VC dimension of an integrate-and-fire neuron model

Abstract

We find the VC dimension of a leaky integrate-and-fire neuron model. The VC dimension quantifies the ability of a function class to partition an input pattern space, and can be considered a measure of computational capacity. In this case, the function class is the class of integrate-and-fire models generated by varying the integration time constant τ and the threshold θ, the input space they partition is the space of continuous-time signals, and the binary partition is specified by whether or not the model reaches threshold and spikes at some specified time. We show that the VC dimension diverges only logarithmically with the input signal bandwidth N, where the signal bandwidth is determined by the noise inherent in the process of spike generation. For reasonable estimates of the signal bandwidth, the VC dimension turns out to be quite small (aoprox. 10). We also extend this approach to arbitrary passive dendritic trees. The main contributions of this work are (1) it offers a novel treatment of the computational capacity of this class of dynamic system; and (2) it provides a framework for analyzing the computational capabilities of the dynamical systems defined by networks of spiking neurons.