monitoring effective connectivity in the preterm brain: a graph...

Research ArticleMonitoring Effective Connectivity in the Preterm BrainA Graph Approach to Study Maturation

M Lavanga12 O De Wel12 A Caicedo12 K Jansen34 A Dereymaeker3

G Naulaers3 and S Van Huffel12

1Department of Electrical Engineering (ESAT) STADIUS Center for Dynamical SystemsSignal Processing and Data Analytics KU Leuven Leuven Belgium2imec Leuven Belgium3Department of Development and Regeneration Neonatal Intensive Care Unit UZ Leuven Leuven Belgium4Department of Development and Regeneration Child Neurology UZ Leuven Leuven Belgium

Correspondence should be addressed to M Lavanga mlavangaesatkuleuvenbe

Received 12 May 2017 Revised 28 July 2017 Accepted 6 September 2017 Published 17 October 2017

Academic Editor Mirjana Popovic

Copyright copy 2017 M Lavanga et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In recent years functional connectivity in the developmental science received increasing attention Although it has been reportedthat the anatomical connectivity in the preterm brain develops dramatically during the last months of pregnancy little is knownabout how functional and effective connectivity change with maturationThe present study investigated how effective connectivityin premature infants evolves To assess it we use EEG measurements and graph-theory methodologies We recorded data from25 preterm babies who underwent long-EEG monitoring at least twice during their stay in the NICU The recordings took placefrom 27 weeks postmenstrual age (PMA) until 42 weeks PMA Results showed that the EEG-connectivity assessed using graph-theory indices moved from a small-world network to a random one since the clustering coefficient increases and the path lengthdecreases This shift can be due to the development of the thalamocortical connections and long-range cortical connections Basedon the network indices we developed different age-predictionmodelsThe best result showed that it is possible to predict the age ofthe infant with a root mean-squared error (radicMSE) equal to 211 weeksThese results are similar to the ones reported in the literaturefor age prediction in preterm babies

1 Introduction

The brain can be seen as a complex network of interactingregions and hierarchical communications which are con-strained by the anatomy but not limited to it [1]Theneuronalclusters can actually work together and communicate toperform a joint task beyond their structural locations Theclinical literature [2] distinguishes this type of connectivityfrom the anatomical one which is often called structuralThe consequence of this functional infrastructure is thegeneration of complex electrophysiological patterns whichare temporally correlated by distant cerebral areas [3]Those spatiotemporal patterns are dynamic they changeaccording to the individual development trajectory [4] Inparticular the last trimester of gestation is a period of brain

development which includes both anatomical rewiring [5]and electrophysiological modifications [6] Different authorsillustrated that the cortical regions undergo differentiationfolding and gyrification while the subcortical areas expe-rience synaptogenesis and myelination as well as neuralpruning to establish thalamocortical connections or longdistance cortical connections [7 8] Based on MRI scansof preterm babies Dubois et al [8] showed that the whitematter volume and the inner cortical surface increase withgestational age Furthermore the same author [8 9] demon-strated that fractional anisotropy (FA) of the brain fiberbundles increases with postbirth maturation although thedifferent connection pathways seem to develop in an asyn-chronous way According to Batalle et al [5] FA is a measureof anatomical directivity that describes the connectivity

HindawiComplexityVolume 2017 Article ID 9078541 13 pageshttpsdoiorg10115520179078541

2 Complexity

strength among brain regions together with the densityand the percentage of connecting streamlines Huppi andDubois [9] eventually argued that diffusion tensor imagingparameters such as anisotropy can be structural markers ofnetwork functional organization [5] A possible implicationis that the temporal correlation among brain regions isalso expected to change since the functional connectivityis related to the structural topology [1] However less isknown about functional connectivity and maturation [10]Therefore functional connectivity in developmental sciencereceived increasing attention in recent years [1] Moreoverdifferent tools to describe functional connectivity becameavailable in the last two decades According to Bullmore andSporns [11] brain networks can be represented as a graphwhere the nodes are brain regions and the edges are theconnection strengths The nodes are defined by the neuralactivity while the edges define the presence of a couplingbetween time series and the associated weight represents thecoupling degree Friston [12] recognizes two main types offunctional connectivity the actual functional connectivitywhich investigates the temporal correlation among signalsand the effective connectivity which measures the causalityof the coupling While the former does not imply anydirectionality the second one highlights which area or nodecauses the activity of other ones Functional connectivitymethods generate undirected graphs while the effectiveconnectivity methods are associated with directed graphs[11] In different studies of the preterm infant brain [5 13]the neural activity has been measured using fMRI Althoughthis neuroimaging technique can investigate the subcorticalstructures it is an expensive method and it is not suitableto measure effective connectivity due to fMRI low temporalresolution [14] In contrast EEG is a suitable measurementfor effective connectivity and has been employed in recentpapers to study the brain connectivitymaturation [14 15]Themain drawbacks of those studies are the limited investigatedmaturation period or the absence of graph-theory metrics toinvestigate the connectivity changes Studies about the longterm development of effective connectivity based on EEGhave not been performed yet In particular there is a lack ofresearch that investigates the effective connectivity frombirthuntil full-term age in premature infants Furthermore in theliterature there are a few works that study the connectivitymaturation in the pretermbrain froma graphpoint of view [513] The main aim of the present study was a thorough inves-tigation of the effective connectivity evolution during thepreterm maturation after birth exploiting different networkmetrics (as in [5]) In particular we measured the effectiveconnectivity by means of transfer entropy [16] and Grangercausality [17] On the obtained directed graph different graphmetrics were computed to track their evolution from 27 to 42postmenstrual age (PMA) weeks The network analysis wascomplemented by a regression study to predict the age of thepatient and assess the predictive power of the connectivityanalysis (as performed in one of our previous studies [15])The paper is organized as follows in Section 2 we introducethe simulated and the real dataset alongside themethods usedto estimate effective connectivity In Section 2 the appliedgraph-theory metrics are also discussed Section 3 reports

1

2

3 4

5

6

Figure 1 The figure displays the graph associated with model (1)

the results while Section 4 includes our discussion on thereported results

2 Methods

21 Data Different datasets were used A simulated datasetwas also employed in order to show how connectivityanalysis performs in a controlled case In particular the mainobjective of the simulationwas to illustrate themeaning of themost common graph indices used in the literature A datasetcomprised of EEG signalswas themain part of thematurationstudy

211 Simulated Data The first experiment of the studyconsisted of a simulation based on a linear Gaussian regres-sion model (derived from [18]) expressed by the followingequations

1199091119905 = 091199091119905minus1 + 091205731199092119905minus1 + 071205731199093119905minus1 + 12058511199051199092119905 = minus091205731199091119905minus1 + 071205731199093119905minus1 + 12058521199051199093119905 = 081205731199091119905minus1 + 071205731199092119905minus1 + 12058531199051199094119905 = minus0251205731199091119905minus1 minus 061199094119905minus1 + 12058541199051199095119905 = 0251205731199091119905minus1 + 091205731199094119905minus1 + 12058551199051199096119905 = 091205731199094119905minus1 + 091199095119905minus1 minus 071205731199096119905minus1 + 1205856119905

(1)

The parameter 120573 is a scalar value that varies between 0and 1 and it influences the strength of the coupling amongthe 119909119894119905 variables The 120585119894119905 variables represent white noise withunit varianceThe associated graph is reported in Figure 1 Inorder to give the reader a clear insight of the graph-theorymeasures the objective of the simulation was twofold firstlywe investigated the influence of the strength of the couplingprovided by the parameter 120573 and show the effect of theweakening of causality among the variables Secondly weinvestigated the influence of the noise using different levelsof signal-to-noise ratio (SNR) which were obtained varyingthe variance of white noise in (1) In the latter objectivetime series were simulated using the AR model (1) andthe coupling was estimated with the effective connectivitymethods discussed in the following paragraphs We alsostudied the impact of filtering on the graph based metrics bycomputing the scores from filtered signals using a samplingfrequency which was set at 200Hz and a band-pass filterbetween 1 and 80Hz was applied on the simulated data Thereason to investigate the impact of filtering is explained in

Complexity 3

Section 23 The band-pass frequencies were obtained from[19]

