms)
dw/w
y1y2
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.21
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Simulations2 neurons
Trajectories
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.22
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Simulations10 neurons
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.23
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Perspectives
Study the weights dynamics
Simulations to test with other plasticity rules
Neurons states from discrete to continuous.
Mean field approximations.
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.24
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
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.
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.25
Introduction to plasticityPhenomenological STDP models
Our modelSimulations and perspectives
Thank you for your attentionDo you have questions ?
Pascal HelsonSpike-Timing-Dependent-Plasticity (STDP) models or how to understand memory.26