{"title": "Measuring Neural Synchrony by Message Passing", "book": "Advances in Neural Information Processing Systems", "page_first": 361, "page_last": 368, "abstract": null, "full_text": "Measuring Neural Synchrony by Message Passing\n\nJustin Dauwels\n\nAmari Research Unit\n\nRIKEN Brain Science Institute\n\nWako-shi, Saitama, Japan\njustin@dauwels.com\n\nFranc\u00b8ois Vialatte, Tomasz Rutkowski, and Andrzej Cichocki\n\nAdvanced Brain Signal Processing Laboratory\n\nRIKEN Brain Science Institute\n\nWako-shi, Saitama, Japan\n\n{fvialatte,tomek,cia}@brain.riken.jp\n\nAbstract\n\nA novel approach to measure the interdependence of two time series is proposed,\nreferred to as \u201cstochastic event synchrony\u201d (SES); it quanti\ufb01es the alignment of\ntwo point processes by means of the following parameters: time delay, variance\nof the timing jitter, fraction of \u201cspurious\u201d events, and average similarity of events.\nSES may be applied to generic one-dimensional and multi-dimensional point pro-\ncesses, however, the paper mainly focusses on point processes in time-frequency\ndomain. The average event similarity is in that case described by two parameters:\nthe average frequency offset between events in the time-frequency plane, and the\nvariance of the frequency offset (\u201cfrequency jitter\u201d); SES then consists of \ufb01ve pa-\nrameters in total. Those parameters quantify the synchrony of oscillatory events,\nand hence, they provide an alternative to existing synchrony measures that quan-\ntify amplitude or phase synchrony. The pairwise alignment of point processes\nis cast as a statistical inference problem, which is solved by applying the max-\nproduct algorithm on a graphical model. The SES parameters are determined from\nthe resulting pairwise alignment by maximum a posteriori (MAP) estimation. The\nproposed interdependence measure is applied to the problem of detecting anoma-\nlies in EEG synchrony of Mild Cognitive Impairment (MCI) patients; the results\nindicate that SES signi\ufb01cantly improves the sensitivity of EEG in detecting MCI.\n\n1 Introduction\n\nSynchrony is an important topic in neuroscience. For instance, it is hotly debated whether the\nsynchronous \ufb01ring of neurons plays a role in cognition [1] and even in consciousness [2]. The syn-\nchronous \ufb01ring paradigm has also attracted substantial attention in both the experimental (e.g., [3])\nand the theoretical neuroscience literature (e.g., [4]). Moreover, medical studies have reported that\nmany neurophysiological diseases (such as Alzheimer\u2019s disease) are often associated with abnor-\nmalities in neural synchrony [5, 6].\n\nIn this paper, we propose a novel measure to quantify the interdependence between point processes,\nreferred to as \u201cstochastic event synchrony\u201d (SES); it consists of the following parameters: time delay,\nvariance of the timing jitter, fraction of \u201cspurious\u201d events, and average similarity of the events. The\npairwise alignment of point processes is cast as a statistical inference problem, which is solved\nby applying the max-product algorithm on a graphical model [7]. In the case of one-dimensional\npoint processes, the graphical model is cycle-free and statistical inference is exact, whereas for\n\n1\n\n\fmulti-dimensional point processes, exact inference becomes intractable; the max-product algorithm\nis then applied on a cyclic graphical model, which not necessarily yields the optimal alignment [7].\nOur experiments, however, indicate that the it \ufb01nds reasonable alignments in practice. The SES\nparameters are determined from the resulting pairwise alignments by maximum a posteriori (MAP)\nestimation.