Statistical Complexity and Nontrivial Collective Behavior in Electroencephalografic Signals
aa r X i v : . [ n li n . AO ] M a y STATISTICAL COMPLEXITY AND NONTRIVIALCOLLECTIVE BEHAVIOR INELECTROENCEPHALOGRAPHIC SIGNALS
M. ESCALONA-MOR ´AN,
1, 2, 3
M. G. COSENZA, R. L ´OPEZ-RUIZ, and P. GARC´IA Grupo de Ingenier´ıa Biom´edica, Facultad de Ingenier´ıa,Universidad de Los Andes, M´erida, Venezuela. Laboratoire Traitement du Signal et de l’Image (LTSI),Universit´e de Rennes 1, Campus Scientifique de Beaulieu,Bˆat. 22, 35042 Rennes Cedex, France. INSERM U642, LTSI, Bˆat. 22, 35042 Rennes Cedex, France Centro de F´ısica Fundamental, Universidad de Los Andes, M´erida, Venezuela. DIIS and BIFI, Facultad de Ciencias,Universidad de Zaragoza, E-50009 Zaragoza, Spain. Laboratorio de Sistemas Complejos,Departamento de F´ısica Aplicada, Facultad de Ingenier´ıa,Universidad Central de Venezuela, Caracas, Venezuela. bstract We calculate a measure of statistical complexity from the global dynamics of electroencephalo-graphic (EEG) signals from healthy subjects and epileptic patients, and are able to establish acriterion to characterize the collective behavior in both groups of individuals. It is found that thecollective dynamics of EEG signals possess relative higher values of complexity for healthy subjectsin comparison to that for epileptic patients. To interpret these results, we propose a model of a net-work of coupled chaotic maps where we calculate the complexity as a function of a parameter andrelate this measure with the emergence of nontrivial collective behavior in the system. Our resultsshow that the presence of nontrivial collective behavior is associated to high values of complexity;thus suggesting that similar dynamical collective process may take place in the human brain. Ourfindings also suggest that epilepsy is a degenerative illness related to the loss of complexity in thebrain.
PACS numbers: 05.45.-a, 05.45.Xt, 05.45.Ra et al. , 2002; Manrubia et al. , 2004].Nontrivial collective behavior is characterized by a well defined evolution of macroscopicquantities coexisting with local chaos. Models based on coupled map networks have beenwidely used in the investigation of collective phenomena that appear in many complexsystems [Kaneko & Tsuda, 2000]. In particular, networks of coupled chaotic maps canexhibit nontrivial collective behavior.A paradigmatic example of a complex system is provided by the human brain. It consistsof a highly interconnected network of millions of neurons. The local dynamics of a neuronin general behaves as a non-linear excitable element [Herz et al. , 2006]. From the signal of asingle neuron it is not possible to understand the highly structured collective behavior andfunctions of the brain.In this paper we investigate the relative complexity of the human brain by considering thecollective dynamics that arise from the local dynamics of groups of neurons, as manifestedin electroencephalographic (EEG) signals. We calculate a measure of complexity from theglobal dynamics of EEG signals from healthy subjects and epileptic patients, and are ableto establish a criterion to characterize the collective behavior in both groups of individuals.It is found that the collective dynamics of EEG signals possess relative higher values ofcomplexity for healthy subjects in comparison to that for epileptic patients. Our results3upport the view that epilepsy is characterized by a loss of complexity in the brain, asindicated by measurements of the dimension correlation [Babloyantz A. & Destexhe A.,1986], algorithmic complexity [Rapp et al. , 1994], and anticipation of seizures [Martinerie etal. , 1998].In order to interpret our results, we propose a model of coupled chaotic maps where wecalculate the measure of complexity as a function of a parameter and relate this measurewith the emergence of nontrivial collective behavior in the system. Our results show thatthe appearance of nontrivial collective behavior is associated to high values of complexity;thus suggesting that similar dynamical collective process may take place in the human brain.Several measures of complexity have been