arXiv:2202.04179v1 [stat.ME] 8 Feb 2022

A Framework for the Time- and Frequency-Domain Assessment of High-OrderInteractions in Brain and Physiological Networks

Luca Faes,1, ∗ Gorana Mijatovic,2 Yuri Antonacci,3 Riccardo Pernice,4 Chiara Bara,4 Laura

Sparacino,4 Marco Sammartino,4 Alberto Porta,5 Daniele Marinazzo,6 and Sebastiano Stramaglia7

1Department of Engineering, University of Palermo, Italy2Faculty of Technical Sciences, University of Novi Sad, Serbia

3Department of Physics and Chemistry ”Emilio Segre”, University of Palermo, Italy4Department of Engineering, University of Palermo, Italy

5Department of Biomedical Sciences for Health, University of Milano, Italy, and Department of Cardiothoracic,Vascular Anesthesia and Intensive Care, IRCCS Policlinico San Donato, Italy

6Department of Data Analysis, University of Ghent, Belgium7Department of Physics, University of Bari Aldo Moro, and INFN Sezione di Bari, Italy

(Dated: February 10, 2022)

While the standard network description of complex systems is based on quantifying the link be-tween pairs of system units, higher-order interactions (HOIs) involving three or more units oftenplay a major role in governing the collective network behavior. This work introduces an approachto quantify pairwise and HOIs for multivariate rhythmic processes interacting across multiple timescales. We define the so-called O-information rate (OIR) as a new metric to assess HOIs for mul-tivariate time series, and propose a framework to decompose the OIR into measures quantifyingGranger-causal and instantaneous influences, as well as to expand all measures in the frequency do-main. The framework exploits the spectral representation of vector autoregressive and state-spacemodels to assess the synergistic and redundant interaction among groups of processes, both in spe-cific bands of biological interest and in the time domain after whole-band integration. Validationof the framework on simulated networks illustrates how the spectral OIR can highlight redundantand synergistic HOIs emerging at specific frequencies, which cannot be detected using time-domainmeasures. The applications to physiological networks described by heart period, arterial pressureand respiration variability measured in healthy subjects during a protocol of paced breathing, andto brain networks described by electrocorticographic signals acquired in an animal experiment dur-ing anesthesia, document the capability of our approach to identify informational circuits relevantto well-defined cardiovascular oscillations and brain rhythms and related to specific physiologicalmechanisms involving autonomic control and altered consciousness. The proposed framework al-lows a hierarchically-organized evaluation of time- and frequency-domain interactions in dynamicnetworks mapped by multivariate time series, and its high flexibility and scalability make it suitablefor the investigation of networks beyond pairwise interactions in neuroscience, physiology and manyother fields.

I. INTRODUCTION

The increasing availability of large-scale and fine-grained recordings of biomedical signals and physiolog-ical time series is nowadays boosting the developmentof new methods for the data-driven modelling of com-plex biological systems. Among them, the network rep-resentation of physiological systems is probably the mostused approach to the description of multivariate time se-ries measured from these systems [1]. Paradigmatic in-stances of this approach come from the neurosciences,where the organizational principles of functional segre-gation and integration in the brain are typically studiedthrough the theoretical and empirical tools of NetworkNeuroscience [2], and from integrative physiology, wherethe reductionist approach of studying the function of anorgan system in isolation is complemented by the holis-tic investigation of collective interactions among diversesystems performed in the field of Network Physiology [3].

∗ [email protected]

Data-driven methods for the inference and analysis ofphysiological networks are based on building a networkmodel out of a set of observed time series, in which nodesrepresent the units composing the observed system (be-ing, e.g., distinct neural populations or organ systems)and connecting edges map functional dependencies be-tween pairs of units (descriptive, e.g., of brain connectiv-ity or cardiovascular interactions) [4, 5]. Nevertheless, inspite of the ubiquitous utilization of pairwise measuresto describe interactions in a network, there is mountingevidence that such measures cannot full capture the in-terplay among the multiple units of a complex system [6].In fact, brain and physiological networks -among others-exhibit collective behaviors which are integrated at differ-ent hierarchical levels, thus displaying interactions thatinvolve more than two network nodes. These so-calledhigh-order interactions (HOIs) occur for instance whenbrain dynamics require the joint examination of multipleunits to be predicted accurately [7, 8], or when cardio-vascular interactions are influenced by the effects of therespiratory activity [9, 10].

The recognized need to study networks beyond the

arX

iv:2

202.

0417

9v1

[st

at.M

E]

8 F

eb 2

022

2

framework of pairwise interactions calls for the theoret-ical definition and practical development of methods toassess HOIs among multiple time series. Various met-rics solidly grounded in the general field of informationtheory have been proposed in recent years for this pur-pose, all attempting to capture the redundant or syner-gistic information shared by groups of random variablesor processes [11–13]. In broad terms, synergy arises fromstatistical interactions that can be found collectively in anetwork but not in parts of it considered separately, whileredundancy refers to group interactions that can be ex-plained by the communication of sub-groups of variables.The most popular measures of synergy and redundancyare those based on the interaction information and par-tial information decomposition of random variables, alsoextended to assess directed interactions in dynamic phys-iological processes [7, 10–12]. A recently-proposed mea-sure is the so-called O-information, a metric capable toreveal synergy- and redundancy-dominated interactionsin a network of multiple interacting variables [13]. Itssymmetric nature, the fact that it scales with the net-work size, and the possibility to compute it for dynamicprocesses make the O-information a very promising toolfor the practical analysis of multivariate physiological dy-namics [14].

A main limitation of the information-theoretic mea-sures proposed so far to investigate HOIs in networksystems is that they characterize the system dynamicswith one single value reflecting the aggregate effect ofinteractions possibly occurring at different time scales.However, the time series measured at the nodes of brainand physiological networks are typically rich of oscilla-tory content: for instance, cardiovascular and electroen-cephalographic (EEG) interactions occur through thecoupling of rhythms in different frequency bands withdifferent physiological meaning [15, 16]. Remarkably,the amplitude of oscillations and the coupling strengthmay vary with frequency, and HOIs can have differentnature for different rhythms because synergistic and re-dundant behaviors may alternate in separate frequencybands [17, 18]. Therefore, there is the need to con-nect the spectral representation of information-theoreticmeasures with the HOI description of complex networksto overcome spectral pairwise approaches [19, 20]. Tothis end, the present study introduces a new frameworkfor the time- and frequency-domain analysis of HOIs inmultivariate stochastic processes mapping the activityof network systems. Building on our recent efforts tocompute multivariate information measures in the fre-quency domain [17, 18], we generalize and extend themin many directions. First, we define a new measure, theO-information rate (OIR), which generalizes the mutualinformation rate (MIR) of bivariate processes using thesame rationale whereby the O-information generalizesthe mutual information (MI) between random variables.Then, we provide both a causal decomposition and aspectral expansion of the OIR, thereby connecting it withwell-known and widely used measures of coupling and

Granger causality formulated in the time and frequencydomains [20]. Causal and spectral measures are definedfrom the vector autoregressive (VAR) formulation of mul-tivariate Gaussian stochastic processes [21], in a way suchthat the spectral integration of each frequency domainmeasure yields the corresponding time domain measure.Further, to allow their closed-form computation, all mea-sures composing the time-and frequency-domain OIR areimplemented exploiting the state-space (SS) representa-tion of VAR processes [22].

In this paper, the proposed framework is first illus-trated on theoretical examples of simulated VAR pro-cesses featuring HOIs of different type and order. Then,it is tested in two practical applications of of brainand physiological networks where HOIs are expectedto play a crucial role in governing collective dynam-ics: beat-to-beat variability series of heart period, ar-terial pressure and respiration measured during a pro-tocol of paced breathing [9], and multi-electrode inva-sive EEG signals acquired in an animal experiment ofaltered consciousness [23]. The time- and frequency-domain measures of bivariate and higher-order interac-tions provided by the framework are collected in theOIR Matlab toolbox, freely available for download atwww.lucafaes.net/OIR.html.