\n\nThe proposed method may be helpful to detect mental disorders such as Alzheimer\u2019s disease, since\nmental disorders are often associated with abnormal blood and neural activity \ufb02ows, and changes in\nthe synchrony of brain activity (see, e.g., [5, 6]). In this paper, we will present promising results on\nthe early prediction of Alzheimer\u2019s disease from EEG signals based on SES.\n\nThis paper is organized as follows.\nIn the next section, we introduce SES for the case of one-\ndimensional point processes. In Section 3, we consider the extension to multi-dimensional point\nprocesses.\nIn Section 4, we use our measure to detect abnormalities in the EEG synchrony of\nAlzheimer\u2019s disease patients.\n\n2 One-Dimensional Point Processes\n\nLet us consider the one-dimensional point processes (\u201cevent strings\u201d) X and X 0 in Fig. 1(a) (ignore\nY and Z for now). We wish to quantify to which extent X and X 0 are synchronized. Intuitively\nspeaking, two event strings can be considered as synchronous (or \u201clocked\u201d) if they are identical apart\nfrom: (i) a time shift \u03b4t; (ii) small deviations in the event occurrence times (\u201cevent timing jitter\u201d); (iii)\na few event insertions and/or deletions. More precisely, for two event strings to be synchronous, the\nevent timing jitter should be signi\ufb01cantly smaller than the average inter-event time, and the number\nof deletions and insertions should comprise only a small fraction of the total number of events.\nThis intuitive concept of synchrony is illustrated in Fig. 1(a). The event string X 0 is obtained from\nevent string X by successively shifting X over \u03b4t (resulting in Y ), slightly perturbing the event\noccurrence times (resulting in Z), and eventually, by adding (plus sign) and deleting (minus sign)\nevents, resulting in X 0. Adding and deleting events in Z leads to \u201cspurious\u201d events in X and X 0\n(see Fig. 1(a); spurious events are marked in red): a spurious event in X is an event that cannot be\npaired with an event in X 0 and vice versa.\nThe above intuitive reasoning leads to our novel measure for synchrony between two event strings,\ni.e., \u201cstochastic event synchrony\u201d (SES); for the one-dimensional case, it is de\ufb01ned as the triplet (\u03b4t,\nst, \u03c1spur), where st is the variance of the (event) timing jitter, and \u03c1spur is the percentage of spurious\nevents\n\n\u03c1spur\n\n4=\n\n,\n\n(1)\n\nnspur + n0\nn + n0\n\nspur\n\nwith n and n0 the total number of events in X and X 0 respectively, and nspur and n0\nspur the total\nnumber of spurious events in X and X 0 respectively. SES is related to the metrics (\u201cdistances\u201d)\nproposed in [9]; those metrics are single numbers that quantify the synchrony between event strings.\nIn contrast, we characterize synchrony by means of three parameters, which allows us to distinguish\ndifferent types of synchrony (see [10]). We compute those three parameters by performing inference\nin a probabilistic model. In order to describe that model, we consider Fig. 1(b), which shows a\nsymmetric procedure to generate X and X 0. First, one generates an event string V of length `,\nwhere the events Vk are mutually independent and uniformly distributed in [0, T0]. The strings Z\nand Z 0 are generated by delaying V over \u2212\u03b4t/2 and \u03b4t/2 respectively and by (slightly) perturbing\nthe resulting event occurrence times (variance of timing jitter equals st/2). The sequences X and\nX 0 are obtained from Z and Z 0 by removing some of the events; more precisely, from each pair\n(Zk, Z 0\nThis procedure amounts to the statistical model:\n\nk is removed with probability ps.