Sensory optimization by stochastic tuning

Abstract

Individually, visual neurons are each selective for several aspects of stimulation, such as stimulus location, frequency content, and speed. Collectively, the neurons implement the visual system's preferential sensitivity to some stimuli over others, manifested in behavioral sensitivity functions. We ask how the individual neurons are coordinated to optimize visual sensitivity. We model synaptic plasticity in a generic neural circuit and find that stochastic changes in strengths of synaptic connections entail fluctuations in parameters of neural receptive fields. The fluctuations correlate with uncertainty of sensory measurement in individual neurons: The higher the uncertainty the larger the amplitude of fluctuation. We show that this simple relationship is sufficient for the stochastic fluctuations to steer sensitivities of neurons toward a characteristic distribution, from which follows a sensitivity function observed in human psychophysics and which is predicted by a theory of optimal allocation of receptive fields. The optimal allocation arises in our simulations without supervision or feedback about system performance and independently of coupling between neurons, making the system highly adaptive and sensitive to prevailing stimulation.

Figure 1. Visual contrast sensitivity in a space-time graph

(A) Human spatiotemporal contrast sensitivity (Kelly function) transformed from the frequency domain to space-time (Kelly, 1979; Nakayama, 1985; Gepshtein et al., 2007). The axes are the temporal and spatial extents of receptive fields. The colored contours (isosensitivity contours) represent contrast sensitivity. The oblique lines represent speeds (constant-speed lines). The lines are parallel to one another in logarithmic coordinates. (B) Spatiotemporal sensitivity function that emerges in the present simulations from independent stochastic fluctuations of receptive fields in multiple motion-sensitive neurons.

Figure 2. Mechanism and dynamics of receptive field size

(A) Basic neural circuit. Input cells ℐ₁ and ℐ₂ receive stimulation from sensory surface x, indicated by the converging lines in corresponding colors. The range of inputs for each cell is its receptive field. ℛ is the readout cell whose activation is mediated by input-readout weights w₁ and w₂. The weights are dynamic: they depend on coincidence of activation of input and readout cells (Equation 7 and Fig. 3A). The weights determine the size of the readout receptive field. (B) Central tendency of readout receptive field (S_r in Equation 15), measured in numerical simulations of the circuit in A. As the input-readout weights are updated, S_r fluctuates on the interval between the smallest and largest input receptive field sizes, marked on the two sides of the plot. The histogram represents probabilities of the different magnitudes of S_r over the course of numerical simulation (Appendix A). The dashed line is the central tendency of S_r: the most likely receptive field size. (C) Variability of S_r. The data points represent average changes of S_r for different magnitudes of S_r. The more S_r is removed from its central tendency (the dashed line copied from B) the larger is its variation, akin to the variation of uncertainty of measurement by a single receptive field captured by Equation 2.

Figure 3. Coincident firing of input and readout cells in the basic circuit

(A) Coincidence rate c_i of spiking for input and readout cells (Equation 14) for all combinations of input-readout weights w₁ and w₂, plotted separately for cells ℐ₁ (left) and ℐ₂ (right). For ℐ₁, all input spikes are accompanied by readout spikes, so that c₁ = 1 for every combination of w₁ and w₂. For ℐ₂, input spikes sometimes do not lead to firing of the readout cell. Since ℐ₁ is more likely to fire together with the readout cell than ℐ₂, weight w₁ is on average larger than weight w₂, and the size of readout receptive field gravitates toward the size of receptive field in ℐ₁ which is smaller than the size of receptive field in ℐ₂. (B) Adaptive response threshold Θ of readout cell for three regimes of measurement. The blue dots represent the values of Θ recorded in 10,000 iterations (every fifth value is shown). The simulation was divided to three periods of equal length, each using a different computation of input cell responses, from left to right: Equation 4, Equation 9, and Equation 8. Threshold Θ fluctuates in the vicinity of a value that is distinct for each method of response computation; it is represented by the red line: the running average of 100 previous magnitudes of Θ.

