Dynamics in cortical activity revealed by resting-state MEG rhythms
J. Mendoza-Ruiz, C. E. Alonso-Malaver, M. Valderrama, O. A. Rosso, J.H. Martínez
DDynamics in cortical activity revealed by resting-state MEG rhythms
J. Mendoza-Ruiz, C. E. Alonso-Malaver, M. Valderrama, O. A. Rosso, and J.H. Martinez
2, 4, a) Department of Statistics, Universidad Nacional, Cr 45 Biomedical Engineering Department, Universidad de los Andes, Cr 1 Instituto de Física, Universidade Federal de Alagoas (UFAL), BR 104 Norte km 97, 57072-970, Maceió, Alagoas,Brasil. Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain. (Dated: July 10, 2020)
The brain is a biophysical system subject to information flows that may be thought of as a many-body architecture witha spatio-temporal dynamics described by its neuronal structures. The oscillatory nature of brain activity allows thesestructures (nodes) to be described as a set of coupled oscillators forming a network where the node dynamics, and that ofthe network topology can be studied. Quantifying its dynamics at various scales is an issue that claims to be exploredfor several brain activities, e.g., activity at rest. The resting-state (RS) associates the underlying brain dynamics ofhealthy subjects that are not actively compromised with sensory or cognitive processes. Studying its dynamics is highlynon-trivial but opens the door to understand the general principles of brain functioning, as well as to contrast a passivenull condition versus the dynamics of pathologies or non-resting activities. Here we hypothesize about how could bethe spatio-temporal dynamics of cortical fluctuations for healthy subjects at RS. To do that, we retrieve the alphabet thatreconstructs the dynamics (entropy/complexity) of magnetoencephalograpy (MEG) signals. We assemble the corticalconnectivity to elicit the dynamics in the network topology. We depict an order relation between entropy and complexityfor frequency bands that is ubiquitous for different temporal scales. We unveiled that the posterior cortex conglomeratesnodes with both stronger dynamics and high clustering for α band. The existence of an order relation between dynamicproperties suggests an emergent phenomenon characteristic of each band. Interestingly, we find the posterior cortex asa domain of dual character that plays a cardinal role in both the dynamics and structure regarding the activity at rest. Tothe best of our knowledge, this is the first study with MEG involving information theory and network science to betterunderstand the dynamics and structure of brain activity at rest for different bands and scales. Studying the RS dynamics is highly non-trivial but opensthe door to understand the general principles of brainfunctioning. A relevant question is how much informationthe cortical fluctuations convey among neural structures.Entropy and complexity are candidates to evaluate the in-formation content of MEG signals of healthy subjects. Wepropose to capture the dynamics of cortical structures andthat of the functional network at different bands for sev-eral time-scales, via ordinal patterns and clustering coeffi-cient. We evidence an order relation between entropy andcomplexity regarding brain rhythms. We unveil the poste-rior cortex as the one that conglomerates structures withhigh levels of both dynamics and clustering. Our resultsconfirm the emergence of certain information processingtypical of each band with topographical localization at theoccipital lobe.
I. INTRODUCTION
The human brain is probably one of the most complex sys-tems we face, being this main reason for its study and fascina-tion about its dynamics and how different activity organizationis carried out. At present days, we could say that the structureof the human brain (anatomy) is well known, however, as far a) Electronic mail: [email protected] as, its dynamics and how it creates thoughts, processes emo-tions, and perceptions, it is not fully understood. The devel-opment of new image acquisition and processing technologieshave made it possible to look inside a living brain and see it atwork. The brain works by generating small electrical imbal-ances in neural membranes. Functional images reveal whichareas are most active. This can be done by directly measuringelectrical activity (EEG), by capturing magnetic fields createdby electrical activity (MEG), or by measuring metabolic sideeffects such as alterations in glucose uptake (PET) and bloodflow (fMRI).The brain is composed of the cerebrum, cerebellum, andbrainstem . The cerebrum is the dominant part of the brainand, it is the large pinky-gray wrinkled structure that formsmore than three-quarters of the brain’s total volume. The cere-brum is divided into left and right hemispheres, which arelinked by a “bridge” of nerve fibers, the corpus callosum. Thecerebellum is located under the cerebrum, its main functionis to coordinate muscle movement, maintain posture, and bal-ance. The brain’s physical structure broadly reflects its mentalorganization. In general, higher mental processes occur in theupper regions, while the brain’s lower regions take care of ba-sic life support .The uppermost brain region, the cerebral cortex, is mostly in-volved in conscious sensations, abstract thought processes,reasoning, planning, working memory, and similar highermental processes. The limbic areas on the brain’s innermostsides, around the brainstem, deal largely with more emotionaland instinctive behaviors and reactions, as well as long-termmemory. The thalamus is a preprocessing and relay center, a r X i v : . [ q - b i o . N C ] J u l primarily for sensory information coming from lower in thebrainstem, bound for the cerebral hemispheres above. Movingdown the brainstem into the medulla are the so-called “vegeta-tive” centers of the brain, which sustain life even if the personhas lost consciousness. The brain’s vertical zonation movesfrom high-level mental activity in the cerebral cortex grad-ually through to more basic or “primitive” lower functions,especially the autonomic centers of the medulla in the lowerbrainstem that deal with vital body functions, such as breath-ing and heartbeat .Structurally, the left and right cerebral hemispheres lookbroadly similar. Functionally, however, speech and language,stepwise reasoning and analysis, and certain communicatingactions are based mainly on the left side in most people.Since nerve fibers cross from left to right at the base of thebrain, this dominant left side receives sensory informationfrom and sends messages to, muscles on the right side of thebody. Meanwhile, the right hemisphere is more concernedwith sensory inputs, auditory and visual awareness, creativeabilities, and spatial-temporal awareness .The cerebral cortex is the outer layer of the brain’s most dom-inant part, the cerebrum. It is the bulging wrinkled surfacewe see when looking at the brain from any angle. It is com-monly known as gray matter from its color, which contrastswith the white matter in the layer below. Bulges and grooveshelp divide the cortex into four paired lobes: frontal, tempo-ral, parietal, and occipital. The main and deepest groove is thelongitudinal fissure that separates the cerebral hemispheres.The highly convoluted sheet of gray matter that constitutesthe cerebral cortex varies in thickness from about 2 mm to 5 mm . Estimates of its cell numbers vary from 10 to more than50 × neurons and about 5 to 10 times this number of glial(supporting) and other cells .The human cortex contains a distinct pattern of neuron types.Cortical neurons receive and send signals to other brain ar-eas, including other parts of the cortex. This to and from ofmessages keeps all parts of the brain aware of what is go-ing on elsewhere. Neurons in the cortex are “head down” –their receiving parts (dendrites) point up to the surface, whilethreads that carry messages to other cells (axons) are orienteddown. Some axons extend below the cortex and form part ofthe “white matter” connective tissue that carries informationto distant brain areas. Other axons travel through the lowerlayers of the cortex to connect with other cortical cells .There are over a thousand types of brain cells, which fall intotwo broad groups: neurons and glial cells. Neurons send elec-trical signals, or “fire”, in response to stimuli. There are about86 × neurons in an average human brain and ten timesas many glial cells. Neurons can be categorized structurallyaccording to the location of the cell body in relation to theaxon and dendrites, and also the number of dendrites and axonbranches. In the cortex, one neuron may receive signals frommany thousands of other neurons via its multitudinous branch-ing dendrites. Signals are conducted to the soma, around this,and then away along the axon–always by the cell membrane.Glial cells give physical support to neurons, but they are alsothought to influence neurons’ electrical activity. They providephysical support for the thin dendrites and axons that wind their way around the neural network and supply nutrition forneurons in the form of sugars and raw materials for growthand repair . Synapses are communication sites where neu-rons pass nerve impulses among themselves. Many neuronsdo not actually touch one another, but pass their signals viachemicals (neurotransmitters) across an incredibly thin gap,called the synaptic cleft. Synapses are divided into types ac-cording to the sites where the neurons almost touch, e.g., thesoma, the dendrites, the axons, and tiny narrow projectionscalled dendritic spines .A nerve impulse or signal can be thought of as a tiny, brief“spike” of electricity traveling through a neuron. At a morefundamental level, it consists of chemical particles movingacross the cell’s outer membrane, from one side to the other.Nerve signals are composed of a series of discrete impulses,also known as action potentials. A single impulse is causedby a traveling “wave” of chemical particles called ions, whichhave electrical charges and are mainly the minerals sodium,potassium, and chloride. In the brain, and throughout thebody, most impulses in most neurons are of the same strength,about 100 mV . They are also of the same duration, aroundone millisecond, but travel at varying speeds. The informationthey convey depends on how frequently they pass in terms ofimpulses per second, where they came from, and where theyare heading .In the normal brain, an action potential travels down the axonto the nerve terminal where a