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Effective Detection of Coupling in Short and Noisy Bivariate Data Joydeep Bhattacharya, Ernesto Pereda, and Hellmuth Petsche
Abstract—In the study of complex systems, one of the primary concerns is the characterization and quantification of interdependencies between different subsystems. In real-life systems, the nature of dependencies or coupling can be nonlinear and asymmetric, rendering the classical linear methods unsuitable for this purpose. Furthermore, experimental signals are noisy and short, which pose additional constraints for the measurement of underlying coupling. We discuss an index based on nonlinear dynamical system theory to measure the degree of coupling which can be asymmetric. The usefulness of this index has been demonstrated by several examples including simulated and real-life signals. This index is found to effectively disclose the nature and the degree of interactions even when the coupling is very weak and data are noisy and of limited length; by this way, new insight into the functioning of the underlying complex system is possible. Index Terms—Asymmetry, interdependency, noise, nonlinearity, state space, time series.
I. INTRODUCTION
O
NE basic problem in the field of experimental signal analysis is: Given two time series, can one tell if they originate from interacting or noninteracting systems? The fact that two signals are statistically not independent suggests the existence of hidden (not known a priori) relation between them. The detection of such hidden interaction or coupling facilitates designing a better control strategy for the evolution of the other system with the knowledge of one system. In nature, interdependency is a rule, rather than an exception [1]. In experimental situations, we do not have any direct access to the underlying dynamical systems but only to the time series or signals (used here interchangeably) of observations. Furthermore, experimental signals are short and likely to be contaminated by measurement noise. The curse of nonstationarity provides constraints for the duration of recording. Thus, robust methods are needed which can successfully unearth the hidden interdependence from noisy and short signals even if the degree of interaction is weak. Manuscript received August 15, 2001; revised February 27, 2002. This work was supported by the Sloan-Swartz Foundation, and Grant P.I. 00/0022-02 of the F.I.S. and Grant P.I. 2000/094 of the Canary Government. This paper was recommended by Associate Editor L. O. Hall. J. Bhattacharya is with the Division of Biology, California Institute of Technology, Pasadena, CA 91125 USA and also with the Commission for Scientific Visualization, Austrian Academy of Sciences, A-1220 Vienna, Austria (e-mail:
[email protected]). E. Pereda was with the Department of System Engineering, Institute of Technology and Renewable Energies (ITER), Poligono Industrial de Granadilla, Tenerife, Spain. He is now with the Departmento de Fisica Basica, Universidad de La Laguna, Tenerife 38320, Spain (e-mail:
[email protected]). H. Petsche is with the Brain Research Institute, University of Vienna, Vienna 1090, Austria (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSMCB.2003.808175
The basic form of interdependence is the linear one. If two and , generated by two Gaussian time series, processes, are linearly correlated in the time domain, the cross correlation or covariance index is able to disclose the linear relationship [2]. Linear associations in the frequency domain are measured by the coherence function, ratio of squares of cross spectral densities divided by the products of the two auto-spectra [2]. The higher the values of these indexes, the higher the degrees of correlations or interactions can be assumed. However, these only hold true if the interacting systems and the nature of interactions are linear. But nonlinear systems are ubiquitous in nature, thus, approaches, radically different from linear system theory-based methods, are needed when dealing with nonlinear dynamical systems as underlying real-life signals. A statistically more rigorous approach to detect interdependence is mutual information [3]. Since this function is not restricted to Gaussian systems, mutual information, intrinsically, is sensitive to all higher order correlations, both linear and nonlinear. Yet mutual information, as a symmetrical measure, cannot detect directional information of coupling unless two systems are coupled with a time delay. Related statistical methods are introduced in [4], [5] by comparing delay vector distribution functions of both system in their reconstructed state spaces. An information theoretic measure, called transfer entropy [6], was recently proposed to measure the statistical coherence between systems with the final aim of quantifying the coupling asymmetry. However, all measures based on information theory call for a huge quantity of noise-free stationary data; this strict condition is seldom met with experimental situations. Recently, another concept of interdependency, known as phase synchronization, was reported [7]. Kurths et al. demonstrated (using a model of coupled chaotic Rössler oscillators) [7], [8] that phase locking, usually found when two periodic oscillators are weakly coupled, can also occur in weakly coupled chaotic and/or noisy oscillators. This phenomenon is observed where a constant (instead of zero) phase difference is maintained between the oscillators yet the amplitudes remain chaotic. The strength of coupling required to achieve phase synchronization depends on the size of mismatch between the natural frequencies of the individual oscillators, and the resulting phase differences is a function of the strength of coupling and the frequency mismatch. Typically for bivariate data, instantaneous phase of individual sequence is computed by analytic signal approach and generalized phase difference sequence can be quantified by several proposed indexes (see [9] for a detailed discussions). This approach has been successful