212 EEG Data The second dataset comprised 25 preterminfants with gestational age (GA) le 32 weeks who wererecruited for a larger EEG study to assess brain developmentand automatically detect quiet sleep epochs [20 21] Eachpatient was enrolled with informed parental consent at theNeonatal Intensive Care Unit (NICU) of the UniversityHospital of Leuven Belgium All the included subjectspresented good outcome at 2 years EEG was recorded with 9electrodes (1198651199011 1198651199012 1198623 1198624 1198793 119879411987411198742 and reference 119862119911)according to the 10ndash20 international system (BRAINRTOSGEquipment Mechelen Belgium) Throughout connectivityanalysis the channel 119862119911 was disregarded The measurementswere performed twice during their stay in the unit resultingin 88 recordings ranging from postmenstrual age of 27 weeksto 42 weeks Mean time of recording EEG was 4 h 55minMonopolar signals were recorded at a sampling frequency of250Hz Two independent EEG readers annotated the datafor the different sleep stages namely quiet sleep (QS) andnonquiet sleep (NQS) epochs The reason for this binarymanual classification is due to the difficulty to discriminateactive sleep from the awake state in the very young child

22 Connectivity Analysis The literature about effective con-nectivity presents differentmethods to assess coupling amongtime series such as directed transfer function (DTF) partialdirected coherence orGranger causality test (GCT) [22] Withthese approaches the coupling is frequently defined as G-causality [23] in the sense that a time series Granger-causesanother one when the past values of a time series significantlyexplain the evolution of the other one Even though Grangerhimself pointed out the limited applicability to the bivariatecase the recently published conditional multivariate Grangercausality (cMVGC) [24] overcomes the problem Accordingto [23 25] these methods are based on the same multivariateor vector autoregressive (VAR) modeling However theycan show different aspects in the causality analysis Whilethe DTF estimates the reachability from one channel toanother (defined also as G-influentiability by [23]) bothPDC and cMVGC look into the direct active link betweentwo channels which is defined as G-connectivity [23] Thosetwo methods investigate the same connectivity model intwo different domains which are the frequency (PDC) andtime (cMVGC) domain Since in this paper we are mainlyinterested in the time-domain coupling we opted for thecMVGC in its autoregressive (AR) and information dynamicsimplementations respectively described in [18 24]

221 Granger Causality Thefirst method employed to assessthe G-connectivity among EEG channels was Granger causal-ity (GC) with a VAR modeling framework In the presentstudy we followed the formulation by [24] A 119901th ordermultivariate autoregressive model VAR(119901) takes the form

U119905 =119901

sum119896=1

A119896 sdot U119905minus119896 + 120598 (119905) (2)

where U119905 is a collection of process 119880119894 while A119896 and 120598(119905)are respectively the regression coefficient matrices and thestochastic process residuals In the conditional scenario [24]where we want to know the pairwise influence of process X119905over Y119905 considering the presence of the other119880119894 variablesU119905can be rewritten as

U119905 = [[[

X119905Y119905Z119905

]]]

(3)

where X119905 and Y119905 are the two time series of interest while Z119905represents the third set of variables involved in the analysisIn our study X119905 and Y119905 can be two variables 119909119894119905 in (1) forexample 1199094119905 and 1199095119905 or two EEG channels for example1198651199011 and 1198623 Consequently Z119905 will be a vector time serieswhich contain the remaining 119909119894119905 in (1) or the remainingEEG channels Based on the VAR(119901) model (2) and the splitdefined in (3) we may actually explain the future of X119905 basedon the following full and reduced regressions

Y119905 =119901

sum119896=1

A119910119910 sdot Y119905minus119896 +119901

sum119896=1

A119910119909 sdot X119905minus119896 +119901

sum119896=1

A119910119911 sdot Z119905minus119896+ 120598119910 (119905)

Y119905 =119901

sum119896=1

A119910119910 sdot Y119905minus119896 +119901

sum119896=1

A119910119911 sdot Z119905minus119896 + 120598119910 (119905)

(4)

In both cases we consider the conditioning variable Z119905although only the first model considers explicitly the influ-ence of X119905 The difference is also highlighted by the tworegression coefficient matrices A119910119910 and A119910119910 In order totest whether the coefficient matrices A119910119909 are significantlydifferent from zero the Granger causality is defined as thelog-likelihood ratio

F119883rarr119884|Z = log10038161003816100381610038161003816Σ119910119910

1003816100381610038161003816100381610038161003816100381610038161003816Σ11991011991010038161003816100381610038161003816 (5)

whereΣ119910119910 and Σ119910119910 are the covariancematrices of the residuals120598119910(119905) and 120598119910(119905) Based on this multivariate framework the G-causality basically quantifies the reduction of the predictionerror if the variable 119883 is added to explanatory variables of119884 one of which is the variable Z [24] Unlike the classicalGranger causality test limited to the bivariate case we definedF119883rarr119884|Z conditional multivariate Granger causality A specialcase of the conditional scenario is the pairwise conditionalG-causality where the pairwise effective couplings among allpair of variables 119880119894 and 119880119895 contained in U119905 are measuredSince we take in account the spurious effect due to thepresence of other variables (ie the coupling between 119880119894 and119880119895 conditioned by the presence of the other variables) thepairwise conditional causalities are defined as

G119894119895 (U) = F119880119894rarr119880119895|U[ij] (6)

which is an 119872 times 119872 matrix where 119872 is the number ofprocesses and contains all the pairwise coupling estimations

4 Complexity

Those quantities may be considered a weighted directedgraph also known as G-causal graph and the matrix G119894119895as the associated adjacency matrix 119860 The G-causality wascomputed with the multivariate Granger causality toolbox[24] implemented in MATLAB (Mathworks Natick MAUSA)

222 Transfer Entropy The second method applied to assesstheG-connectivity among EEG channels was transfer entropyAccording to the information dynamics framework by [16]the expected Kullback-Leibler divergence defines the transferentropy (TE) from process 119883 to process 119884 as follows

119879119883rarr119884 = sum119901 (119884119905X119898119905 Y119899119905 ) log 119901 (119884119905 | X119898119905 Y119899119905 )119901 (119884119905 | Y119899119905 ) (7)

where 119901(119884119905 | X119898119905 Y119899119905 ) is the conditional probability that 119884 attimes 119905 is explained by past values of119883 and 119884 with respectivememory order 119898 and 119899 119901(119884119905 | Y119899119905 ) is the conditionalprobability that 119884 at times 119905 is only explained by past valuesof119884 119901(119884119905X119898119905 Y119899119905 ) is the joint probability distribution amongthe three variables 119884119905 X119898119905 Y119899119905 Transfer entropy inherentlyimplies directionality (ie effective connectivity) since it isasymmetric and contains transition probabilities [16 26]In order to estimate the information dynamics couplingMontalto et al [18] pointed out that TE is equivalent to thedifference of two conditional entropies (CE)

119879119883rarr119884|Z = 119867(119884119905 | Y119899119905 Z119901119905 ) minus 119867(119884119905 | Y119899119905 X119898119905 Z119901119905 ) (8)

With respect to (7) the variable Z119901119905 is here introduced Ina multivariate analysis we cannot discriminate the exclusiverelationship between two processes [17 24] However it ispossible to investigate the contribution of time series 119883 inthe evolution of time series 119884 with respect to all the otheragents involved in the analysis Consequently we can actuallydefine Z119901119905 as a vector variable that does not contain neither119884 nor 119883 as past values and 119879119883rarr119884|Z illustrates the transfer ofinformation from119883 to119884 taking into account other time seriesinvolved If we assume that 119883 119884 Z have a joint Gaussiandistribution the two CE in (8) can be expressed as linearregression of past values of the vector variables involved inthe multivariate system as follows

119884119905 = 119860119906119881119906 + 120598 (119905) 119884119905 = 119860119903119881119903 + 120598 (119905) (9)

The first equation explains 119884119905 with a regression on thevector 119881119906 = [119881119909 119881119910 119881119911] where 119881119909 119881119910 119881119911 approximaterespectively X119898119905 Y

119899119905 Z119901119905 with a vector of size 119901 as follows

119881119909 = [119883119905minus1 119883119905minus2 119883119905minus119901] 119881119910 = [119884119905minus1 119884119905minus2 119884119905minus119901] 119881119911 = [Z119905minus1Z119905minus2 Z119905minus119901]

(10)

