A theory of brain-computer interface learning via low-dimensional control.

A remarkable demonstration of the flexibility of mammalian motor systems is primates' ability to learn to control brain-computer interfaces (BCIs). This constitutes a completely novel motor behavior, yet primates are capable of learning to control BCIs under a wide range of conditions. BCIs with carefully calibrated decoders, for example, can be learned with only minutes to hours of practice. With a few weeks of practice, even BCIs with randomly constructed decoders can be learned. What are the biological substrates of this learning process? Here, we develop a theory based on a re-aiming strategy, whereby learning operates within a low-dimensional subspace of task-relevant inputs driving the local population of recorded neurons. Through comprehensive numerical and formal analysis, we demonstrate that this theory can provide a unifying explanation for disparate phenomena previously reported in three different BCI learning tasks, and we derive a novel experimental prediction that we verify with previously published data. By explicitly modeling the underlying neural circuitry, the theory reveals an interpretation of these phenomena in terms of biological constraints on neural activity.

Probabilistic population codes and the exponential family of distributions.

Many experiments have shown that human behavior is nearly Bayes optimal in a variety of tasks. This implies that neural activity is capable of representing both the value and uncertainty of a stimulus, if not an entire probability distribution, and can also combine such representations in an optimal manner. Moreover, this computation can be performed optimally despite the fact that observed neural activity is highly variable (noisy) on a trial-by-trial basis. Here, we argue that this observed variability is actually expected in a neural system which represents uncertainty. Specifically, we note that Bayes' rule implies that a variable pattern of activity provides a natural representation of a probability distribution, and that the specific form of neural variability can be structured so that optimal inference can be executed using simple operations available to neural circuits.

The rat approximates an ideal detector of changes in rates of reward: implications for the law of effect.

Rats responded on 2 levers delivering brain stimulation reward on concurrent variable interval schedules. Following many successive sessions with unchanging relative rates of reward, subjects adjusted to an eventual change slowly and showed spontaneous reversions at the beginning of subsequent sessions. When changes in rates of reward occurred between and within every session, subjects adjusted to them about as rapidly as they could in principle do so, as shown by comparison to a Bayesian model of an ideal detector. This and other features of the adjustments to frequent changes imply that the behavioral effect of reinforcement depends on the subject's perception of incomes and changes in incomes rather than on the strengthening and weakening of behaviors in accord with their past effects or expected results. Models for the process by which perceived incomes determine stay durations and for the process that detects changes in rates are developed.

Efficient computation and cue integration with noisy population codes.

The brain represents sensory and motor variables through the activity of large populations of neurons. It is not understood how the nervous system computes with these population codes, given that individual neurons are noisy and thus unreliable. We focus here on two general types of computation, function approximation and cue integration, as these are powerful enough to handle a range of tasks, including sensorimotor transformations, feature extraction in sensory systems and multisensory integration. We demonstrate that a particular class of neural networks, basis function networks with multidimensional attractors, can perform both types of computation optimally with noisy neurons. Moreover, neurons in the intermediate layers of our model show response properties similar to those observed in several multimodal cortical areas. Thus, basis function networks with multidimensional attractors may be used by the brain to compute efficiently with population codes.

Retinal ganglion cells act largely as independent encoders.

Correlated firing among neurons is widespread in the visual system. Neighbouring neurons, in areas from retina to cortex, tend to fire together more often than would be expected by chance. The importance of this correlated firing for encoding visual information is unclear and controversial. Here we examine its importance in the retina. We present the retina with natural stimuli and record the responses of its output cells, the ganglion cells. We then use information theoretic techniques to measure the amount of information about the stimuli that can be obtained from the cells under two conditions: when their correlated firing is taken into account, and when their correlated firing is ignored. We find that more than 90% of the information about the stimuli can be obtained from the cells when their correlated firing is ignored. This indicates that ganglion cells act largely independently to encode information, which greatly simplifies the problem of decoding their activity.

Intrinsic dynamics in neuronal networks. I. Theory.

Many networks in the mammalian nervous system remain active in the absence of stimuli. This activity falls into two main patterns: steady firing at low rates and rhythmic bursting. How are these firing patterns generated? Specifically, how do dynamic interactions between excitatory and inhibitory neurons produce these firing patterns, and how do networks switch from one firing pattern to the other? We investigated these questions theoretically by examining the intrinsic dynamics of large networks of neurons. Using both a semianalytic model based on mean firing rate dynamics and simulations with large neuronal networks, we found that the dynamics, and thus the firing patterns, are controlled largely by one parameter, the fraction of endogenously active cells. When no endogenously active cells are present, networks are either silent or fire at a high rate; as the number of endogenously active cells increases, there is a transition to bursting; and, with a further increase, there is a second transition to steady firing at a low rate. A secondary role is played by network connectivity, which determines whether activity occurs at a constant mean firing rate or oscillates around that mean. These conclusions require only conventional assumptions: excitatory input to a neuron increases its firing rate, inhibitory input decreases it, and neurons exhibit spike-frequency adaptation. These conclusions also lead to two experimentally testable predictions: 1) isolated networks that fire at low rates must contain endogenously active cells and 2) a reduction in the fraction of endogenously active cells in such networks must lead to bursting.

