Statistical dependencies in the responses of sensory neurons govern both the amount of stimulus information conveyed and the means by which downstream neurons can extract it. Although a variety of measurements indicate the existence of such dependencies [1] [2] [3] , their origin and importance for neural coding are poorly understood. Here we analyse the functional significance of correlated firing in a complete population of macaque parasol retinal ganglion cells using a model of multi-neuron spike responses 4, 5 . The model, with parameters fit directly to physiological data, simultaneously captures both the stimulus dependence and detailed spatio-temporal correlations in population responses, and provides two insights into the structure of the neural code. First, neural encoding at the population level is less noisy than one would expect from the variability of individual neurons: spike times are more precise, and can be predicted more accurately when the spiking of neighbouring neurons is taken into account. Second, correlations provide additional sensory information: optimal, model-based decoding that exploits the response correlation structure extracts 20% more information about the visual scene than decoding under the assumption of independence, and preserves 40% more visual information than optimal linear decoding 6 . This model-based approach reveals the role of correlated activity in the retinal coding of visual stimuli, and provides a general framework for understanding the importance of correlated activity in populations of neurons. How does the spiking activity of a neural population represent the sensory environment? The answer depends critically on the structure of neuronal correlations, or the tendency of groups of neurons to fire temporally coordinated spike patterns. The statistics of such patterns have been studied in a variety of brain areas, and their significance in the processing and representation of sensory information has been debated extensively 2, 3, [7] [8] [9] [10] [11] [12] [13] . Previous studies have examined visual coding by pairs of neurons 11 and the statistics of simultaneous firing patterns in larger neural populations 14, 15 . However, no previous approach has addressed how correlated spiking activity in complete neural populations depends on the pattern of visual stimulation, or has answered the question of how such dependencies affect the encoding of visual stimuli. Here we introduce a model-based methodology for studying this problem. We describe the encoding of stimuli in the spike trains of a neural population with a generalized linear model (Fig. 1a) , a generalization of the well-known linear-nonlinear-Poisson (LNP) cascade model 4, 5, 16, 17 . In this model, each cell's input is described by a set of linear filters: a stimulus filter, or spatio-temporal receptive field; a post-spike filter, which captures dependencies on spike-train history (for example, refractoriness, burstiness and adaptation); and a set of coupling filters, which capture dependencies on the recent spiking of other cells. For each neuron, the summed filter responses are exponentiated to obtain an instantaneous spike rate. This is equivalent to exponentiating the filter outputs and then multiplying; the exponentiated post-spike and coupling filters (as plotted in Fig. 1 ) may therefore be interpreted as spike-induced gain adjustments of the neuron's firing rate. Although this model is strictly phenomenological, its components can be loosely compared to biophysical mechanisms: the stimulus filter approximates the spatio-temporal integration of light in the outer retina and passive dendritic filtering; the post-spike filter mimics voltage-activated currents following a spike; coupling filters resemble synaptic or electrical interactions between cells (and can mimic the effects of shared input noise); and the exponential nonlinearity implements a 'soft threshold', converting membrane potential to instantaneous spike probability. Note that the post-spike and coupling filters, which allow stochastic spiking in one cell to affect subsequent population activity, give rise to shared, non-Poisson variability in the model response. We fit the model to data recorded in vitro from a population of 27 ON and OFF parasol ganglion cells (RGCs) in a small patch of isolated macaque monkey retina, stimulated with 120-Hz spatio-temporal binary white noise. The receptive fields of each of the two cell types formed a complete mosaic covering a small region of visual space (Fig. 1b) , indicating that every parasol cell in this region was recorded 15, 18 . Such complete recordings, which have not been achieved elsewhere in the mammalian nervous system, are essential for understanding visual coding in neural populations. The model contains many parameters that specify the shapes of all filters, but fitting by maximizing likelihood remains highly tractable 5 . A penalty on coupling filters was used to obtain a minimally sufficient set of coupling filters, which yields an estimate of the network's functional connectivity 19, 20 . Figure 1 shows the estimated filters describing input to example ON and OFF cells. The stimulus filters exhibit centre-surround receptive field organization consistent with previous characterizations of parasol cells. Post-spike filters show the time course of recovery from refractoriness after a spike, and coupling filters show the effects of spikes from nearby cells: for the ON cell (top), spikes in neighbouring ON cells elicit a large, transient excitation (increasing the instantaneous spike rate by a factor of three), whereas spikes in nearby OFF cells elicit suppression. These effects are reversed in the OFF cell, which is excited/suppressed by spikes in neighbouring OFF/ ON cells. Both populations exhibit approximate nearest-neighbour connectivity, with coupling strength falling as a function of distance
Based on LNP, another well-known model, the generalized linear model (GLM) REF is proposed.
Spatio-temporal correlations and visual signalling in a complete neuronal population
Nature
Nature
Biology Medicine