Benefits of pathway splitting in sensory coding

Abstract

In many sensory systems, the neural signal splits into multiple parallel pathways. For example, in the mammalian retina, ~20 types of retinal ganglion cells transmit information about the visual scene to the brain. The purpose of this profuse and early pathway splitting remains unknown. We examine a common instance of splitting into ON and OFF neurons excited by increments and decrements of light intensity in the visual scene, respectively. We test the hypothesis that pathway splitting enables more efficient encoding of sensory stimuli. Specifically, we compare a model system with an ON and an OFF neuron to one with two ON neurons. Surprisingly, the optimal ON-OFF system transmits the same information as the optimal ON-ON system, if one constrains the maximal firing rate of the neurons. However, the ON-OFF system uses fewer spikes on average to transmit this information. This superiority of the ON-OFF system is also observed when the two systems are optimized while constraining their mean firing rate. The efficiency gain for the ON-OFF split is comparable with that derived from decorrelation, a well known processing strategy of early sensory systems. The gain can be orders of magnitude larger when the ecologically important stimuli are rare but large events of either polarity. The ON-OFF system also provides a better code for extracting information by a linear downstream decoder. The results suggest that the evolution of ON-OFF diversification in sensory systems may be driven by the benefits of lowering average metabolic cost, especially in a world in which the relevant stimuli are sparse.

Keywords: ON–OFF; efficient coding; optimality; parallel pathways; retina; sensory processing.

Figure 1.

Modeling framework. A stimulus s is encoded by a system of two cell types: one ON and OFF (as shown) or two ON cells. Each cell has a binary response nonlinearity with thresholds θ₁ or θ₂. During each coding window of duration T, the stimulus is constant and the spike count is drawn from a Poisson distribution. A measure of coding efficiency here is the mutual information between the stimulus s and the spike count responses of the two cells k₁ and k₂, I(s; k₁, k₂).

Figure 2.

Output states in the binary neuron model. An arbitrary stimulus distribution P(s) can be mapped into a uniform distribution so that the optimal thresholds can be depicted in terms of the fraction of stimuli below threshold, ranging from 0 to 1. A, Possible output states for the two neurons in the ON–OFF system, depending on the stimulus value. 0 denotes silence, 1 denotes one or more spikes in the coding window. B, Same as A but for two ON cells.

Figure 3.

Mutual information with binary neurons and Poisson output noise. A, The mutual information for an ON–OFF and an ON–ON system is identical when constraining the maximum expected spike count, N_max. Inset, The ON–ON system uses more spikes to convey the same information. The total mean is normalized by N_max. B, The thresholds of the response functions for the optimal ON–OFF and ON–ON systems when constraining N_max. Right, Optimal response functions in the limits of low and high N_max. C, The ON–OFF system transmits more information than the ON–ON system when the total mean spike count is constrained. Inset, the fraction of total mean spike count assigned to the ON cell with the smaller threshold. D, The thresholds of the response functions for the optimal ON–OFF and ON–ON systems when constraining the total mean spike count. Right, Optimal response functions in the limits of low and high mean spike count (compare with B). E, The optimal ON–ON system transmits more information than a system of identical ON cells. F, The thresholds of the response functions for the optimal ON–ON and identical 2ON systems as a function of the total mean spike count. G, The ratio of mutual information transmitted by the optimal ON–OFF versus ON–ON systems in C. H, The ratio of mean spike counts required to convey the same information by the ON–ON versus ON–OFF systems. I, The ratio of mutual information from the optimal ON–ON versus identical 2ON systems in E.

Figure 4.

Mutual information with a non-monotonic response function. A, The mutual information for three different systems: (1) one ON and one OFF cell; (2) two ON cells; and (3) one U-cell with a U-shaped nonlinearity and an ON cell. Although the first two systems divide the stimulus range into three regions, the last divides it into a total of four regions and transmits a maximum of log₂(4) = 2 bits of information. Inset, The total mean spike count (normalized by N_max). B, The optimal thresholds of the response functions for the three systems in A when constraining N_max. Right, Optimal response functions in the limits of low and high N_max.

Figure 5.

Mutual information with sigmoidal response functions and Poisson output noise. A, The sigmoidal response model (see Materials and Methods) is illustrated for an ON cell showing the maximum expected spike count N_max, the gain (red dashed), and the threshold (green dashed) of the cell. The schematic shows the effect of varying the gain for a fixed threshold. B, The ratio of mutual information transmitted by the optimal ON–OFF versus ON–ON systems about a stimulus drawn from a Laplace distribution. The total mean spike count of each system, N_mean, was constrained. C, The optimal thresholds of the sigmoidal response functions for the ON–ON system when constraining the total mean spike count. Below a critical value of the gain, the two thresholds are identical. D, The optimal thresholds of the sigmoidal response functions for the ON–OFF system when constraining the total mean spike count. The two thresholds are different except in the limit of small gain and large mean spike count.

Figure 6.

