A symbolic information approach to characterize response-related differences in cortical activity during a Go/No-Go task
Helena B. Lucas, Steven L. Bressler, Fernanda S. Matias, Osvaldo A. Rosso
AA symbolic information approach to characterize response-related differences incortical activity during a Go/No-Go task
Helena B. Lucas, ∗ Steven L. Bressler, Fernanda S. Matias, † and Osvaldo A. Rosso ‡ Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o, Alagoas 57072-970 Brazil. Center for Complex Systems and Brain Sciences, Dept. of Psychology,Florida Atlantic University, Boca Raton, FL 33431, USA
How the brain processes information from external stimuli in order to perceive the world andact on it is one of the greatest questions in neuroscience. To address this question different timeseries analyzes techniques have been employed to characterize the statistical properties of brainsignals during cognitive tasks. Typically response-specific processes are addressed by comparing thetime course of average event-related potentials in different trials type. Here we analyze monkeyLocal Field Potentials data during visual pattern discrimination called Go/No-Go task in the lightof information theory quantifiers. We show that the Bandt-Pompe symbolization methodology tocalculate entropy and complexity of data is a useful tool to distinguish response-related differencesbetween Go and No-Go trials. We propose to use an asymmetry index to statistically validatetrial type differences. Moreover, by using the multi-scale approach and embedding time delays todownsample the data we can estimate the important time scales in which the relevant informationis been processed.
PACS numbers:
I. INTRODUCTION
The brain is one of the biophysical systems that cap-ture with greater attention our interest in understandinghow it operates dynamically and organizationally [1]. Ina general description, one can think of the brain as an effi-cient complex network composed of many parts or struc-tures and with an intense flow of information betweenthese structures/zones. Moreover, this exchange of infor-mation could be considered non-linear and determine thecorresponding brain behavior. In particular, each one ofthese brain regions has its own structures and functions.They are also interconnected and share information be-tween them, forming in this way an integrated networkthat determines the dynamics and behavioral response ofthe brain.Taking these characteristics of the brain such as in-terconnections and information flow between differentbrain areas, measurement of time series activity at dif-ferent regions and under external stimuli could bring usinformation about the corresponding dynamics, as wellas about the information transfer. Integrated processesat the visual and motor systems during visual discrimi-nation tasks have been extensively studied by analyzingmonkey Local Field Potentials (LFP) data [2–6]. In par-ticular, the averaged evoked response potential (ERP) ofmonkey LFP has been employed to estimate response-related differences between Go and No-Go trials [7]. Theauthors have shown that response-specific processing be-gan around 150 ms post-stimulus in widespread cortical ∗ helena.bordini@fis.ufal.br † fernanda@fis.ufal.br ‡ [email protected] areas. The time course of task-related activity was exam-ined to identify the cortical locations and the time win-dows that exhibit a significant difference between eachresponse type. Here we propose that analysis of the sta-tistical properties of the concatenated trials could alsobe useful and eventually could provide more informa-tion than comparing the activation response by averagingbrain activity.To do this type of study, we chose to work with timecausal quantifiers based on Information Theory (Shan-non entropy, MPR-statistical complexity, and entropy-complexity plane) [8–12]. These quantifiers are evaluatedusing the Bandt-Pompe symbolization methodology [13],which includes naturally the time causal ordering pro-vided by the time-series data in the corresponding as-sociate probability distribution function (PDF). Thesetools have been employed to analyze brain signals inplenty of studies: to estimate time differences duringphase synchronization [14], to show that complexity ismaximized close to criticality in cortical states [15], todistinguish cortical states using EEG data [16] as well asneuronal activity [17, 18].The manuscript is organized in the following way: inSec. II we introduce the Information Theory quantifiers,as well as in Sec. III, in which the Bandt-Pompe method-ology for their evaluation is presented. The experimen-tal data corresponding to Local Field Potentials (LFP),are presented in Sec. IV, at four cortical-deep electrodes,in a monkey brain, and under visual stimulus conditionGo/No-Go. In Sec. V, we report our analysis and results.Finally, concluding remarks and a brief discussion of thesignificance of our findings for neuroscience are presentedin Sec. VI. a r X i v : . [ q - b i o . N C ] J a n II. INFORMATION-THEORY QUANTIFIERS:PERMUTATION ENTROPY AND STATISTICALCOMPLEXITY
In the characterization of dynamical systems (systemsthat evolve with time) the main objective is to infer someproperties and behavior of the system under study. Inparticular, the starting point is a set of M discrete mea-sures taken at regular intervals of the same representativevariable of the system, that we denote by the correspond-ing time series X ( t ). In the second step, we need to as-sociate a probability distribution function P to the timeseries, and with it, we will able to evaluate the corre-sponding Information Theory quantifiers.There is no unique answer for the best procedure toassociate a time series with a PDF, and in fact, dif-ferent proposals can be found. However, we start froma sequence/measurement taken in causality order, thesecharacteristics are something to be preserved in the de-termination of the PDF. For this reason, we follow theprocedure proposed by Bandt-Pompe [13] for the asso-ciation of time causal PDF to the time series.Let X ( t ) ≡ { x t ; t = 1 , , . . . , M } , be the time seriesrepresenting a set of M measurements of the observable X and let a probability distribution function be givenby P ≡ { p j ; j = 1 , , . . . , N } , with (cid:80) Nj =1 p j = 1, and0 ≤ p j ≤
1, where N is the number of possible states ofthe system.The first Information Theory quantifier that we intro-duce is the Shannon’s logarithmic information measure[8], defined by: S [ P ] = − N (cid:88) j =1 p j ln( p j ) . (1)This functional is equal to zero when we are able to cor-rectly predict the outcome every time. For example, forlinearly increasing time series all probabilities are zerobut one which is equal to 1. The corresponding PDF willbe P = { p k = 1 and p j = 0 , ∀ j (cid:54) = k, j = 1 , . . . , N − } ,then S [ P ] = 0. By contrast, the entropy is maximizedfor the uniform distribution P e = { p j = 1 /N, ∀ j =1 , , . . . , N } , being S max = S [ P e ] = ln( N ). We definethe normalized Shannon entropy by H [ P ] = S [ P ] S max = S [ P ]ln( N ) , (2)and 0 ≤ H [ P ] ≤
1, which give a measure of the informa-tion content of the corresponding PDF ( P ).The second Information Theory based quantifier thatwe introduced is the Statistical Complexity, defined byfunctional product form [19] C [ P, P e ] = H [ P ] · Q J [ P, P e ] . (3)Where, H [ P ] is the normalized Shannon entropy (seeEq. (2)) and Q J [ P, P e ] represent the disequilibrium, t X ( t ) π j P ( π j )
012 021 102 120 201 210(b)(c) (a)
FIG. 1: Characterizing the symbolic information ap-proach. (a) The six symbols associated to permutations π j for ordinal patterns of length D = 3, and embeddingtime τ = 1. (b) Simple example of a time series given by X ( t ) = { , , , , , , , , } , M = 9 and, (c) its own non-normalized probability density function (PDF). which is defined in terms of the Jensen–Shannon diver-gence [20] as: Q J [ P, P e ] = Q J [ P, P e ] , (4)where J [ P, P e ] = S (cid:20) ( P + P e )2 (cid:21) − S [ P ]2 − S [ P e ]2 , (5)and Q is a normalization constant (0 ≤ Q J ≤ J [ P, P e ],that is Q = 1 /J [ P , P e ]. In this way, also the statisticalcomplexity is a normalized quantity, 0 ≤ C [ P, P e ] ≤ H , thecorresponding complexity varies in a range of values givenby C min and C max , and these values depend only on thedimension of the PDF considered and the functional formchosen for the entropy [21]. III. THE BANDT-POMPE SYMBOLIZATIONMETHOD