\n\nk), either Zk or Z 0\n\np(x, x0, b, b0, v, \u03b4t, st, `) = p(x|b, v, \u03b4t, st)p(x0|b0, v, \u03b4t, st)p(b, b0|`)p(v|`)p(`)p(\u03b4t)p(st),\n\n(2)\n\nwhere b and b0 are binary strings that indicate whether the events in X and X 0 are spurious (Bk = 1\nif Xk is spurious, Bk = 0 otherwise; likewise for B0\nk); the length ` has a geometric prior p(`) =\n(1 \u2212 \u03bb)\u03bb` with \u03bb \u2208 (0, 1), and p(v|`) = T \u2212`\n\n0 . The prior on the binary strings b and b0 is given by\n\np(b, b0|`) = (1 \u2212 ps)n+n0\n\np2`\u2212n\u2212n0\n\ns\n\n= (1 \u2212 ps)n+n0\n\np\n\nntot\nspur\ns\n\n,\n\n(3)\n\n2\n\n\fwith ntot\n\nspur = nspur + n0\n\nk=1 bk = ` \u2212 n0 the number of spurious events in X, and likewise n0\n\nspur = 2` \u2212 n \u2212 n0 the total number of spurious events in X and X 0, nspur =\nspur, the number of spurious\n\nevents in X 0. The conditional distributions in X and X 0 are equal to:\n\nPn\n\np(x|b, v, \u03b4t, st) =\n\np(x0|b0, v, \u03b4t, st) =\n\nn\n\nYk=1(cid:18)N(cid:0)xk \u2212 vik ; \u2212\nYk=1(cid:18)N(cid:0)x0\n\nk \u2212 vi0\nk ;\n\n\u03b4t\n2\n\nn0\n\n\u03b4t\n2\n\n,\n\n,\n\nst\n\n2(cid:1)(cid:19)1\u2212bk\n2(cid:1)(cid:19)1\u2212b0\n\n,\n\nk\n\nst\n\n(4)\n\n(5)\n\nwhere Vik is the event in V that corresponds to Xk (likewise Vi0\n), and N (x; m, s) is a univariate\nGaussian distribution with mean m and variance s. Since we do not wish/need to encode prior\ninformation about \u03b4t and st, we adopt improper priors p(\u03b4t) = 1 = p(st).\nEventually, marginalizing (2) w.r.t. v results in the model:\n\nk\n\np(x, x0, b, b0, \u03b4t, st, `) =Z p(x, x0, b, b0, v, \u03b4t, st, `)dv \u221d \u03b2ntot\n\nspur\n\nnnon-spur\n\nYk=1\n\nN (x0\nj0\n\nk\n\n\u2212 xjk ; \u03b4t, st),\n\n(6)\n\nwith (xjk , x0\nj0\n\nk\n\n) the pairs of non-spurious events, nnon-spur = n + n0 \u2212 ` the total number of non-\n\nT0\n\nspurious event pairs, and \u03b2 = psq \u03bb\n\n; in the example of Fig. 1(b), J = (1, 2, 3, 5, 6, 7, 8),\nJ 0 = (2, 3, 4, 5, 6, 7, 8), and nnon-spur = 7.\nIn the following, we will denote model (6) by\np(x, x0, j, j0, \u03b4t, st) instead of p(x, x0, b, b0, \u03b4t, st, `), since for given x, x0, b, and b0 (and hence given\nn, n0, and nnon-spur), the length ` is fully determined, i.e., ` = n + n0 \u2212 nnon-spur; moreover, it is more\nnatural to describe the model in terms of J and J 0 instead of B and B0 (cf. RHS of (6)). Note that\nB and B0 can directly be obtained from J and J 0.\nIt also noteworthy that T0, \u03bb and ps do not need to be speci\ufb01ed individually, since they appear in (6)\nonly through \u03b2. The latter serves in practice as a knob to control the number of spurious events.\n\nX\n\nY\n\nZ\n\nX 0\n\n\u03b4t\n\n(a) Asymmetric procedure\n\nI\nB\n\nX\n\nZ\n\nV\n\nZ 0\n\nX 0\n\nB0\nI 0\n\n2\n0\n\n3\n4\n00\n\n5\n1\n\n6\n0\n\n9\n7\n8\n000\n\n0\n\n\u03b4t\n2\n\n\u03b4t\n2\n\nT0\n\n1\n1\n\n000\n2\n4\n\n3\n\n0\n6\n\n000\n9\n7\n\n8\n\n(b) Symmetric procedure\n\nFigure 1: One-dimensional stochastic event synchrony.\n\nGiven event strings X and X 0, we wish to determine the parameters \u03b4t and st, and the hidden\nvariables B and B0; the parameter \u03c1spur (cf. (1)) can obtained from the latter :\n\n\u03c1spur\n\n4\n\n= Pn\n\nk=1 bk +Pn0\n\nn + n0\n\nk=1 b0\n\nk\n\n.\n\n(7)\n\nThere are various ways to solve this inference problem, but perhaps the most natural one is cyclic\nmaximization: \ufb01rst one chooses initial values \u02c6\u03b4(0)\n, then one alternates the following two\nupdate rules until convergence (or until the available time has elapsed):\n\nand \u02c6s(0)\n\nt\n\nt\n\n(\u02c6j(i+1), \u02c6j0(i+1)) = argmax\n\nb,b0\n\np(x, x0, j, j0, \u02c6\u03b4(i)\n\nt\n\n, \u02c6s(i)\nt )\n\n(\u02c6\u03b4(i+1)\n\nt\n\n, \u02c6s(i+1)\n\nt\n\n) = argmax\n\np(x, x0, \u02c6j(i+1), \u02c6j0(i+1), \u03b4t, st).