II. FRAMEWORK TO MEASUREHIGH-ORDER INTERACTIONS IN

MULTIVARIATE PROCESSES

This section presents the framework to measure dy-namic interactions among Q stationary stochastic pro-cesses Y = Y1, . . . , YQ, grouped in M blocks X =X1, . . . , XM which can be thought as descriptive ofthe activity of a network formed by M dynamic systems

(the ith block has dimension Mi, so that Q =∑Mi=1Mi).

To highlight the dynamic nature of the process Xi, wedenote as Xi,n, Xk

i,n = [Xi,n−1 · · ·Xi,n−k], and X−i,n =

limk→∞Xki,n the random variables that sample the pro-

cess at the present time n, over the past k lags, and overthe whole past history, respectively.

In the following, interactions are characterized pro-viding definitions of high-order measures as well as oftheir causal decomposition and spectral expansion, anddescribing the approach implemented for their computa-tion. While the subsections are self-explanatory, we re-fer the reader to the supplemental material for detailedmathematical treatments.

A. O-information rate

We start recalling the concept of information rate,which quantifies the time density of the average infor-mation in a stochastic process. For a generic processXi, the entropy rate is defined as the conditional en-tropy of the present state given the past history, i.e.

3

HXi= H(Xi,n|X−i,n) [24]. If two processes Xi and

Xj are considered, the definition of MI and the useof basic information rules [24] lead to derive MIR asIXi;Xj

= HXi+HXj

−HXi,Xj.

While the MIR is a dynamic measure of pairwise inter-dependence, multivariate measures involving more thantwo processes can be used to assess HOIs. Here, follow-ing recent works [13, 14], we measure the organizationalstructure of a group of stochastic processes introducingthe so-called O-information rate (OIR). Specifically, theOIR of N processes taken from the set X1, . . . , XM isdefined via the recursion

ΩX2 = 0, (1a)

ΩXN = ΩXN−j

+ ∆XN−j ;Xj

, N ≥ 3 (1b)

where XN = Xi1 , ..., XiN , i1, . . . , iN ∈1, . . .M, N ≤ M , is the analyzed group of pro-cesses, XN

−j = XN\Xj is the subset where Xj is removed(j ∈ i1, . . . iN), and where the variation of the OIRobtained with the addition of Xj to XN

−j is the quantity

∆XN−j ;Xj

= (2−N)IXj ;XN−j

+

N∑m=1m6=j

IXj ;XN−mj

, (2)

with XN−mj = XN\Xm, Xj. The OIR ΩXN is a sym-

metric measure capturing the balance between high- andlow-order statistical constraints in the dynamic interac-tions occurring within XN : ΩXN > 0 reflects a dom-inance of low-order constraints, also known as redun-dancy, while ΩXN < 0 indicates that high-order con-straints prevail, denoting synergy. In turn, the sign ofthe OIR increment defined in (2) detects the informa-tional character of the circuits which link the jth pro-cess with the remaining N − 1 processes of XN : theinformation that Xj shares with XN

−j is redundant when∆XN

−j ;Xj> 0, while it is synergistic when ∆XN

−j ;Xj< 0.

Note that when N = 3 processes X3 = Xi, Xk, Xjare considered, substituting (1a) in (1b) yields ΩX3 =∆Xi,Xk;Xj

, which expanded with (2) gives a dynamicversion of the well-known interaction information [25],i.e. ΩXi,Xk,Xj = IXi;Xj

+ IXk;Xj− IXi,Xk;Xj

.Now we move to provide a causal decomposition of

the OIR increment ∆XN−j ;Xj

. To this end, we note that

this measure is obtained inserting N different MIR valuesin (2), i.e. the MIRs between the processes Z1 = Xj

and Z2 = XN−mj where Z1 is fixed and Z2 varies with

m = 0, 1, . . . , N,m 6= j (Z2 = XN−j when m = 0). The

MIR IZ1;Z2can be formulated according to the expansion

[20]

IZ1;Z2 = TZ1→Z2 + TZ2→Z1 + IZ1 •Z2, (3)

where TZi→Zj = I(Zj,n;Z−i,n|Z−j,n) is the transfer en-

tropy (TE) from Zi to Zj (i, j = 1, 2), and IZ1 •Z2=

I(Z1,n;Z2,n|Z−1,n, Z−2,n) represents the instantaneous in-

formation shared between Z1 and Z2. The TE is a well-known measure of directed information transfer between

two stochastic processes [26], while the instantaneoustransfer is a symmetric measure of information shared atzero lag, quantified after removing the common informa-tion with the past states of the processes. The substitu-tion of (3) in (2) allows to decompose the OIR incrementas

∆XN−j ;Xj

= ∆XN−j→Xj

+ ∆Xj→XN−j

+ ∆XN−j

•Xj, (4)

where the terms ∆XN−j→Xj

and ∆Xj→XN−j

are derived

from the transfer entropies and quantify the informa-tional character of the directed information transfer fromXN−j to Xj and from Xj to XN

−j , and the term ∆XN−j

•Xjis

derived from the information shared instantaneously be-tween Xj and XN

−j and quantifies its informational char-acter; the informational character of each term is redun-dant when the term is positive, and synergistic when theterm is negative.

B. Linear parametric formulation

This section reports the linear parametric formulationof the OIR decomposition, which exploits the knowledgethat this formulation captures all of the entropy differ-ences relevant to the various information measures whenthe observed processes have a joint Gaussian distribution[27]. The linear parametric representation of the originalvector Y is provided by the vector autoregressive (VAR)model

Yn =

p∑k=1

AkYn−k + Un, (5)

where p is the model order, Yn = [Y1,n · · ·YQ,n]ᵀ is a Q-dimensional vector collecting the present state of all pro-cesses, Ak is theQ×Qmatrix of the model coefficients re-lating the present with the past of the processes assessedat lag k, and Un = [U1,n · · ·UQ,n]ᵀ is a vector of Q zero-mean white and uncorrelated noises with Q×Q positivedefinite covariance matrix ΣU = E[UnU

ᵀn ]. While the

VAR model (25) provides a global representation of theoverall multivariate process, to describe the linear inter-actions relevant to the subset of processes Z = Z1, Z2for which the MIR decomposition (3) is sought we needto define a reduced VAR model involving only those pro-cesses. This reduced model is formulated as

Zn =

∞∑k=1

BkZn−k +Wn, (6)

where Zn and Wn are columns vectors of dimension R =R1 + R2 (R1 = Mj is the dimension of Z1 = Xj andR2 is the dimension of Z2 = XN

−mj), and Bk is an R ×R coefficient matrix. Note that the coefficients Bk andinnovations Wn generally differ from those defined for theoverall model, and that the order of the reduced modelis typically infinite [22]; for this reason, it is strongly

4

advised to identify the reduced VAR model (26) in closedform using state space models, as reported in Sect. II C.

The linear parametric representation (26) can be usedto perform MIR and OIR decomposition in the frequencydomain. To this end, the Fourier transfrom (FT) of (26)is taken to derive

Z(ω) = [IR −∞∑k=1

Bke−jωk]−1W (ω) = H(ω)W (ω), (7)

where Z(ω) and W (ω) are the Fourier transforms of Znand Wn, ω ∈ [−π, π] is the normalized angular frequency

(ω = 2π ffs

with f ∈ [− fs2 ,fs2 ], being fs the sampling

frequency of the processes), j =√−1 and IR is the R-

dimensional identity matrix. The R × R matrix H(ω)contains the transfer functions relating the FTs of theinnovation processes in W to the FTs of the processes inZ. This matrix, as well as the R×R power spectral den-sity (PSD) matrix of the process defined as the FT of theautocorrelation function (SZ(ω) = FRZ(k),RZ(k) =E[ZnZ

ᵀn]), can be factorized in four blocks to evidence

the spectral properties related to the internal dynamicsof Z1 and Z2 through the Ri×Ri diagonal blocks SZi

(ω)and Hii(ω), or to the causal interactions between Z1 andZ2 through the Ri×Rj off-diagonal blocks SZiZj

(ω) andHij(ω), i, j ∈ 1, 2. Then, using spectral factorizationto express the PSD of Z as SZ(ω) = H(ω)ΣWH∗(ω),where ΣW = E[WnW