The second equation explains119884119905 with a regression on the vec-tor 119881119903 = [119881119910 119881119911] which only contains 119881119910 119881119911 Equations (9)

are usually referred to as full and restricted regressions Thereader can easily recognize the same structure of Grangercausality equations (4) which is asymptotically equivalentto transfer entropy [16] At the light of the previous results119879119883rarr119884|Z can be rewritten as

119879119883rarr119884|Z = 12 log 120590119903

120590119906 (11)

where 120590119903 and 120590119906 are the variances of the white noise residualsin (9) Except for a scalar factor (11) is the same expressionof (5) It is important to notice that also TE estimates theG-connectivity in a conditioned framework In the presentstudy the TE entropy has been computed using the MUTEtoolbox [18] implemented in MATLAB (Mathworks NatickMA USA)

223 Graph-Theory Indices The effective connectivity meth-ods generate an adjacencymatrix119860 that describes a weighteddirected graph [24] called G-causal graph Unlike a binarynetwork those graphs tend to be complete or full without aspecific thresholding and the adjacencymatrix is asymmetricSince the maximal number of couplings is1198722 minus119872 (where119872is the number of signals) graph-theorymeasures can be usedto summarize the causal connectivity [27] Although thereis no minimal theoretical number of graph nodes 8 EEGchannels can be considered quite limited for a graph analysisHowever there are studies in the neural processing literaturewhere graph theory was applied on a limited number of timeseries in order to give a concise view of a high-density (orcomplete) network [27 28] in particular if the sources-levelis concerned Bullmore and Sporns [11] illustrated a list ofmeasures to describe the graph topology such as the averagecharacteristic path length the clustering coefficient and thediameter The path length represents the minimum numberof edges to reach each other node in the graph starting fromone specific node The average path length is the mean ofthe nodesrsquo shortest paths and can be considered a measureof network integration capacity [5] The latter can be alsoinvestigated by means of the clustering coefficient In caseof a weighted network a clustering coefficient is defined asaverage intensity of all triangles around each node [29] Acommon overall measure is the average of nodes clusteringcoefficients defined in a similar way to the average pathlength The last index to evaluate the integration capacityof the graph is the diameter Let the node eccentricity bethe maximum graph distance from one node to any othernode in the network [30]The diameter is then defined as themaximum eccentricity in the graph All those indices havebeen computed thanks to the brain connectivity toolbox [31]implemented in MATLAB (Mathworks Natick MA USA)In addition we computed also the causal density as thesum of all significant couplings of the adjacency matrix asdefined by [32] Alongside those networkmetrics the spectralindices such as the spectral radius the spectral gap and thealgebraic connectivity were considered The spectral radiusand the spectral indiceswere themaximal absolute eigenvalueand the difference between the first and the second absoluteeigenvalues of the adjacencymatrix respectively [33 34]The

Complexity 5

Table 1 An overview of all graph metrics that have been used in the study

Overview of graph indicesPath length Mean of the nodesrsquo shortest pathsClustering coefficient Mean of the nodesrsquo triangles intensity around each nodeDiameter Maximum graph eccentricityCausal density Sum of all significant couplings in 119860Spectral radius 1205821(119860) first eigenvalue of adjacency matrix 119860Spectral gap 1205821(119860) minus 1205822(119860) the difference between the first two eigenvalues of matrix 119860Algebraic connectivity 120582119872minus1(119871 sym) the second smallest eigenvalue of the Laplacian matrix 119871 sym

spectral radius commonly known as ldquoPage Rankrdquo representsthe dominance degree of a node in the network the higherthe value the higher the centrality of the dominant nodein the network In other words the dominant node behavesas the center of a hub [33] Another way to look into thechange of dominance is the spectral gap if the differencebetween the first and second eigenvalues in the graph matrixdecreases the node with highest eigenvalue (the spectralradius) is less dominant with respect to the other nodes inthe network However Estrada and Hatano [34] argued thatthe spectral gap behaves as a measure of clustering in theundirected graph in case of lower values of spectral gapthe graph can present small clusters in the network The lastspectral measure is the algebraic connectivity which is thesecond smallest eigenvalue of the Laplacian matrix 119871 sym andit illustrates how easily a graph can be divided into clusters[30] In case of a directed graph the Laplacian matrix can beobtained as follows

119871dir = 119863 minus 119860

119871 sym = 12 (119871dir + 119871119879dir) = 119863 minus 119860 + 119860119879

2 (12)

where 119860 is the adjacency matrix obtained by the effectiveconnectivity tools described above and119863 is the degreematrixof the associated graph [35] An overview of all graph indicesis reported in Table 1

23 Algorithmic Pipeline and Statistical Analysis Accordingto different authors [19 36] filtering could add spuriousconnectivity in the effective connectivity analysis or makethe estimation of the underlying Granger causality VARmodel unstable Although a theoretical invariance of causalityestimation has been demonstrated under filtering the G-causality works in practice if and only if the data arestationary The use of filtering as mean to reach stationaritywith filtering has already been investigated by [36] Howeverthree main reasons can undermine this approach The firstreason is the increase of the estimated VAR model orderdue to the fitting of filtered process of theoretical infiniteorder with a numerical finite order This could lead to apoor and not robust parameter estimation or even unstableVAR model which is the second reason to avoid band-passfilteringThe last reason is the ill-conditioning of the Toeplitzmatrix of the autocorrelation sequence Γ119896 = cov(U119905U119905minus119896)which is necessary for VAR model estimation All the related

theoretical details are explained in [36] The practical down-sides of the band-pass filtering are the dramatic increase ofthe VAR model order compared to the unprocessed data orthe increase of false positive detection of connectivity linkswith both the GC [36] and the PDC [19] In particular theamount of false detection increases with narrowing of thefiltering frequency band or the increase of the filter order Ingeneral there is no distinction between FIR and IIR filteringfor both the authors However in both studies [19 36] notchfiltering and differentiation (or high pass filtering at 1Hz)were pointed out as methods to keep the VAR model orderlow and reduce the false detection even compared to unpro-cessed data Those approaches help to achieve stationarityor keep the VAR model order low for nearly nonstationaryprocesses like EEG In addition the presence of trends orseasonality can add unit-roots to the time series (poles on theunit circle or outside it in the complex plane) which violatesthe hypothesis of covariance stationarity In the presenceof unit-roots the impulse response of the VAR modelwould be oscillatory or diverge to infinity Consequentlyit is suggested to eliminate trends and seasonality by first-order differentiation or differentiation at various lags (notchfiltering) Theoretical and numerical details of the differenttype of filtering are reported in [24 36] Given the statedliterature results on the one hand we decided to investigatethe impact of filtering on the simulated data and on theother hand we applied only notch filtering at 50Hz and100Hz and differentiation on the EEG data in order to reducenonstationarities in the time series However the EEG canbe affected by muscle artifacts which can spread out amongthe different electrodes and bias the connectivity analysis Tomitigate this effect we applied canonical correlation analysis(CCA) to remove the artifacts caused by the change in theEMG signals [37] To investigate the effect of the CCA on theanalysis we compared the output of the connectivity analysisin two scenarios the first one did not consider the usage ofCCA the second one appliesCCAand reconstructed the EEGremoving the 3 sources with lowest autocorrelation In thefirst case the authors segmented the EEG in 5 s windows andcomputed the effective coupling with both listed methods Inthe second scenario CCA was applied on 5 s EEG segmentsHowever before performing any connectivity analysis werecombined the segments in 30 s intervals in order to increasethe rank of the reconstructed EEG matrix This step wasrequired since both GC and TE request data that do notpresent collinearity (ie the time series matrix cannot be

