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127 nips-2007-Measuring Neural Synchrony by Message Passing

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Author: Justin Dauwels, François Vialatte, Tomasz Rutkowski, Andrzej S. Cichocki

Abstract: A novel approach to measure the interdependence of two time series is proposed, referred to as “stochastic event synchrony” (SES); it quantiﬁes the alignment of two point processes by means of the following parameters: time delay, variance of the timing jitter, fraction of “spurious” events, and average similarity of events. SES may be applied to generic one-dimensional and multi-dimensional point processes, however, the paper mainly focusses on point processes in time-frequency domain. The average event similarity is in that case described by two parameters: the average frequency offset between events in the time-frequency plane, and the variance of the frequency offset (“frequency jitter”); SES then consists of ﬁve parameters in total. Those parameters quantify the synchrony of oscillatory events, and hence, they provide an alternative to existing synchrony measures that quantify amplitude or phase synchrony. The pairwise alignment of point processes is cast as a statistical inference problem, which is solved by applying the maxproduct algorithm on a graphical model. The SES parameters are determined from the resulting pairwise alignment by maximum a posteriori (MAP) estimation. The proposed interdependence measure is applied to the problem of detecting anomalies in EEG synchrony of Mild Cognitive Impairment (MCI) patients; the results indicate that SES signiﬁcantly improves the sensitivity of EEG in detecting MCI.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 SES may be applied to generic one-dimensional and multi-dimensional point processes, however, the paper mainly focusses on point processes in time-frequency domain. [sent-5, score-0.083]

2 The average event similarity is in that case described by two parameters: the average frequency offset between events in the time-frequency plane, and the variance of the frequency offset (“frequency jitter”); SES then consists of ﬁve parameters in total. [sent-6, score-0.528]

3 Those parameters quantify the synchrony of oscillatory events, and hence, they provide an alternative to existing synchrony measures that quantify amplitude or phase synchrony. [sent-7, score-0.73]

4 The pairwise alignment of point processes is cast as a statistical inference problem, which is solved by applying the maxproduct algorithm on a graphical model. [sent-8, score-0.175]

5 The SES parameters are determined from the resulting pairwise alignment by maximum a posteriori (MAP) estimation. [sent-9, score-0.094]

6 The proposed interdependence measure is applied to the problem of detecting anomalies in EEG synchrony of Mild Cognitive Impairment (MCI) patients; the results indicate that SES signiﬁcantly improves the sensitivity of EEG in detecting MCI. [sent-10, score-0.371]

7 For instance, it is hotly debated whether the synchronous ﬁring of neurons plays a role in cognition [1] and even in consciousness [2]. [sent-12, score-0.086]

8 The synchronous ﬁring paradigm has also attracted substantial attention in both the experimental (e. [sent-13, score-0.059]

9 Moreover, medical studies have reported that many neurophysiological diseases (such as Alzheimer’s disease) are often associated with abnormalities in neural synchrony [5, 6]. [sent-18, score-0.31]

10 The pairwise alignment of point processes is cast as a statistical inference problem, which is solved by applying the max-product algorithm on a graphical model [7]. [sent-20, score-0.175]

11 Our experiments, however, indicate that the it ﬁnds reasonable alignments in practice. [sent-22, score-0.028]

12 The SES parameters are determined from the resulting pairwise alignments by maximum a posteriori (MAP) estimation. [sent-23, score-0.056]

13 The proposed method may be helpful to detect mental disorders such as Alzheimer’s disease, since mental disorders are often associated with abnormal blood and neural activity ﬂows, and changes in the synchrony of brain activity (see, e. [sent-24, score-0.459]

14 In this paper, we will present promising results on the early prediction of Alzheimer’s disease from EEG signals based on SES. [sent-27, score-0.087]

15 In the next section, we introduce SES for the case of onedimensional point processes. [sent-29, score-0.022]

16 In Section 3, we consider the extension to multi-dimensional point processes. [sent-30, score-0.022]

17 In Section 4, we use our measure to detect abnormalities in the EEG synchrony of Alzheimer’s disease patients. [sent-31, score-0.383]

18 2 One-Dimensional Point Processes Let us consider the one-dimensional point processes (“event strings”) X and X in Fig. [sent-32, score-0.061]

