The spatial receptive fields of simple cells in mammalian striate cortex have been reasonably well described physiologically and can be characterized as being localized, oriented, and ban@ass, comparable with the basis functions of wavelet transforms. Previously, we have shown that these receptive field properties may be accounted for in terms of a strategy for producing a sparse distribution of output activity in response to natural images. Here, in addition to describing this work in a more expansive fashion, we examine the neurobiological implications of sparse coding. Of particular interest is the case when the code is overcomplete--i.e., when the number of code elements is greater than the effective dimensionality of the input space. Because the basis functions are non-orthogonal and not linearly independent of each other, sparsifying the code will recruit only those basis functions necessary for representing a given input, and so the input-output function will deviate from being purely linear. These deviations from linearity provide a potential explanation for the weak forms of non-linearity observed in the response properties of cortical simple cells, and they further make predictions about the expected interactions among units in

Natural images contain characteristic statistical regularities that set them apart from purely random images. Understanding what these regularities are can enable natural images to be coded more efficiently. In this paper, we describe some of the forms of structure that are contained in natural images, and we show how these are related to the response properties of neurons at early stages of the visual system. Many of the important forms of structure require higher-order (i.e. more than linear, pairwise) statistics to characterize, which makes models based on linear Hebbian learning, or principal components analysis, inappropriate for finding efficient codes for natural images. We suggest that a good objective for an efficient coding of natural scenes is to maximize the sparseness of the representation, and we show that a network that learns sparse codes of natural scenes succeeds in developing localized, oriented, bandpass receptive fields similar to those in the mammalian striate cortex.

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The images we typically view, or natural scenes, constitute a minuscule fraction of the space of all possible images. It seems reasonable that the visual cortex, which has evolved and developed to effectively cope with these images, has discovered efficient coding strategies for representing their structure. Here, we explore the hypothesis that the coding strategy employed at the earliest stage of the mammalian visual cortex maximizes the sparseness of the representation. We show that a learning algorithm that attempts to find linear sparse codes for natural scenes will develop receptive fields that are localized, oriented, and bandpass, much like those in the visual system. These receptive fields produce a more efficient image representation for later stages of processing because sparseness reduces the entropies of individual outputs, which in turn reduces the redundancy due to complex statistical dependencies among unit activities. The spatial receptive fields of simple ...

This article will pose the following challenge: that despite four decades of research characterizing the response properties of V1 neurons, we still do not have a decent picture of how V1 really operates---i.e., how a population of its neurons represents natural scenes under realistic viewing conditions. We identify five problems with the current view that stem largely from biases in the design and execution of experiments, in addition to the contributions of non-linearities in the cortex that are not well understood. Our purpose is to open the window to new theories, a number of which we describe along with some proposals for testing them.

tation for later stages of processing because it possesses a higher degree of statistical independence among its outputs. We start with the basic assumption that an image, I(x; y), can be represented in terms of a linear superposition of (not necessarily orthogonal) basis functions, OE i (x; y): I(x; y) = X i a i OE i (x; y) : (1) The image code is determined by the choice of basis functions, OE i . The coefficients, a i , are dynamic variables that change from one image to the next. The goal of efficient coding is to find a set of OE i that forms a complete code (i.e., spans the image space) and results in the coefficient values being as statistically independent as possible over an ensemble of natural images. The

Glass patterns are texture stimuli made by pairing randomly placed dots with partners at specific offsets. The strong percept of global form that arises from the sparse local orientation cues has made these patterns the subject of psychophysical investigations, yet neuronal responses to Glass patterns have not been studied. We measured the responses of neurons in macaque striate cortex (V1) to dynamic, translational Glass patterns as a function of dot separation and dot-pair orientation. Responses were selective, but were on average more than an order of magnitude weaker than responses to sinusoidal gratings. Response and selectivity were greatest when the dot-pair orientation matched that of the preferred grating and when dot separation was between one-quarter and one-half of the spatial period of the optimal grating; changing the dot-pair separation or inverting the contrast of one of the dots radically changed the orientation selectivity. We computed the expected responses for a receptive field model to translational Glass patterns and found that the complexity of our V1 tuning curves could be understood in terms of the responses of linear filters to pairs of dots. This modeling connects our understanding of V1 receptive fields as rectified, quasi-linear filters with results from psychophysical studies of Glass patterns. Our results provide a basis for studying how subsequent visual areas integrate weak, local signals into global form percepts. Key words: Glass patterns; macaque monkey; primary visual cortex; V1; random dots; orientation selectivity; linear filter; spatial frequency