neurotransmitter is released. Atthe postsynaptic membrane, the neurotransmitter produces achange in the membrane conductance and transmembrane po-tential. If the signal has an excitatory effect on the neuronit leads to a local reduction of the transmembrane potential(depolarization) and it is called an excitatory postsynaptic po-tential (EPSP), typically located in the dendrites. If the signalhas an inhibitory effect on the neuron it leads to local hyper-polarization, also called an inhibitory postsynaptic potential(IPSP), typically located on the cell body of the neuron. Thecombination of EPSPs and IPSPs induces currents that flowwithin and around the neuron with a potential field sufficientto be recorded on the scalp. These ionic currents are in therange of nano ampere (10 − A ). When it circulates, generatemagnetic induction that can be measured on the scale of fem-toteslas (10 − T ) .The electroencephalography (EEG) and the magnetoen-cephalography (MEG) are the most extended non-invasivephysiologic techniques that allow one to measure electricaland magnetic activity generated by the brain on the scalp sur-face. EEG and MEG are different but complementary tech-niques recorded at the scalp for observing brain electromag-netic activity (see the review by Sylain Baillet ).On the one hand, EEG measures the electrical activity in or-der of microvolts ( µ V ), due to the difference of extracellularpotential from cortical groups of pyramidal neurons. On theother hand, MEG measures the magnetic activity, in the scaleof femtoteslas ( f T ) of large axons in the same group of neu-rons. These signals recorded on the scalp have a poor rela-tionship with the spiking activity of individual neurons. Theyare spatial-temporally smoothed versions of cortical neuronalactivity under an area of approximately 10 cm and, can ac-curately detect brain activity at the time resolution of a sin-gle millisecond (ms). Therefore, mental states would emergefrom the dynamical interaction between multiple physical andfunctional levels, giving origin to the oscillatory rhythmicbrain activity. They are of functional importance to under-stand how information is processed in the brain, highlightingoscillations bands like: delta δ ∈ [ . − ) Hz , θ ∈ ( − ) Hz , α ∈ ( − ) Hz , β ∈ ( − ) Hz and, γ ≥ Hz .If we have an observational set of measures (MEG time se-ries, X ( t ) ) from a dynamical system whose evolution can betracked through time, a natural question arises: How much in-formation these observations encoding about the dynamics ofthe underlying system (i.e., the brain dynamics). We adoptedan Information Theory point of view . The information con-tent of the system is typically evaluated via a probability dis-tribution function (PDF) P describing the apportionment ofthe observable quantity (the time series X ( t ) ). Quantify-ing the information content of a given observable is there-fore largely tantamount to characterizing its PDF. This is of-ten done with a wide family of measures called InformationTheory Quantifiers (ITQ). The ITQ can characterize relevantproperties of the PDF associated with the time series, and rep-resent metrics of the space of PDF’s for the data set, allow-ing to compare different sets and classify them according tothe properties of the underlying processes (deterministic vsstochastic) .The temporal brain dynamics is our focus of interest and, themeasured data are the time series X ( t ) of MEG. Metrics thattake the temporal order of observations explicitly into accountare of our interest. That is, our approach is fundamentally a“causal" (the data sequence determines the PDF) rather than“statistical" one (the correlation between successive values aredestroyed or not consider in the construction of the PDF). Wefollow the Bandt and Pompe methodology which is a sim-ple and robust symbolic procedure that takes into account thetime causality of X ( t ) (causal coarse-grained methodology)by comparing neighboring values in the time series (permuta-tion BP-PDF).The ITQ selected are the Shannon Entropy , S [ P ] and,the effective Statistical Complexity Measure , C [ P ] , whichwill be evaluated using permutation BP-PDF. Entropy is a ba-sic quantity with multiple field-specific interpretations; for in-stance, it has been associated with the disorder, state-spacevolume, and lack of information . When dealing with in-formation content, the Shannon entropy is often considered asthe foundational and most natural one . In contrast to in-formation content, there is not a universal definition of com-plexity. Between two special instances of perfect order andhigh entropy, a wide range of possible degrees of physicalstructure exists that should be reflected in the features of theunderlying probability distribution P . One would like to as-sume that the degree of correlational structures would be ade-quately captured by some functional C [ P ] in the same way thatShannon’s entropy S [ P ] “captures" randomness. The ordi-nal structures present in the process are not quantified by ran-domness measures, and consequently, measures of structuralcomplexity are necessary for a better understanding (charac-terization) of the system dynamics represented by their time series . The opposite extremes of perfect order and maximalrandomness are very simple to describe because they do nothave any structure. The complexity should be zero in thesecases. At a given distance from these extremes, a wide rangeof possible ordinal structures exists.Complexity can be characterized by a certain degree of organi-zation, structure, memory, regularity, symmetry, and patterns . The complexity measure does much more than satisfy theboundary conditions of vanishing in the high- and low-entropylimits. In particular, the maximum complexity occurs in theregion between the system’s perfectly ordered state and theperfectly disordered one. Complexity allows us to detect es-sential details of the dynamics, and more importantly to char-acterize the correlational structures of the orderings presentin the time series. A suitable measure of complexity can bedefined as the product of a measure of information and a mea-sure of disequilibrium (i.e., some kind of distance from theequilibrium PDF P e to the accessible actual states of the sys-tem P ) . In particular, Rosso and coworkers introducedan effective Statistical Complexity Measure (SCM) C [ P ] , thatcan detect essential details of the underlying dynamical pro-cesses.Dynamics is of high relevance, but more recently, the structureand function of the brain have begun to be investigated using“Network Science” , offering a large number of quantita-tive tools and properties, thus greatly enriching the set of ob-jective descriptors of brain structure and function available toneuroscientists . The link between “network science” and“neuroscience” has shed light on how the entangled anatomyof the brain is, and how cortical activations may synchronizeto generate the so-called functional brain networks. Withinthis context, complexity appears to be the bridge between thetopological and dynamical properties of biological systemsand more specifically, the interplay between the organizationand dynamics of functional networks. Particularly, how corti-cal activations can be understood as an output of a network ofdynamical systems that are intimately related to the processesoccurring in the brain.Early neuroimaging and electrophysiological studies wereusually aimed at identifying task-specific areas of activation,as well as local patterns varying over time. Nowadays, thereexists a common knowledge that task-related brain activityis temporally multiscale and spatially extended, in the sameway as networks of coordinated brain areas are continuouslyformed and destroyed. The studies of functional brain activityhave focused on identifying the specific nodes forming thesenetworks and on characterizing the metrics of connectivitybetween them. The underlying hidden hypothesis being thateach node, which constitutes a coarse-grained representationof a given Region of Interest (ROI), makes a unique contribu-tion to the whole. In this way, functional neuroimaging ini-tially integrated the two basic ingredients of early neuropsy-chology: the localization of cognitive functions in specializedbrain modules and the role of connection fibers in the integra-tion of them.From the structural point of view, the human brain can be un-derstood as a network of cells forming a massively parallelsystem, organized to carry out three major functions: a) com-putation, b) information storage and transport, and c) commu-nication among computational structures. The brain archivesimpressively high levels of computational resources by adopt-ing efficient architectures, involving the timing of signals andthe representation of information with energy-efficient codes,distributing signals appropriately in space and time. Corti-cal activations allow being studied in a two-face direction de-scribed above: by looking at the dynamics of cortical sig-nals, or by taking into account the dynamics occurring in thebrain connectivity. Brain dynamics characterization has beenused to explore the reorganization of networks in mild cog-nitive impairment ; the modularity of connectivity patternsin epilepsy brain networks and normal subjects ; the effectsof memory in brain networks in old and young individuals,the interchange of information between brain hemispheres inresting-state ; the characterization of visuomotor/imaginarymovement in EEG ; to name a few. From the perspective ofITQ, they were used to characterize and classify EEG recordsfrom control and epileptic patients ; it has also been ap-plied to differentiate processing information zones for sub-jects with Alzheimer at several frequency bands ; to dis-criminate imagined and non-imagined tasks in motor cortexarea and its relation with rhythmic oscillations ; to evi-dence the irreversibility aspect of EEG at resting-state andepilepsy ; to unveil a relationship between the dynamics ofelectrophysiological signals and the brain network structure ; and even to propose a new ordinal-structure methodologyto better account for the information transit between brain sig-nals .While the applicability of these methodologies spans over awide range of neural phenomena, these applications are alsoof importance when concerning the activation of healthy sub-jects that are not actively compromised with the sensory orcognitive processes. This fact