(A) Schematic of the Goris modulated Poisson model [1], in which spike counts (y) arise from a Poisson process with rate given by the product of two components: a stimulus-dependent 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.
Simulation-based validation of CMP inference.
(A) From top to bottom: true and inferred stimulus-driven components for stimulus 1 and stimulus 3; true and inferred gain processes for two representative trials from each stimulus condition; corresponding true and inferred firing rates; and observed spike counts. (B) Comparison of true and inferred gain hyperparameters ℓg, ρg, q and stimulus-drive hyperparameters .
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. [36]).
(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), cross-trial gain mean, cross-trial gain variance 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) Comparison of pre-stimulus and post-stimulus cross-trial gain variance across V1 neurons. Boxes show the distribution across neurons, and black dots indicate the mean. (D) 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: Goris-independent 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 across the neuronal population (top) and gain auto-correlation (bottom). (E) Distribution of the average stimulus-drive variance across neurons. (F) Scatter plots of versus ρg across neurons, with linear regression fits (black) and slopes annotated. ***p < 0.001.
Comparison of the average test data log-likelihood improvement of each model relative to the baseline Poisson model, across 654 V1 neurons and all 72 grating orientations in recordings from the macaque primary visual cortex (data from [36]).
From left to right: the proposed CMP model, the CMP-RBF model, the CMP-Matérn 0 model, the CMP-Matérn 1 model, the Goris-independent gain model, the Goris-constant gain model, and the Poisson-GP model. These results demonstrate that the CMP model systematically outperforms all other models in fitting held-out data.
Comparison of the performance of the proposed CMP framework with an alternative model in which, instead of sharing a common set of gain hyperparameters ρg, ℓg, q across stimuli (as in the CMP framework), each stimulus condition has its own independent gain hyperparameters .
Results are shown for a sample neuron from the macaque primary visual cortex dataset (data from [36]). (A) Comparison of the inferred time-domain activity. From top to bottom: the inferred stimulus-driven components at three selected orientation settings (0◦, 125◦, 295◦); inferred gain processes for two trials per orientation; inferred firing rates from the proposed CMP model (solid lines) and the alternative model with independent gain hyperparameters per stimulus (dashed lines); and the observed spike raster. (B) Comparison of the log-likelihood improvement relative to the baseline Poisson model for the proposed CMP model (left) versus the alternative independent-gain model (right). These results show that the performance of both models is very similar, corroborating that the proposed CMP modeling framework robustly captures trial-to-trial variability across different stimuli using a shared set of hyperparameters.
Comparison of CMP model inference at different grid sizes dt for a sample neuron from the macaque primary visual cortex dataset (data from [36]).
(A) Comparison of the inferred gain hyperparameters ℓg, ρg, q across three sufficiently fine grid resolutions: dt = 0.02 s, dt = 0.05 s, and dt = 0.1 s. These results indicate that the inference procedure is robust to moderate changes in grid size once the temporal discretization is fine enough to approximate the underlying continuous-time model. (B) Comparison of the test data log-likelihood gain relative to the baseline Poisson model at two different grid resolutions (dt = 0.02 s and dt = 0.1 s). From left to right: the proposed CMP model, CMP-RBF model, CMP-Matérn 0 model, CMP-Matérn 1 model, Goris-independent gain model, Goris-constant gain model, and the Poisson-GP model. These results show that the CMP consistently outperforms alternative models over this range.
Performance comparisons of different models on neuronal population data recorded from various areas along the visual hierarchy: the lateral geniculate nucleus (LGN), V1, V2, and MT, in response to drifting sinusoidal gratings of the preferred size and speed, varying either in spatial frequency (12 spatial frequencies, ranging from 0 to 10 cycles/deg) or in drift direction (16 equally spaced directions) (data from [5]).
(A) Log-likelihood improvement of each model relative to the baseline Poisson model, averaged across held-out data for each population. Rows from top to bottom: LGN, V1, V2, and MT. The number of neurons in each population is indicated in parentheses. From left to right: proposed CMP model, CMP-RBF model, CMP-Matérn 0 model, CMP-Matérn 1 model, Goris-independent gain model, Goris-constant gain model, and the Poisson-GP model. (B) Comparison of the log-likelihood of the proposed CMP model with the CMP-RBF model (left), the CMP-Matérn 0 model (middle), and the CMP-Matérn 1 model (right) for individual neurons. Colors indicate brain areas: LGN (yellow), V1 (green), V2 (blue), and MT (purple). (C) Comparison of the inferred Fano factor as a function of bin size, averaged across each population (rows from top to bottom: LGN, V1, V2, and MT). Curves show the real data (black), the proposed CMP model (red), CMP-RBF model (magenta), CMP-Matérn 0 model (dark green), CMP-Matérn 1 model (light green), Goris-independent gain model (dark blue dashed lines), Goris-constant gain model (light blue), and the baseline Poisson model (gray).
Comparison of the performance of the Goris model with and without a smoothness prior on the stimulus drive (data from [5]).
(A) Log-likelihood improvement of each model relative to the baseline Poisson model, averaged across held-out data for each population. Rows from top to bottom: LGN, V1, V2, and MT, with the number of neurons in each population indicated in parentheses. From left to right: the proposed CMP model, the Poisson-GP model, the Goris-independent gain model without a GP prior on the stimulus drive, and the Goris-constant gain model without a GP prior on the stimulus drive. (B) Comparison of the log-likelihood of the proposed CMP model versus the Goris-independent gain model without a GP prior on the stimulus drive (left), and versus the Goris-constant gain model without a GP prior on the stimulus drive (right), for individual neurons. (C) Comparison of the log-likelihood of the Poisson-GP model versus the Goris-independent gain model without a GP prior (left), and versus the Goris-constant gain model without a GP prior (right), for individual neurons. (D) Comparison of the log-likelihood of the Goris-independent gain model with a GP prior on the stimulus drive versus without a GP prior (left), and the Goris-constant gain model with a GP prior versus without a GP prior (right), for individual neurons. Colors indicate the areas: LGN (yellow), V1 (green), V2 (blue), and MT (purple). These results show that including a smoothness prior on the stimulus drive generally improves the model’s ability to fit held-out data, though the proposed CMP framework still achieves systematically higher performance across all areas. This highlights the importance of jointly modeling smooth stimulus tuning and structured gain fluctuations for accurately capturing neural variability.
