Understanding spike-triggered covariance using Wiener theory for receptive field identification

Abstract

Receptive field identification is a vital problem in sensory neurophysiology and vision. Much research has been done in identifying the receptive fields of nonlinear neurons whose firing rate is determined by the nonlinear interactions of a small number of linear filters. Despite more advanced methods that have been proposed, spike-triggered covariance (STC) continues to be the most widely used method in such situations due to its simplicity and intuitiveness. Although the connection between STC and Wiener/Volterra kernels has often been mentioned in the literature, this relationship has never been explicitly derived. Here we derive this relationship and show that the STC matrix is actually a modified version of the second-order Wiener kernel, which incorporates the input autocorrelation and mixes first- and second-order dynamics. It is then shown how, with little modification of the STC method, the Wiener kernels may be obtained and, from them, the principal dynamic modes, a set of compact and efficient linear filters that essentially combine the spike-triggered average and STC matrix and generalize to systems with both continuous and point-process outputs. Finally, using Wiener theory, we show how these obtained filters may be corrected when they were estimated using correlated inputs. Our correction technique is shown to be superior to those commonly used in the literature for both correlated Gaussian images and natural images.

Figure 1

(A) Input–output data records were generated by feeding GWN (x[t]) through a second-order Volterra system defined by a first- and second-order kernel (K₁ and K₂) and thresholding the continuous output to generate the binary output, y[t]. (B) Equivalent Wiener-Bose cascade model of the system shown in (A) composed of three linearly independent filters followed by static quadratic nonlinearities.

Figure 2

(A–D) Results of four estimation methods to characterize system in Figure 1. Note (ii) shows gray confidence bounds for largest and smallest eigenvalues (see section on PDMs). (E) ROC plots of prediction results using only second-order kernels (Equation 2) before they were decomposed into linear filters and static nonlinearities (Equation 12). (F) ROC plots showing prediction results using estimated linear filters and static nonlinearities. Notice that ROC results for STC and STC-M_Φ filters are practically identical because the M_Φ modification has minimal effect on the obtained filters when using uncorrelated inputs. In both (E) and (F), the light blue line (TPR = FPR) indicates a model with no predictive power. (G) Projection of spike-triggering stimuli (blue) and non-spike-triggering stimuli (red) onto the first two PDMs (shown in Diii). Note that separation is imperfect even without noise. The third PDM is needed to achieve optimal results. WIEN = Wiener kernel. PDM = principal dynamic mode. All column axes are equivalent.

Figure 3

(A) Schematic representation of Adelson-Bergen energy model. (B) Recovered STA from energy model contains no information. (C) STC eigenvalues and obtained filters. Note only two filters contain information. (C) Wiener kernel eigenvalues and recovered filters.

Figure 4

(A) Schematic of sample nonlinear Gabor model containing three underlying linear filters. (B–D) Same as in Figure 3. Note STC was only able to detect two of the three ground truth linear filters. ANF = associated nonlinear function.

Figure 5

Predictive power of traditional STC versus PDM method. (A) Predictive power as a function of included filters. For STC, the first included filter was the STA. Notice that including the fourth significant STC filter actually slightly reduced predictive power. Error bars show SEM. (B) Predictive power for T = 30 trials. All predictive power results are evaluated on testing set data.

Figure 6

(A) Convergence and overfitting analysis. Top shows the predictive power of both training (solid lines) and testing (dashed lines) sets for data of various lengths ranging from 20 to 3,000 spikes (200 to 15,000 bins). Bottom indicates the performance of the testing set, normalized by its maximal (steady-state) performance. All points in (A) and (B) were averaged over N = 30 trials. (B) Eigenvalue distribution for a sample system as amount of output spikes are increased. (C) Model robustness measured by predictive power as input noise is increased. % Input noise = (input noise power/total input power) × 100. Same layout as (A). (D) Recovered PDMs of sample system in Figure 1 when the input was composed of 60% noise. Error bars show in (A) and (C) show SD.

Figure 7

(A) A correlated Gaussian input was put through sample system and thresholded as in Figure 1. (B–D) Results of three different correction techniques for I/O data of (A) (see text). (E and F) Predictive power of obtained kernels (E) and linear filters (F). (G) Projection f data onto the first two filters of filter set III is able to almost perfectly separate the data.

Figure 8

Results of three different recovery methods onto correlated Gaussian images (A) and natural images (B). (i) Sample input. (ii–iv) Obtained filters of three different methods (see text). (v and vi) Predictive power of kernels and linear filter models, respectively. ROC = receiver operative curve. PDM = principal dynamic mode.