The Structure of Multi-Neuron Firing Patterns in Primate Retina

Pairwise synchrony. A, Receptive fields of 118 ON and 175 OFF parasol RGCs simultaneously recorded in one retina. Ellipses represent 1 SD of the Gaussian fit to the spatial profile of the receptive field. Rectangles represent outline of electrode array. Scale bar, 200 μm. B, Sample cross-correlation functions for pairs of ON and OFF parasol cells shown shaded in black in A. The firing rate of one cell is shown as a function of time relative to the time of a spike in the second cell. The synchrony index for these pairs is S = 1.57 and 0.59, respectively. C, Synchrony index S as a function of distance between receptive fields of ON and OFF parasol cells. For comparison, the black bar near the origin represents the modal separation between cells in the mosaic (see Materials and Methods).

Triplet synchrony. A, Cross-correlation functions for each pair from a group of three adjacent ON parasol cells (labeled A, B, C in Fig. 2A). Each cell pair exhibits strong synchrony (S = 1.90, 2.05, and 1.64 from left to right). B, Three-dimensional cross-correlation for triplet synchrony (left) and maximum entropy pairwise prediction (right). Note that maximum entropy predictions computed for different time offsets are not statistically independent.

Pattern index for multi-neuron firing. A, Pattern index Q (Eq. 6) for firing pattern (1, 1, 1) in groups of three cells as a function of the geometric mean of the distances between each pair in the group. Black symbols, Pattern index using statistical independence as the null model. Red symbols, Pattern index using pairwise constrained maximum entropy as the null model. B, Pattern index for all eight firing patterns in groups of three cells. Symbol colors as in A. C, Pattern index for groups of six cells. Red and black symbols as in A. Blue symbols, Pattern index obtained using the pairwise–adjacent maximum entropy model. For the latter, analysis was restricted to local groups of cells conforming to selection criteria described in Materials and Methods and containing at least one non-adjacent cell pair.

Sample firing pattern distributions. A–C show firing pattern distributions for three ON parasol cells. Each distribution is shown as a histogram, with the firing pattern indicated by binary digits on the abscissa representing a spike or no spike in each cell, and the probability indicated on the ordinate (note logarithmic scale). Firing pattern distributions were obtained from raw data (A), the pairwise model (B), and the statistically independent model (C) (the latter obtained by multiplying the marginal probability distributions for each cell; see Eq. 5). The likelihood of the data obtained from the pairwise model was 0.99944, from the independent model was 0.94959, and from the empirical model was 0.99955. Distributions in B and C were fitted to interleaved recordings from the same cells, distinct from the data in A (data not shown). D–F, Same as A–C but for five ON parasol cells. The likelihood of the data obtained from the pairwise model was 0.99728, from the independent model was 0.92107, and from the empirical model was 0.99836.

Likelihood test of pairwise and adjacent models under constant, spatially uniform illumination. A, Likelihood of observed data under an assumption of statistical independence as a function of likelihood obtained from an empirical model based on a repeated measurement, for groups of three, four, five, six, and seven cells. Diagonal gray line near top indicates equality (note different scales on abscissa and ordinate); large departures from this line indicate substantial failures of statistical independence. B, Likelihood of observed data in the pairwise maximum entropy model as a function of empirical likelihood. Symbol colors and equality line same as in A. C, Likelihood of observed data in the pairwise–adjacent maximum entropy model as a function of empirical likelihood. Symbol colors and equality line same as in A. All likelihood analysis was restricted to local groups of cells conforming to selection criteria described in Materials and Methods. For C, analysis was further restricted to groups that included at least one nonadjacent cell pair.

Likelihood test of pairwise and adjacent models with a fine-grained random visual stimulus. Panels and symbols as in Figure 6.

Predicted and observed pairwise synchrony index. Left panels show the synchrony index (Eq. 4) in pairs of parasol cells as a function of distance between their receptive fields (see Fig. 2). Black points indicate the observed synchrony index, and red points indicate the predictions obtained from the maximum entropy pairwise–adjacent model fitted to groups of n = 7 cells. The black bar near the origin represents the modal separation between cells in the mosaic. Right panels show the comparison between data and model predictions for each cell pair tested. In all panels, large open symbols represent cell pairs that are adjacent in the mosaic; small symbols represent cell pairs that are not.

Sensitivity of maximum entropy analysis. Each panel shows the sensitivity of the maximum entropy analysis procedure for detecting a hypothetical non-pairwise, non-adjacent common input. The hypothetical input occurs at a rate r and causes a spike in each of n = 7 RGCs with a probability p. Results are shown for common input over a range of values of r and p added to simulations obtained with either the pairwise (A) or pairwise–adjacent (B) model fitted to data. In each case, the abscissa indicates the average evoked rate in the simulated RGCs, which is determined by r and p. The ordinate indicates the fraction of the departures from statistical independence accounted for by the pairwise or pairwise–adjacent model. Each gray trace shows the results obtained from a single group of ON parasol cells; red traces indicate the average across all 20 groups. In each panel, the rate of common input r required to reproduce the average value observed in the original data was converted to the equivalent evoked rate and is indicated by a dashed line.