6 Complexity

rank-deficient [24]) The window length was suggested as afurther step to keep time series stationarity [24] For eachrecording we computed the adjacency connectivity matrixfor EEG segments and we averaged over the QS epochs andNQS epochs This averaging step did not consider couplingvalues that were not statistically significant According to[18 24] the GC and TE follow an asymptotic 119865-distributionlike1198772 statisticsTherefore an 119865-test was implemented to testthe significance of coupling among EEG channels and all thecouplings with119901 gt 005were set to zero Consequently 352 =88lowast2lowast2 coupling graphs were obtained where 88 is the totalnumber of recordings 2 is the employed causality methods2 is the considered number of sleep states On the averagematrices we computed the network indices described abovewhich results in a tensor 88 times 7 times 4 where 7 is the numberof graph features and 4 is the number of combinationsconsidering the number of connectivity methods and sleepstates involved (TE in QS TE in NQS GC in QS and GC inNQS) For each feature we evaluated the maturation trend inthree distinctive age groups (le31isin (31minus37)ge37 PMAweeks)as median (IQR) where IQR is InterQuartile Range Besideswe computed the Pearson correlation coefficient betweenthe variable age and each single feature and its statisticalsignificance were computedThose results were meant to givea general overview of the feature maturation trend for eachconnectivity method for each sleep state with or withoutCCA preprocessing In addition we computed an ordinaryleast squares (OLS) regression for each single feature in caseof GC during QS and the associated confidence interval at95 in order to give a visual representation of any networkindex prediction power In particular we split randomly thesingle graph index dataset 100 times in 70 training set and30 testing set and we computed the prediction error onthe test set as root mean-squared error radicMSE as shown by[38] Furthermore each slice of the tensor (a matrix 88 times 7)was used to predict the patientrsquos age at the moment of therecording with a multivariate linear regression Similar to thesingle feature approach we randomly split the dataset 100times as in [38] andwe assessedradicMSE in the test set At eachiteration 1198772 index and the 119865-test statistics were computed

3 Results

In the following paragraphs the results obtained in both thesimulated and the real dataset are discussed In particular wewill show how the network indices behave in both examplesthe simulated data and the EEG maturation dataset The lastpart summarizes the predictive power of those graph metricsto infer the postmenstrual age of the subject

31 Simulation Study Figures 2(a) 2(b) and 2(c) displaythe integration graph indices for the AR model defined in(1) whose network is depicted in Figure 1 In the originalmodel (120573 = 1) the graph presents two distinct clusterswhich are loosely connected by the edges between node 1and nodes 4 and 5 This two-hubs structure is reflected byFigures 2(a) and 2(b) When the intracluster connectivity ishigh (120573 = 1) the clustering coefficient reaches its highest

level while the path length is at its lowest level Thoseresults are in line with a rich-club or small-world network[11] However when the coefficient 120573 starts decreasing theclustering coefficient proportionally decreases while the pathlength increasesThe spectral radius decreases with vanishingvalues of 120573 as the clustering coefficient does Figures 2(d)2(e) and 2(f) display the ratio between the network indicesestimated from the simulated time series via transfer entropyand the original measures (in particular in (d) CCTECCorigin (e) lengthoriglengthTE in (f) 120582TE120582orig) Different noiselevels have been used The results are quite straightforwardfor the clustering coefficient and the spectral radius thehigher the signal-to-noise ratio the higher the values of twoindices are and the closer they are to the original values(Figures 2(d) and 2(f)) In the case of the path length theabsolute value is decreasing with higher SNR However theestimated value becomes similar to the original one for verylow noise variance (Figure 2(e)) Figures 2(d) 2(e) and 2(f)show also the effect on band-pass filtering on the graphindices estimationThemost remarkable results are related tothe path length and the clustering coefficient The estimatedclustering coefficient tends to underestimate the original onewhile the estimated path length tends to be persistently higherthan the original one On the contrary the ratio for thespectral radius in case of filtering behaves similar to the oneobtained using the raw data

32 EEG Data Graph Indices Figure 3 shows graph metricsfor the adjacency matrices obtained using the EEG measure-ments and Granger causality in QS (GC in QS) Figures 3(a)3(b) 3(c) and 3(d) show the scatter plots with the fittedOLS regression model while Figures 3(e) and 3(f) displaythe clustering coefficient and path length dynamics in threedistinct age groups As already mentioned in Section 23Figure 3 gives a visual representation of the trends for thegraph features while Tables 2 and 3 provide a completeoverview for all coupling methods and all sleep states Eachsingle feature has a significant trend with age although thePearson correlation coefficient 120588 () increases when CCA isused as a preprocessing step Specifically the trend for theclustering coefficient the spectral radius and the spectralgap is negative while the path length is increasing with ageThis result is persistent in each method and each sleep stateThe connectivity weakening for GC in QS is also reportedin Figure 4 which shows the average connectivity graph forthree distinct age groups The three panels show how thecoupling among time series decreases by the reduction inarrows width and the color shift from red to blue Table 4reports the results for age prediction with a multivariatelinear model combining all the network features All themodels can predict the age of the infant recording with aradicMSE between 2 and 3 weeks and the CCA models alwaysoutperform the model without CCA as a preprocessing stepFurthermore the explained variance (1198772) is higher with themodels that include CCA It is also interesting to noticethat best prediction results are obtained with GC duringQS (radicMSEsimple = 252 PMA weeks radicMSECCA = 210PMA weeks) Table 4 does not report the results for one

Complexity 7

020406080

100120140160180200

Path

leng

th (a

u)

Path length

Coupling strength (au)02 04 06 08 1

(a)

0

005

01

015

02

025

03

CC (a

u)

Clustering coefficient


(b)

0

02

04

06

08

1

12

Spec

tral

radi

us (a

u)

Spectral radius


(c)

SNR (db)

0

01

02

03

04

05

06

07

Ratio

CC

(nu

)

Ratio between estimated CC

minus20 0 20 40 60 80 100 120 140

Raw dataFiltered data

and the original CC

(d)

0

02

04

06

08

1

12

Ratio

pat

h le

ngth

(nu

)

Ratio between estimated path lengthand the original path length


SNR (db)minus20 0 20 40 60 80 100 120 140

(e)

0010203040506070809

1

Ratio

spec

tral

radi

us (n

u)

Ratio between estimated spectral radiusand the original spectral radius


SNR (db)minus20 0 20 40 60 80 100 120 140

(f)

Figure 2 The figure shows the results for the simulation dataset (a b c) show how the graph indices behave for different level of couplingin model (1) (d e f) investigate how the transfer entropy can estimate the network indices for different level of SNR In particular the figurecompares the two cases when the data is filtered and when the raw data is used

single model estimation but the median and InterQuartileRange (IQR) of the evaluation parameters for 100 bootstrapiterations In each single iteration the model proved to besignificant as reported by the 119901 value column in Table 4 (119901 lt001)

4 Discussion

In the present study we quantified the effective brainconnectivity in preterm infants to track their maturationAlthough there are some studies that investigate connectivityin the neonates [4 5] and its change in the first days oflife (short maturation period) [14] this is the first studyto track maturity using effective EEG-based connectivity inpreterm patients on a wide maturation period from birth tofull-term age To the best of our knowledge MRI [8] andfMRI [10] have been the leading method to track maturityHowever fMRI is only suitable for functional connectivityas discussed above In this article we compared two well-known methods to estimate coupling between processes like

GC and TE Based on the obtained connectivity matriceswe estimated integration and spectral network indices fordirected weighted graphsThose features were used to predictthe age of the patient during the recordingThe same processwas applied on a simulated dataset to investigate how thenetwork measures behaved in a controlled case

41 Simulated Dataset According to [11] graphs with highclustering coefficient and low path length behave like a rich-club network while graphs with low clustering coefficientand high path length denote a random network where thenumber of edges for each node is normally distributedFigure 1 portraits two club networks where the nodes areconnected to each other with a short distance This leads toa high clustering coefficient and low path length when thecoupling coefficient is equal to 1 However when 120573 decreasesthe intracluster connectivity weakens and the graph becomesmore similar to a random network This type of graph ischaracterized by low clustering coefficient and high pathlength indeed the nodes are less connected among each

8 Complexity

Clustering coefficient (au)

10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x

DataRegression lineConfidence interval

0005 001 0015 002

Linear regression R2 = 04685－３ = 24101

(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)

Spectral radius (au)

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th

PMA (weeks)le31 ge37(31ndash37)

(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)

Figure 3 The figure shows the results for EEG data (a b c d) show OLS regression between 4 main graph indices versus the age (PMA inweeks) for GC in QSThe grey area is the confidence interval at 95 On the top of the panel the associated 1198772 andradicMSE in PMA weeks onthe test set (e f) show the trend of the clustering coefficient and the path length in three distinct age groups The results are reported aboutGC in QS

Complexity 9

T3T4

Fp1

C3

O1

Effective connectivity graph

0 01 02 03 04 05 06 07 08 09 1

０－ le 31 weeks ０－ isin (31ndash37) weeks ０－ ge 37 weeks

Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4

Figure 4 The figure shows the average connectivity graph (GC for three different age groups) The strength of the coupling among theelectrodes is decoded by the color (the closer to the red color the higher the coupling) and by the width of the arrowThe connectivity valueshave been normalized between 0 and 1 for the three groups together The panels clearly show the weakening of the coupling among EEGchannels with maturation The consequence is the increase of path length and the decrease of the clustering coefficient (Figure 3)

