{"title": "Information-Geometric Decomposition in Spike Analysis", "book": "Advances in Neural Information Processing Systems", "page_first": 253, "page_last": 260, "abstract": null, "full_text": "Information-geometric decomposition \n\nspike analysis \n\n. In \n\nHiroyuki Nakahara; Shun-ichi Amari \n\nLab. for Mathematical Neuroscience, RIKEN Brain Science Institute \n\n2-1 Hirosawa, Wako, Saitama, 351-0198 Japan \n\n{him, amari} @brain.riken.go.jp \n\nAbstract \n\nWe present an information-geometric measure to systematically \ninvestigate neuronal firing patterns, taking account not only of \nthe second-order but also of higher-order interactions. We begin \nwith the case of two neurons for illustration and show how to test \nwhether or not any pairwise correlation in one period is significantly \ndifferent from that in the other period. In order to test such a hy(cid:173)\npothesis of different firing rates, the correlation term needs to be \nsingled out 'orthogonally' to the firing rates, where the null hypoth(cid:173)\nesis might not be of independent firing. This method is also shown \nto directly associate neural firing with behavior via their mutual \ninformation, which is decomposed into two types of information, \nconveyed by mean firing rate and coincident firing, respectively. \nThen, we show that these results, using the 'orthogonal' decompo(cid:173)\nsition, are naturally extended to the case of three neurons and n \nneurons in general. \n\n1 \n\nIntroduction \n\nBased on the theory of hierarchical structure and related invariant decomposition \nof interactions by information geometry [3], the present paper briefly summarizes \nmethods useful for systematically analyzing a population of neural firing [9]. \n\nMany researches have shown that the mean firing rate of a single neuron may carry \nsignificant information on sensory and motion signals. Information conveyed by \npopulational firing, however, may not be only an accumulation of mean firing rates. \nOther statistical structure, e.g., coincident firing [13, 14], may also carry behavioral \ninformation. One obvious step to investigate this issue is to single out a contribution \nby coincident firing between two neurons, i.e., the pairwise correlation [2, 6]. \n\nIn general, however, it is not sufficient to test a pairwise correlation of neural firing, \nbecause there can be triplewise and higher correlations. For example, three variables \n(neurons) are not independent in general even when they are pairwise independent. \n\nWe need to establish a systematic method of analysis, including these higher-order \n\n\u2022 also affiliated with Dept. of Knowledge Sci., Japan Advanced Inst. of Sci. & Tech. \n\n\fcorrelations [1, 5,7, 13] . We propose one approach, the information-geometric mea(cid:173)\nsure that uses the dual orthogonality of the natural and expectation parameters in \nexponential family distributions [4]. We represent a neural firing pattern by a binary \nrandom vector x. The probability distribution of firing patterns can be expanded \nby a log linear model, where the set {p( x)} of all the probability distributions forms \na (2n - I)-dimensional manifold 8 n. Each p(x) is given by 2n probabilities \npi1\u00b7\u00b7\u00b7in=Prob{X1=i1,\u00b7\u00b7\u00b7,Xn=in}, ik=O,I, subjectto L