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in the analysis of neurophysiological data in clinical [10], [11], and in cognitive [12], [13] environment. However, the method is primarily suitable for systems with continuous flow-like dynamics with some degrees of oscillation and is insensitive to asymmetry in coupling between systems.1 The topic of coupled complex systems was given a new dimension by the fact that two irregularly oscillating coupled chaotic systems can show synchronized behavior if the coupling strength is properly adjusted [15]. Strong mutual coupling of chaotic oscillators leads to identical synchronization, complete coincidence of variables of interacting systems, though individually varying chaotically. For the special case, if the two systems (e.g., and ) are coupled in a unidirectional fashion (one is driver and the other is response) in such a way that the state of the response system can be uniquely determined by the state of the driving system, induced synchronized behavior is termed as generalized synchrony [16], [17]. When two systems are generally synchronized, a topological equivalence exists between their individual state spaces. Indeed, with unidirectionally coupled generally synchronized chaotic systems, the state of the response system is a nonlinear continuous function of the state of the driving system; the presence of such function maps neighboring states of the driving system to similar neighboring states of the response system. Several methods based on this preserving property of closeness of neighborhoods—an indicator for the existence of underlying function—yield indirect evidence of generalized synchronization. Mutual false nearest neighbor technique was introduced [16] to establish continuity which finally indicates generalized synchrony. A set of statistics (continuity and differentiability) has been discussed in [18], [19] to characterize the nature of synchronizing manifold by detecting the existence of predictability and functionality between two time series from unidirectionally coupled deterministic systems. A method of mutual prediction [20] is also based on the underlying smoothness in the mapping function and one-step ahead zeroth-order prediction errors are compared to detect the directional coupling information. A measure of dynamical interdependence is derived by defining the predictability of one each system based on the knowledge of the other system; this method is applied to analyze seizure electroencephalogram (EEG) signals [21]. All these approaches have been proposed within a purely deterministic framework which is a rather strict condition. The weakest formulation can be: Given two time series preferably from stochastic systems, can we detect any sort of underlying weak interdependency? In this paper, a measure based on mean Euclidean distances between reference vectors and its original and mutual neighboring vectors is discussed. If the two signals are related by some function (whether continuous or not), the topological geometries of the two trajectories in the neighborhood sense are similar [22]. By carefully constructing the local neighborhoods, we show in this paper that it is indeed possible to provide a measure for both the relative strength and the directionality of coupling between the two systems even when the coupling is very weak and the observed signals are very short and noisy. 1A variant of this method is introduced in [14] for experimental detection of directionality of weak coupling between coupled self-sustained oscillators.
The paper is organized as follows. In Section II, we describe the procedure to obtain the proposed measure based on state space reconstruction. In Section III, we study the properties of the proposed measure by applying it to simulated systems, uni/bi-directionally coupled chaotic Henon maps. Analysis of real-life signals, such as cardio-respiratory signals and multivariate EEG signals, are presented in Section IV. Comparisons with other relevant measures are included, whenever appropriate. In Section V, we discuss the main results and provide intuitive justification for the superior performances of the proposed measure over available measures in case of weak coupling. The limitations of the proposed method are also mentioned. II. METHODS Let the two time series and be observed separately from two individual systems (or two different subsystems and , respectively. Initially, both of one complex system) time series are normalized to zero mean and unit variance. The corresponding state spaces are reconstructed using time delay embedding technique [23] so that the state vectors are obtained from consecutive scalar values of the time series, i.e., , where is the embedding dimension and is the time delay between succescan be sive elements of a state vector. Any state vector considered as a point in the -dimensional space. Next, let us -dimensional neighboring points ( state form a cloud of and say that the vectors of ) around this chosen vector . The time indexes of these neighbors are neighborhood is formed on the basis of Euclidean distances, closest neighbors of are those vectors i.e., whose Euclidean distances to are smaller than the disand other state space vectors. The average tances between Euclidean distances (or the average radius of this point-cloud) and its original neighbors are given by between (1) vectors neighboring to In the other state space, are found and denotes the time indexes of is formed these vectors. Next, another point cloud around , which bear the same temporal by its mutual neighbors . The average radius of indexes of the nearest neighbors of the cloud formed by mutual neighbors is defined as (2) If the two systems are strongly synchronized, both sets of neigh, bors, original and mutual, coincide and whereas for independent systems, mutual neighbors are more . Thus, the spread, which leads to strength of interdependence is reflected by the similarities (or dissimilarities) between these two kinds of clouds formed by original and mutual neighbors, respectively. In order to quantify the degree of similarity between these two point clouds, the
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following index (which was termed similarity index [24]) is computed:
where the average radius of cloud formed by the original neighbors in state space is given by
(3)
(8)
is the total number of state vectors and is the numbers of data points). In a similar way, is computed. For identical synchronization, both these indexes take the maximum value of one, whereas for independent systems, the indexes are zero or vanishingly small. On the other is assumed to have greater hand, if than vice versa. In this way, the directional influence on influence can be assessed. A related measure [25], [26] with similar properties like similarity index is defined by
We will demonstrate by several simulated and real-life examples that the modified index [(6) and (7)] can successfully disclose the strength of the coupling in both directions from a limited number data points, even contaminated by noise. Since only neighbors are considered for the formation of clouds, the computational time of is less than that of , where the radii of clouds are compared with the average radius of the whole state space. is close to by construction, true comparison can As be made only between these two indexes, although the results are also included, wherever appropriate. It should based on be stressed that is not normalized, thus, should be treated as a regulatory statistic or comparative measure without highlighting the absolute values.
