{"title": "Statistical Modeling of Cell Assemblies Activities in Associative Cortex of Behaving Monkeys", "book": "Advances in Neural Information Processing Systems", "page_first": 945, "page_last": 952, "abstract": null, "full_text": "Statistical Modeling of Cell-Assemblies \n\nActivities in Associative Cortex of \n\nBehaving Monkeys \n\nItay Gat and Naftali Tishby \nInstitute of Computer Science and \nCenter for Neural Computation \n\nHebrew University, Jerusalem 91904, Israel * \n\nAbstract \n\nSo far there has been no general method for relating extracellular \nelectrophysiological measured activity of neurons in the associative \ncortex to underlying network or \"cognitive\" states. We propose \nto model such data using a multivariate Poisson Hidden Markov \nModel. We demonstrate the application of this approach for tem(cid:173)\nporal segmentation of the firing patterns, and for characterization \nof the cortical responses to external stimuli. Using such a statisti(cid:173)\ncal model we can significantly discriminate two behavioral modes \nof the monkey, and characterize them by the different firing pat(cid:173)\nterns, as well as by the level of coherency of their multi-unit firing \nactivity. \nOur study utilized measurements carried out on behaving Rhesus \nmonkeys by M. Abeles, E. Vaadia, and H. Bergman, of the Hadassa \nMedical School of the Hebrew University. \n\n1 \n\nIntroduction \n\nHebb hypothesized in 1949 that the basic information processing unit in the cortex \nis a cell-assembly which may include thousands of cells in a highly interconnected \nnetwork[l]. The cell-assembly hypothesis shifts the focus from the single cell to the \n\n* {itay,tishby }@cs.huji.ac.il \n\n945 \n\n\f946 \n\nGat and Tishby \n\ncomplete network activity. This view has led several laboratories to develop technol(cid:173)\nogy for simultaneous multi-cellular recording from a small region in the cortex[2, 3]. \nThere remains, however, a large discrepancy between our ability to construct neural(cid:173)\nnetwork models and their correspondence with such multi-cellular recordings. To \nsome extent this is due to the difficulty in observing simultaneous activity of any \nsignificant number of individual cells in a living nerve tissue. Extracellular elec(cid:173)\ntrophysiological measurements have so far obtained simultaneous recordings from \njust a few randomly selected cells (about 10), a negligibly small number compared \nto the size of the hypothesized cell-assembly. It is quite remarkable therefore, that \nsuch local measurements in the associative cortex have yielded so much information, \nsuch as synfire chains [2], multi-cell firing correlation[6], and statistical correlation \nbetween cell activity and external behavior. However, such observations have so \nfar relied mostly on the accumulated statistics of cell firing over a large number of \nrepeated experiments, to obtain any statistically significant effect. This is due to \nthe very low firing rates (about 10Hz) of individual cells in the associative cortex, \nas can be seen in figure 1. \n\n30~--------------------------~~------------------------~ \n\nO~--------------------------~----------------------------r \n\n111 1' \n\nI I ' \n\n\" '1 \n\nI \u2022 \u2022 \n\nI \n\nI \n\n\" \nI \n\n\" \n\n, \u2022 \u2022 \n\n\" \n, \nI \n\nI \n\nI \n\nI \n\n, \n\nI \n\n\u2022 \n\n. , \n\nI t ' t \n\n, ' , ,II , \u2022\u2022\u2022\u2022\u2022 , ' \" I \" \n\nI I II \" I \n\n\" , I \n\nI \n\nI , \n\nI \n\nI I I \n\n\u2022 \n\n\" \n\n\u2022 \nI \n\n111 11 . ' I \nI \n\n'\" \nI 1\" \n\n\u2022 \n\nI \n\nI \n\n\" . I I \n\nI I \n\nI \n\nI \n\n, \nI \n\" I ' \" \n\nI , I \n\n, \n\nI \n\nI \" . \" II I \n\nI \" ' \" \" . \" . .11 \n\nI I ,. \n\n\" \n\nI \n\n\u2022 \u2022 \u2022 ' \" \n\n\"\n\nI \n\n., \" ' \" \u2022 \n\nI I . \n\nI ' I \n\nI \n\nI \n\nI I .,1 I 110'\" \n\nI , , , \n\n, . , I \n\nI \n\ntI, \n\nI \n\nI \n. , \" \n\n\" \n\n, \n\nI I . ' , \" \n\nI \n\nI \n\" \n\n, \n\nI \n\nI \n\nI \n\n. , , \" \n\nI 111 \" ' \" \n\nI . I \n\n' \" \n\n\" \n\nI \n' \"1 \n\nI \n\nI \n\n\" \n\n\" \n\n\" \n\n\u2022 \u2022 , \n\n\" \n\nI \n\nI \u2022 \u2022 I \n\n\" \n\n\u2022 \n\nI I ' \u2022 , . \" ' I , \n, . \n\" \"1 ' \n\n.. \n\n\u2022 \n, \n\n, \n\nI \n\nI \n\n, . , ' I ' \n\n',.',',,\"'1 \" \n\n'\" \n\n:\" \n\n' I ' I ' \nI', \n\n\" \n\n, \n\u2022 \n\n',' \n\n\" \n\nI I I I \n\nI \n\nI I \n\nI \n\n, \n\n\u2022 \n\nI \n\nI \n\n, \n\nI \n\nI \n\nI \n\n1 '1 1,\n\n1111 \n1111 . \" .. \n\nI \n\nI \nI ' \" ' , \n\n1111 \n\n\" . , \u2022\u2022 ,'\" \n, \n\nI \n\nI \n\nI \n\" \nI,,, .. ,,, \u2022 \u2022 , \n\nI It , \n\n\" \n\nI II \n\nI I I ,.11 \" . \" l t l III I . \" , \n\nI.\" \u2022\u2022 \" .. , \"\" \". , \n\n111 \u2022\u2022 ,11' \n\nI \n\n' \" ' \u2022 \u2022 \" \n\nI , I \n\n\" \n\n, I \nI \n\u2022 \n\n, . \n\n.1111 I \n\nI\n\nI \n\nI \n\u2022 \n\nI \n\nI \n\nt \n'1\" \n\nI\n\nI \n\n\u2022 \n\nI . \n\nI III I ' \" \n\n\" \n\n\" \n\nI \n. , \n\nI \n\nI \n\n. , . , \n\" \nI \n\nI \n\nI \n\n\" \n\n'0 ' I \n\nI \n\nII \u2022 \u2022 I I I 1 , . , . ' \n\n\u2022\u2022 , . I \" , I \n\n\u2022 1\" \n\nII \n\nI \n\nI \nI \" \n\n\u2022 ' \" \n\nI \n\n\u2022 \n\n, \" \n\n, \n\nI \n\n, \u2022 \u2022 I \n\nI \n\nI \n\n' I I \n\n, \n\nI \n\u2022 \n\nII \nI \n\n. . . . . . . . I I , \nI \n\nI \n\n\u2022 \u2022 \u2022 III \" . \" .. ' , . , II \n\n, \n\nI\n\n. \n\n, \n\n\" \n\n, \n\n.. \n\nI \n\" \n. , ' , . , I , II , . , It, \" \" \" \" \n.1 \n\n\" \u2022 \u2022 ' \" \" ' \n\n'1 ' I . I I \n\nI \n\nI \n\nI \n\n\" . \" , \n\u2022 \n\nI \n\n\" \n\n' \" , \n,: , . , . \",.' .,' \n\n\" \" \" \n\nI \n\n\u2022 \u2022 11 \n\n,\"\\.'''' '.\" ,'\" \n\n\u2022 , . . . . \u2022 \n,. \n\nI \n\n\" \n\nI \n\n1111.,1\", \" ' \" , I I I \" \nI \n' \" \nI \u2022\n\n\u2022 \n\u2022 \u2022 \u2022 , ' \n\nI . \" \n\n\" \n\nI \n\nI \n\n.f! '\" \" \n\n\" \" \" I \n., \u2022 \u2022 , \n\n. , \n\u2022 \n, \u2022\u2022 \",. . \" . I I \n\n, . \" . \n\n'\"~ It \n.. \n\n\u2022 \n\nI \n\nI \n\n, \n\nI \n\nI \n\nI \n\nI \n\nII , . \n\n\u2022 \" \" \" \n\n,\",:,: ::::,::: ;:;i:~:;,;> ,:,~:::: \",:,> :,~~i:',i: ::;'~', ,: ,.:, ::, ,::::: :\\:~~:\";;:i;.:'\u00b7~~>::/~;:',:,: ',~'::,.,:':;~::' ~;:C:'~' ::.: ~: \"~;':::: ':, \n\n.. \" : ,:~',\"',\" .. ,:: '~::,\" \",',::,:1 ,',,: ',\"\",:~' \".':'\" :/:\",:' ': ,:,:, ,: ':' ,\" ,', :':':;:~3,,-::-,:::::~::~: '::: :::; ,:':.