proposed in the literature. Here we employ theconcept of statistical complexity, introduced by Lopez-Ruiz et al. [1995]. This quantity isbased on the statistical description of a system at a given scale, and it has been shown to becapable of discerning among different macroscopic structures emerging in complex systems[S´anchez & Lopez-Ruiz, 2005]. The amount of complexity C is obtained by computingthe product between the entropy H , and a sort of distance to the equipartition state inthe system, named the disequilibrium D . Thus, the statistical complexity is defined as[Lopez-Ruiz et al. , 1995] C = H · D = − K R X s =1 p s log p s · R X s =1 (cid:16) p s − R (cid:17) , (1)where H and D are, respectively, the entropy and the disequilibrium; p s represents theprobability associated to the state s ; R is the number of states, and K is a positive normal-ization constant. Note that p s may vary for different levels of observation, reflected in R .The quantity C can quantify relative values of complexity in a specific system at a givenlevel of description.The EEG data base used in this study consists of records from 40 individuals in an agerange between 22 and 48 years old. These individuals are classified into four groups: (I)a group of 10 healthy subjects; (II) a group of 10 epileptic patients receiving treatmentwith Phenobarbital for at least 18 months; (III) a group consisting of 18 epileptic patientsthat have not yet received medical treatment; and (IV) a group of 2 epileptic patients whoexperienced spontaneous seizures during the EEG recording. All the epileptic patients inthe data base were diagnosed generalized epilepsy manifested through tonic-clonic seizures.The record of the EEG signal from each individual was carried out over 19 channels4onnected to scalp electrodes according to the international 10 −
20 system [Jasper, 1958].The potentials were measured with respect to a reference level consisting of both ears short-circuited. The signal was digitalized at a sampling frequency of 256 Hz and A/D conversionof 12 bits, and filtered to bandwidth between 0 . t can be described by, the instantaneous meanfield h t , defined as h t = 1 M M X j =1 e jt , (2)where e jt is the real value registered by electrode j at discrete time t , j = 1 , . . . , M ; and M = 19 is the number of electrodes.The probability distribution of the mean field values corresponding to a given EEG signalis constructed from the time series of h t calculated for that signal. We define the numberof states R as a partition consisting of R equal size segments on the range of values of h t .Next, the probability p s associated to the s state is calculated. Here we set R = 10 for allthe EEG signals. Then, Eq. (1) is used to calculate the statistical complexity C associatedto the mean field for each EEG.Figure 1 shows the statistical complexity of the mean field of the EEG signal for allthe individuals in each group from the data base. Figure 1 indicates that the complexitymeasure for the global dynamics of the EEG signal is higher in healthy subjects (groupI) in relation to that of epileptic patients (groups II, III, and IV). The lowest levels ofcomplexity correspond to the patients undergoing epileptic seizures (group IV). This measureof complexity allows us to discern between healthy subjects and epileptic patients. AmongEEG signals from epileptic patients, those signals from patients under treatment seem topossess slightly greater complexity than those from patients without treatment. However,the quantity C is not very efficient for distinguishing between the different groups of epilepticpatients. The spread of the values of the complexity found in group I indicates a greater5ariability in the statistical properties of EEG signals from the healthy subjects. IVIIIIII C Individuals
FIG. 1: Statistical complexity of healthy subjects (group I), epileptic patients subject to treatment(group II), epileptic patients without treatment (group III), and epileptic patients undergoingseizures (group IV). Vertical lines indicate the limits for each group. The order of the individualsin each group does not have any meaning.
