{"title": "Visual Encoding with Jittering Eyes", "book": "Advances in Neural Information Processing Systems", "page_first": 1137, "page_last": 1144, "abstract": null, "full_text": "Visual Encoding with Jittering Eyes\n\nDepartment of Cognitive and Neural Systems\n\nMichele Rucci(cid:3)\n\nBoston University\nBoston, MA 02215\n\nrucci@cns.bu.edu\n\nAbstract\n\nUnder natural viewing conditions, small movements of the eye and body\nprevent the maintenance of a steady direction of gaze. It is known that\nstimuli tend to fade when they are stabilized on the retina for several sec-\nonds. However, it is unclear whether the physiological self-motion of the\nretinal image serves a visual purpose during the brief periods of natural\nvisual \ufb01xation. This study examines the impact of \ufb01xational instability\non the statistics of visual input to the retina and on the structure of neural\nactivity in the early visual system. Fixational instability introduces \ufb02uc-\ntuations in the retinal input signals that, in the presence of natural images,\nlack spatial correlations. These input \ufb02uctuations strongly in\ufb02uence neu-\nral activity in a model of the LGN. They decorrelate cell responses, even\nif the contrast sensitivity functions of simulated cells are not perfectly\ntuned to counter-balance the power-law spectrum of natural images. A\ndecorrelation of neural activity has been proposed to be bene\ufb01cial for\ndiscarding statistical redundancies in the input signals. Fixational insta-\nbility might, therefore, contribute to establishing ef\ufb01cient representations\nof natural stimuli.\n\n1\n\nIntroduction\n\nModels of the visual system often examine steady-state levels of neural activity during\npresentations of visual stimuli. It is dif\ufb01cult, however, to envision how such steady-states\ncould occur under natural viewing conditions, given that the projection of the visual scene\non the retina is never stationary.\nIndeed, the physiological instability of visual \ufb01xation\nkeeps the retinal image in permanent motion even during the brief periods in between\nsaccades.\n\nSeveral sources cause this constant jittering of the eye. Fixational eye movements, of which\nwe are not aware, alternate small saccades with periods of drifts, even when subjects are\ninstructed to maintain steady \ufb01xation [8]. Following macroscopic redirection of gaze, other\nsmall eye movements, such as corrective saccades and post-saccadic drifts, are likely to\noccur. Furthermore, outside of the controlled conditions of a laboratory, when the head\nis not constrained by a bite bar, movements of the body, as well as imperfections in the\nvestibulo-ocular re\ufb02ex, signi\ufb01cantly amplify the motion of the retinal image. In the light of\n\n(cid:3)Webpage: www.cns.bu.edu/(cid:24)rucci\n\n\fthis constant jitter, it is remarkable that the brain is capable of constructing a stable percept,\nas \ufb01xational instability moves the stimulus by an amount that should be clearly visible (see,\nfor example, [7]).\n\nLittle is known about the purposes of \ufb01xational instability. It is often claimed that small\nsaccades are necessary to refresh neuronal responses and prevent the disappearance of a\nstationary scene, a claim that has remained controversial given the brief durations of natural\nvisual \ufb01xation (reviewed in [16]). Yet, recent theoretical proposals [1, 11] have claimed\nthat \ufb01xational instability plays a more central role in the acquisition and neural encoding\nof visual information than that of simply refreshing neural activity. Consistent with the\nideas of these proposals, neurophysiological investigations have shown that \ufb01xational eye\nmovements strongly in\ufb02uence the activity of neurons in several areas of the monkey\u2019s brain\n[5, 14, 6]. Furthermore, modeling studies that simulated neural responses during free-\nviewing suggest that \ufb01xational instability profoundly affects the statistics of thalamic [13]\nand thalamocortical activity [10].