Mathematical Analysis of Learning Behavior of Neuronal Models

165 In this paper, we wish to analyze the convergence behavior of a number of neuronal plasticity models. Recent neurophysiological research suggests that the neuronal behavior is adaptive. In particular, memory stored within a neuron is associated with the synaptic weights which are varied or adjusted to achieve learning. A number of adaptive neuronal models have been proposed in the literature. Three specific models will be analyzed in this paper, specifically the Hebb model, the Sutton-Barto model, and the most recent trace model. In this paper we will examine the conditions for convergence, the position of convergence and the rate at convergence, of these models as they applied to classical conditioning. Simulation results are also presented to verify the analysis. INTRODUCTION A number of static models to describe the behavior of a neuron have been in use in the past decades. More recently, research in neurophysiology suggests that a static view may be insufficient. Rather, the parameters within a neuron tend to vary with past history to achieve learning. It was suggested that by altering the internal parameters, neurons may adapt themselves to repetitive input stimuli and become conditioned. Learning thus occurs when the neurons are conditioned. To describe this behavior of neuronal plasticity, a number of models have been proposed. The earliest one may have been postulated by Hebb and more recently by Sutton and Barto 1. We will also introduce a new model, the most recent trace (or MRT) model in this paper. The primary objective of this paper, however, is to analyze the convergence behavior of these models during adaptation. The general neuronal model used in this paper is shown in Figure 1. There are a number of neuronal inputs x,(t), i = 1, ... , N. Each input is scaled by the corresponding synaptic weights w,(t), i = 1, ... , N. The weighted inputs are arithmetically summed. N y(t) = L x,(t)w,(t) 9(t) (1) ,=1 where 9(t) is taken to be zero.