{"title": "Receptive Fields without Spike-Triggering", "book": "Advances in Neural Information Processing Systems", "page_first": 969, "page_last": 976, "abstract": "Stimulus selectivity of sensory neurons is often characterized by estimating their receptive field properties such as orientation selectivity. Receptive fields are usually derived from the mean (or covariance) of the spike-triggered stimulus ensemble. This approach treats each spike as an independent message but does not take into account that information might be conveyed through patterns of neural activity that are distributed across space or time. Can we find a concise description for the processing of a whole population of neurons analogous to the receptive field for single neurons? Here, we present a generalization of the linear receptive field which is not bound to be triggered on individual spikes but can be meaningfully linked to distributed response patterns. More precisely, we seek to identify those stimulus features and the corresponding patterns of neural activity that are most reliably coupled. We use an extension of reverse-correlation methods based on canonical correlation analysis. The resulting population receptive fields span the subspace of stimuli that is most informative about the population response. We evaluate our approach using both neuronal models and multi-electrode recordings from rabbit retinal ganglion cells. We show how the model can be extended to capture nonlinear stimulus-response relationships using kernel canonical correlation analysis, which makes it possible to test different coding mechanisms. Our technique can also be used to calculate receptive fields from multi-dimensional neural measurements such as those obtained from dynamic imaging methods.", "full_text": "Receptive Fields without Spike- Triggering\n\nJakob H Macke\n\nj a k o b@ t u e bi n g e n . mpg . de\n\nG \u00a8unther Zeck\n\nz e c k @ n e u r o . mpg . de\n\nMax Planck Ins titute for B iological Cybernetics\n\nMax Planck Ins titute of Neurobiology\n\nS pemanns tras s e 41\n\n72076 T \u00a8ubingen, Germany\n\nAm Klopfers pitze 1 8\n\n8 21 5 2 Martins ried, Germany\n\nMatthias Bethge\n\nmbe t h g e @ t u e bi n g e n . mp g . de\n\nMax Planck Ins titute for B iological Cybernetics\n\nS pemanns tras s e 41\n\n72076 T \u00a8ubingen, Germany\n\nAbstract\n\nS timulus s electivity of s ens ory neurons is often characterized by es timating their\nreceptive \ufb01eld properties s uch as orientation s electivity. Receptive \ufb01elds are us u-\nally derived from the mean (or covariance) of the s pike-triggered s timulus ens em-\nble. This approach treats each s pike as an independent mes s age but does not take\ninto account that information might be conveyed through patterns of neural activ-\nity that are dis tributed acros s s pace or time. Can we \ufb01nd a concis e des cription for\nthe proces s ing of a whole population of neurons analogous to the receptive \ufb01eld\nfor s ingle neurons ? Here, we pres ent a generalization of the linear receptive \ufb01eld\nwhich is not bound to be triggered on individual s pikes but can be meaningfully\nlinked to dis tributed res pons e patterns . More precis ely, we s eek to identify thos e\ns timulus features and the corres ponding patterns of neural activity that are mos t\nreliably coupled. We us e an extens ion of revers e-correlation methods bas ed on\ncanonical correlation analys is . The res ulting population receptive \ufb01elds s pan the\ns ubs pace of s timuli that is mos t informative about the population res pons e. We\nevaluate our approach us ing both neuronal models and multi-electrode recordings\nfrom rabbit retinal ganglion cells . We s how how the model can be extended to\ncapture nonlinear s timulus -res pons e relations hips us ing kernel canonical correla-\ntion analys is , which makes it pos s ible to tes t different coding mechanis ms . Our\ntechnique can als o be us ed to calculate receptive \ufb01elds from multi-dimens ional\nneural meas urements s uch as thos e obtained from dynamic imaging methods .