ᵀn ] and ∗ stands for conjugate trans-

pose, and expanding this factorization to evidence thePSD of Zi in terms of internal dynamics and causal in-teractions from Zj , logarithmic spectral measures of thetotal coupling between Zi on Zj and of the causal cou-pling from Zj on Zi can be computed as [19]

fZi;Zj(ω) = log

|SZi(ω)||SZj (ω)||SZ(ω)|

, (8)

fZj→Zi(ω) = log|SZi

(ω)||Hii(ω)ΣWiH

∗ii(ω)|

. (9)

Moreover, a spectral measure fZi •Zj (ω) can be definedsubtracting the two causal measures (38) from the cou-pling measure (36) so as to satisfy in the frequency do-main a decomposition similar to the time-domain decom-position (3), i.e.

fZ1;Z2(ω) = fZ1→Z2(ω) + fZ2→Z1(ω) + fZ1 •Z2(ω). (10)

Importantly, the spectral measures in (40) and the time-domain measures are tightly linked to the similar mea-sures given in the time domain in (3). In fact, it can beshown (see, e.g., [20]) that integration over the whole fre-quency axis of the spectral coupling measure (36) returns,with proper scaling, the MIR between the two processes,

IZ1;Z2 =1

4π

∫ π

−πfZ1;Z2(ω) dω, (11)

and that the same relation holds integrating fZ1→Z2(ω),

fZ2→Z1(ω) and fZ1 •Z2

(ω) to get respectively TZ1→Z2,

TZ2→Z1, and IZ1 •Z2

. This spectral integration prop-erty gives to the spectral measures fZ1;Z2

(ω) andfZ1→Z2

(ω),fZ2→Z1(ω), the information-theoretic mean-

ing of density of information shared between the twoprocesses, or transferred from one process to the other,at the angular frequency ω. We note that, while thecoupling measure is always non-negative, the two causalmeasures can take negative values at some frequencies ifthe process Z is not strictly causal (i.e. if the innovationcovariance ΣW is not block-diagonal). On the contrary,the measure fZ1 •Z2(ω) can take negative values even forstrictly causal processes.

The spectral integration property can be exploited notonly to compute the time-domain measures in (3) as theaverage of the spectral measures in (40), but also toachieve a causal decomposition of the OIR formulatedfor spectral functions. Indeed, it is easy to show that thefrequency-specific OIR increment defined in analogy to(2) as

δXN−j ;Xj

(ω) = (2−N)fXj ;XN−j

(ω) +

N∑m=1m6=j

fXj ;XN−mj

(ω),

(12)satisfies the spectral integration property, i.e. ∆XN

−j ;Xj=

(1/4π)∫ π−π δXN

−j ;Xj(ω) dω, and can also be expanded

through a causal decomposition similar to (23) as

δXN−j ;Xj

(ω) = δXN−j→Xj

(ω) + δXj→XN−j

(ω) + δXN−j

•Xj(ω),

(13)where the three terms on the r.h.s. of (44) are obtainedexpanding fXj ;XN

−j(ω) and fXj ;XN

−mj(ω) in (43) accord-

ing to (40). Moreover, the spectral OIR increment (43)can be used to compute recursively a frequency-domainversion of the OIR, in analogy to (21), as

νXN (ω) = νXN−j

(ω) + δXN−j ;Xj

(ω), (14)

which again satisfies the spectral integration property,i.e. ΩXN = (1/4π)

∫ π−π νXN (ω) dω. Therefore, the spec-

tral versions of the HOI measures defined in this sectioncan be meaningfully interpreted as densities of the syner-gistic/redundant character of the information shared be-tween multiple stochastic processes. As shown in the the-oretical examples of Sect. III and practical applicationsof Sect. IV, the evaluation of these measures within spe-cific frequency bands allows to assess the informationalcharacter of specific oscillations within circuits of nodesof the analyzed network.

C. Framework Implementation

This section reports the time- and frequency-domaincomputation of the OIR and of the terms of its decom-position performed within the framework of state space

5

(SS) models. The advantage of using SS models is thatthis class of models is closed under the definition of re-duced models, i.e. models which contain only some ofthe original analyzed processes. In other words, while areduced VAR model like that formulated in (26) is gen-erally of infinite order and thus very difficult to identifyfrom finite-length time series, SS models can be reducedmaintaining their form and can be therefore identifiedkeeping high computational reliability.

Here, we follow the SS modeling approach of [22] tocompute all the MIR terms needed to derive the OIR (21)and to perform the related causal decomposition (23) andspectral expansion (44). First, we describe the originalprocess Y obeying the VAR representation (25) using theSS model

Sn+1 = ASn + KUn, (15a)

Yn = CSn + Un, (15b)

where Sn = [Y ᵀn−1 · · ·Y

ᵀn−p]

ᵀ is the pQ-dimensionalstate process and the SS parameters (A,C,K,V)are given by the matrices C = [A1 · · ·Ap], A =[C; IQ(p−1)0Q(p−1)×Q], K = [IQ0Q×Q(p−1)]

ᵀ, and V =E[UnU

ᵀn ] = ΣU . Then, to represent the R-dimensional

process Z = Z1, Z2 formed by taking from Y thesubset of processes indexed by the elements of r =r1, r2 ⊂ 1, . . . , Q (where ri contains the Ri indicesof Zi, i = 1, 2), we replace (26) with a reduced SSmodel with state equation (29a) and observation equa-tion Zn = C(r,:)Sn + Wn. This model has parame-ters (A,C(r,:),KVKᵀ,V(r,r),KV(:,r)), where the super-scripts (r,:), (:,r), and (r,r) denote selection of the rowsand/or columns with indices r in a matrix. To exploitthe reduced SS model for the Granger-causal analysis ofZ it is necessary to lead its form back to that of (29),which reads [22]

Sn+1 = ASn + KWn, (16a)

Zn = CSn +Wn. (16b)

The parameters of the model (31) are (A, C, K, V), ofdimension pQ × pQ,R × pQ, pQ × R,R × R; while thestate and observation matrices are easily determined asA = A and and C = C(r,:), the gain K and the reducedinnovation covariance V = E[WnW

ᵀn ] = ΣW must be

obtained by solving a discrete algebraic Riccati equation(DARE) (see refs. [22, 28] for detailed derivations).

After its identification, the reduced model (31) can beanalyzed in the frequency domain to compute the spec-tral measures described in Sect. II B. To this end, the FTof (31a) is computed to derive the PSD of the state pro-cess, S(ω), which is then substituted in the FT of (31b)to obtain [22]

Z(ω) =(IR+C[IpQ−Ae−jω]−1Ke−jω

)W (ω) = H(ω)W (ω),

(17)The transfer function matrix H(ω) is then used to de-rive the PSD matrix of the reduced process as SZ(ω) =

H(ω)VH∗(ω). All these spectral matrices are finallyemployed as described in Sect. II B to compute thefrequency-domain decomposition of the MIR and OIRmeasures, and to derive the corresponding time-domainmeasures through spectral integration.

The flowchart of the calculations implemented for thetime- and frequency-domain computation of the OIR in-crement ∆XN

−j ;Xjis depicted in Fig. 1. The procedure,

which is described here making reference to the equa-tions implemented in the various steps and to the corre-sponding codes of the OIR Matlab toolbox, starts with avector process Y organized in the blocks X1, . . . , XM,and from a set of indices identifying the group of blockprocesses XN to analyze and the target process Xj .The first steps are to identify the VAR model fittingthe whole process Y (oir idVAR.m, oir mosVAR.m, eq.25) and to convert the VAR parameters into SS param-eters (oir ar2ss.m, eq. 29). The SS parameters, to-gether with the set of numbers indexing XN inside Y , arepassed as inputs to the iteration computing the terms ofthe causal decomposition and frequency expansion of theMIR IZ1;Z2

(oir mir.m). Such iteration performs, fixingZ1 = Xj and varying Z2 = XN

−mj (oir subindexes.m),the extraction of the reduced model describing Z =Z1, Z2 and its conversion to SS form (oir submodel.m,eq. 31), followed by the computation of the frequency-domain coupling and causality measures (oir fdGC.m,eqs. 34, 36, 38) and their integration to the time domain(eq. 41a). All MIR measures are then combined in thefrequency domain (eq. 43) and in the time domain (eq.2) to get the desired OIR increments (oir deltaO.m).These increments constitute the output of our algorithm,and can be easily exploited by the time-domain (eq. 21)and frequency-domain (eq. 45) recursions to compute theOIR of any group of N blocks extracted from the originalvector process.

