{"title": "Analyzing Auditory Neurons by Learning Distance Functions", "book": "Advances in Neural Information Processing Systems", "page_first": 1481, "page_last": 1488, "abstract": "", "full_text": "Analyzing Auditory Neurons by Learning\n\nDistance Functions\n\nInna Weiner1\n\nTomer Hertz1,2\n\nIsrael Nelken2,3\n\nDaphna Weinshall1,2\n\n1School of Computer Science and Engineering,\n\n2The Center for Neural Computation, 3Department of Neurobiology,\n\nThe Hebrew University of Jerusalem, Jerusalem, Israel, 91904\n\nweinerin,tomboy,daphna@cs.huji.ac.il,israel@md.huji.ac.il\n\nAbstract\n\nWe present a novel approach to the characterization of complex sensory\nneurons. One of the main goals of characterizing sensory neurons is\nto characterize dimensions in stimulus space to which the neurons are\nhighly sensitive (causing large gradients in the neural responses) or al-\nternatively dimensions in stimulus space to which the neuronal response\nare invariant (de\ufb01ning iso-response manifolds). We formulate this prob-\nlem as that of learning a geometry on stimulus space that is compatible\nwith the neural responses: the distance between stimuli should be large\nwhen the responses they evoke are very different, and small when the re-\nsponses they evoke are similar. Here we show how to successfully train\nsuch distance functions using rather limited amount of information. The\ndata consisted of the responses of neurons in primary auditory cortex\n(A1) of anesthetized cats to 32 stimuli derived from natural sounds. For\neach neuron, a subset of all pairs of stimuli was selected such that the\nresponses of the two stimuli in a pair were either very similar or very\ndissimilar. The distance function was trained to \ufb01t these constraints. The\nresulting distance functions generalized to predict the distances between\nthe responses of a test stimulus and the trained stimuli.\n\n1\n\nIntroduction\n\nA major challenge in auditory neuroscience is to understand how cortical neurons represent\nthe acoustic environment. Neural responses to complex sounds are idiosyncratic, and small\nperturbations in the stimuli may give rise to large changes in the responses. Furthermore,\ndifferent neurons, even with similar frequency response areas, may respond very differently\nto the same set of stimuli. The dominant approach to the functional characterization of\nsensory neurons attempts to predict the response of the cortical neuron to a novel stimulus.\nPrediction is usually estimated from a set of known responses of a given neuron to a set of\nstimuli (sounds). The most popular approach computes the spectrotemporal receptive \ufb01eld\n(STRF) of each neuron, and uses this linear model to predict neuronal responses. However,\nSTRFs have been recently shown to have low predictive power [10, 14].\n\nIn this paper we take a different approach to the characterization of auditory cortical neu-\nrons. Our approach attempts to learn the non-linear warping of stimulus space that is in-\n\n\fduced by the neuronal responses. This approach is motivated by our previous observations\n[3] that different neurons impose different partitions of the stimulus space, which are not\nnecessarily simply related to the spectro-temporal structure of the stimuli. More speci\ufb01-\ncally, we characterize a neuron by learning a pairwise distance function over the stimulus\ndomain that will be consistent with the similarities between the responses to different stim-\nuli, see Section 2. Intuitively a good distance function would assign small values to pairs\nof stimuli that elicit a similar neuronal response, and large values to pairs of stimuli that\nelicit different neuronal responses.