It is important to say that other choices for the partition of states of the EEG signal arepossible, and hence different values for the H and D measures can be obtained, withoutaffecting the overall behavior of the statistical quantity C . For instance, Rosso et al. [2003]divided the time axis of EEG signals into non-overlapping temporal windows on which thewavelet energy at different resolution levels can be calculated.In order to give an interpretation to the above results, we propose a dynamical model. Weconsider a system of N interacting nonlinear, heterogeneous elements forming a network,where the state of element i ( i = 1 , , . . . , N ) at time t is denoted by x it . The evolutionof the state of each element is assumed to depend on its own local dynamics and on itsinteraction with the network, whose intensity is described by the coupling parameter ǫ .Then, we consider a network of maps subjected to a global interaction as follows [Cisneros et al. , 2002] x it +1 = (1 − ǫ ) f i ( x it ) + ǫN N X j =1 f j ( x jt ) , (3)where the function f i ( x it ) describes the local dynamics of map i . The usual homogeneousglobally coupled map system [Kaneko, 1990] corresponds to having the same local function6or all the elements, i.e., f i ( x it ) = f ( x it ). As local chaotic dynamics we choose the logarithmicmap f ( x ) = b + ln | x | , x ∈ ( −∞ , ∞ ), where b is a real parameter. This map does not belongto the standard class of universality of unimodal or bounded maps. Robust chaos occursin the parameter interval b ∈ [ − , f i ( x it ) = b i + ln | x it | , where the values b i are uniformly distributed at randomin the interval [ − , h t = 1 N N X j =1 f j ( x jt ) . (4)Figure 2(a) shows the bifurcation diagram of the mean field h t of the globally coupledheterogeneous map network, Eq. (3), with b i ∈ [ − ,
1] and N = 10 , as a function of thecoupling strength ǫ . For each value of ǫ , the mean field was calculated at each time stepduring a run of 10 iterates starting from random initial conditions on the local maps, uni-formly distributed on the interval x i ∈ [ − , transients. The localmaps are chaotic and desynchronized (see Fig.2(b)). However, the mean field in Fig. 2(a) re-veals the existence of global periodic attractors for some intervals of the coupling parameter.This is the phenomenon of nontrivial collective behavior where macroscopic order coexistswith local disorder in a system of interacting dynamical elements. Different collective statesemerge as a function of the coupling ǫ . In this representation, collective periodic states at agiven value of the coupling appear as sets of vertical segments which correspond to intrinsicfluctuations of the periodic orbits of the mean field. Increasing the system size N does notdecrease the amplitude of the collective periodic orbits. Moreover, when N is increased thewidths of the segments that make a periodic orbit in the bifurcation diagrams such as inFig. 2(a) shrink, indicating that the global periodic attractors become better defined in thelarge system limit.Figure 2(c) shows the complexity C of the mean field as a function of ǫ . Here, theobservation level was set at R = 15. When the value of ǫ is small, the bifurcation diagramof h t shows a period-one collective attractor, which implies that for this parameter rangethe system follows the standard statistical behavior of uncorrelated disordered variables thatyield a single period for its mean field. At the chosen level of resolution, the complexity7 IG. 2: (a) Bifurcation diagram of h t as a function of ǫ for a coupled heterogeneous map network,Eq. (4), with b i ∈ [ − ,
1] and N = 10 . (b) Bifurcation diagram of a local map x it versus ǫ , exposingthe underlying chaotic dynamics. (c) Complexity C as a function of ǫ . measure considers the macroscopical variable as laying in a single state, thus giving C = 0.The complexity C remains zero up to a critical value of the coupling ǫ c ≃ .