where
(4) Here, the radii of the clouds formed by the mutual neighbors are compared with the average radii formed by all possible state has been found to be more robust vectors. This latter index and easier to interpret than [25]. From a practical point of view, the mutual neighbors, by definition, cannot be closer to the reference vector than its original neighbors. If the signals are noisy, and if the communication channels between the two systems are noisy, the mutual neighbors spread further in the state space. Thus, for weak coupling and noisy time series, the influences of one system on the other as manifested by the above mutual neighbors are obscured and the detection of a hidden coupling becomes difficult. In order to emphasize the contribution of one system explaining the dynamics of the other system, its own influence in terms of original neighbors has to be diminished. In terms of neighborhood, this , goal can be achieved as follows. For each reference vector numbers of neighbors bearing the time indexes we consider . Out of these state vectors, only the farthest vectors are chosen to form the original cloud. The average radius of the modified original cloud is given by (5) neighbors of time indexes In a similar way, are considered for the state vector . Out of these neighbors, only the time indexes of the closest neighbors . Thus, the are used to construct the mutual neighbors of average radius of the cloud formed by mutual neighbors remains the same as (2). We propose the following index to detect the strength of influence of on (6) In an analogous way, the influence of
on
is measured by (7)
III. SIMULATED SIGNALS A. Unidirectionally Coupled Systems As the first simulated example, we consider two unidirectionally coupled Henon maps [20], [26]
(9) (10) represented as systems and , respectively. Here, system drives system with coupling strength . Initially, we consider and vary the couboth systems to be identical pling strength from zero to one. For each coupling strength, and consists of only 300 the simulated series data points (initial 5000 data points are discarded as transients). In all analysis related to unidirectionally coupled Henon maps, and embedding parameters are fixed as follows: . and as a function of . Fig. 1(a) shows The obtained values are averaged over ten realizations. Above a , both critical threshold of coupling strength and are equally high, which indicates strong synchronization. For weaker coupling, the asymmetry is very promias found by others [25], [26]. nent, i.e., is driven by , the state space of represents only Since the dynamics of and has no information about the temporal represents the dynamics of , whereas the state space of has more informacombined dynamics of and . Thus, than vice versa. However, once the two systems tion about are identically synchronized, unidirectional coupling cannot be distinguished from bidirectional coupling by using state space . reconstruction based method. This is evident for Fig. 1(b) shows the results for unidirectionally coupled non. Since the two sysidentical systems tems are not same, identical synchronization is impossible so
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XjY
YjX
Fig. 1. E ( ) (‘o’) and E ( ) (‘x’) for unidirectionally coupled (a) identical (b = b = 0:3) and (b) nonidentical (b = 0:3 and b = 0:1) Henon maps. The small vertical lines denote the standard deviations. E shows the asymmetry in coupling.
both and will be high although not equal for higher coupling strength. Here also, the strong asymmetry holds for weaker coupling indicating that the systems are asymmetrically coupled. The local hump corresponding to the zone is most in likely due to the minima of the largest sub-Lyapunov exponent [20]. Next, we concentrate on the zone of very weak coupling . For comparative purpose, [(3)], and [(4)] are , and also computed along with ( sample points for all computations) and the results are shown in Fig. 2(a)–(c), respectively. When the coupling is very weak, ambivalent values of and lead to wrong conclusions about always the directional information, yet the modified index . This maintains the correct asymmetry, demonstrates that the proposed index can successfully detect the directional information even if the coupling is very weak. To study the sensitivity of these indexes on neighborhood and are computed for identically coupled ( size, varies from 0 to 0.2) Henon maps [(9) and (10)] with different and numbers of nearest neighbors as follows: ; results are presented in Fig. 3. The embedding diand , respecmension and time delay are set as tively. produces confusing results in terms of asymmetry with and is able to detect the asymmetry with higher number of nearest neighbors. The correct detection of asymmetry by is not possible if the coupling strength is very weak. However, only the modified index is found to be robust against the neighborhood sizes and it always truly identifies the hidden asymmetry in all cases with significance. Since experimental or real-life signals are very short sometimes, any index for detection of hidden relationships should be
Fig. 2. Performances of (a) S , (b) H , and (c) E for a zone of weak coupling (0 0:2). For all cases, only 300 data points are considered and the results are averaged over 10 realizations. Symbols as in Fig. 1. E consistently detects the asymmetry whereas S and H produce partially confusing results.
