Neural codes are inevitably shaped by various kinds of biological constraints, \emph{e.g.} noise and metabolic cost. Here we formulate a coding framework which explicitly deals with noise and the metabolic costs associated with the neural representation of information, and analytically derive the optimal neural code for monotonic response functions and arbitrary stimulus distributions. For a single neuron, the theory predicts a family of optimal response functions depending on the metabolic budget and noise characteristics. Interestingly, the well-known histogram equalization solution can be viewed as a special case when metabolic resources are unlimited. For a pair of neurons, our theory suggests that under more severe metabolic constraints, ON-OFF coding is an increasingly more efficient coding scheme compared to ON-ON or OFF-OFF. The advantage could be as large as one-fold, substantially larger than the previous estimation. Some of these predictions could be generalized to the case of large neural populations. In particular, these analytical results may provide a theoretical basis for the predominant segregation into ON- and OFF-cells in early visual processing areas. Overall, we provide a unified framework for optimal neural codes with monotonic tuning curves in the brain, and makes predictions that can be directly tested with physiology experiments.

Abstract Subjective expected utility theory assumes that decision-makers possess unlimited computational resources to reason about their choices; however, virtually all decisions in everyday life are made under resource constraints---i.e. decision-makers are bounded in their rationality. Here we experimentally tested the predictions made by a formalization of bounded rationality based on ideas from statistical mechanics and information-theory. We systematically tested human subjects in their ability to solve combinatorial puzzles under different time limitations. We found that our bounded-rational model accounts well for the data. The decomposition of the fitted model parameter into the subjects' expected utility function and resource parameter provide interesting insight into the subjects' information capacity limits. Our results confirm that humans gradually fall back on their learned prior choice patterns when confronted with increasing resource limitations.

Subjective expected utility theory assumes that decision-makers possess unlimited computational resources to reason about their choices; however, virtually all decisions in everyday life are made under resource constraints - i.e. decision-makers are bounded in their rationality. Here we experimentally tested the predictions made by a formalization of bounded rationality based on ideas from statistical mechanics and information-theory. We systematically tested human subjects in their ability to solve combinatorial puzzles under different time limitations. We found that our bounded-rational model accounts well for the data. The decomposition of the fitted model parameter into the subjects' expected utility function and resource parameter provide interesting insight into the subjects' information capacity limits. Our results confirm that humans gradually fall back on their learned prior choice patterns when confronted with increasing resource limitations.

How does the human visual system compute the speed of a coherent motion stimulus that contains motion energy in different spatiotemporal frequency bands? Here we propose that perceived speed is the result of optimal integration of speed information from independent spatiotemporal frequency tuned channels. We formalize this hypothesis with a Bayesian observer model that treats the channel activity as independent cues, which are optimally combined with a prior expectation for slow speeds. We test the model against behavioral data from a 2AFC speed discrimination task with which we measured subjects' perceived speed of drifting sinusoidal gratings with different contrasts and spatial frequencies, and of various combinations of these single gratings. We find that perceived speed of the combined stimuli is independent of the relative phase of the underlying grating components, and that the perceptual biases and discrimination thresholds are always smaller for the combined stimuli, supporting the cue combination hypothesis. The proposed Bayesian model fits the data well, accounting for perceptual biases and thresholds of both simple and combined stimuli. Fits are improved if we assume that the channel responses are subject to divisive normalization, which is in line with physiological evidence. Our results provide an important step toward a more complete model of visual motion perception that can predict perceived speeds for stimuli of arbitrary spatial structure.

How does neural population process sensory information? Optimal coding theories assume that neural tuning curves are adapted to the prior distribution of the stimulus variable. Most of the previous work has discussed optimal solutions for only one-dimensional stimulus variables. Here, we expand some of these ideas and present new solutions that define optimal tuning curves for high-dimensional stimulus variables. We consider solutions for a minimal case where the number of neurons in the population is equal to the number of stimulus dimensions (diffeomorphic). In the case of two-dimensional stimulus variables, we analytically derive optimal solutions for different optimal criteria such as minimal L2 reconstruction error or maximal mutual information. For higher dimensional case, the learning rule to improve the population code is provided.

A common challenge for Bayesian models of perception is the fact that the two fundamental Bayesian components, the prior distribution and the likelihood function, are formally unconstrained. Here we argue that a neural system that emulates Bayesian inference is naturally constrained by the way it represents sensory information in populations of neurons. More specifically, we show that an efficient coding principle creates a direct link between prior and likelihood based on the underlying stimulus distribution. The resulting Bayesian estimates can show biases away from the peaks of the prior distribution, a behavior seemingly at odds with the traditional view of Bayesian estimation, yet one that has been reported in human perception. We demonstrate that our framework correctly accounts for the repulsive biases previously reported for the perception of visual orientation, and show that the predicted tuning characteristics of the model neurons match the reported orientation tuning properties of neurons in primary visual cortex. Our results suggest that efficient coding is a promising hypothesis in constraining Bayesian models of perceptual inference.

In this work we study how the stimulus distribution influences the optimal coding of an individual neuron. Closed-form solutions to the optimal sigmoidal tuning curve are provided for a neuron obeying Poisson statistics under a given stimulus distribution.

A common challenge for Bayesian models of perception is the fact that the two fundamental Bayesian components, the prior distribution and the likelihood function, areformally unconstrained. Here we argue that a neural system that emulates Bayesian inference is naturally constrained by the way it represents sensory information inpopulations of neurons. More specifically, we show that an efficient coding principle creates a direct link between prior and likelihood based on the underlying stimulus distribution. The resulting Bayesian estimates can show biases awayfrom the peaks of the prior distribution, a behavior seemingly at odds with the traditional view of Bayesian estimation, yet one that has been reported in human perception. We demonstrate that our framework correctly accounts for the repulsive biases previously reported for the perception of visual orientation, and show that the predicted tuning characteristics of the model neurons match the reported orientation tuning properties of neurons in primary visual cortex. Our results suggest that efficient coding is a promising hypothesis in constraining Bayesianmodels of perceptual inference.

We propose an extended probabilistic model for human perception. We argue that in many circumstances, human observers simultaneously evaluate sensory evidence under different hypotheses regarding the underlying physical process that might have generated the sensory information. Within this context, inference can be optimal if the observer weighs each hypothesis according to the correct belief in that hypothesis. But if the observer commits to a particular hypothesis, the belief in that hypothesis is converted into subjective certainty, and subsequent perceptual behavior is suboptimal, conditioned only on the chosen hypothesis. We demonstrate that this framework can explain psychophysical data of a recently reported decision-estimation experiment. The model well accounts for the data, predicting the same estimation bias as a consequence of the preceding decision step. The power of the framework is that it has no free parameters except the degree of the observer's uncertainty about its internal sensory representation. All other parameters are defined by the particular experiment which allows us to make quantitative predictions of human perception to two modifications of the original experiment.

We extend a previously developed Bayesian framework for perception to account for sensory adaptation. We first note that the perceptual effects ofadaptation seems inconsistent with an adjustment of the internally represented prior distribution. Instead, we postulate that adaptation increases the signal-to-noise ratio of the measurements by adapting the operational range of the measurement stage to the input range. We show that this changes the likelihood function in such a way that the Bayesian estimator model can account for reported perceptual behavior. In particular, wecompare the model's predictions to human motion discrimination data and demonstrate that the model accounts for the commonly observed perceptual adaptation effects of repulsion and enhanced discriminability.