Pi1\u00b7\u00b7\u00b7in=1 \n\nil ,\"',in \n\nand expansion in log p( x) is given by \n\nlogp(x) = L BiXi + L BijXiXj + L BijkXiXjXk\u00b7\u00b7\u00b7 + B1 ... nX1 ... Xn - 'Ij;, \n\ni 0, i,j = 0, 1. Among four probabilities, {POO ,P01,P10,Pl1}, \nonly three are free. The set of all such distributions of x forms a three-dimensional \nmanifold 8 2. Any three of Pij can be used as a coordinate system of 8 2. \nThere are many different coordinate systems of 8 2 . The coordinates of the expec(cid:173)\ntation parameters, called 17-coordinates, 'TI = (171,172,1712), is given by \n\n17i = Prob {Xi = I} = E [Xi], \n\ni = 1,2, \n\n173 = 1712 = E [X1 X2] = P12, \n\nwhere E denotes the expectation and 17i and 1712 correspond to the mean firing rates \nand the mean coincident firing, respectively. \n\nAs other coordinate systems, we can also use the triplet, (171,172, Cov [Xl, X 2]) , where \nCov [Xl , X 2] is the covariance,and/or the triplet (171,172, p), where p is the correlation \ncoefficient (COR), p = J \n\n, often called N-JPSTH [2]. \n\n'112 -,/11 '12 \n\n'/11 (l - '7d'72 (1 - '72) \n\n\fWhich quantity would be convenient to represent the pairwise correlational com(cid:173)\nponent? It is desirable to define the degree of the correlation independently from \nthe marginals (171,172), To this end, we use the 'orthogonal' coordinates (171 , 172 , B), \noriginating from information geometry of 8 2 , so that the coordinate curve of B is \nalways orthogonal to those of 171 and 172. \nThe orthogonality of two directions in 8 2 (8 n in general) is defined by the Rieman(cid:173)\nnian metric due to the Fisher information matrix [8, 4]. Denoting any coordinates \nin 8 n by ~ = (6, ... , ~n)' the Fisher information matrix G is given by \n\n(1) \n\nwhere l (x;~) = logp (x; ~). The orthogonality between ~i and ~j is defined by \n9ij(~) = O. In case of 8 2 , we desire to have E [tel (X ;171 ,172, B) 8~il(x;171'172,B)] = \no (i = 1, 2). When B is orthogonal to (171, 172), we say that B represents pure \ncorrelations independently of marginals. Such B is given by the following theorem. \n\nTheorem 1. \n\nThe coordinate \n\nB = log PuPoo \nP01PlO \n\n(2) \n\nis orthogonal to the marginals 171 and 172 . \n\nWe have another interpretation of B. Let's expand p(x) by logp(x) = L;=l BiXi + \nB12X1X2 - 'IjJ. Simple calculation lets us get the coefficients, B1 = log Pia, \nB2 = \npaa \n'IjJ = -logpoo, and B = B12 (as Eq 2). The triplet () = (B1' B2, B12 ) forms \nlog EQl, \npaa \nanother coordinate system, called the natural parameters, or B-coordinates. We \nremark that B12 is 0 when and only when Xl and X 2 are independent. \nThe triplet \n\nC == (171,172,B12 ) \n\nforms an 'orthogonal' coordinate system of 8 2 , called the mixed coordinates [4]. \n\nWe use the Kullback-Leibler divergence (KL) to measure the discrepancy between \ntwo probabilities p(x) and q(x) , defined by D[p:q] = LxP(x)log~t~}. In the \nfollowing, we denote any coordinates of p by e etc (the same for q). Using the \n\northogonality between 17- and B-coordinates, we have the decomposition in the KL. \n\nTheorem 2. \n\nD [p : q] = D [p : r*] + D [r* : q], \n\n(3) \nwhere r* and r** are given by Cr > = (17f, 17~, Bj) and Cr \u00bb = (17f, 17g, B~), respectively. \nThe squared distance ds 2 between two nearby distributions p(x , ~) and