{"title": "Learning Exact Patterns of Quasi-synchronization among Spiking Neurons from Data on Multi-unit Recordings", "book": "Advances in Neural Information Processing Systems", "page_first": 76, "page_last": 82, "abstract": null, "full_text": ".. \n\nLearning Exact Patterns of Quasi-synchronization \n\namong Spiking Neurons \n\nfrom Data on Multi-unit Recordings \n\nLaura Martignon \nMax Planck Institute \n\nfor Psychological Research \n\nAdaptive Behavior and Cognition \n\n80802 Munich, Germany \n\nlaura@mpipf-muenchen.mpg.de \n\nGustavo Deco \nSiemens AG \n\nCentral Research \nOtto Hahn Ring 6 \n\n81730 Munich \n\ngustavo.deco@zfe.siemens.de \n\nKathryn Laskey \n\nDept. of Systems Engineering \n\nand the Krasnow Institute \nGeorge Mason University \n\nFairfax, Va. 22030 \nklaskey@gmu.edu \n\nEilon Vaadia \n\nDept. of Physiology \n\nHadassah Medical School \n\nHebrew University of Jerusalem \n\nJerusalem 91010, Israel \n\neilon@hbf.huji.ac.il \n\nAbstract \n\nThis paper develops arguments for a family of temporal log-linear models \nto represent spatio-temporal correlations among the spiking events in a \ngroup of neurons. The models can represent not just pairwise correlations \nbut also correlations of higher order. Methods are discussed for inferring \nthe existence or absence of correlations and estimating their strength. \nA frequentist and a Bayesian approach to correlation detection are \ncompared. The frequentist method is based on G 2 statistic with estimates \nobtained via the Max-Ent principle. In the Bayesian approach a Markov \nChain Monte Carlo Model Composition (MC3) algorithm is applied to \nsearch over connectivity structures and Laplace's method is used to \napproximate their posterior probability. Performance of the methods was \ntested on synthetic data. The methods were applied to experimental data \nobtained by the fourth author by means of measurements carried out on \nbehaving Rhesus monkeys at the Hadassah Medical School of the Hebrew \nUniversity. As conjectured, neural connectivity structures need not be \nneither hierarchical nor decomposable. \n\n\fLearning Quasi-synchronization Patterns among Spiking Neurons \n\n77 \n\n1 INTRODUCTION \n\nHebb conjectured that information processing in the brain is achieved through the \ncollective action of groups of neurons, which he called cell assemblies (Hebb, 1949). His \nfollowers were left with a twofold challenge: \n\u2022 \n\u2022 \n\nto define cell assemblies in an unambiguous way. \nto conceive and carry out the experiments that demonstrate their existence. \n\nCell assemblies have been defined in various sometimes conflicting ways, both in terms of \nanatomy and of shared function. One persistent approach characterizes the cell assembly \nby near-simultaneity or some other specific timing relation in the firing of the involved \nneurons. If two neurons converge on a third one, their synaptic influence is much larger \nfor near-coincident firing, due to \nthe dendrite \n(Abeles, 1991; Abeles et al. 1993). Thus syn-jiring is directly available to the brain as a \npotential code. \n\nthe spatio-temporal summation \n\nin \n\nThe second challenge has led physiologists to develop methods \nto observe the \nsimultaneous activity of individual neurons to seek evidence for spatio-temporal patterns. \nIt is now possible to obtain multi-unit recordings of up to 100 neurons in awake behaving \nanimals. In the data we analyze, the spiking events (in the 1 msec range) are encoded as \nsequences of O's and 1 's, and the activity of the whole group is described as a sequence of \nbinary configurations. This paper presents a statistical model in which the parameters \nrepresent spatio-temporal firing patterns. We discuss methods for estimating these \npararameters and drawing inferences about which interactions are present. \n\n2 PARAMETERS FOR SPATIO-TEMPORAL FIRING PATTERNS \n\nThe term spatial correlation has been used to denote synchronous firing of a group of \nneurons, while the term temporal correlation has been used to indicate chains of firing \nevents at specific temporal intervals. Terms like \"couple\" or \"triplet\" have been used to \ndenote spatio-temporal patterns of two or three neurons (Abeles et al., 1993; GrOn, 1996) \nfrring simultaneously or in sequence. Establishing the presence of such patterns is not \nstraightforward. For example, three neurons may fire together more often than expected \nby chancel without exhibiting an authentic third order interaction. This phenomenon may \nbe due, for instance, to synchronous frring of two couples out of the three neurons. \nAuthentic triplets, and, in general, authentic n-th order correlations, must therefore be \ndistinguished from correlations that can be explained in terms of lower order interactions. \nIn what follows, we present a parameterized model that represents a spatio-temporal \ncorrelation by a parameter that depends on the involved neurons and on a set of time \nintervals, where synchronization is characterized by all time intervals being zero. \nAssume that the sequence of configurations !:t = ( x (W'\u00b7 .. , x ( N.l J ) of N neurons forms a \nMarkov chain of order r. Let 8 be the time step, and denote the conditional distribution \nfor :!t given previous configurations by p(:!t I :!(t-oJ' :!(t-2oJ , ,,\u00b7':!(t-roJ ). We \nassume that all transition probabilities are strictly positive and expand the logarithm of \nthe conditional distribution as: \n\nI that is to say, more often than predicted by the null hypothesis of independence. \n\n\f78 \n\nL. Martignon, K. Laskey, G. Deco and E. Vaadia \n\np(!t I !(t-O ) '!(t-2o ) ,\u00b7\u00b7\u00b7'!(t-ro}) = czp{ (}o + L (J A X A) \n\nAe=: \n\n(1) \n\nwhere each A \nleast one pair of the form (i, t). Here X A = \n\nis a subset of pairs of subscripts of the form (i, t - sO) that includes at \nII x(i t-m 1\u00bb denotes the event that all \n\nl$J$k \n\ni ' \n\nJ \n\nneurons in A are active. The set \n::: c 2 A of all subsets for which () A is non-zero is \ncalled the interaction structure for the distribution p. The effect () A \nis called the \ninteraction strength for the interaction on subset A. Clearly, () A = 0 is equivalent to \nA e::: and is taken to indicate absence of an order-I A I interaction among neurons in \nA . We denote the structure-specific vector of non-zero interaction strengths by (J s. \nConsider a set A of N binary neurons and denote by p the probability distribution on \nthe binary configurations of A. \n\nDEFINITION 1: We say that neurons (i1,i2 , .. ..ik ) exhibit a spatio-temporal \npattern if there is a set of time intervals m I8,m28, ... ,m k8 with at least one \nmi = 0, such that () A \"# 0 in Equation (1), where \nA = {( i1,t- m18J,..J ik ,t- m k8)). \nDEFINITION 2: A subset (i1.i2 , ... , ik ) of neurons exhibits a synchronization or \nspatial correlation if (J A * a for A = {( iI ' 0 J, ... , ( ik, 0)) . \n\nIn the case of absence of any temporal dependencies the configurations are independent and \nwe drop the time index: \n\np(!) = czp{(}o + I.(}AXA) \n\nwhere A is any nonempty subset of A and X A = n Xi . \n\nieA \n\n(2) \n\nOf course (2) is unrealistic. Temporal correlation of some kind is always present, one \nsuch example being the refractory period after firing. Nevertheless, (2) may be adequate in \ncases of weak temporal correlation. Although the models (1) and (2) are statistical not \nphysiological, it is an established conjecture that synaptic connection between two \nneurons will manifest as a non-zero (J A for the corresponding set A in the temporal \nmodel (1). Another example leading to non-zero (J A will be simultaneous activation cf \nthe neurons in A due to a common input, as illustrated in Figure 1 below. Such a (J A \nwill appear in model (1) with time intervals equal to O. An attractive feature of our \nmodels is that it is capable of distinguishing between cases a. and b. of Figure 1. This \ncan be seen by extending the model (2) to include the external neurons (H in case a., H,K \nin case b.) and then marginalizing. An information-theoretic argument supports the \n\nchoice of (J A * a