\n\n1\n\nIntroduction\n\nVis ual input to the retina cons is ts of complex light intens ity patterns . The interpretation of thes e\npatterns cons titutes a challenging problem: for computational tas ks like object recognition, it is not\nclear what information about the image s hould be extracted and in which format it s hould be repre-\ns ented. S imilarly, it is dif\ufb01cult to as s es s what information is conveyed by the multitude of neurons\nin the vis ual pathway. Right from the \ufb01rs t s ynaps e, the information of an individual photoreceptor\nis s ignaled to many different cells with different temporal \ufb01ltering properties , each of which is only\na s mall unit within a complex neural network [ 20] . Even if we leave the dif\ufb01culties impos ed by\nnonlinearities and feedback as ide, it is hard to judge what the contribution of any particular neuron\nis to the information trans mitted.\n\n1\n\n\fThe prevalent tool for characterizing the behavior of s ens ory neurons , the s pike triggered average,\nis bas ed on a quas i-linear model of neural res pons es [ 1 5 ] . For the s ake of clarity, we cons ider an\nidealized model of the s ignaling channel\n\n.\n\n.\n\n.\n\n.\n\n.\n\n.\n\ny = W x + \u03be ,\n\n(1 )\n, yN ) T denotes the vector of neural res pons es , x the s timulus parameters , W =\nwhere y = ( y1 ,\n, wN ) T the \ufb01lter matrix with row \u2018 k \u2019 containing the receptive \ufb01eld wk of neuron k , and \u03be\n( w1 ,\nis the nois e. The s pike-triggered average only allows des cription of the s timulus -res pons e function\n(i. e.\nIn order to unders tand the collective behavior of a\nneuronal population, we rather have to unders tand the behavior of the matrix W, and the s tructure\nof the nois e correlations \u03a3 \u03be : B oth of them in\ufb02uence the feature s electivity of the population.\n\nthe wk ) of one s ingle neuron at a time.\n\nCan we \ufb01nd a compact des cription of the features that a neural ens emble is mos t s ens itive to? In\nthe cas e of a s ingle cell, the receptive \ufb01eld provides s uch a des cription: It can be interpreted as the\n\u201cfavorite s timulus \u201d of the neuron, in the s ens e that the more s imilar an input is to the receptive \ufb01eld,\nthe higher is the s piking probability, and thus the \ufb01ring rate of the neuron. In addition, the receptive\n\ufb01eld can eas ily be es timated us ing a s pike-triggered average, which, under certain as s umptions ,\nyields the optimal es timate of the receptive \ufb01eld in a linear-nonlinear cas cade model [ 1 1 ] .\n\nIf we are cons idering an ens emble of neurons rather than a s ingle neuron, it is not obvious what to\ntrigger on: This requires as s umptions about what patterns of s pikes or modulations in \ufb01ring rates\nacros s the population carry information about the s timulus . Rather than addres s ing the ques tion\n\u201cwhat features of the s timulus are correlated with the occurence of s pikes \u201d, the ques tion now is :\n\u201cWhat s timulus features are correlated with what patterns of s piking activity? \u201d [ 1 4] . Phras ed in the\nlanguage of information theory, we are s earching for the s ubs pace that contains mos t of the mu-\ntual information between s ens ory inputs and neuronal res pons es . B y this dimens ionality reduction\ntechnique, we can \ufb01nd a compact des cription of the proces s ing of the population.\n\nAs an ef\ufb01cient implementation of this s trategy, we pres ent an extens ion of revers e-correlation meth-\nods bas ed on canonical correlation analys is . The res ulting population receptive \ufb01elds (PRFs ) are not\nbound to be triggered on individual s pikes but are linked to res pons e patterns that are s imultaneous ly\ndetermined by the algorithm.\n\nWe calculate the PRF for a population cons is ting of uniformly s paced cells with center-s urround\nreceptive \ufb01elds and nois e correlations , and es timate the PRF of a population of rabbit retinal ganglion\ncells from multi-electrode recordings .\nIn addition, we s how how our method can be extended to\nexplore different hypothes es about the neural code, s uch as s pike latencies or interval coding, which\nrequire nonlinear read out mechanis ms .