Efficient coding models predict that the optimal code for natural images is a population of oriented Gabor receptive fields. These results match response properties of neurons in primary visual cortex, but not those in the retina. Does the retina use an optimal code, and if so, what is it optimized for? Previous theories of retinal coding have assumed that the goal is to encode the maximal amount of information about the sensory signal. However, the image sampled by retinal photoreceptors is degraded both by the optics of the eye and by the photoreceptor noise. Therefore, de-blurring and de-noising of the retinal signal should be important aspects of retinal coding.

Efficient coding models predict that the optimal code for natural images is a population oforiented Gabor receptive fields. These results match response properties of neurons in primary visual cortex, but not those in the retina. Does the retina use an optimal code, and if so, what is it optimized for? Previous theories of retinal coding have assumed that the goal is to encode the maximal amount of information about the sensory signal. However, the image sampled by retinal photoreceptors is degraded both by the optics of the eye and by the photoreceptor noise.

Biological sensory systems are faced with the problem of encoding a high-fidelity sensory signal with a population of noisy, low-fidelity neurons. Thisproblem can be expressed in information theoretic terms as coding and transmitting a multidimensional, analog signal over a set of noisy channels. Previously, we have shown that robust, overcomplete codes can be learned by minimizing the reconstruction error with a constraint onthe channel capacity. Here, we present a theoretical analysis that characterizes the optimal linear coder and decoder for one-and twodimensional data.The analysis allows for an arbitrary number of coding units, thus including both under-and over-complete representations, and provides a number of important insights into optimal coding strategies. In particular, we show how the form of the code adapts to the number of coding units and to different data and noise conditions to achieve robustness. Wealso report numerical solutions for robust coding of highdimensional imagedata and show that these codes are substantially more robust compared against other image codes such as ICA and wavelets.

Linear implementations of the efficient coding hypothesis, such as independent component analysis (ICA) and sparse coding models, have provided functional explanations for properties of simple cells in V1 [1, 2]. These models, however, ignore the nonlinear behavior of neurons and fail to match individual and population properties of neural receptive fields in subtle but important ways. Hierarchical models, including Gaussian Scale Mixtures [3, 4] and other generative statistical models [5, 6], can capture higher-order regularities in natural images and explain nonlinear aspects of neural processing such as normalization and context effects [6,7]. Previously, it had been assumed that the lower level representation is independent of the hierarchy, and had been fixed when training these models. Here we examine the optimal lower-level representations derived in the context of a hierarchical model and find that the resulting representations are strikingly different from those based on linear models. Unlike the the basis functions and filters learned by ICA or sparse coding, these functions individually more closely resemble simple cell receptive fields and collectively span a broad range of spatial scales. Our work unifies several related approaches and observations about natural image structure and suggests that hierarchical models might yield better representations of image structure throughout the hierarchy.

Biological sensory systems are faced with the problem of encoding a high-fidelity sensory signal with a population of noisy, low-fidelity neurons. This problem can be expressed in information theoretic terms as coding and transmitting a multidimensional, analog signal over a set of noisy channels. Previously, we have shown that robust, overcomplete codes can be learned by minimizing the reconstruction error with a constraint on the channel capacity. Here, we present a theoretical analysis that characterizes the optimal linear coder and decoder for one-and twodimensional data. The analysis allows for an arbitrary number of coding units, thus including both under-and over-complete representations, and provides a number of important insights into optimal coding strategies. In particular, we show how the form of the code adapts to the number of coding units and to different data and noise conditions to achieve robustness. We also report numerical solutions for robust coding of highdimensional image data and show that these codes are substantially more robust compared against other image codes such as ICA and wavelets.

Linear implementations of the efficient coding hypothesis, such as independent componentanalysis (ICA) and sparse coding models, have provided functional explanations for properties of simple cells in V1 [1, 2]. These models, however, ignore the nonlinear behavior of neurons and fail to match individual and population properties of neural receptive fields in subtle but important ways. Hierarchical models, including Gaussian ScaleMixtures [3, 4] and other generative statistical models [5, 6], can capture higher-order regularities in natural images and explain nonlinear aspectsof neural processing such as normalization and context effects [6,7]. Previously, it had been assumed that the lower level representation isindependent of the hierarchy, and had been fixed when training these models. Here we examine the optimal lower-level representations derived in the context of a hierarchical model and find that the resulting representations are strikingly different from those based on linear models. Unlikethe the basis functions and filters learned by ICA or sparse coding, these functions individually more closely resemble simple cell receptive fields and collectively span a broad range of spatial scales. Our work unifies several related approaches and observations about natural image structure and suggests that hierarchical models might yield better representations of image structure throughout the hierarchy.

The representation of acoustic signals at the cochlear nerve must serve a wide range of auditory tasks that require exquisite sensitivity in both time and frequency. Lewicki (2002) demonstrated that many of the filtering properties of the cochlea could be explained in terms of efficient coding of natural sounds. This model, however, did not account for properties such as phase-locking or how sound could be encoded in terms of action potentials. Here, we extend this theoretical approach with algorithm for learning efficient auditory codes using a spiking population code. Here, we propose an algorithm for learning efficient auditory codes using a theoretical model for coding sound in terms of spikes.

It has been suggested that the primary goal of the sensory system is to represent input in such a way as to reduce the high degree of redundancy. Givena noisy neural representation, however, solely reducing redundancy is not desirable, since redundancy is the only clue to reduce the effects of noise. Here we propose a model that best balances redundancy reductionand redundant representation. Like previous models, our model accounts for the localized and oriented structure of simple cells, but it also predicts a different organization for the population. With noisy, limited-capacity units, the optimal representation becomes an overcomplete, multi-scalerepresentation, which, compared to previous models, is in closer agreement with physiological data. These results offer a new perspective on the expansion of the number of neurons from retina to V1 and provide a theoretical model of incorporating useful redundancy into efficient neural representations.

The representation of acoustic signals at the cochlear nerve must serve a wide range of auditory tasks that require exquisite sensitivity in both time and frequency. Lewicki (2002) demonstrated that many of the filtering properties of the cochlea could be explained in terms of efficient coding of natural sounds. This model, however, did not account for properties such as phase-locking or how sound could be encoded in terms of action potentials. Here, we extend this theoretical approach with algorithm for learning efficient auditory codes using a spiking population code. Here, we propose an algorithm for learning efficient auditory codes using a theoretical model for coding sound in terms of spikes.

We present a hierarchical Bayesian model for learning efficient codes of higher-order structure in natural images. The model, a nonlinear generalization of independent component analysis, replaces the standard assumption of independence for the joint distribution of coefficients with a distribution that is adapted to the variance structure of the coefficients of an efficient image basis. This offers a novel description of higherorder image structure and provides a way to learn coarse-coded, sparsedistributed representations of abstract image properties such as object location, scale, and texture.