Figure 1.
Simple model of a population code.
A. Schematics of our model with two pools with neurons each. Correlation within Pool 1 is
for all pairs; correlation within Pool 2 is
for all pairs; correlation between the two pools is
for all pairs. Firing probability in a single window of time for Pool 1 is
for Target and
for Distracter; firing probabilities are the opposite for Pool 2. B. Probability contours (lightest shade represents highest probability) for Target stimulus (red) and Distracter (blue) stimuli in the case of independent neurons (left). Correlation can shrink the distribution along the line separating them and extend the distribution perpendicular to their separation (right). Variances along the two principle axes are denoted by
and
; the angle between the long axis and the horizontal line is denoted by
. Variances along the axes of Pool 1 and 2 are denoted by
and
, respectively; the variance across Pools 1 and 2 is denoted by
.
Figure 2.
Positive correlation can dramatically suppress the error.
A. Probability of discrimination error for a 2-Pool model of a neural population, as a function of the number of neurons, , for independent (dashed; all
) and correlated (circles) populations; parameters are
,
for both, and
,
in the correlated case. Numerical (circles) and analytic (solid line) results are compared. B. Suppression factor due to correlation, defined as the ratio between the error probability of independent and correlated populations, as a function of the number of neurons,
; numeric (circles) and analytic (solid line) results. C. Error probability as a function of the cross-pool correlation,
, for independent (dashed line) and correlated (circles,
) populations; analytic results for correlated population (solid line).
. D. Error probability as a function of the correlation within Pool 1,
, for independent (dashed line) and correlated (circles,
,
) populations; analytic results for correlated population (solid line).
. E. Probability contours for three examples of neural populations; independent (green cross,
,
,
), near lock-in correlation (pink dot,
,
), and uneven correlation (blue diamond,
,
,
). Colored symbols correspond to points on plots in previous panels.
Figure 3.
A. Probability of error as a function of the cross-pool correlation, , for a small neural population (circles,
neurons,
,
,
), with analytic result for correlated population (solid line) and independent population (dashed line) for the sake of comparison. B. Probability of error versus
for populations of different sizes (colors); independent population (dashed lines) and analytic results for correlated population (solid lines). C. Probability of error versus
for a neural population with responses differing by an average of 2 spikes (
neurons,
,
,
); numeric solutions (circles), analytic result (solid line), and independent comparison population (dashed line). D. Probability of error versus
for populations having different sizes but with
held constant at 2 spikes (colors); independent population (dashed lines) and analytic results for correlated population (solid lines).
Figure 4.
Heterogeneous neural populations.
A, B. Histogram of the error suppression (error in the homogeneous, 2-Pool model divided by the error in the fully heterogeneous model) for variability values and
, respectively. All suppression values are greater than one. C. Value of the error suppression (geometric mean) versus the degree of population variability;
neurons,
,
,
,
. (With these parameters, correlation suppresses the error probability by a factor of 4350 relative to the matched independent population.)
Figure 5.
Number of encoded stimuli for independent versus correlated populations.
A, B. Schematics of the optimal arrangement of the probability distributions for independent (A) and correlated (B) populations. Each set of contours represents the log probability distribution of neural activity given a stimulus (hotter colors indicate higher probability). Spacing is set by the criterion that adjacent pairs of distributions have a discrimination error threshold . C. Number of stimuli encoded at low error, per neuron, versus
, for correlated (thin dashed line for
, thick dashed line for
) and independent (solid lines) populations, for different values of the error criterion,
(colors). D. Number of encoded stimuli per neuron, for correlated (thin dashed line for
, thick dashed line for
) and independent (solid lines) populations, versus
, for different values of the number of neurons,
(colors).
Figure 6.
Coding capacity of heterogeneous populations.
A. Number of encoded stimuli versus , for an independent population divided into different numbers of pools,
(colors); the error criterion is
. B. Ratio of the number of encoded stimuli in a correlated population and the number of encoded stimuli in a matched independent population, for different numbers of pools
(colors). C. Optimal pool size,
, versus error criterion,
, for correlated (dashed line,
) and independent (solid line) populations. D. Optimal capacity per neuron,
, versus error criterion,
, for correlated (dashed line,
) and independent (solid line) populations.
Figure 7.
Schematics of an experimental test of high-fidelity correlated coding.
A. Representation of a population of 50 neurons recorded under two stimulus conditions. Each cell displays firing rates and
in response to the two stimuli, respectively; the color scale shows the difference in rates,
. B. The population is divided into two groups, depending on whether their cells fire more significantly in response to the first (preferred) or the second (anti-preferred) stimulus. C. Matrix of correlation values among all pairs of neurons (red = large, blue = small, black = average), divided into preferred and anti-preferred groups. Although the overall correlation is stronger for neurons with the same stimulus tuning (average correlation of pref-pref = 0.206, anti-anti = 0.217, and pref-anti = 0.111), a subset of neurons (Pool 1 and Pool 2) are identified which have the pattern of correlation favorable to lock-in. D. Matrix of pairwise correlations after re-labeling cells in order to sort out Pools 1 and 2. Now the favorable pattern of correlation is visible.
Figure 8.
Illustration of a proposed decoding mechanism and circuit.
A. The decoding mechanism is illustrated in the case of a two-pool model, in which denotes the spike count in Pool
. The stimulus to be decoded elicits the distribution of activities represented by the yellow-red contour lines; other distributions, in blue-grey, flank it and result from different stimuli. Optimal decision boundaries (dashed lines), defined by simple inequalities, are implemented by the read-out circuit. B. The read-out circuit is a two-layer perceptron. In its first layer, excitatory and inhibitory inputs from both pools are non-linearly summed into two intermediary read-out neurons; the synaptic weights and thresholds (equivalently, baselines) are chosen such that the two intermediary neurons implement the inequalities
and
, respectively. Their two outputs are then summed non-linearly in turn, so that the ‘decoder neuron’ is active only if both inequalities are satisfied.
Figure 9.
Robustness to parameter variations.
A. Probability of error as a function of the cross-pool correlation for populations with
neurons and different angles
of their probability distributions in the space of
; parameters are (
,
,
) with
set to give the chosen angle (Eq. (62)). B. Probability of error as a function of angle for fixed difference in spike count,
, intersects the error criterion
at two angles, which defines the angular bandwidth. C. Angular bandwidth plotted as a function of within pool correlation,
, for different values of the difference in spike count,
.