19 We wish to quantify to which extent X and X are synchronized. [sent-34, score-0.036]

20 Intuitively speaking, two event strings can be considered as synchronous (or “locked”) if they are identical apart from: (i) a time shift δt ; (ii) small deviations in the event occurrence times (“event timing jitter”); (iii) a few event insertions and/or deletions. [sent-35, score-0.66]

21 More precisely, for two event strings to be synchronous, the event timing jitter should be signiﬁcantly smaller than the average inter-event time, and the number of deletions and insertions should comprise only a small fraction of the total number of events. [sent-36, score-0.53]

22 This intuitive concept of synchrony is illustrated in Fig. [sent-37, score-0.269]

23 The event string X is obtained from event string X by successively shifting X over δt (resulting in Y ), slightly perturbing the event occurrence times (resulting in Z), and eventually, by adding (plus sign) and deleting (minus sign) events, resulting in X . [sent-39, score-0.537]

24 Adding and deleting events in Z leads to “spurious” events in X and X (see Fig. [sent-40, score-0.286]

25 1(a); spurious events are marked in red): a spurious event in X is an event that cannot be paired with an event in X and vice versa. [sent-41, score-0.897]

26 The above intuitive reasoning leads to our novel measure for synchrony between two event strings, i. [sent-42, score-0.434]

27 SES is related to the metrics (“distances”) proposed in [9]; those metrics are single numbers that quantify the synchrony between event strings. [sent-45, score-0.449]

28 In contrast, we characterize synchrony by means of three parameters, which allows us to distinguish different types of synchrony (see [10]). [sent-46, score-0.538]

29 First, one generates an event string V of length , where the events Vk are mutually independent and uniformly distributed in [0, T0 ]. [sent-50, score-0.299]

30 The strings Z and Z are generated by delaying V over −δt /2 and δt /2 respectively and by (slightly) perturbing the resulting event occurrence times (variance of timing jitter equals st /2). [sent-51, score-0.652]

31 The sequences X and X are obtained from Z and Z by removing some of the events; more precisely, from each pair (Zk , Zk ), either Zk or Zk is removed with probability ps . [sent-52, score-0.057]

32 Since we do not wish/need to encode prior information about δt and st , we adopt improper priors p(δt ) = 1 = p(st ). [sent-56, score-0.247]

33 In the following, we will denote model (6) by p(x, x , j, j , δt , st ) instead of p(x, x , b, b , δt , st , ), since for given x, x , b, and b (and hence given n, n , and nnon-spur ), the length is fully determined, i. [sent-62, score-0.494]

34 It also noteworthy that T0 , λ and ps do not need to be speciﬁed individually, since they appear in (6) only through β. [sent-67, score-0.077]

35 The latter serves in practice as a knob to control the number of spurious events. [sent-68, score-0.166]

36 I B 2 0 34 00 78 9 00 0 5 6 1 0 X Z X V T0 0 δt 2 Y Z δt δt 2 Z X B 1 0 0 0 0 0 00 I X 1 2 3 4 6 7 89 (a) Asymmetric procedure (b) Symmetric procedure Figure 1: One-dimensional stochastic event synchrony. [sent-69, score-0.144]

37 Given event strings X and X , we wish to determine the parameters δt and st , and the hidden variables B and B ; the parameter ρspur (cf. [sent-70, score-0.451]

38 (1)) can obtained from the latter : n k=1 bk + n+n ρspur = n k=1 bk . [sent-71, score-0.316]

39 3 Multi-Dimensional Point Processes In this section, we will focus on the interdependence of multi-dimensional point processes. [sent-75, score-0.103]

40 As a concrete example, we will consider multi-dimensional point processes in time-frequency domain; the proposed algorithm, however, is not restricted to that particular situation, it is applicable to generic multi-dimensional point processes. [sent-76, score-0.083]

41 2 and [17]); each bump is described by ﬁve parameters: time X, frequency F , width ∆X, height ∆F , and amplitude W . [sent-81, score-0.214]

42 The resulting bump models Y = ((X1 , F1 , ∆X1 , ∆F1 , W1 ), . [sent-82, score-0.14]

43 , (Xn , Fn , ∆Xn , ∆Fn , Wn )), representing the most prominent oscillatory activity, are thus 5-dimensional point processes. [sent-88, score-0.06]

44 Our extension of stochastic event synchrony to multi-dimensional point processes (and bump models in particular) is derived from the following observation (see Fig. [sent-89, score-0.614]