III. THEORETICAL EXAMPLES

In this section, the framework for the computation ofpairwise and higher-order interactions in the time andfrequency domains is illustrated making use of theoreti-cal examples of simulated multivariate VAR models forwhich the various measures are computed directly fromthe known model parameters. These simulations are ex-ploited to show how our measures can be used: (a) tohighlight the emergence of patterns of interaction amonggroups of processes which cannot be traced from pairwiseconnections; (b) to dissect pairwise and higher-order in-teractions into causal components which can be relatedto the topological structure of the underlying network;(c) to ascribe interactions to specific oscillations confinedwithin specific frequency bands; (d) to evidence the pres-ence of circuits dominated by synergy or redundancy, oreven by simultaneous synergistic and redundant behav-iors coexisting at different frequencies. Detailed equa-tions and parameter settings are provided for the two

6

VAR

Y = Y1, Y2, , YQ

A1, , Ap,

ΣU Submodels

oir_idVAR.moir_mosVAR.m oir_ar2SS.m

Frequencydomain

measures

Selectionof indexes

oir_submodel.m oir_fdGC.m

oir_deltaO.m

oir_MIR.m

SS

Eq. (13)

oir_subindexes.m

X = X1, X2, , XM

computation OIR

Eq. (2)

computationOIR

SpectralIntegration

Eq. (10)

Eq. (16)

Eq. (11)

r1,r2

Eq. (15)Eq. (5)

A, C,K, V

A, C,K, V

= IZ Z 1 2+ IZ Z 2 1

+ IZ Z 2 1

= fZ Z 1 2+ fZ Z 2 1

+ fZ Z 2 1

Repeat varying Z2 X-mj, m = 0,1,...N; m jN

Δ

δX ; X -j j

= N δX X -j j

+N δX X -jj

+N δX X -j jN

X ; X -j j=

NX X -j j

+N X X -jj

+N X X -j jNΔ Δ Δ

Block indexes:

i1, , iN

j i1, , iN

fZ ; Z 1 2

IZ ; Z 1 2

FIG. 1. Schematic description of the algorithmic implementation of the framework for OIR computation. Blocks representthe operations applied to the input stochastic processes and/or to their parametric representations for deriving the differenttime- and frequency-domain interaction measures. The equations and the functions of the OIR Matlab Toolbox implementedfor each step are reported in red close to the corresponding block.

simulations in the supplemental material.

A. Simulation 1

The first simulation reproduces the trivariate systemproposed in [17], adapted to generate realistic cardiovas-cular and respiratory dynamics. The activity of this sys-tem is mapped by a trivariate VAR process defined as in(25) fed by independent Gaussian innovations, for whichthe parameters are set as illustrated in Fig. 2a. The vec-tor process is studied keeping the three scalar processesseparate (M = Q = 3, X = Y ), and assuming samplingfrequency fs = 1 (spectral functions are described com-pletely in the frequency range 0−0.5 Hz). The coefficientmatrix A is designed to mimic the dynamics of respira-tion (X1), arterial pressure (X2) and heart period (X3)variability, generating self-dependencies for the processesX1 and X2 through the coefficients a11,k and a22,k, andimposing causal effects along the directions X1 → X2,X1 → X3 and X2 → X3 through the coefficients a21,k,a31,k and a32. Self-dependencies are set to induce oscil-lations in the respiratory band (∼ 0.35 Hz) for X1 andin the low-frequency band (∼ 0.1 Hz) for X1 and par-ticularly for X2, while causal effects are set to realize ahigh-pass filter from X1 to X2, a low-pass filter from X1

to X3 and an all-pass configuration from X2 to X3 (spec-tral transfer functions are shown in Fig. 2a, right); low-and high-pass filtering are achieved through FIR filtersof order 20 with cut-off frequency of 0.2 Hz.

The application of our framework to the VAR param-eters describing the simulated process leads to the spec-tral functions depicted in Fig. 2b,c. The PSD pro-files (Fig. 2b, diagonal plots) highlight oscillations at∼ 0.1 Hz and ∼ 0.35 Hz for the three processes. Thecausal coupling between pairs of processes (Fig. 2b,off-diagonal plots) evidences the presence of information

flows originating from the first process (nonzero profilesof fX1→X2

,fX1→X3and fX2→X3

) and the absence of in-formation flowing back towards it (fX3→X2

= fX2→X1=

fX3→X1= 0 at each frequency). Note that, given the

unidirectional coupling and the absence of instantaneousinteractions, in virtue of (40) the three nonzero causalcoupling measures are equivalent to the spectral measuresof total coupling fX1;X2

, fX1;X3and fX2;X3

(red curvesin Fig. 2b); whole-band integration of such measures by(41a) leads to the MIR quantifying the total informationshared between pairs of processes, whose values resultIX1;X2

= TX1→X2= 0.28 nats, IX1;X3

= TX1→X3= 0.05

nats and IX2;X3= TX2→X3

= 0.24 nats. Then, computa-tion of the MIR between one process and the remainingtwo leads to obtain the OIR via (2), which for this simu-lation is ΩX1;X2;X3 = 0.019 nats, denoting a small redun-dant interaction among the three processes. Importantly,the spectral expansion (Fig. 2c) reveals that this smallOIR value is the balance between a synergistic interactionat low frequencies (ΩX1;X2;X3 = −0.15 nats in the band0.04−0.12 Hz) and a redundant interaction at higher fre-quencies (ΩX1;X2;X3 = +0.33 nats in the band 0.31−0.39Hz). We also highlight that the causal decompositionof the OIR νX1;X2;X3 = δX1;X2,X3 reveals the unidirec-tional nature of the OIR increment (i.e., δX1;X2,X3 =δX1→X2,X3 and δX2,X3→X1 = δX1 •X2,X3

= 0). The op-posite OIR values observed in the two frequency bandscan be explained by the simulation design (see Fig. 2a):synergy and redundancy arise respectively because theflow of information from X1 to X3 is entirely mediatedby X2 at the respiratory frequency (the path X1 → X3 isblocked by H31 at ∼ 0.35 Hz), and because such flow oc-curs via the independent paths X1 → X3 and X2 → X3

at lower frequencies (the path X1 → X2 is blocked byH21 at ∼ 0.1 Hz).

7

2

0

2

0

2

0

2

0

2

0

1 2X Xf 1 3X Xf

2 3X Xf

3 2X Xf 3 1X Xf

2 1X Xf

3 1X Xf

2 1X Xf

3 2X Xf

0

2

a)

b)

0.5

Freq. [Hz]

0.350.10.5

Freq. [Hz]

0.350.10.5

Freq. [Hz]

0.350.1

1 2 3; ;X X X

2 3 1;X X X

1 2 3;X X X

2 3 1;X X X

c)

0.5

Freq. [Hz]

0.350.1-0.2

0

0.2

0.4

0.6

X2

a31,k

a22,k

a11,k

a32a21,k

X1 X3

1XS

2XS

3XS

Freq. [Hz]

0.4

0.6H31 H21

0.0

1.0H32

0.50.2 0.50.2 0.50.2Freq. [Hz] Freq. [Hz]

FIG. 2. Theoretical simulation of cardiovascular interactions.(a) Connectivity structure of the simulated VAR process (left)and of its spectral transfer functions (right); (b) power spec-tral density of the three processes (diagonal) and componentsof the causal decomposition of the spectral coupling betweeneach pair of processes (off-diagonal); (c) spectral profiles ofthe O-information rate of the three processes and of the com-ponents of its causal decomposition.