capable of handling also limited amount of data. Fig. 4 shows the performances of computed for the coupled identical Henon maps, in terms of the detection of asymmetry represented by , as a function of the number of the ratio time samples for four different coupling strengths. The condileading the tion of true asymmetry is ratio to be greater than one. The proposed index is capable of detecting asymmetries even if the time series are as short as 200 data points. Furthermore, experimental signals are most likely contaminated by noise, so the applied index has to be robust against noise too. Noise can be broadly of two types: 1) additive measurement noise and 2) intrinsic noise. As intrinsic noise has the capacity to change the inherent dynamics and can induce premature bifurcations, we only consider time series contaminated by measurement noise which does not perturb the inherent and for short dynamics. Fig. 5 shows performances of (200 time samples) and noisy bivariate data sequences generated by (9) and (10). Independent realizations of white noise and . The cousequences are added separately to . is found to be very sensitive to pling strength is noise when the coupling is weak whereas consistently shows the correct asymmetry even for very large noise intensity. Thus,
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Fig. 3. Influence of different choices of neighborhood size (R) on three indexes [(a)–(c) S , (d)–(f) H , and (g)–(i) E ], computed from unidirectionally coupled identical Henon maps with coupling strength fixed at = 0:2. From top to bottom, the profiles are for 8, 16, and 24 nearest neighbors, respectively. Results are averaged over ten realizations. Symbols are the same as in Fig. 1.
Fig. 4. Effect of sample size on the performances of E computed from unidirectionally coupled identical Henon maps for four different coupling strengths: = 0:1 (solid), = 0:2 (dashed), = 0:4 (dash-dotted), and = 0:6 (dotted), respectively. The ratio, E ( )=E ( ), is plotted as a function of sample size; for true detection of asymmetry, the ratio will be 1. E is found to be applicable for very short data of 200 points only. Symbols are the same as in Fig. 1.
XjY
YjX
is found to be robust against measurement noise even if the data sequence is of very limited size. In experimental situations, sudden changes in the dynamics of interacting systems can be caused by change in the coupling strength which essentially renders the data to be notoriously nonstationary. In order to study such transient dynamical phenomena, identical Henon maps [(9) and (10)] are considered which are coupled only during a single epoch and otherwise un-
coupled as studied recently [27]. The coupling strength is set as follows: for and and for . All these indexes ( , and ) are, principally, suitable for an epoch of data whereby averaging is done in reconstructed state space to produce a single number for an is entire epoch; in order to get an instantaneous response, and ) with same time computed for each vector pair ( indexes and the averaging operator [summation in (6) and (7)] is skipped; the result is shown in Fig. 6. For uncoupled regions, and fluctuates around some baseline both , there is a sharp increase in level. At time sample , and at , there is a sharp decrease in falling off again to baseline level. The interval where is significantly higher than the baseline level corresponds nicely with the coupling interval. Thus, is potentially able to detect a change in the coupling between the interacting systems with a high temporal resolution. B. Bidirectionally Coupled Systems As a second example, we consider bidirectionally coupled Henon maps [28], [29]
(11)
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Fig. 5. Effect of measurement noise on the performances of (a) and (b) E , computed from unidirectionally coupled identical Henon maps with coupling strength fixed at = 0:1. Independent realizations of white noise sequences are added to both time series. Noise intensity is finally averaged with respect to signal intensity. Only 300 data points are considered and results are averaged over ten realizations. Symbols as the same as in Fig. 1.
and . An asymmetry is found by the computed for this bivariate time series; typically, and which leads to the claim that heart has stronger influence on respiratory system than vice versa. This asymmetry is also found by although to lesser extent ( and ). Thus, the information flow from the heart to the respiratory system is significantly higher than in the opposite direction, which was reported earlier although less conspicuously [6]. In order to detect the nature of interactions (linear or nonlinear), the study is repeated with surrogate data. Two different kind of surrogates are constructed: 1) Fourier-based multivariate surrogates [31] and 2) time-shifted surrogates [32]. In the Fourier-based surrogates, the null hypothesis is that each signal is generated by linear Gaussian stochastic dynamical process observed through a (possibly nonlinear) measurement function which is instantaneous, invertible, static and time invariant, and further, the interaction between signals is linear. Systems in this class are strictly speaking nonlinear (and nonGaussian) though the nonlinearity (and non-Gaussianity) is not in the inherent dynamics but generated by observable process. These surrogates have the same mean, variance, auto correlation, and cross correlation properties as the original signal (see [33] for the method to generate Fourier-based multivariate surrogates). Thus, multivariate surrogate data of this kind preserve only linear correlations between signals while devoid of nonlinear correlations, if any. Thirty-nine surrogates are used in this study and is computed for the set of surrogates. If values of the surrogates are significantly lower than that of the original signals, the associated null hypothesis can be rejected. The result shown in Fig. 8(a) indicates that the coupling between the heart rate and the breath rate is significantly higher (probability, ) than the values of surrogates, rejecting the hypothesis of linearly coupled Gaussian systems. The asymmetry, as reported above, can provide information about the directional flow of information, but also can be affected by the different dynamical properties of each signal, in particular their intrinsic dimensionality [26]. A different classes of Fourier-based surrogates are proposed in [34] to circumvent values of
XY
YX
Fig. 6. Time dependence of E (E ( j ) by thick line, and E ( j ) by thin line) in unidirectionally coupled nonidentical Henon maps [(9), (10)]. The coupling parameter = 0 except for the interval between 200th and 500th time samples where = 0:5. The results are averaged over ten realizations followed by 20-point smoothing. E shows a clear and sharp change corresponding to the onset and end of coupling zone.