allows to contrast results frompathologies or non-resting activities versus, a passive null con-dition also called Resting-State (RS). The RS becomes use-ful to observe the underlying brain dynamics under normalM/EEG, specifically for α rhythm, which accounts for therange of frequencies with the highest energy in the spectrum . The dynamics and structure of RS have been exploredall over the last decade. However, the relation between dy-namics and topology of the brain at rest is still poorly un-derstood. It is in this scenario where this work lies. Wehypothesized about how could be the information content ofthe MEG time series for a healthy group of subjects at RS.Specifically, how the spatio-temporal complexity of corticalfluctuations behaves. To do that, we captured the dynamicalproperties of oscillations at the level of ROIs. We measurethe dynamical properties of both the entropy and complexitythrough symbolic representation and computed structural fea-tures extracted from the connectivity pattern. We evidencedentropy-complexity correlations at different temporal scales.Our results depict an order relation among dynamical param-eters for several frequency bands. We unveiled the occipitallobe as the one with higher levels of complexity in the α band.We detected cortical regions of high levels of network cluster-ing around the same occipital lobe. This fact highlights therole of occipital lobe where cortical regions influence both the dynamics (entropy/complexity) and the structure (clustering).Our manuscript is organized as follows: Section II describesthe MEG data set from the Human Connectome Project . Sec-tion III is divided into three parts. The first accounts for theextraction of the information content from cortical X ( t ) , thesecond is about the computation of the permutation entropy( H ) and statistical complexity ( C ) for X ( t ) , the last one fordescribing the statistical coherence to gather functional net-works, and network properties, i.e., clustering ( c w ). SectionIV introduces the results of the multiscale analysis in severalfrequency bands, with special attention on the α rhythm. InSection V we related the results with structure and dynamics.Section VI is for conclusions, II. DATASET
Data set consists of MEG time series X ( t ) . In this pre-liminary study, 190 magnetometers were located in the scalpsof 40 healthy adult subjects (21 male, 19 female). The rangeof age was chosen to represent adults beyond the age of ma-jor neurodevelopmental changes, and before the onset of neu-rodegenerative conditions. Cortical signals were collected us-ing a MAGNES 3600 system (4D Neuroimaging San Diego)housed in a magnetically shielded room, with subjects lay-ing down with open eyes, instructed to relax, and fixationmaintained on a projected red crosshair on a dark background.Magnetic field measures were taken in empty rooms to avoidexcessive environmental noise and to estimate noise base lev-els. Data preprocessing was conducted to evaluate signals ac-curacy, identify low-quality sensors, avoid sharp increases innoise levels, and to confirm rough linearity in the range ofsampling frequency. Correlations among neighbor sensors,variance ratio, and z-score values were conducted to detectflawed sensors. The procedure allowed us to identify somelow-quality channels, which were then used in an Iterative In-dependent Component Analysis to identify other low-qualitysignals. The cardiac, ocular, and muscular activity were fil-tered from brain signals to remove artifacts. 20 consecutivetrials of 1018 samples each, were taken shaping the time seriesfor all individuals. Each X ( t ) is of M = Hz . The experiment is reportedin the Human Connectome Project. See detailed explanationsin Van Essen’s, and Larson’s work . We tested weaklystationarity of signals with the Augmented Dicky-Fuller rou-tine before using the band-pass filter to gather the bands( θ , α , β , and γ ). All computations were performed with thestatistical software R . 3.6.1. III. METHODS
1. Bandt-Pompe methodology for PDF
We use the Bandt-Pompe methodology to associate a timecausality probability distribution to a time series under study.This methodology takes into account a suitable partition basedon ordinal patterns obtained by comparing neighboring seriesvalues. For a given time series X ≡ { x t , t = , . . . , T ; x t ∈ R } of length T , we identified the M = T − ( D − ) overlappingsegments X s = ( x s , x s + , x s + , . . . , x s +( D − ) ) (1)of length D ∈ N , D ≥ r , r , r , . . . , r ( D − ) such that x s + r ≤ x s + r ≤ x s + r ≤ · · · ≤ x s + r ( D − ) (2)The corresponding D -tuples (or words) π =( r , r , r , . . . , r ( D − ) ) are symbols corresponding the originalsegments, and can be assumed any of the D ! possible permu-tations of the set { , , , . . . , ( D − ) } . Then the permutationentropy (Shannon entropy) of the Bandt-Pompe PDF ( Π ( D ) )is defined as S [ Π ( D ) ] = − ∑ { π } p ( π ) ln ( p ( π )) , (3)where { π } represents the summation over all the D ! possi-ble permutations of order D . p ( π ) ≥ π and clearly sat-isfied ∑ { π } p ( π ) =
1. The BP-PDF, Π ( D ) is invariant beforemonotonic transformations of the time series values. The op-timal value of the pattern length D is strongly related with thephenomenology of the event under study and the availabilityof the data, however, as a rule of thumb, we choose the max-imum D value such that T (cid:29) D ! in order to obtain a goodstatistic.Let’s see an example of the procedure to obtain the BP-PDF.Figure. 1.(a), depicts the finite collection of samples: X ( t ) = {− . , , , , , , , , , } ,the time series length is T = D =
3, we can form M = T − ( D − ) = D =
3, then we have 6 possible patterns, given by: π = { x ≤ x ≤ x } , π = { x ≤ x ≤ x } , π = { x ≤ x ≤ x } , π = { x ≤ x ≤ x } , π = { x ≤ x ≤ x } , π = { x ≤ x ≤ x } . Fig. 1.(b), shows the eight obtained 3-tuples.The pattern probability are: p ( π ) = p ( π ) = / p ( π ) = p ( π ) = p ( π ) = p ( π ) = /
8, which conforms the obtainedBP-PDF Π ( D = ) showed in Fig. 1.(c).
2. Dynamics
The randomness in the system is measured trough Normal-ized Permutation Entropy defined as H [ P ] = S [ P ] S max = − S max N ∑ j = p j ln ( p j ) , (4)where P = { p j = , · · · , N } denote the PDF representing theactual state of the system. S max = S [ P e ] = ln N = ln ( D ! ) is anormalization constant, obtained from the equilibrium prob-ability distribution P e = { p j = / N , ∀ j = , · · · , N } . In thisway, 0 ≤ H [ P ] ≤
1, being the extreme values for a completeordered and uncorrelated random system. H [ P ] allows com-parisons between the proportion of information contained in a Figure 1. Ordinal patterns extraction procedure for D =
3. (a) Orig-inal signal X ( t ) , with length T =
9, (b) Ordinal patterns extractedfrom X ( t ) , (c) Ordinal patterns probability distribution (BP-PDF). system with different amount of states.A second measure that allows characterizing the emergence ofnew properties and organization in the system is the statisticalcomplexity, such that 0 ≤ C [ P ] ≤ C [ P ] = H [ P ] · Q [ P , P e ] , (5)where Q [ P , P e ] stands for normalized Jensen-Shannon diver-gence, a non-euclidean distance between observed and uni-form probability distributions: Q [ P , P e ] = Q (cid:26) S (cid:20) ( P + P e ) (cid:21) − S [ P ] − S [ P e ] (cid:27) , (6)with Q as a normalization constant Q = − (cid:26)(cid:18) N + N (cid:19) ln ( N + ) − ( N ) + ln ( N ) (cid:27) − . (7)Note also, the global character of the both introduced quanti-fiers, entropy (eq. (4)) and complexity (eq. (5)). That is, theirvalues do not change with different orderings of the compo-nent of the PDF.The statistical complexity quantifies the existence of correla-tional structures, giving a measure of the complexity of a timeseries. In the case of perfect order ( H [ P ] = H [ P ] = C [ P ] =
0, meaning that the signal possessesno structure. In-between these two extreme instances, a largerange of possible stages of physical structure may be realizedby a dynamical system .So, the C [ P ] quantifies the disorder, but also the degree of cor-relational structures in the time series. It is evident that thepresent statistical complexity is not a trivial function of theentropy, due to its dependence with two PDF, and it has con-sequences in the ranges that this information quantifier cantake. For a given entropy H value, the statistical complexity C runs on a precise range limited by a minimum C min and amaximum C max values. These extreme values depend only onthe probability space dimension and, of course, on the func-tional form adopted for the entropy and disequilibrium .The temporal evolution of entropy H [ P ] and complexity C [ P ] ,can be analyzed using a two-dimensional diagram calledentropy-complexity plane H × C . In accordance with the sec-ond law of thermodynamics, the entropy grows monotonicallywith time. Then, the entropy H can be regarded as an arrowof time , allowing this form to follow the time evolution of adynamical system or its changes of behavior with the differentcontrol parameters .The H × C -plane, as a diagnostic tool, has shown to be par-ticularly efficient at distinguishing between the deterministicchaotic and stochastic nature of time series since the quan-tifiers have distinctive behaviors for different types of dy-namics. In particular, Rosso and co-workers showed thatchaotic maps have intermediate entropy H , while complexity C reaches larger values, close to those of the limit C max . More-over, similar behavior is still observed when the time series arecontaminated with small or moderate amount of additive un-correlated or correlated noise As we mention previously, ordinal patterns represent a naturalalphabet from the time series and their probability distribution Π ( D ) ≡ P allows the computation of some measures to quan-tify the randomness and complexity in the related system.Dynamical properties were computed for all signals. To con-trol the experiment, each of one was contrasted with 50 sur-rogate versions via the Iterative Amplitude Adjusted FourierTransform (IAAFT) . H × C -plane was generated for alldimensions D to evaluate potential correlations between bothparameters in all bands. Due to the intimate closeness of α rhythm with RS, we focused on quantifies H and C at corti-cal ROI, as well as how they are distributed at local level inthe scalp for this band, in particular at an intermediate obser-vational scale D =