Mutual information with response functions and noise from retinal data. A, Two sigmoidal nonlinearities for an ON cell (green) and an OFF cell (orange), describing the firing rate as a function of stimulus with the maximum expected spike count N_max, the gain β, and the threshold θ. The shaded curve denotes the Laplace stimulus probability distribution. B, The information transmitted by the ON–OFF and ON–ON systems, using nonlinearities measured from salamander (left) and macaque (right) ganglion cells (Pitkow and Meister, 2012). The maximum spike count was constrained. C, The total mean spike count used by the ON–OFF relative to the ON–ON system to transmit the maximal information for the two types of nonlinearities in B. D, The information transmitted by the two systems using a constraint on the total mean spike count. E, The total mean spike count that would be needed by the ON–OFF and the ON–ON systems to transmit the same information as the ON–OFF system shown in D.

Figure 7.

Input noise increases the benefits of ON–OFF splitting. A, Top, Input noise is added to the stimulus before the nonlinearities. Bottom, Distribution of stimulus (Laplace) and input noise (Gaussian with different SDs). B, Ratio of information transmitted by the ON–OFF versus ON–ON systems when constraining the total mean spike count. Input noise increases the benefit of the ON–OFF system but non-monotonically. In this and subsequent panels, the dashed green line demonstrates the results for a Gaussian stimulus distribution with input noise SD 1 (compare to the full green line for the Laplace stimulus distribution with SD 1). C, The ratio of total mean spike count needed to transmit a given information by the ON–ON versus the ON–OFF system, computed for different values of the noise SD. For instance, to transmit 0.9 bits (dashed black line), the ON–ON scheme needs twice more spikes at input noise SD 0.2 and 2.75 times more spikes at input noise SD 0.3. D, The optimal thresholds in the ON–OFF system get pushed toward the tails of the distribution as the input noise increases. E, The optimal thresholds in the ON–ON system are also distributed at the two tails of the distribution for large spike counts but cluster at one end of the stimulus distribution for low spike counts.

Figure 8.

Linear decoding of stimuli from cell responses increases the benefits of ON–OFF splitting. A, B, The SNR of the stimulus reconstruction for two stimulus distributions (Gaussian and Laplace) and subject to two different constraints (maximum expected spike count and total mean spike count). C, D, The ratio of SNR curves shown in A and B under the two different constraints. E, F, The optimal thresholds (shown as the fraction of stimuli below threshold) for the response functions under the two different constraints (compare with Fig. 3B,D). G, The ratio of total mean spike count needed by the ON–ON versus the ON–OFF system to achieve the same SNR. Larger benefits for ON–OFF are observed for the Laplace distribution.

Figure 9.

Schematic of linear decoding of time-varying stimuli from cell responses. A stimulus with temporal correlations was first passed through a biphasic filter and then encoded by an ON–OFF or ON–ON system. The resulting time series of spike counts were convolved with decoding filters to produce the reconstructed stimulus estimate. Examples of the decoding filters for each system (ON–OFF and ON–ON) and the resulting reconstructed stimuli are shown for N_max = 10. Stimulus (black), stimulus estimate from ON–OFF (blue), and stimulus estimate from ON–ON (red).

Figure 10.

Decoding of temporally correlated stimuli benefits from ON–OFF splitting. A, B, The SNR from linear reconstruction of a dynamic stimulus while constraining the maximum expected spike count (A) or the total mean spike count (B). C, D, The SNR achieved by the ON–OFF versus the ON–ON system under the conditions of A and B. E, F, The optimal thresholds (shown as the fraction of stimuli below threshold) for the response functions of the ON–OFF and ON–ON systems of A and B. G, The ratio of total spike counts used by the ON–ON versus the ON–OFF system to achieve the same SNR.

Figure 11.

Comparing the effects of ON–OFF splitting and decorrelation. A, B, Two neurons encode almost identical stimulus variables. As shown in the previous figures, ON–OFF coding enhances the transmitted information by ∼25%. C, Here the two stimulus variables are first decorrelated by lateral inhibition to statistical independence and then encoded by two neurons. Under best conditions, this transmits 2 bits, a 26% increase over the log₃(2) bits in B.

Figure 12.

Sparse stimulus distributions enhance the advantage of ON–OFF splitting. A, Left, A stimulus distribution consisting of three events, s = −1, 0, 1, that occur with different probabilities p₋₁, p₀, p₁ = p₋₁, indicated by the bars. Right, Schematic responses of an ON–OFF system (top) and an ON–ON system (bottom) to such a stimulus. B, The mutual information for the ON–OFF and ON–ON systems as a function of the total mean spike count. Here the sparseness of the distribution is 1/p₁ = 100. Given a desired level of information (horizontal dashed line), the ON–ON system needs 95 times higher firing rate than the ON–OFF system for the sparse stimulus in A (compare with Fig. 3). C, The ratio of the overall mean spike count used by the ON–ON versus the ON–OFF system as a function of the mutual information. The vertical dashed line corresponds to the dashed line in B.