Structures of Neural Correlation and How They Favor Coding

Abstract

The neural representation of information suffers from "noise"-the trial-to-trial variability in the response of neurons. The impact of correlated noise upon population coding has been debated, but a direct connection between theory and experiment remains tenuous. Here, we substantiate this connection and propose a refined theoretical picture. Using simultaneous recordings from a population of direction-selective retinal ganglion cells, we demonstrate that coding benefits from noise correlations. The effect is appreciable already in small populations, yet it is a collective phenomenon. Furthermore, the stimulus-dependent structure of correlation is key. We develop simple functional models that capture the stimulus-dependent statistics. We then use them to quantify the performance of population coding, which depends upon interplays of feature sensitivities and noise correlations in the population. Because favorable structures of correlation emerge robustly in circuits with noisy, nonlinear elements, they will arise and benefit coding beyond the confines of retina.

Figure 1. Spiking Responses, Tuning Curves, and Noise Statistics in Retinal Direction-Selective Neurons

(A) Spike raster of a “quadruplet” of neurons, four direction-selective retinal ganglion cells with preferred directions aligned along the four cardinal directions (temporal, red; superior, green; nasal, blue; inferior, yellow). For each stimilus direction (black arrows), five trials of spiking responses are displayed. The top panel exhibits the receptive fields of this example quadruplet, as obtained from white noise stimulation and reverse correlation; the ellipses illustrate the shapes of the receptive fields. (B) The preferred directions of all 90 recorded direction-selective cells (gray arrows), from seven experiments, point along the four cardinal directions (S, superior; I, inferior; T, temporal; N, nasal; same color coding as in A). One example quadruplet of neurons is indicated by arrows in colors, and their tuning curves are represented in polar plots (solid lines in the top right panel, peaks normalized to 1, dotted lines show the mean ± SD). (C) Same tuning curves as in (B), in a Cartesian plot and unnormalized. (D) Spike counts from the example quadruplet in (A), (B), and (C) in the presence of moving bar stimuli: each quadrant exhibits the responses of two cells whose preferred directions differ by 90°; one point per trial. Although stimulus directions were separated by 10° in experiments, here, for the sake of clarity, we display every other instance (stimulus directions separated by 20°). Colors label the direction of motion of the bar in the stimulus, as illustrated in the top right quadrant: the orange bar and red bar illustrate the stimulus and the corresponding arrow its direction; the color wheel indicates the directions of motion corresponding to each choice of color. (E) Schematic tuning curves and measured noise correlations in pairs of simultaneously recorded cells as a function of the direction of motion in the stimulus, for two neurons with preferred directions that differ by 0° (left panel), 90° (middle panel), and 180° (right panel). The top panels present schematic tuning curves (in Cartesian plots, blue and green curves) for these three configurations (illustrated in the insets), as well as the geometric means of these two quantities (thin, gray curves). The bottom panels display the measured correlations, where the mean correlations are plotted as solid lines and the standard deviations appears as shaded areas.

Figure 2. Improvement of the Coding Performance by Correlations in Populations of Retinal Direction-Selective Neurons

(A and B) The coding error (in angular degrees, A) and the percent improvement in resolution due to noise correlations (B), as functions of the number of simultaneously recorded neurons. Different colors correspond to different experiments; each point represents an average over up to 50 distinct groups of cells selected randomly within each experiment; error bars indicate standard errors of the mean.

Figure 3. Improvement of the Coding Performance in Quadruplets and Octuplets, with Stimulus-Dependent versus Stimulus-Independent Correlations

(A) Histograms of percent improvement of the coding resolution from the full data, in which correlations depend upon the stimulus, for quadruplets (left panel) and octuplets (right panel) recorded simultaneously. Small gray triangles indicate the means of these empirical distributions, quoted above them. Different colors correspond to different experiments. (B) Same as (A) but with histograms derived in the case in which noise correlations in the data were replaced by their average over all stimuli and hence became stimulus independent. The effect of the stimulus dependence of the correlations on coding can be seen by comparing (A) and (B).

Figure 4. Functional Models of Response and Noise Statistics of Direction-Selective Neurons

(A) Illustration of the models defined in Equation 2: the variability of neural response and the correlation among the responses of neurons can originate from additive input noise (Model I), input gain modulation (Model II), or output gain modulation (Model III). (B) Examples of model fits of the tuning curves of two neurons (model, solid curves; data, dashed curves). (C) Examples of model fits of the variances and covariance in the activity of the same two neurons as in (B) (model, solid curves; data, dashed curves).

Figure 5. Behavior of the Coding Error and the Coding Improvement as a Function of Correlated Noise

(A) Percent improvement in the accuracy of direction coding as a function of the magnitude of the secondary noise, in Model I (left panel), Model II (middle panel), and Model III (right panel), for the stimulus-dependent correlations generated by the models (red curves), as compared to the mean-matched stimulus-independent case (green curves). For independent neurons (blue curves), the percent improvement is constant and vanishing. (B) Deterioration of the coding accuracy by the secondary sources of noise, as described by the error normalized by the error in the noiseless case with σ₁ = σ₂ = σ₃ = 0. In the presence of the stimulus-dependent correlations generated by Model I (left panel), Model II (middle panel), and Model III (right panel), the coding accuracy is largely insensitive to the secondary noise (red curves), whereas in the cases of stimulus-independent correlation (green curves) and independent neurons (blue curves), noise is detrimental to the coding accuracy. For a sufficiently strong non-linearity, g, input-gain modulation can improve the coding accuracy (dashed line, middle panel).