\" ::': :::</'T: .. ~':,::\"~::\",:'::':: ',:'::' ~: ,'J \n\nI \n\nI \n\n\u2022 \n\n, \n\n'I 01\" '~. ~: \n\n\"I \" ' \n\n\u2022 \n\n-2000 \n\no \n\nMiIliSec \n\n2000 \n\nFigure 1: An example of firing times of a single unit. Shown are 48 repetitions of \nthe same trial, aligned by the external stimulus marker, and drawn horizontally one \non top of another. The accumulated histogram estimates the firing rate in 50msec \nbins, and exhibits a clear increase of activity right after the stimulus. \n\nClearly, simultaneous measurements of the activity of 10 units contain more infor(cid:173)\nmation than single unit firing and pairwise correlations. The goal of the present \nstudy is to develop and evaluate a statistical method which can better capture the \nmulti- unit nature of this data, by treating it as a vector stochastic process. The \nfiring train of each of these units is conventionally modeled as a Poisson process \nwith a time-dependent average firing rate[2]. Estimating the firing rate parameter \nrequires careful averaging over a sliding window. The size of this window should be \nlong enough to include several spikes, and short enough to capture the variability. \n\n\fModeling Cell-Assemblies Activities in Associative Cortex of Behaving Monkeys \n\n947 \n\nWithin such a window the process is characterized by a vector of average rates, and \npossibly higher order correlations between the units. \n\nThe next step, in this framework, is to collect such vector-frames into statistically \nsimilar clusters, which should correspond to similar network activity, as reflected \nby the firing of these units. Furthermore, we can facilitate the well-established \nformulation of Hidden-Markov-Models[7] to estimate these \"hidden\" states of the \nnetwork activity, similarly to the application of such models to other stochastic \ndata, e.g. speech. The main advantage of this approach is its ability to characterize \nstatistically the multi-unit process, in an unsupervised manner, thus allowing for \nfiner discrimination of individual events. In this report we focus on the statistical \ndiscrimination of two behavioral modes, and demonstrate not only their distinct \nmulti-unit firing patterns, but also the fact that the coherency level of the firing \nactivity in these two modes is significantly different. \n\n2 Origin of the data \n\nThe data used for the present analysis was collected at the Hadassa Medical School, \nby recording from a Rhesus monkey Macaca Mulatta who was trained to perform a \nspatial delayed release task. In this task the monkey had to remember the location \nfrom which a stimulus was given and after a delay of 1-32 seconds, respond by \ntouching that location. Correct responses were reinforced by a drop of juice. After \ncompletion of the training period, the monkey was anesthetized and prepared for \nrecording of electrical activity in the frontal cortex. After the monkey recovered \nfrom the surgery the activity of the cortex was recorded, while the monkey was \nperforming the previously learned routine. Thus the recording does not reflect \nthe learning process, but rather the cortical activity of the well trained monkey \nwhile performing its task. During each of the recording sessions six microelectrodes \nwere used simultaneously. With the aid of two pattern detectors and four window(cid:173)\ninscriminates, the activity