The two models define the spike train emitted by a neuron as the result of a two-stage process: an initial linear filtering stage, which governs the membrane potential, followed by a nonlinear spike generation mechanism. The initial linear stage consists of three filters: (1) a stimulus filter, or linear receptive field, which captures the effects of the stimulus, (2) a spike-history filter, which captures the influence of past spikes (e.g., the refractory period), and (3) a cross-neuron filter, which captures the effects of spikes in other neurons on the response. This architecture captures both stimulus-dependent and noise-dependent correlations between neurons. Stimulus-dependent correlations arise from overlap in individual stimulus filters, whereas noise-dependent correlations arise from the cross-neuron filters. The spike-history filter captures a wide array of biologically relevant dynamical behaviors of individual neurons, such as spike rate adaptation, facilitation, bursting, and bistability.

The primary difference between the models arises in the nonlinear, probabilistic spiking stage. In the IF model, spikes occur whenever voltage crosses a fixed threshold, after which voltage resets instantaneously to zero, and response variability results from an injected Gaussian white noise current. In the GLM model, voltage is converted to an instantaneous probability of spiking via a fixed, accelerating nonlinear function [see also Truccolo et al, 2004]. This model can be considered an approximation to integrate-and-fire, where instead of spiking at a fixed threshold, the probability of spiking increases exponentially as a function of voltage. We have shown in recent work [Pillow et al, NIPS 2003; Paninski et al, Neural Comp 2004; Paninski, Network 2004] that these models can be tractably and efficiently fit to data using maximum likelihood estimation. Specifically, the log-likelihood functions of the models are concave, meaning that gradient ascent techniques can be used to efficiently find the optimal estimate of the model parameters (i.e. filters, and the reversal and conductance parameters for the IF model). The technique extends straightforwardly to multi-cell data.

We apply these models to probe the origin of correlations in simultaneously-recorded responses of pairs of macaque retinal ganglion cells, stimulated with a 120-Hz spatially and chromatically varying binary white noise stimulus (i.e. spatial and chromatic flicker). We show that the IF and GLM models reproduce the detailed individual spike train statistics, as well as correlations between spike trains of different cells. We analyze the relative contribution of signal and noise to correlated firing by comparing the performance of the model fit with and without the cross-neuron input. We find that in some cell pairs, the models predict nearly uncorrelated responses without cross-neuron input, but a substantial correlation in spike trains with cross-neuron terms present. We provide a detailed comparison of the performance of the two models and a discussion of how they can be used to investigate the coding significance of correlated firing.