\n\n\u03b4t,st\n\n(8)\n\n(9)\n\n3\n\n\fThe update (9) is straightforward, it amounts to the empirical mean and variance, computed over\nthe non-spurious events. The update (8) can readily be carried out by applying the Viterbi algorithm\n(\u201cdynamic programming\u201d) on an appropriate trellis (with the pairs of non-spurious events (xjk , x0\n)\nj0\nas states), or equivalently, by applying the max-product algorithm on a suitable factor graph [7]; the\nprocedure is similar to dynamic time warping [8].\n\nk\n\n3 Multi-Dimensional Point Processes\n\nIn this section, we will focus on the interdependence of multi-dimensional point processes. As a\nconcrete example, we will consider multi-dimensional point processes in time-frequency domain;\nthe proposed algorithm, however, is not restricted to that particular situation, it is applicable to\ngeneric multi-dimensional point processes.\n\n1, F 0\n\n1, \u2206X 0\n\nSuppose that we are given a pair of (continuous-time) signals, e.g., EEG signals recorded from two\ndifferent channels. As a \ufb01rst step, the time-frequency (\u201cwavelet\u201d) transform of each signal is approx-\nimated as a sum of (half-ellipsoid) basis functions, referred to as \u201cbumps\u201d (see Fig. 2 and [17]); each\nbump is described by \ufb01ve parameters: time X, frequency F , width \u2206X, height \u2206F , and amplitude\nW . The resulting bump models Y = ((X1, F1, \u2206X1, \u2206F1, W1), . . . , (Xn, Fn, \u2206Xn, \u2206Fn, Wn))\nn0)), representing the most\nand Y 0 = ((X 0\nprominent oscillatory activity, are thus 5-dimensional point processes. Our extension of stochastic\nevent synchrony to multi-dimensional point processes (and bump models in particular) is derived\nfrom the following observation (see Fig. 3): bumps in one time-frequency map may not be present\nin the other map (\u201cspurious\u201d bumps); other bumps are present in both maps (\u201cnon-spurious bumps\u201d),\nbut appear at slightly different positions on the maps. The black lines in Fig. 3 connect the centers\nof non-spurious bumps, and hence, visualize the offset between pairs of non-spurious bumps. We\nquantify the interdependence between two bump models by \ufb01ve parameters, i.e., the parameters\n\u03c1spur, \u03b4t, and st introduced in Section 2, in addition to:\n\n1), . . . , (X 0\n\nn0 , \u2206X 0\n\nn0, \u2206F 0\n\n1, \u2206F 0\n\n1, W 0\n\nn0 , W 0\n\nn0 , F 0\n\n\u2022 \u03b4f : the average frequency offset between non-spurious bumps,\n\u2022 sf : the variance of the frequency offset between non-spurious bumps.\n\nWe determine the alignment of two bump models in addition to the 5 above parameters by an infer-\nence algorithm similar to the one of Section 2, as we will explain in the following; we will use the\nnotation \u03b8 = (\u03b4t, st, \u03b4f , sf ). Model (6) may naturally be extended in time-frequency domain as:\n\np(y, y0, j, j0, \u03b8) \u221d \u03b2ntot\n\nspur\n\nnnon-spur\n\nYk=1\n\nk0 \u2212 xk\n\u2206xk + \u2206x0\nk0\n\nN(cid:16) x0\n\n\u00b7 p(\u03b4t)p(cid:0)st(cid:1)p(\u03b4f )p(cid:0)sf(cid:1),\n\n; \u03b4t, st(cid:17) N(cid:16) f 0\n\nk0 \u2212 fk\n\u2206fk + \u2206f 0\nk0\n\n; \u03b4f , sf(cid:17)\n\n(10)\n\nk0 \u2212 xk in time and offset f 0\n\nk0 \u2212 fk in frequency are normalized by the width and\nwhere the offset x0\nheight respectively of the bumps; we will elaborate on the priors on the parameters \u03b8 later on. In\nprinciple, one may determine the sequences J and J 0 and the parameters \u03b8 by cyclic maximization\nalong the lines of (8) and (9). In the multi-dimensional case, however, the update (8) is no longer\ntractable: one needs to allow permutations of events, the indices jk and j0\nk0 are no longer necessarily\nmonotonically increasing, and as a consequence, the state space becomes drastically larger. As a\nresult, the Viterbi algorithm (or equivalently, the max-product algorithm applied on cycle-free factor\ngraph of model (10)) becomes impractical.