\n\n2 From reverse correlation to canonical correlation\n\nm .\nWe regard the s timulus at time t as a random variable Xt \u2208 R\nFor s implicity, we as s ume that the s timulus cons is ts of Gaus s ian white nois e, i. e. E( X) = 0 and\nCov( X) = I.\n\nn , and the neural res pons e as Yt \u2208 R\n\nThe s pike-triggered average a of a neuron can be motivated by the fact that it is the direction in\ns timulus -s pace maximizing the correlation-coef\ufb01cient\n\n\u03c1 =\n\n\ufffd\n\nCov( aT X, Y1 )\nVar( aT X) Var( Y1 )\n\n.\n\n(2)\n\nbetween the \ufb01ltered s timulus aT X and a univariate neural res pons e Y1 .\nIn the cas e of a neural\npopulation, we are not only looking for the s timulus feature a, but als o need to determine what\npattern of s piking activity b it is coupled with. The natural extens ion is to s earch for thos e vectors\na1 and b 1 that maximize\n\n.\n\n(3 )\n\n\ufffd\n\n\u03c11 =\n\nCov( aT\nVar( aT\n\n1 X, b T\n1 Y)\n1 X) Var( b T\n\n1 Y)\n\nWe interpret a1 as the s timulus \ufb01lter whos e output is maximally correlated with the output of the\n\u201cres pons e \ufb01lter\u201d b 1 . Thus , we are s imultaneous ly s earching for features of the s timulus that the\nneural s ys tem is s elective for, and the patterns of activity that it us es to s ignal the pres ence or abs ence\n\n2\n\n\fof this feature. We refer to the vector a1 as the (\ufb01rs t) population receptive \ufb01eld of the population,\nand b 1 is the response feature corres ponding to a1 . If a hypothetical neuron receives input from the\npopulation, and wants to decode the pres ence of the s timulus a1 , the weights of the optimal linear\nreadout [ 1 6] could be derived from b 1 .\n\nCanonical Correlation Analys is (CCA) [ 9]\nthat\nmaximize (3 ): We denote the covariances of X and Y by \u03a3 x , \u03a3 y , the cros s -covariance by \u03a3 x y , and\nthe whitened cros s -covariance by\n\nis an algorithm that \ufb01nds the vectors a1 and b 1\n\nC = \u03a3 ( \u2212 1 / 2 )\n\nx\n\n\u03a3 x y \u03a3 ( \u2212 1 / 2 )\n\ny\n\n.\n\n(4)\n\nLet C = UDV T denote the s ingular value decompos ition of C, where the entries of the diagonal\nmatrix D are non-negative and decreas ing along the diagonal. Then, the k -th pair of canonical\nvariables is given by ak = \u03a3\nvk , where uk and vk are the k -th column\nthe k -th diagonal\nvectors of U and V , res pectively. Furthermore, the k -th s ingular value of C, i. e.\nentry of D is the correlation-coef\ufb01cient \u03c1k of aT\nj X\nare uncorrelated for i \ufffd= j.\n\nk Y. The random variables aT\n\nuk and b k = \u03a3\n\ni X and aT\n\nk Xand b T\n\n( \u2212 1 / 2 )\nx\n\n( \u2212 1 / 2 )\ny\n\nImportantly, the s olution for the optimization problem in CCA is unique and can be computed ef-\n\ufb01ciently via a s ingle eigenvalue problem. The population receptive \ufb01elds and the characteris tic\npatterns are found by a joint optimization in s timulus and res pons e s pace. Therefore, one does\nnot need to know\u2014or as s ume\u2014a priori what features the population is s ens itive to, or what s pike\npatterns convey the information.\n\nThe \ufb01rs t K PRFs form a bas is for the s ubs pace of s timuli that the neural population is mos t s ens itive\nto, and the individual bas is vectors ak are s orted according to their \u201cinformativenes s \u201d [ 1 3 , 1 7] .\n\nThe mutual information between two one-dimens ional Gaus s ian Variables with correlation \u03c1 is given\nby MIG a us s = \u2212 1\n2 log( 1 \u2212 \u03c12 ) , s o maximizing correlation coef\ufb01cients is equivalent to maximizing\nmutual information [ 3 ] . As s uming the neural res pons e Y to be Gaus s ian, the s ubs pace s panned by\nthe \ufb01rs t K vectors BK = ( b 1 ,\n, b K ) is als o the K-s ubs pace of s timuli that contains the maximal\namount of mutual information between s timuli and neural res pons e. That is\n\n.\n\n.\n\n.\n\n`\n\n\u00b4\n\n\u201c\n\n\u201c\n\ndet\n\nT\n\nB\n\n\u03a3 y B\n\n\u201d\n\n\u201d\n\nBK = argmax\n\nB \u2208 R n \u00d7 k\n\ndet\n\nB T\n\n\u03a3 y \u2212 \u03a3 T\n\nx y \u03a3\n\n( \u2212 1 )\nx\n\n\u03a3 x y\n\nB\n\n.