p(x,~, +d~) \nis given by the quadratic form of d~, \n\nD [q : p] = D [q : r**] + D [r** : p] , \n\nds2 = L \n\ni,jE(1,2,3) \n\n9ij(~)d~id~j, \n\nwhich is approximately twice the KL, i.e. , ds 2 ~ 2D [P(x , ~) : p(x,~ + ~)]. \nNow suppose ~ is the mixed coordinates C. Then, the Fisher information matrix \n\nand we have ds2 = dsi + ds~, where dsi = \n\nis of the form gfj = \n\n[ gll \ngf2 \no \n\ngl2 0 1 \n\ng~2 0 \n0 \n\ng~3 \n\ng~3(dB3)2, ds~ = Li,j E(1,2) 9fjd17id17j, corresponding to Eq. 3. \n\n\f\u2022 \n\n\u2022 \n\nN \n\nl..#{x\u00b7 = I} and 8 = \n\nThis decomposition comes from the choice of the orthogonal coordinates and gives \nus the merits of simple procedure in statistical inference. First, let us estimate \nthe parameter TI = (1}1,1}2) and B from N observed data Xl, ... , XN. The maxi(cid:173)\nmum likelihood estimator (mle) ( , which is asymptotically unbiased and efficient, \nlog fh?(1-=-fh-.ib+~12) using \nis easily obtained by 1)' . = \nfj12 = tt#{XIX2 = I}. The covariance of estimation error, f::J.TI and f::J.B, is given \nasymptotically by Cov [ ~~ ] = ttGZ1. Since the cross terms of G or G-1 vanish \nfor the orthogonal coordinates, we have Cov [f::J.TI, f::J.B] = 0, implying that the es(cid:173)\ntimation error f::J.TI of marginals and that of interaction are mutually independent. \nSuch a property does not hold for other non-orthogonal parameterization such as \nthe COR p, the covariance etc. Second, in practice, we often like to compare many \nspike distributions, q(x(t)) (i.e, (q(t)) for (t = 1\", T), with a distribution in the \ncontrol period p( x) , or (P. Because the orthogonality between TI and B allows us to \ntreat them independently, these comparisons become very simple. \n\n(1]1-1]12)(1]2-1]12) , \n\nThese properties bring a simple procedure of testing hypothesis concerning the null \nhypothesis \n\nHo : B = Bo \n\n(4) \nwhere Bo is not necessarily zero, whereas Bo = 0 corresponds to the null hypothesis \nof independent firing, which is often used in literature in different setting. Let the \nlog likelihood of the models Ho and HI be, respectively, \n\nagainst \n\nlo = maxlogp(Xl ' ... , XN ; TI , Bo) and h = maxlogp(Xl' ... , XN; TI, B). \n\nTI,e \n\nTI \n\nThe likelihood ratio test uses the test statistics A = 2log ~. By the mle with respect \nto TI and e, which can be performed independently, we have \n\nlo = logp(x ,r\"Bo), \n\n(5) \nwhere r, are the same in both models. A similar situation holds in the case of testing \nTI = Tlo against TI =I Tlo for unknown B. \nUnder the hypothesis H o, A is approximated for a large N as \n\nA = 2 t log P(Xi;~' B~) ';::;j N gi3 (8 - BO)2 '\" X2(1). \n\ni=l \n\np(Xi; TI, B) \n\n(6) \n\nThus, we can easily submit our data to a hypothetical testing of significant coinci(cid:173)\ndent firing against null hypothesis of any correlated firing, independently from the \nmean firing rate modulation1 . \n\nWe now turn to relate the above approach with another important issue, which is \nto relate such a coincident firing with behavior. Let us denote by Y a variable of \ndiscrete behavioral choices. The MI between X = (X1 ,X2 ) and Y is written by \n\nJ(X, Y) = Ep(x ,y) \n\np(x,y)] \n\n[ \nlog p(x)p(y) = Ep(Y) [D [P(Xly) : p(X)]]. \n\nUsing the mixed coordinates for p(Xly) and p(X) , we have D [P(Xly) : p(X)] \nD [\u00ab(Xly) : \u00ab(X)] = D [\u00ab(Xly) : ('J + D [(I : \u00ab(X)J, where (' = ('(X,y) \n((1 (Xly), (2 (X Iy), (3 (X)) = (1}1 (Xly), 1}2(Xly), B3(X)). \n\n1 A more proper formulation in this hypothetical testing can be derived, resulting in \n\nusing p value from X2 (2) distribution , but we omit it here due to the limited space [9] \n\n\fTheorem 3. \n\nJ(X, Y) = It (X, Y) + h(X, Y) , \n\n(7) \n\nwhere It (X, Y), h(X, Y) are given by \nIt (X, Y) = Ep(Y) [D [\u00ab(Xly) : ('(X,y)]] ,h(X, Y) = Ep(Y) [D [('(X,y) : \u00ab(X)]] . \n\nObviously, the similar result holds with respect to p(YIX). By this theorem, J is the \nsum of the two terms: It is by modulation of the correlation components of X, while \nh is by modulation of the marginals of X. This observation helps us investigate the \nbehavioral significance by modulating either coincident firing or mean firing rates. \n\n0 . 1 ,-----~-~--~-~-____, \n\n0.1 ,-------~--~-~-____, \n\n~ A (al \n\nu; \n~ 0.05~ .\u2022.\u2022\u2022.\u2022 __ I~) \u2022 \u2022 _ ~, .\u2022 .\u2022 .\u2022.\u2022 .\u2022.\u2022 .\u2022 ~C) \u2022 \u2022 .\u2022 .\u2022 .\u2022 .\u2022 .\u2022 \n\n/ \n\n'J, \n\n- ........................ _12 . \n\n100 \n\n.. \n300 \n\n-\n\n500 \n\no \na \n\n\\ll \n~0.05 \n\nB \n\n. , .. , .. . , . .. , .. \". , .. ' .. , .. \" .\n!-\nW : \n\n~ \n\n... . . , .. . . , .. , _ .# ' ...... .. , .. . . ' .. -\" .. . , \u2022\u2022 , \n\n~ \n\n. .. ' .. , .\n112 \n'J, \n\n_________ \n\n.. __ '_12 '- - - .-- - --- -\"\" -_ .. \n\n\u00b00:------=:-:10:::-0 ~~~~3=00=---~---:-:500 \n\n00 \n\n100 \n\n300 \n\nlime Ims) \n\n500 \n\n00 \n\n100 \n\n\"300 \n\nlime Ims) \n\n500 \n\nFigure 1: Demonstration of information-geometric measure in two neuron case, \nusing simulated neural data, where two behavioral choices (sl, s2) are assumed. \nA,B. (1]1 , 1]2 , 1]12) with respect to sl, s2. C,D . COR,B, computed by using \", \nL-iP(Si)\",(Si) with P(Si) = 1/2 (i = 1, 2). E. p-values. F. MI. \n\nFig 1 succinctly demonstrates results in this section. Figs 1 A, B are supposed to \nshow mean firing rates of two neurons and mean coincident firing for two different \nstimuli (sl, s2). The period (a) is assumed as the control period, i.e. , where no \nstimuli is shown yet, whereas the stimulus is shown in the periods (b,c). Fig 1 C, \nD gives COR, B. They look to change similarly over periods, which is reasonable \nbecause both COR and B represent the same correlational component, but indeed \nchange slightly differently over periods (e.g., the relative magnitudes between the \nperiods (a) and (c) are different for COR and B) , which is also reasonable because \nboth represent the correlational component as in different coordinate systems. Using \nB in Fig 1 D, Fig 1 E shows p-values derived from X2 (1) (i.e., P > 0.95 in Fig 1 E is \n'a significance with P < 0.05') for two different null hypotheses, one of the averaged \nfiring in the control period (by solid line) and the other of independent firing (by \ndashed line) , which is of popular use in literature. \n\nIn general, it becomes complicated to test the former hypothesis, using COR. This \nis because the COR, as the coordinate component, is not orthogonal to the mean \nfiring rates so that estimation errors among the COR and mean firing rates are \nentangled and that the proper metric