of up to 11 single units (neurons) was concomitantly \nrecorded. The recorded data contains the firing times of these units, the behavioral \nevents of the monkey, and the electro-occulogram (EOG)[5, 2,4]. \n\n2.1 Behavioral modes \n\nTo understand the results reported here it is important to focus on the behavioral \naspect of these experiments. The monkey was trained to perform a spatial delayed \nresponse task during which he had to alternate between two behavioral modes. The \nmonkey initiated the trial, by pressing a central key, and a fixation light was turned \non in front of it. Then after 3-6 seconds a visual stimulus was given either from the \nleft or from the right. The stimulus was presented for 100 millisec. After a delay \nthe fixation light was dimmed and the monkey had to touch the key from which the \nvisual stimulus came (\"Go\" mode), or keep his hand on the central key regardless \nof the external stimulus (\"No-Go\" mode). For the correct behavior the monkey was \nrewarded with a drop of juice. After 4 correct trials all the lights in front of the \nmonkey blinked (this is called \"switch\" henceforth), signaling the monkey to change \nthe behavioral mode - so that if started in the \"Go\" mode he now had to switch to \n\"No-Go\" mode, or vice versa. \n\n\f948 \n\nGat and Tishby \n\nThere is a clear statistical indication, based on the accumulated firing histograms, \nthat the firing patterns are different in these two modes. One of our main exper(cid:173)\nimental results so far is a more quantitative analysis of this observation, both in \nterms of the firing patterns directly, and by using a new measure of the coherency \nlevel of the firing activity. \n\n1 \n\nc5 \n\n.. t.' .. , ... ' \n\n5 \n\"-.... 1 . . '. \n.'.11. i.' .. , ' .. ' \n, \n\u00b7 .. ' ... .. lrll\u00b7.I\u00b7' 'I.\". \u00b7 ...... f .. .. . . '\nI'JI.II.'\" \nJ \u2022 J . 11.1 .. f' .', .I~ LUI'. , U. J. '. '. [ '. \" \n~ ... I .. ,., .... . \n. II\u00b7 . lit r 1I11b, n I' r Illlmlt I H' I I' '11 H. I , ~ II\u00b7 \u00b711 .. 11 \u00b7'\u00b7111 \n\u00b7 \u2022.\n. !.'.!!.'~I.l .... I.\" .\".'.f 11., .. 11 ... ' .1 .. II,' \n\u00b7 -' I. . I .ll.. .~. .. \n\n. '. \n\n. ' . \n\n' .\n\n5 \n\nII \"1 ' .. ' II \n\n.. ' . f ..\n\n. , . \n. .. l .. '\n. '~I\n. '. '\n\n'1' 1'1 II ill'lilli (,','lIlili I'li 11'11'1 f 1 ' ..\n\n.. ll'.\u00b7I'1I111nnHil . J ... .\nI. I, , . .II, .\n\n.' . !!J \n. ' . 1 . . '.) '. '.! ,I .. '. ,. !.' .' . \n~ 1\u00b711\u00b7 \u00b71' .1 \n, ... 1,1 .. 11' .... ' ... \\ . 1 )11 . .1.1,' .. 1 \n.I .. 1, .1 , . 1 !. '. \nI . .lIJ! .\" .... ',l'.II. !II. .. 1 . 'L .I.I \n\n. , 'I~ II ,\n\n.I . .:11 \n\n'.. . ' .. '. ! '. U .1.1 . . ., ,1 I! I I I,. .! ., .. I , I. , . .' . '. II \"I'! I . II. \n\n\u00b7 .. llI.IlJlli I.. ,,1.1)1 ... 1 l.I,I.I.i'lIHH\"I\u00b7 . J .,-,.L'.I,ILf ~\"\"'I!. \n1 . L . , L.l1 1, \n. ~ \n\n: , \n., \n.:- .. '. \" _ _ . _ _ l .. - -\n\n'\" \n- - .. . _ .' '\n\n: \n(, , \" \n\n, !, \n\nII' \n\n. _ . t \u00b7 _ . . . . . \u2022 . .\n\n- - \u2022\u2022 \n\n. -\n\n1 \n\n. \n\n\u2022 \n\" \n. . . . . , \n\n: \n,! \n\n.\n\n.\n\n\u2022\n\n. _ ' . _\n\nSvvitch \n\nc5 \n\n. _ \n\n'\n\n. \n\n. _ \n\n. _ \n\n. \n\n5 \n\n5 \n\nc5 \n\n\"7 \n\n. '1'1 .. 11 \n\n.'! .11' ,-' .. , 1.1.1 .'