5. Remaining bands are compared at theSupplementary Material.
3. Structure
The statistical coherence C i j ( f ) quantifies the intra-frequency relation between two signals assigning 0 (1) whenthey evolves autonomous (synchronized) at a frequency f . In Eq. (8), S i j ( f ) stands for the cross-spectral densitybetween signals X i ( t ) and X j ( t ) , and S ii ( f ) and S j j ( f ) , fortheir autospectral densities, respectively. C i j ( f ) = | S i j ( f ) | S ii ( f ) S j j ( f ) . (8)The pondered interaction w i j between two signals at a specificband is taken as the average of C i j ( f ) along with its respec-tive frequencies. The pairwise interrelation among n signalsunveils the connectivity structure among cortical ROIs. This connectivity is associated with the adjacency matrix W , W = w i j i (cid:54) = j , . (9) W is a n × n positive and symmetric matrix representing thefunctional network of cortical ROIs, where rows and columns i , j represent the respective nodes, and w i j its link weight. Thelink weight w i j exclusively depends on the activity correlationat certain frequencies. The set { w i j } measures the statisticalinterrelations among pairwise nodes without taking into ac-count causal relations.The network science studies the way by which nodes and theirlinks are organized. It provides a series of associated metricsallowed to quantify the network structure by taking into ac-count their strongest links. To do that, each network is filteredby removing spurious or weakest connections. Therefore, thetopological properties of the resulting graph depend on howmuch connectivity of the inferred network is maintained. Fol-lowing an efficient cost optimization filter, we threshold thematrices by optimizing the balance between the network effi-ciency and its wiring cost .We retrieve the minimum spanning tree of each graph by pre-serving its stronger links. This way, the connection densityhighlights the properties of the network while preserving itssparsity and maximizing the balance between the network ef-ficiency and the link density . Thus we quantify networkfeatures to understand what would be the role of nodes in theprocess of segregation of information and to detect the ROIsof cardinal importance in the cortical wiring. Specifically, wecapture the node clustering and the eigenvector centrality.Clustering is defined as the capacity of a network to formsconnected groups of three nodes. That is the fraction of tri-angles potentially connected around a node. The higher theclustering, the higher the segregation of information in the net-work . The segregation is then associated with the presenceof specialized processing in specific and densely regions of in-terconnected nodes. Eq. (10) stands for the weighted variantof the local clustering c w ( i ) = k i ( k i − ) ∑ j , k ∈ n ( w i j w jk w ki ) / , (10)that takes into account the number of links k i of a node i , andthe weights among nodes i , j , k . A node with high clus-tering is assumed to be one with a high probability of formingtriads with its first neighbors, also linked among them. Oth-erwise, its clustering index is low. The mean clustering indexrepresents a network that, in average, presents clustered con-nectivity.Another role a node has is that by which it facilitates com-munication among specialized regions promoting a functionalintegration of information. Eigenvector centrality ec ( i ) is amicroscale measure that captures the node centrality by con-sidering the global information of W . While c w ( i ) is estimatedby considering its first neighbors, ec ( i ) made use of the impor-tance of all nodes in the networks using spectral methods. W is positive and real-squared, so there exist n real non-negativeeigenvalues. W can be written as W = UKU T , where U is a Figure 2. Entropy probability distributions for D = D = D = D = D = θ , (b) α , (c) β , (d) γ . Entropyis inversely proportional to higher D . matrix of eigenvectors and K a diagonal matrix of its eigen-values. The Perron-Frobenius theorem states that W has aunique largest eigenvalue with an associated eigenvector hav-ing all positive entries. ec is then the normalized version ofthis eigenvector, where ec ( i ) is the centrality of the node i .It implies that a node with a higher ec ( i ) is that one that isconnected to highly connected nodes, also called hubs. c w and ec were identified for frontal, occipital, parietal, andtemporal lobes. For a better comparison, we normalized c w re-spect to the average of a set of 50 surrogate matrices. We takethe mean value of clustering c w ( i ) for all over the 40 subjects.We repeat the process by using the 50 randomized versions c ∗ w ( i ) per subject. The c w n ( i ) is the normalized version ofclustering respect to its rewired versions c w n ( i ) = c w ( i ) / c ∗ w ( i ) .This normalization allows us to focus on structural changes ofnetworks, avoiding variations of the average weights of theconnectivity . In this vein, we keep c w n ( i ) ≥
1, since thisdoes not come from a random hidden structure. Normalizednetwork features were topographically associated with eachlobe, in particular for the α band. Finally, we performed aprincipal component analysis (PCA) between { H ∧ C } and { c w ∧ ec } to scrutinize potential relationships between dynam-ics and structure for each lobe. IV. RESULTS
We introduce results in the following way. We show thebehavior of entropy H and complexity C for all bands at em-bedding dimensions D ranging from 3 to 7. We make knownthe order relation of dynamic parameters in the H × C -planein terms of frequencies and scales. Subsequent results focusedon the α band due to its importance for RS. We focused on thetopographic distribution of H , C for D =
5. Finally, we also in-troduced the structural parameters c w and ec for each lobe and Figure 3. Complexity probability distributions. Similar conventionsof Fig. 2. Complexities of both (a) θ and (b) α decrease inverselyproportional to the dimension D . A change of order occurs whencomplexities decrease for low scales D = , , β and (d) γ . its topographical distribution.Preliminary exploration of signals showed some differenti-ation in the autocorrelation function (ACF) in terms of fre-quency bands. The higher a band, the faster its ACF tends tozero. Oscillatory and sign alternation trait of ACF evidencesome type of seasonal behavior. Partial autocorrelation func-tion (PACF) takes into account inter-lag effects in contrast toACF. The ACF of signals also evidenced a similar behaviorobtained with PACF. This fact suggests that cortical dynam-ics might change on frequency bands for RS. Although thestatistical viewpoint indicates differences in bands, we are in-terested in the causal perspective of the signals’ dynamics. Inother words, we want to evidence whether such differencesare captured through spatio-temporal patterns associated witheach rhythm A. Dynamics
To capture dynamical parameters, we computed the BP-PDF for filtered signals at several dimensions D . The corre-sponding normalized Shannon entropy H and statistical com-plexity C were evaluated. In Fig. 2 and Fig. 3 the histogramPDF of obtained values for entropy H and complexiry C areshown. For a single band, PDFs of entropy are inversely or-dered respect to the dimensions D . The bigger the observa-tional scale D , the less the entropy. For θ band, entropy rangesspan around a order phase from ( H D = ≤ .
2) to an entropyvalue ( H D = ≤ . α band, it goes from( H D = ≈ .
2) up to ( H D = ≤ . β band extents its entropiesfrom ( H D = ≈ .
3) to ( H D = ≈ . D =
3, and the richness of possiblepatterns have to be re-distributed in few degrees of freedom.
Figure 4. H × C -plane for dimensions D = D = D = D = D = θ (red), α (blue), β (yellow), γ (green). Gray curves represent the limits C min and C max . There is an order relation among H and C for all frequencies. In contrast to H , statistical complexity C behaves in a par-ticular way. In principle, Fig. 3.(a) and (b), show a similarorganization pattern as for H . The observational scale D ap-pears to be inversely ordered respect to the complexity levels.This behavior is kept in cortical activity at frequency bands θ and α . For instance, α complexities spans over a range of( C D = ≈ .
