A stochastic calculus approach of Learning in Spike Models

Transcription

1 A stochastic calculus approach of Learning in Spike Models Adriana Climescu-Haulica Laboratoire de Modélisation et Calcul Institute d Informatique et Mathématiques Appliquées de Grenoble 51, rue des Mathématiques, Grenoble cedex 9 France adriana.climescu@imag.fr Abstract Some aspects of the Hebb s rule are formalized by means of a SRM 0 model where refractoriness and external input are neglected. Using tools from stochastic calculus theory it is shown explicitly that Hebb s rule is a 0-1 rule, based on a local learning window. Assuming 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 natural space-time generalization. 1

2 1 The model We consider a neuron i that receives spike input from N neurons. The membrane potential u i (t) is described by a SRM 0 model (Gerstner): u i (t) = 1 N N j=1 0 w ij (t s)ɛ(s)s j (t s)ds where refractoriness and external input are neglected. w ij (t) is the weight of the (i, j) synapse at time t ɛ(s) is the response kernel modelling the postsynaptic potential S j is defined as S j (t) = f F j δ(t t f j ) where F j is the set of spiking times of the neuron j. Using the change of variable t s = x the membrane potential is expressed as a stochastic integral with respect to a Poisson process Π j (t) = f F j 1 {t t f j } u i (t) = 1 N 1 N N j=1 t 0 N j=1 w ij (x)ɛ(t x)dπ j (x) f F j w ij (t f j )ɛ(t tf j ) 2

3 Figure 1: N spike neurons as input 3

4 2 0-1 Hebbian Learning rule Proposition There is a vectorial stochastic process m(t) = (m 1 (t), m 2 (t),..., m N (t)) such that the conditional probability P ( Π j (t) A u i [0, t] = h[0, t] ) = λ A ( m j (h[0, t]) ) = { 1 if mj (h[0, t]) A 0 if m j (h[0, t]) / A where each component of the stochastic process m(t) is constructed by the relation m j (h[0, t]) = k=1 1 λ j k H j (I[0, t]), e j k L 2 [ ν j (t)] h[0, t], ej k L 2 [ ν j (t)] with H j = Ru j the square root functional associated with the covariance functional of the membrane potential the kernel R j u(h)(t) = t 0 C u (t, x)h(x)d ν j (x) = H j Hj (h)(t) λ j k and ej k Ru. j C u (t, x) = t x 0 w ij (s)ɛ(t s)w ij (s)ɛ(x s)dν j (s) are the eigenvalues and eigenvectors of the covariance functional 4

5 Definition An interval W L = [n D, n] is a counting window learning set associated with the spike train S j if there is a spike time t f i for the neuron i such that and Π j (t f i ) = n Π j (t f i ) Π j(t f i t) = D 1. The Hebb s rule become P ( Π j (t) W L u i [0, t] = h[0, t] ) = { 1 if mj (h[0, t]) W L 0 if m j (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 m j (u i [0, t]) W L then the synapse (i, j) is strengthened If m j (u i [0, t]) / W L then the synapse (i, j) vanishes. 5

6 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 with { n D m j (u i [0, t]) n (1) Π j (t f i ) = n Π j (t f i ) S j(t f i t) = D 1 As it is known that the stochastic process m j (t) can be assimilated with a Poisson process, the study of the inequations (1) reduces to the study of an equation k=1 1 λ j k H j (I[0, t]), e j k L 2 [ ν j (t)] u i[0, t], e j k L 2 [ ν j (t)] = l with l an integer value of the interval [n D, n]. In order to determine w ij (t) from the above equation, an extraction algorithm is needed. The learning inequalities reflects the existence of a delay on which the learning can be accomplished. 6

7 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 u i (t, x) = w ij (s, y)ɛ(t s, x y)dπ j (s, y) [O,t] D x with Π j (t, x) = δ(t t f j )δ(x xf j ) f F j Definition An interval W L = [n D, n] is a counting window learning set associated with the spike train S j if there is a spike time t f i and a spike place for the neuron i such that x f i and Π j (t f i, xf i ) = n Π j (t f i, xf i ) Π j(t f i t, xf i x) = D 1. The Hebb rule become If m j (u i ([0, t] D x )) W L then the synapse (i, j) is strengthened If m j (u i ([0, t] D x )) / W L then the synapse (i, j) vanishes. The above formulation exhibits the local aspect in time and space of the Hebb rule. 7

8 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 appliqué aux problèmes de détection des signaux aléatoires, Ph.D. Thesis, EPFL, Lausanne, 1999 [2] W. Gerstner and W.M. Kistler, Spiking Neurons Models. Single Neurons, Populations, Plasticity, Cambridge University Press,