as a natural indicator of an order-I A I interaction among the neurons \nin A. Assume that we are in the case of no temporal correlation. The absence cf \ninteraction of order I A I \n\n\fLearning Quasi-synchronization Patterns among Spiking Neurons \n\n79 \n\nH \n\na. \n\nFigure 1 \n\nb. \n\namong neurons in A should be taken to mean that the distribution is determined by the \nmarginal distributions on proper subsets of A. A well established criterion for selecting \na distribution among those matching the lower order marginals fIxed by proper subsets 0.1 \n\n.Y \n.30 \n.4U \nclose to pnor \n\n4.0853 \nNo \n2.35 \n4.7 \n\nu.lI \nU.64 \n0.64 \n\nof A \n\n(best \nIOOmodels) \n.89 \n0.32 \nQ]8 \nclose to pnor \n\n(}A \n\n0.47 \n2.3U \n2.30 \n\n\fL. Martignon, K. Laskey, G. Deco and E. Vaadia \n\n82 \n\nCluster \nA \n\n5,6 \n4,7 \n1,4,5,6 \n1,3,4,6,7 \n\nPosterior prob. \n\nof A \n(frequency) \n\n.79 \n.246 \n0.18 \n0.24 \n\nof A \n\n(best \nmodels) \n0.96 \n0.18 \n0.13 \n0.17 \n\nPosterior prob. \n\nMAP estimate ot \n\n100 \n\nfJ, \n\n1.00 \n0.93 \n1.06 \n2.69 \n\nStandard \ndeviation of \nfJ, \n\nSlgnincance \n\n0.27 \n0.34 \n0.36 \n0.13 \n\n1.82 \n2.68 \nNo \nNo \n\nTable2:results for post-ready signal data. Effects with posterior prob >0.1 \n\nAnother set of data from 5 simulated neurons was provided by the fourth author for a \ndouble-check of the methods. Only second order correlations had been simulated: a \nsynapse lasting 2 msec, an inhibitory common input, and two excitatory common inputs. \nThe Bayesian method was very accurate, detecting exactly the simulated interactions. \nThe frequentist method made one mistake. Other data sets with temporal correlations \nhave also been analyzed. By means of the frequentist approach on shifted data, temporal \ntriplets have been detected and even fourth order correlations. Temporal correlograms are \ncomputed for shifts of up to 50 msec (Martignon-Deco, 1997). \n\nReferences \n\nHebb, D. (1949) The Organization of Behavior. New York: Wiley, 1949. \n\nAbeles, M .(l991)Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge: Cambridge University Press, \n\n1991. \n\nAbeles, M., H. Bergman, E. Margalit, and E. Vaadia. (1993) \"SpatiotemporaI Firing Patterns in the Frontal \n\nCortex of Behaving Monkeys.\" Journal of Neurophysiology 70, 4:, 1629-1638. \n\nGriin S. (1996) Unitary Joint-Events in Multiple-Neuron Spiking Activity-Detection, Signijicance and \n\nInterpretation. Verlag Harry Deutsch, Frankfurt. \n\nMartignon L. and Deco G. (1997) \"Neurostatistics of Spatio-Temporal Patterns of Neural Activation: the \n\nfrequentist approach\" Technical Report, MPI-ABC no.3. \n\nDeco G. and Martignon L. (1997) \"Higher-order Phenomena among Spiking Events of Groups of Neurons\" \n\nPreprint. \n\nBishop, Y., S. Fienberg, and P. Holland (1975) Discrete Multivariate Analysis. Cambridge, MA: MIT Press. \n\nMartignon L,.v.Hasseln H. Griin S, Aertsen A, Palm G.(1995) \"Detecting Higher Order Interactions among \n\nthe Spiking Events of a Group of Neurons\" Biol.Cyb. 73, 69-81 . \n\nKass, . and Raftery A. (1995) \"Bayes factors\"Journal of the American Statistical Association 90, no. 430:, \n\n773-795. \n\nTierney, L., and J. B. Kadane (J 986) \"Accurate Approximations for Posterior Moments and Marginal \n\nDensities.\" Journal of the American Statistical Association 81, 82-86 \n\nLaskey K., and Martignon L.( 1997) \"Neurostatistics of Spatio-temporal Patterns of Neural Activation: the \n\nBayesian Approach\", in preparation \n\nLaskey K., and Martignon, L.(1996) \"Bayesian Learning of Log-linear Models for Neural Connectivity\" \nProceedings of the XII Conference on Uncertainty in Artijiciallntelligence, Horvitz E. ed., \nMorgan-Kaufmann, San Mateo. \n\n\f", "award": [], "sourceid": 1274, "authors": [{"given_name": "Laura", "family_name": "Martignon", "institution": null}, {"given_name": "Kathryn", "family_name": "Laskey", "institution": null}, {"given_name": "Gustavo", "family_name": "Deco", "institution": null}, {"given_name": "Eilon", "family_name": "Vaadia", "institution": null}]}