2) up to a complexity degree ( C D = ≤ . D = , , β band, breaking it up the order previously maintained. For β , scales D = { , , } , are now directly proportional to theircomplexity levels, for instance, (cid:104) C D = (cid:105) β = . (cid:104) C D = (cid:105) β = . (cid:104) C D = (cid:105) β = . γ band magnifiesthis new ordering by expanding the distances among PDFs of C for different D -scales. Table I summarizes the mean valuesfor H and C for all bands and D ’s.Previous results depict the independent organization of H and C based on D and bands, but nothing says about a potentialrelationship between both parameters. For a single value of H , C may spans over a range limited by C min and C max . Thenwe ask for how the relationship of the entropy and complex-ity evolves for different bands? The entropy-complexity plane H × C allows following the time evolution of a dynamical sys-tem for different D dimensions.A negative relation between H and C occurs for D = H and C are now in the region of positive relationshipsfor D = D =
5, now locate it on the deterministic regionof the H × C -plane (Fig. 4.(c)). This tendency is near the val-ues of maximum complexity for D = D =
7. Note that although the slope ofthe relationship changes from negative (for D =
3) to positive(since D =
4) the order relation among bands is maintainedfor all dimensions. It may suggest the existence of a band-dynamics of its own that is well captured by the Bandt-Pompemethodology. We chose D = D =
3) and an oversampling of symbols due to D = H (and C ) for each ROI,respect to the empirical distribution of those captured fromrandom versions of signals via the Kolmogorov-Smirnov test, Figure 5. Topographical distribution of (cid:104) H (cid:105) and (cid:104) C (cid:105) for α with D =
5. Dashed lines divide the cortical surface into lobes. Node sizes areproportional to the average of entropy and complexity. High dynamical activity are located in posterior parietal cortex and occipital lobe.
Bonferroni corrected. The nonlinear character of a signal isassumed when the comparison rejects the null hypothesis thatthe dynamical parameters came up from time series with un-correlated amplitudes. Nonlinearity was detected for entropyin θ , α and γ oscillations with p − value ≤ . β oscillationsdo not reject the null hypothesis. All bands reject the null hy-pothesis regarding complexity. With the previous, the infor-mation processing at the cortical level suggests to be drivenby a nonlinear system presents in the dynamical properties ofMEG time series.Due to the relevance of α band for RS, we focus our attentionon the topography distribution of H and C in the scalp so asto unveil cortical regions of high levels of dynamics. Specifi-cally, we take the average of entropy (cid:104) H (cid:105) and (cid:104) C (cid:105) and observetheir spatial distribution on scalp (Fig. 5). Remaining bandsare also showed in Fig 8 in Supplementary Material. Notethat H × C -plane has a positive correlation for D =
5, thennodes of higher entropy match with ROIs of higher complex-ity. Parieto-occipital lobe contains the majority of ROIs withhigh levels of H and C . These regions exhibit an incrementof cortical activity under passive attention processes and theyare also associated with visual stimuli perception and spatialrecognition . These results are in accordance with RS exper-iments, which highlight the occipital lobe as the one with themajor activity. In the global perspective, entropy ranges be-tween zero and one. Its mean value for D = (cid:104) H D = (cid:105) alpha = . (cid:104) H (cid:105) and (cid:104) C (cid:105) for remaining bands in Table. I. B. Structure
The dynamic analysis is performed on the activity of in-dependent ROIs’ activations. However, this approximationcannot take into account the statistical dependencies of ROIs’activations. In this sense, another spatial pattern may alsoarise due to these interdependencies of cortical brain signals.Hence, we reconstructed the functional network of the α ac-tivity in order to capture topological features of relevance forinformation processing in RS. We computed the clusteringand eigenvector centrality for all subjects and take their av-erage value c w and ec , respectively. We assessing both nor-malized c w n and ec n by accounting for counterparts from 50rewired networks per subject. Network features were con-trasted with those obtained from randomized matrices usinga Mann-Whitnney test at the node level. Regarding the clus-tering index, p − values were significantly less than 0.05 foroccipital and posterior-parietal regions, and an important por-tion of the temporal lobe. There are no significant differencesin eigenvector centrality. A normalized quantity higher thanone implies a network parameter that is extracted from the real0 Figure 6. Normalized network features. They are allocated to corti-cal lobes: frontal (orange), occipital (green), parietal (cyan), tempo-ral (purple). (a) Clustering c w n is higher than one for all lobes, (b)Eigenvector centrality ec n is lower than one. nature of the system. Otherwise, it would entail a networkfeature that is captured from a random configuration. We al-located features to each cortical lobe as it is shown in Fig. 6.Boxplots of Fig. 6. (a) indicate that clustering architecturearises from the structural organization of functional networkssince normalized values are greater than one. Notice how theoccipital lobe accumulates ROIs with the highest clusteringindex. Interestingly, this region also intensifies its dynamicsfor α band. On the other hand, Fig. 6. (b) suggests that eigen-vector are similar to those extracted from random versions ofnetworks ec ≤ ec ∗ .Having said so, the natural step is to examine how the topo-graphical distribution of the clustering architecture is at thenode level. Fig. 7 depicts the functional network for α high-lighting which nodes belong to each lobe. We highlight thenodes with a high level of clustering. They are dispersedmainly in occipital and temporal lobes, specialized regionsthat process information regarding attention, sensory and vi-sual stimuli, and auditive perception . Several posterior re-gions of the cortex reveal ROIs with active roles for the seg-regation of information due to its high clustering. The factthat those regions reside in different lobes may reflect someintegration type of information concerning the α band. Theconnectivity maps are shown in Fig. 9 of Supplementary Ma-terial for additional bands. V. STRUCTURE AND DYNAMICS
The occipital lobe engages the presence of high dynamicsactivity ( H , C ) and a relevant role of structure ( c w ). For thisreason, we now explore whether this lobe associates somequantifiable relationship among its dynamical parameters andnetwork features. A Principal Component Analysis (PCA)was performed with five variables: entropy H , complexity C ,eigenvector centrality ec , strength s , and clustering c w . Eachvariable contains 190 observations. The average along 40 sub-jects. We incorporate the cortical lobe as a supplementaryvariable related to each ROI. The factorial plane preserves the91 . . . Figure 7. Functional network for α band. Dashed lines divide cortexsurface into lobes. Node sizes are proportional to clustering c w . Highclustering ROIs are located mainly in temporal and occipital lobes.links are drawn with the same width for simplicity. at Supplementary Material. Dynamical parameters are bestexplained since they preserve a variability greater than 99 %.See the row Total variability for H (99.34), and C (99.16) ofTable. II. The network features ec and s are the best associatedwith the first component with 86 .
53% and 66 .
01% of the ex-plained variability. In contrast, ( H , C ) are more related to thesecond component with a 59 .
33 % and 49 .
