Predictive information in a sensory population

Abstract

Guiding behavior requires the brain to make predictions about the future values of sensory inputs. Here, we show that efficient predictive computation starts at the earliest stages of the visual system. We compute how much information groups of retinal ganglion cells carry about the future state of their visual inputs and show that nearly every cell in the retina participates in a group of cells for which this predictive information is close to the physical limit set by the statistical structure of the inputs themselves. Groups of cells in the retina carry information about the future state of their own activity, and we show that this information can be compressed further and encoded by downstream predictor neurons that exhibit feature selectivity that would support predictive computations. Efficient representation of predictive information is a candidate principle that can be applied at each stage of neural computation.

Keywords: information theory; neural coding; retina.

Fig. 1.

Information about position of a moving bar. (A) Trajectory $x_t$ and spiking responses recorded from several cells simultaneously (36); responses in a single small window of time $\mathrm{\Delta }\tau$ can be expressed as binary words $w_t$. (B) Information that N-cell words provide about bar position (Eq. 1), as a function of the delay $\mathrm{\Delta }t$, averaged over many N-cell groups. Estimation errors and SEMs over groups both are negligible (∼0.01 bits/spike); we stop at $N=7$ to avoid undersampling. (C) Normalized autocorrelation of the trajectory, $C_{xx}$, vs. time delay, $t-t^{\prime }$. The peak has been shifted to align with the peak information in B.

Fig. 2.

Bounds on predictive information. (A) Any prediction strategy defines a point in the plane I_future vs. I_past. This plane is separated into allowed and forbidden regions by a bound, $I_{\text{future}}*\left(I_{\text{past}}\right)$, shown for the sensory world of a moving bar following the stochastic trajectories of Fig. 1. (B) We can capture the same information about the past in different ways, illustrated by the black “error ellipses” in the position/velocity plane. If we know that the trajectory is inside one of these ellipses, we have captured I_past = 0.42 bits. However, points inside these ellipses propagate forward along different trajectories, and after $\mathrm{\Delta }\tau =1/60\hspace{0.17em}\text{s}$, these trajectories arrive at the points shown in purple and green. Using the same number of bits to make more accurate statements about position leads to more predictive information (purple; $0.40\hspace{0.17em}\text{bits}$) than if we use these bits to make more accurate statements about velocity (green; $0.18\hspace{0.17em}\text{bits}$).

Fig. 3.

Direct measures of the predictive information in neural responses. (A) Many independent samples of the trajectory $x_t$ converge onto one of several common futures, two of which are shown here (red and blue). The time of convergence is indicated by the vertical dashed line. (B) Mean spike rates of a single neuron in response to the stimuli in A. Shaded regions are ±1 SEM. (C) Information about the common future for one group of five cells, as a function of the time, $\mathrm{\Delta }t$, until convergence. Solid line shows the bound on $I_{\text{future}}\left(\mathrm{\Delta }t\right)$ for this group’s $I_{\text{past}}$. (D) Information about the future vs. information about the past, for many groups of different size, N with Δt = 1/60 s; group A as in C. Error bars include contributions from the variance across groups and the SD of the individual information estimates. Solid line is the bound from Fig. 2A.

Fig. 4.

Mutual information between past and future neural responses. (A) Conditional distribution $P\left(w_{t+\mathrm{\Delta }t}|w_t\right)$, at time $\mathrm{\Delta }t=1/60\hspace{0.17em}\text{s}$, for the group of four cells with the maximum information (1.1 bits/spike), in response to a natural movie. The prior distribution of words, $P\left(w\right)$, is shown adjacent to the conditional. Probabilities are plotted on a log scale; blank bins indicate zero samples. (B) Distributions of $I\left(W_t;W_{t+\mathrm{\Delta }t}\right)$ for $N=2$, $N=4$, and $N=9$ cells, with $\mathrm{\Delta }t=1/60\hspace{0.17em}\text{s}$. (C) Information between words as a function of $\mathrm{\Delta }t$. Inset shown information vs. N at $\mathrm{\Delta }t$ marked by arrows. (D) Information between words for groups of $N=9$, as a function $\mathrm{\Delta }t$ for different classes of stimuli: a natural movie, the moving bar from Fig. 1, and a random flickering checkerboard refreshed at 30 fps. Shaded regions indicate $\pm 1$ SD across different groups of cells.

Fig. 5.

Predictor neurons. (A) Maximum efficiency $I\left({\sigma }_t^{\text{out}};W_{t+\mathrm{\Delta }t}\right)/I\left(W_t;W_{t+\mathrm{\Delta }t}\right)$ as a function of the output firing rate, for 150 four-cell groups. Average over all groups is indicated by the dashed line; solid black line indicates perfect capture of all of the predictive information. (B) Efficiency of a perceptron rule relative to the best possible rule, for the same groups as in A. (C) The information that ${\sigma }^{\text{out}}$ provides about the visual stimulus grows with the predictive information that it captures. Results shown are the means over all possible output rules, for 150 four-cell input groups; error bars indicate SDs across the groups. (D) Average velocity triggered on a spike of the predictor neuron for one four-cell group; light gray lines show the triggered averages for the input spikes; the predictor neuron selects for a long epoch of constant velocity. In A–C, Δt = 1/60 s; in D, Δt = 1/30 s.