\n\nThis paper summarizes an alternative theory for the existence of \ufb01xational instability. In-\nstead of regarding the jitter of visual \ufb01xation as necessary for refreshing neuronal responses,\nit is argued that the self-motion of the retinal image is essential for properly structuring neu-\nral activity in the early visual system into a format that is suitable for processing at later\nstages. It is proposed that \ufb01xational instability is part of a strategy of acquisition of visual\ninformation that enables compact visual representations in the presence of natural visual\ninput.\n\n2 Neural decorrelation and \ufb01xational instability\n\nIt is a long-standing proposal that an important function of early visual processing is the\nremoval of part of the redundancy that characterizes natural visual input [3]. Less redundant\nsignals enable more compact representations, in which the same amount of information can\nbe represented by smaller neuronal ensembles. While several methods exist for eliminating\ninput redundancies, a possible approach is the removal of pairwise correlations between\nthe intensity values of nearby pixels [2]. Elimination of these spatial correlations allows\nef\ufb01cient representations in which neuronal responses tend to be less statistically dependent.\n\nAccording to the theory described in this paper, \ufb01xational instability contributes to decor-\nrelating the responses of cells in the retina and the LGN during viewing of natural scenes.\nThis theory is based on two factors, which are described separately in the following sec-\ntions. The \ufb01rst component, analyzed in Section 2.1, is the spatially uncorrelated input\nsignal that occurs when natural scenes are scanned by jittering eyes. The second factor\nis an ampli\ufb01cation of this spatially uncorrelated input, which is mediated by cell response\ncharacteristics. Section 2.2 examines the interaction between the dynamics of \ufb01xational\ninstability and the temporal characteristics of neurons in the Lateral Geniculate Nucleus\n(LGN), the main relay of visual information to the cortex.\n\n2.1\n\nIn\ufb02uence of \ufb01xational instability on visual input\n\nTo analyze the effect of \ufb01xational instability on the statistics of geniculate activity, it is\nuseful to approximate the input image in a neighborhood of a \ufb01xation point x0 by means\nof its Taylor series:\n\nI(x) (cid:25) I(x0) + rI(x0) (cid:1) (x (cid:0) x0)T + o(jx (cid:0) x0j2)\n\n(1)\nIf the jittering produced by \ufb01xational instability is suf\ufb01ciently small, high-order derivatives\ncan be neglected, and the input to a location x on the retina during visual \ufb01xation can be\napproximated by its \ufb01rst-order expansion:\n\nS(x; t) (cid:25) I(x) + (cid:24)T (t) (cid:1) rI(x) = I(x) + ~I(x; t)\n\n(2)\n\n\fwhere (cid:24)(t) = [(cid:24)x(t); (cid:24)y(t)] is the trajectory of the center of gaze during the period of\n\ufb01xation, t is the time elapsed from \ufb01xation onset, I(x) is the visual input at t = 0, and\n~I(x; t) = @I(x)\n@y (cid:24)y(t) is the dynamic \ufb02uctuation in the visual input produced\nby \ufb01xational instability.\n\n@x (cid:24)x(t) + @I(x)\n\nEq. 2 allows an analytical estimation of the power spectrum of the signal entering the eye\nduring the self-motion of the retinal image. Since, according to Eq. 2, the retinal input\nS(x; t) can be approximated by the sum of two contributions, I and ~I, its power spectrum\nRSS consists of three terms:\n\nRSS(u; w) (cid:25) RII + R ~I ~I + 2RI ~I\n\nwhere u and w represent, respectively, spatial and temporal frequency.\nFixational instability can be modeled as an ergodic process with zero mean and uncorre-\nlated components along the two axes, i.e., h(cid:24)iT = 0 and R(cid:24)x(cid:24)y (t) = 0. Although not\nnecessary for the proposed theory, these assumptions simplify our statistical analysis, as\nRI ~I is zero, and the power spectrum of the visual input is given by:\n\n(3)\nwhere RII is the power spectrum of the stimulus, and R ~I ~I depends on both the stimulus\nand \ufb01xational instability.