Session W7: Nonlinear Dynamics of Neural Systems: A Statistical Mechanics of the Brain

Representing Information with Correlated Neural Populations

Using a multi-electrode array, we can record simultaneously from up to 50 ganglion cells, the output neurons of the retina, while stimulating with varied visual images generated on a computer monitor. We find that nearby ganglion cells, the output neurons of the retina, have spatial receptive fields that overlap significantly, leading to correlated firing and redundancy in the visual information that cells encode. Although the strength of correlations among pairs of cells is weak ($\sim$10\%), the effect in larger populations is dramatic: patterns of spiking and silence in groups of just 10 cells can occur with a probability $\sim$100,000-fold different from that predicted from statistical independence. We show that these strong network correlations can be explained by a model that includes all pairwise correlations, but no higher-order statistics. This model is identical to the Ising model, and predicts that larger populations may exhibit a form of freezing transition that allows for robust error correction. We have begun to explore these error-correcting properties in simple visual discrimination tasks using large populations of ganglion cells. We find that the correlations among neurons can dramatically reduce the discrimination error.   

Exploring the network structure of concerted activity in the primate retina using maximum entropy methods

All visual signals in the brain originate in the electrical activity of retinal ganglion cells (RGCs). Standard models implicitly assume that RGCs signal information independently of one another. However, several studies have demonstrated that significant concerted activity in pairs of RGCs may fundamentally alter visual signals. We recorded the electrical activity of several hundred RGCs in peripheral monkey retina. The regular mosaic organization of RGCs indicated that we recorded from nearly every cell in a region of the visual field. In the presence of constant illumination, pairs of RGCs fired synchronously several-fold more often than expected by chance, indicating significant network interactions. Synchrony was localized and universal amongst cells of the same type indicating that it arises from local and highly stereotyped circuitry. To test whether concerted firing can be explained by known pairwise interactions, we used a maximum entropy approach borrowed from statistical mechanics to predict concerted activity. The model accurately reproduced the data. This suggests that network interactions in the primate retina are well approximated by a nearest neighbor Ising model and concerted activity can be understood based on local interactions within a neural population.   

A continuous phase transition in neocortical slice networks

Recent experiments demonstrate that activity in neocortical circuits can propagate in the form of avalanches whose sizes follow a power law distribution, suggesting that these circuits operate near a continuous phase transition point. Computational models indicate that this critical point may be optimal for information processing. However, the existence of a power law is not sufficient to establish critical behavior because stochastic processes that do not undergo phase transitions can also produce power laws. We recorded power law distributions of neuronal avalanches from neocortical slices and then pharmacologically perturbed network activity. We then measured deviations from power law behavior. Here we show that these deviations covaried systematically with a control parameter, as would be expected for a continuous phase transition. A critical branching model captures this transition, while stochastic models do not. Our findings imply that the physical theory underlying continuous phase transitions can be fruitfully applied to neocortical circuits.   

Patterns of Neural Activity in Networks with Complex Connectivity

An understanding of emergent dynamics on complex networks requires investigating the interplay between the intrinsic dynamics of the node elements and the connectivity of the network in which they are embedded. In order to address some of these questions in a specific scenario of relevance to the dynamical states of neural ensembles, we have studied the collective behavior of excitable model neurons in a network with small-world topology. The small-world network has local lattice order, but includes a number of randomly placed connections that may provide connectivity shortcuts. This topology bears a schematic resemblance to the connectivity of the cerebral cortex, in which neurons are most strongly coupled to nearby cells within fifty to a hundred micrometers, but also make projections to cells millimeters away. We find that the dynamics of this small-world network of excitable neurons depend mostly on both the density of shortcuts and the delay associated with neuronal projections. In the regime of low shortcut density, the system exhibits persistent activity in the form of propagating waves, which annihilate upon collision and are spawned anew via the re-injection of activity through shortcut connections. As the density of shortcuts reaches a critical value, the system undergoes a transition to failure. The critical shortcut density results from matching the time associated with a recurrent path through the network to an intrinsic recovery time of the individual neurons. Furthermore, if the delay associated with neuronal interactions is sufficiently long, activity reemerges above the critical density of shortcuts. The activity in this regime exhibits long, chaotic transients composed of noisy, large-amplitude population bursts.   

Optimal processing and the statistics of visual input signals

Sensory information processing can be seen as a statistical estimation problem, where relevant features are extracted from a raw stream of sensory input containing an imperfect representation of those features. Broadly speaking, the optimal solution to the feature extraction problem depends on the statistical structure of those input signals. Here we study the statistics of natural visual input signals, and the optimal solution to the problem of visual motion detection. Motion detection is a biologically important feature estimation problem, as many animals use vision to estimate their motion through space. Many years ago, Reichardt and Poggio drew attention to two important aspects of this problem: First, computing motion from an array of photosensors is an irreducibly nonlinear operation, and second, biological versions of this operation seem mathematically tractable. To paraphrase, the problem is interesting but not hopelessly complicated. In this spirit I will discuss motion estimation in the visual system of the blowfly, with an emphasis on performance under natural conditions. As noted above, the array of photoreceptors in the retina implicitly contains data on self motion, but this relation is noisy, indirect and ambiguous due to photon shot noise and optical blurring, and also as a result of the structure of the natural environment. Further, natural variations in the visual signal to noise ratio are enormous, and nonlinear operations are especially susceptible to noise. One can therefore reasonably hope that animals have evolved interesting optimization strategies to deal with large variations in signal quality. I will present experimental data, both from sampling natural probability distributions, and from motion sensitive neurons in the fly brain, that illustrate some of these solutions and that suggest that the fly indeed approaches optimality. The implications of these findings and their possible generalizations will be discussed.