\n\nThis approach has a number of potential advantages: First, it allows us to aggregate infor-\nmation from a number of neurons, in order to learn a good distance function even when the\nnumber of known stimuli responses per neuron is small, which is a typical concern in the\ndomain of neuronal characterization. Second, unlike most functional characterizations that\nare limited to linear or weakly non-linear models, distance learning can approximate func-\ntions that are highly non-linear. Finally, we explicitly learn a distance function on stimulus\nspace; by examining the properties of such a function, it may be possible to determine the\nstimulus features that most strongly in\ufb02uence the responses of a cortical neuron. While\nthis information is also implicitly incorporated into functional characterizations such as the\nSTRF, it is much more explicit in our new formulation.\n\nIn this paper we therefore focus on two questions: (1) Can we learn distance functions\nover the stimulus domain for single cells using information extracted from their neuronal\nresponses?? and (2) What is the predictive power of these cell speci\ufb01c distance functions\nwhen presented with novel stimuli? In order to address these questions we used extracellu-\nlar recordings from 22 cells in the auditory cortex of cats in response to natural bird chirps\nand some modi\ufb01ed versions of these chirps [1]. To estimate the distance between responses,\nwe used a normalized distance measure between the peri-stimulus time histograms of the\nresponses to the different stimuli.\n\nOur results, described in Section 4, show that we can learn compatible distance functions on\nthe stimulus domain with relatively low training errors. This result is interesting by itself as\na possible characterization of cortical auditory neurons, a goal which eluded many previous\nstudies [3]. Using cross validation, we measure the test error (or predictive power) of\nour method, and report generalization power which is signi\ufb01cantly higher than previously\nreported for natural stimuli [10]. We then show that performance can be further improved\nby learning a distance function using information from pairs of related neurons. Finally, we\nshow better generalization performance for wide-band stimuli as compared to narrow-band\nstimuli. These latter two contributions may have some interesting biological implications\nregarding the nature of the computations done by auditory cortical neurons.\n\nRelated work Recently, considerable attention has been focused on spectrotemporal re-\nceptive \ufb01elds (STRFs) as characterizations of the function of auditory cortical neurons\n[8, 4, 2, 11, 16].\nThe STRF model is appealing in several respects: it is a conceptu-\nally simple model that provides a linear description of the neuron\u2019s behavior. It can be\ninterpreted both as providing the neuron\u2019s most ef\ufb01cient stimulus (in the time-frequency\ndomain), and also as the spectro-temporal impulse response of the neuron [10, 12]. Finally,\nSTRFs can be ef\ufb01ciently estimated using simple algebraic techniques.\n\nHowever, while there were initial hopes that STRFs would uncover relatively complex\nresponse properties of cortical neurons, several recent reports of large sets of STRFs of\ncortical neurons concluded that most STRFs are somewhat too simple [5], and that STRFs\nare typically rather sluggish in time, therefore missing the highly precise synchronization\nof some cortical neurons [11]. Furthermore, when STRFs are used to predict neuronal\nresponses to natural stimuli they often fail to predict the correct responses [10, 6]. For\nexample, in Machens et al. only 11% of the response power could be predicted by STRFs\non average [10]. Similar results were also reported in [14], who found that STRF models\n\n\faccount for only 18 \u2212 40% (on average) of the stimulus related power in auditory cortical\nneural responses to dynamic random chord stimuli. Various other studies have shown that\nthere are signi\ufb01cant and relevant non-linearities in auditory cortical responses to natural\nstimuli [13, 1, 9, 10]. Using natural sounds, Bar-Yosef et. al [1] have shown that auditory\nneurons are extremely sensitive to small perturbations in the (natural) acoustic context.