Table 2 The main integration and spectral features in three discrete time points The table shows the indices for both sleep states (QS =quiet sleep NQS = nonquiet sleep) and they were computed on the transfer entropy connectivity graph The results are reported as median(IQR) where IQR stands for InterQuartile Range The symbol 120588 stands for the Pearson correlation coefficient while represents a significantcorrelation with 119901 le 001 The values 10minus3 or 10minus2 mean the reported results are multiplied by a factor 10minus3 or 10minus2

Network indices transfer entropy in three age groupsMedian (IQR) PMA weeks le31 isin (31ndash37) ge37 120588 ()Clustering coefficient

QS 025 (008) 021 (005) 017 (002) minus53NQS 025 (006) 020 (006) 018 (002) minus49

Path lengthQS 373 (30) 389 (22) 407 (10) 59NQS 371 (20) 391 (25) 404 (14) 54

Spectral radiusQS 18 (07) 16 (05) 12 (01) minus49NQS 18 (05) 15 (04) 13 (02) minus48

Spectral gapQS 15 (09) 12 (04) 10 (02) minus51NQS 15 (06) 12 (04) 11 (02) minus48

Network indices transfer entropy CCAMedian (IQR) le31 isin (31ndash37) ge37 120588 ()Clustering coefficient

QS (10minus3) 982 (61) 621 (33) 498 (09) minus57NQS (10minus3) 913 (35) 629 (20) 590 (10) minus49


Spectral radiusQS (10minus2) 820 (47) 468 (31) 363 (6) minus55NQS (10minus2) 694 (28) 474 (16) 430 (7) minus48

Spectral gapQS (10minus2) 584 (31) 393 (20) 330 (6) minus52NQS (10minus2) 584 (31) 393 (20) 330 (6) minus49

10 Complexity

Table 3 The main integration and spectral features in three discrete time points The table shows the indices for both sleep states (QS =quiet sleep NQS = nonquiet sleep) and they were computed on the Granger causality connectivity graph The results are reported as median(IQR) where IQR stands for InterQuartile Range The symbol 120588 stands for the Pearson correlation coefficient while represents a significantcorrelation with 119901 le 001 The values 10minus3 or 10minus2 mean the reported results are multiplied by a factor 10minus3 or 10minus2

Network indices Granger causalityMedian (IQR) PMA weeks le31 isin (31ndash37) ge37 120588 ()Clustering coefficient





Network indices Granger causality CCAMedian (IQR) PMA weeks le31 isin (31ndash37) ge37 120588 ()Clustering coefficient





Table 4 Multivariate regression model performances The table shows the error on the test set (Error) 1198772 and the 119865-statistics (F-stat) andthe 119901 value obtained with the different connectivity methods in the different sleep states The results are reported as median (IQR) whereIQR stands for InterQuartile Range over the 100 random splits of the dataset The labels reported are TE = transfer entropy GC = Grangercausality QS = quiet sleep and NQS = nonquiet sleep

Multivariate regression performancesMedian (IQR) Error (weeks) 1198772 F-stat 119901 valueSimple filtering

TE QS 254 (041) 057 (007) 1164 (346) 119901 lt 001 lowast 100TE NQS 288 (039) 040 (007) 623 (172) 119901 lt 001 lowast 100GC QS 252 (037) 052 (007) 1001 (264) 119901 lt 001 lowast 100GC NQS 279 (053) 044 (007) 720 (209) 119901 lt 001 lowast 100

CCATE QS 223 (029) 063 (006) 1290 (314) 119901 lt 001 lowast 100TE NQS 254 (051) 057 (006) 1036 (266) 119901 lt 001 lowast 100GC QS 210 (038) 067 (005) 1596 (364) 119901 lt 001 lowast 100GC NQS 235 (042) 063 (004) 1338 (261) 119901 lt 001 lowast 100

Complexity 11

other in a more homogeneous network This result is alsosupported by the direct proportionality between spectralradius and coupling coefficient Another interesting point isrelated to the filtering Figure 2 illustrates clearly how theclustering coefficient and the path length estimation canbe highly affected by the filtering Those results are in linewith the analysis by [36] which demonstrated that carelessfiltering can add spurious connectivity in the time courses Inour simulation the effect of filtering weakens the intraclusterconnectivity (adding intercluster connectivity)The net effectis a decrease in clustering coefficient and an increase in pathlength Therefore we decided only to apply a notch filter anddifferentiation as preprocessing steps on the EEG

42 EEG Data In the literature a number of studies can befound to have assessed the brain maturation in children andadolescents by graph theory [1 4 10] and a fewpapers focusedon preterm brain maturation by network metrics [5 13] Thefirst result obtained in our study is the change in effectiveconnectivity with age Although Schumacher et al [14] used adifferentmethod they also concluded that there is a change ineffective connectivity mainly driven by postnatal maturationIn addition to that we have been able to demonstrate theexistence of a relationship between connectivity and PMAIn this study we also observed a change in graph parametersthat suggest that the EEG-scalp network moved from a rich-club network to a more random network The integrationand spectral indices decreased with age except for the pathlength which reflects a segregation of nodes due to a highergraph distance as well as less intense triangle patterns aroundthe nodes themselves Hub-networks have high clusteringcoefficients since they have central club nodes which aresurrounded by triangle patterns On the contrary a randomnetwork presents nodes which are connected to any othernode in the network with a weak coupling The net effectis a low clustering coefficient and high path length Thisemergence of a normal-distributed network is also confirmedby the decrease of the spectral gap spectral radius andthe algebraic connectivity The latter two indices emphasizehow easily the graph can be clustered and a negative trendwould suggest the absence of modules or groups in thegraph On the contrary a negative trend of the formerindex should suggest an increase of modularity as shownby [34] However since the spectral gap is also inverselyproportional to the path length its reduction just showsthat the spectral radius is less dominant with respect to theother eigenvalues [33] and it becomes another measure ofmodularity like the other two spectral indices The resultsof the multivariate model further highlight the shift froma rich-club network at younger age to a more random oneat full-term age In particular the lowest radicMSE on the testset is around 211 weeks compared to other studies [15 39]Finally it is important to notice that those negative trends forgraph features are consistent for each effective connectivitymethod and each sleep state At first sight the results weobtained seem to be in contradiction with maturation trendsthat can be found in children or adolescents where a shiftfrom random to a rich-club network has been discovered

However it should be taken in account that on one sideonly 9 electrodes on the scalp were used due to the smallsize of the preterm brain and on the other hand thereis a fast development of the brain during this monitoringperiod with different trajectories for the different cerebralregions [40] This composite evolution is mainly driven bythe different disappearance timing of cortical subplate in thevarious brain areas [40] In particular two main changestook place The first one is the faster development of thethalamocortical connections compared to the corticocorticalones [5] The shift from one type of connections to theother happens only at later age [41] therefore the maturationprocess might lead to ldquoseparationrdquo of the nodes as reflectedby the results in this study In simple terms the EEG-scalp connections became weaker in favour of a connectionstrengthening between the cortical and subcortical areasThe second important change is the negative correlationbetween age and short-range corticocortical connections asshown by [5] via fMRI In [5] the study results showedthat long distance connections develop faster than the shortones Furthermore the development is characterized by astrengthening of the former connections and weakeningof the short-range couplings [1] Consequently the EEGelectrodesnodes (which measure short-range connectivity)tend to separate each other with an increase of the path lengthand a reduction of the clustering coefficient This hypothesisis also supported by the decrease in fronto-frontal andoccipito-occipital functional coupling measured by fMRI [5]This segregation can be emphasized by the fact that thereare a few electrodes on infant scalp However a studywith high-density EEG on preterm infants [13] found anincreased modularity on the scalp EEG network and areduced clustering coefficient in the postcentral networkwhile the clustering coefficient increases in the precentralnetwork This result could confirm the segregation or themore local integration of the brain network due to thepruning of short-range connections as also shown by [42] inthe comparison between adults and children It is importantto notice that both models with and without CCA foundthe same trends but source filtering increased the predictionpower of the model It is possible that the EMG artifactsdisturbed the connectivity analysis and biased the predictionmodel in the first considered scenario