\nTo determine R ~I ~I (u; w), from Eq. 2 follows that\n\nRSS (cid:25) RII + R ~I ~I\n\n~I(u; w) = iuxI(u)(cid:24)x(w) + iuyI(u)(cid:24)y(w)\n\nand under the assumption of uncorrelated motion components, approximating the power\nspectrum via \ufb01nite Fourier Transform yields:\n\nR ~I ~I (u; w) = lim\nT !1\n\n<\n\n1\nT\n\nj ~IT (u; w)j2 >(cid:24);I= R(cid:24)(cid:24)(w)RII (u)juj2\n\n(4)\n\nwhere ~IT is the Fourier Transform of a signal of duration T , and we have assumed identi-\ncal second-order statistics of retinal image motion along the two Cartesian axes. As shown\nin Fig. 1 is clear that the presence of the term u2 in Eq. 4 compensates for the scaling\ninvariance of natural images. That is, since for natural images RII (u) / u(cid:0)2, the prod-\nuct RII (u)juj2 whitens RII by producing a power spectrum R ~I ~I that remains virtually\nconstant at all spatial frequencies.\n\n2.2\n\nIn\ufb02uence of \ufb01xational instability on neural activity\n\nThis section analyzes the structure of correlated activity during \ufb01xational instability in a\nmodel of the LGN. To delineate the important elements of the theory, we consider linear\napproximations of geniculate responses provided by space-time separable kernels. This\nassumption greatly simpli\ufb01es the analysis of levels of correlation. Results are, however,\ngeneral, and the outcomes of simulations with space-time inseparable kernels and different\nlevels of recti\ufb01cation (the most prominent nonlinear behavior of parvocellular geniculate\nneurons) can be found in [13, 10].\n\nMean instantaneous \ufb01ring rates were estimated on the basis of the convolution between the\ninput I and the cell spatiotemporal kernel h(cid:11):\n\n(cid:11)(t) = h(cid:11)(x; t) ? I(x; t) = Z t\n\n(cid:0)1\n\n0 Z 1\n\n(cid:0)1Z 1\n\nh(cid:11)(x0; y0; t0)I(x (cid:0) x0; y (cid:0) y0; t (cid:0) t0) dx0 dy0 dt0\n\nwhere h(cid:11)(x; t) = g(cid:11)(t)f(cid:11)(x). Kernels were designed on the basis of data from neurophys-\niological recordings to replicate the responses of parvocellular ON-center cells in the LGN\n\n\f100\n\nr\ne\nw\no\nP\n\n10\u22122\n\n10\u22124\n\nR~I ~I\nRII\n\n1\n\n10\n\nSpatial Frequency (cycles/deg)\n\nFigure 1: Fixational instability introduces a spatially uncorrelated component in the vi-\nsual input to the retina during viewing of natural scenes. The graph compares the power\nspectrum of natural images (RII) to the dynamic power spectrum introduced by \ufb01xational\ninstability (R ~I ~I). The two curves represent radial averages evaluated over 15 pictures of\nnatural scenes.\n\nof the macaque. The spatial component f(cid:11)(x) was modeled by a standard difference of\nGaussian [15]. The temporal kernel g(cid:11)(t) possessed a biphasic pro\ufb01le with positive peak\nat 50 ms, negative peak at 75 ms, and overall duration of less than 200 ms [4].\n\nIn this section, levels of correlation in the activity of pairs of geniculate neurons are sum-\nmarized by the correlation pattern ^c(cid:11)(cid:11)(x):\n\n^c(cid:11)(cid:11)(x) = h(cid:11)y(t)(cid:11)z(t)i(cid:12)(cid:12)(cid:12)(cid:12)T ;I\n\n(5)\n\n^c(cid:11)(cid:11)(x) = c(cid:11)(cid:11)(x; t)(cid:12)(cid:12)(cid:12)(cid:12)t=0\n\nwhere (cid:11)y(t) and (cid:11)z(t) are the responses of cells with receptive \ufb01elds centered at y and z,\nand x = y (cid:0) z is the separation between receptive \ufb01eld centers. The average is evaluated\nover time T and over a set of stimuli I.\nWith linear models, ^c(cid:11)(cid:11)(x) can be estimated on the basis of the input power spectrum\nRSS(u; w):\n\nand c(cid:11)(cid:11)(x; t) = F (cid:0)1fR(cid:11)(cid:11)g\n\n(6)\n\nwhere R(cid:11)(cid:11) = jH(cid:11)j2RSS(u; w) is the power spectrum of LGN activity (H(cid:11)(u; w) is the\nspatiotemporal Fourier transform of the kernel h(cid:11)(x; t)), and F (cid:0)1 represents the inverse\nFourier transform operator.