\nClearly, these non-linearities cannot be suf\ufb01ciently explained using linear models such as\nthe STRF.\n\n2 Formalizing the problem as a distance learning problem\n\nOur approach is based on the idea of learning a cell-speci\ufb01c distance function over the space\nof all possible stimuli, relying on partial information extracted from the neuronal responses\nof the cell. The initial data consists of stimuli and the resulting neural responses. We use\nthe neuronal responses to identify pairs of stimuli to which the neuron responded similarly\nand pairs to which the neuron responded very differently. These pairs can be formally\ndescribed by equivalence constraints. Equivalence constraints are relations between pairs\nof datapoints, which indicate whether the points in the pair belong to the same category or\nnot. We term a constraint positive when they points are known to originate from the same\nclass, and negative belong to different classes. In this setting the goal of the algorithm is to\nlearn a distance function that attempts to comply with the equivalence constraints.\n\nThis formalism allows us to combine information from a number of cells to improve\nthe resulting characterization. Speci\ufb01cally, we combine equivalence constraints gathered\nfrom pairs of cells which have similar responses, and train a single distance function for\nboth cells. Our results demonstrate that this approach improves prediction results of the\n\u201cweaker\u201d cell, and almost always improves the result of the \u201cstronger\u201d cell in each pair.\nAnother interesting result of this formalism is the ability to classify stimuli based on the\nresponses of the total recorded cortical cell ensemble. For some stimuli, the predictive\nperformance based on the learned inter-stimuli distance was very good, whereas for other\nstimuli it was rather poor. These differences were correlated with the acoustic structure of\nthe stimuli, partitioning them into narrowband and wideband stimuli.\n\n3 Methods\n\nExperimental setup Extracellular recordings were made in primary auditory cortex of\nnine halothane-anesthetized cats. Anesthesia was induced by ketamine and xylazine and\nmaintained with halothane (0.25-1.5%) in 70% N2O using standard protocols authorized\nby the committee for animal care and ethics of the Hebrew University - Haddasah Medical\nSchool. Single neurons were recorded using metal microelectrodes and an online spike\nsorter (MSD, alpha-omega). All neurons were well separated. Penetrations were performed\nover the whole dorso-ventral extent of the appropriate frequency slab (between about 2 and\n8 kHz). Stimuli were presented 20 times using sealed, calibrated earphones at 60-80 dB\nSPL, at the preferred aurality of the neurons as determined using broad-band noise bursts.\nSounds were taken from the Cornell Laboratory of Ornithology and have been selected\nas in [1]. Four stimuli, each of length 60-100 ms, consisted of a main tonal component\nwith frequency and amplitude modulation and of a background noise consisting of echoes\nand unrelated components. Each of these stimuli was further modi\ufb01ed by separating the\nmain tonal component from the noise, and by further separating the noise into echoes and\nbackground. All possible combinations of these components were used here, in addition\nto a stylized arti\ufb01cial version that lacked the amplitude modulation of the natural sound.\nIn total, 8 versions of each stimulus were used, and therefore each neuron had a dataset\nconsisting of 32 datapoints. For more detailed methods, see Bar-Yosef et al. [1].