\n\n(5 )\n\nThus ,\nin terms of dimens ionality reduction, CCA optimizes the s ame objective as oriented PCA\n[ 5 ] .\nIn contras t to oriented PCA, however, CCA does not require one to know explicitly how the\nres pons e covariance \u03a3 y = \u03a3 s + \u03a3 \u03be s plits into s ignal \u03a3 s and nois e \u03a3 \u03be covariance. Ins tead, it us es the\ncros s -covariance \u03a3 x y which is directly available from revers e correlation experiments . In addition,\nCCA not only returns the mos t predictable res pons e features b 1 ,\n. b K but als o the mos t predictive\ns timulus components AK = ( a1 ,\nFor general Y and for s timuli X with elliptically contoured dis tribution, MIG a us s \u2212 J( AT X) pro-\nvides a lower bound to the mutual information between AT X and B T Y, where\n\n. aK ) .\n\n.\n\n.\n\n.\n\n.\n\nT\n\nJ( A\n\nX) =\n\n1\n2\n\nlog( det( 2 \u03c0eA\n\nT\n\n\u03a3 x A) ) \u2212 h( A\n\nT\n\nX)\n\n(6)\n\nis the Negentropy of AT X, and h( AT X)\nits differential entropy. S ince for elliptically contoured\ndis tributions J( AT X) does not depend on A, the PRFs can be s een as the s olution of a variational\napproach, maximizing a lower bound to the mutual information. Maximizing mutual information\ndirectly is hard, requires extens ive amounts of data, and us ually multiple repetitions of the s ame\ns timulus s equence.\n\n3 The receptive \ufb01eld of a population of neurons\n\n3.1 The effect of tuning functions and noise correlations\n\nTo illus trate the relations hip between the tuning-functions of individual neurons and the PRFs [ 22] ,\nwe calculate the \ufb01rs t PRF of a s imple one-dimens ional population model cons is ting of center-\n\n3\n\n\fs urround neurons . Each tuning function is modeled by a \u201cDifference of Gaus s ians \u201d (DOG)\n\n\u201e\n\n\u201c\n\n\u00ab\n\n\u201d\n\n \n\n\u201e\n\n!\n\n\u00ab\n\nf ( x) = exp\n\n\u2212\n\n1\n2\n\nx \u2212 c\n\n\u03c3\n\n2\n\n\u2212 A exp\n\n\u2212\n\n2\n\n1\n2\n\nx \u2212 c\n\n\u03b7\n\n(7)\n\nwhos e centers c are uniformly dis tributed over the real axis . The width \u03b7 of the negative Gaus s ian is\ns et to be twice as large as the width \u03c3 of the pos itive Gaus s ian. If the area of both Gaus s ians is the\ns ame ( A = 1 ) , the DC component of the DOG-\ufb01llter is zero, i. e.\nthe neuron is not s ens itive to the\nmean luminance of the s timulus . If the ratio between both areas becomes s ubs tantially unbalanced,\nthe DC component will become the larges t s ignal ( A \u2248 0) .\n\nIn addition to the parameter A, we will s tudy the length s cale of nois e correlations \u03bb [ 1 8 ] . S peci\ufb01-\ncally, we as s ume exponentially decaying nois e correlation with \u03a3 \u03be ( s ) = exp( \u2212 | s | / \u03bb) .\n\nAs this model is invariant under s patial s hifts , the \ufb01rs t PRF can be calculated by \ufb01nding the s patial\nfrequency at which the S NR is maximal. That is , the \ufb01rs t PRF can be us ed to es timate the pas s band\nof the population trans fer function. The S NR is given by\n\n\u201e\n\n\u201e\n\n\u00ab\u00ab\n\nS NR( \u03c9) =\n\n2\n\n\u03c9\n\n2\n\n1 + \u03bb\n2 \u03bb\n\n\u2212 \u03c9 2 \u03c3 2\n\ne\n\n+ A\n\n2\n\ne\n\n\u2212 \u03b7 2 \u03c9 2\n\n\u2212 2 Ae\n\n\u03c3 2 + \u03b7 2\n\n2\n\n\u03c9 2\n\n\u2212\n\n2\n\n.\n\n(8 )\n\nThe pas s band of the \ufb01rs t population \ufb01lter moves as a function of both parameters A and \u03bb. It equals\nlarge imbalance) and s mall \u03bb (i. e. s hort correlation length). In\nthe DC component for s mall A (i. e.\nthis cas e, the mean intens ity is the s timulus property that is mos t faithfully s ignaled by the ens emble.\n\nA\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n \n\n \n\n0.5\n\n1\n\u03bb\n\n1.5\n\n2\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n\nFigure 1 : S patial frequency of the \ufb01rs t PRF for the model des cribed above. \u03bb is the length-s cale of\nthe nois e correlations , A is the weight of the negative Gaus s ian in the DOG-model. The region in\nthe bottom left corner (bounded by the white line) is the part of the parameter-s pace in which the\nPRF equals the DC component.