!:!:':::. ,'\" \" [II'.', ,1' .. 11,'1 .. J . II \u00b7 ;,;;,;,;,,:';: \n\n: : i \n(' \n\n, ,ill . . IlII,: 1. 1 .. .. '\n\n1 ' .. \n\n.. \n\nI \n\n' \n\n.\n\n'\n\n: \n\n: \n\n1 \n\n\" \n\n5 \n\"I. . q .. ,.'.~II,li , . , ,, \n\n..... 'Ii . , .. ~II~! \n. I \n\u00b7L, \"\",'I~.I .\n\n\u00b7 ,II \nIH ' ; \n\n\" , /\u00b7fl \u00b7 ;'\n\u00b7 _. \u2022. . r - - .\u2022 \u2022 to \n\n\u2022 -\n\n.\n\nII ,. Ll.'~'i1I1f1t i\" ,'.1.1'.1.\nI;R.,.1~11f,r\u00b7lrll\u00b7lIi\"'ill'- 'lIil\u00b7lill \u00b7 1t-1' , I.I:.IJ.'I;~;;I\";;~I:'l.llI:I\n. \n; \" \n\n.11 J. ,,'\" \n\n'\" \n\nI \n\nI, \n\n\u2022 ~ \u2022\u2022 \u2022\n\n.ILU 1.1 \u2022\u2022 ! \", I,. 'f \"! '\" . '\u00b7f ,-11 II' II . H '\u00b71, I. U JI . I U\u00b7 HI\u00b7 r . l '. J .. 11. \n\u00b7 J. !. i .-' .. 1~I.i .. .. ... .\n\n. .' .!' .. \n\n, _. \u2022\u2022 - . \u2022 -\n\n. 1 .\n\n- . -\n\nI \n\nI \n\n\u2022 \n\n. \n\n\u2022\n\n'\n\n-\n\n-\n\n, \n\n. \n\n\u2022 _. . : \u2022 -\n\n\" \"i .... ,l;'-,I, .. ,I,.,'.I.:.I\",l.,' !;I' :\n[I ..\u2022 ,! ... 1 .'1 ,~I .'. \n'.1 ... .' .. I . . ... ' .(11 ... 1 .. ~'.I .. I \n\n\u2022 \u2022\u2022 \u2022\u2022 .! . . _. -! .. . \n\n\u2022 \n\n1\n\n\"\n\n\"' 1 .. III.'.lJ.,.m\u00b7'H ii,'I1'\" \n: ; \n.I,J) \n',\" \n\n.! . ~ .. '.II.1.L \n:,I I\" \n\n~::,\n\n'-'I l.' .. ' .1.1. I L .I J \n\nI \n' .. - _ ... \n\nI \n\nII \n\nSvv'itch \n\nFigure 2: Multi-unit firing trains and their statistical segmentation by the model. \nShown are 4 sec. of activity, in two trials, near the \"switch\". Estimated firing rates \nfor each channel are also plotted on top of the firing spikes. The upper example is \ntaken from the training data, while the lower is outside of the training set. Shown \nare also the association probabilities for each of the 8 states of the model. The \nmonkey's cell-assembly clearly undergoes the state sequence \"1\", \"5\", \"6\", \"5\" in \nboth cases. Similar sequence was observed near the same marker in many (but not \nall) other instances of the same event during that measurement day. \n\n2.2 Method of analysis \n\nAs was indicated before, most of the statistical analysis so far was done by accu(cid:173)\nmulating the firing patterns from many trials, aligned by external markers. This \nsupervised mode of analysis can be understood from figure 1, where 48 different \n\n\fModeling Cell-Assemblies Activities in Associative Cortex of Behaving Monkeys \n\n949 \n\n\"Go\" firing trains of a single unit are aligned by the marker. There is a clear in(cid:173)\ncrease in the accumulated firing rate following the marker, indicating a response of \nthis unit to the stimulus. In contrast, we would like to obtain, in an unsupervised \nself organizing manner, a statistical characterization of the multi-unit firing activity \naround the marked stimuli, as well as in other unobserved cortical processes. We \nclaim to achieve this goal through characteristic sequences of Markov states. \n\n3 Multivariate Poisson Hidden Markov Model \n\nThe following statistical assumptions underlie our model. Channel firing is dis(cid:173)\ntributed according to a Poisson distribution. The distances between spikes are \ndistributed exponentially and their number in each frame, n, depends only on the \nmean firing