\n\n\fData representation We used the \ufb01rst 60 ms of each stimulus. Each stimulus was rep-\nresented using the \ufb01rst d real Cepstral coef\ufb01cients. The real Cepstrum of a signal x was\ncalculated by taking the natural logarithm of magnitude of the Fourier transform of x and\nthen computing the inverse Fourier transform of the resulting sequence. In our experiments\nwe used the \ufb01rst 21-30 coef\ufb01cients. Neuronal responses were represented by creating Peri-\nStimulus Time Histograms (PSTHs) using 20 repetitions recorded for each stimuli. Re-\nsponse duration was 100 ms.\n\nObtaining equivalence constraints over stimuli pairs The distances between responses\nwere measured using a normalized \u03c72 distance measure. All responses to both stimuli (40\nresponses in total) were superimposed to generate a single high-resolution PSTH. Then, this\nPSTH was non-uniformly binned so that each bin contained at least 10 spikes. The same\nbins were then used to generate the PSTHs of the responses to the two stimuli separately.\nFor similar responses, we would expect that on average each bin in these histograms would\n1,ri\ncontain 5 spikes. Formally, let N denote the number of bins in each histogram, and let ri\n2\ndenote the number of spikes in the i\u2019th bin in each of the two histograms respectively. The\n2)2\ndistance between pairs of histograms is given by: \u03c72(ri\n2)/2 /(N \u2212 1).\nIn order to identify pairs (or small groups) of similar responses, we computed the normal-\nized \u03c72 distance matrix over all pairs of responses, and used the complete-linkage algorithm\nto cluster the responses into 8 \u2212 12 clusters. All of the points in each cluster were marked\nas similar to one another, thus providing positive equivalence constraints. In order to obtain\nnegative equivalence constraints, for each cluster ci we used the 2 \u22123 furthest clusters from\nit to de\ufb01ne negative constraints. All pairs, composed of a point from cluster ci and another\npoint from these distant clusters, were used as negative constraints.\n\n2) = PN\n\n1, ri\n\n(ri\n(ri\n\n1\u2212ri\n1+ri\n\ni=1\n\nDistance learning method In this paper, we use the DistBoost algorithm [7], which is\na semi-supervised boosting learning algorithm that learns a distance function using unla-\nbeled datapoints and equivalence constraints. The algorithm boosts weak learners which\nare soft partitions of the input space, that are computed using the constrained Expectation-\nMaximization (cEM) algorithm [15]. The DistBoost algorithm, which is brie\ufb02y summa-\nrized in 1, has been previously used in several different applications and has been shown\nto perform well [7, 17].\n\nEvaluation methods\nIn order to evaluate the quality of the learned distance function,\nwe measured the correlation between the distances computed by our distance learning al-\ngorithm to those induced by the \u03c72 distance over the responses. For each stimulus we\nmeasured the distances to all other stimuli using the learnt distance function. We then com-\nputed the rank-order (Spearman) correlation coef\ufb01cient between these learnt distances in\nthe stimulus domain and the \u03c72 distances between the appropriate responses. This proce-\ndure produced a single correlation coef\ufb01cient for each of the 32 stimuli, and the average\ncorrelation coef\ufb01cient across all stimuli was used as the overall performance measure.\n\nParameter selection The following parameters of the DistBoost algorithm can be \ufb01ne-\ntuned: (1) the input dimensionality d = 21-30, (2) the number of Gaussian models in\neach weak learner M = 2-4, (3) the number of clusters used to extract equivalence con-\nstraints C = 8-12, and (4) the number of distant clusters used to de\ufb01ne negative constraints\nnumAnti = 2-3. Optimal parameters were determined separately for each of the 22 cells,\nbased solely on the training data. Speci\ufb01cally, in the cross-validation testing we used a\nvalidation paradigm: Using the 31 training stimuli, we removed an additional datapoint\nand trained our algorithm on the remaining 30 points. We then validated its performance\nusing the left out datapoint. The optimal cell speci\ufb01c parameters were determined using\nthis approach.