A stochastic calculus approach of Learning in Spike Models

A stochastic calculus approach of Learning in Spike Models Adriana Climescu-Haulica

Laboratoire de Mod´elisation et Calcul

Institute d’Informatique et Math´ematiques Appliqu´ees de Grenoble 51, rue des Math´ematiques, 38041-Grenoble cedex 9 France

email: [email protected]

Abstract

Some aspects of the Hebb’s rule are formalized by means of a SRM0

model where refractoriness and external input are neglected. Us-ing tools from stochastic calculus theory it is shown explicitly that Hebb’s rule is a 0-1 rule, based on a local learning window. As-suming the knowledge of the membrane potential, some learning inequalities are proposed, reflecting the existence of a delay on which the learning can be accomplished. The model allows a nat-ural space-time generalization.

1 The model

We consider a neuron i that receives spike input from N neurons. The membrane potential ui(t) is described by a SRM0 model (Gerstner):

ui(t) = 1 N N X j=1 Z ∞ 0 wij(t−s)(s)Sj(t−s)ds

where refractoriness and external input are neglected.

• wij(t) is the weight of the (i, j) synapse at time t

• (s) is the response kernel modelling the postsynaptic potential

• Sj is defined as

Sj(t) = X

f∈Fj

δ(t−tf_j)

where Fj is the set of spiking times of the neuron j.

Using the change of variable t−s = xthe membrane potential is expressed as a stochastic integral with respect to a Poisson process

Πj(t) = X f∈Fj 1_{t≥tf j} ui(t) = 1 N N X j=1 Z t 0 wij(x)(t−x)dΠj(x) 1 N N X j=1 X f∈Fj wij(t f j)(t−t f j)

2 0-1 Hebbian Learning rule Proposition

There is a vectorial stochastic process m(t) = (m1(t), m2(t), . . . , mN(t))

such that the conditional probability

P ( Πj(t) ∈ A | ui[0, t] = h[0, t] ) = λA( mj(h[0, t]) ) =

1 if mj(h[0, t]) ∈ A

0 if mj(h[0, t]) ∈/ A

where each component of the stochastic process m(t) is constructed by the relation mj(h[0, t]) = ∞ X k=1 1 λj_khHj(I[0, t]), e j kiL2[ √ νj(t)]hh[0, t], e j kiL2[ √ νj(t)] with Hj = p

Rju the square root functional associated with the covariance

functional of the membrane potential

Rj_u(h)(t) = Z t 0 Cu(t, x)h(x)d q νj(x) =HjHj∗(h)(t) the kernel Cu(t, x) = Z t∧x 0 wij(s)(t−s)wij(s)(x−s)dνj(s)

λj_k and ej_k are the eigenvalues and eigenvectors of the covariance functional

Definition

An interval W L = [n−D, n] is a counting window learning set asso-ciated with the spike train Sj if there is a spike time tf_i for the neuron i

such that Πj(t f i) = n and Πj(t f i)−Πj(t f i −∆t) = D ≥ 1.

The Hebb’s rule become

P ( Πj(t) ∈ W L | ui[0, t] = h[0, t] ) =

1 if mj(h[0, t]) ∈ W L

0 if mj(h[0, t]) ∈/ W L

• With an a priori interpretation, the Hebb rule is a detection criterion: between the N neurons firing the neuron i which one become wired with i?

If mj(ui[0, t]) ∈ W L then the synapse (i, j) is strengthened If mj(ui[0, t]) ∈/ W L then the synapse (i, j) vanishes.

3 Learning Inequations

With an a posteriori interpretation, the Hebb rule allows the knowledge of the synaptic weights modified by a learning process.

Assume that the synapse (i, j) was strengthened. Then

n−D ≤ mj(ui[0, t]) ≤ n (1) with ( Πj(t f i) = n Πj(t f i)−Sj(t f i −∆t) = D ≥ 1

As it is known that the stochastic process mj(t) can be assimilated with a

Poisson process, the study of the inequations (1) reduces to the study of an equation ∞ X k=1 1 λj_khHj(I[0, t]), e j kiL2[ √ νj(t)]hui[0, t], e j kiL2[ √ νj(t)] = l

with l an integer value of the interval [n−D, n].

In order to determine wij(t) from the above equation, an extraction

algo-rithm is needed.

• The learning inequalities reflects the existence of a delay on which the learning can be accomplished.

4 Generalization for space-time approach

On this framework, the membrane potential can be modelled as a stochastic integral with respect to a space-time Poisson process

ui(t, x) = Z Z [O,t]×Dx wij(s, y)(t−s, x−y)dΠj(s, y) with Πj(t, x) = X f∈Fj δ(t−tf_j)δ(x−xf_j) Definition

An interval W L = [n−D, n] is a counting window learning set asso-ciated with the spike train Sj if there is a spike time tf_i and a spike place

xf_i for the neuron i such that Πj(t f i, x f i) =n and Πj(t f i, x f i)−Πj(t f i −∆t, x f i −∆x) = D ≥ 1.

The Hebb rule become

If mj (ui([0, t]×Dx)) ∈ W L then the synapse (i, j) is strengthened If mj (ui([0, t]×Dx)) ∈/ W L then the synapse (i, j) vanishes.

5 Conclusions

• The stochastic integral with respect to a Poisson process is a natural framework for spike response models.

• The Hebb rule is expressed as a detection criterion depending on the membrane potential.

• As kernels of the covariance functional of the membrane potential, the synaptic weights are involved in learning inequalities. In order to extract them, an algorithmic solution is needed.

• Space-time spiking neurons can be modelled by means of a space-time Poisson processes. The above remarks apply to space-time model as well.

References

[1] A. Climescu-Haulica, Calcul stochastique appliqu´e aux probl`emes de d´etection des signaux al´eatoires, Ph.D. Thesis, EPFL, Lausanne, 1999 [2] W. Gerstner and W.M. Kistler, Spiking Neurons Models. Single