5 Conclusions

In the present study we investigated effective EEG-basedbrain connectivity in premature infants whose PMA rangedfrom 27 to 42 weeks Results showed that the EEG-graphschanged with age in terms of topology In particular theclustering coefficient and the spectral radius decreased withmaturation while the path length increasedThis perspectivesuggests that the EEG graph shifted from a small-world net-work to a random network This apparent nodesrsquo segregationcan be a consequence of the thalamocortical connectionsdevelopment and the strengthening of the long-range corticalconnections The lowest age-prediction error was 211 PMAweeks (obtained with GC in QS) which is in line withliterature results Application of source filtering methods

12 Complexity

like CCA can improve the performance of the connectivityanalysis

Disclosure

This paper reflects only the authorsrsquo views and the Union isnot liable for any use that may be made of the containedinformation

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper The received fundingstated in the Acknowledgments does not lead to any conflictsof interest

Acknowledgments

This research is supported byBijzonderOnderzoeksfondsKULeuven (BOF) The Effect of Perinatal Stress on the LaterOutcome in Preterm Babies (no C2415036) iMinds Med-ical Information Technologies (SBO 2016) Belgian FederalScience Policy Office IUAP no P719(DYSCO ldquoDynam-ical Systems Control and Optimizationrdquo 2012ndash2017) Bel-gian Foreign Affairs-Development Cooperation (VLIR UOSPrograms (2013ndash2019)) and EU the research leading tothese results has received funding from the EuropeanResearch Council under the European Unionrsquos SeventhFramework Programme (FP72007ndash2013)ERC AdvancedGrant BIOTENSORS (no 339804) A Caicedo is a postdoc fellow at Fonds voor Wetenschappelijk Onderzoek-Vlaanderen (FWO) supported by Flemish government MLavanga is a SB PhD fellow at Fonds voor WetenschappelijkOnderzoek-Vlaanderen (FWO) supported by Flemish gov-ernment

References

[1] D Goldenberg and A Galvan ldquoThe use of functional andeffective connectivity techniques to understand the developingbrainrdquoDevelopmental Cognitive Neuroscience vol 12 article no265 pp 155ndash164 2015

[2] E W Lang A M Tome I R Keck J M Gorriz-Saez andC G Puntonet ldquoBrain connectivity analysis A short surveyrdquoComputational Intelligence and Neuroscience vol 2012 ArticleID 412512 21 pages 2012

[3] J L Vincent G H Patel M D Fox et al ldquoIntrinsic functionalarchitecture in the anaesthetized monkey brainrdquo Nature vol447 no 7140 pp 83ndash86 2007

[4] J D Power D A Fair B L Schlaggar and S E Petersen ldquoThedevelopment of human functional brain networksrdquoNeuron vol67 no 5 pp 735ndash748 2010

[5] D Batalle E J Hughes H Zhang et al ldquoEarly developmentof structural networks and the impact of prematurity on brainconnectivityrdquo NeuroImage vol 149 pp 379ndash392 2017

[6] M Andre M-D Lamblin A M drsquoAllest et al ldquoElectroen-cephalography in premature and full-term infants Develop-mental features and glossaryrdquo Neurophysiologie Clinique vol40 no 2 pp 59ndash124 2010

[7] E Takahashi R D Folkerth A M Galaburda and P E GrantldquoEmerging cerebral connectivity in the human fetal Brain anMR tractography studyrdquo Cerebral Cortex vol 22 no 2 pp 455ndash464 2012

[8] J Dubois M Benders A Cachia et al ldquoMapping the earlycortical folding process in the pretermnewborn brainrdquoCerebralCortex vol 18 no 6 pp 1444ndash1454 2008

[9] P S Huppi and J Dubois ldquoDiffusion tensor imaging of braindevelopmentrdquo Seminars in Fetal and Neonatal Medicine vol 11no 6 pp 489ndash497 2006

[10] M C Stevens ldquoThe developmental cognitive neuroscience offunctional connectivityrdquo Brain and Cognition vol 70 no 1 pp1ndash12 2009

[11] E Bullmore and O Sporns ldquoComplex brain networks graphtheoretical analysis of structural and functional systemsrdquoNature Reviews Neuroscience vol 10 no 3 pp 186ndash198 2009

[12] K J Friston ldquoFunctional and effective connectivity in neu-roimaging a synthesisrdquo Human Brain Mapping vol 2 no 1-2pp 56ndash78 1994

[13] A Omidvarnia P Fransson M Metsaranta and S VanhataloldquoFunctional bimodality in the brain networks of preterm andterm human newbornsrdquo Cerebral Cortex vol 24 no 10 pp2657ndash2668 2014

[14] E M Schumacher T A Stiris and P G Larsson ldquoEffectiveconnectivity in long-term EEG monitoring in preterm infantsrdquoClinical Neurophysiology vol 126 no 12 pp 2261ndash2268 2015

[15] M Lavanga O De Wel A Caicedo Dorado K JansenA Dereymaeker G Naulaers et al ldquoLinear and nonlinearfunctional connectivity methods to predict brain maturationin preterm babiesrdquo in Proceedings of the 8th InternationalWorkshop on Biosignal Interpretation pp 37ndash40 2016

[16] T Schreiber ldquoMeasuring information transferrdquo Physical ReviewLetters vol 85 no 2 pp 461ndash464 2000

[17] C W J Granger ldquoInvestigating causal relations by econometricmodels and cross-spectral methodsrdquo Econometrica vol 37 no3 pp 424-238 1969

[18] A Montalto L Faes and D Marinazzo ldquoMuTE a MATLABtoolbox to compare established and novel estimators of themultivariate transfer entropyrdquo PLoS ONE vol 9 no 10 ArticleID e109462 2014

[19] E Florin J Gross J Pfeifer G R Fink and L TimmermannldquoThe effect of filtering on Granger causality based multivariatecausality measuresrdquo NeuroImage vol 50 no 2 pp 577ndash5882010

[20] A Dereymaeker K Pillay J Vervisch et al ldquoAn automatedquiet sleep detection approach in preterm infants as a gatewayto assess brain maturationrdquo International Journal of NeuralSystems vol 27 no 06 p 1750023 2017

[21] N Koolen A Dereymaeker O Rasanen et al ldquoData-drivenmetric representing the maturation of preterm EEGrdquo in Pro-ceedings of the 37th Annual International Conference of the IEEEEngineering in Medicine and Biology Society EMBC rsquo15 pp1492ndash1495 IEEE Milan Italy August 2015

[22] K J Blinowska ldquoReview of the methods of determination ofdirected connectivity from multichannel datardquoMedical Biolog-ical Engineering Computing vol 49 no 5 pp 521ndash529 2011

[23] L A Baccala andK Sameshima ldquoCausality and influentiabilitythe need for distinct neural connectivity conceptsrdquo Journal ofChemistry pp 424ndash435 2014

[24] L Barnett and A K Seth ldquoThe MVGC multivariate grangercausality toolbox a new approach to granger-causal inferencerdquoJournal of Neuroscience Methods vol 223 pp 50ndash68 2014

Complexity 13

[25] K Sameshima D Y Takahashi and L A Baccala ldquoOn the sta-tistical performance ofGranger-causal connectivity estimatorsrdquoBrain Informatics vol 2 no 2 pp 119ndash133 2015

[26] R Vicente M Wibral M Lindner and G Pipa ldquoTransferentropymdashamodel-free measure of effective connectivity for theneurosciencesrdquo Journal of Computational Neuroscience vol 30no 1 pp 45ndash67 2011

[27] A K Seth ldquoCausal connectivity of evolved neural networksduring behaviorrdquoNetwork Computation in Neural Systems vol16 no 1 pp 35ndash54 2015

[28] F D Fallani and F Babiloni ldquoThe graph theoretical approach inbrain functional networks theory and applicationsrdquo SynthesisLectures on Biomedical Engineering vol 5 no 1 pp 1ndash92 2010

[29] G Fagiolo ldquoClustering in complex directed networksrdquo PhysicalReview EmdashStatistical Nonlinear and SoftMatter Physics vol 76no 2 part 2 Article ID 026107 2007

[30] J Bondy and U Murty Graph Theory with ApplicationsSpringer Berlin Heidelberg Germany 1976

[31] M Rubinov and O Sporns ldquoComplex network measures ofbrain connectivity uses and interpretationsrdquo NeuroImage vol52 no 3 pp 1059ndash1069 2010