\nTo evaluate R(cid:11)(cid:11), substitution of RSS from Eq. 3 and separation of spatial and temporal\nelements yield:\n\nR(cid:11)(cid:11) (cid:25) jG(cid:11)j2jF(cid:11)j2RII + jG(cid:11)j2jF(cid:11)j2R ~I ~I = RS\n\n(cid:11)(cid:11) + RD\n\n(cid:11)(cid:11)\n\n(7)\n\nwhere F(cid:11)(u) and G(cid:11)(w) represent the Fourier Transforms of the spatial and temporal\nkernels. Eq. 7 shows that, similar to the retinal input, also the power spectrum of geniculate\n(cid:11)(cid:11) depends on\nactivity can be approximated by the sum of two separate elements. Only RD\n(cid:11)(cid:11), is determined by the power spectrum of the\n\ufb01xational instability. The \ufb01rst term, RS\n\n\fstimulus and the characteristics of geniculate cells but does not depend on the motion of\nthe eye during the acquisition of visual information.\nBy substituting in Eq. 6 the expression of R(cid:11)(cid:11) from Eq. 7, we obtain\n\nc(cid:11)(cid:11)(x; t) (cid:25) cS\n\n(cid:11)(cid:11)(x; t) + cD\n\n(cid:11)(cid:11)(x; t)\n\n(8)\n\nwhere\n\n(cid:11)(cid:11)(x; t) = F (cid:0)1fRS\ncS\n\n(cid:11)(cid:11)(u; w)g and cD\n\n(cid:11)(cid:11)(x; t) = F (cid:0)1fRD\n\n(cid:11)(cid:11)(u; w)g\n\n(cid:11)(cid:11) to the pattern of correlated activity\nEq. 8 shows that \ufb01xational instability adds the term cD\n(cid:11)(cid:11) that would obtained with presentation of the same set of stimuli without the self-motion\ncS\nof the eye.\nWith presentation of pictures of natural scenes, RII (w) = 2(cid:25)(cid:14)(w), and the two input\nsignals RS\n(cid:11)(cid:11) provide, respectively, a static and a dynamic contribution to the\nspatiotemporal correlation of geniculate activity. The \ufb01rst term in Eq. 8 gives a correlation\npattern:\n\n(cid:11)(cid:11) and RD\n\n(cid:11)(cid:11)(x) = kSF (cid:0)1\n^cS\n\nS fjF(cid:11)j2RS\n\nII (u)g\n\nwhere kS = jG(0)j2.\nBy substituting R ~I ~I from Eq. 4, the second term in Eq. 8 gives a correlation pattern:\n\n^cD\n(cid:11)(cid:11)(x) = kDFS\n\n(cid:0)1fjF(cid:11)j2RS\n\nII (u)juj2g\n\n(9)\n\n(10)\n\nwhere kD = FT\n\nis a constant given by the temporal dynamics\n\n(cid:0)1fjG(cid:11)(w)j2R(cid:24)(cid:24)(w)g(cid:12)(cid:12)(cid:12)(cid:12)t=0\n\nof cell response and \ufb01xational instability. F (cid:0)1\nFourier Transform in time and space.\n\nT\n\nand F (cid:0)1\n\nS indicate the operations of inverse\n\nTo summarize, during the physiological instability of visual \ufb01xation, the structure of cor-\nrelated activity in a linear model of the LGN is given by the superposition of two spatial\nterms, each of them weighted by a coef\ufb01cient (kS and kD) that depends on dynamics:\n\n^c(cid:11)(cid:11)(x) = kSFS\n\n(cid:0)1f(jF(cid:11)j2RS\n\nII (u)g + kDFS\n\n(cid:0)1fjF(cid:11)j2RS\n\nII (u)juj2g\n\n(11)\n\nWhereas the stimulus contributes to the structure of correlated activity by means of the\npower spectrum RS\n,\nII, the contribution introduced by \ufb01xational instability depends on RS\n~I ~I\na signal that discards the broad correlation of natural images. Since in natural images, most\npower is concentrated at low spatial frequencies, the uncorrelated \ufb02uctuations in the input\nsignals generated by \ufb01xational instability have small amplitudes. That is, RD\nII provides less\npower than RS\nII. However, geniculate cells tend to respond more strongly to changing stim-\nuli than stationary ones, and kD is larger than kS. Therefore, the small input modulations\nintroduced by \ufb01xational instability are ampli\ufb01ed by the dynamics of geniculate cells.\n\nFig. 2 shows the structure of correlated activity in the model when images of natural scenes\nare examined in the presence of \ufb01xational instability. In this example, \ufb01xational instability\nwas assumed to possess Gaussian temporal correlation, R(cid:24)(cid:24)(w), with standard deviation\n(cid:27)T = 22 ms and amplitude (cid:27)S = 12 arcmin. In addition to the total pattern of correlation\ngiven by Eq. 11, Fig. 2 also shows the patterns of correlation produced by the two compo-\n(cid:11)(cid:11) was strongly in\ufb02uenced by the broad spatial correlations\nnents ^cS\nof natural images, ^cD\n(cid:11)(cid:11), due to its dependence on the whitened power spectrum R ~I ~I, was\ndetermined exclusively by cell receptive \ufb01elds. Due to the ampli\ufb01cation factor kD, ^cD\n(cid:11)(cid:11)\nprovided a stronger contribution than ^cS\n(cid:11)(cid:11) and heavily in\ufb02uenced the global structure of\ncorrelated activity.