with and . The coupling strengths ( and ) are varied from 0 (uncoupled) to 0.33 (before the onset of identical synchronization). The proposed index , with and , is computed in both directions for 50 possible values of coupling within this range. The results are shown in Fig. 7. Only 500 data points are considered for computation, which is an order of magnitude less than used in other studies. The index successfully reflects the coupling strengths in both directions and till strong synchronization results. For , the system is more strongly coupled with than vice versa, and by Taken’s theorem again, the dynamics of is more explained by the dynamics of than vice versa. A similar conclusion, but in the opposite direction, holds for the and . These coupling strengths and differences between the two indexes reflects the asymmetry in the degree of coupling (see Fig. 7). IV. REAL-LIFE SIGNALS A. Cardio-Respiratory Signals As a first example of real-life systems, we consider the benchmark bivariate signals of breath rate and heart rate variations in a human subject with symptoms of sleep apnea [30]. The embedding parameters used for both the signals are:
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Fig. 7. Performances of E for bidirectionally coupled Henon maps [(11)]. For visual clarity, only a small range of values (0–0.4) are mapped to gray scale. Note the clear asymmetry in the right figure, which portrays the different degrees of influences exerted by one system to the other for specific zones of coupling.
Fig. 8. Statistical results of E for bivariate physiological data on comparison with those of their 39 different realizations of two types of surrogates: (a) Fourier surrogates and (b) time-shifted surrogates. See text for details. Original values are indicated by “X” and the distribution of values of surrogates are indicated as box plot. Both original indexes are found to be much higher than the values of their Fourier surrogates thus rejecting the null hypothesis of stochastic linear Gaussian processes. The influence of heart rate on breath rate is stronger than vice versa and cannot be explained by time-shifted surrogates either. Arrowhead indicates direction of influence.
this problem, however, which is a complicated procedure. Instead, we generate time shifted signals from one original signal and treat them as new surrogates [32]; interdependency measure is calculated between the other original signal and the timeshifted surrogates. Since the intrinsic dimensionality of the first original signal is preserved in its time-shifted surrogates, any solely due to the difasymmetry in the original values of ferences in intrinsic dimension, will also be preserved in the values of surrogates, otherwise the null hypothesis of differences in complexity as the prime reason explaining any reported asymmetry can be rejected. Fig. 8(b) shows the results. It is clear that the degree of influence of heart rate on breath rate is still sighigher than their surrogates counterpart, nificantly thus, providing firm evidence that although these two signals, heart rate and breath rate, are generated by two different systems of different complexity, heart rate exerts more influence on breath rate than vice versa and the nature of influence is inherently nonlinear. B. Neuro-Electrical Signals Hz) Neuronal oscillations in the -band (frequency and its synchronization between distant cortical areas are proposed to provide a platform for general cognitive integration [35], [36]. The detection of the -band synchronization, there-
fore, has immense importance for understanding and characterization of cognitive functioning of human brain. Furthermore, it has been reported that the degree of -band synchronization can be correlated with the maturity of brain [37] or with professional training [12]. Here we consider multivariate EEG signals recorded from 20 human subjects (ten professional musicians with mean age of 25.4 y and ten nonmusicians with mean age of 25.7 y) while listening to a piece of music (computer music composed by G. Martin) and at resting condition with eyes closed. From each individual, signals from 19 electrodes located on scalp according to the standard 10–20 electrode placement system were obtained; the electrodes are numbered as 1 to 19, from left frontal to right occipital; the recording time was 90 s. As a preprocessing, baseline drifts were removed by subtracting a second-order polynomial which was followed by digital bandpass filtering in the -band using sixth order Butwas terworth filter with cutoff frequencies 30 and 50 Hz. computed for all possible electrode combinations ( pairs of electrodes) with a nonoverlapping window of 6 s and were chosen as duration. The embedding parameters 10 and 2, respectively. Fig. 9(a) shows the profiles of absolute values of averaged over all possible electrode combinations for both groups (musicians and nonmusicians) while listening to music. In both groups, midline electrodes (Fz, Cz, Pz)
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Fig. 9. (a) Interdependency as measured by E for musicians (solid line) and nonmusicians (dotted line), averaged over nonoverlapped windows and over subjects within each group, for all possible combinations for electrode while listening to music. (b) Differences in interdependencies between listening to music and at resting condition. Both groups presented higher synchronization during listening to music as compared to rest, yet the effect was stronger in musicians.