\n\n3.2 The receptive \ufb01eld of an ensemble of retinal ganglion cells\n\nWe mapped the population receptive \ufb01elds of rabbit retinal ganglion cells recorded with a whole-\nmount preparation. We are not primarily interes ted in prediction performance [ 1 2] , but rather in\ndimens ionality reduction: We want to characterize the \ufb01ltering properties of the population.\n\nThe neurons were s timulated with a 1 6 \u00d7 1 6 checkerboard cons is ting of binary white nois e which\nwas updated every 20ms . The experimental procedures are des cribed in detail in [ 21 ] . After s pike-\ns orting, s pike trains from 32 neurons were binned at 2 0ms res olution, and the res pons e of a neuron\nto a s timulus at time t was de\ufb01ned to cons is t of the the s pike-counts in the 1 0 bins between 40ms\nand 240ms after t. Thus , each population res pons e Yt is a 3 20 dimens ional vector.\n\nFigure 3 . 2 A) dis plays the \ufb01rs t 6 PRFs , the corres ponding patterns of neural activity (B ) and their\ncorrelation coef\ufb01cients \u03c1k (which were calculated us ing a cros s -validation procedure). It can be s een\nthat the PRFs look very different to the us ual center-s urrond s tructure of retinal ganglion. However,\none s hould keep in mind that it is really the s pace s panned by the PRFs that is relevant, and thus be\ncareful when interpreting the actual \ufb01lter s hapes [ 1 5 ] .\n\nFor comparis on, we als o plotted the s ingle-cell receptive \ufb01elds in Figure 3 . 2 C), and their projections\ninto the s paced s panned by the \ufb01rs t 6 PRFs . Thes e plots s ugges t that a s mall number of PRFs might\n\n4\n\n\f\ufffd\n\nbe s uf\ufb01cient to approximate each of the receptive \ufb01elds . To determine the dimens ionality of the\nrelevant s ubs pace, we analyzed the correlation-coef\ufb01cients \u03c1k . The Gaus s ian Mutual Information\nMIG a us s = \u2212 1\nis an es timate of the information contained in the s ubs pace\n2\ns panned by the \ufb01rs t K PRFs . B as ed on this meas ure, a 1 2 dimens ional s ubs pace accounts for 90%\nof the total information.\n\nK\nk = 1 log( 1 \u2212 \u03c12\nk )\n\nIn order to link the empirically es timated PRFs with the theoretical analys is in s ection 3 . 1 , we\ncalculated the s pectral properties of the \ufb01rs t PRF. Our analys is revealed that mos t of the power is in\nthe low frequencies , s ugges ting that the population is in the parameter-regime where the s ingle-cell\nreceptive \ufb01elds have power in the DC-component and the nois e-correlations have s hort range, which\nis certainly reas onable for retinal ganglion cells [ 4] .\n\n0.51\n\n0.44\n\n0.38\n\n0.35\n\n0.29\n\n0.27\n\nA)\n\nB )\n\nx\ne\nd\nn\n\ni\n \n\nn\no\nr\nu\ne\nN\n\n5\n10\n15\n20\n25\n30\n\n5\n10\n15\n20\n25\n30\n\n5\n10\n15\n20\n25\n30\n\n5\n10\n15\n20\n25\n30\n\n5\n10\n15\n20\n25\n30\n\n \n\n0.2\n\n0\n\n\u22120.2\n\n160\n\n220\n\n5\n10\n15\n20\n25\n30\n220\n\n \n40\n\n40\n\n160\n\nTime \u2192\n\n220\n\n40\n\n160\n\n220\n\n40\n\n160\n\n220\n\n40\n\n160\n\n220\n\n40\n\n160\n\n \n \n \n \n \n\nC)\nF\nR\n\n \n \n \n \n \n\nF\nR\n\n \n.\nj\no\nr\nP\n\n \n \n \n \n \n\nF\nR\n\n \n \n \n \n \n\nF\nR\n\n \n.\nj\n\no\nr\nP\n\nFigure 2: The population receptive \ufb01elds of a group of 32 retinal ganglion cells : A) the \ufb01rs t 6 PRFs ,\nas s orted by the correlation coef\ufb01cient \u03c1k B ) the res pons e features bk coupled with the PRFs . Each\nrow of each image corres ponds to one neuron, and each column to one time-bin. B lue color denotes\nenhanced activity, red s uppres s ed. It can be s een that only a s ubs et of neurons contributed to the \ufb01rs t\n6 PRFs . C) The s ingle-cell receptive \ufb01elds of 2 4 neurons from our population, and their projections\ninto the s pace s panned by the 6 PRFs .\n\n5\n\n\fA)\n\nk\n\n\u03c1\n \ns\nt\n\ni\n\nn\ne\nc\ni\nf\nf\n\ne\no\nc\n \ns\nn\no\n\ni\nt\n\nl\n\na\ne\nr\nr\no\nC\n\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0\n\u22120.1\n\n1\n\n5\n\n10 15 20\n\n30\nPRF index\n\n40\n\n50\n\nB )\n\nI\n\nM\n\n \nf\n\n \n\no\ne\ng\na\n\nt\n\nn\ne\nc\nr\ne\nP\n\n100\n90\n80\n\n60\n\n40\n\n20\n\n0\n\n1\n\n5\n\n10\n\n15\n\n20\n\nDimensionality of subspace\n\n30\n\n40\n\n50\n\nFigure 3 : A) Correlation coef\ufb01cients \u03c1k for the PRFs . Es timates and error-bars are calculated us ing\na cros s -validation procedure. B ) Gaus s ian-MI of the s ubs pace s panned by the \ufb01rs t K PRFs .