among them is rather difficult to compute. \nOnce using B, this testing becomes simple due to orthogonality between B and mean \nfiring rates. \n\nNotably, we would draw completely different conclusions on significant coincident \nfiring given each null hypothesis in Fig 1 E. This difference may be striking when we \nare to understand the brain function with these kinds of data. Fig 1 F shows the MI \n\n\fbetween firing and behavior, where behavioral event is with respect to stimuli, and \nits decomposition. There is no behavioral information conveyed by the modulation \nof coincident firing in the period (b) (i.e., h = 0 in the period (b)). The increase \nin the total MI (i.e., I) in the period (c), compared with the period (b), is due not \nto the MI in mean firing (h) but to the MI correlation (h). Thus, with a great \nease, we can directly inspect a function of neural correlation component in relation \nto behavior. \n\n3 Three neuron case \n\nWith more than two neurons, we need to look not only into a pairwise interaction \nbut also into higher-order interactions. Our results in the two neuron case are \nnaturally extended to n neuron case and here, we focus on three neuron case for \nillustration. \nFor three neurons X = (X1,X2,X3), we let p(x), x = (X1,X2,X3), be their joint \nprobability distribution and put Pijk = Prob {Xl = i, X2 = j, X3 = k}, i, j, k = 0,1. \nThe set of all such distributions forms a 7-dimensional manifold 8 3 due to \"L.Pijk = \n1. The 1]-coordinates 'fI = ('fI1; 'fI2; 'fI3) = (1]1,1]2,1]3; 1]12,1]23,1]13; 1]123) is defined by \n1]i = E [Xi] \nTo single out the purely triplewise correlation, we utilize the dual orthogo(cid:173)\nnality of 8- and 1]-coordinates. By using expansion of log p( x) = \"L. 8iXi + \n\"L.8ijXiXj + 8123X1X2X3 -\n(()1;()2;()3) = \n(81,82,83; 812 ,823 ,813 ; 8123 ). \nIt's easy to get the expression of these coefficients \n(e.g. 8 = log P111 PIOO POIOP001). Information geometry gives the following theorem. \n\n'ljJ, we obtain 8-coordinates, \n\n(i, j = 1,2, 3; i i- j), \n\n(i = 1,2,3), \n\n1Jij = E [XiXj] \n\n,123 \n\nP110PIOIP0l1POOO \n\n1]123 = E [X1X2X3]. \n\n() = \n\n8123 represents the pure triplewise interaction in the sense that \nTheorem 4. \nit is orthogonal to any changes in the single and pairwise marginals, i.e., 'fIl and 'fI2. \n\nWe use the following two mixed coordinates to utilize the dual orthogonality, \n\n(I = ('fIl; ()2; ()3), (2 = ('fIl; 'fI2; ()3). \n\nHere (2 is useful to single out the triple wise interaction (()3 = 8123), while (I is to \nsingle out the pairwise and triplewise interactions together (()2; ()3). Note that 8123 is \nnot orthogonal to {8ij }. In other words, except the case of no triple wise interaction \n(8123 = 0), 8ij do not directly represent the pairwise correlation of two random \nvariables Xi, X j . The case of independent firing is given by 1]ij = 1]i1]j, 1]123 = 1]11]21]3 \nor equivalently by ()2 = 0, ()3 = o. \nThe decomposition in the KL is now given as follows. \n\nTheorem 5. \nD [p : q] = D [p : p] + D [p : q] = D [p : fi] + D [p : q] = D [p : p] + D [p : fi] + D [p : q] . \n(8) \nwhere, using the mixed coordinates, we have (g = ('fIi; 'fI~; ()\u00a7), (f = ('fIi; ()~; ()\u00a7). \nA hypothetical testing is formulated similarly to the two neuron case. We can