High-Fidelity Coding with Correlated Neurons

Positive correlations in the activity of neurons are widely observed in the brain. Previous studies have shown these correlations to be detrimental to the fidelity of population codes or at best marginally favorable compared to independent codes. Here, we show that positive correlations can enhance coding performance by astronomical factors. Specifically, the probability of discrimination error can be suppressed by many orders of magnitude. Likewise, the number of stimuli encoded--the capacity--can be enhanced by similarly large factors. These effects do not necessitate unrealistic correlation values and can occur for populations with a few tens of neurons. We further show that both effects benefit from heterogeneity commonly seen in population activity. Error suppression and capacity enhancement rest upon a pattern of correlation. In the limit of perfect coding, this pattern leads to a `lock-in' of response probabilities that eliminates variability in the subspace relevant for stimulus discrimination. We discuss the nature of this pattern and suggest experimental tests to identify it.

neuron i, the variance of the population activity is (1) where brackets indicate an average over trials. The …rst sum on the right-handside pertains to ‡uctuations in single-neuron responses and is non-vanishing in both independent and correlated cases. The second sum on the right-handside pertains to correlations among neurons. For negative correlations (anticorrelations), this sum is negative and, hence, the distribution of neural activity is more narrowly peaked than in the independent case. By contrast, positive correlations broaden the distribution. In the anti-correlated case, distributions of population activity corresponding to di¤erent stimuli tend to be well separated, while in the positively correlated case, overlaps tend to be greater. Therefore, homogeneous populations with positive correlation have worse coding performance than corresponding independent populations and, consequently, require more neurons to achieve a low rate of coding errors.
We can understand the hindrance of coding performance from positive correlations in an alternate, simple fashion. A homogeneous population with positive correlation behaves, e¤ectively, as a smaller population. In the limiting case of a perfectly correlated population in which all neurons respond identically, the entire population behaves as one, big neuron. Hence, we expect such positively correlated populations to code information with less 'resolution' and, consequently, to commit coding errors more often than corresponding independent populations do.

High-…delity coding bare bones
In the companion paper we demonstrate, quantitatively and with the use of simple models, that positive correlation can suppress coding errors and enhance coding capacity massively. The basic mechanism behind this e¤ect was noted by a number of authors [7,15,1,13,2] and is simple to understand: positive correlations can deform the shape of probability distributions of neural activity in such a way as to sharpen the distinction between nearby probability distributions (Fig. 1B). Put di¤erently, while positive correlations have a broadening e¤ect overall, they can nonetheless suppress the tails of probability distributions along relevant directions, thereby reducing the unfavorable e¤ect of neural variability.
The same idea can be expressed in a more general fashion: the structure of correlation can be such that it relegates noise into a non-informative mode of the neural population response. A simple example provides a nice illustration ( [11]; a similar argument is presented in Ref. [1]). Consider two neurons with responses r 1 = m 1 + 1 ; (2) We assume that the mean responses, m 1 and m 2 , are di¤erent, such that m 1 (m 2 ) is large (small) in response to the Target stimulus, and vice versa for the Distracter stimulus. The additive variabilities, 1 and 2 , are highly correlated, such that 1 2 . Then the informative mode, is close to noiseless, while all the noise is relegated to the uninformative mode, Our results can also be viewed in terms of a similar mechanism: informative and uninformative modes correspond to combinations of pool spike counts, the k i s, and given patterns of positive correlations relegate variability to the uninformative modes. In the simplest, symmetric, 2-pool model, correlation sharpens the response distributions along the informative mode, k 1 k 2 , while it blurs them along the uninformative mode, k 1 + k 2 . Clearly, it is a signature of correlated coding that informative modes can be identi…ed only when simultaneous activities of the neurons in the population are considered.

Learning in read-out circuits
The fact that realistic values of correlation can yield highly accurate coding in small population suggests a picture in which individual neurons in a 'higherlevel'area read out information from a very small subset of the neurons in the 'lower-level' area. The question then becomes: how does the brain …nd the appropriate neurons form which to extract relevant information? This question is unsolved, but experiments on brain-machine interfaces demonstrate that the brain has a truly remarkable ability to change its circuitry in sensory-motor pathways to activate the relevant motor neurons. In an experiment, 100 neurons in primary motor cortex were recorded and their responses were used to drive movements of cursors or even robotic arms with simple cosine tuning functions [14]. Under these circumstances, monkeys were able to achieve high performance in directing movements. Most impressively, the authors showed that they could re-arrange the tuning curves used to translate neural activity into movements of the robotic arm, and monkeys could change their entire sensory-motor pathway in order to …re those particular neurons in the right pattern to achieve the desired movement [6]. In many cases, the tuning curves were completely inverted, and yet the monkeys re-learned how to …re those neurons appropriately. Analogous results have been reproduced by another lab [5]. A related example comes from experiments in which human subject wear inverting prism glasses. Initially, the world appears upside-down, resulting in profound motor de…cits and disorientation. But after about a week, subjects regain their coordination, evidently requiring a complete remapping (inversion) of visual stimuli to motor outputs [12].
So, while the brain faces great di¢ culties in obtaining useful information encoded by sensory circuits and must be subject to certain limits in accomplishing these, it is clear that the brain has a remarkable ability to surmount these di¢culties in many situations. We currently have very little understanding of how the brain manages this, and hence we really don't know at this point what the limitations are.