Figure 4. Stable-state behavior of the basic circuit

(A) Circuit dynamics led to remarkably stable outcomes, illustrated here in the space of input-readout weights w₁ and w₂. The plot summarizes the results of multiple numerical simulations with different starting pairs of weights (w₁, w₂). Each arrow represents the mean direction and magnitude of weight change measured at the arrow origin. The region to which the weights tended to converge is marked by a gray outline, magnified in panel B. (The two other outlines, in pink and green, are explained in panel B.) (B) Enlargement of a region of the weight space in panel A. The gray outline marks the same region in (w₁, w₂) as the gray outline in panel A. The outline is superimposed on a histogram of weights: a grid of (gray) disks of which the sizes represent how often the simulation yielded the pairs of weights (w₁, w₂) at corresponding disk locations. When circuit activity was controlled by stimulus location only, weight w₁ tended to be larger than weight w₂, and so the size of the readout receptive field tended toward the size of smaller input receptive field, as reported in Fig. 2B–C. When circuit activity was controlled only by stimulus frequency content, the weights reversed: w₂ tended to exceed w₁, and so the size of the readout receptive field tended toward the size of the larger input receptive field (cell ℐ₂), summarized by the histogram in pink. When circuit activity was controlled by both stimulus location and frequency content, the weights had intermediate values, summarized by the histogram in green, yielding readout receptive fields of intermediate size.

Figure 5. Lawful fluctuations of receptive field size

Variability of readout receptive field size (S_r) for different regimes of measurement: measurement of stimulus location alone (gray), stimulus frequency content alone (pink), and jointly stimulus location and frequency content (green). The data points represent average changes of S_r for different magnitudes of S_r (as in Fig. 2C). In all cases, the more S_r was removed from its central tendency (the dashed line) the larger was its variation.

Figure 6. Stochastic tuning of readout receptive fields in space-time

The coordinates in every panel represent the temporal and spatial extents, T and S, of a receptive field. (A) Preferred size of readout receptive field. The white cross indicates the starting receptive field size X₀ = (T₀, S₀). The green cross indicates a neutral point, at which the weights of all the input cells to the readout cell are equal to one another. (The white and green crosses have the same locations in all panels.) The map in the background is the probability of the different sizes of readout receptive fields over the course of simulation (10,000 iterations), indicating that the size of readout receptive field tends to drift toward a certain spatial-temporal size (“preferred size") marked by the intersection of white dotted lines at ${\mathbf{X}}_r^{*}=(T_r^{*},S_r^{*})$. (B–D) Effect of the prevailing speed of stimulation on the readout receptive field. Simulations were performed as for panel A, but using biased stimulus distributions, characterized by different prevailing speeds ν_e (Equation 24) represented by the stimulus probability distributions plotted in green at top-right of each panel (0.2 deg/s in B, 1.0 deg/s in C, and 5.0 deg/s in D).

Figure 7. Consequences of stochastic tuning for single receptive fields

Coordinates of every panel represent the temporal and spatial extents of receptive fields relative to the center of the region of interest marked by the white cross. (A) Effect of measurement uncertainty. Initially all receptive fields (N = 1, 000) have the same parameters X₀ = (T₀, S₀) marked by the white cross (at the same location in all panels). Red dots represent the final sizes of receptive fields (“end points"), each after 700 iterations by Equation 11. The large yellow circumference contains the region of permitted fluctuations (Equation 12). The contour plot in the background represents the measurement uncertainty function (Equation 3) whose minimum is marked by the gray asterisk. The three trajectories composed by gray arrows illustrate 20 updates of three model neurons (arbitrarily selected for this illustration). The lengths of arrows are proportional to measurement uncertainties at arrow origins, and arrow directions are sampled from an isotropic probability distribution (R in Equation 11). The inset is a normalized histogram of all end points, indicating that receptive field sizes tend to drift toward lower measurement uncertainty. (B–D) Effect of stimulation. Results of simulations of stochastic tuning performed as in A, but at three different prevailing speeds of stimulation (Equation 24). The three columns show results for different prevailing speeds, increasing from B to D. The direction of receptive field drift depends on the prevailing stimulus speed, indicated by the directed yellow markers in top plots, and by the high concentration of end points in the histograms in bottom plots. Intersections of the white grid lines in bottom panels mark preferred locations of receptive fields, as in Fig. 6. In the yellow directed markers (also used in Fig. 8A), the initial location of receptive fields is represented by a small disk and the direction of receptive field drift is represented by a line to the mean end point of receptive field fluctuations.

