Significance: Prediction is an essential part of life. However, are we really “good” at making predictions? More specifically, are pieces of our brain close to being optimal predictors? To assess the efficiency of prediction, we need to measure the information that neurons carry about the future of our sensory experiences. We show how to do this, at least in simplified contexts, and find that groups of neurons in the retina indeed are close to maximally efficient at separating predictive information from the nonpredictive background. Efficient coding of predictive information is a principle that can be applied at every stage of neural computation.

The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics. We construct this relationship for populations of up to N = 160 neurons in a small patch of the vertebrate retina, using a combination of direct and model-based analyses of experiments on the response of this network to naturalistic movies. We see signs of a thermodynamic limit, where the entropy per neuron approaches a smooth function of the energy per neuron as N increases. The form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point. We suggest further tests of criticality, and give a brief discussion of its functional significance. Significance The activity of a brain—or even a small region of a brain devoted to a particular task—cannot be just the summed activity of many independent neurons. Here we use methods from statistical physics to describe the collective activity in the retina as it responds to complex inputs such as those encountered in the natural environment. We find that the distribution of messages that the retina sends to the brain is very special, mathematically equivalent to the behavior of a material near a critical point in its phase diagram.

We study the classical Heisenberg model for spins on a pyrochlore lattice interacting via long-range dipole-dipole forces and nearest-neighbor exchange. Antiferromagnetic exchange alone is known not to induce ordering in this system. We analyze low-temperature order resulting from the combined interactions, both by using a mean-field approach and by examining the energy cost of fluctuations about an ordered state. We discuss behavior as a function of the ratio of the dipolar and exchange interaction strengths and find two types of ordered phase. Below a certain value of this ratio, we find that the system orders in a four-sublattice Néel state. For interaction strengths above this critical ratio, the system orders with an incommensurate wave vector. We relate our results to the recent experimental work and reproduce and extend the theoretical calculations on the pyrochlore compound, ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7,$ by N. P. Raju, M. Dion, M. J. P. Gingras, T. E. Mason, and J. E. Greedan, Phys. Rev. B 59, 14 489 (1999).

We have used a combination of theory and experiment to assess how information is represented in a realistic cortical population response, examining how motion direction and timing is encoded in groups of neurons in cortical area MT. Combining data from several single-unit experiments, we constructed model population responses in small time windows and represented the response in each window as a binary vector of 1s or 0s signifying spikes or no spikes from each cell. We found that patterns of spikes and silence across a population of nominally redundant neurons can carry up to twice as much information about visual motion than does population spike count, even when the neurons respond independently to their sensory inputs. This extra information arises by virtue of the broad diversity of firing rate dynamics found in even very similarly tuned groups of MT neurons. Additionally, specific patterns of spiking and silence can carry more information than the sum of their parts (synergy), opening up the possibility for combinatorial coding in cortex. These results also held for populations in which we imposed levels of nonindependence (correlation) comparable to those found in cortical recordings. Our findings suggest that combinatorial codes are advantageous for representing stimulus information on short time scales, even when neurons have no complicated, stimulus-dependent correlation structure.

Neuronal activity in the brain is variable, yet both perception and behavior are generally reliable. How does the brain achieve this? Here, we show that the conjunctive coding of multiple stimulus features, commonly known as nonlinear mixed selectivity, may be used by the brain to support reliable information transmission using unreliable neurons. Nonlinearly mixed feature representations have been observed throughout primary sensory, decision-making, and motor brain areas. In these areas, different features are almost always nonlinearly mixed to some degree, rather than represented separately or with only additive (linear) mixing, which we refer to as pure selectivity. Mixed selectivity has been previously shown to support flexible linear decoding for complex behavioral tasks. Here, we show that it has another important benefit: in many cases, it makes orders of magnitude fewer decoding errors than pure selectivity even when both forms of selectivity use the same number of spikes. This benefit holds for sensory, motor, and more abstract, cognitive representations. Further, we show experimental evidence that mixed selectivity exists in the brain even when it does not enable behaviorally useful linear decoding. This suggests that nonlinear mixed selectivity may be a general coding scheme exploited by the brain for reliable and efficient neural computation. Neurons in the brain are unreliable, while both perception and behavior are generally reliable. In this work, we study how the neural population response to sensory, motor, and cognitive features can produce this reliability. Across the brain, single neurons have been shown to respond to particular conjunctions of multiple features, termed nonlinear mixed selectivity. In this work, we show that populations of these mixed selective neurons lead to many fewer decoding errors than populations without mixed selectivity, even when both neural codes are given the same number of spikes. We show that the reliability benefits from mixed selectivity are quite general, holding under different assumptions about metabolic costs and neural noise as well as for both categorical and sensory errors. Further, previous theoretical work has shown that mixed selectivity enables the learning of complex behaviors with simple decoders. Through the analysis of neural data, we show that the brain implements mixed selectivity even when it would not serve this purpose. Thus, we argue that the brain also implements mixed selectivity to exploit its general benefits for reliable and efficient neural computation.

Significance: Supervised learning algorithms can learn subtle features that distinguish one class of input examples from another. We explore a supervised training framework in which mechanical metamaterials physically learn to distinguish different classes of forces by exploiting plasticity and nonlinearities in the material. After a period of training with examples of forces, the material can respond correctly to previously unseen novel forces that share spatial correlation patterns with the training examples. Such generalization can allow mechanical parts of microelectronics and adaptive robotics to learn to distinguish patterns of force stimuli on the fly. Our work shows how learning and generalization are not restricted to software algorithms, but can naturally emerge from plasticity and nonlinearities in elastic materials.