A Quick Study of Oscillations in a BCM Neuron

Objective The purpose of this study is to examine oscillations in the activity of a BCM neuron, as the neuron adapts to an input. We want to explore how the various aspects of the oscillations (frequency, amplitude, etc...) depend on the parameters of the BCM model. 1 The Theory This study is to organize my thoughts on the subject, and to present the material in a way which highlights the main ideas and goals of the topic. It is by no means complete, but in writing it, I hope to gain some significant insights. In this study we will look at the simplified system of one neuron, with a single input and a single output (see Figure 1). We model the activity of the neuron (c) as merely c = m · d (1.1) where d is the input strength d and m is the " efficiency " of the neuron (most commonly refered to as the weight). This choice is used because it is the simplest one we can make. Later we may choose to use a slightly more complicated expression, such as c = g(m · d) (with g being a sigmoidal function), in order to preserve some physical reasonability for especially large inputs. d m c Figure 1: A Single Neuron Our task now is to obtain a rule for neuron learning, or in terms of the math, an expression for the time derivative of the neuron weight. In this study we will look at two learning rules associated with the BCM model. ˙ m = ηc(c − θ)d ≡ ηφ 1 (c, θ)d (1.2) and ˙ m = ηc(c − θ)d θ ≡ ηφ 2 (c, θ)d (1.3) where θ is given by the expression θ = 1 τ t −∞ c 2 (t)e −(t−t)/τ dt (1.4)