B. Simulation 2

The second simulation illustrates the possibility offeredby our framework to quantify higher-order spectral in-teractions among multiple blocks of processes whose dy-namics resemble those of neurophyiological signals. Thesimulation extends previous simulations of VAR pro-cesses [18, 29] to the analysis of Q = 10 processes or-ganized in M = 5 blocks, with connectivity structureorganized as in Fig. 3a. The network is designed tosimulate three autonomous vector processes X1, X2 andX3 which generate, through their own subnetwork in-teractions, a stochastic oscillation resembling the brainα rhythm (∼ 10 Hz) which is transmitted to the cen-tral node X4; such node is a sink for the α waves butalso acts as a source of oscillatory activity in the β band(∼ 25 Hz), which is transmitted back to X1 through thepassive block X5. The presence of the two simulatedrhythms and their transmission through the network isdocumented by the power spectra SXi and by the pair-wise coupling measures fXi;Xj

reported respectively inred and gray in Fig. 3b; integration of the coupling mea-sures leads to detect significant MIR values between eachpair of processes except X2 and X3.

The analysis of higher-order interactions was per-formed computing the spectral OIR for all multiplets of

order N = 3, 4, 5 (Fig. 3c) as well as the correspondingtime-domain OIR values obtained integrating the spec-tral measures over all frequencies or within the α (8-12Hz) or β (18-30 Hz) bands (Fig. 3d). This analysis al-lows to evidence patterns of interaction which cannot beinferred from lower-order pairwise links. In particular,the presence of independent sources sending informationto a common target originates synergistic modes of in-teraction characterized by negative profiles of the OIR;this is the case for the multiplets including two or threeof the source processes X1, X2, X3 and one between X4

and X5 (e.g., νX1,X2,X4and νX1,X2,X3,X4

, red and violetnegative OIRs in Fig. 3c). On the contrary, chains ofinteractions including three or more block processes de-termine redundant modes of dependence characterized bypositive OIR values; this occurs when one or two of thesources X1, X2, X3 and both the driven processes X4 andX5 are included in the analyzed multiplet (e.g., νX1,X4,X5

and νX1,X2,X4,X5, green and cyan positive OIRs in Fig.

3c). We note also that the OIR is uniformly null for thetriplet with independent processes X1, X2, X3 (grayline in Fig. 3c, left panel). The computation of thetime-domain OIR puts in evidence the purely synergisticor redundant nature of the interactions occurring withinthe multiplets of order 3 and 4, as documented in Fig.3d by the clearly negative or positive values of the OIRs.Interestingly, the integration within a specific frequencyband (α or β) leads to infer which is the rhythm mostlyassociated with the interactions, which in this simulationoccur dominantly in the α band for the synergistic modeswith negative OIR, and in both bands with prevalence ofβ for the redundant modes with positive OIR.

The analysis of the highest-order multiplet incorporat-ing all processes puts clearly in evidence that synergyand redundancy are related to the simulated α and βrhythms, respectively. Indeed, the spectral OIR νX5 dis-plays a negative peak at ∼ 10 Hz and a positive peakat ∼ 25 Hz (Fig. 3c, right panel), and the integrationof this spectral function within the α and β bands evi-dences clearly negative and positive values (grey bars atthe right of Fig. 3d). This mode is an example of howthe coexistence of synergy and redundancy at differentfrequencies may mask their time domain detection, as inthis case the whole-band integration of the spectral OIRgives small negative values which could be difficult toassess in practice.

IV. APPLICATION TO PHYSIOLOGICALNETWORKS

This section reports the application of the frameworkfor the analysis of multivariate interactions in the timeand frequency domain to two different physiological net-works, i.e. cardiovascular and respiratory interactionsduring paced breathing, and neural interactions fromECoG signals in the anesthetized macaque monkey. Fulldetails about the analyzed datasets and complete results

8

a) c)

10 Hz

Y4

Y3

Y1

Y2

Y8

25 Hz

X2

X4

X3

X5X1

10 Hz

Y5

10 Hz Y7

Y6

b) d)

ΩX

, X

, X

, X

, X

12

34

5

ΩX

, X

, X

, X

1

23

4 ΩX

, X

, X

, X

1

23

5

ΩX

, X

, X

, X

2

34

5

ΩX

, X

, X

, X

1

34

5

ΩX

, X

, X

, X

1

24

5

1Ω

X ,

X ,

X

24

ΩX

, X

, X

2

34

ΩX

, X

, X

1

25

ΩX

, X

, X

2

35

ΩX

, X

, X

1

34

ΩX

, X

, X

1

35

ΩX

, X

, X

3

45

ΩX

, X

, X

1

45

ΩX

, X

, X

2

45

Ω X

3 Ω X

5Ω X

4

α band

whole band β band

fX ; X 2 3

SX 5

Freq. [Hz]

500

Freq. [Hz]

500

Freq. [Hz]

500

Freq. [Hz]

500

Freq. [Hz]

500

fX ; X 1 2fX ; X 1 3

fX ; X 1 4fX ; X 1 5

fX ; X 2 4fX ; X 2 5

fX ; X 3 4fX ; X 3 5

fX ; X 4 5

SX 4SX 3

SX 2SX 1

νX , X , X , X , X1 2 3 4 5

-3

-2

-1

0

1

-0.8

-0.4

0

0.4

0.8

-1

-0.5

0

0.5

ν X

3 ν X

5ν X

4

-0.2

-0.1

0.1

0

40

Freq. [Hz]

10 20 30

Freq. [Hz]

10 20 30 40

Freq. [Hz]

10 20 30 40

ΩX

, X

, X

1

23

νX , X , X , X 2 3 4 5

νX , X , X , X 1 2 3 4

νX , X , X , X 1 2 3 5

νX , X , X , X 1 3 4 5

νX , X , X , X 1 2 4 5

νX , X , X

νX , X , X 1 2 3

νX , X , X 3 4 5

νX , X , X 2 4 5

1 4 5

νX , X , X νX , X , X νX , X , X νX , X , X

1 3 4

2 3 5

1 2 5

2 3 4

νX , X , X 1 3 5

1νX , X , X

2 4

FIG. 3. Theoretical simulation of neurophysiological interactions. (a) Connectivity structure of the simulated VAR process,featuring 10 scalar processes grouped in 5 blocks; (b) power spectral densities (red) and spectral coupling functions (gray)between each pair of block processes; (c) spectral profiles of the O-information rate computed for multiplets of three (left),four (middle) and five (right) block processes; (d) time-domain O-information rate obtained integrating the spectral measurerelevant to each multiplet over the whole frequency axis (left bars), inside the α band (8-12 Hz, middle bars), or inside the βband (18-30 Hz, right bars).

are provided in the supplemental material.

A. Cardiovascular and respiratory interactionsduring paced breathing

The analyzed dataset refers to beat-to-beat variabil-ity series of respiration (RESP, process X1), systolicarterial pressure (SAP, process X2) and heart period(HP, processX3), synchronously measured in a groupof 18 young healthy subjects monitored in the restingsupine position during an experimental protocol con-sisting of four phases: spontaneous breathing (SB) andcontrolled breathing at 10, 15, and 20 breaths/minute(CB10, CB15, CB20) [9]. The HP, SAP and RESP timeseries were extracted respectively from the electrocardio-gram, noninvasive arterial blood pressure and nasal respi-ration flow as the sequences of the duration of the cardiaccycle (R-R interval), of the local maximum of the bloodpressure signal within each detected cardiac cycle, and ofthe value of the respiration signal sampled at the onset ofeach cardiac cycle. This measurement convention impliesthat instantaneous influences can be described as causaleffects from RESP to SAP and HP and from SAP to HP

(directions X1 → X2, X1 → X3, X2 → X3) [30].