\n\n(cid:11)(cid:11). Whereas ^cS\n\n(cid:11)(cid:11) and ^cD\n\nTo examine the relative in\ufb02uence of the two terms ^cS\nrelated activity, Fig. 3 shows their ratio at separation zero, (cid:26)DS = ^cD\n\n(cid:11)(cid:11) on the structure of cor-\n(cid:11)(cid:11)(0), with\n\n(cid:11)(cid:11) and ^cD\n\n(cid:11)(cid:11)(0)=^cS\n\n\f1\n\nn\no\n\ni\nt\n\n0.8\n\nStatic\nDynamic\nTotal\n\nl\n\na\ne\nr\nr\no\nC\nd\ne\nz\n\n \n\ni\nl\n\na\nm\nr\no\nN\n\n0.6\n\n0.4\n\n0.2\n\n0\n0\n\n1\n4\nCell RF Separation (deg.)\n\n3\n\n2\n\n5\n\nFigure 2: Patterns of correlation obtained from Eq. 11 when natural images are examined in\nthe presence of \ufb01xational instability. The three curves represent the total level of correlation\n(cid:11)(cid:11)(x) that would be present if the same images were examined\n(Total), the correlation ^cS\nin the absence of \ufb01xational instability (Static), and the contribution ^cD\n(cid:11)(cid:11)(x) of \ufb01xational\ninstability (Dynamic). Data are radial averages evaluated over pairs of cells with the same\nseparation jjxjj between their receptive \ufb01elds.\n\npresentation of natural images and for various parameters of \ufb01xational instability. Fig. 3\n(a) shows the effect of varying the spatial amplitude of the retinal jitter. In order to remain\nwithin the range of validity of the Taylor approximation in Eq. 2, only small amplitude val-\nues are considered. As shown by Fig. 3 (a), the larger the instability of visual \ufb01xation, the\nlarger the contribution of the dynamic term ^cD\n(cid:11)(cid:11). Except for very small\nvalues of (cid:27)S, (cid:26)DS is larger than one, indicating that ^cD\n(cid:11)(cid:11) in\ufb02uences the structure of corre-\nlated activity more strongly than ^cS\n(cid:11)(cid:11). Fig. 3 (b) shows the impact of varying (cid:27)T , which\nde\ufb01nes the temporal window over which \ufb01xational jitter is correlated. Note that (cid:26)DS is a\nnon-monotonic function of (cid:27)T . For a range of (cid:27)T corresponding to intervals shorter than\nthe typical duration of visual \ufb01xation, ^cD\n(cid:11)(cid:11). Thus, \ufb01xational\ninstability strongly in\ufb02uences correlated activity in the model when it moves the direction\nof gaze within a range of a few arcmin and is correlated over a fraction of the duration\nof visual \ufb01xation. This range of parameters is consistent with the instability of \ufb01xation\nobserved in primates.\n\n(cid:11)(cid:11) is signi\ufb01cantly larger than ^cS\n\n(cid:11)(cid:11) with respect to ^cS\n\n3 Conclusions\n\nIt has been proposed that neurons in the early visual system decorrelate their responses\nto natural stimuli, an operation that is believed to be bene\ufb01cial for the encoding of visual\ninformation [2]. The original claim, which was based on psychophysical measurements of\nhuman contrast sensitivity, relies on an inverse proportionality between the spatial response\ncharacteristics of retinal and geniculate neurons and the structure of natural images. How-\never, data from neurophysiological recordings have clearly shown that neurons in the retina\nand the LGN respond signi\ufb01cantly to low spatial frequencies, in a way that is not compat-\nible with the requirements of Atick and Redlich\u2019s proposal. During natural viewing, input\nsignals to the retina depend not only on the stimulus, but also on the physiological instabil-\nity of visual \ufb01xation. The results of this study show that when natural scenes are examined\n\n\fc\ni\nt\n\ni\n\nt\n\na\nS\n/\nc\nm\na\nn\ny\nD\no\ni\nt\na\nR\n\n \n\n6\n\n5\n\n4\n\n3\n\n2\n\n1\n\n0\n0\n\n3\n\n2.5\n\n2\n\n1.5\n\n1\n\n0.5\n\nc\ni\nt\n\ni\n\nt\n\na\nS\n/\nc\nm\na\nn\ny\nD\no\n\n \n\ni\nt\n\na\nR\n\n10\n\n5\n\nS (arcmin)\n(a)\n\n15\n\n0\n0\n\n100\n\n200\n\n300\n\nT (ms)\n(b)\n\nFigure 3: In\ufb02uence of the characteristics of \ufb01xational instability on the patterns of corre-\nlated activity during presentation of natural images. The two graphs show the ratio (cid:26)DS\n(cid:11)(cid:11) in Eq. 8. Fixational instability was as-\nbetween the peaks of the two terms ^cD\nsumed to possess a Gaussian correlation with standard deviation (cid:27)T and amplitude (cid:27)S. (a)\nEffect of varying (cid:27)S ((cid:27)T = 22 ms). (b) Effect of varying (cid:27)T ((cid:27)S = 12 arcmin).