45 3): bumps in one time-frequency map may not be present in the other map (“spurious” bumps); other bumps are present in both maps (“non-spurious bumps”), but appear at slightly different positions on the maps. [sent-90, score-0.236]

46 3 connect the centers of non-spurious bumps, and hence, visualize the offset between pairs of non-spurious bumps. [sent-92, score-0.064]

47 We quantify the interdependence between two bump models by ﬁve parameters, i. [sent-93, score-0.257]

48 , the parameters ρspur , δt , and st introduced in Section 2, in addition to: • δf : the average frequency offset between non-spurious bumps, • sf : the variance of the frequency offset between non-spurious bumps. [sent-95, score-0.719]

49 We determine the alignment of two bump models in addition to the 5 above parameters by an inference algorithm similar to the one of Section 2, as we will explain in the following; we will use the notation θ = (δt , st , δf , sf ). [sent-96, score-0.674]

50 In principle, one may determine the sequences J and J and the parameters θ by cyclic maximization along the lines of (8) and (9). [sent-98, score-0.069]

51 As a result, the Viterbi algorithm (or equivalently, the max-product algorithm applied on cycle-free factor graph of model (10)) becomes impractical. [sent-100, score-0.044]

52 We solve this problem by applying the max-product algorithm on a cyclic factor graph of the system at hand, which will amount to a suboptimal but practical procedure to obtain pairwise alignments of multi-dimensional point processes (and bump models in particular). [sent-101, score-0.37]

53 To this end, we introduce a representation of model (10) that is naturally represented by a cyclic graph: for each pair of events Yk and Yk , we introduce a binary variable Ckk that equals one if Yk and Yk form pair of nonspurious events and is zero otherwise. [sent-102, score-0.335]

54 Since each event in Y associated to at most one event in Y , we have the constraints: n n C1k = S1 ∈ {0, 1}, k =1 n C2k = S2 ∈ {0, 1}, . [sent-103, score-0.288]

55 , k =1 Cnk = Sn ∈ {0, 1}, k =1 4 (11) and similarly, each event in Y is associated to at most one event in Y , which is expressed by a similar set of constraints. [sent-106, score-0.288]

56 On the other hand, we have prior knowledge about st and sf . [sent-111, score-0.468]

57 Indeed, we expect a bump in one time-frequency map to appear in the other map at about the same frequency, but there may be some timing offset between both bumps. [sent-112, score-0.326]

58 8s), since the former is much closer in frequency than the latter. [sent-120, score-0.052]

59 As a consequence, we a priori expect smaller values for sf than for st . [sent-121, score-0.468]

60 We encode this prior information by means of conjugate priors for st and sf , i. [sent-122, score-0.468]

61 4 (each edge represents a variable, each node corresponds to a factor of (14), as indicated by the arrows at the right hand side; we refer to [7] for an introduction to factor graphs). [sent-126, score-0.04]

62 We omitted the edges for the (observed) variables Xk , Xk , Fk , Fk , ∆Xk , ∆Xk , ∆Fk , and ∆Fk in order not to clutter the ﬁgure. [sent-127, score-0.029]

63 Time-frequency map Time-frequency map ↓ ↓ Bump model Bump model ⇔ Figure 2: Two-dimensional stochastic event synchrony. [sent-128, score-0.2]

64 5 10 t [s] 15 20 (b) Non-spurious bumps (ρspur = 27%); the black lines connect the centers of non-spurious bumps. [sent-134, score-0.09]

65 n k =1 cnk = δ[bn + − 1] µ↓ µ↑ µ↑ µ↓ = C11 N = . [sent-145, score-0.027]

66 ttt N cnn ˆ θ(k) = θ = (δt , st , δf , sf ) N ˆ θ(k) xn −xn ∆xn +∆xn ; δ t , st N fn −fn ∆fn +∆fn ; δ f , sf p(δt , st , δf , sf ) = p(δt )p(st )p(δf )p(sf ) Figure 4: Factor graph of model (14). [sent-160, score-1.558]

67 From c, we obtain the estimate ρspur as: ˆ ˆ n n n + k=1 ˆk b n + n − 2 k=1 k =1 ckk ˆ = . [sent-161, score-0.556]

68 ˆ (17) θ ˆ The estimate θ(i+1) (17) is available in closed-form; indeed, it is easily veriﬁed that the point esˆ(i+1) and δ (i+1) are the (sample) mean of the timing and frequency offset respectively, ˆ timates δt f (i+1) computed over all pairs of non-spurious events. [sent-163, score-0.204]