\n\nWe solve this problem by applying the max-product algorithm on a cyclic factor graph of the system\nat hand, which will amount to a suboptimal but practical procedure to obtain pairwise alignments\nof multi-dimensional point processes (and bump models in particular). To this end, we introduce a\nrepresentation of model (10) that is naturally represented by a cyclic graph: for each pair of events\nYk and Y 0\nk0 form pair of non-\nspurious events and is zero otherwise. Since each event in Y associated to at most one event in Y 0,\nwe have the constraints:\n\nk0 , we introduce a binary variable Ckk0 that equals one if Yk and Y 0\n\nC1k0\n\n4\n\n= S1 \u2208 {0, 1},\n\nn0\n\nXk0=1\n\nn0\n\nXk0=1\n\nC2k0\n\n4\n\n= S2 \u2208 {0, 1}, . . . ,\n\nn0\n\nXk0=1\n\nCnk0\n\n4\n\n= Sn \u2208 {0, 1},\n\n(11)\n\n4\n\n\fand similarly, each event in Y 0 is associated to at most one event in Y , which is expressed by a similar\nset of constraints. The sequences S and S0 are related to the sequences B and B0 (cf. Section 2):\nBk = 1 \u2212 Sk and B0\nk. In this representation, the global statistical model (10) can be cast\nas:\n\nk = 1 \u2212 S0\n\nn0\n\nn\n\np(y, y0, b, b0, c, \u03b8) \u221d\n\n(\u03b2\u03b4[bk \u2212 1] + \u03b4[bk])\n\n(\u03b2\u03b4[b0\n\nk \u2212 1] + \u03b4[b0\n\nk])\n\nn\n\nn0\n\nYk=1\nYk0=1 N(cid:16) x0\nYk=1\nYk=1(cid:0)\u03b4[bk +\nXk0=1\n\nn0\n\nn\n\n\u00b7\n\n\u00b7\n\nk0 \u2212 xk\n\u2206xk + \u2206x0\nk0\n\nckk0 \u2212 1](cid:1)\n\nYk0=1\n; \u03b4t, st(cid:17) N(cid:16) f 0\nYk0=1(cid:0)\u03b4[b0\nXk=1\n\nk0 +\n\nn0\n\nn\n\nckk0 \u2212 1](cid:1).\n\nk0 \u2212 fk\n\u2206fk + \u2206f 0\nk0\n\n; \u03b4f , sf(cid:17)!ckk0\n\np(\u03b4t)p(cid:0)st(cid:1)p(\u03b4f )p(cid:0)sf(cid:1)\n\n(12)\n\nSince we do not need to encode prior information about \u03b4t and \u03b4f , we choose improper priors\np(\u03b4t) = 1 = p(\u03b4f ). On the other hand, we have prior knowledge about st and sf . Indeed, we expect\na bump in one time-frequency map to appear in the other map at about the same frequency, but there\nmay be some timing offset between both bumps. For example, bump nr. 1 in Fig. 3(a) (t = 10.7s)\nshould be paired with bump nr. 3 (t = 10.9s) and not with nr. 2 (t = 10.8s), since the former is much\ncloser in frequency than the latter. As a consequence, we a priori expect smaller values for sf than\nfor st. We encode this prior information by means of conjugate priors for st and sf , i.e., scaled\ninverse chi-square distributions.\n\nA factor graph of model (14) is shown in Fig. 4 (each edge represents a variable, each node corre-\nsponds to a factor of (14), as indicated by the arrows at the right hand side; we refer to [7] for an\nintroduction to factor graphs). We omitted the edges for the (observed) variables Xk, X 0\nk0,\nk0 , Fk, F 0\n\u2206Xk, \u2206X 0\n\nk0 in order not to clutter the \ufb01gure.\n\nk0 , \u2206Fk, and \u2206F 0\nTime-frequency map\n\n\u2193\n\nBump model\n\n\u21d4\n\nTime-frequency map\n\n\u2193\n\nBump model\n\nFigure 2: Two-dimensional stochastic event synchrony.