rate A, through the distribution \n\nPA(n) = \n\ne-AA n \n\nI \nn. \n\n. \n\n(1) \n\nThe estimation of the parameter A is performed in each channel, within a sliding \nwindow of 500ms length, every lOOms. These overlapping windows introduce corre(cid:173)\nlations between the frames, but generate less noisy, smoother, firing curves. These \ncurves are depicted on top of the spike trains for each unit in figure 2. \n\nThe multivariate Poisson process is taken as a Maximum Entropy distribution with \ni.i.d. Poisson prior, subject to pairwise channel correlations as additional con(cid:173)\nstraints, yielding the following parametric distribution \n\nPA(nl,n2, ... ,nd) = II PA,(ni) exp[ - LAij(ni - Ad(nj - Aj) - AO]' \n\nd \n\nij \n\n(2) \n\nThe Aij are additional Lagrange multipliers, determined by the observed pairwise \ncorrelation E[( ni - Ad( nj - Aj)), while AO ensures the normalization. In the anal(cid:173)\nysis reported here the pairwise correlation term has not been implemented. \n\nThe statistical distance between a frame and the cluster centers is determined by \nthe probability that this frame is generated by the centroid distribution. This \nprobability is asymptotically fixed by the empirical information divergence (KL \ndistance) between the processes[8, 9]. For I-dimensional Poisson distributions the \ndivergence is simply given by \n\n(3) \n\nThe uncorrelated multi-unit divergence is simply the sum of divergences for all the \nunits. Using this measure, we can train a multivariate Poisson Hidden Markov \nModel, where each state is characterized by such a vector Poisson process. This is \na special case of a method called distributional clustering, recently developed in a \nmore general setup[IO]. \n\nThe clustering provides us with the desired statistical segmentation of the data into \nstates. The probability of a frame, Xt, to belong to a given state, Sj, is determined \nby the probability that the vector firing pattern is generated by the state centroid's \n\n\f950 \n\nGat and Tishby \n\ndistribution. Under our model assumptions this probability is a function solely of \nthe empirical divergences, Eq.(3), and is given by \n\n(4) \n\nwhere f3 determines the \"cluster-hardness\". These state probability curves are plot(cid:173)\nted in figure 2 in correspondence with the spike trains. The most probable state at \neach instance determines the most likely segmentation of the data, and the frames \nare labeled by this most probable state number. These labels are also shown on top \nof the spike trains in figure 2. \n\n4 Experimental results \n\nWe used about 6000 seconds of recordings done during a single day. It is important \nto note that this was an exceptionally good day in terms of the measurement quality. \nDuring that period the monkey performed 60 repetitions of his trained routine, in \nsets of 4 trials of \"Go\" mode, followed by 4 trials in the \"No-Go\" mode. We selected \nthe 8 most active recorded units for our modeling. The training of the models was \ndone on the first 4000 seconds of recording, 2000 seconds for each mode, while the \nrest was used for testing. \n\n4.1 The nature of the segmentation \n\nAny method can segment the data in some way, but the point is to obtain