The analysis was performed on stationary segmentsof the time series including 256 heartbeats, selectedby visual inspection for each subject and experimen-tal condition [9]. The pre-processing consisted on de-trending and mean removal for each time series. TheVAR model fitting the three series was identified throughthe ordinary least squares method, selecting the orderp in the range 3-14 by means of the Akaike Informa-tion Criterion [21]. The analysis was focused on de-composing the OIR of the three processes in OIR in-crements obtained when the HP process is added tothe bivariate process RESP,SAP. Specifically, start-ing from the estimated VAR parameters, we computedδX1,X2→X3

(f), δX3→X1,X2(f) and δX1,X2 •X3(f) from the

terms of the spectral decomposition (10), then derivingνX1,X2,X3

(f) = δX1,X2;X3(f) via (13,14). From these

spectral measures, time-domain measures were obtainedthrough integration over the whole frequency axis orwithin the low frequency range (LF, 0.04-0.12 Hz) andthe high frequency range (HF, ±0.04 Hz around the res-piratory frequency fRESP ). Given the possibility to as-cribe instantaneous effects to specific causal directions(see above), the analysis is performed summing the in-

9

CB10SB CB15 CB20

TOT LF HF

* *

*

*

**

b)

a) SB CB10 CB15 CB20

Freq. [Hz] 0.1 0.2 0.3 0.4

Freq. [Hz] 0.1 0.2 0.3 0.4

Freq. [Hz] 0.1 0.2 0.3 0.4

Freq. [Hz] 0.1 0.2 0.3 0.4

3.0

2.0

1.0

0.0

1.5

0.5

-0.5

1.5

1.0

0.5

0.0

δ X ,

X

X

31

2

0.15 0.01

0

-0.01

0.10

0 0

δ X

X

,X2

31

δ X ;

X ,

X2

31

X

; X

,X 2

31

Δ X ,

X

X 3

12

Δ X

X ,X 2

31

Δ X

; X

,X 2

31

Δ X ,

X

X 3

12

Δ X

X ,X 2

31

Δ X

; X

,X 2

31

Δ X ,

X

X 3

12

Δ X

X ,X 2

31

Δ

FIG. 4. OIR decomposition of cardiovascular interactionsduring controlled breathing (CB). (a) Average spectral pro-files across subjects (line: median; shades: 1st-3rd quar-tiles) of the OIR increment obtained with the addition ofHP to SAP,RESP (upper panels) and of its decomposi-tion in causal terms (middle and lower panels) computedduring spontaneous breathing (SB) and CB at 10, 15 and20 breaths/min. (b) Time-domain values of the mean OIRincrements obtained integrating the spectral measures overthe whole frequency axis (TOT), in the range 0.04-0.12 Hz(LF) or in the range fRESP ± 0.04 Hz (HF); asterisks denotestatistically significant difference between the CB conditioncompared with SB (Wilcoxon signed-rank test: black, uncor-rected; red, Bonferroni-Holm correction for multiple compar-isons).

formation shared instantaneously between RESP,SAPand HP to the information transferred from RESP,SAPto HP, i.e. computing the spectral and time domain mea-sures δX1,X2

.→X3(f) = δX1,X2→X3(f) + δX1,X2 •X3(f) and

∆X1,X2.→X3

= ∆X1,X2→X3 + ∆X1,X2 •X3 .

The results of OIR computation and decompositionare reported in Fig. 4, showing the grand average of thefrequency-domain measures as well as the whole-band,LF and HF time-domain average measures. Spectralanalysis was performed assuming the series as uniformlysampled with sampling frequency equal to the inverse ofthe mean HP. The spectral OIR and most of the termsof its decomposition exhibit prominent peaks, which arewell-defined at the frequency of the paced breathing dur-ing the CB conditions and are less narrow-banded duringSB (Fig. 4a). This behavior reflects the fact that pacedbreathing regularizes the RESP signal around the im-

posed rhythm and enforces synchronous oscillations atthe same frequency in the HP and SAP time series, de-termining increased spectral content and spectral cou-pling in the HF band [9]. The positive values of thetime-domain OIR (Fig. 4b, left) document that this syn-chronized interaction is dominantly redundant, confirm-ing previous findings [12]. Looking at the spectral pro-files of Fig. 4a, the peak values of the OIR show a ten-dency to increase while moving from SB to CB10, andto decrease progressively during CB15 and CB20; thesetrends confirm from the perspective of HOIs results ob-tained on the same data using information-theoretic mea-sures of cardiorespiratory coupling [31]. The dominanceof redundancy in the HF band of the spectrum (Fig.4b, right) suggests that the main underlying physiolog-ical mechanism is the mechanical influence of RESP onSAP variability, transmitted to HP through the barore-flex feedback [32]; the OIR component directed from HPto SAP,RESP, which tends to be less redundant atincreasing the frequency of paced breathing, is of moredifficult interpretation and is likely dominated by the me-chanical feedforward effects from HP to SAP [33]. Thedominance of redundant mechanisms around the respi-ratory frequency impacts substantially the whole-bandtime-domain OIR, which show comparable values acrossthe analyzed conditions (Fig. 4b, left). On the otherhand, the measures integrated within the LF band varysignificantly moving from spontaneous to paced breath-ing (Fig. 4b, middle): the information transfer fromSAP,RESP to HP becomes mostly synergistic dur-ing CB10, and during CB15 and CB20 returns progres-sively to the redundant values observed at SB; the infor-mation transfer along the direction HP→ SAP,RESPis prevalently synergistic at rest and shifts to redun-dant values during CB. The shift to synergy observedat CB10 for ∆X1,X2→X3 suggests that, when the respira-tory activity slows down and tends to overlap with theMayer waves typically observed in SAP and HP [34], thebaroreflex (SAP→HP) and respiratory sinus arrhythmia(RESP→HP) mechanisms operate independently in de-termining the variability of heart rate.

B. Neural interactions from ECoG signals in theanesthetized macaque monkey

The second practical application refers to monkey elec-trocorticographic (ECoG) signals downloaded from thepublic server neurotycho.org. The analyzed dataset wasrecorded with a sampling frequency of 1000 Hz in onemacaque monkey using 128 electrodes, placed in pairswith an inter-electrode distance of 5 mm to cover thefrontal, parietal, temporal and occipital lobes of the lefthemisphere [23]. Specifically, we considered two five-minutes recording sessions during which the blindfoldedmonkey was seated in a primate chair with tied hands,first in a resting state (REST) and then after injec-tion of a sedative inducing anesthesia (ANES). From

10

the 128 electrodes, a subset of 20 was selected as de-picted in Figure 5a to cover, considering ten bipolarECoG signals obtained taking the differential activity be-tween close electrodes, the following five brain regionsof the default mode network, i.e. the pre-frontal cortex(X1 = [Y1, Y2]), parietal cortex (X2 = [Y3, Y4]), temporalcortex (X3 = [Y5, Y6]), low visual cortex (X4 = [Y7, Y8]),and high visual cortex (X5 = [Y9, Y10]). The ten bipo-lar signals were band-pass filtered between 0.5 and 200Hz, downsampled to fs = 250 Hz, epoched to extract∼ 160 trials lasting 2 sec for each condition, and finallynormalized to zero mean and unit variance within eachtrial. Then, a VAR model was fitted on the Q = 10 sig-nals of each trial using least squares identification andsetting the model order according to the Bayesian Infor-mation Criterion (BIC) [21]. From the VAR parameters,the analysis of high-order interactions was performed forthe M = 5 blocks computing the spectral OIR for allmultiplets of order N = 3, 4, 5. Time-domain OIR values(Ω) were then obtained integrating the spectral measuresν(f) within the δ (0.2-3 Hz), θ (4-7 Hz), α (8-12 Hz), β(12-30 Hz) and γ (31-70 Hz) frequency bands, as well ascumulatively between 0 and 70 Hz.