12 %, respectively.Entropy and complexity are positively related to the secondcomponent, while network parameters are inversely related toit. This indicates linear independence between dynamics andstructure.Notice how Fig. 10 shows how most of the occipital and pari-etal nodes are located at the right-hand side of the plane, ac-cording to the results introduced above. Specifically, the oc-cipital lobe concentrates nodes with both high dynamics andstructural levels. The fact that PCA considers a linear combi-nation of variables and the absence of such a relationship be-tween both classes might hint a nonlinear relationship worth-while to explore in future works.We performed a complimentary analysis. As entropy leadsto complexity, and clustering accounts for significant valuesfor all lobes, we measure the interaction of H and c w . Weperformed linear fits taken into account the node stratificationfor each lobe. Linear fits are far from explaining the entropyvariability throughout a linear relation with c w . R values arelow. Results for linear fits are shown in Fig. 11 and TableIII of Supplementary Material. Nonetheless, these approxi-1mations are preliminary ones and do not imply the absenceof a potential relationship. More studies must be done in or-der to verify any interrelation between introduced dynamicalparameters and topology features. VI. CONCLUSIONS
We have introduced the first study that characterizes the dy-namics and structure of cortical fluctuations of healthy sub-jects in RS for different frequency bands. The cohort of thiswork stands out for being in an age range higher than the ageof major neurodevelopmental changes, and before the onsetof neurodegenerative conditions. We use ordinal patterns as arobust methodology to extract the alphabet that constructs theinformation content of MEG signals. By using informationtheory, we capture the complexity and entropy as dynamic pa-rameters. We found that the clustering coefficient plays an im-portant role in the topology of the functional network createdfrom the synchronization of the signals’ powers at differentfrequencies. We highlight the relevance of the occipital lobeas the domain that concentrates cortical structures that play adual role in both the dynamics and the network structure forthe α band.Ordinal patterns capture information in high-order dimensionsof a nonlinear system. The time scales at which the amountof information is captured from a dynamic system are thoseembedding dimensions. In this sense, the amount of infor-mation contained in the MEG signals increases as we observethe system in low dimensionality. The lower the dimension-ality, the higher the amount of information contained in MEGsignals. The larger the frequency band, the more entropy inthe signals. However, cortical signals have entropy values be-low 0.6, which implies that cortical fluctuations are moder-ately ordered for either all observation scales and bands. Al-though our study focuses on RS, it highlights that similar re-sults were obtained at the microscale level. The work of Mon-tani developed models of spiking neurons with a diversityof different neurocomputational properties of biological neu-rons. Similarly, they find that the degree of order decreases asentropy increases.Resting brain activity assumes an interdependence of multi-ple states that evolve in a non-random manner , whichimplies that there is a complex dynamic that shows the richtemporal structure of cortical activations in RS . We thencapture this complexity as another dynamic parameter of theMEG signals and associate it with their levels of randomnessusing the H × C -plane. Importantly, we found an order rela-tion in the dynamics of cortical fluctuations with marked gapsbetween each oscillation rhythm. The faster the oscillations,the information is processed with higher levels of complexityand entropy.Echegoyen’s recent work discriminates MEG signals fromsubjects with mild cognitive impairment and Alzheimer’s indifferent frequency bands. However, it does not show an or-der relation between each rhythm and the entropy ranges. Atthe same time, Baravalle demonstrated that an order rela-tion appears for slow rhythms. However, this order relation changes for fast oscillations, specifically in EEG of subjectsperforming a visuomotor task. In contrast, our results showthat the H × C -plane differentiates the dynamic of MEG sig-nals with a marked order associated with each band, whichimplies that each rhythm has a preference to occupy specificstates of the system. Interestingly, this order relation remainsindependent of the observation scale, suggesting the existenceof some consistency in the local dynamics associated with theRS.The α rhythm reaches the highest power in RS and associatesthe posterior zone of the cortex. In this sense, the strong dy-namic of the occipital lobe is well captured by the methodol-ogy used in this work. We evaluated the importance of corticallobes that contain ROIs with high levels of information pro-cessing. We find the posterior-parietal and occipital lobes asthose that accumulate cortical regions where information pro-cessing associates high levels of entropy and complexity forthe α band. These results are consistent with those of the re-cent work by Quintero-Quiroz , who analyzed the dynamicsof EEG signals in RS. They found that the posterior zone ofthe cerebral cortex concentrates ROIs with high entropy com-pared to the other zones. It is also interesting to note that thisregion marks the difference between the levels of entropy andcomplexity of MEG signals in healthy subjects with differentcognitive reserves .Overall, the methodology used discriminates spatiotemporalpatterns for different frequencies. The θ band showed a distri-bution in the posterior cortex and some regions of the temporallobe that is associated with the processing of information fromvisual stimuli. The β band highlights regions of the frontallobe, and the γ accentuates some areas of the cortical periph-ery. Interestingly, although this work focused on MEG in RS,the work of Baravalle and coworkers highlights regions ofhigh entropy in areas similar to those found in this work butusing EEG of subjects with visuomotor tasks. A comparisonbetween findings allows us to find certain similarities betweenthese two works. On the one hand, we work with MEG of RSwith open eyes. On the other hand, they work on a visuomotortask of healthy subjects. Therefore, we speculate that visualactivity might have a central role in the topographic distribu-tion of ROIs with high dynamics in the different frequencybands.Regarding network analysis, we find that clustering is of car-dinal importance, while the role of other network parameters,such as eigenvector centrality or strength, is still unclear. Highclustering levels are due to the high density of connectionsbetween functionally related regions of the network in the α band. It also is related to the phenomenon of functional seg-regation, in which specialized regions carry out informationprocessing. In this sense, we find the occipital lobe and sometemporal ROIs as regions that concentrate a high level of clus-tering. Interestingly, clustering has formerly been associatedwith the robustness of a functional network in RS. Specifi-cally, the clustering of functional networks is bound-up to therobustness when occurring loss of information between hemi-spheres in the α band . Although the work shows the im-portance of the occipital lobe in information processing fordynamics ( H , C ) and structure ( c w ), no statistical relationship2was found between these parameters.It is well known that the MEG technique is extremely sen-sitive to diverse factors like skin conductance, arousal, eyemovements, among others that may add noise and lead to ar-tifacts. For this reason, a preprocessing pipeline is continu-ally performed for filtering and cleaning raw data. However,some preprocessing techniques depend on both parametersthat must be tuned ad hoc , and filters that can mitigate thepresence of noise but profoundly influence the results. In thissense, although a high sampling frequency in the signals leadsto a high temporal resolution, it can also induce drawbackswhen capturing ordinal patterns. An oversampling could hidethe real dynamics of a signal, promoting the existence of as-cending and descending patterns mostly associated with peri-odic time series. The excess of these types of patterns woulddecrease the richness of other states that coexist in the signals,creating very ordered probability distributions that lead to lowentropy.Increasing the observation window increases the possiblestates of the system, but does not eliminate the low entropy ofthe distributions. As a consequence, complexity approachesthe theoretical maximum levels for large embedding dimen-sions. This fact may explain why the dynamical parameters inthe frequency bands radically change when increasing the ob-servation dimension. Because the Bandt and Pompe method-ology is sensitive to the observational scale, oversampling canstrongly influence both the observed patterns captured and thesystem dynamics. We take D = .Beyond being barriers, these observations can be taken aschallenges useful for future studies. For example, insteadof taking the amplitudes of the time series, we can considerthe patterns obtained by taking the relative differences be-tween the local maxima, when working with highly sampledsignals. This procedure could avoid oversampling and cap-ture the richness of the patterns. On the other hand, althoughwe did not evidence a statistical relationship between dynam-ics and structure, another study found a relation between en-tropy/complexity and clustering/strength for the cognitive re-serve phenomenon. This fact suggests the need for furtherstudies with other perspectives to give plausible explanationsabout this potential relationship in RS (e.g., using nonlinearcorrelations for connectivity matrices such as information en-tropy or synchronization likelihood).Even so, the potential of the Bandt-Pompe methodology isdemonstrated independent of potential drawbacks in the sig-nals. The methodology has the potential to detect changes inthe topographic distribution of regions with dynamic activity,as well as characterizing rhythms at which the brain operatesin RS. Although many questions remain open, to the best ofour knowledge, this is one of the first studies with MEG sig-nals that shed light on the understanding of the dynamics andstructure around RS activity using a technique that involvesinformation theory, symbolic dynamics, and network science. Converging evidence suggests that Bandt-Pompe methodol-ogy is robust to noise and efficient for its ability to extractknowledge of the dynamics of a system. In this case, wedemonstrate the existence of an order relationship present inthe RS dynamics and strongly marked in the frequency bands.It should be noted that such a relationship is ubiquitous amongdifferent observational scales. Our evidence highlights thecardinal role of the posterior region of cortex in the α band,particularly the occipital lobe. This region conglomeratesneural structures in charge of information processing whoseplay a dual role in dynamics-structure, which might be con-sidered as a fingerprint of the RS. ACKNOWLEDGMENTS
J.H.M. thanks to Colciencias Call
SUPPLEMENTARY MATERIAL
In the present work, we have focused on the dominant roleof the α band in RS, including the adjacent frequency bands( θ and β bands). Given the relevance of the γ band (highfrequencies) in resting-state for visuomotor tasks or for braindisorders, this frequency band has also been included in ouranalysis . This section introduces results for remain-ing embedding dimensions and frequency bands that were nottaken into account in the main document. Respecting dynam-ics approach, the analysis was carried out considering an em-bedding of D = D = D =
7. In the main text, α band is describedfor being this the one that is associated with RS. Here, Fig.8 shows dynamical parameters H and C for remaining fre-quency bands. Cortical lobes are highlighted. θ band revealshigher activity the most in the occipital lobe and few sectorsof the temporal lobe. These regions are associated with vi-sual and auditory stimuli. As it was mentioned in the maintext, α associates posterior-parietal and occipital lobes as ac-tivated regions. Cortical activity under β band is located inthe frontal lobe and some ROIs in the anterior parietal lobe,which is related to motor processes during RS. γ band focusesthe activity in the outer periphery of frontal, occipital, andsmall regions of temporal lobes. Table. I deploys mean valuesof H and C for remaining frequency bands and dimensions.Regarding the architecture of functional networks, normalizedfeatures of eigenvector centrality ec and weighted clusteringcoefficient c w were described for α in the main text. Theyappeared to be conglomerated around the occipital lobe andpart of the temporal one. Here, Fig. 9 depicts the topograph-ical distribution of clustering. Networks related to θ and α bands show a similarity in their cortical distribution of clus-tering. ROIs, of high functional segregation, are located at the3 Figure 8. (cid:104) H (cid:105) and (cid:104) C (cid:105) for bands. Columns (a) for θ , (b) for α , (c) for β , (d) for γ . Upper panel for (cid:104) H (cid:105) . Bottom panel for (cid:104) C (cid:105) . Circles sizesare proportional to the dynamic parameter of each band. Color bar relates the dynamical parameters all along the frequency bands.Table I. Mean values (cid:104) H (cid:105) and (cid:104) C (cid:105) for bands and dimensions D .Parameter Band Dimension ( D )3 4 5 6 7 (cid:104) H (cid:105) θ α β γ (cid:104) C (cid:105) θ α β γ occipital and temporal lobes. These lobes are also involvedin cortical activity associated with dynamic parameters (Fig.8). Network related to β band shows greater clustering levelsin occipital lobes. Surprisingly, only the left-temporal regionaccumulates ROIs of high clustering. Finally, γ band networkreproduces a topographical distribution similar to β band. Af-ter the analysis of dynamics and structure, we focused ourattention on inspecting a potential relationship between them.Firstly, we bet for a PCA for dynamic parameters { H , C } , andstructural features ec , s , c w . Table. II describes the first andsecond components of PCA. They explain less than 100% ofthe variability related to dynamical parameters, and more than80% from topological features. The first of two componentsexplain the 91 .