reliable \npredictions using this segmentation. As always, there is some arbitrariness in the \nchoice of the number of states (or clusters), which ideally is determined by the \ndata. Here we tested only 8 and 15 states, and in most cases 8 were sufficient for \nour purposes. Since we used \"fuzzy\", or \"soft\" clustering, each frame has some \nprobability of belonging to any of the clusters. Although in most cases the most \nlikely state is clearly defined, the complete picture is seen only from the complete \nassociation distribution. Notice, e.g., in the lower segment of figure 2, where a most \nlikely state \"7\" \"pops up\" between states \"6\" and \"5\", but is clearly not significant, \nas seen from the corresponding probability curve. \n\n4.2 Characterization of events by state-sequences \n\nThe first test of the segmentation is whether it is correlated with the external \nmarkers in any way. Since the markers were not used in any way during the training \nof the model (clustering), such correlations is a valid test of consistency. Moreover, \none would like this correspondence to the markers to hold also outside of the training \ndata. An exhaustive statistical examination of this question has not been made, \nas yet, but we could easily find many instances of similar state sequences near the \nsame external marker, both within and outside of the training data. In figure 2 we \nbring a typical example to this effect. The next step is to train smallieft-to-right \nMarkov models to spot these events more reliably. \n\n\fModeling Cell-Assemblies Activities in Associative Cortex of Behaving Monkeys \n\n951 \n\nGo-Mode \n\nNo Go-Mode \n\n0 \n\n0 \n\n0 0 \n0 0 0 0 \n0 \n\n0 \n\nt:I \n\n0 \n\n0 0 0 \nt:I D 0 D \n...... 0 0 \nt:I 0 0 0 \n0 \n\nn \n\n0 \n0 D 0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n.. \nIl .. .. \n;:I -U \n\nCJ \n\n1::1 \n\n0 0 0 0 0 \n\n0 \n\nD \n\n0 \n\n1::1 0 0 \n0 0 0 0 0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n\nt:I D 0 D \n0 0 0 \n\n0 \n\n...... D 0 \n\n0 \n\n0 0 0 \n0 0 D \n\n0 \n\n0 0 0 ~ \n\n0 \n\nD \n\nn \n\n0 \n\n0 \n\n0 \n\n0 \n\nn \n\n0 \n\n0 \n\n0 \n\n0 0 ,..., \n0 n \n0 \n\n\"\" \n\n0 \n\n0 \n\nUnits \n\nUnits \n\nFigure 3: Average firing rates for each unit in each state, for the \"Go\" and \u00abNo-\nGo\" modes. Notice that while no single unit clearly discriminates the two modes, \ntheir overall statistical discrimination is big enough that on average 100 frames are \nenough to determine the correct mode, more than 95% of the time_ \n\n4.3 Statistical Inference of \"Go\" and \"No-Go\" modes \n\nNext we examined the statistical difference between models trained on the \"Go\" \nvs. \"No-Go\" modes. Here we obtained a highly significant difference in the cluster \ncentroid's distributions, as shown in figure 3. The average statistical divergence be(cid:173)\ntween different clusters within each mode were 9.18 and 9.52 (natural logarithm) ,in \n\u00abGo\" and \"No-Go\" respectively, while in between those modes the divergence was \nmore than 35. \n\n4.4 Behavioral mode and the network firing coherency \n\nIn addition to the clearly different cluster centers in the two modes, there is another \ninteresting and unexpected difference_ We would