Figure 6. Impact of Different Structures of Noise Correlation upon Population Coding

(A) Left panel: two model direction-selective neurons respond to different stimuli (dashed lines) according to tuning curves (solid gray curves), f₁(θ) and f₂(θ), with two direction preferences that differ by 90°. Right panel: the two tuning curves are represented as a solid gray line parametrized by the stimulus direction, θ. In the space of the two-cell output, this gray line forms an informative subspace: the location of the pair response along the gray line yields information about the stimulus presented. More precisely, for each stimulus, θ, the tangent vector, $(f_1^{\prime }(\theta ),f_2^{\prime }(\theta )),$ defines the informative direction (arrows in colors corresponding to the stimulus values in the left panel). (B) For each stimulus presented, noise correlation distorts the cloud of two-cell responses about the mean over trials; depending upon the geometry of this distortion with respect to the informative direction, it can either benefit or harm the coding accuracy. Positive correlation in the pair (c > 0) favors the reliability of coding with respect to the independent case (c = 0), while negative correlation (c < 0) is detrimental. Specifically, when c > 0, the responses for nearby stimulus directions overlap less, and, hence, coding is more reliable. (Conversely, if the two tuning curves have similar preference, c < 0 is favorable whereas c > 0 is detrimental; illustration not shown.) More precisely, coding is favored if the eigenvector of the covariance matrix parallel to the tangent vector, $(f_1^{\prime }(\theta ),f_2^{\prime }(\theta )),$ comes with a small eigenvalue: correlation then relegates the noise in the orthogonal, uninformative direction. Ellipses are contours of equal probability, drawn at 2.5 standard deviations. (C) Responses of a pair of neurons with 90° difference in preferred direction; each point is a trial, colors represent different stimuli (direction of moving bar). Data were simulated using the measured tuning functions and covariance matrices. Three different scenarios are visualized: two independent neurons (left panel), two neurons with stimulus-independent correlations (middle panel), two neurons with the full stimulus-dependent correlation present in data (right panel); ellipses are contours of equal probability for each stimulus, at 2 standard deviations; thick black lines illustrate eigenvectors of the covariance matrix (for one example stimulus, in red). The ellipses are reoriented by noise correlations so that their long axis moves away from the informative direction along which stimuli are coded (thick gray line, given by the tuning functions). Insets: arrangements of the preferred directions and sketches of the dependence of the correlation upon stimulus. (D) Same as (C) but for two pairs of neurons, corresponding to half of an octuplet. For the sake of illustration, spike counts of neurons with similar preferred directions were summed. The effects are similar to (C), and stronger.

Figure 7. Effect of Correlation in Small Populations of Direction-Selective Neurons

(A) Illustration of the model of direction-selective neurons, for a given stimulus, in which there are m neurons whose preferred directions point along each of the four cardinal directions. The correlation structure is parametrized by c₀, c_π/2, and c_π, the values of correlations in pairs of neurons whose preferred directions differ by 0°, 90°, and 180°. (B) Profiles of the percent improvement in coding accuracy as functions of c_π/2 and c_π, for different values of m; in the cases m = 2, 3, and 5, c₀ was set to 0.2, comparable to the observed values. Black dots represent the values observed in the data; the green dot represents the average over all data. The gray area represents forbidden values of the correlations (with negative eigenvalues of the covariance matrix).

Figure 8. Beyond the Retina: Model of Coding with a Population of Broadly Tuned, Correlated Neurons

(A) Illustration of the model of a population of N₀ broadly tuned, correlated neurons. The population is divided into N pools, each of which contains m neurons with similar stimulus preference. (B) Measured stimulus-dependent correlations in pairs of neurons. Top panel: illustration of the tuning curves. Bottom panels: three examples of measured correlations in pairs of direction-selective cells with 0° difference in preferred direction (squares) and with 90° difference (triangles), from given groups of three cells. Solid lines are smoothed versions of the raw correlation values. For a range of stimulus directions, the correlation between neurons whose preferred directions differ by 90° exceeds the correlation between neurons with similar preferred directions; the examples were chosen from three different experiments. (C) Shapes of the correlation function used in the model; c(θ) is the correlation between two neurons with preferred stimuli that differ by θ. Different colors correspond to different parameter choices, as labeled in the legend. (D) Percent improvement of the coding accuracy as a function of the number of neurons in the population, N₀. Color coding as in (C). (E) Optimal number of pools, N, as a function of the number of neurons in the population, N₀. Color coding as in (C). (F) Illustration of the optimal configurations in a correlated population of different sizes (total number of neurons, N₀). For small populations, neurons cover the stimulus range uniformly, i.e., there are as many pools as there are neurons (N = N₀) and their separation decreases as N₀ increases. When the separation becomes comparable to the “correlation length” set by the shape of the correlation function, the number of pools, N, stops growing with increasing N₀; instead, each pool becomes more populated.