Recent psychophysical experiments indicate that humans perform near-optimal Bayesian inference in a wide variety of tasks, ranging from cue integration to decision making to motor control. This implies that neurons both represent probability distributions and combine those distributions according to a close approximation to Bayes' rule. At first sight, it would seem that the high variability in the responses of cortical neurons would make it difficult to implement such optimal statistical inference in cortical circuits. We argue that, in fact, this variability implies that populations of neurons automatically represent probability distributions over the stimulus, a type of code we call probabilistic population codes. Moreover, we demonstrate that the Poisson-like variability observed in cortex reduces a broad class of Bayesian inference to simple linear combinations of populations of neural activity. These results hold for arbitrary probability distributions over the stimulus, for tuning curves of arbitrary shape and for realistic neuronal variability.

/pdf/bayesian-inference-with-probabilistic-population-codes-38n29soax4.pdf

Bayesian inference with probabilistic population codes.

We study the impact of correlated neuronal firing rate variability on the accuracy with which an encoded quantity can be extracted from a population of neurons. Contrary to widespread belief, correlations in the variabilities of neuronal firing rates do not, in general, limit the increase in coding accuracy provided by using large populations of encoding neurons. Furthermore, in some cases, but not all, correlations improve the accuracy of a population code.

/pdf/the-effect-of-correlated-variability-on-the-accuracy-of-a-4guz23yv43.pdf

The effect of correlated variability on the accuracy of a population code

https://homepages.cwi.nl/~sbohte/publication/backprop.pdf

Error-backpropagation in temporally encoded networks of spiking neurons

Physical variables such as the orientation of a line in the visual field or the location of the body in space are coded as activity levels in populations of neurons. Reconstruction or decoding is an inverse problem in which the physical variables are estimated from observed neural activity. Reconstruction is useful first in quantifying how much information about the physical variables is present in the population and, second, in providing insight into how the brain might use distributed representations in solving related computational problems such as visual object recognition and spatial navigation. Two classes of reconstruction methods, namely, probabilistic or Bayesian methods and basis function methods, are discussed. They include important existing methods as special cases, such as population vector coding, optimal linear estimation, and template matching. As a representative example for the reconstruction problem, different methods were applied to multi-electrode spike train data from hippocampal place cells in freely moving rats. The reconstruction accuracy of the trajectories of the rats was compared for the different methods. Bayesian methods were especially accurate when a continuity constraint was enforced, and the best errors were within a factor of two of the information-theoretic limit on how accurate any reconstruction can be and were comparable with the intrinsic experimental errors in position tracking. In addition, the reconstruction analysis uncovered some interesting aspects of place cell activity, such as the tendency for erratic jumps of the reconstructed trajectory when the animal stopped running. In general, the theoretical values of the minimal achievable reconstruction errors quantify how accurately a physical variable is encoded in the neuronal population in the sense of mean square error, regardless of the method used for reading out the information. One related result is that the theoretical accuracy is independent of the width of the Gaussian tuning function only in two dimensions. Finally, all the reconstruction methods considered in this paper can be implemented by a unified neural network architecture, which the brain feasibly could use to solve related problems.

/pdf/interpreting-neuronal-population-activity-by-reconstruction-4kltooosz2.pdf

Interpreting neuronal population activity by reconstruction : unified framework with application to hippocampal place cells

The problem of predicting the position of a freely foraging rat based on the ensemble firing patterns of place cells recorded from the CA1 region of its hippocampus is used to develop a two-stage statistical paradigm for neural spike train decoding. In the first, or encoding stage, place cell spiking activity is modeled as an inhomogeneous Poisson process whose instantaneous rate is a function of the animal’s position in space and phase of its theta rhythm. The animal’s path is modeled as a Gaussian random walk. In the second, or decoding stage, a Bayesian statistical paradigm is used to derive a nonlinear recursive causal filter algorithm for predicting the position of the animal from the place cell ensemble firing patterns. The algebra of the decoding algorithm defines an explicit map of the discrete spike trains into the position prediction. The confidence regions for the position predictions quantify spike train information in terms of the most probable locations of the animal given the ensemble firing pattern. Under our inhomogeneous Poisson model position was a three to five times stronger modulator of the place cell spiking activity than theta phase in an open circular environment. For animal 1 (2) the median decoding error based on 34 (33) place cells recorded during 10 min of foraging was 8.0 (7.7) cm. Our statistical paradigm provides a reliable approach for quantifying the spatial information in the ensemble place cell firing patterns and defines a generally applicable framework for studying information encoding in neural systems.

https://www.jneurosci.org/content/jneuro/18/18/7411.full.pdf

A Statistical Paradigm for Neural Spike Train Decoding Applied to Position Prediction from Ensemble Firing Patterns of Rat Hippocampal Place Cells

In perceptual systems, a stimulus parameter can be extracted by determining the center-of-gravity of the response profile of a population of neural sensors. Likewise at the motor end of a neural system, center-of-gravity decoding, also known as vector decoding, generates a movement direction from the neural activation profile. We evaluate these schemes from a statistical perspective, by comparing their statistical variance with the minimum variance possible for an unbiased parameter extraction from the noisy neuronal ensemble activation profile. Center-of-gravity decoding can be statistically optimal. This is the case for regular arrays of sensors with gaussian tuning profiles that have an output described by Poisson statistics, and for arrays of sensors with a sinusoidal tuning profile for the (angular) parameter estimated. However, there are also many cases in which center-of-gravity decoding is highly inefficient. This includes the important case where sensor positions are very irregular. Finally, we study the robustness of center-of-gravity decoding against response nonlinearities at different stages of an information processing hierarchy. We conclude that, in neural systems, instead of representing a parameter explicitly, it is safer to leave the parameter coded implicitly in a neuronal ensemble activation profile.

Herman P. Snippe

Papers

Parameter extraction from population codes: a critical assessment