Neural Representations of Probability


Improved probabilistic inference as a general learning mechanism with action video games

C. S. Green and A. Pouget and D. Bavelier

Curr Biol  20  1573-9  (2010)

Action video game play benefits performance in an array of sensory, perceptual, and attentional tasks that go well beyond the specifics of game play [1-9]. That a training regimen may induce improvements in so many different skills is notable because the majority of studies on training-induced learning report improvements on the trained task but limited transfer to other, even closely related, tasks ([10], but see also [11-13]). Here we ask whether improved probabilistic inference may explain such broad transfer. By using a visual perceptual decision making task [14, 15], the present study shows for the first time that action video game experience does indeed improve probabilistic inference. A neural model of this task [16] establishes how changing a single parameter, namely the strength of the connections between the neural layer providing the momentary evidence and the layer integrating the evidence over time, captures improvements in action-gamers behavior. These results were established in a visual, but also in a novel auditory, task, indicating generalization across modalities. Thus, improved probabilistic inference provides a general mechanism for why action video game playing enhances performance in a wide variety of tasks. In addition, this mechanism may serve as a signature of training regimens that are likely to produce transfer of learning.


Cortical circuits for perceptual inference

K. Friston and S. Kiebel

Neural Netw  22  1093-104  (2009)

This paper assumes that cortical circuits have evolved to enable inference about the causes of sensory input received by the brain. This provides a principled specification of what neural circuits have to achieve. Here, we attempt to address how the brain makes inferences by casting inference as an optimisation problem. We look at how the ensuing recognition dynamics could be supported by directed connections and message-passing among neuronal populations, given our knowledge of intrinsic and extrinsic neuronal connections. We assume that the brain models the world as a dynamic system, which imposes causal structure on the sensorium. Perception is equated with the optimisation or inversion of this internal model, to explain sensory input. Given a model of how sensory data are generated, we use a generic variational approach to model inversion to furnish equations that prescribe recognition; i.e., the dynamics of neuronal activity that represents the causes of sensory input. Here, we focus on a model whose hierarchical and dynamical structure enables simulated brains to recognise and predict sequences of sensory states. We first review these models and their inversion under a variational free-energy formulation. We then show that the brain has the necessary infrastructure to implement this inversion and present stimulations using synthetic birds that generate and recognise birdsongs.


Bayesian spiking neurons II: learning

S. Deneve

Neural Comput  20  118-45  (2008)

In the companion letter in this issue ("Bayesian Spiking Neurons I: Inference"), we showed that the dynamics of spiking neurons can be interpreted as a form of Bayesian integration, accumulating evidence over time about events in the external world or the body. We proceed to develop a theory of Bayesian learning in spiking neural networks, where the neurons learn to recognize temporal dynamics of their synaptic inputs. Meanwhile, successive layers of neurons learn hierarchical causal models for the sensory input. The corresponding learning rule is local, spike-time dependent, and highly nonlinear. This approach provides a principled description of spiking and plasticity rules maximizing information transfer, while limiting the number of costly spikes, between successive layers of neurons.


Bayesian spiking neurons I: inference

S. Deneve

Neural Comput  20  91-117  (2008)

We show that the dynamics of spiking neurons can be interpreted as a form of Bayesian inference in time. Neurons that optimally integrate evidence about events in the external world exhibit properties similar to leaky integrate-and-fire neurons with spike-dependent adaptation and maximally respond to fluctuations of their input. Spikes signal the occurrence of new information-what cannot be predicted from the past activity. As a result, firing statistics are close to Poisson, albeit providing a deterministic representation of probabilities.


Neural substrates of reliability-weighted visual-tactile multisensory integration

M. S. Beauchamp and S. Pasalar and T. Ro

Front Syst Neurosci  4  25  (2010)

As sensory systems deteriorate in aging or disease, the brain must relearn the appropriate weights to assign each modality during multisensory integration. Using blood-oxygen level dependent functional magnetic resonance imaging of human subjects, we tested a model for the neural mechanisms of sensory weighting, termed "weighted connections." This model holds that the connection weights between early and late areas vary depending on the reliability of the modality, independent of the level of early sensory cortex activity. When subjects detected viewed and felt touches to the hand, a network of brain areas was active, including visual areas in lateral occipital cortex, somatosensory areas in inferior parietal lobe, and multisensory areas in the intraparietal sulcus (IPS). In agreement with the weighted connection model, the connection weight measured with structural equation modeling between somatosensory cortex and IPS increased for somatosensory-reliable stimuli, and the connection weight between visual cortex and IPS increased for visual-reliable stimuli. This double dissociation of connection strengths was similar to the pattern of behavioral responses during incongruent multisensory stimulation, suggesting that weighted connections may be a neural mechanism for behavioral reliability weighting.