6% of the variability, see the axis of Fig. 10that shows the factorial plane of PCA. In Fig. 10, sensors
Figure 9. Functional networks for frequency bands. (a) θ , (b) α , (c) β , (d) γ . Node sizes are proportional to (cid:104) c w (cid:105) . Table II. Variability % explained by PCA.Component Parameter
H C ec s c w st nd located in the occipital lobe are colored by green. They arelocated on the right-hand side of the first component. Most ofthe parietal ROIs (cyan) are also located in the same sector ofthe factorial plane. Both lobes concentrate cortical regions ofhigh dynamics and clustering in accordance with the directionof vectors, which are disposed of respect to the positive partof the first component. Sensors at the frontal lobe do not ex-hibit any relationship with dynamics or structural parameters.In a second approximation, we performed linear fits betweenthe entropy and clustering index. Fig. 11 shows linear fits forall lobes in the α band. Each lobe accounts for a canonical y = b + m x model of its own. For instance, the occipital lobeaccounts for: H = . − . c w . Table. III, introduces thestatistics of linear models performed on each lobe. Models donot explain enough entropy variability ( R ). DATA AVAILABILITY
The experimental MEG data set correspond to “HumanConnectome Project”. See detailed explanations in Van Es-
Table III. Linear model statistics between H and c w for each lobe.Region Statistic R p-value Frontal 0.19 0.005Occipital 0.11 0.002Parietal 0.24 0.005Temporal 0.04 0.063 sen’s, and Larson’s work . The data that support the find-ings of this study are available within the article.
REFERENCES R. Carter, S. Aldrige, M. Page, and S. Parker,
The human brain book , 3rded. (Penguin Random Hause, DK London, 2019). A. Jackson and D. Bolger, “The neurophysiological bases of EEG and EEGmeasurement: A review for the rest of us,” Psychophysiology , 1061–1071 (2014). M. Cohen, “Where does EEG come from and what does it mean?” Trendsin neurosciences , 208–218 (2017). M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounas-maa, “Magnetoencephalography - theory, instrumentation, and applicationsto noninvasive studies of the working human brain,” Reviews of modernPhysics , 413–497 (1993). S. Baillet, “Magnetoencephalography for brain electrophysiology andimaging,” Nature neuroscience , 327–339 (2017). C. S. Herrmann, D. Strüber, R. F. Helfrich, and A. . K. Engel, “EEG oscil-lations: from correlation to causality,” International Journal of Psychophys-iology , 12–21 (2016). R. M. Gray,
Entropy and Information Theory , 2nd ed. (Springer Science &Business Media, 2011). O. A. Rosso, H. A. Larrondo, M. T. Martín, A. Plastino, and M. Fuentes,“Distinguishing noise from chaos,” Physical Review Letters , 154102(2007). C. Bandt and B. Pompe, “Permutation entropy: a natural complexity mea-sure for time series,” Physical Review Letters , 174102 (2002). C. E. Shannon, “A mathematical theory of communication,” Bell SystemTechnical Journal , 379–423, 623–656 (1948). C. E. Gray and W. Weaver,
The Mathematical theory of communication (University of Illinois Press, 1949). P. W. Lamberti, M. T. Martín, A. Plastino, and O. A. Rosso, “Intensiveentropic non-triviality measure,” Physica A: Statistical Mechanics and itsApplications , 119–131 (2004). J. B. Brissaud, “The meanings of entropy,” Entropy , 68–96 (2005). D. P. Feldman and J. P. Crutchfield, “Measures of statistical complexity:Why?” Physics Letters A , 244–252 (1998). D. P. Feldman, C. S. McTague, and J. P. Crutchfield, “The organizationof intrinsic computation: Complexity-entropy diagrams and the diversityof natural information processing,” Chaos: An Interdisciplinary Journal ofNonlinear Science , 043106 (2008). R. Lopez-Ruiz, H. L. Mancini, and X. Calbet, “A statistical measure ofcomplexity,” Physics Letters A , 321–326 (1995). L. da F. Costa, F. A. Rodrigues, G. Travieso, and P. R. V. Boas, “Charac-terization of complex networks: A survey of measurements,” Advances inPhysics , 167–242 (2007). E. Bullmore and O. Sporns, “Complex brain networks: graph theoreti-cal analysis of structural and functional systems,” Nature Reviews Neu-roscience , 186–198 (2009). M. Rubinov and O. Sporns, “Complex network measures of brain connec-tivity: uses and interpretations,” NeuroImage , 1059–1069 (2010). J. M. Buldú, R. Bajo, N. C. F. Maestú, I. Leyva, P. Gil, I. Sendiña-Nadal,J. A. Almendral, A. Nevado, F. del Pozo, and S. Boccaletti, “Reorgani-zation of funtional networks in mild cognitive impairment,” Plos One ,e19584 (2011). Figure 11. Linear fits for cortical lobes in α . (a) Frontal, (b) occipital, (c) parietal, (d) temporal lobe. Points outside the confidence intervalsshow that models cannot explain the variability in entropy H through with c w , as explanatory variable. M. Chavez, M. Valencia, V. Navarro, V. Latora, and J. Martinerie, “Func-tional modularity of background activity in normal and epileptic brain net-works,” Physical Review Letters , 118701 (2010). J. H. Martínez, J. Buldú, D. Papo, F. D. V. Fallani, and M. Chavez, “Roleof inter-hemispheric connections in functional brain networks,” ScientificReports , 10246 (2018). R. Baravalle, N. Guisande, M. Granado, O. A. Rosso, and F. Montani,“Characterization of visuomotor/imaginary movements in EEG: An infor-mation theory and complex network approach,” Frontiers in Physics , 115(2019). O. A. Rosso, M. T. Martín, A. Figliola, K. Keller, and A. Plastino, “EEGanalysis using wavelet-based informational tools,” Journal of NeuroscienceMethods , 163–182 (2006). O. A. Rosso, W. Hyslop, R. Gerlach, R. L. L. Smith, J. Rostas, andM. Hunter, “Quantitative EEG analysis of the maturational changes asso-ciated with childhood absence epilepsy,” Physica A: Statistical Mechanicsand its Applications , 184–189 (2005). O. A. Rosso, A. Mendes, J. A. Rostas, M. Hunter, and P. Moscato, “Dis-tinguishing childhood absence epilepsy patients from controls by the anal-ysis of their background brain electrical activity,” Journal of NeuroscienceMethods , 461–468 (2009). O. A. Rosso, A. Mendes, R. Berreta, J. A. Rostas, M. Hunter, andP. Moscato, “Distinguishing childhood absence epilepsy patients from con-trols by the analysis of their background brain activity (ii): A combinato-rial optimization approach for electrode selection,” Journal of NeuroscienceMethods , 257–267 (2009). F. Redelico, F. Traversaro, M. C. Garcéda, W. Silva, O. A. Rosso, andM. Risk, “Classification of normal and pre-ictal EEG signals using permu-tation entropies and a generalized linear model as a classifier,” Entropy ,72 (2017). I. Echegoyen, D. López-Sanz, J. H. Martínez, F. Maestú, and J. M. Buldú,“Pertmutation entropy and statistical complexity in mild cognitive impair-ment and alzheimer disease: An analysis based on frequency bands,” En-tropy , 116 (2020). R. Baravalle, O. A. Rosso, and F. Montani, “Discriminating imagined andnon-imagined task in the motor cortex area: Entropy-complexity plane witha wavelet decomposition,” Physica A: Statistical Mechanics and its Appli-cations , 27–39 (2018). R. Baravalle, O. A. Rosso, and F. Montani, “Rhythmic activities of thebrain: Quantifying the high complexity of beta and gamma oscillations dur-ing visuomotor tasks,” Chaos , 075513 (2018). R. Baravalle, O. A. Rosso, and