Natural sounds exhibit complex statistical regularities at multiple scales. Acoustic eventsunderlying speech, for example, are characterized by precise temporal and frequency relationships, but they can also vary substantially according to the pitch, duration, and other high-level properties of speech production. Learning this structure from data while capturing the inherent variability is an important first step in building auditory processing systems, as well as understanding the mechanisms of auditory perception. Here we develop Hierarchical Spike Coding, a two-layer probabilistic generative model for complex acoustic structure. The first layer consists of a sparse spiking representation that encodes the sound using kernelspositioned precisely in time and frequency. Patterns in the positions of first layer spikes are learned from the data: on a coarse scale, statistical regularities areencoded by a second-layer spiking representation, while fine-scale structure is captured by recurrent interactions within the first layer. When fit to speech data, the second layer acoustic features include harmonic stacks, sweeps, frequency modulations, and precise temporal onsets, which can be composed to represent complex acoustic events. Unlike spectrogram-based methods, the model gives a probability distribution over sound pressure waveforms. This allows us to use the second-layer representation to synthesize sounds directly, and to perform model-based denoising, on which we demonstrate a significant improvement over standard methods.

Efficient coding provides a powerful principle for explaining early sensory coding. Most attempts to test this principle have been limited to linear, noiseless models, and when applied to natural images, have yielded oriented filters consistent with responses in primary visual cortex. Here we show that an efficient coding model that incorporates biologically realistic ingredients - input and output noise, nonlinear response functions, and a metabolic cost on the firing rate - predicts receptive fields and response nonlinearities similar to those observed in the retina. Specifically, we develop numerical methods for simultaneously learning the linear filters and response nonlinearities of a population of model neurons, so as to maximize information transmission subject to metabolic costs. When applied to an ensemble of natural images, the method yields filters that are center-surround and nonlinearities that are rectifying. The filters are organized into two populations, with On-and Off-centers, which independently tile the visual space. As observed in the primate retina, the Off-center neurons are more numerous and have filters with smaller spatial extent. In the absence of noise, our method reduces to a generalized version of independent components analysis, with an adapted nonlinear "contrast" function; in this case, the optimal filters are localized and oriented.

Efficient coding provides a powerful principle for explaining early sensory coding.

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.

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.

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.

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.

We present a hierarchical Bayesian model for learning efficient codes of higher-order structure in natural images. The model, a nonlinear generalization ofindependent 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 imagestructure and provides a way to learn coarse-coded, sparsedistributed representationsof abstract image properties such as object location, scale, and texture.