Single Neuron Models

Abstract

Neuron models seem to come in three types: binary, threshold, and dynamical. We indicate some of the potential problems for probabilists associated with each type. In each case we first describe the deterministic model and its characteristics and then indicate how introducing noise, or stochasticity, into the model affects its behavior. 2.1 Binary neurons In 1943 McCulloch and Pitts [15] introduced a simple binary model neuron that has been used in a variety of contexts, for example as the computational element in some neural network models. This model is an extreme simplification of a neuron, and thus is of little interest to neuroscientists on its own, although neural networks built from it do display interesting behavior. The simplification consists of collapsing all of the complicated electro-chemical dynamics of neurons into two states, firing and not firing. In our descriptions of more complicated single neuron models in the following sections, we will describe these dynamics in more detail. The McCullough-Pitts neuron model consists of a weighted sum of inputs, I i , and a single binary output, y (Figure 2.1). The weights W i are scaled between 1 and 1, with negative weights associated with inhibitory inputs and positive weights associated with excitatory inputs. An arbitrary threshold is designated at which the neuron 'fires' a spike, or a '1' output. The model proceeds in discrete time steps. Inputs from all sources are assumed to occur at a discrete moment in time, and a 0 or 1 is recorded, as in equations (2.1) and (2.2). The system is cycled indefinitely. The model can be adapted easily to sum inputs occurring over continuous or discrete time, although this is typically not done for this simple model. If inputs do sum over