presented higher values of which is most likely due to 1) the higher number of neighboring electrodes for midline than for lateral electrodes and 2) their higher involvement in the information transfer between the two hemispheres [12]. Fig. 9(b) shows the mean absolute differences between listening to music and resting condition for each group. There are several substantial points to be mentioned. First, both groups showed enhanced interactions between multiple cortical areas while listening to music as compared to resting condition. Second, the enhancement was higher for musicians and further, midline cortical areas again emerged as the most influential. We would like to note that the averaged profile of electrode regions as influencing other regions is similar to the profile when considering individual electrode regions being influenced by other regions—a possible signature of dense reciprocal and symmetric interactions between multiple cortical areas. To obtain an idea of the overall topographic pattern of changes in the degree of synchronization as measured by , we applied rank-sum Wilcoxon test between the two groups of subjects and the associated electrode pairs in which ) different (higher or significantly (probability, lower) degrees of synchronization while listening to music are connected by lines (Fig. 10). It is explicit that the degree of the -band synchronization in musicians was significantly higher than in nonmusicians and a strong network between cortical regions was formed in the former group while listening to music. In musicians, the stronger enhancement of the degree of the -band synchronization is most likely due to 1) their higher ability of binding various acoustical attributes, 2) their higher involvement of short term memory in modeling music, and 3) the retrieval of larger numbers of musical patterns from their long-term memory [24], [12]. We also studied the problem of hemispheric asymmetry in these two groups (see Fig. 11). It turned out that the left hemisphere in musicians presented significantly higher synchronization than the right during listening to music, in contrast to nonmusicians. Such an asymmetry in the functioning of the two hemispheres in musicians was earlier reported, based
Fig. 10. Significant probability mapping showing the comparison (in the degrees of -band synchronization) between musicians and nonmusicians during listening to music. The left column shows the increase for musicians and the right for nonmusicians. For better visual clarity, intra- and inter-hemispheric connections are displayed separately. It is evident that musicians showed significantly higher synchronization over multiple cortical areas as compared with nonmusicians while listening to music.
on anatomical findings, among others, that the left planum temporale is usually larger in musicians [38]. Generally, the left hemisphere is more responsible for specialized, prepositional, and analytic processing of information, whereas the right hemisphere is more adapted for the perception of appositional and holistic processing. Therefore, the result raises the possibility that being musically trained has neurological concominants enabling the musicians to employ distinctly different strategies for the analytical appreciation of music as compared with those of the nonmusicians.
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Fig. 11. Absolute mean values of E within each hemisphere for two groups. Results were averaged over all subjects within each group and all possible electrode combinations within each hemisphere. Left hemispheric dominance in the degrees of the -band synchronization is clear in musicians whereas right hemisphere is found to be dominant in nonmusicians.
Thus, this index was found to be useful in the detection of hidden functional interdependencies between multiple cortical areas, which are essential for the performance of higher cognitive functioning. V. DISCUSSION In this paper, we described an index for quantifying the relationships between bivariate time series and applied this technique to various examples including simulated and real-life time series. It has been found that the proposed index is capable of revealing asymmetries in interactions even if the numbers of data points are very limited. First, we considered unidirectionally coupled chaotic Henon maps as a benchmark simulation example. Since the aim of the present study is to establish a measure which is applicable for very short data, most of the simulations, unless stated otherwise, were performed with as few as 300 data points, which is an order of magnitude less is found than the number of points used in other studies. to change systematically when the coupling strength increases while preserving the asymmetrical coupling. However, once the interacting systems are completely synchronized, asymmetry is lost and values are high in both ways. In experimental situation, the weak coupling can be hidden whereas the case for strong coupling is easier to detect just by normal visual inspection. Therefore, the study was repeated to detect zones of very , and were presented for weak coupling and the results of comparative purpose. Only the proposed index was found to unearth the hidden asymmetry, however weak, in a consistent way, whereas the other measures produced confusing results for extremely weak coupling. The robustness of this proposed index was also demonstrated against measurement noise and it was found that was capable to detect the asymmetry in coupling even if the time series were as short as 200 data points and signal-to-noise ratio (SNR) as low as 3 dB. The proposed index was also found to be robust against the neighborhood sizes. In order to address the sudden change in coupling strength,