04. The onsetof the complexity at the value ǫ c resembles a first order phase transition. As the periodicityof the collective orbit increases, more states are occupied by the probability distribution ofthe mean field h t . The probability distribution of h t corresponding to a periodic collectivestate is not uniform and consists of a set of distinct “humps”. A nonuniform probabilitydistribution and few occupied states lead to larger values of the complexity C , as observedfor the period-two collective orbit. When the system enters chaotic collective motion, morestates are occupied by the probability distribution of the mean field and therefore thisprobability becomes more uniform. As a consequence, the complexity decreases.8he emergence of ordered collective behavior in the coupled map network, Eq. (3), cannotbe attributed to the existence of windows of periodicity nor to chaotic band splitting in thelocal dynamics. Figure 2 shows that higher values of complexity are associated to theoccurrence of nontrivial collective behavior in a network of interacting dynamical elements.We have obtained similar results for different network topologies and different local mapdynamics. This result adds support to the concept of complexity as an emergent behavior;in this case the ordered collective behavior is not present at the local level. Furthermore,these results suggest that similar dynamical collective processes may take place in the humanbrain. From the dynamics of a single neuron as an excitable element it is not possible ingeneral to characterize the collective behavior of the brain, as the collective behavior of thecoupled map network cannot be inferred from the knowledge of the dynamics of a singlemap. Thus, the lower values of complexity found in the global dynamics of the EEG signalsfrom epileptic patients may be associated to a decrease in the ability to generate collectiveorganization and functions in the brain affected by such physiological condition.As we have mentioned, the measure of the statistical complexity depends on the level ofobservation (number of states R ) considered for the computations. Figure 3(a) shows thestatistical complexity as a function of the number of states for the mean field value from anEEG signal of a healthy subject. In Fig. 3(b) a similar plot of h t is shown for the value ofthe coupling parameter ǫ = 0 .
0 15 30 45 60 75 90
Complexity (C) number of states (R) 0 0.1
0 500 1000 1500
Complexity (C) number of states (R) ba FIG. 3: (a) Complexity C as a function of R for the mean field from EEG signals for a healthysubject. (b) C as a function of R for the coupled heterogeneous map network, Eq. (3), ǫ = 0 . Figure 3 shows that the measures of the complexity in both systems tend to constant,9symptotic values when the number of states R is increased. For the coupled map network,the number of states used in the computations was R = 15, while for the EEG signalswe employed R = 1000; these values lie in the corresponding asymptotic regime for eachsystem.In the heterogeneous coupled maps network, the size of the system N can be relatedto the minimum number of states or level of description R c that should be used in thecalculation of C . The width of a vertical segment in the bifurcation diagram of h t at thelevel of description R c is given by R − c . In addition, we have observed that this width isproportional to the statistical dispersion of the points inside the segment, which in turndecreases with the size of the system following the law of large numbers as N / . Then,1 R c ∼ √ N ⇒ N ∼ R c , (5)thus, for a given system size N , it is possible to estimate the minimum number of statesthat should be considered for the computation of the statistical complexity in this system.Correspondingly, for the resolution level R = 15 that we employed in Fig. 2, the minimumsystem size that could have been used is N ∼ et al. [2002] who showed that the prediction error used to measurethe mutual information transfer between a local and a global variable in a similar networkdecreased when nontrivial collective behavior arises in the system. Thus, a high level ofcomplexity can also be associated to an increase in the mutual information transfer betweenmacroscopic and microscopic variables in a system.Finally, our intention is not to describe the human as network of interacting chaoticelements, but to show that the collective properties related to the concept of complexity are10ualitatively similar, and that a rise in complexity can be associated with the emergency ofcollective ordered behavior in different systems. Acknowledgments
M. G. C. acknowledges support from Consejo de Desarrollo Cient´ıfico, Human´ıstico yTecnol´ogico (C.D.C.H.T.), Universidad de Los Andes (ULA), Venezuela, under grant C-1579-08-05-B. M.E-M acknowledges support from C.D.C.H.T.-ULA under grant I-1075-07-02-B. R.L-R. acknowledges support from Postgrado en F´ısica Fundamental, Universidadde Los Andes, Venezuela, during his visit there in July 2008. P. G. thanks C.D.C.H.T,Universidad Central de Venezuela, for support. The authors thank Hospital Luis GomezLopez, in Barquisimeto, Venezuela, for providing the clinical data base and diagnoses.11 eferences
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