\n\n4 Nonlinear extensions using Kernel Canonical Correlation Analysis\n\nThus far, our model is completely linear: We as s ume that the s timulus is linearly related to the\nneural res pons es , and we als o as s ume a linear readout of the res pons e.\nIn this s ection, we will\nexplore generalizations of the CCA model us ing Kernel CCA: B y embedding the s timulus -s pace\nnonlinearly in a feature s pace, nonlinear codes can be des cribed.\n\nKernel methods provide a framework for extending linear algorithms to the nonlinear cas e [ 8 ] . After\nprojecting the data into a feature s pace via a feature maps \u03c6 and \u03c8, a s olution is found us ing linear\nIn the cas e of Kernel CCA [ 1 , 1 0, 2, 7] one s eeks to \ufb01nd a linear\nmethods in the feature s pace.\n\u02c6X = \u03c6( X) and \u02c6Y = \u03c8( Y) , rather than between X and\nrelations hip between the random variables\nY. If an algorithm is purely de\ufb01ned in terms of dot-products , and if the dot-product in feature s pace\nk( s , t) = \ufffd \u03c8( s ) , \u03c8( t) \ufffd can be computed ef\ufb01ciently, then the algorithm does not require explicit\ncalculation of the feature maps \u03c6 and \u03c8. This \u201ckernel-trick\u201d makes it pos s ible to work in high-\nIt is worth mentioning that the s pace of patterns Y its elf\n(or in\ufb01nite)-dimens ional feature s paces .\ndoes not have to be a vector s pace. Given a data-s et x 1 .\n. x n , it s uf\ufb01ces to know the dot-products\nbetween any pair of training points , Ki j : = \ufffd \u03c8( yi ) , \u03c8( yj ) \ufffd .\n\n.\n\nThe kernel function k( s , t) can be s een as a s imiliarity meas ure.\nIt incorporates our as s umptions\nabout which s pike-patterns s hould be regarded as s imilar \u201cmes s ages \u201d. Therefore, the choice of the\nkernel-function is clos ely related to s peci\ufb01ng what the s earch-s pace of potential neural codes is . A\nnumber of dis tance- and kernel-functions [ 6, 1 9] have been propos ed to compute dis tances between\ns pike-trains . They can be des igned to take into account precis ely timed pattern of s pikes , or to be\ninvariant to certain trans formations s uch as temporal jitter.\n\nWe illus trate the concept on s imulated data: We will us e a s imilarity meas ure bas ed on the metric\nD interval\n[ 1 9] to es timate the receptive \ufb01eld of a neuron which does not us e its \ufb01ring rate, but rather\nthe occurrence of s peci\ufb01c inters pike intervals to convey information about the s timulus . The metric\nD interval\nbetween two s pike-trains is es s entially the cos t of matching their intervals by s hifting, adding\nor deleting s pikes .\nIn theory, this function is not guaranteed\nto be pos itive de\ufb01nite, which could lead to numerical problems , but we did not encounter any in\nour s imulation. ) If we cons ider coding-s chemes that are bas ed on patterns of s pikes , the methods\ndes cribed here become us eful even for the analys is of s ingle neurons . We will here concentrate on a\ns ingle neuron, but the analys is can be extended to patterns dis tributed acros s s everal neurons .\n\n(We s et k( s , t) = exp( \u2212 D( s , t) .\n\nOur hypothetical neuron encodes information in a pattern cons is ting of three s pikes : The relative\ntiming of the s econd s pike is informative about the s timulus : The bigger the correlation between\nreceptive \ufb01eld and s timulus \ufffd r, s t \ufffd , the s horter is the interval. If the receptive \ufb01eld is very dis s imilar\nto the s timulus , the interval is long. While the timing of the s pikes relative to each other is precis e,\nthere is jitter in the timing of the pattern relative to the s timulus . Figure 4 A) is a ras ter plot of\ns imulated s pike-trains from this model, ordered by \ufffd r, s t \ufffd . We als o included nois e s pikes at random\ntimes .