[32] A B Barrett L Barnett and A K Seth ldquoMultivariate Grangercausality and generalized variancerdquo Physical Review E Statisti-cal Nonlinear and Soft Matter Physics vol 81 no 4 041907 14pages 2010

[33] A Yu M Lu and F Tian ldquoOn the spectral radius of graphsrdquoLinear Algebra and its Applications vol 387 pp 41ndash49 2004

[34] E Estrada and N Hatano ldquoCommunicability angle and thespatial efficiency of networksrdquo SIAM Review vol 58 no 4 pp692ndash715 2016

[35] F Chung ldquoLaplacians and the Cheeger inequality for directedgraphsrdquo Annals of Combinatorics vol 9 no 1 pp 1ndash19 2005

[36] L Barnett and A K Seth ldquoBehaviour of Granger causalityunder filtering theoretical invariance andpractical applicationrdquoJournal of Neuroscience Methods vol 201 no 2 pp 404ndash4192011

[37] W DeClercq A Vergult B Vanrumste W van Paesschenand S van Huffel ldquoCanonical correlation analysis applied toremove muscle artifacts from the electroencephalogramrdquo IEEETransactions onBiomedical Engineering vol 53 no 12 pp 2583ndash2587 2006

[38] H Peng K Li B Li H Ling W Xiong andW Hu ldquoPredictingimage memorability by multi-view adaptive regressionrdquo inProceedings of the the 23rd ACM international conference pp1147ndash1150 Brisbane Australia October 2015

[39] J M OrsquoToole G B Boylan S Vanhatalo and N J StevensonldquoEstimating functional brain maturity in very and extremelypreterm neonates using automated analysis of the electroen-cephalogramrdquoClinical Neurophysiology vol 127 no 8 pp 2910ndash2918 2016

[40] I Kostovic andM Judas ldquoThe development of the subplate andthalamocortical connections in the human foetal brainrdquo ActaPaediatrica vol 99 no 8 pp 1119ndash1127 2010

[41] E J Meijer K H M Hermans A Zwanenburg et alldquoFunctional connectivity in preterm infants derived from EEGcoherence analysisrdquo European Journal of Paediatric Neurologyvol 18 no 6 pp 780ndash789 2014

[42] D A Fair N U Dosenbach J A Church A L Cohen SBrahmbhatt F M Miezin et al ldquoDevelopment of distinct con-trol networks through segregation and integrationrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 104 no 33 pp 13507ndash13512 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

strength among brain regions together with the densityand the percentage of connecting streamlines Huppi andDubois [9] eventually argued that diffusion tensor imagingparameters such as anisotropy can be structural markers ofnetwork functional organization [5] A possible implicationis that the temporal correlation among brain regions isalso expected to change since the functional connectivityis related to the structural topology [1] However less isknown about functional connectivity and maturation [10]Therefore functional connectivity in developmental sciencereceived increasing attention in recent years [1] Moreoverdifferent tools to describe functional connectivity becameavailable in the last two decades According to Bullmore andSporns [11] brain networks can be represented as a graphwhere the nodes are brain regions and the edges are theconnection strengths The nodes are defined by the neuralactivity while the edges define the presence of a couplingbetween time series and the associated weight represents thecoupling degree Friston [12] recognizes two main types offunctional connectivity the actual functional connectivitywhich investigates the temporal correlation among signalsand the effective connectivity which measures the causalityof the coupling While the former does not imply anydirectionality the second one highlights which area or nodecauses the activity of other ones Functional connectivitymethods generate undirected graphs while the effectiveconnectivity methods are associated with directed graphs[11] In different studies of the preterm infant brain [5 13]the neural activity has been measured using fMRI Althoughthis neuroimaging technique can investigate the subcorticalstructures it is an expensive method and it is not suitableto measure effective connectivity due to fMRI low temporalresolution [14] In contrast EEG is a suitable measurementfor effective connectivity and has been employed in recentpapers to study the brain connectivitymaturation [14 15]Themain drawbacks of those studies are the limited investigatedmaturation period or the absence of graph-theory metrics toinvestigate the connectivity changes Studies about the longterm development of effective connectivity based on EEGhave not been performed yet In particular there is a lack ofresearch that investigates the effective connectivity frombirthuntil full-term age in premature infants Furthermore in theliterature there are a few works that study the connectivitymaturation in the pretermbrain froma graphpoint of view [513] The main aim of the present study was a thorough inves-tigation of the effective connectivity evolution during thepreterm maturation after birth exploiting different networkmetrics (as in [5]) In particular we measured the effectiveconnectivity by means of transfer entropy [16] and Grangercausality [17] On the obtained directed graph different graphmetrics were computed to track their evolution from 27 to 42postmenstrual age (PMA) weeks The network analysis wascomplemented by a regression study to predict the age of thepatient and assess the predictive power of the connectivityanalysis (as performed in one of our previous studies [15])The paper is organized as follows in Section 2 we introducethe simulated and the real dataset alongside themethods usedto estimate effective connectivity In Section 2 the appliedgraph-theory metrics are also discussed Section 3 reports

1

2

3 4

5

6

Figure 1 The figure displays the graph associated with model (1)

the results while Section 4 includes our discussion on thereported results

2 Methods

21 Data Different datasets were used A simulated datasetwas also employed in order to show how connectivityanalysis performs in a controlled case In particular the mainobjective of the simulationwas to illustrate themeaning of themost common graph indices used in the literature A datasetcomprised of EEG signalswas themain part of thematurationstudy

211 Simulated Data The first experiment of the studyconsisted of a simulation based on a linear Gaussian regres-sion model (derived from [18]) expressed by the followingequations

1199091119905 = 091199091119905minus1 + 091205731199092119905minus1 + 071205731199093119905minus1 + 12058511199051199092119905 = minus091205731199091119905minus1 + 071205731199093119905minus1 + 12058521199051199093119905 = 081205731199091119905minus1 + 071205731199092119905minus1 + 12058531199051199094119905 = minus0251205731199091119905minus1 minus 061199094119905minus1 + 12058541199051199095119905 = 0251205731199091119905minus1 + 091205731199094119905minus1 + 12058551199051199096119905 = 091205731199094119905minus1 + 091199095119905minus1 minus 071205731199096119905minus1 + 1205856119905

(1)

The parameter 120573 is a scalar value that varies between 0and 1 and it influences the strength of the coupling amongthe 119909119894119905 variables The 120585119894119905 variables represent white noise withunit varianceThe associated graph is reported in Figure 1 Inorder to give the reader a clear insight of the graph-theorymeasures the objective of the simulation was twofold firstlywe investigated the influence of the strength of the couplingprovided by the parameter 120573 and show the effect of theweakening of causality among the variables Secondly weinvestigated the influence of the noise using different levelsof signal-to-noise ratio (SNR) which were obtained varyingthe variance of white noise in (1) In the latter objectivetime series were simulated using the AR model (1) andthe coupling was estimated with the effective connectivitymethods discussed in the following paragraphs We alsostudied the impact of filtering on the graph based metrics bycomputing the scores from filtered signals using a samplingfrequency which was set at 200Hz and a band-pass filterbetween 1 and 80Hz was applied on the simulated data Thereason to investigate the impact of filtering is explained in

Complexity 3





U119905 =119901

sum119896=1

A119896 sdot U119905minus119896 + 120598 (119905) (2)


U119905 = [[[

X119905Y119905Z119905

]]]

(3)


Y119905 =119901

sum119896=1

A119910119910 sdot Y119905minus119896 +119901

sum119896=1

A119910119909 sdot X119905minus119896 +119901

sum119896=1

A119910119911 sdot Z119905minus119896+ 120598119910 (119905)

Y119905 =119901

sum119896=1

A119910119910 sdot Y119905minus119896 +119901

sum119896=1

A119910119911 sdot Z119905minus119896 + 120598119910 (119905)

(4)


F119883rarr119884|Z = log10038161003816100381610038161003816Σ119910119910

1003816100381610038161003816100381610038161003816100381610038161003816Σ11991011991010038161003816100381610038161003816 (5)


G119894119895 (U) = F119880119894rarr119880119895|U[ij] (6)


4 Complexity







119884119905 = 119860119906119881119906 + 120598 (119905) 119884119905 = 119860119903119881119903 + 120598 (119905) (9)




(10)



119879119883rarr119884|Z = 12 log 120590119903

120590119906 (11)