\n\n\fAlgorithm 1 The DistBoost Algorithm\nInput:\n\nData points: (x1, ..., xn), xk \u2208 X\nA set of equivalence constraints: (xi1 , xi2 , yi), where yi \u2208 {\u22121, 1}\nUnlabeled pairs of points: (xi1 , xi2 , yi = \u2217), implicitly de\ufb01ned by all unconstrained pairs of points\n\n\u2022 Initialize W 1\n\n= 1/(n2) i1, i2 = 1, . . . , n (weights over pairs of points)\n\ni1 i2\n\nwk = 1/n\n\nk = 1, . . . , n (weights over data points)\n\n\u2022 For t = 1, .., T\n\n1. Fit a constrained GMM (weak learner) on weighted data points in X using the equivalence constraints.\n2. Generate a weak hypothesis \u02dcht\n\n: X \u00d7 X \u2192 [\u22121, 1] and de\ufb01ne a weak distance function as\n\nht(xi, xj ) = 1\n\n2 \u201c1 \u2212 \u02dcht(xi, xj )\u201d \u2208 [0, 1]\n\n3. Compute rt =\n\n,xi2\nhypothesis only if rt > 0.\n\n(xi1\n\n,yi=\u00b11)\n\nP\n\nW t\n\ni1 i2\n\nyiht(xi1 , xi2 ), only over labeled pairs. Accept the current\n\n4. Choose the hypothesis weight \u03b1t = 1\n5. Update the weights of all points in X \u00d7 X as follows:\n\n2 ln( 1+rt\n1\u2212rt\n\n)\n\nW t+1\ni1 i2\n\n= ( W t\n\nW t\n\ni1 i2\n\ni1 i2\n\nexp(\u2212\u03b1tyi\u02dcht(xi1 , xi2 ))\nexp(\u2212\u03b1t)\n\nyi \u2208 {\u22121, 1}\nyi = \u2217\n\n6. Normalize: W t+1\ni1 i2\n\n=\n\nW\n\nt+1\ni1 i2\n\nn\n\nPi1 ,i2=1\n\nW\n\nt+1\ni1 i2\n\n7. Translate the weights from X \u00d7 X to X : wt+1\n\nk = Pj W t+1\n\nkj\n\nt=1 \u03b1tht(xi, xj )\n\nOutput: A \ufb01nal distance function D(xi, xj ) = PT\n4 Results\n\nCell-speci\ufb01c distance functions We begin our analysis with an evaluation of the \ufb01tting\npower of the method, by training with the entire set of 32 stimuli (see Fig. 1). In gen-\neral almost all of the correlation values are positive and they are quite high. The average\ncorrelation over all cells is 0.58 with ste = 0.023.\nIn order to evaluate the generalization potential of our approach, we used a Leave-One-\nOut (LOU) cross-validation paradigm. In each run, we removed a single stimulus from the\ndataset, trained our algorithm on the remaining 31 stimuli, and then tested its performance\non the datapoint that was left out (see Fig. 3). In each histogram we plot the test correlations\nof a single cell, obtained when using the LOU paradigm over all of the 32 stimuli. As can\nbe seen, on some cells our algorithm obtains correlations that are as high as 0.41, while\nfor other cells the average test correlation is less then 0.1. The average correlation over all\ncells is 0.26 with ste = 0.019.\nNot surprisingly, the train results (Fig. 1) are better than the test results (Fig. 3). Inter-\nestingly, however, we found that there was a signi\ufb01cant correlation between the training\nperformance and the test performance C = 0.57, p < 0.05 (see Fig. 2, left).\n\nBoosting the performance of weak cells\nIn order to boost the performance of cells with\nlow average correlations, we constructed the following experiment: We clustered the re-\nsponses of each cell, using the complete-linkage algorithm over the \u03c72 distances with 4\nclusters. We then used the F 1\nscore that evaluates how well two clustering partitions are\nin agreement with one another (F 1\nP +R , where P denotes precision and R denotes\nrecall.). This measure was used to identify pairs of cells whose partition of the stimuli\nwas most similar to each other. In our experiment we took the four cells with the lowest\n\n= 2\u2217P \u2217R\n\n2\n\n2\n\n\fAll