Let X ( t ) ≡ { x t ; t = 1 , . . . , M } denote a time seriesmeasured from the dynamics state of a dynamical system. M is the number of data (time series length) measured atregular equal spaced times. We define the symbol vector (cid:126)π ( D ) by (cid:126)π ( D ) = ( π , π , π , . . . , π D − ) , (6)in such a way that every element of (cid:126)π ( D ) is unique ( π j (cid:54) = π k for every j (cid:54) = k ).Given the parameters D ∈ N with D ≥
2, the em-bedding dimension which represent the quantity of infor-mation to be included and; the parameter τ ∈ N with τ ≥
1, the embedding time (time delay), we divide thetime series X ( t ) in overlapping vectors of length D , (cid:126)Y ( D,τ ) s = ( x s , x s + τ , x s +2 τ , . . . , x s +( D − τ ) , (7)with s = 1 , , . . . , M − ( D − τ . The index s controlsthe beginning of each vector and τ the overlap degreebetween vectors. The vector (cid:126)Y ( D,τ ) s can be mapped to asymbol vector (cid:126)π ( D ) s . This mapping is such that preservesthe desired relation between the elements x s ∈ (cid:126)Y ( D,τ ) s and all s ∈ { , , . . . , M − ( D − τ } that share this pat-tern (also called motif) and are mapped to the same (cid:126)π ( D ) .We define the mapping overlapping vectors of length D , (cid:126)Y ( D,τ ) s (cid:55)→ (cid:126)π ( D ) s , (8)by ordering the observations x s ∈ (cid:126)Y ( D,τ ) s in increasingorder. That is, from the vector (cid:126)Y ( D,τ ) s (see eq. (7) welooking for the vector (cid:126)π ( D ) s = ( r , r , r , . . . , r ( D − ) , (9)which represent the permutation pattern, given by π ( D ) s = [0 , , , . . . , ( D − , (10)such that x s + r < x s + r < x s + r < . . . < x s + r D − . (11)In order to get an unique result, we set r j < r j +1 if x s + r j = x s + r j +1 .The Bandt-Pompe probability distribution function(BP-PDF) of ordinal patterns Π D is obtained by the fre-quency histogram of symbols associated to a given timeseries (cid:126)Y ( D,τ ) s . The BP-PDF is the relative frequency ofsymbols in the series against the D ! possible patterns,that is p ( π ( D ) s ) = (cid:93) { (cid:126)Y ( D,τ ) s is of type π ( D ) s } [ M − ( D − τ ] , (12)in which the symbol (cid:93) indicate number, and s = { , , . . . , M − ( D − τ } . Note that p ( π ( D ) s ) satis-fies the probability condition 0 ≤ p ( π ( D ) s ) ≤ (cid:80) D ! j =1 p ( π ( D ) s ) = 1. An important property of the BP-PDF is its invariant monotonic transformations. (a)(b) ∆ t (ms)
100 400
ImageWindow
FIG. 2: Schematic representation of the experimental setup.(a) Electrodes position in the monkey brain: primary mo-tor cortex (channel 1, violet), primary somatosensory cortex(channel 2, red), posterior parietal areas (site 3 in orange and4 in blue). (b) Temporal organization of the Go/No-Go task.We analyze the statistical properties of the time series in dif-ferent intervals of the trial by separating it in 11 windows( W i ) of 90 ms each, starting every 50 ms. Each trial goesfrom t = 0 to t = 600 ms (see more details in Sec. IV). Let consider an example: given the time series X ( t ) = { , , , , , , , , } , with M = 9, we evaluate the BP-PDF with D = 3 and τ = 1. In Fig. 1.a, we representthe 6 ordinal patterns, corresponding to D = 3, and inthe Fig. 1.b, the time series considered is represented asfunction of time. Then we have N (cid:48) = M − ( D − τ = 7embedding vectors: (cid:126)Y (3 , = (1 , , (cid:55)→ (cid:126)π (3)1 = (0 , , (cid:55)→ π = [021]; (cid:126)Y (3 , = (10 , , (cid:55)→ (cid:126)π (3)2 = (2 , , (cid:55)→ π = [210]; (cid:126)Y (3 , = (6 , , (cid:55)→ (cid:126)π (3)3 = (1 , , (cid:55)→ π = [120]; (cid:126)Y (3 , = (2 , , (cid:55)→ (cid:126)π (3)4 = (0 , , (cid:55)→ π = [012]; (cid:126)Y (3 , = (4 , , (cid:55)→ (cid:126)π (3)5 = (2 , , (cid:55)→ π = [201]; (cid:126)Y (3 , = (8 , , (cid:55)→ (cid:126)π (3)6 = (1 , , (cid:55)→ π = [102]; (cid:126)Y (3 , = (2 , , (cid:55)→ (cid:126)π (3)7 = (2 , , (cid:55)→ π = [201].Then the pattern probability are p ( π ) = p ( π ) = p ( π ) = p ( π ) = p ( π ) = 1 / p ( π ) = 2 /