69 (i+1) and sf ˆ are obtained simi- ˆ Update (16), i. [sent-165, score-0.221]

70 , ﬁnding the optimal pairwise alignment C for given values θ(i) of the parameters θ, is less straightforward: it involves an intractable combinatorial optimization problem. [sent-167, score-0.094]

71 We attempt to solve that problem by applying the max-product algorithm to the (cyclic) factor graph depicted ˆ in Fig. [sent-168, score-0.044]

72 Let us ﬁrst point out that, since the alignment C is computed for given θ = θ(i) , (i) ˆ the (upward) messages along the edges θ are the point estimate θ (cf. [sent-170, score-0.2]

73 (16)); equivalently, for the purpose of computing (16), one may remove the θ edges and the two bottom nodes in Fig. [sent-171, score-0.029]

74 The other messages in the graph are iteratively updated according to the generic max-product update rule [7]. [sent-173, score-0.085]

75 The messages ¯ µ↑(ckk ) and µ↑ (ckk ) propagate upward along the edges ckk towards the Σ-nodes connected to the edges Bk and Bk respectively (see Fig. [sent-175, score-0.702]

76 4, left hand side); the messages µ↓(ckk ) and µ↓ (ckk ) ¯ propagate downward along the edges ckk from the Σ-nodes connected to the edges Bk and Bk respectively. [sent-176, score-0.694]

77 After initialization (18) of the messages µ↑(ckk ) and µ↑ (ckk ) (k = 1, 2, . [sent-177, score-0.061]

78 At last, one computes the marginals p(ckk ) (23), and from the latter, one may determine the decisions ckk by greedy decimation. [sent-184, score-0.556]

79 The sampling frequency was 200 Hz, and the signals were bandpass ﬁltered between 4 6 Initialization µ↑(ckk ) = µ↑ (ckk ) ∝ N ckk xk − xk fk − fk ; δ t , st N ; δ f , sf ∆xk + ∆xk ∆fk + ∆fk (18) Iteratively compute messages until convergence A. [sent-186, score-2.076]

80 The subjects comprised two study groups: the ﬁrst consisted of a group of 22 patients diagnosed as suffering from MCI, who subsequently developed mild AD. [sent-190, score-0.067]

81 We computed a large variety of synchrony measures for both data sets; the results are summarized in Table 2. [sent-193, score-0.297]

82 We report results for global synchrony, obtained by averaging the synchrony measures over 5 brain regions (frontal, temporal left and right, central, occipital). [sent-194, score-0.34]

83 For SES, the bump models were clustered by means of the aggregation algorithm described in [17]. [sent-195, score-0.14]

84 The strongest observed effect is a signiﬁcantly higher degree of background noise (ρspur ) in MCI patients, more speciﬁcally, a high number of spurious, non-synchronous oscillatory events (p = 0. [sent-196, score-0.171]

85 We veriﬁed that the SES measures are not correlated (Pearson r) with other synchrony measures (p > 0. [sent-198, score-0.325]

86 10); in contrast to the other measures, SES quantiﬁes the synchrony of oscillatory events (instead of more conventional amplitude or phase synchrony). [sent-199, score-0.494]

87 Interestingly, we did not observe a signiﬁcant effect on the timing jitter st of the non-spurious events (p = 0. [sent-203, score-0.54]

88 In other words, AD seems to be associated with a signiﬁcant increase of spurious background activity, while the non-spurious activity remains well synchronized. [sent-205, score-0.208]

89 Moreover, only the non-spurious activity slows down (p = 0. [sent-206, score-0.042]

90 5(c)), the average frequency of the spurious activity is not affected in MCI patients (see Fig. [sent-208, score-0.31]

91 012∗ References [16] [18] [20] Measure Granger coherence Partial Coherence PDC DTF ffDTF dDTF p-value 0. [sent-216, score-0.032]

92 030∗ Measure Kullback-Leibler R´ nyi e Jensen-Shannon Jensen-R´ nyi e IW I p-value 0. [sent-222, score-0.048]

93 00021∗∗ Table 2: Sensitivity of synchrony measures for early prediction of AD (p-values for Mann-Whitney test; * and ** indicate p < 0. [sent-238, score-0.297]

94 N k , S k , and H k are three measures of nonlinear interdependence [15]. [sent-241, score-0.109]

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