Spatio-temporal correlations and visual signalling in a complete neuronal population

Abstract

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, 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. 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. 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.

Figure 1. Multi-neuron encoding model and fitted parameters

a, Model schematic for two coupled neurons: each neuron has a stimulus filter, a post-spike filter and coupling filters that capture dependencies on spiking in other neurons. Summed filter output passes through an exponential nonlinearity to produce the instantaneous spike rate. b, Mosaics of 11 ON and 16 OFF retinal ganglion cell receptive fields, tiling a small region of visual space. Ellipses represent 1 s.d. of a Gaussian fit to each receptive field centre; the square grid indicates stimulus pixels. c–e, Parameters for an example ON cell. c, Temporal and spatial components of centre (red) and surround (blue) filter components, the difference of which is the full stimulus filter. d, Exponentiated post-spike filter, which may be interpreted as multiplying the spike rate after a spike at time zero. It produces a brief refractory period and gradual recovery (with a slight overshoot). e, Connectivity and coupling filters from other cells in the population. The black filled ellipse is this cell’s RF centre, and blue and red lines show connections from neighbouring OFF and ON cells, respectively (line thickness indicates coupling strength). Below, exponentiated coupling filters show the multiplicative effect on this cell’s spike rate after a spike in a neighbouring cell. f–h, Analogous plots for an example OFF cell.

Figure 2. Analysis of response correlations

a–c, Example CCFs of retinal responses, and simulated responses of the full and uncoupled models, for two ON cells (a), two OFF cells (b) and an ON–OFF pair (c). The baseline is subtracted so that units are in spikes per s above (or below) the cell’s mean rate. d, Receptive field mosaic overlaid with arbitrary labels. Dark grey indicates cells shown in Fig. 1; light grey indicates cells used for triple correlations (h, i). e, CCFs between all ON pairs, where the i,jth plot shows the CCF between cell i and cell j. The grey box indicates the CCF plotted in a. f, g, CCFs between all OFF–OFF pairs (f) and all ON–OFF pairs (g; abscissa height is 30 Hz). h, Third-order (triplet) CCF between three adjacent ON cells, showing the instantaneous spike rate of cell 5 as a function of the relative spike time in cells 4 and 8 (left, RGCs; middle, full model; right, uncoupled model). i, Analogous triplet CCF for OFF cells 15, 16 and 22. j, Comparison of the triplet CCF peak in RGC and model responses (full model, black; uncoupled, grey) for randomly selected triplets of adjacent ON (open) and OFF (filled) cells.

Figure 3. Spike-train prediction comparison

a, Raster of responses of an ON RGC to 25 repeats of a novel 1-s stimulus (top), and responses of uncoupled (middle) and full (bottom) models to the same stimulus. b, PSTH of the RGC (black), uncoupled (blue) and coupled (red) model; both models account for ~84% of the variance of the true PSTH. c, PSTH prediction by full and uncoupled models, showing that coupling confers no advantage in predicting average responses. d, Log-likelihood of novel RGC spike responses under full and uncoupled models; the full model provides 8% more information about novel spike trains. e, Magnified 150-ms portion of RGC raster and PSTH (grey box in a). Red dots highlight RGC spike times on selected individual trials, which are replotted in f. f, Single-trial spike-train prediction using the coupled model. The top half of each plot shows the population activity on a single trial: true spike times of the cell (red dots), coupled ON cells (light grey dots) and coupled OFF cells (dark grey dots; each line in the raster shows the spike times of a different cell). The bottom half of each plot shows a raster of 50 predicted responses of the cell in question, using both the stimulus and coupled responses (shown above) to predict spike trains. The red trace shows the single-trial rate prediction (population-conditioned PSTH), compared with true PSTH of the cell (black trace, identical in all plots). g, Correlation coefficient of true spike trains with the PSTH (ordinate) and with population-conditioned predictions (abscissa); the full model predicts single-trial responses with higher accuracy than the true PSTH.

Figure 4. Decoding performance comparison

a, Shown is a Bayesian decoding schematic: to estimate an unknown stimulus segment from a set of observed spike times (highlighted in boxes), the stimulus prior distribution p(s) is multiplied by the model-defined likelihood p(r|s) to obtain the posterior p(s|r). The posterior mean is the Bayes’ least-squares stimulus estimate. b, Log of the SNR for linear decoding, as well as for Bayesian decoding under the Poisson, uncoupled and full models. The full model preserves 20% more information than the uncoupled model, which indicates that there is additional sensory information available from the population response when correlations are taken into account. Error bars show 95% confidence intervals based on 2,000 bootstrap resamplings of 3,000 decoded stimulus segments. c, Log SNR decomposed as a function of temporal frequency for various decoding methods (Poisson omitted for clarity).