7. Fig. 1.cshows the non-normalized histogram patterns.
IV. EXPERIMENTAL TIME SERIES
Local Field Potentials were recorded surface-to-depthwith bipolar Teflon-coated platinum micro-electrodesfrom 15 distributed sites located in the right hemisphereof the adult male rhesus macaque monkeys (GE). Here weanalyze the four sites shown in Fig. 2(a). Site 1 is in theprimary motor cortex, site 2 in the primary somatosen-sory cortex, and sites 3 and 4 are in the posterior parietalcortex. LFPs time series were sampled at 200 points/s(every 5 ms), and collected from 100 ms before to 500 msafter stimulus onset (see Fig. 3 for illustrative examplesof LFPs). Therefore we define our trial time from t = 0to t = 600 ms, which means that the stimulus appears at t = 100 ms.The monkey was highly trained to perform a visualpattern discrimination task called Go/No-Go. On eachtrial, the monkey depressed a hand lever and kept itpressed during a random interval ranging from 0 .
12 to2 . t = 600 ms). On No-Go trials the monkey should notrelease the lever. Experiments were performed at theLaboratory of Neuropsychology at the National Instituteof Mental Health (USA), and animal care was in accor-dance with institutional guidelines at the time.We separate each trial in 11 windows ( W i ) of 90 mseach, starting every 50 ms. For example, the first window W ranges from 0 to 90 ms, W is from 50 to 140 ms, W is from 100 to 190 ms (as illustrated in Fig. 2(b) forthe even windows). The average response for Go trials(mean and standard deviation over sessions) occurs at t = 349 ± ± W i window of all 359 Go trials and, separately,we concatenate all 351 No-Go trials. Figures 3(a) and (b)show the first 20 Go trials (whereas figures 3(c) and (d)show the first 20 No-Go trials) of each region of interestduring W and W . We use this concatenated time seriesto extract the symbols and calculate the PDF in order toobtain the entropy and the complexity.The first window has been previously analyzed in thelight of the entropy-complexity plane by Montani etal. [14] and by using Granger causality measures [4, 22]to infer the connectivity between these regions. The en-tire task has been analyzed using coherence and Grangercausality and separating Go and No-Go trials for sites1, 2, and 3 by Zhang et al. [23]. The same sites and afew others have been analyzed in the light of EvokedResponse Potentials (ERP) to study stimulus-evokedactivation onset, stimulus-specific processing, stimuluscategory-specific processing, and response-specific pro-cessing by Ledberg et al. [7]. (a) W Go (b) W No-Go (c) W Go (d) W No-Go t (ms) V ( m V ) FIG. 3: Illustrative examples of the time series from monkey-LFP taken over concatenated first 20 trials for the analyzedregions primary motor cortex; somatosensory cortex; parietalcortices. Same color code for the signals from each region asdepicted in Fig 2(a). Time window W (0 < t <
90) ms) (a)under Go condition and (b) under No-Go condition. Timewindow W (300 < t <
390 ms) (c) Go trials and (d) forNo-Go trials. (b) W
18 36 54 72 90-600-3000300600 18 36 54 72 90 (a) W t (ms) t (ms) V ( m V ) GoNo-Go GoNo-Go
FIG. 4: Comparison between Go and No-Go trials. Timecourse of average event-related potentials (ERP) in region 2:(a) over all corresponding trials for time window W (pre-stimulus) and (b) over all corresponding trials for time win-dow W (post-stimulus). Error bars are the standard devia-tion of the mean. V. RESULTS
Here we employ information theory quantifiers as auseful tool to study response-specific processing in brainsignals during a visual-motor task. We have separatelyanalyzed the monkey-LFP time series of all Go responsetrials, as well as, of all No-Go response trials for the four
FIG. 5: Response-specific differences in region 2 captured bythe Information theory quantifiers. Entropy as a function ofthe embedding time (or time delay) τ for (a) W and (b) W .Complexity as a function of the embedding time τ for (c) W and (d) W . brain regions shown in Fig. 2(a): (1) the primary mo-tor cortex, (2) the somatosensory cortex, (3) and (4) theparietal cortex (see more details in Sec. IV). Illustrativeexamples of the time series for each one of the four sitesduring W (from trial time t = 0 to 90 ms) and W (fromtrial time t = 300 to 390 ms) for the first 20 trials of Goand No-Go conditions are shown in Fig. 3. The activity ofthe motor cortex is clearly different from the other threeregions. Moreover, in a naive comparison, just by eyeinspection, it seems that Go and No-Go trials are moresimilar during W than during W in the four channels.Our first hypothesis is related to the fact that for thefirst two time windows ( W and W ) one should not findany