Complexity 5




119871dir = 119863 minus 119860

119871 sym = 12 (119871dir + 119871119879dir) = 119863 minus 119860 + 119860119879

2 (12)




6 Complexity


3 Results





Complexity 7

020406080

100120140160180200

Path

leng

th (a

u)

Path length


(a)

0

005

01

015

02

025

03

CC (a

u)



(b)

0

02

04

06

08

1

12

Spec

tral

radi

us (a

u)

Spectral radius


(c)

SNR (db)

0

01

02

03

04

05

06

07

Ratio

CC

(nu

)


minus20 0 20 40 60 80 100 120 140


and the original CC

(d)

0

02

04

06

08

1

12

Ratio

pat

h le

ngth

(nu

)



SNR (db)minus20 0 20 40 60 80 100 120 140

(e)

0010203040506070809

1

Ratio

spec

tral

radi

us (n

u)



SNR (db)minus20 0 20 40 60 80 100 120 140

(f)



4 Discussion




8 Complexity


10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x


0005 001 0015 002


(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)


15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th


(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)


Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3





U119905 =119901

sum119896=1

A119896 sdot U119905minus119896 + 120598 (119905) (2)


U119905 = [[[

X119905Y119905Z119905

]]]

(3)


Y119905 =119901

sum119896=1

A119910119910 sdot Y119905minus119896 +119901

sum119896=1

A119910119909 sdot X119905minus119896 +119901

sum119896=1

A119910119911 sdot Z119905minus119896+ 120598119910 (119905)

Y119905 =119901

sum119896=1

A119910119910 sdot Y119905minus119896 +119901

sum119896=1

A119910119911 sdot Z119905minus119896 + 120598119910 (119905)

(4)


F119883rarr119884|Z = log10038161003816100381610038161003816Σ119910119910

1003816100381610038161003816100381610038161003816100381610038161003816Σ11991011991010038161003816100381610038161003816 (5)


G119894119895 (U) = F119880119894rarr119880119895|U[ij] (6)


4 Complexity







119884119905 = 119860119906119881119906 + 120598 (119905) 119884119905 = 119860119903119881119903 + 120598 (119905) (9)




(10)



119879119883rarr119884|Z = 12 log 120590119903

120590119906 (11)



Complexity 5




119871dir = 119863 minus 119860

119871 sym = 12 (119871dir + 119871119879dir) = 119863 minus 119860 + 119860119879

2 (12)




6 Complexity


3 Results





Complexity 7

020406080

100120140160180200

Path

leng

th (a

u)

Path length


(a)

0

005

01

015

02

025

03

CC (a

u)



(b)

0

02

04

06

08

1

12

Spec

tral

radi

us (a

u)

Spectral radius


(c)

SNR (db)

0

01

02

03

04

05

06

07

Ratio

CC

(nu

)


minus20 0 20 40 60 80 100 120 140


and the original CC

(d)

0

02

04

06

08

1

12

Ratio

pat

h le

ngth

(nu

)



SNR (db)minus20 0 20 40 60 80 100 120 140

(e)

0010203040506070809

1

Ratio

spec

tral

radi

us (n

u)



SNR (db)minus20 0 20 40 60 80 100 120 140

(f)



4 Discussion




8 Complexity


10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x


0005 001 0015 002


(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)


15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th


(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)


Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity







119884119905 = 119860119906119881119906 + 120598 (119905) 119884119905 = 119860119903119881119903 + 120598 (119905) (9)




(10)



119879119883rarr119884|Z = 12 log 120590119903

120590119906 (11)



Complexity 5




119871dir = 119863 minus 119860

119871 sym = 12 (119871dir + 119871119879dir) = 119863 minus 119860 + 119860119879

2 (12)




6 Complexity


3 Results





Complexity 7

020406080

100120140160180200

Path

leng

th (a

u)

Path length


(a)

0

005

01

015

02

025

03

CC (a

u)



(b)

0

02

04

06

08

1

12

Spec

tral

radi

us (a

u)

Spectral radius


(c)

SNR (db)

0

01

02

03

04

05

06

07

Ratio

CC

(nu

)


minus20 0 20 40 60 80 100 120 140


and the original CC

(d)

0

02

04

06

08

1

12

Ratio

pat

h le

ngth

(nu

)



SNR (db)minus20 0 20 40 60 80 100 120 140

(e)

0010203040506070809

1

Ratio

spec

tral

radi

us (n

u)



SNR (db)minus20 0 20 40 60 80 100 120 140

(f)



4 Discussion




8 Complexity


10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x


0005 001 0015 002


(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)


15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th


(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)


Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5




119871dir = 119863 minus 119860

119871 sym = 12 (119871dir + 119871119879dir) = 119863 minus 119860 + 119860119879

2 (12)




6 Complexity


3 Results





Complexity 7

020406080

100120140160180200

Path

leng

th (a

u)

Path length


(a)

0

005

01

015

02

025

03

CC (a

u)



(b)

0

02

04

06

08

1

12

Spec

tral

radi

us (a

u)

Spectral radius


(c)

SNR (db)

0

01

02

03

04

05

06

07

Ratio

CC

(nu

)


minus20 0 20 40 60 80 100 120 140


and the original CC

(d)

0

02

04

06

08

1

12

Ratio

pat

h le

ngth

(nu

)



SNR (db)minus20 0 20 40 60 80 100 120 140

(e)

0010203040506070809

1

Ratio

spec

tral

radi

us (n

u)



SNR (db)minus20 0 20 40 60 80 100 120 140

(f)



4 Discussion




8 Complexity


10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x


0005 001 0015 002


(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)


15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th


(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)


Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity


3 Results





Complexity 7

020406080

100120140160180200

Path

leng

th (a

u)

Path length


(a)

0

005

01

015

02

025

03

CC (a

u)



(b)

0

02

04

06

08

1

12

Spec

tral

radi

us (a

u)

Spectral radius


(c)

SNR (db)

0

01

02

03

04

05

06

07

Ratio

CC

(nu

)


minus20 0 20 40 60 80 100 120 140


and the original CC

(d)

0

02

04

06

08

1

12

Ratio

pat

h le

ngth

(nu

)



SNR (db)minus20 0 20 40 60 80 100 120 140

(e)

0010203040506070809

1

Ratio

spec

tral

radi

us (n

u)



SNR (db)minus20 0 20 40 60 80 100 120 140

(f)



4 Discussion




8 Complexity


10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x


0005 001 0015 002


(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)


15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th


(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)


Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7

020406080

100120140160180200

Path

leng

th (a

u)

Path length


(a)

0

005

01

015

02

025

03

CC (a

u)



(b)

0

02

04

06

08

1

12

Spec

tral

radi

us (a

u)

Spectral radius


(c)

SNR (db)

0

01

02

03

04

05

06

07

Ratio

CC

(nu

)


minus20 0 20 40 60 80 100 120 140


and the original CC

(d)

0

02

04

06

08

1

12

Ratio

pat

h le

ngth

(nu

)



SNR (db)minus20 0 20 40 60 80 100 120 140

(e)

0010203040506070809

1

Ratio

spec

tral

radi

us (n

u)



SNR (db)minus20 0 20 40 60 80 100 120 140

(f)



4 Discussion




8 Complexity


10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x


0005 001 0015 002


(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)


15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th


(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)


Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity


10

15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 4138 minus 74422x


0005 001 0015 002


(a)

PMA

(wee

ks)

Path length (au)

15

20

25

30

35

40

45

50y = minus637 + 989x


34 36 38 4 42 44 46


(b)


15

20

25

30

35

40

45

50

PMA

(wee

ks)

y = 41 minus 9709x


002 004 006 008 01 012 014


(c)

Spectral gap (au)

15

20

25

30

35

40

45

50PM

A (w

eeks

)

y = 4165 minus 12922x


002 004 006 008 01


(d)

4

42

44

46

48

5

52

Path

leng

th


(e)

0006

0008

001

0012

0014

0016

0018

Clus

terin

g co

effici

ent


(f)


Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 9

T3T4

Fp1

C3

O1


0 01 02 03 04 05 06 07 08 09 1


Fp2

O2

C4T3T4

Fp1

C3

O1

Fp2

O2

C4T3 T4

Fp1

C3

O1

Fp2

O2

C4













10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




10 Complexity
















Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 11




5 Conclusions


12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




12 Complexity


Disclosure




Acknowledgments


References

























Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 13


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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