\n\n6\n\n\fA)\n\nC)\n\nD)\n\nB )\n\ni\n\ns\nn\na\nr\nt\n \n\ni\n\ne\nk\np\nS\n\n0\n\n50\n\n100\n\nTime \u2192\n\n150\n\n200\n\nFigure 4: Coding by s pike patterns : A) Receptive \ufb01eld of neuron des cribed in S ection 4. B ) A\ns ubs et of the s imulated s pike-trains , s orted with res pect to the s imilarity between the s hown s timulus\nand the receptive \ufb01eld of the model. The interval between the \ufb01rs t two informative s pikes in each\ntrial is highlighted in red. C) Receptive \ufb01eld recovered by Kernel CCA, the correlation coef\ufb01cient\nbetween real and es timated receptive \ufb01eld is 0. 93 . D) Receptive \ufb01eld derived us ing linear decoding,\ncorrelation coef\ufb01cient is 0. 02 .\n\nUs ing thes e s pike-trains , we tried to recover the receptive \ufb01eld r without telling the algorithm what\nthe indicating pattern was . Each s timulus was s hown only once, and therefore, that every s pike-\npattern occurred only once. We s imulated 5 000 s timulus pres entations for this model, and applied\nKernel CCA with a linear kernel on the s timuli, and the alignment-s core on the s pike-trains . B y\nus ing incomplete Choles ky decompos itions [ 2] , one can compute Kernel CCA without having to\ncalculate the full kernel matrix. As many kernels on s pike trains are computationally expens ive, this\ntrick can res ult in s ubs tantial s peed-ups of the computation. The receptive \ufb01eld was recovered (s ee\nFigure 4), des pite the highly nonlinear encoding mechanis m of the neuron. For comparis on, we als o\ns how what receptive \ufb01eld would be obtained us ing linear decoding on the indicated bins .\n\nAlthough this neuron model may s eem s lightly contrived,\nin\nprinciple, receptive \ufb01elds can be es timated even if the \ufb01ring rate gives no information at all about\nthe s timulus , and the encoding is highly nonlinear. Our algorithm does not only look at patterns that\noccur more often than expected by chance, but als o takes into account to what extent their occurrence\nis correlated to the s ens ory input.\n\nit is a good proof of concept that,\n\n5 Conclusions\n\nWe s et out to \ufb01nd a us eful des cription of the s timulus -res pons e relations hip of an ens emble of\nneurons akin to the concept of receptive \ufb01eld for s ingle neurons . The population receptive \ufb01elds are\nfound by a joint optimization over s timuli and s pike-patterns , and are thus not bound to be triggered\nby s ingle s pikes .\n\nWe es timated the PRFs of a group of retinal ganglion cells , and found that the \ufb01rs t PRF had mos t\ns pectral power in the low-frequency bands , cons is tent with our theoretical analys is . The s timulus\nwe us ed was a white-nois e s equence\u2014it will be interes ting to s ee how the informative s ubs pace and\nits s pectral properties change for different s timuli s uch as colored nois e. The ganglion cell layer of\nthe retina is a s ys tem that is relatively well unders tood at the level of s ingle neurons . Therefore,\nour res ults can readily be compared and connected to thos e obtained us ing conventional analys is\ntechniques . However, our approach has the potential to be es pecially us eful in s ys tems in which the\nfunctional s igni\ufb01cance of s ingle cell receptive \ufb01elds is dif\ufb01cult to interpret.\n\n7\n\n\fWe us ually as s umed that each dimens ion of the res pons e vector Y repres ents an electrode-recording\nfrom a s ingle neuron. However, the vector Y could als o repres ent any other multi-dimens ional mea-\ns urement of brain activity: For example, imaging modalities s uch as voltage-s ens itive dye imaging\nyield meas urements at multiple pixels s imultaneous ly. Data from electro-phys iological data, e. g. lo-\ncal \ufb01eld potentials , are often analyzed in frequency s pace, i. e. by looking at the energy of the s ignal\nin different frequency bands . This als o res ults in a multi-dimens ional repres entation of the s ignal.\nUs ing CCA, receptive \ufb01elds can readily be es timated from thes e kinds of repres entations without\nlimiting attention to s ingle channels or extracting neural events .