The results of OIR computation are reported in Fig.5b, showing the grand average of the spectral OIR for fivemultiplets selected as the most representative of the an-alyzed interactions, together with the time-domain OIRobtained through whole-band and band-specific integra-tion. The positive values of the OIR functions and of theintegrated measures, observed for all multiplets in bothconditions and increasing with the order of the multiplet,indicate that the analyzed system is dominated by redun-dancy. Moreover, the redundancy level is modulated bythe experimental condition to an extent that depends onthe analyzed multiplet and spectral band. Indeed, con-sidering the multiplets of order 3 and 4 which involve theprefrontal cortex X1 (1st and 3rd row of panels in Fig.5b), a significant increase of the OIR is observed whilemoving from REST to ANES; such increase is driven bythe rise of a peak in the OIR at ∼ 2 Hz (δ band) to-gether with an increased contribution within the γ band.On the other hand, the multiplets formed by signals fromthe parietal, temporal and visual cortices (2nd and 4throw of panels in Fig. 5b) display a drop of redundancyin the α and β bands during ANES compared to REST.These two opposite behaviors are summarized by the OIRencompassing all five regions (5th row of panels in Fig.5b), which during ANES displays significantly higher lev-els of redundancy in the δ and γ bands (and in the wholeband), and significantly lower redundancy in the θ, α,and β bands.

Our results indicate that the activity relevant to the αand β rhythms observed during the relaxed awake statedisappears during anesthesia, leaving place to dominantinteractions within the δ and γ bands. The redundancyobserved at REST for the α waves is significant for themultiplets involving signals from the visual cortex, inagreement with the knowledge that these waves can be

a)

Y1

X1

Y2

Y4

Y3

X2

Y5Y6 Y9

Y10

Y7

Y8X4

X3

X5

b) c)

-0.2

0

0.2

0.4

*

0

1

2

3*

*

*

**

0

0.5

1

* * **

0

0.5

1

*

*

* * **

0

1

2

T α β γ θ δ

0.2

0.4

* * *

0

2

4

6

0

0.4

0.8

1.2

Freq. [Hz] Freq. [Hz]

RESTANES

10-4

×

10-4

×

10-3

×

10-3

×

10-3

×

0 20 40 60 0 20 40 60

ΩX , X , X 1 2 3

ΩX , X , X 2 3 5

ΩX , X , X , X 1 2 3 4

ΩX , X , X , X 2 3 4 5

ΩX , X , X , X , X1 2 3 4 5

ν X , X

, X

1

23

ν X , X

, X

2

35

ν X , X

, X

, X

1

23

4ν X

, X

, X

, X

2

34

5ν X

, X

, X

, X

, X

12

34

5

-0.2

0

0.2

0.4

0.6

0

0

0.2

0.4

0.6

FIG. 5. OIR analysis of neurophysiological interactions in theanesthetized monkey. (a) ECoG electrode montage highlight-ing the positions of the selected electrodes acquiring the bipo-lar signals Y1, . . . , Y10 grouped in the blocks X1, . . . , X5 cov-ering five regions of the left hemisphere. (b) Average spectralprofiles across trials (line: median; shades: 1st-3rd quartiles)of the OIR computed for five representative multiplets duringrelaxation (REST) and anesthesia (ANES). (c) Time-domainvalues of the mean OIR obtained by integrating the spectralmeasures over the whole frequency axis (T) or within the δ,θ, α, β and γ bands; asterisks denote statistically significantdifference between REST and ANES (Wilcoxon signed-ranktest with Bonferroni correction for multiple comparisons ).

predominantly recorded from the occipital lobes duringwakeful relaxation with closed eyes [35]. On the other

11

hand, the higher redundancy reported in the δ band canbe related to the slow wave oscillations (0.1-4 Hz) typi-cally observed under anesthesia [36]. Moreover, the factthat higher δ redundancy is observed only for multipletsincluding frontal cortex signals supports the knowledgethat the slow oscillations are a manifestation of a cou-pling between the anterior and posterior axes of the brain[37]. Anesthesia evokes also an increase of redundancy re-lated to γ oscillations, which are associated with differentcognitive functions [38].

Overall, these results agree with those in [23] andsupport the integration theory according to which theconscious state is generated by highly integrated neu-ral interactions that disappear in the unconscious state[39]. A recent study comparing resting wakefulness withpropofol-induced anaesthesia in human fMRI data hasshown how the anterior-posterior disconnection occur-ring during anesthesia is associated with a decrease ofIntegrated Information within the default mode networkin the left hemisphere [40]. Importantly, the conceptsof Integration Information and that of redundancy areinterrelated, as explained in [41] where it is highlightedthat a drop of Integrated Information corresponds to anincrease of redundancy. Thus, our results support thetheory of an anterior-posterior disconnection during anes-thesia, which in our case can be ascribed to the significantincrease of the OIR documented when the frontal cortexis considered in the analyzed multiplet.

V. CONCLUSION

This work opens the way to the combined information-theoretic and spectral evaluation of hierarchically-

organized interactions in dynamic networks mapped bymultivariate stochastic processes. The proposed frame-work is highly flexible and scalable as it provides princi-pled measures of both pairwise and higher-order interac-tions among scalar or vector processes, defined in bothtime and frequency domains in a way such that the tworepresentations are connected in a straightforward way.Moreover, it allows to decompose symmetric measuresinto components reflecting Granger-causal and instanta-neous influences, and to estimate them with high com-putational reliability within the framework of vector au-toregressive and state space models.

The application of the new framework to biomedicaltime series illustrates its capability to capture the bal-ance between redundancies and synergies among arbi-trarily large groups of nodes of brain and physiologi-cal networks. Moreover, it highlights the importance ofstudying these features within specific frequency bandsof biological interest to elicit interactions which may beotherwise hidden if investigated only in the time domain.The generality of the information-theoretic grounds andof the parametric implementation of the proposed ap-proach makes it suitable for the assessment of pairwiseand higher-order interactions even beyond the domain ofbiomedical time series, to analyze virtually any type ofdynamic network (e.g., electronic, climatologic, social, orfinancial) with node activity described by rhythmic pro-cesses.

[1] A.-L. Barabasi, Philosophical Transactions of the RoyalSociety A: Mathematical, Physical and Engineering Sci-ences 371, 20120375 (2013).

[2] D. S. Bassett and O. Sporns, Nature neuroscience 20,353 (2017).

[3] A. Bashan, R. P. Bartsch, J. W. Kantelhardt, S. Havlin,and P. C. Ivanov, Nature communications 3, 1 (2012).

[4] M. Rubinov and O. Sporns, Neuroimage 52, 1059 (2010).[5] K. Lehnertz, T. Brohl, and T. Rings, Frontiers in Phys-

iology 11, 1694 (2020).[6] F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lu-

cas, A. Patania, J.-G. Young, and G. Petri, Physics Re-ports (2020).

[7] S. Stramaglia, G.-R. Wu, M. Pellicoro, and D. Mari-nazzo, Physical Review E 86, 066211 (2012).

[8] S. Stramaglia, L. Angelini, G. Wu, J. M. Cortes, L. Faes,and D. Marinazzo, IEEE Transactions on Biomedical En-gineering 63, 2518 (2016).

[9] A. Porta, T. Bassani, V. Bari, G. D. Pinna, R. Maestri,and S. Guzzetti, IEEE Transactions on Biomedical En-gineering 59, 832 (2011).

[10] A. Porta, V. Bari, B. De Maria, A. C. Takahashi,

S. Guzzetti, R. Colombo, A. M. Catai, F. Raimondi, andL. Faes, IEEE Transactions on Biomedical Engineering64, 2628 (2017).

[11] J. T. Lizier, N. Bertschinger, J. Jost, and M. Wibral,“Information decomposition of target effects from multi-source interactions: Perspectives on previous, currentand future work,” (2018).

[12] L. Faes, A. Porta, G. Nollo, and M. Javorka, Entropy19, 5 (2017).

[13] F. E. Rosas, P. A. Mediano, M. Gastpar, and H. J.Jensen, Physical Review E 100, 032305 (2019).

[14] S. Stramaglia, T. Scagliarini, B. C. Daniels, and D. Mari-nazzo, Frontiers in Physiology 11, 1784 (2021).

[15] A. Porta and L. Faes, Proceedings of the IEEE 104, 282(2015).

[16] B. He, L. Astolfi, P. A. Valdes-Sosa, D. Marinazzo, S. O.Palva, C.-G. Benar, C. M. Michel, and T. Koenig, IEEETransactions on Biomedical Engineering 66, 2115 (2019).

[17] L. Faes, R. Pernice, G. Mijatovic, Y. Antonacci, J. C.Krohova, M. Javorka, and A. Porta, Philosophical Trans-actions of the Royal Society A 379, 20200250 (2021).