exam(cid:173)\nine a significance of the triplewise interaction by A2 = 2ND [p : p] ~ N g~7 (~) (8f23-\n8i23)2 ~ X2(1). For a significance of triplewise and pairwise interactions together, \nwe have Al = 2ND [p : fi] ~ N \"L.J,j=4 gfj(f)((f - (f)((f - (f) ~ X2(4). \nFor the decomposition of the MI between firing X and behavior Y, we have \n\nTheorem 6. \n\nJ(X, Y) = h (X, Y) + h(X, Y) = h(X, Y) + J4(X, Y) \n\n(9) \n\n\fwhere \nh(X, Y) = Ep(Y) [D [(I(X ly ) : ( I(X,y)] ] , h(X, Y) = Ep(Y) [D [(I (X,y) : ( I(X)]] , \n\nh(X, Y) = Ep(Y) [D [(2(X ly ) : ( 2(X,y)] ] , 14 (X, Y) = Ep(Y) [D [(2(X,y) : ( 2(X)]], \n\nBy t he first equality, I is decomposed into two parts: II is conveyed by the pairwise \nand triplewise interactions of firing, and h by the mean firing rate modulation. \nBy the second equality, I is decomposed differently: h, conveyed by t he triplewise \ninteraction, and 14 , by the other terms. \n\n0.04 \n'\" \n0.02 \n\n(a) \n\n(e) \n\n(d) \n\nr:- ~ : --\n\n8 ~~I----'\n\n[: \n\n100 \n\n00 \n\n100 \n\n500 \n\n300 \n\nit \u2022 .. \n~,-~ ~ ~~j ~.~~ \n\n-2 \n0 \n\n-0. 10 \n\n300 \n\n300 \n\n500 \n\n500 \n\n700 \n\n700 \n\n100 \n\n1- _____ _ \n\nN205 \n\nE \n\n- - -i: -. -. --, --) ----\n\n700 \n\n0 \n\n100 \n\n95 N~05011-F- - - - - - - - - - - .':,f\u00b7- - - - \u2022 - - - - - - - - .- - - -I\u00b795 \n\n300 \n\n500 \n\n700 \n\nx \n\nt. \nI ~....,:..;..\".u,,'rV! '/ J.~'\" \noo'-'--\"1\"\"oo:\"-'-'-'-'-\"'-'----::3~00c--~---'c5-:-:00-~---=1l700 \n\n.\" ,t: t' ..... ~I/\"'\\:\"\"''! ~I \n~ \n\ntime (ms) \n\n...... \n\n0 \n\n1 00 \n\n300 \ntime (ms) \n\n500 \n\n700 \n\nFigure 2: Demonstration in three neuron case. A '11 = ('111> '112, '113) ~ ('T/i ,'T/ij,'T/ijk) \nfrom top to bottom, since we treated a homogeneous case in this simulation for \nsimplicity. B. COR. C. (}12,(}13 , (}23' D (}12 3 . E p-value,...., X2 (1). F p-value,...., X2(4). \n\nWe emphasize that all the above decompositions come from the choice of the 'or(cid:173)\nthogonal' coordinates. Fig 2 highlights some of the results in this section. Fig 2 A \nshows the mean firing rates (see legend). The period (a) is assumed as the control \nperiod. Fig 2 B indicates that COR changes only in the periods (c,d), while Fig 2 \nC indicates that (}123 changes only in the period (d). Taken together, we observe \nthat the triplewise correlation (}123 can be modulated independently from COR. Fig \n2 E indicates the p-value from X2(1) against the null hypothesis of the activity in \nthe control period. The triple wise coincident firing becomes significant only in the \nperiod (d). Fig 2 F indicates the p-value from X2(4) . The coincident firing, taking \nthe triplewise and pairwise interaction together, becomes significant in both periods \n(c,d). We cannot observe these differences in modulation of pairwise and triplewise \ninteractions over periods (c, d), when we inspect only COR. \n\nRemark: For a general n neuron case, we can use the k-cut mixed coordinates, \n(k = ('111 ' ... , '11k; 0k+l, ... , On) = ('I1k- ; 0k+)' Using the orthogonality between 'I1k(cid:173)\nand 0k+, the similar results hold. To meet the computational complexity involved \nin this general case, some practical difficulties should be resolved in practice [9] . \n\n4 Discussions \n\nWe presented the information-geometric measures to analyze spike firing