Figure 8. Stochastic tuning of motion-sensitive cells across parameter space (T, S)

Parameters T and S correspond to the temporal and spatial extents of receptive fields. (A) Local tendencies of receptive field fluctuations. The small directed markers represent mean directions of receptive field fluctuation (Fig. 7). The inset on top right contains one such marker magnified in linear coordinates, as in Fig. 7B–D. (The shape of local boundary Ω is different in the inset and in the main figure because of the logarithmic coordinates in latter.) The markers in red represent a set of adjacent pathlines (see text). Measurement uncertainty is displayed in the background as a contour plot. The gray curve represents optimal conditions (“optimal set") of speed measurement derived as in Gepshtein et al. (2007). The directed markers across point to the optimal set. If not for conservation of receptive field size (Equation 12), the local tendencies from the locations in red would converge on the white segment of the optimal set. (B–C) Results of stochastic tuning. The heat maps are normalized histograms of end-point densities of receptive field tuning. In B, the histogram is computed for the conditions highlighted in red in panel A. In C, the histogram is computed for the entire parameter space. (The focus of high density and the white segment in panel B are slightly misaligned because of an asymmetry of cell distribution within the group of highlighted pathlines.)

Figure 9. Illustration of stimulus bias in the basic circuit

Panels A–C illustrate outcomes of simulations of the basic circuit using three different distributions of stimulus speed. The stimulus distributions are represented by green curves in the bottom panels, with the mean stimulus speed increasing from A to C. Middle panels (ω_i) illustrate tuning functions (in red) for three input cells centered on tuning speeds ν_i, i ∈ {1, 2, 3}. In the top panels, the blue boxplots represent input-readout weights w_i from over 5, 000 iterations, using stimuli sampled from the speed distributions in the corresponding bottom panels. The top panels also contain plots (in black) of input-readout weights observed for stimuli sampled from a uniform distribution of speed (the same for all top panels). (In the boxplots, the boxes mark the 25th and 75th percentiles, and the whiskers mark the 10th and 90th percentiles, of the distribution of weights.)

Figure 10. Illustration of simulated spiking activity in input and readout cells

Input responses were modeled using a homogeneous Poisson spike generator. Blue and green marks indicate the timing of spikes in input cells ℐ₁ and ℐ₂ within 200 ms after stimulus presentation. In this illustration, the normalized response rates of input cells are r₁ = 0.6 and r₂ = 0.9 and the maximal firing rate r_max is 200. Red bars mark the timing of readout cell spikes. The gray regions enclose input spikes coincident with readout spikes. Since cell ℐ₂ tended to respond when cell ℐ₁ responded, thus activating the readout cell, many spikes of cell ℐ₁ were followed by readout spikes, but spikes of cell ℐ₂ often elicited no readout spikes. The effect of spike coincidence on input-readout weights is represented by the circles in top two rows. The filled and unfilled circles stand, respectively, for increments and decrements of weight. Coincidence rates (Equation 14) for this illustration are c₁ = 12/18 = 0.67 and c₂ = 14/26 = 0.54.

Figure 11. Steady-state parameter distributions

The red line is a prediction of receptive field parameter distribution (Equation 22) in a system with receptive field fluctuations constrained by a reflective boundary. X stands for receptive field size, here normalized to interval [−1 1]. The uncertainty function U(X) is represented by the green curve. The four black curves are results of computational experiments with different fluctuation rates γ. The curves are histograms of parameter distributions, p(X), each after 1, 000 iterations.