cells\n\n\u22121\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\nCell 13\n\nCell 18\n\n30\n\n25\n\n20\n\n15\n\n10\n\n5\n\n0\n\u22121\n\n30\n\n25\n\n20\n\n15\n\n10\n\n5\n\n0\n\u22121\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\nFigure 1: Left: Histogram of train rank-order correlations on the entire ensemble of cells. The\nrank-order correlations were computed between the learnt distances and the distances between the\nrecorded responses for each single stimulus (N = 22 \u2217 32). Center: train correlations for a \u201cstrong\u201d\ncell. Right: train correlations for a \u201cweak\u201d cell. Dotted lines represent average values.\n\n0.8\n\n0.6\n\nn\no\n\ni\nt\n\nl\n\na\ne\nr\nr\no\nc\n \nt\ns\ne\nT\n\n0.4\n\n0.2\n\n0\n0\n\n0.2\n\n0.4\n\n0.6\n\nTrain correlation\n\nn\no\n\ni\nt\n\nl\n\na\ne\nr\nr\no\nC\n\n \nr\ne\nd\nr\nO\n\u2212\nk\nn\na\nR\n\n \n\n \nt\ns\ne\nT\nn\na\ne\nM\n\n0.5\n\n0.4\n\n0.3\n\n0.2\n\n0.1\n\n0\n\n16 20\n\n0.8\n\n1\n\nOriginal constraints\nIntersection of const.\n\n18 14\n\n25 38\n\nCell number\n\n37 19\n\nFigure 2: Left: Train vs. test cell speci\ufb01c correlations. Each point marks the average correlation of a\nsingle cell. The correlation between train and test is 0.57 with p = 0.05. The distribution of train and\ntest correlations is displayed as histograms on the top and on the right respectively. Right: Test rank-\norder correlations when training using constraints extracted from each cell separately, and when using\nthe intersection of the constraints extracted from a pair of cells. This procedure always improves the\nperformance of the weaker cell, and usually also improves the performance of the stronger cell\n\nscore to retrieve the\nperformance (right column of Fig 3), and for each of them used the F 1\nmost similar cell. For each of these pairs, we trained our algorithm once more, using the\nconstraints obtained by intersecting the constraints derived from the two cells in the pair,\nin the LOU paradigm. The results can be seen on the right plot in Fig 2. On all four cells,\nthis procedure improved LOUT test results.\nInterestingly and counter-intuitively, when\ntraining the better performing cell in each pair using the intersection of its constraints with\nthose from the poorly performing cell, results deteriorated only for one of the four better\nperforming cells.\n\n2\n\nStimulus classi\ufb01cation The cross-validation results induced a partition of the stimulus\nspace into narrowband and wideband stimuli. We measured the predictability of each stim-\nulus by averaging the LOU test results obtained for the stimulus across all cells (see Fig. 4).\nOur analysis shows that wideband stimuli are more predictable than narrowband stimuli,\ndespite the fact that the neuronal responses to these two groups are not different as a whole.\nWhereas the non-linearity in the interactions between narrowband and wideband stimuli\nhas already been noted before [9], here we further re\ufb01ne this observation by demonstrating\na signi\ufb01cant difference between the behavior of narrow and wideband stimuli with respect\nto the predictability of the similarity between their responses.\n\n5 Discussion\n\nIn the standard approach to auditory modeling, a linear or weakly non-linear model is \ufb01tted\nto the data, and neuronal properties are read from the resulting model. The usefulness of\nthis approach is limited however by the weak predictability of A1 responses when using\nsuch models. In order to overcome this limitation, we reformulated the problem of char-\n\n\f15\n\n10\n\n5\n\n0\n\u22121\n\n15\n\n10\n\n5\n\n0\n\u22121\n\n15\n\n10\n\n5\n\n0\n\u22121\n\n15\n\n10\n\n5\n\n0\n\u22121\n\nCell 49\n\nCell 15\n\nCell 52\n\nCell 2\n\nCell 25\n\nCell 37\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\nCell 38\n\nCell 24\n\nCell 19\n\nCell 3\n\nCell 1\n\nCell 16\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\nCell 