\n\nAcknowledgments\n\nWe would like to thank A Gretton and J Eichhorn for us eful dis cus s ions , and F J\u00a8akel, J B utler and S Liebe for\ncomments on the manus cript.\n\nReferences\n\n[ 1 ] S . Akaho. A kernel method for canonical correlation analys is . In International Meeting ofPsychometric\n\nSociety, Osaka, 2001 .\n\n[ 2] F. R. B ach and M. I. Jordan. Kernel independent component analys is . Journal ofMachine Learning\n\nResearch, 3 : 1 : 48 , 2002.\n\n[ 3 ] G. Chechik, A. Globers on, N. Tis hby, and Y. Weis s . Information B ottleneck for Gaus s ian Variables . The\n\nJournal ofMachine Learning Research, 6: 1 65 \u20131 8 8 , 2005 .\n\n[ 4] S . Devries and D. B aylor. Mos aic Arrangement of Ganglion Cell Receptive Fields in Rabbit Retina.\n\nJournal ofNeurophysiology, 78 (4): 2048 \u20132060, 1 997.\n\n[ 5 ] K. Diamantaras and S . Kung. Cros s -correlation neural network models . Signal Processing, IEEE Trans-\n\nactions on, 42(1 1 ): 3 21 8 \u20133 223 , 1 994.\n\n[ 6] J. Eichhorn, A. Tolias , A. Zien, M. Kus s , C. E. Ras mus s en, J. Wes ton, N. Logothetis , and B . S ch \u00a8olkopf.\nPrediction on s pike data us ing kernel algorithms . In S . Thrun, L. S aul, and B . S ch \u00a8olkopf, editors , Advances\nin Neural Information Processing Systems 1 6. MIT Pres s , Cambridge, MA, 2004.\n\n[ 7] K. Fukumizu, F. R. B ach, and A. Gretton. S tatis tical cons is tency of kernel canonical correlation analys is .\n\nJournal ofMachine Learning Research, 2007.\n\n[ 8 ] T. Hofmann, B . S ch \u00a8olkopf, and A. S mola. Kernel methods in machine learning. Annals ofStatistics (in\n\npress), 2007.\n\n[ 9] H. Hotelling. Relations between two s ets of variates . Biometrika, 28 : 3 21 \u20133 77, 1 93 6.\n\n[ 1 0] T. Melzer, M. Reiter, and H. B is chof. Nonlinear feature extraction us ing generalized canonical correlation\nanalys is . In Proc. ofInternational Conference on Arti\ufb01cial Neural Networks (ICANN), pages 3 5 3 \u20133 60, 8\n2001 .\n\n[ 1 1 ] L. Panins ki. Convergence properties of three s pike-triggered analys is techniques . Network, 1 4(3 ): 43 7\u201364,\n\nAug 2003 .\n\n[ 1 2] J. W. Pillow, L. Panins ki, V. J. Uzzell, E. P. S imoncelli, and E. J. Chichilnis ky. Prediction and decoding\nof retinal ganglion cell res pons es with a probabilis tic s piking model. J Neurosci, 25 (47): 1 1 003 \u20131 3 , 2005 .\n[ 1 3 ] J. W. Pillow and E. P. S imoncelli. Dimens ionality reduction in neural models : an information-theoretic\n\ngeneralization of s pike-triggered average and covariance analys is . J Vis, 6(4): 41 4\u201328 , 2006.\n\n[ 1 4] M. J. S chnitzer and M. Meis ter. Multineuronal \ufb01ring patterns in the s ignal from eye to brain. Neuron,\n\n3 7(3 ): 499\u20135 1 1 , 2003 .\n\n[ 1 5 ] O. S chwartz, J. W. Pillow, N. C. Rus t, and E. P. S imoncelli. S pike-triggered neural characterization. J\n\nVis, 6(4): 48 4\u20135 07, 2006.\n\n[ 1 6] H. S . S eung and H. S ompolins ky. S imple models for reading neuronal population codes . Proc Natl Acad\n\nSci U S A, 90(22): 1 0749\u20135 3 , 1 993 .\n\n[ 1 7] T. S harpee, N. Rus t, and W. B ialek. Analyzing neural res pons es to natural s ignals : maximally informative\n\ndimens ions . Neural Comput, 1 6(2): 223 \u20135 0, 2004.\n\n[ 1 8 ] H. S ompolins ky, H. Yoon, K. Kang, and M. S hamir. Population coding in neuronal s ys tems with corre-\n\nlated nois e. Phys Rev E Stat Nonlin Soft Matter Phys, 64(5 Pt 1 ): 05 1 904, 2001 .\n\n[ 1 9] J. Victor. S pike train metrics . Curr Opin Neurobiol, 1 5 (5 ): 5 8 5 \u201392, 2005 .\n[ 20] H. W\u00a8as s le. Parallel proces s ing in the mammalian retina. Nat Rev Neurosci, 5 (1 0): 747\u20135 7, 2004.\n[ 21 ] G. M. Zeck, Q. Xiao, and R. H. Mas land. The s patial \ufb01ltering properties of local edge detectors and\n\nbris k-s us tained retinal ganglion cells . Eur J Neurosci, 22(8 ): 201 6\u201326, 2005 .\n\n[ 22] K. Zhang and T. S ejnows ki. Neuronal Tuning: To S harpen or B roaden? , 1 999.\n\n8\n\n\f", "award": [], "sourceid": 234, "authors": [{"given_name": "Guenther", "family_name": "Zeck", "institution": null}, {"given_name": "Matthias", "family_name": "Bethge", "institution": null}, {"given_name": "Jakob", "family_name": "Macke", "institution": null}]}