Figures and data
(A) Schematic of the Goris modulated Poisson model [1], which assumes that spike counts (y) are generated by a Poisson process whose rate is the product of two components: a stimulusdependent drive f (s) and a stimulus-independent gain g. The gain g is a scalar and is independently drawn across trials. (B, C) Two continuous-time extensions of the Goris model. (B) Constant gain model: assumes that the sampling interval of the gain equals the trial duration. Left: stimulus-induced rate, gain process, and instantaneous firing rate for three example trials. Middle: gain autocorrelation function. Right: Fano factor as a function of bin size (dashed line indicates the baseline Poisson model). (C) Independent gain model: assumes that the sampling interval of the gain is much shorter than the length of a single trial. Panels are organized as in (B). (D) Schematic of the proposed Continuous Modulated Poisson (CMP) model. Here, spike trains y(t) are generated by a Poisson process whose rate is the product of a time-varying stimulus drive fs(t) and a continuous-time stochastic gain process g(t). The gain process g(t) follows a Gaussian process (GP) with temporal correlations governed by an exponentiated power law kernel, and is independently drawn across trials. (E) Panels are organized as in (B), for the CMP model.
Illustration of different settings of the proposed Exponentiated Power Law (EPL) covariance function with marginal variance ρg = 0.1.
(A) EPL auto-covariance functions with different settings of length-scale (top row: ℓg = 0.02, bottom row: ℓg = 0.1) and power-law exponent (columns left to right: q = 0.5, q = 1, q = 2). Colored traces below each auto-covariance show three sample gain processes g(t), obtained by sampling from the corresponding GP and exponentiating. (B) Fano Factor curves showing how Fano Factor changes as a function of the bin size used to count spikes, for EPL covariances in the top and bottom rows in (A), respectively.
Performance comparisons of different models on neural population data recorded from macaque primary visual cortex under 72 drifting sinusoidal gratings (data from Graf et al. [34]).
(A) Top to bottom: Tuning curve of a sample neuron; for three selected orientations (00,1050,2800), the PSTH, inferred stimulus drive, inferred gain (trial 1), inferred firing rate (trial 1), observed spike raster (trial 1), inferred gain (trial 2), inferred firing rate (trial 2), observed spike raster (trial 2), and Fano factor vs. bin size curves for real data (black), CMP model (red), Goris-independent gain model (dark blue dashed), Goris-constant gain model (light blue), and Poisson-GP model (gray). (B) EPL gain covariance (top) and autocorrelation (bottom) under the CMP model for the neuron in panel A. (C) Average test log-likelihood improvement of each model relative to the baseline Poisson model, computed on held-out data across all 654 V1 neurons and 72 stimulus orientations.
Performance comparisons of different models on neuronal population data recorded from different areas along the visual hierarchy—the lateral geniculate nucleus (LGN), V1, V2 and MT—under drifting sinusoidal gratings of preferred size and speed, varying in spatial frequency (12 frequencies from 0 to 10 cycles/deg) or in drift direction (16 directions) (data from Goris et al. [1]).
(A) Log-likelihood improvement of each model relative to the baseline Poisson model, averaged across neurons in each population. Rows from top to bottom: LGN, V1, V2, MT. The number of neurons per area is indicated in parentheses. (B) Log-likelihood comparison of the CMP model versus the Goris independent gain model (left), Goris constant gain model (middle), and Poisson-GP model (right) for individual neurons. Colors indicate cortical areas: LGN (yellow), V1 (green), V2 (blue), MT (purple). (C) Fano factor vs. bin size curves averaged across each population. Black: real data; red: CMP; dark blue dashed: Gorisindependent gain; light blue: Goris-constant gain; gray: Poisson-GP.
Comparison of inferred CMP hyperparameters across visual areas: the lateral geniculate nucleus (LGN), V1, V2, and MT (data from [1]).
Rows top to bottom: LGN (yellow), V1 (green), V2 (blue), and MT (purple). The number of neurons per area is indicated in parentheses. (A) Distribution of gain power-law exponent q across neurons. Triangles indicate the median. (B) Distribution of gain length scale ℓg across neurons. (C) Distribution of gain variance ρg across neurons. (D) Median gain covariance 