54\n\nCell 20\n\nCell 13\n\nCell 17\n\nCell 51\n\nCell 18\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\nCell 14\n\n15\n\n10\n\n5\n\n0\n\u22121\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\nCell 36\n\nCell 21\n\nCell 48\n\n15\n\n10\n\n5\n\n15\n\n10\n\n5\n\nAll cells\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n0\n\u22121\n\n1\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\n\u22121\n\n\u22120.5\n\n0\n\n0.5\n\n1\n\nFigure 3: Histograms of cell speci\ufb01c test rank-order correlations for the 22 cells in the dataset. The\nrank-order correlations compare the predicted distances to the distances between the recorded re-\nsponses, measured on a single stimulus which was left out during the training stage. For visualization\npurposes, cells are ordered (columns) by their average test correlation per stimulus in descending\norder. Negative correlations are in yellow, positive in blue.\n\nacterizing neuronal responses of highly non-linear neurons. We use the neural data as a\nguide for training a highly non-linear distance function on stimulus space, which is com-\npatible with the neural responses. The main result of this paper is the demonstration of the\nfeasibility of this approach.\n\nTwo further results underscore the usefulness of the new formulation. First, we demon-\nstrated that we can improve the test performance of a distance function by using constraints\non the similarity or dissimilarity between stimuli derived from the responses of multiple\nneurons. Whereas we expected this manipulation to improve the test performance of the\nalgorithm on the responses of neurons that were initially poorly predicted, we found that it\nactually improved the performance of the algorithm also on neurons that were rather well\npredicted, although we paired them with neurons that were poorly predicted. Thus, it is\npossible that intersecting constraints derived from multiple neurons uncover regularities\nthat are hard to extract from individual neurons.\n\nSecond, it turned out that some stimuli consistently behaved better than others across the\nneuronal population. This difference was correlated with the acoustic structure of the stim-\nuli: those stimuli that contained the weak background component (wideband stimuli) were\ngenerally predicted better. This result is surprising both because background component\nis substantially weaker than the other acoustic components in the stimuli (by as much as\n35-40 dB). It may mean that the relationships between physical structure (as characterized\nby the Cepstral parameters) and the neuronal responses becomes simpler in the presence\nof the background component, but is much more idiosyncratic when this component is ab-\nsent. This result underscores the importance of interactions between narrow and wideband\nstimuli for understanding the complexity of cortical processing.\n\nThe algorithm is fast enough to be used in near real-time. It can therefore be used to guide\nreal experiments. One major problem during an experiment is that of stimulus selection:\nchoosing the best set of stimuli for characterizing the responses of a neuron. The distance\nfunctions trained here can be used to direct this process. For example, they can be used to\n\n\fMain\n\nNatural\n\n)\nz\nH\nk\n(\n \ny\nc\nn\ne\nu\nq\ne\nr\nF\n\n)\nz\nH\nk\n(\n \ny\nc\nn\ne\nu\nq\ne\nr\nF\n\n20\n\n10\n\n0\n\n20\n\n10\n\n0\n\n)\nz\nH\nk\n(\n \ny\nc\nn\ne\nu\nq\ne\nr\nF\n\n)\nz\nH\nk\n(\n \ny\nc\nn\ne\nu\nq\ne\nr\nF\n\n20\n\n10\n\n0\n\n20\n\n10\n\n0\n\n100\n\n50\n\nTime (ms)\n\nEcho\n\n50\n\nTime (ms)\n\n100\n\n50\n\nTime (ms)\n\n100\n\nBackground\n\n50\n\nTime (ms)\n\n100\n\nNarrowband\n\nWideband\n\n0.8\n\n0.7\n\n0.6\n\n0.5\n\n0.4\n\n0.3\n\n0.2\n\n0.1\n\n0\n\n \n\n \n\nNarrow band\nWide band\n\n0.1\n0.6\nA1 Stimuli Mean Test Rank\u2212Order Correlation\n\n0.4\n\n0.5\n\n0.2\n\n0.3\n\nFigure 4: Left: spectrograms of input stimuli, which are four different versions of a single natural\nbird chirp. Right: Stimuli speci\ufb01c correlation values averaged over the entire ensemble of cells. The\npredictability of wideband stimuli is clearly better than that of the narrowband stimuli.