Intrinsic dynamics in neuronal networks. II. experiment.

Neurons in many regions of the mammalian CNS remain active in the absence of stimuli. This activity falls into two main patterns: steady firing at low rates and rhythmic bursting. How these firing patterns are maintained in the presence of powerful recurrent excitation, and how networks switch between them, is not well understood. In the previous paper, we addressed these issues theoretically; in this paper we address them experimentally. We found in both studies that a key parameter in controlling firing patterns is the fraction of endogenously active cells. The theoretical analysis indicated that steady firing rates are possible only when the fraction of endogenously active cells is above some threshold, that there is a transition to bursting when it falls below that threshold, and that networks becomes silent when the fraction drops to zero. Experimentally, we found that all steadily firing cultures contain endogenously active cells, and that reducing the fraction of such cells in steadily firing cultures causes a transition to bursting. The latter finding implies indirectly that the elimination of endogenously active cells would cause a permanent drop to zero firing rate. The experiments described here thus corroborate the theoretical analysis.

Reading population codes: a neural implementation of ideal observers.

Many sensory and motor variables are encoded in the nervous system by the activities of large populations of neurons with bell-shaped tuning curves. Extracting information from these population codes is difficult because of the noise inherent in neuronal responses. In most cases of interest, maximum likelihood (ML) is the best read-out method and would be used by an ideal observer. Using simulations and analysis, we show that a close approximation to ML can be implemented in a biologically plausible model of cortical circuitry. Our results apply to a wide range of nonlinear activation functions, suggesting that cortical areas may, in general, function as ideal observers of activity in preceding areas.

Narrow versus wide tuning curves: What's best for a population code?

Neurophysiologists are often faced with the problem of evaluating the quality of a code for a sensory or motor variable, either to relate it to the performance of the animal in a simple discrimination task or to compare the codes at various stages along the neuronal pathway. One common belief that has emerged from such studies is that sharpening of tuning curves improves the quality of the code, although only to a certain point; sharpening beyond that is believed to be harmful. We show that this belief relies on either problematic technical analysis or improper assumptions about the noise. We conclude that one cannot tell, in the general case, whether narrow tuning curves are better than wide ones; the answer depends critically on the covariance of the noise. The same conclusion applies to other manipulations of the tuning curve profiles such as gain increase.

Population coding in the retina.

Recent advances in multi-electrode recording have brought us closer to understanding how visual information is encoded by populations of retinal ganglion cells. By monitoring the visual responses of many ganglion cells at once, it is now possible to examine how ganglion cells act together to encode a visual scene.

Coding strategies in monkey V1 and inferior temporal cortices.

We would like to know whether the statistics of neuronal responses vary across cortical areas. We examined stimulus-elicited spike count response distributions in V1 and inferior temporal (IT) cortices of awake monkeys. In both areas, the distribution of spike counts for each stimulus was well described by a Gaussian distribution, with the log of the variance in the spike count linearly related to the log of the mean spike count. Two significant differences in response characteristics were found: both the range of spike counts and the slope of the log(variance) versus log(mean) regression were larger in V1 than in IT. However, neurons in the two areas transmitted approximately the same amount of information about the stimuli and had about the same channel capacity (the maximum possible transmitted information given noise in the responses). These results suggest that neurons in V1 use more variable signals over a larger dynamic range than IT neurons, which use less variable signals over a smaller dynamic range. The two coding strategies are approximately as effective in transmitting information.

Statistically efficient estimation using population coding.

Coarse codes are widely used throughout the brain to encode sensory and motor variables. Methods designed to interpret these codes, such as population vector analysis, are either inefficient (the variance of the estimate is much larger than the smallest possible variance) or biologically implausible, like maximum likelihood. Moreover, these methods attempt to compute a scalar or vector estimate of the encoded variable. Neurons are faced with a similar estimation problem. They must read out the responses of the presynaptic neurons, but, by contrast, they typically encode the variable with a further population code rather than as a scalar. We show how a nonlinear recurrent network can be used to perform estimation in a near-optimal way while keeping the estimate in a coarse code format. This work suggests that lateral connections in the cortex may be involved in cleaning up uncorrelated noise among neurons representing similar variables.