Inference and computation with population codes

A. Pouget and P. Dayan and R. S. Zemel

Annu Rev Neurosci  26  381-410  (2003)

In the vertebrate nervous system, sensory stimuli are typically encoded through the concerted activity of large populations of neurons. Classically, these patterns of activity have been treated as encoding the value of the stimulus (e.g., the orientation of a contour), and computation has been formalized in terms of function approximation. More recently, there have been several suggestions that neural computation is akin to a Bayesian inference process, with population activity patterns representing uncertainty about stimuli in the form of probability distributions (e.g., the probability density function over the orientation of a contour). This paper reviews both approaches, with a particular emphasis on the latter, which we see as a very promising framework for future modeling and experimental work.


Probabilistic population codes and the exponential family of distributions

J. Beck and W. J. Ma and P. E. Latham and A. Pouget

Prog Brain Res  165  509-19  (2007)

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.


Multisensory integration in macaque visual cortex depends on cue reliability

M. L. Morgan and G. C. Deangelis and D. E. Angelaki

Neuron  59  662-73  (2008)

Responses of multisensory neurons to combinations of sensory cues are generally enhanced or depressed relative to single cues presented alone, but the rules that govern these interactions have remained unclear. We examined integration of visual and vestibular self-motion cues in macaque area MSTd in response to unimodal as well as congruent and conflicting bimodal stimuli in order to evaluate hypothetical combination rules employed by multisensory neurons. Bimodal responses were well fit by weighted linear sums of unimodal responses, with weights typically less than one (subadditive). Surprisingly, our results indicate that weights change with the relative reliabilities of the two cues: visual weights decrease and vestibular weights increase when visual stimuli are degraded. Moreover, both modulation depth and neuronal discrimination thresholds improve for matched bimodal compared to unimodal stimuli, which might allow for increased neural sensitivity during multisensory stimulation. These findings establish important new constraints for neural models of cue integration.


Dynamic reweighting of visual and vestibular cues during self-motion perception

C. R. Fetsch and A. H. Turner and G. C. DeAngelis and D. E. Angelaki

J Neurosci  29  15601-12  (2009)

The perception of self-motion direction, or heading, relies on integration of multiple sensory cues, especially from the visual and vestibular systems. However, the reliability of sensory information can vary rapidly and unpredictably, and it remains unclear how the brain integrates multiple sensory signals given this dynamic uncertainty. Human psychophysical studies have shown that observers combine cues by weighting them in proportion to their reliability, consistent with statistically optimal integration schemes derived from Bayesian probability theory. Remarkably, because cue reliability is varied randomly across trials, the perceptual weight assigned to each cue must change from trial to trial. Dynamic cue reweighting has not been examined for combinations of visual and vestibular cues, nor has the Bayesian cue integration approach been applied to laboratory animals, an important step toward understanding the neural basis of cue integration. To address these issues, we tested human and monkey subjects in a heading discrimination task involving visual (optic flow) and vestibular (translational motion) cues. The cues were placed in conflict on a subset of trials, and their relative reliability was varied to assess the weights that subjects gave to each cue in their heading judgments. We found that monkeys can rapidly reweight visual and vestibular cues according to their reliability, the first such demonstration in a nonhuman species. However, some monkeys and humans tended to over-weight vestibular cues, inconsistent with simple predictions of a Bayesian model. Nonetheless, our findings establish a robust model system for studying the neural mechanisms of dynamic cue reweighting in multisensory perception.


Probabilistic interpretation of population codes

R. S. Zemel and P. Dayan and A. Pouget

Neural Comput  10  403-30  (1998)

We present a general encoding-decoding framework for interpreting the activity of a population of units. A standard population code interpretation method, the Poisson model, starts from a description as to how a single value of an underlying quantity can generate the activities of each unit in the population. In casting it in the encoding-decoding framework, we find that this model is too restrictive to describe fully the activities of units in population codes in higher processing areas, such as the medial temporal area. Under a more powerful model, the population activity can convey information not only about a single value of some quantity but also about its whole distribution, including its variance, and perhaps even the certainty the system has in the actual presence in the world of the entity generating this quantity. We propose a novel method for forming such probabilistic interpretations of population codes and compare it to the existing method.


