Guide Advances in Neural Population Coding

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The horizontal line indicates the weight value of zero. Middle : The intensity map of the same receptive field. The bright and dark colors indicate positive and negative weight values, respectively, and the medium gray color indicates zero. Superimposed is the outline of the center subregion the contour defined by the half-height from the peak along with the average number of pixels cones photoreceptors inside the contour.

Bottom : The half-height contours of the entire neural population which displays their tiling in the visual field.

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Two neurons are highlighted for clarity one of which corresponds to the neuron shown above. The pixel lattice is depicted by the orange grid. This can be expressed in the spectral domain as the transition from low-pass to band-pass filtering cf. Figure 6. As a result, the overlap of the central region of the receptive fields is very large at the lower SNR, implying that neighboring neurons are transmitting information about a highly overlapped region of pixels at the expense of transmitting independent information. This overlap, however, is optimal for counteracting the high level of sensory noise and encoding the underlying original signal cf.

Figure 4.

In the periphery condition, a similar adaptive change was observed but to a lesser extent. The shape of the receptive field looks similar across all sensory SNRs. As was seen in the spectral analysis Figure 6 , the degree of adaptation is limited by the highly convergent cone-to-RGC ratio. In this article we presented a simple theoretical model of optimal population coding that incorporates several key aspects of sensory systems.

The model is analytically well characterized Figure 6 ; see also Text S1 , Figures S1 — S5 and scales to systems with high input dimensionality Figures 4 — 5. We found that the optimal code conveys significantly more information about the underlying environmental signal compared to a traditional redundancy reduction model. It has long been argued that some redundancy should be useful [4] , [25] , [27] , [29] — [32] , [56] — [59].

Here we provide a simple and quantitative model that optimally incorporates redundancy in a neural population under a wide range of settings. In contrast to earlier studies [24] — [27] , [56] , [60] , the proposed model allows for an arbitrary number of neurons in a population, providing previously unavailable insights and predictions: the degree to and the mechanisms by which the error can be minimized with different input-to-output cell ratios Figure 6 ; the conditions in which the redundancy reduction model is near-optimal Figure 5 ; the degree of adaptation of receptive fields at different eccentricities to different light levels Figure 8.

We observed that the optimal receptive fields are non-unique, as in other models [8] , [25] , [59] — [61] , and found that the additional constraint of spatial locality of the computation [25] , but not previously examined constraints such as sparse weights [41] or sparse responses [7] , [8] , yielded receptive fields similar to those found in the retina Figure 7. A number of other studies have also investigated different optimal coding models that extended the basic idea of redundancy reduction, but with different assumptions and conditions. A commonly assumed objective is information maximization , which maximizes the number of discriminable states about the environmental signal in the neural code [6] , [25] , [27] , [56] , [57] , [59] , [62] — [64] , whereas the present study assumed error minimization , which minimizes the MSE of reconstruction from the neural code [24] , [31].

These objectives can be interpreted as different mathematical approaches to the same general goal some predictions from these different objectives are qualitatively similar [24] , [62] ; an equivalence can be established between the two under some settings [65]. Recently, Doi et al. Some consequences that arise from the choice of the objective are worth mentioning. One is that de-blurring emerges from error minimization but not from those information maximization models [25] , [27] , [59] , because the error is defined with respect to the original signal prior to blurring.

In [25] , [27] , [59] , the information is defined with respect to the original signal, but it is equivalent to the information about the blurred signal under the model assumptions eq.

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Another is that, in the limit of zero sensory noise, the optimal neural transform for information maximization is whitening i. Although this assumption may be valid in some specific settings such as in the fovea [25] , there are many settings in which this assumption is not valid, such as in the periphery Figure 2. By being able to vary the cell ratio to match the conditions of the system of interest, the proposed model showed that the retinal transform of sensory signals and the resulting redundancy in neural representations vary with the retinal eccentricity.

Another common assumption related to the cell ratio is that neural encoding is the inverse of the data generative process [7] , [8] , where individual neurons are noiseless and represent independent features or intrinsic coordinates of the signal space. In this view, the number of neurons should match the intrinsic dimensionality of the signal. In contrast, in the proposed model the number of neurons may be seen as a parameter for total neural capacity and can be varied independently of the signal's intrinsic dimensionality. Consequently, it is even possible that, while representing an identical signal source, two neurons in the proposed model adaptively change what they represent by changing their receptive fields with different sensory or neural noise levels Figures S3—S4; notably, two neurons can have identical receptive fields in some extreme cases.

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While the current study is based on several simplifying assumptions such as linear neurons with white gaussian neural noise, some recent studies have incorporated more realistic neural properties to investigate the optimality of retinal coding, so it is important to contrast these with the proposed model.

Borghuis et al. This is consistent with the prediction of the proposed model under the retinal conditions they studied i. However, the model presented here predicts that the spacing is not optimal in all conditions Figure 8. Also note that the center-surround structure in their study was assumed, and did not emerge as a result of an optimization as presented here.

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Like in the previous study [57] , the center-surround receptive fields were measured, not derived. In addition, their analysis assumed zero sensory noise, which as we have shown here can play a significant role in the form of retinal codes. While they did not assume additional resource constraints or examine different cone-to-RGC ratios systematically, their predictions in certain conditions are consistent with those presented here.