statistically significant difference between the Go andNo-Go time series in any region since the visual stimulusdid not appear or has not been processed yet. However,before the end of the trial, we should be able to distin-guish both cases. For example, in Fig. 4(a) and (b) wecompare the average activity of all Go and all No-Go tri-als of the primary somatosensory area (region 2). Thedifference between the types of trials is clearly larger for W than for W . This approach has been employed byLedberg et al. [7] along the whole trial to estimate howlong it takes for each region to start to show significantdifferences between Go and No-Go conditions. In fact,Fig. 4(b) is comparable to Fig. 12(G) in Ref. [7]. Theyhave shown that this area starts to process response-specific information after t = 300 ms which coincideswith the beginning of W . In what follows, we show thatresponse-related differences can also be verified with theinformation theory quantifiers.To characterize different cortical states during the cog-nitive task we have calculated the entropy and complexity (a) (b)(c) (d)(e) (f)FIG. 6: Quantifying the statistical differences between Goand No-Go trials. Asymmetry index for entropy A ( H ) (leftcolumn) and complexity A ( C ) (right column) for region 2as a function of the time delay τ for three consecutive timewindows (a, b) W , (c, d) W and (e, f) W . We use W and W to calculate the average value µ A ( C ) ( µ A ( H ) ) andits standard deviation σ A ( C ) ( σ A ( H ) ). Dashed lines represent A ( H ) = µ A ( H ) ± σ A ( H ) and A ( C ) = µ A ( C ) ± σ A ( C ) . for each W i in different time scales by changing τ from1 up to 60, which corresponds to downsample the seriesfrom every 5 ms up to every 300 ms. In Fig. 5 we show H versus τ and C versus τ for region 2 during W and W . As expected, there is no difference between Go/No-Go trials for W . However, for W both the entropy andthe complexity are clearly different when comparing Goand No-Go trials. In particular, both conditions havean increase in the minimal complexity and a decreasein the maximal entropy when compared with the firstwindow, but changes in the Go condition are much morepronounced. This is a good indication that we can use H and C together with the average potential to infer if theregion is processing response-specific information. In allregions, differences between response types, when theyexist, are much more pronounced for τ up to 15 (75 ms).In order to quantify the response-specific difference,and statistically validate it, we define an asymmetry in-dex for entropy A ( H ) and complexity A ( C ) respectivelyby: A ( H ) = H Go − H NoGo H Go + H NoGo (13)and A ( C ) = C Go − C NoGo C Go + C NoGo . (14)We calculate A ( H ) and A ( C ) for each τ , region and win-dow. For each region we use W and W to calculate theaverage value µ A ( C ) ( µ A ( H ) ) and its standard deviation σ A ( C ) ( σ A ( H ) ). We consider that the difference in the en-tropy (complexity) between Go and No-Go trials is signif-icant if A ( H ) > µ A ( H ) ± σ A ( H ) ( A ( C ) > µ A ( C ) ± σ A ( C ) ),see dashed lines in Fig. 6.In Fig. 6 we show these asymmetries as a function ofthe time delay in region 2 for three consecutive time win-dows: W , W and W . From W to W the results arevery similar and it is not possible to verify significantdifferences between trial types with neither entropy norcomplexity during these windows. However, at W (fromtrial time t = 250 to 340 ms) the response-specific differ-ences in complexity start to appear for intermediate val-ues of time delay. This also corroborates the importanceof using not only the entropy. At W the asymmetry canbe verified for both H and C at all time scales. In par-ticular, the Go/No-Go difference at the somatosensoryregion is larger for W than for all others W i .By using the average event-related potentials time se-ries, Ledberg et al. [7] have found that the significantseparation between Go and No-Go trials just started af-ter t >
300 ms for site 2 which is in agreement withour results since W goes up to 340 ms. Their findingsalso corroborate the fact that our asymmetry indices aremuch larger in W which comprises the largest intervalof significant differences in the average activity.The advantage of using the multi-scale approach canbe verified for example for region 2 during W and W .The largest differences for complexity between Go andNo-Go trials does not occur for τ = 1 but for 3 < τ < W (and τ = 3 , , , W ). This strongly suggeststhat the relevant information for the task is related to atime scale from 15 to 30 ms.In Fig. 7 we show the complexity asymmetry index versus time delay for W n and W n − for sites 1, 3 and 4,where W n is the first window showing significant asym-metry index for each region. Regarding these plots, onecan see that the parietal areas present specific-responsedifferences already at the interval 150 < t <