Neural representation of probabilistic information

M. J. Barber and J. W. Clark and C. H. Anderson

Neural Comput  15  1843-64  (2003)

It has been proposed that populations of neurons process information in terms of probability density functions (PDFs) of analog variables. Such analog variables range, for example, from target luminance and depth on the sensory interface to eye position and joint angles on the motor output side. The requirement that analog variables must be processed leads inevitably to a probabilistic description, while the limited precision and lifetime of the neuronal processing units lead naturally to a population representation of information. We show how a time-dependent probability density rho(x; t) over variable x, residing in a specified function space of dimension D, may be decoded from the neuronal activities in a population as a linear combination of certain decoding functions phi(i)(x), with coefficients given by the N firing rates a(i)(t) (generally with D << N). We show how the neuronal encoding process may be described by projecting a set of complementary encoding functions phi;(i)(x) on the probability density rho(x; t), and passing the result through a rectifying nonlinear activation function. We show how both encoders phi;(i)(x) and decoders phi(i)(x) may be determined by minimizing cost functions that quantify the inaccuracy of the representation. Expressing a given computation in terms of manipulation and transformation of probabilities, we show how this representation leads to a neural circuit that can carry out the required computation within a consistent Bayesian framework, with the synaptic weights being explicitly generated in terms of encoders, decoders, conditional probabilities, and priors.


Generating neural circuits that implement probabilistic reasoning

M. J. Barber and J. W. Clark and C. H. Anderson

Phys Rev E Stat Nonlin Soft Matter Phys  68  041912  (2003)

We extend the hypothesis that neuronal populations represent and process analog variables in terms of probability density functions (PDFs). Aided by an intermediate representation of the probability density based on orthogonal functions spanning an underlying low-dimensional function space, it is shown how neural circuits may be generated from Bayesian belief networks. The ideas and the formalism of this PDF approach are illustrated and tested with several elementary examples, and in particular through a problem in which model-driven top-down information flow influences the processing of bottom-up sensory input.


Bayesian inference with probabilistic population codes

W. J. Ma and J. M. Beck and P. E. Latham and A. Pouget

Nat Neurosci  9  1432-8  (2006)

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.


Matching behavior and the representation of value in the parietal cortex

L. P. Sugrue and G. S. Corrado and W. T. Newsome

Science  304  1782-7  (2004)

Psychologists and economists have long appreciated the contribution of reward history and expectation to decision-making. Yet we know little about how specific histories of choice and reward lead to an internal representation of the "value" of possible actions. We approached this problem through an integrated application of behavioral, computational, and physiological techniques. Monkeys were placed in a dynamic foraging environment in which they had to track the changing values of alternative choices through time. In this context, the monkeys' foraging behavior provided a window into their subjective valuation. We found that a simple model based on reward history can duplicate this behavior and that neurons in the parietal cortex represent the relative value of competing actions predicted by this model.


A biophysically based neural model of matching law behavior: melioration by stochastic synapses

A. Soltani and X.-J. Wang

J Neurosci  26  3731-44  (2006)

In experiments designed to uncover the neural basis of adaptive decision making in a foraging environment, neuroscientists have reported single-cell activities in the lateral intraparietal cortex (LIP) that are correlated with choice options and their subjective values. To investigate the underlying synaptic mechanism, we considered a spiking neuron model of decision making endowed with synaptic plasticity that follows a reward-dependent stochastic Hebbian learning rule. This general model is tested in a matching task in which rewards on two targets are scheduled randomly with different rates. Our main results are threefold. First, we show that plastic synapses provide a natural way to integrate past rewards and estimate the local (in time) "return" of a choice. Second, our model reproduces the matching behavior (i.e., the proportional allocation of choices matches the relative reinforcement obtained on those choices, which is achieved through melioration in individual trials). Our model also explains the observed "undermatching" phenomenon and points to biophysical constraints (such as finite learning rate and stochastic neuronal firing) that set the limits to matching behavior. Third, although our decision model is an attractor network exhibiting winner-take-all competition, it captures graded neural spiking activities observed in LIP, when the latter were sorted according to the choices and the difference in the returns for the two targets. These results suggest that neurons in LIP are involved in selecting the oculomotor responses, whereas rewards are integrated and stored elsewhere, possibly by plastic synapses and in the form of the return rather than income of choice options.


Operant matching is a generic outcome of synaptic plasticity based on the covariance between reward and neural activity

Y. Loewenstein and H. S. Seung

Proc Natl Acad Sci U S A  103  15224-9  (2006)

The probability of choosing an alternative in a long sequence of repeated choices is proportional to the total reward derived from that alternative, a phenomenon known as Herrnstein's matching law. This behavior is remarkably conserved across species and experimental conditions, but its underlying neural mechanisms still are unknown. Here, we propose a neural explanation of this empirical law of behavior. We hypothesize that there are forms of synaptic plasticity driven by the covariance between reward and neural activity and prove mathematically that matching is a generic outcome of such plasticity. Two hypothetical types of synaptic plasticity, embedded in decision-making neural network models, are shown to yield matching behavior in numerical simulations, in accord with our general theorem. We show how this class of models can be tested experimentally by making reward not only contingent on the choices of the subject but also directly contingent on fluctuations in neural activity. Maximization is shown to be a generic outcome of synaptic plasticity driven by the sum of the covariances between reward and all past neural activities.