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was computed for each vector by skipping the averaging operator, which made it possible to obtain as a function of time, thereby, making it suitable for detection of changes in coupling with a good temporal resolution. However, this method needs to be modified in order not to include future state vectors before making it useful for real-time applications. A set of bidirectionally coupled Henon map was also analyzed and could detect the asymmetry in the strength of coupling, albeit bidirectional, till the zone of identical synchronization. The new index was then applied to two sets of physiological signals: heart-respiratory signals from a sleep apneic subject, and multivariate EEG signals from healthy subjects during music perception. A strong asymmetry was found as an influence by heart on respiration, which was further tested by different surrogates. A new type of time-shifted surrogate was briefly discussed which is suitable to address the problem of mismatch of intrinsic dimensionality. The aforesaid coupling in the direction from heart to breath was found to be significantly nonlinear. Next, multivariate EEG signals were recorded from two broad groups of subjects, musicians and nonmusicians, while listening to music and at rest. Analysis of synchronization or interdependency is of particular interest in neuroscience since there are widespread evidences that during cognition, multiple cortical areas may not only become coactive, but also functionally correlated [39]. In this paper, emphasis was put on the -band, since neuronal oscillations and synchronization in the 30–70 Hz range play a pivotal role in the higher brain functioning including object recognition, conscious perception, attention, memory etc. [40]–[43]. Subjects with professional training in music showed global enhancement of -band interdependency as compared with naive listeners during music perception as reported earlier [12]. Furthermore, musicians showed left-hemispheric dominance, which is also consistent with recent imaging study [44]. Thus, the proposed index is able to extract valuable information from real-life signals providing meaningful insights into the underlying dynamics. This new index has excelled and mainly while dealing with noisy and weakly coupled signals containing limited data points. For completely deterministic series and strong coupling, performances of all these indexes are found to be equivalent, which can be easily understood by taking into account that, in such cases, the cloud of mutual neighbors is close to the reference vector. At the same time, the average distance of the nearest neighbors to the reference vector is not very different to the average distance between the reference vector and all the other vectors (especially if is great and the signal contains limited points) so that (4) and (6) are equivalent. Even in this case, the use of the proposed index [(6)] instead of available index [(4)] would present a practical advantage because its calculation is less costly in computational terms and further, the underlying asymmetry, if any, is more emphasized in terms of the new index. The advantage of for weak coupling is closely related to the way in which it is defined. In this case, mutual neighbors are approximately the same to that of the randomly chosen neighbors, and is close to 0. In this situation, neither the denominator nor the numerator in (6) contains any relevant information about the dynamics of the neighborhood of the reference vector. However, nearest neighbors instead of the by picking up the whole cloud, scans state vectors that are dynamically related
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to the reference vector. More importantly, the response system has a larger effective dimension on this length scale [22], thus . This relagiving rise to the fact that [25], is masked by the tionship, which should also hold for fact mentioned above that in the calculation of this index all the state vectors are considered, which is very likely the factor responsible for the contradictory behavior of at extremely weak coupling. As for the case of noisy signals, it is likely that, due to the measurement noise, some of the true nearest neighbors of a given reference vector are scattered away from its neighborhood. As a consequence, there are geometrical nearest neighbors, which are not actually dynamical neighbors. By considering nearest neighbors, these scattered vectors, the which for a not very high SNR cannot be scattered far from the reference vector, are taken into account in the calculation of . The above reasoning indicates that, although the definition of might seem rather ad hoc, it presents some advantages as compared to the other existing interdependence indexes based on the concept of mutual neighbors especially while dealing with short noisy experimental signals. Another situation in which the strategy of discarding some of the nearest neighbors of the reference vectors might pay off is the case where two systems and are related to each other due to the existence of a common driver system . In such a case, it could be useful to assess the influence of by excluding from the calculation and those of the corresponding coupling index between nearest neighbors of that are also nearest neighbors of . Then, it is possible to compare the value of the index calculated with and without such vectors and, in case this difference is statistically significant, we can conclude that the and are, at least partially, the interdependence between result of the influence of a common driver. The problem of elucidating such question, which has been recently addressed by using entropy-related measures of interdependence [6], is certainly of importance in practice, for instance, in certain neurological disorders such as epilepsy, where the onset of a seizure is marked by the increasing influence of the epileptic focus on the neighboring brain areas. There are a few critical and unresolved points which need to be mentioned. Although has been found to perform better than , its main drawback is that , like , is not normalized. In order to circumvent this problem, was used as a regulatory or comparative measure in this paper instead of as an absolute measure. Secondly, the state space reconstruction lies at the heart of this method and one has to be careful in choosing embedding parameters ( and ) in order to have a valid reconstruction; we refer the interested readers to [45], [46] for practical discussions on the selection of embedding parameters. It is beyond the scope of the present paper to try innumerous combinations on embedding parameters; the to find the sensitivity of embedding parameters used in this study were chosen after fulfilling the criteria of a valid embedding. On the other hand, was found to be fairly robust against the choice of number of neighbors. Thirdly, like other measures based on state space reconstruction, is likely to be sensitive to the excited degrees of freedom operating at length scales set by radius of the chosen neighborhood. Thus, any difference in the intrinsic dimensionality (as decided by the inherent degrees of freedom)