\n\n(cid:11)(cid:11) and ^cS\n\nwith jittering eyes, as occurs under natural viewing conditions, \ufb01xational instability tends\nto decorrelate cell responses even if the contrast sensitivity functions of individual neurons\ndo not counterbalance the power spectrum of visual input.\n\nThe theory described in this paper relies of two main elements. The \ufb01rst component is\nthe presence of a spatially uncorrelated input signal during presentation of natural visual\nstimuli (R ~I ~I in Eq. 3). This input signal is a direct consequence of the scale invariance\nof natural images. It is a property of natural images that, although the intensity values of\nnearby pixels tend to be correlated, changes in intensity around pairs of pixels are uncorre-\nlated. This property is not satis\ufb01ed by an arbitrary image. In a spatial grating, for example,\nintensity changes at any two locations are highly correlated. During the instability of vi-\nsual \ufb01xation, neurons receive input from the small regions of the visual \ufb01eld covered by the\njittering of their receptive \ufb01elds. In the presence of natural images, although the inputs to\ncells with nearby receptive \ufb01elds are on average correlated, the \ufb02uctuations in these input\nsignals produced by \ufb01xational instability are not correlated. Fixational instability appears\nto be tuned to the statistics of natural images, as it introduces a spatially uncorrelated signal\nonly in the presence of visual input with a power spectrum that declines as u(cid:0)2 with spatial\nfrequency.\n\nThe second element of the theory is the neuronal ampli\ufb01cation of the spatially uncorre-\nlated input signal introduced by the self-motion of the retinal image. This ampli\ufb01cation\noriginates from the interaction between the dynamics of \ufb01xational instability and the tem-\nporal sensitivity of geniculate units. Since R ~I ~I attenuates the low spatial frequencies of\nthe stimulus, it tends to possess less power than RII. However, in Eq. 11, the contribu-\ntions of the two input signals are modulated by the multiplicative terms kS and kD, which\ndepend on the temporal characteristics of cell responses (both kS and kD) and \ufb01xational\ninstability (kD only). Since geniculate neurons respond more strongly to changing stimuli\nthan to stationary ones, kD tends to be higher than kS. Correspondingly, in a linear model\nof the LGN, units are highly sensitive to the uncorrelated \ufb02uctuations in the input signals\nproduced by \ufb01xational instability.\n\nThe theory summarized in this study is consistent with the strong modulations of neural\nresponses observed during \ufb01xational eye movements [5, 14, 6], as well as with the results\n\ns\ns\n\fof recent psychophysical experiments aimed at investigating perceptual in\ufb02uences of \ufb01x-\national instability [12, 9]. It should be observed that, since patterns of correlations were\nevaluated via Fourier analysis, this study implicitly assumed a steady-state condition of\nvisual \ufb01xation. Further work is needed to extend the proposed theory in order to take into\naccount time-varying natural stimuli and the nonstationary regime produced by the occur-\nrence of saccades.\n\nAcknowledgments\n\nThe author thanks Antonino Casile and Gaelle Desbordes for many helpful discussions.\nThis material is based upon work supported by the National Institute of Health under Grant\nEY15732-01 and the National Science Foundation under Grant CCF-0432104.\n\nReferences\n[1] E. Ahissar and A. Arieli. Figuring space by time. Neuron, 32(2):185\u2013201, 2001.\n[2] J. J. Atick and A. Redlich. What does the retina know about natural scenes? Neural Comp.,\n\n4:449\u2013572, 1992.\n\n[3] H. B. Barlow. The coding of sensory messages. In W. H. Thorpe and O. L. 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Elsevier Science, 1990.\n\n\f", "award": [], "sourceid": 2894, "authors": [{"given_name": "Michele", "family_name": "Rucci", "institution": null}]}