240 ms( W ), whereas the motor cortex just shows significantdifferences for t >
200 ms ( W ). It is worth mentioningthat Ledberg et al. [7] have reported that the response-specific differences in the average potential start only for t >
250 ms at parietal areas 3 and 4. This means that byusing the information theory quantifiers we can capture (a) (b)(c) (d)(e) (f)FIG. 7: Determining the time interval in which the signifi-cant separation between Go and No-Go trials can be observedfor each region. Asymmetry index for complexity A ( C ) as afunction of the time delay τ for the first window showing sig-nificant asymmetry index and their previous one: channel 1(a) W , (b) W ; channel 3 (c) W , (d) W ; channel 4 (e) W ,(f) W . significant differences a little earlier in time than withthe average activity.Finally, we can use the multi-scale entropy-complexityplane as an extra way to visualize the separation betweenGo and No-Go trials at specific time windows (see coloredlines in Fig.8). Again, the most pronounced difference inthe plane happens for region 2 and W (Fig.8(b)). Wecan also use the C × H plane to see differences amongtime windows: by comparing colored lines with grey linesfor W in the plots, which could mean that the brain isprocessing information related to the task even if it isnot differentiating Go and No-Go trials. Moreover, wecan compare differences among regions: the statisticalproperties of the motor cortex area is clearly differentfrom the others. For regions 2, 3 and 4, the system starts (a) (b)(c) (d)FIG. 8: Multi-scale Complexity-Entropy plane for the first 15 time delays τ . For each region we depict C versus H values forthe first window W (Go and No-Go trials in grey) and a post-stimulus window that shows an illustrative separation betweenGo and No-Go trials in the plane. (a) Region 1: W . (b) Region 2: W . (c) Region 3: W . (d) Region 4: W . Solid lines inblack represent the maximum and minimum complexity values for a fixed value of the entropy. close to a totally disordered regime ( H = 1 and C = 0)and presents an increase in complexity in the directionof C max . The motor area, however, is never close to theborder neither to the most disordered state.It has been shown by Zhang et al. [23] that for the No-Go trials the oscillatory activity at the beta band reap-pears at the end of trials. They suggested that since forthe No-Go condition the lever pressure was maintaineduntil the end of the recorded time period, the beta re-bound reflects the resumption of the same network of thebeginning of the trial in support of sensorimotor integra-tion, or preparation for the next trial. In Fig.8(a) we cansee for the motor area that despite statistical propertiesare not the same at W and W , the No-Go curves at W are more similar to W than the Go plots, which should be related to the rebound effect. VI. CONCLUSION
To summarize, we have shown that information theoryquantifiers (such as Shannon entropy, MPR-statisticalcomplexity, and multi-scale entropy-complexity plane)are a useful tool to characterize the information processin the brain signals. On one hand, we can determine cor-tical regions in which the response-specific informationis processed. On the other hand, we can estimate timeintervals and time scales in which these differences aremore pronounced.By introducing the asymmetry index we can staticallyquantify the differences between Go and No-Go trials.First, by combining entropy and complexity we can de-termine in which interval of the trial it is possible toverify any difference between trial types. For some in-tervals, only one of the two measures presents a signifi-cant difference between Go and No-Go trials which meansthat it is important to calculate both. Second, by us-ing different values of the time embedding we can findresponse-specific differences earlier in time than by us-ing the average potential. We can also estimate whatare the important time scales of the information processby verifying which time delays maximizes the asymmetryindex.This means that the information theory quantifiers andtheir asymmetry index can be employed together with average event-related potentials to characterize response-specific brain activity. Our results open new avenues inthe investigation of response-specific or stimulus-specificbrain activity. Moreover, the method is potentially usefulto quantify other features of the task such as differencesbetween intervals of time and between regions.