\n\n\ufb01nd surprising stimuli: either stimuli that are very different in terms of physical structure\nbut that would result in responses that are similar to those already measured, or stimuli that\nare very similar to already tested stimuli but that are predicted to give rise to very different\nresponses.\n\nReferences\n[1] O. Bar-Yosef, Y. Rotman, and I. Nelken. Responses of Neurons in Cat Primary Auditory Cortex to Bird Chirps: Effects of\n\nTemporal and Spectral Context. J. Neurosci., 22(19):8619\u20138632, 2002.\n\n[2] D. T. Blake and M. M. Merzenich. Changes of AI Receptive Fields With Sound Density. J Neurophysiol, 88(6):3409\u20133420,\n\n2002.\n\n[3] G. Chechik, A. Globerson, M.J. Anderson, E.D. Young, I. Nelken, and N. Tishby. Group redundancy measures reveal\n\nredundancy reduction in the auditory pathway. In NIPS, 2002.\n\n[4] R. C. deCharms, D. T. Blake, and M. M. Merzenich. Optimizing Sound Features for Cortical Neurons.\n\nScience,\n\n280(5368):1439\u20131444, 1998.\n\n[5] D. A. Depireux, J. Z. Simon, D. J. Klein, and S. A. Shamma. Spectro-Temporal Response Field Characterization With\n\nDynamic Ripples in Ferret Primary Auditory Cortex. J Neurophysiol, 85(3):1220\u20131234, 2001.\n\n[6] J. J. Eggermont, P. M. Johannesma, and A. M. Aertsen. Reverse-correlation methods in auditory research. Q Rev Biophys.,\n\n16(3):341\u2013414, 1983.\n\n[7] T. Hertz, A. Bar-Hillel, and D. Weinshall. Boosting margin based distance functions for clustering. In ICML, 2004.\n\n[8] N. Kowalski, D. A. Depireux, and S. A. Shamma. Analysis of dynamic spectra in ferret primary auditory cortex. I. Charac-\n\nteristics of single-unit responses to moving ripple spectra. J Neurophysiol, 76(5):3503\u20133523, 1996.\n\n[9] L. Las, E. A. Stern, and I. Nelken. Representation of Tone in Fluctuating Maskers in the Ascending Auditory System. J.\n\nNeurosci., 25(6):1503\u20131513, 2005.\n\n[10] C. K. Machens, M. S. Wehr, and A. M. Zador. Linearity of Cortical Receptive Fields Measured with Natural Sounds. J.\n\nNeurosci., 24(5):1089\u20131100, 2004.\n\n[11] L. M. Miller, M. A. Escabi, H. L. Read, and C. E. Schreiner. Spectrotemporal Receptive Fields in the Lemniscal Auditory\n\nThalamus and Cortex. J Neurophysiol, 87(1):516\u2013527, 2002.\n\n[12]\n\nI. Nelken. Processing of complex stimuli and natural scenes in the auditory cortex. Current Opinion in Neurobiology,\n14(4):474\u2013480, 2004.\n\n[13] Y. Rotman, O. Bar-Yosef, and I. Nelken. Relating cluster and population responses to natural sounds and tonal stimuli in\n\ncat primary auditory cortex. Hearing Research, 152(1-2):110\u2013127, 2001.\n\n[14] M. Sahani and J. F. Linden. How linear are auditory cortical responses? In NIPS, 2003.\n\n[15] N. Shental, A. Bar-Hilel, T. Hertz, and D. Weinshall. Computing Gaussian mixture models with EM using equivalence\n\nconstraints. In NIPS, 2003.\n\n[16] F. E. Theunissen, K. Sen, and A. J. Doupe. Spectral-Temporal Receptive Fields of Nonlinear Auditory Neurons Obtained\n\nUsing Natural Sounds. J. Neurosci., 20(6):2315\u20132331, 2000.\n\n[17] C. Yanover and T. Hertz. Predicting protein-peptide binding af\ufb01nity by learning peptide-peptide distance functions.\n\nIn\n\nRECOMB, 2005.\n\n\f", "award": [], "sourceid": 2893, "authors": [{"given_name": "Inna", "family_name": "Weiner", "institution": null}, {"given_name": "Tomer", "family_name": "Hertz", "institution": null}, {"given_name": "Israel", "family_name": "Nelken", "institution": null}, {"given_name": "Daphna", "family_name": "Weinshall", "institution": null}]}