patterns, \nusing two and three neuron cases for illustration. The choice of 'orthogonal' co(cid:173)\nordinates provides us with a simple, transparent and systematic procedure to test \nsignificant firing patterns and to directly relate such a pattern with behavior. We \nhope that this method simplifies and strengthens experimental data analysis. \n\n\fAcknowledgments \n\nHN thanks M. Tatsuno, K. Siu and K. Kobayashi for their assistance. HN is sup(cid:173)\nported by Grants-in-Aid 13210154 from the Ministry of Edu. Japan. \n\nReferences \n\n[1] M. Abeles, H. Bergman, E. Margalit, and E. Vaadia. Spatiotemporal firing \npatterns in the frontal cortex of behaving monkeys. J Neurophysiol, 70(4):1629-\n38.,1993. \n\n[2] A. M. H. J. Aertsen, G. 1. Gerstein, M. K. Habib, and G. Palm. Dynamics of \nneuronal firing correlation: Modulation of \"effective connectivity\". Journal of \nNeurophysiology, 61(5):900- 917, May 1989. \n\n[3] S. Amari. Information geometry on hierarchical decomposition of stochastic \ninteractions. IEEE Transaction on Information Theory, pages 1701- 1711,2001. \n[4] S. Amari and H. Nagaoka. Methods of Information Geometry. AMS and Oxford \n\nUniversity Press, 2000. \n\n[5] S. Griin. Unitary joint-events in multiple-neuron spiking activity: detection, \nsignificance, and interpretation. Verlag Harri Deutsch, Reihe Physik, Band 60. \nThun, Frankfurt/Main, 1996. \n\n[6] H. Ito and S. Tsuji. Model dependence in quantification of spike interdepen(cid:173)\n\ndence by joint peri-stimulus time histogram. Neural Computation, 12:195- 217, \n2000. \n\n[7] L. Martignon, G. Deco, K. Laskey, M. Diamond, W. A. Freiwald, and E. Vaa(cid:173)\n\ndia. Neural coding: Higher-order temporal patterns in the neurostatistics of \ncell assemblies. Neural Computation, 12(11):2621- 2653, 2000. \n\n[8] H. Nagaoka and S. Amari. Differential geometry of smooth families of proba(cid:173)\n\nbility distributions. Technical report , University of Tokyo, 1982. \n\n[9] H. Nakahara and S. Amari. Information geometric measure for neural spikes. \n\nin prepration. \n\n[10] H. Nakahara, S. Amari, M. Tatsuno, S. Kang, K. Kobayashi, K. Anderson, \nE. Miller, and T. Poggio. Information geometric measures for spike firing. \nSociety for Neuroscience Abstracts, 27:821.46 (page.2178), 2001. \n\n[11] M. W . Oram, N. G. Hatsopoulos, B. J. Richmond, and J . P. Donoghue. Excess \nsynchrony in motor cortical neurons provides redundant direction information \nwith that from coarse temporal measures. J Neurophysiol., 86(4):1700- 1716, \n2001. \n\n[12] S. Panzeri and S. R. Schultz. A unified approach to the study of temporal, \n\ncorrelational, and rate coding. Neural Computation, 13(6):1311-49., 2001a. \n\n[13] A. Riehle, S. Griin, M. Diesmann, and A. Aertsen. Spike synchronization and \nrate modulation differentially involved in motor cortical function. Science, \n278:1950- 1953, 12 Dec 1997. \n\n[14] E. Vaadia, I. Haalman, M. Abeles, H. Bergman, Y. Prut, H. Slovin, and \nA. Aertsen. Dynamics of neuronal interactions in monkey cortex in relation to \nbehavioural events. Nature, 373:515- 518, 9 Feb 1995. \n\n\f", "award": [], "sourceid": 2006, "authors": [{"given_name": "Hiroyuki", "family_name": "Nakahara", "institution": null}, {"given_name": "Shun-ichi", "family_name": "Amari", "institution": null}]}