Some differences are significant, for example, in their model different types of receptive fields were derived under different sensory and neural SNRs. Further investigations are necessary to bring clarity to these differences. Overall, it is fair to say that there is no model that incorporates all aspects of retinal coding with realistic assumptions, and developing such a model is an open problem for future research.


We would point out, however, that there are advantages to simpler models, especially if they can account for important aspects of sensory coding. Some issues that arise with more realistic and more complex models are whether they can be analytically characterized, scale to biologically relevant high-dimensional problems, or provide insights beyond simpler models.

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The proposed model may be seen as a first-order approximation to a complex sensory system and can be used as a base model for developing and comparing to models with more realistic properties. Moreover, the optimization of the model is convex, implying that the optimal solution is guaranteed and can be obtained with standard algorithms.

The proposed model made a novel prediction that the change of receptive field structure and organization with different light levels is much greater in the fovea than in the periphery of the macaque midget RGCs Figure 8. This prediction has not been tested directly because, to the best of our knowledge, all physiological measurements from RGCs with different light levels have carried out either in cat [15] , [54] , [55] or rabbit [67] retinas, where the reported adaptive changes were marginal.

This observation seems to be consistent with our prediction for the periphery, where the cone-to-RGC ratio is high. Note that in the cat retina, the cone-to-RGC ratios specifically with respect to the most numerous beta RGCs range from 30 to across eccentricity [70] ; in the rabbit retina, we estimate the ratio to be greater than , according to the cone density [71] , receptive field sizes, and their tiling [67].

Note also that some studies have reported larger changes in receptive fields sizes [15] , [54] , but these were measured between scotopic and photopic conditions. Like previous approaches, here we have only considered cone photoreceptors which implicitly assumes photopic conditions.

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To include scotopic conditions, one would need to model the rod system [72] , [73] , which has yet to be incorporated into an efficient coding framework. The proposed model incorporated a broad range of properties and constraints for sensory systems. It is an abstract model and hence predictions can be made for a wide range of sensory systems by incorporating system-specific conditions.

Although we have only modeled conditions for the midget RGCs in the macaque retina, the same framework could be applied to other cell types e.

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  6. The model can also be applied to other sensory systems, as nothing in the proposed model is specific to the retina. Auditory systems have been approached in the same framework of efficient coding [74] — [77] , but the factors introduced in this study have not fully been incorporated into previous models. For example, the cell ratio of sensory units inner hair cells to encoding units auditory nerve fibers is [78] , i. Further, the auditory signal is filtered by the head-related transfer function [79] , which could be modeled by the linear distortion in the proposed framework.

    Olfactory systems have also been studied in an efficient coding framework e. It is possible that the optimal redundancy computed with the proposed model may provide insights into olfactory coding beyond decorrelation [81]. Finally, the sensory SNR models the varied intensity of environmental signals relative to the background noise, and the neural SNR models the neural capacity, both of which are broadly relevant. The application of the proposed model to different retinal conditions and other sensory modalities would be a powerful way to investigate common principles of sensory systems.

    We define the linear gaussian model Figure 3 , a functional model of neural responses on which both the proposed and whitening models are constructed. The observed signal is generated by 1 where is the original signal, is a linear distortion in the sensing system such as optical blur in vision or the head-related transfer function in audition, and is the sensory noise with variance , where denotes the -dimensional identity matrix. The covariance of the original signal is defined by. We assume that the original signal is zero mean but need not be gaussian as in [85].

    The sensory SNR is measured in dB, where denotes the trace of a matrix. The neural representation is generated by 2 where is the encoding matrix whose row vectors are the encoding filters or linear receptive fields , and is the neural noise with variance. The neural SNR is also measured in dB, where is the covariance of the observed signal, and is the covariance of the encoded signal,. We set the neural SNR to 10 dB so that its information capacity, 1. The reconstruction of the original signal from the neural representation is computed by a linear transform 3 that minimizes the MSE 4 where indicates sample average and L 2 -norm, given the covariances of signal and noise components in the neural representation i.

    In other words, the decoding matrix is the Wiener filter which estimates the original signal from its degraded version with the linear transform and additive correlated gaussian noise [24] , [47]. The proposed, optimal encoding, , achieves the theoretical limit of the MSE under the linear gaussian model subject to the neural capacity constraint. This constraint can be defined either for the neural population, i. Note eq.

    Importantly, the minimum MSEs under those two conditions are identical [47]. The difference between the two solutions is only in the left orthogonal matrix of the singular value decomposition of the encoding matrix, 7 where is some M -dimensional orthogonal matrix, is a unique diagonal matrix whose diagonal elements are the modulation transfer function or the gain in the spectrum domain of the encoding, and is the eigenvector matrix of the original signal covariance. To summarize, the minimum value of MSE, the coordinates of the encoding , and its power spectrum are uniquely determined and in common with the optimization problems with total or individual power constraints.

    For the derivation of , readers should refer to [47]. The whitening matrix, , removes all the second-order regularities, both of the signal statistics and of the signal blur [48] , and the resulting covariance is the identity matrix with a scaling factor c , 8. This scaling is computed such that the neural capacity constraint is satisfied just as in the proposed model i.