{"title": "Modeling Surround Suppression in V1 Neurons with a Statistically Derived Normalization Model", "book": "Advances in Neural Information Processing Systems", "page_first": 153, "page_last": 159, "abstract": null, "full_text": "Modeling Surround Suppression in VI Neurons \nwith a Statistically-Derived Normalization Model \n\nEero P. Simoncelli \n\nCenter for Neural Science, and \n\nCourant Institute of Mathematical Sciences \n\nNew York University \n\neero.simoncelli@nyu.edu \n\nOdelia Schwartz \n\nCenter for Neural Science \n\nNew York University \nodelia@cns.nyu.edu \n\nAbstract \n\nWe examine the statistics of natural monochromatic images decomposed \nusing a multi-scale wavelet basis. Although the coefficients of this rep(cid:173)\nresentation are nearly decorrelated, they exhibit important higher-order \nstatistical dependencies that cannot be eliminated with purely linear pro(cid:173)\nc~ssing. In particular, rectified coefficients corresponding to basis func(cid:173)\ntions at neighboring spatial positions, orientations and scales are highly \ncorrelated. A method of removing these dependencies is to divide each \ncoefficient by a weighted combination of its rectified neighbors. Sev(cid:173)\neral successful models of the steady -state behavior of neurons in primary \nvisual cortex are based on such \"divisive normalization\" computations, \nand thus our analysis provides a theoretical justification for these models. \nPerhaps more importantly, the statistical measurements explicitly specify \nthe weights that should be used in computing the normalization signal. \nWe demonstrate that this weighting is qualitatively consistent with re(cid:173)\ncent physiological experiments that characterize the suppressive effect \nof stimuli presented outside of the classical receptive field. Our obser(cid:173)\nvations thus provide evidence for the hypothesis that early visual neural \nprocessing is well matched to these statistical properties of images. \n\nAn appealing hypothesis for neural processing states that sensory systems develop in re(cid:173)\nsponse to the statistical properties of the signals to which they are exposed [e.g., 1, 2]. \nThis has led many researchers to look for a means of deriving a model of cortical process(cid:173)\ning purely from a statistical characterization of sensory signals. In particular, many such \nattempts are based on the notion that neural responses should be statistically independent. \n\nThe pixels of digitized natural images are highly redundant, but one can always find a \nlinear decomposition (i.e., principal component analysis) that eliminates second-order cor-\n\nResearch supported by an Alfred P. Sloan Fellowship to EPS, and by the Sloan Center for Theoretical \nNeurobiology at NYU. \n\n\f154 \n\nE. P Simoncelli and 0. Schwartz \n\nrelation. A number of researchers have used such concepts to derive linear receptive fields \nsimilar to those determined from physiological measurements [e.g., 16,20]. The principal \ncomponents decomposition is, however, not unique. Because of this, these early attempts \nrequired additional constraints, such as spatial locality and/or symmetry, in order to achieve \nfunctions approximating cortical receptive fields. \n\nMore recently, a number of authors have shown that one may use higher-order statisti(cid:173)\ncal measurements to uniquely constrain the choice of linear decomposition [e.g., 7, 9]. \nThis is commonly known as independent components analysis. Vision researchers have \ndemonstrated that the resulting basis functions are similar to cortical receptive fields, in \nthat they are localized in spatial position, orientation and scale [e.g., 17, 3]. The associ(cid:173)\nated coefficients of such decompositions are (second-order) decorrelated, highly kurtotic, \nand generally more independent than principal components. \n\nBut the response properties of neurons in primary visual cortex are not adequately described \nby linear processes. Even if one chooses to describe only the mean firing rate of such \nneurons, one must at a minimum include a rectifying, saturating nonlinearity. A number of \nauthors have shown that a gain control mechanism, known as divisive normalization, can \nexplain a wide variety of the nonlinear behaviors of these neurons [18, 4, II, 12,6]. In most \ninstantiations of normalization, the response of each linear basis function is rectified (and \ntypically squared) and then divided by a uniformly weighted sum of the rectified responses \nof all other neurons. PhYSiologically, this is hypothesized to occur via feedback shunting \ninhibitory mechanisms [e.g., 13, 5]. Ruderman and Bialek [19] have discussed divisive \nnormalization as a means of increasing entropy. \n\nIn this paper, we examine the joint statistics of coefficients of an orthonormal wavelet im(cid:173)\nage decomposition that approximates the independent components of natural images. We \nshow that the coefficients are second-order decorrelated, but not independent. In partic(cid:173)\nular, pairs of rectified responses are highly correlated. These pairwise dependencies may \nbe eliminated by dividing each coefficient by a weighted combination of the rectified re(cid:173)\nsponses of other neurons, with the weighting determined from image statistics. We show \nthat the resulting model, with all parameters determined from the statistics of a set of im(cid:173)\nages, can account for recent physiological observations regarding suppression of cortical \nresponses by stimuli presented outside the classical receptive field. These concepts have \nbeen previously presented in [21, 25]. \n\n1 Joint Statistics of Orthonormal Wavelet Coefficients \n\nMulti-scale linear transforms such as wavelets have become popular for image representa(cid:173)\ntion. 'TYpically, the basis functions of these representations are localized in spatial position, \norientation, and spatial frequency (scale). The coefficients resulting from projection of \nnatural images onto these functions are essentially uncorrelated. In addition, a number \nof authors have noted that wavelet coefficients have significantly non-Gaussian marginal \nstatistics [e.g., 10,14]. Because of these properties, we believe that wavelet bases provide \na close approximation to the independent components decomposition for natural images. \nFor the purposes of this paper, we utilize a typical separable decomposition, based on \nsymmetric quadrature mirror filters taken from [23]. The decomposition is constructed by \nsplitting an image into four subbands (lowpass, vertical, horizontal, diagonal), and then \nrecursively splitting the lowpass subband. \n\nDespite the decorrelation properties of the wavelet decomposition, it is quite evident that \nwavelet coefficients are not statistically independent [26, 22]. Large-magnitude coefficients \n(either positive or negative) tend to lie along ridges with orientation matching that of the \nsubband. Large-magnitude coefficients also tend to occur at the same relative spatialloca(cid:173)\ntions in subbands at adjacent scales, and orientations. To make these statistical relationships \n\n\f", "award": [], "sourceid": 1533, "authors": [{"given_name": "Eero", "family_name": "Simoncelli", "institution": null}, {"given_name": "Odelia", "family_name": "Schwartz", "institution": null}]}