Acknowledgments
The authors thank FAPEAL, UFAL, CNPq (grant432429/2016-6) and CAPES (grant 88881.120309/2016-01) for financial support. [1] O. Sporns, ed.,
Networks of the brain (The MIT Press(Cambridge), 2011).[2] S. L. Bressler, R. Coppola, and R. Nakamura, Nature , 153 (1993).[3] H. Liang, M. Ding, R. Nakamura, and S. L. Bressler,Neuroreport , 2875 (2000).[4] A. Brovelli, M. Ding, A. Ledberg, Y. Chen, R. Nakamura,and S. L. Bressler, Proc. Natl. Acad. Sci. USA , 9849(2004).[5] R. F. Salazar, N. M. Dotson, S. L. Bressler, and C. M.Gray, Science , 1097 (2012).[6] N. M. Dotson, R. F. Salazar, and C. M. Gray, The Jour-nal of Neuroscience , 13600 (2014).[7] A. Ledberg, S. L. Bressler, M. Ding, R. Coppola, andR. Nakamura, Cerebral cortex , 44 (2007).[8] C. Shannon and W. Weaver, The mathematical theoryof communication (Champaign, IL: University of IllinoisPress, 1949).[9] P. W. Lamberti, M. T. Mart´ın, A. Plastino, and O. A.Rosso, Physica A: Statistical Mechanics and its Applica-tions , 119 (2004).[10] O. A. Rosso, H. A. Larrondo, M. T. Mart´ın, A. Plas-tino, and M. Fuentes, Physical Review Letters , 154102(2007).[11] L. Zunino, M. C. Soriano, and O. A. Rosso, Phys. Rev.E , 046210 (2012).[12] H. Xiong, P. Shang, J. He, and Y. Zhang, Nonlinear Dy-namics , 1673–1687 (2020).[13] C. Bandt and B. Pompe, Physical review letters , 174102 (2002).[14] F. Montani, O. A. Rosso, F. S. Matias, S. L. Bressler,and C. R. Mirasso, Phil. Trans. R. Soc. A , 20150110(2015).[15] N. Lotfi, T. Feliciano, L. A. Aguiar, T. P. L. Silva, T. T.Carvalho, O. A. Rosso, M. Copelli, F. S. Matias, andP. V. Carelli, arXiv preprint arXiv:2010.04123 (2020).[16] O. Rosso, M. Martin, A. Figliola, K. Keller, and A. Plas-tino, Journal of neuroscience methods , 163 (2006).[17] F. Montani, R. Baravalle, L. Montangie, and O. A.Rosso, Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences ,20150109 (2015).[18] F. Montani, E. B. Deleglise, and O. A. Rosso, PhysicaA: Statistical Mechanics and its Applications , 58(2014).[19] H. M. R. L´opez-Ruiz and X. Calbet, Physics Letters A , 321 (1995).[20] I. Grosse, P. Bernaola-Galv´an, P. Carpena, R. Rom´an-Rold´an, J. Oliver, and H. E. Stanley, Phys. Rev. E ,041905 (2002).[21] A. P. M.T. Martin and O. Rosso, Physica A: StatisticalMechanics and its Applications , 439 (2006).[22] F. S. Matias, L. L. Gollo, P. V. Carelli, S. L. Bressler,M. Copelli, and C. R. Mirasso, NeuroImage , 411(2014).[23] Y. Zhang, Y. Chen, S. L. Bressler, and M. Ding, Neuro-science156