Probabilistic reasoning by neurons

T. Yang and M. N. Shadlen

Nature  447  1075-80  (2007)

Our brains allow us to reason about alternatives and to make choices that are likely to pay off. Often there is no one correct answer, but instead one that is favoured simply because it is more likely to lead to reward. A variety of probabilistic classification tasks probe the covert strategies that humans use to decide among alternatives based on evidence that bears only probabilistically on outcome. Here we show that rhesus monkeys can also achieve such reasoning. We have trained two monkeys to choose between a pair of coloured targets after viewing four shapes, shown sequentially, that governed the probability that one of the targets would furnish reward. Monkeys learned to combine probabilistic information from the shape combinations. Moreover, neurons in the parietal cortex reveal the addition and subtraction of probabilistic quantities that underlie decision-making on this task.


Representation of confidence associated with a decision by neurons in the parietal cortex

R. Kiani and M. N. Shadlen

Science  324  759-64  (2009)

The degree of confidence in a decision provides a graded and probabilistic assessment of expected outcome. Although neural mechanisms of perceptual decisions have been studied extensively in primates, little is known about the mechanisms underlying choice certainty. We have shown that the same neurons that represent formation of a decision encode certainty about the decision. Rhesus monkeys made decisions about the direction of moving random dots, spanning a range of difficulties. They were rewarded for correct decisions. On some trials, after viewing the stimulus, the monkeys could opt out of the direction decision for a small but certain reward. Monkeys exercised this option in a manner that revealed their degree of certainty. Neurons in parietal cortex represented formation of the direction decision and the degree of certainty underlying the decision to opt out.


Probabilistic population codes for Bayesian decision making

J. M. Beck and W. J. Ma and R. Kiani and T. Hanks and A. K. Churchland and J. Roitman and M. N. Shadlen and P. E. Latham and A. Pouget

Neuron  60  1142-52  (2008)

When making a decision, one must first accumulate evidence, often over time, and then select the appropriate action. Here, we present a neural model of decision making that can perform both evidence accumulation and action selection optimally. More specifically, we show that, given a Poisson-like distribution of spike counts, biological neural networks can accumulate evidence without loss of information through linear integration of neural activity and can select the most likely action through attractor dynamics. This holds for arbitrary correlations, any tuning curves, continuous and discrete variables, and sensory evidence whose reliability varies over time. Our model predicts that the neurons in the lateral intraparietal cortex involved in evidence accumulation encode, on every trial, a probability distribution which predicts the animal's performance. We present experimental evidence consistent with this prediction and discuss other predictions applicable to more general settings.


Spiking networks for Bayesian inference and choice

W. J. Ma and J. M. Beck and A. Pouget

Curr Opin Neurobiol  18  217-22  (2008)

Systems neuroscience traditionally conceptualizes a population of spiking neurons as merely encoding the value of a stimulus. Yet, psychophysics has revealed that people take into account stimulus uncertainty when performing sensory or motor computations and do so in a nearly Bayes-optimal way. This suggests that neural populations do not encode just a single value but an entire probability distribution over the stimulus. Several such probabilistic codes have been proposed, including one that utilizes the structure of neural variability to enable simple neural implementations of probabilistic computations such as optimal cue integration. This approach provides a quantitative link between Bayes-optimal behaviors and specific neural operations. It allows for novel ways to evaluate probabilistic codes and for predictions for physiological population recordings.


Synaptic computation underlying probabilistic inference

A. Soltani and X.-J. Wang

Nat Neurosci  13  112-9  (2010)

We propose that synapses may be the workhorse of the neuronal computations that underlie probabilistic reasoning. We built a neural circuit model for probabilistic inference in which information provided by different sensory cues must be integrated and the predictive powers of individual cues about an outcome are deduced through experience. We found that bounded synapses naturally compute, through reward-dependent plasticity, the posterior probability that a choice alternative is correct given that a cue is presented. Furthermore, a decision circuit endowed with such synapses makes choices on the basis of the summed log posterior odds and performs near-optimal cue combination. The model was validated by reproducing salient observations of, and provides insights into, a monkey experiment using a categorization task. Our model thus suggests a biophysical instantiation of the Bayesian decision rule, while predicting important deviations from it similar to the 'base-rate neglect' observed in human studies when alternatives have unequal prior probabilities.