F. Montani, “Causal shannon-fisher charac-terization of motor/imagery movements in EEG,” Entropy , 660 (2018). M. Zanin, B. Güntekin, T. Aktürk, and L. Hanoglu, “Time reversivilityof resting-state activity in the healthy brain and pathology,” Frontiers inPhysiology , 1619 (2020). J. H. Martínez, J. L. Herrera-Diestra, and M. Chavez, “Detection of timereversivility in time series by ordinal patterns analysis,” Chaos , 123111(2018). J. Martínez, M. E. López, P. Ariza, M. Chavez, J. A. Pineda-Pardo,D. López-Sanz, P. Gil, F. Maestú, and J. M. Buldú, “Functional brain net-works reveal the existence of cognitive reserve and the interplay betweennetwork topology and dynamics,” Scientific Reports , 10525 (2018). I. Echegoyen, V. Vera-Ávila, R. Sevilla-Escoboza, J. H. Martínez, and J. M.Buldú, “Ordinal synchronization: Using ordinal patterns to capture inter-dependencies between time series,” Chaos, Solitons & Fractals , 8–18(2019). C. Stam, “Nonlinear dynamical analysis of EEG and MEG: review of anemerging field,” Clinical Neurophysiology , 2266–2301 (2005). J. R. Barry, A. R. Clarke, S. J. Johnstone, C. A. Magee, and J. A. Rushby,“EEG differences between eyes-closed and eyes-open resting conditions,”Clinical Neurophysiology , 2765–2773 (2007). A. Khanna, A. Pascual-Leonea, C. M. Michel, and F. Farzan, “Microstatesin resting-state EEG: Current status and future directions,” Neuroscience &Biobehavioral Reviews , 105–113 (2015). C. Quintero-Quiroz, L. Montesano, A. J. Pons, M. C. Torrent, J. García-Ojalvo, and C. Masoller, “Differentiating resting brain states using ordinalsymbolic analysis,” Chaos , 106307 (2018). H. C. Project, “Wu-minn hcp MEG initial data release: Reference manual,”(2014). D. C. V. Essen, K. Ugurbil, E. Auerbach, D. Barch, T. E. J. Behrens, R. Bu-cholz, A. Chang, L. Chen, M. Corbetta, S. W. Curtiss, et al. , “The humanconnectome project: a data acquisition perspective,” Neuroimage , 2222–2231 (2012). L. J. Larson-Prior, R. Oostenveld, S. D. Penna, G. Michalareas, F. Prior,A. Babajani-Feremi, J. M. Schoffelen, L. Marzetti, F. de Pasquale, F. D.Pompeo, J. Stout, M. Woolrich, Q. Luo, R. Bucholz, P. Fries, V. Pizzella,G. L. Romani, M. Corbetta, and A. Z. Snyder, “Adding dynamics to thehuman connectome project with MEG,” Neuroimage , 190–201 (2013). W. A. Fuller,
Introduction to statistical time series , Vol. 428 (John Wiley &Sons, 2009). M. Cohen,
Analyzing neural time series data: theory and practice (MITpress, 2014). J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctua-tions in semiconductor lasers with optical feedback,” Physical Review A , 013819 (2010). O. A. Rosso, L. D. Micco, H. A. Larrondo, M. T. Martín, and A. Plas-tino, “Generalized statistical complexity measure,” International Journal ofBifurcation and Chaos , 775–785 (2010). M. T. Martín, A. Plastino, and O. A. Rosso, “Generalized statistical com-plexity measures: Geometrical and analytical properties,” Physica A: Sta-tistical Mechanics and its Applications , 439––462 (2006). A. R. Plastino and A. Plastino, “Symmetries of the fokker-plank equa-tion and fisher-frieden arrow of time,” Physical Review E , 4423—-4326(1996). O. A. Rosso, L. D. Micco, H. A. Larrondo, M. T. Martín, and A. Plas-tino, “Generalized statistical complexity measure,” International Journal ofBifurcation and Chaos , 775—-785 (2010). O. A. Rosso, L. C. Carpi, P. M. Saco, M. G. Ravetti, A. Plastino, and H. A.Larrondo, “Causality and the entropy–complexity plane: Robustness andmissing ordinal patterns,” Physica A: Statistical Mechanics and its Appli-cations , 42–55 (2012). O. A. Rosso, L. C. Carpi, P. M. Saco, M. G. Ravetti, H. A. Larrondo, andA. Plastino, “The amigó paradigm of forbidden/missing patterns: a detailedanalysis,” European Physic Journal B , 419 (2012). J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, “Testingfor nonlinearity in time series: the method of surrogate data,” Physica D:Nonlinear Phenomena , 77–94 (1992). K. T. Dolan and M. L. Spano, “Surrogate for nonlinear time series analysis,”Physical Review E , 046128 (2001). T. Schreiber and A. Schmitz, “Surrogate time series,” Physica D: NonlinearPhenomena , 346–382 (2000). G. C. Carter, “Coherence and time delay estimation,” Proceedings of theIEEE , 236–255 (1987). M. A. Kramer, “An introduction to field analysis techniques: The powerspectrum and coherence,” The Science of Large Data Sets: Spikes, Fields,and Voxels. Short Course by the Society for Neuroscience. , 18–25(2013). F. D. V. Fallani, V. Latora, and M. Chavez, “A topological criterion forfiltering information in complex brain networks,” PLoS computational bi-ology , e1005305 (2017). J. P. Onnela, J. Saramäki, J. Kertész, and K. Kaski, “Intensity and co-herence of motifs in weighted complex networks,” Physical Review E ,065103 (2005). S. Boccaletti, V. Latora, Y. Moreno, M. Chávez, and D.-U. Hwang, “Com-plex networks: Structure and dynamics,” Physics reports , 175–308(2006). J. Huang, “Overview of cerebral function,” (2019), [Online; accessed 10-November-2019]. F. Montani, R. Baravalle, L. Montangie, and O. A. Rosso, “Causal informa-tion quantification of prominent dynamical features of biological neurons,”Phil. Trans. R. Soc. A , 1–9 (2015). T. Kenet, D. Bibitchkov, M. Tsodyks, A. Grinvald, and A. Arieli, “Spon-taneously emerging cortical representations of visual attributes,” Nature ,954–956 (2003). J. M. Beggs and D. Plenz, “Neuronal avalanches in neocortical circuits,” J.Neuroscience , 11167–11177 (2003). J. M. Beggs and D. Plenz, “Fractal dynamics in physiology: Alterationswith disease and aging,” J. Neuroscience , 11167–11177 (2002). H. Azami, K. Smith, A. Fernandez, and J. Escudero, “Evaluation of resting-state magnetoencephalogram complexity in alzheimer’s disease with mul-tivariate multiscale permutation and sample entropies,” 37th Annual Inter-national Conference of the IEEE Engineering in Medicine and Biology So-ciety (EMBC),Milan,2015 , 7422–7425 (2015). E. Shumbayawonda, A. Fernández, M. P. Hughes, and D. Abásolo, “Per-mutation entropy for the characterisation of brain activity recorded withmagnetoencephalograms in healthy ageing,” Entropy , 141 (2017). M. X. Huang, C. W. Huang, D. L. Harrington, S. Nichols, A. Robb-Swan,A. Angeles-Quinto, L. Le, C. Rimmele, A. Drake, T. Song, J. W. Huang,R. Clifford, Z. Ji, C. K. Cheng, I. Lerman, K. A. Yurgil, R. R. Lee, andD. G. Baker, “Marked Increases in Resting-State MEG Gamma-Band Ac-tivity in Combat-Related Mild Traumatic Brain Injury,” Cerebral Cortex ,283–295 (2019). T. Grent-’t Jong, J. Gross, J. Goense, M. Wibral, R. Gajwani, A. I. Gumley,S. M. Lawrie, M. Schwannauer, F. Schultze-Lutter, T. Navarro Schröder,D. Koethe, F. M. Leweke, W. Singer, and P. J. Uhlhaas, “Resting-stategamma-band power alterations in schizophrenia reveal e/i-balance abnor-malities across illness-stages,” , e37799 (2018). R. Zhou, J. Wang, W. Qi, F.-Y. Liu, M. Yi, H. Guo, and Y. Wan, “Ele-vated resting state gamma oscillatory activities in electroencephalogram ofpatients with post-herpetic neuralgia,” Frontiers in Neuroscience , 750(2018). C. Andreou, G. Nolte, G. Leicht, N. Polomac, I. L. Hanganu-Opatz,M. Lambert, A. K. Engel, and C. Mulert, “Increased Resting-State Gamma-Band Connectivity in First-Episode Schizophrenia,” Schizophrenia Bulletin41