\n\nWe determine the alignment C = (C11, C12, . . . , Cnn0 ) and the parameters \u03b8 = (\u03b4t, st, \u03b4f , sf ) by\nmaximum a posteriori (MAP) estimation:\n\n(\u02c6c, \u02c6\u03b8) = argmax\n\np(y, y0, c, \u03b8),\n\nc,\u03b8\n\n(13)\n\nwhere p(y, y0, c, \u03b8) is obtained from (14) by marginalizing over b and b0:\n\np(y, y0, c, \u03b8) \u221d\n\nn\n\nn0\n\nckk0(cid:3) + \u03b4(cid:2)\n\nn0\n\nYk=1(cid:16)\u03b2\u03b4(cid:2)\nXk0=1\nYk0=1 N(cid:16) x0\n\nk0 \u2212 xk\n\u2206xk + \u2206x0\nk0\n\nn0\n\nXk0=1\n\nckk0 \u2212 1(cid:3)(cid:17)\n\n; \u03b4t, st(cid:17) N(cid:16) f 0\n\nk0 \u2212 fk\n\u2206fk + \u2206f 0\nk0\n\nn0\n\nn\n\nn\n\nYk0=1(cid:16)\u03b2\u03b4(cid:2)\nXk=1\n; \u03b4f , sf(cid:17)!ckk0\n\nXk=1\n\nckk0 \u2212 1(cid:3)(cid:17)\nckk0(cid:3) + \u03b4(cid:2)\np(\u03b4t)p(cid:0)st(cid:1)p(\u03b4f )p(cid:0)sf(cid:1).\n\n(14)\n\n1\n\n2\n\n3\n\n5\n\n10\nt [s]\n\n15\n\n20\n\n]\nz\nH\n\n[\n\nf\n\n30\n\n25\n\n20\n\n15\n\n10\n\n5\n\n0 0\n\n5\n\n10\nt [s]\n\n15\n\n20\n\n(a) Bump models of two EEG channels.\n\n(b) Non-spurious bumps (\u03c1spur = 27%);\nthe\nblack lines connect the centers of non-spurious\nbumps.\n\nFigure 3: Spurious and non-spurious activity.\n\n5\n\n\u00b7\n\nn\n\nYk=1\n\n]\nz\nH\n\n[\n\nf\n\n30\n\n25\n\n20\n\n15\n\n10\n\n5\n0 0\n\n\f\u03b2\n\nB1\n\n\u00af\u03a3\n\nB0\n2\n\n\u03b2\n\n\u00af\u03a3\n\nB2\n\n\u03b2\n\n\u00af\u03a3\n\n. . .\n\n\u03b2\n\nB0\nn0\n\n\u03b2\n\nBn\n\n\u00af\u03a3\n\n\u00af\u03a3\n\n\u03b4[bn] + \u03b2\u03b4[bn \u2212 1]\n\n\u03b4[bn +Pn0\n\nk0=1 cnk0 \u2212 1]\n\n\u03b2\n\nB0\n1\n\n\u00b5\u21930\n\n\u00af\u03a3\n\n\u00b5\u21910\n\n\u00b5\u2191\n\n\u00b5\u2193\n\n=\n\n=\n\n. . .\n\n=\n\n=\n\n=\n\n. . .\n\n=\n\n. . .\n\n=\n\n=\n\n. . .\n\n=\n\nC11 C12\n\nC1n0\n\nC21\n\nC22\n\nC2n0\n\nCn1 Cn2\n\nCnn0\n\nttt\n\nN\n\n. . .\n\nN\n\nN\n\nN\n\n. . .\n\nN\n\nN\n\nN\n\n. . .\n\nN\n\nN\n\n\u02c6\u03b8(k)\n\n=\n\n\u03b8 = (\u03b4t, st, \u03b4f , sf )\n\n\u02c6\u03b8(k)\n\n N(cid:16) x0\n\nn0 \u2212xn\n\u2206xn+\u2206x0\n\nn0\n\n; \u03b4t, st(cid:17) N(cid:16) f 0\n\nn0 \u2212fn\n\u2206fn+\u2206f 0\nn0\n\n; \u03b4f , sf(cid:17)!cnn0\n\np(\u03b4t, st, \u03b4f , sf ) = p(\u03b4t)p(st)p(\u03b4f )p(sf )\n\nFigure 4: Factor graph of model (14).\n\nFrom \u02c6c, we obtain the estimate \u02c6\u03c1spur as:\n\n\u02c6\u03c1spur = Pn\n\nk=1\n\n\u02c6bk +Pn0\n\nn + n0\n\nk=1\n\n\u02c6b0\nk0\n\n=\n\nn + n0 \u2212 2Pn\n\nk=1Pn0\n\nn + n0\n\nk0=1 \u02c6ckk0\n\n.\n\n(15)\n\nThe MAP estimate (13) is intractable, and we try to obtain (13) by cyclic maximization: \ufb01rst, the\nparameters \u03b8 are initialized: \u02c6\u03b4(0)\nf = s0,f , then one alternates the\nfollowing two update rules until convergence (or until the available time has elapsed):\n\nt = \u02c6s0,t, and \u02c6s(0)\n\nt = 0 = \u03b4(0)\n\nf , \u02c6s(0)\n\n\u02c6c(i+1) = argmax\n\np(y, y0, c, \u02c6\u03b8(i))\n\nc\n\n\u02c6\u03b8(i+1) = argmax\n\np(y, y0, \u02c6c(i+1), \u03b8).\n\n\u03b8\n\n(16)\n\n(17)\n\nThe estimate \u02c6\u03b8(i+1) (17) is available in closed-form; indeed, it is easily veri\ufb01ed that the point es-\ntimates \u02c6\u03b4(i+1)\nare the (sample) mean of the timing and frequency offset respectively,\ncomputed over all pairs of non-spurious events. The estimates \u02c6s(i+1)\nare obtained simi-\nlarly.\n\nand \u02c6\u03b4(i+1)\n\nand \u02c6s(i+1)\n\nf\n\nf\n\nt\n\nt\n\nUpdate (16), i.e., \ufb01nding the optimal pairwise alignment C for given values \u02c6\u03b8(i) of the parameters \u03b8,\nis less straightforward: it involves an intractable combinatorial optimization problem. We attempt\nto solve that problem by applying the max-product algorithm to the (cyclic) factor graph depicted\nin Fig. 4 [7]. Let us \ufb01rst point out that, since the alignment C is computed for given \u03b8 = \u02c6\u03b8(i),\nthe (upward) messages along the edges \u03b8 are the point estimate \u02c6\u03b8(i) (cf. (16)); equivalently, for the\npurpose of computing (16), one may remove the \u03b8 edges and the two bottom nodes in Fig. 4; the\nN -nodes then become leaf nodes. The other messages in the graph are iteratively updated according\nto the generic max-product update rule [7].