between two systems would likely effect the asymmetry as reported by . Recently, a measure called synchronization likelihood is introduced [27] which claimed to avoid this bias introduced by the degrees of freedom of the interacting systems but this measure is symmetric, thus being unable to detect any directional information. We addressed this problem by using time-shifted surrogates of real-life signals, although we feel that more systematic study is needed to settle this issue. In summary, we have shown that by means of a careful choice of the state space vectors for the calculation of coupling between two signals, it is possible to define a measure, based on nonlinear dynamical system theory, which can successfully disclose the nature and strength of hidden coupling, even if weak, from noisy and short signals. In consequence, we expect that the proposed index will be useful in the practical applications in different scientific fields dealing with the analysis of real-life multivariate experimental signals. ACKNOWLEDGMENT The authors wish to thank U. Feldmann for pointing out an error in their earlier computations. J. Bhattacharya also wishes to thank U. Feldmann for her critical but useful comments. Multivariate Fourier surrogates were generated by TISEAN [47]. REFERENCES [1] L. Glass, “Synchronization and rhythmic processes in physiology,” Nature, vol. 410, pp. 277–284, 2001. [2] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures. New York: Wiley, 2000. [3] C. E. Shannon and W. Weaver, The Mathematical Theory of Communication. Chicago, IL: Univ. of Illinois Press, 1949. [4] H. Kantz, “Quantifying the closeness of fractal measures,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat., vol. 49, pp. 5091–5097, 1994. [5] C. Diks, W. van Zwet, F. Takens, and J. DeGoede, “Detecting differences between delay vector distributions,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat., vol. 53, pp. 2169–2176, 1996. [6] T. Schreiber, “Measuring information transfer,” Phys. Rev. Lett., vol. 85, no. 2, pp. 461–464, 2000. [7] M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett., vol. 76, no. 11, pp. 1804–1807, 1996. [8] , “Phase synchronization in driven and coupled chaotic oscillators,” IEEE Trans. Circuits Syst. I, vol. 44, pp. 874–881, Oct. 1997. [9] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization—A Universal Concept in Nonlinear Sciences, Cambridge, U.K.: Cambridge Univ. Press, 2001. [10] P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, phase locking A. A. Schnitzler, and H.-K. Freund, “Detection of : from noisy data: Application to magnetoencephalography,” Phys. Rev. Lett., vol. 81, no. 15, pp. 3291–3294, 1996. [11] F. Mormann, K. Lehnertz, P. David, and C. E. Elger, “Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients,” Phys. D, vol. 144, pp. 358–369, 2000. [12] J. Bhattacharya and H. Petsche, “Musicians and the Gamma Band—A secret affair?,” NeuroReport, vol. 12, pp. 371–374, 2001. [13] J. Bhattacharya, H. Petsche, U. Feldmann, and B. Rescher, “Electroencephalograph gamma-band phase synchronization between posterior and frontal cortex during mental rotation in humans,” Neurosci. Lett., vol. 311, pp. 29–32, 2001. [14] M. G. Rosenblum and A. S. Pikovsky, “Detecting direction of coupling in interacting oscillators,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat., vol. 64, p. 045 202, 2001. [15] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, no. 8, pp. 821–824, 1990. [16] N. F. Rulkov, M. M. Suschik, L. S. Tsimring, and H. D. I. Abarbanel, “Generalized synchronization of chaos in directionally coupled chaotic systems,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat., vol. 51, no. 2, pp. 980–994, 1995.
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Joydeep Bhattacharya was born in Rishra, India, in 1972. He received the B.E. degree in electronics engineering from Calcutta University, Calcutta, India, in 1994 and the Ph.D. degree from Indian Institute of Technology (IIT), Kharagpur, in 2000. From 1994 to 1998, he was a Project Officer in IIT. During 1998–1999, he was with Max-PlanckInstitute for Physics of Complex Systems, Dresden, Germany. Since 1999, he has been a member of the Research Staff in the Austrian Academy of Sciences. Currently, he is in the Division of Biology, California Institute of Technology as a Sloan-Swartz Fellow. His broad research interests include physiological signal analysis, nonlinear dynamics and chaos, and cognitive neuroscience. Dr. Bhattacharya was the recipient of the German Academic Exchange Fellowship and the International Federation of Clinical Neurophysiology Young Scientist Fellowship.
Ernesto Pereda was born in Madrid, Spain, in 1973. He received the B.Sc. degree in physics and the Ph.D. degree in applied physics, both from the University of La Laguna, Tenerife, Spain, in 1996 and 2001, respectively. After one year as a Postdoctoral Fellow with the Institute of Technology and Renewable Energy, Tenerife, he became an Assistant Professor in the Department of Basic Physics at the University of La Laguna. He was associated shortly with Max-Planck-Institute for Physics of Complex Systems, Dresden, Germany. His main research interests are complex time-series analysis and the application of dynamical systems theory to characterize experimental signals.
Hellmuth Petsche was born in Scheibbs, Austria, in 1923. He received the M.D. degree from the University of Vienna, Vienna, Austria. He was formerly the Head of the Institute of Neurophysiology, University of Vienna, and has been an Emeritus Professor at the university since 1990. He is also a Senior Scientist at the Institute of Brain Research, University of Vienna, and at the Austrian Academy of Sciences. He has been associated with the field of neurophysiology and especially with EEG over five decades and published more than 200 papers. His earlier work was on the origin of EEG and its anatomical background and presently his interest is EEG in mental processes in particular with respect to music and visual art. Dr. Petsche is a Full Member of the Austrian Academy of Sciences. He served on the editorial board of several journals including Clinical Neurophysiology, Brain Topography, and the International Journal of Psychophysiology. He has coauthored and edited six books including the latest monograph, EEG and Thinking (together with S.C. Etlinger), Austrian Academy of Sciences, 1998.