like to call this firing coherency \nlevel, and it characterize the spread of the data around the cluster centers. The \naverage divergence between the frames and their most likely state is consistently \nmuch higher in the \"No-Go\" mode than in the \"Go\" mode (figure 4). This is in \nagreement with the assumption that correct performance of the \u00abNo-Go\" paradigm \nrequires little attention, and therefore the brain may engage in a variety of processes. \n\nAcknowledglnents \n\nSpecial thanks are due to Moshe Abeles for his continuous encouragement and \nsupport, and for his important comments on the manuscript. We would also like \nto thank Hagai Bergman, and Eilon Vaadia for sharing their data with us, and for \nnumerous stimulating and encouraging discussions of our approach. This research \nwas supported in part by a grant from the Unites States Israeli Binational Science \nFoundation (BSF). \n\n\f952 \n\nGat and Tishby \n\n10.---~-------.-------,--------.-------.-------,----, \n\n9.5 \n\nGo. \n.. \n\nNoGo \n\n~ \n~ \nis \n~ 0g \n0Q \n!! \ntil \n\n9 \n\n8.5 \n\n8 \n\n7.5 \n\n+ + .. + + + \n\n8 clusters \n\n+ + + + + + \n\n.. + .. + + + \n\n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \u2022\u2022\u2022 \n\n\u2022 \n\n\u2022\u2022\u2022 \n\u2022 \n\n\u2022\u2022 \n\n15 clusters \n\n+ + + + + + \n\n\u2022 \n\u2022\u2022\u2022\u2022 \u2022 \n\n7 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \n\n6.5~--~------~--____ ~L ________ ~ ______ ~ ______ ~ __ ~ \n\n2 \n\n3 \n\nTrial Number \n\n2 \n\n3 \n\nFigure 4: Firing coherency in the two behavioral modes at different clustering trials. \nThe \"No-Go\" average divergence to the cluster centers is systematically higher than \nin the \"Go\" mode. The effect is shown for both 8 and 15 states, and is even more \nprofound with 8 states. \n\nReferences \n\n[1] D. O. Hebb, The Organization of Behavior, Wiley, New York (1949) \n[2] M. Abeles, Corticonics, (Cambridge University Press, 1991) \n[3] J. Kruger, Simultaneous Individual Recordings From Many Cerebral Neurons: \nTechniques and Results, Rev. Phys. Biochem. Pharmacol.: 98:pp. 177-233 \n(1983) \n\n[4] M. Abeles, E. Vaadia, H. Bergman, Firing patterns of single unit in the pre(cid:173)\n\nfrontal cortex and neural-networks models., Network 1 (1990) \n\n[5] M. Abeles, H. Bergman, E. Margalit and E. Vaadia, Spatio Temporal Fir(cid:173)\ning Patterns in the Frontal Cortex of Behaving Monkeys., Hebrew University \npreprint (1992) \n\n[6] E. Vaadia, E. Ahissar, H. Bergman, and Y. Lavner, Correlated activity of \nneurons: a neural code for higher brain functions in: J.Kruger (ed), Neural \nCooperativity pp. 249-279, (Springer-Verlag 1991). \n\n[7] A. B. Poritz, Hidden Markov Models: A Guided tour,(ICASSP 1988 New York). \n[8] T.M. Cover and J.A. Thomas, Information Theory, (Wiley, 1991). \n[9] J. Ziv and N. Merhav, A Measure of Relative Entropy between Individual Se(cid:173)\n\nquences, Technion preprint (1992) \n\n[10J N. Tishby and F. Pereira, Distributional Clustering, Hebrew University preprint \n\n(1993). \n\n\f", "award": [], "sourceid": 685, "authors": [{"given_name": "Itay", "family_name": "Gat", "institution": null}, {"given_name": "Naftali", "family_name": "Tishby", "institution": null}]}