\n\nThe resulting inference algorithm for computing (16) is summarized in Table 1. The messages\n\u00b5\u2191(ckk0 ) and \u00b5\u21910(ckk0 ) propagate upward along the edges ckk0 towards the \u00af\u03a3-nodes connected to\nthe edges Bk and B0\nk0 respectively (see Fig. 4, left hand side); the messages \u00b5\u2193(ckk0 ) and \u00b5\u21930(ckk0 )\npropagate downward along the edges ckk0 from the \u00af\u03a3-nodes connected to the edges Bk and B0\nk0\nrespectively. After initialization (18) of the messages \u00b5\u2191(ckk0 ) and \u00b5\u21910(ckk0 ) (k = 1, 2, . . . , n; k0\n= 1, 2, . . . , n0), one alternatively updates (i) the messages \u00b5\u2193(ckk0 ) (19) and \u00b5\u21930(ckk0 ) (20), (ii) the\nmessages \u00b5\u2191(ckk0 ) (21) and \u00b5\u2191 0(ckk0 ) (22), until convergence; it is noteworthy that, although the\nmax-product algorithm is not guaranteed to converge on cyclic graphs, we observed in our experi-\nments (see Section 4) that alternating the updates (19)\u2013(22) always converged to a \ufb01xed point. At\nlast, one computes the marginals p(ckk0 ) (23), and from the latter, one may determine the decisions\n\u02c6ckk0 by greedy decimation.\n\n4 Diagnosis of MCI from EEG\n\nWe analyzed rest eyes-closed EEG data recorded from 21 sites on the scalp based on the 10\u201320\nsystem. The sampling frequency was 200 Hz, and the signals were bandpass \ufb01ltered between 4\n\n6\n\n\fInitialization\n\n\u00b5\u2191(ckk0 ) = \u00b5\u21910(ckk0 ) \u221d N(cid:16) x0\n\nk0 \u2212 xk\n\u2206xk + \u2206x0\nk0\n\n; \u03b4t, st(cid:17) N(cid:16) f 0\n\nk0 \u2212 fk\n\u2206fk + \u2206f 0\nk0\n\n; \u03b4f , sf(cid:17)!ckk0\n\nIteratively compute messages until convergence\nA. Downward messages:\n\n(cid:18) \u00b5\u2193(ckk0 = 0)\n\u00b5\u2193(ckk0 = 1) (cid:19) \u221d(cid:18) max (\u03b2, max`06=k0 \u00b5\u2191(ck`0 = 1)/\u00b5\u2191(ck`0 = 0))\n(cid:18) \u00b5\u21930(ckk0 = 0)\n\u00b5\u21930(ckk0 = 1) (cid:19) \u221d(cid:18) max (\u03b2, max`6=k \u00b5\u21910(c`k0 = 1)/\u00b5\u21910(c`k0 = 0))\n\n1\n\n1\n\nB. Upward messages:\n\n(18)\n\n(19)\n\n(20)\n\n(21)\n\n(22)\n\n(23)\n\n(cid:19)\n(cid:19)\n; \u03b4f , sf(cid:17)!ckk0\n; \u03b4f , sf(cid:17)!ckk0\n; \u03b4f , sf(cid:17)!ckk0\n\nk0 \u2212 xk\n\u2206xk + \u2206x0\nk0\n\nk0 \u2212 xk\n\u2206xk + \u2206x0\nk0\n\n\u00b5\u2191(ckk0 ) \u221d \u00b5\u21930(ckk0 ) N(cid:16) x0\n\u00b5\u21910(ckk0 ) \u221d \u00b5\u2193(ckk0 ) N(cid:16) x0\np(ckk0 ) \u221d \u00b5\u2193(ckk0 )\u00b5\u21930(ckk0 ) N(cid:16) x0\n\nk0 \u2212 xk\n\u2206xk + \u2206x0\nk0\n\nk0 \u2212 fk\n\u2206fk + \u2206f 0\nk0\n\nk0 \u2212 fk\n\u2206fk + \u2206f 0\nk0\n\n; \u03b4t, st(cid:17) N(cid:16) f 0\n; \u03b4t, st(cid:17) N(cid:16) f 0\n; \u03b4t, st(cid:17) N(cid:16) f 0\n\nk0 \u2212 fk\n\u2206fk + \u2206f 0\nk0\n\nMarginals\n\nTable 1: Inference algorithm.\n\nand 30Hz. The subjects comprised two study groups: the \ufb01rst consisted of a group of 22 patients\ndiagnosed as suffering from MCI, who subsequently developed mild AD. The other group was a\ncontrol set of 38 age-matched, healthy subjects who had no memory or other cognitive impairments.\nPre-selection was conducted to ensure that the data were of a high quality, as determined by the\npresence of at least 20s of artifact free data. We computed a large variety of synchrony measures\nfor both data sets; the results are summarized in Table 2. We report results for global synchrony,\nobtained by averaging the synchrony measures over 5 brain regions (frontal, temporal left and right,\ncentral, occipital). For SES, the bump models were clustered by means of the aggregation algorithm\ndescribed in [17].\nThe strongest observed effect is a signi\ufb01cantly higher degree of background noise (\u03c1spur) in MCI\npatients, more speci\ufb01cally, a high number of spurious, non-synchronous oscillatory events (p =\n0.00021). We veri\ufb01ed that the SES measures are not correlated (Pearson r) with other synchrony\nmeasures (p > 0.10); in contrast to the other measures, SES quanti\ufb01es the synchrony of oscillatory\nevents (instead of more conventional amplitude or phase synchrony). Combining \u03c1spur with ffDTF\nyields good classi\ufb01cation of MCI vs. Control patients (see Fig.5(a)). Interestingly, we did not ob-\nserve a signi\ufb01cant effect on the timing jitter st of the non-spurious events (p = 0.91). In other words,\nAD seems to be associated with a signi\ufb01cant increase of spurious background activity, while the\nnon-spurious activity remains well synchronized. Moreover, only the non-spurious activity slows\ndown (p = 0.0012; see Fig.5(c)), the average frequency of the spurious activity is not affected in MCI\npatients (see Fig.5(c)). In future work, we will verify those observations by means of additional data\nsets.\n\nMeasure\np-value\n\nReferences\n\nMeasure\np-value\n\nReferences\n\nMeasure\np-value\n\nReferences\n\nMeasure\np-value\n\nReferences\n\nMeasure\np-value\n\nReferences\n\nMeasure\np-value\n\nCross-correlation\n\nCoherence\n\nPhase Coherence\n\nCorr-entropy\n\nWave-entropy\n\n0.012\u2217\n[20]\n\nffDTF\n0.0012\u2217\u2217\n\nIW\n0.060\n\ndDTF\n0.030\u2217\n\nI\n\n0.080\n[22]\n\n0.028\u2217\n\n0.060\n[16]\n\nGranger coherence\n\nPartial Coherence\n\n0.15\n\nKullback-Leibler\n\n0.072\n\nN k\n\n0.032\u2217\n\n0.16\n\nR\u00b4enyi\n0.076\n\nSk\n0.29\n[15]\n\n0.72\n\nPDC\n0.60\n\n[13]\n\n0.27\n[18]\n\nDTF\n0.34\n\nJensen-Shannon\n\nJensen-R\u00b4enyi\n\n0.084\n[23]\n\nH k\n0.090\n\n0.12\n\nS-estimator\n\n0.33\n[21]\n\nHilbert Phase\n\nWavelet Phase\n\nEvolution Map\n\nInstantaneous Period\n\n0.15\n\nst\n0.91\n\n0.082\n\n0.072\n\n0.020\u2217\n\n[24]\n\n[19]\n\n\u03c1spur\n\n0.00021\u2217\u2217\n\nTable 2: Sensitivity of synchrony measures for early prediction of AD (p-values for Mann-Whitney\ntest; * and ** indicate p < 0.05 and p < 0.005 respectively). N k, Sk, and H k are three measures\nof nonlinear interdependence [15].\n\n7\n\n\f0.45\n\n0.4\n\n0.35\n\nr\nu\np\ns\n\u03c1\n\n0.3\n\n0.25\n\n0.2\n\n0.15\n\n0.1\n0.045\n\n0.05\n\nMCI\nCTR\n\n0.055\n\n0.06\n\n19\n\n18\n\n17\n\nr\nu\np\ns\nf\n\n16\n\n15\n\n14\n\n13\n\n12\n\nCTR\n\nMCI\n\nr\nu\np\ns\n-\nn\no\nn\nf\n\n19\n\n18\n\n17\n\n16\n\n15\n\n14\n\n13\n\n12\n\nCTR\n\nMCI\n\nF 2\nij\n\n(a) \u03c1spur vs. ffDTF\n\n(b) Av. frequency of the spuri-\nous activity (p = 0.87)\n\n(c) Av. frequency of the non-\nspurious activity (p = 0.0019)\n\nFigure 5: Results.\n\nReferences\n[1] F. Varela, J. P. Lachaux, E. Rodriguez, and J. Martinerie, \u201cThe Brainweb: Phase Synchronization and\n\nLarge-Scale Integration\u201d, Nature Reviews Neuroscience, 2(4):229\u201339, 2001.\n\n[2] W. Singer, \u201cConsciousness and the Binding Problem,\u201d Annals of the New York Academy of Sciences,\n\n929:123\u2013146, April 2001.\n\n[3] M. Abeles, H. Bergman, E. Margalit, and E. 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Varela, \u201cMeasuring Phase Synchrony in Brain Sig-\n\nnals,\u201d Human Brain Mapping 8:194208 (1999).\n\n8\n\n\f", "award": [], "sourceid": 565, "authors": [{"given_name": "Justin", "family_name": "Dauwels", "institution": null}, {"given_name": "Fran\u00e7ois", "family_name": "Vialatte", "institution": null}, {"given_name": "Tomasz", "family_name": "Rutkowski", "institution": null}, {"given_name": "Andrzej", "family_name": "Cichocki", "institution": null}]}