[18] Y. Antonacci, L. Minati, D. Nuzzi, G. Mijatovic, R. Per-

12

nice, D. Marinazzo, S. Stramaglia, and L. Faes, IEEEAccess (2021).

[19] J. Geweke, Journal of the American statistical associa-tion 77, 304 (1982).

[20] D. Chicharro, Biological cybernetics 105, 331 (2011).[21] L. Faes, S. Erla, and G. Nollo, Computational and math-

ematical methods in medicine 2012 (2012).[22] L. Barnett and A. K. Seth, Physical Review E 91, 040101

(2015).[23] T. Yanagawa, Z. C. Chao, N. Hasegawa, and N. Fujii,

PloS one 8, e80845 (2013).[24] M. Cover Thomas and A. Thomas Joy, New York: Wiley

3, 37 (1991).[25] W. McGill, Psychometrika 19, 97 (1954).[26] T. Schreiber, Physical review letters 85, 461 (2000).[27] A. B. Barrett, L. Barnett, and A. K. Seth, Physical

Review E 81, 041907 (2010).[28] L. Faes, D. Marinazzo, and S. Stramaglia, Entropy 19,

408 (2017).[29] L. Faes and G. Nollo, Biological cybernetics 107, 217

(2013).[30] L. Faes, S. Erla, A. Porta, and G. Nollo, Philosophi-

cal Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences 371, 20110618 (2013).

[31] A. Porta, S. Guzzetti, N. Montano, M. Pagani,V. Somers, A. Malliani, G. Baselli, and S. Cerutti, Med-ical and Biological Engineering and Computing 38, 180(2000).

[32] J. Krohova, L. Faes, B. Czippelova, Z. Turianikova,N. Mazgutova, R. Pernice, A. Busacca, D. Marinazzo,S. Stramaglia, and M. Javorka, Entropy 21, 526 (2019).

[33] M. Javorka, B. Czippelova, Z. Turianikova, Z. Lazarova,I. Tonhajzerova, and L. Faes, Medical & biological engi-neering & computing 55, 179 (2017).

[34] C. Julien, Cardiovascular research 70, 12 (2006).[35] S. Palva and J. M. Palva, Trends in neurosciences 30,

150 (2007).[36] S. Chauvette, S. Crochet, M. Volgushev, and I. Timo-

feev, Journal of Neuroscience 31, 14998 (2011).[37] M. Murphy, B. A. Riedner, R. Huber, M. Massimini,

F. Ferrarelli, and G. Tononi, Proceedings of the NationalAcademy of Sciences 106, 1608 (2009).

[38] P. Fries, Annual review of neuroscience 32, 209 (2009).[39] B. J. Baars, Trends in cognitive sciences 6, 47 (2002).[40] A. I. Luppi, P. A. Mediano, F. E. Rosas, J. Allanson, J. D.

Pickard, R. L. Carhart-Harris, G. B. Williams, M. M.Craig, P. Finoia, A. M. Owen, et al., BioRxiv (2020).

[41] P. A. Mediano, F. Rosas, R. L. Carhart-Harris,A. K. Seth, and A. B. Barrett, arXiv preprintarXiv:1909.02297 (2019).

[42] T. M. Cover, Elements of information theory (John Wi-ley & Sons, 1999).

[43] T. E. Duncan, SIAM Journal on Applied Mathematics19, 215 (1970).

[44] L. Barnett, A. B. Barrett, and A. K. Seth, Physicalreview letters 103, 238701 (2009).

[45] B. D. Anderson and M. Gevers, Automatica 18, 195(1982).

[46] A. G. Nedungadi, M. Ding, and G. Rangarajan, Biolog-ical cybernetics 104, 197 (2011).

[47] S. L. Marple Jr and W. M. Carey, “Digital spectral anal-ysis with applications,” (1989).

[48] R. Pernice, L. Sparacino, V. Bari, F. Gelpi, B. Cairo,G. Mijatovic, Y. Antonacci, D. Tonon, G. Rossato, M. Ja-

vorka, A. Porta, and L. Faes, Autonomic Neuroscience,Basic and Clinical , under revision (2022).

[49] G. Nollo, L. Faes, B. Pellegrini, A. Porta, and R. An-tolini, in Computers in Cardiology 2000. Vol. 27 (Cat.00CH37163) (IEEE, 2000) pp. 143–146.

[50] Y. Nagasaka, K. Shimoda, and N. Fujii, PloS one 6,e22561 (2011).

13

Supplemental material: Framework for the Time- and Frequency-Domain Assessmentof High-Order Interactions in Brain andPhysiological Networks

FRAMEWORK TO MEASURE HIGH-ORDER INTERACTIONS IN MULTIVARIATE PROCESSES

A. O-information rate

Considering a generic stationary stochastic process Xi composed by the random variables Xi,n (where n ∈ N is thetemporal index), the following definitions of entropy rate are equivalent under the assumption of stationarity [42]:

HXi= limk→∞

1

kH(Xk

i,n) = H(Xi,n|X−i,n), (18)

where H(Xki,n) is the entropy of Xk

i,n = [Xi,n−1 · · ·Xi,n−k] and H(Xi,n|X−i,n) is the conditional entropy of Xi,n given

X−i,n = limk→∞Xki,n. With our notation, H(•) denotes the entropy of a random variable, and H(•) denotes the entropy

rate of a random process. As (18) holds also for groups of processes, the definition of mutual information (MI) [42]can be exploited to define the MI rate (MIR) between the processes Xi and Xj as [43]

IXi;Xj= limk→∞

1

kI(Xk

i,n;Xkj,n) = HXi

+HXj−HXi,Xj

. (19)

The O-information rate (OIR) of a set of N stochastic processes XN = X1, ..., XN is defined elaborating the entropyrates of subsets of XN according to

ΩXN = (N − 2)HXN +

N∑i=1

[HXi −HXN−i

], (20)

where XN−i = XN\Xi is the subset of processes where Xi is removed. The OIR defined in (20) is a symmetric measure

which generalizes to stochastic processes the O-information measure recently proposed to assess the ”organizationstructure” of a group of random variables [13]. For a bivariate process (N = 2), it is easy to show from (20) that ΩX2 =0. ForN = 3 processes, the OIR is equivalent to the interaction information rate, i.e. ΩX3 = IXj ;Xi

+IXj ;Xk−IXj ;Xi,Xk

(with i, j, k = 1, 2, 3), a measure which generalizes to stochastic processes the concept of interaction information[25]. Then, applying to information rates the extensions provided in [13] for information quantities, one can derivethe iterative definition of the OIR given in Eqs. 1 and 2 of the main paper (see also [7]), which can be used to derive,for any order N ≥ 3, the OIR of order N given any OIR of order N − 1 and the corresponding OIR increment:

ΩXN = ΩXN−j

+ ∆XN−j ;Xj

, (21a)

∆XN−j ;Xj

= (2−N)IXj ;XN−j

+

N∑m=1m 6=j

IXj ;XN−mj

. (21b)

B. Causal decomposition of the O-information rate

Given two processes Xi and Xj , the MIR (19) can be formulated according to the expansion (see, e.g., [20])

IXi;Xj= TXj→Xi

+ TXi→Xj+ IXi •Xj

, (22)

where TXj→Xi= I(Xi,n;X−j,n|X

−i,n) and TXi→Xj

= I(Xj,n;X−i,n|X−j,n) are the transfer entropy from Xj to Xi and

from Xi to Xj , and IXi •Xj = I(Xi,n;Xj,n|X−i,n, X−j,n) represents the instantaneous information shared between Xi

and Xj (where I(·; ·|·) denotes conditional MI for three random variables). Since the decomposition (22) is equallyvalid for groups of stochastic processes, it can be applied to any MIR term appearing in the OIR increment definedin 21b, which can be therefore rewritten in the form of Eq. 4 of the main paper as

∆XN−j ;Xj

= ∆XN−j→Xj

+ ∆Xj→XN−j

+ ∆XN−j

•Xj, (23)

14

where the three terms

∆XN−j→Xj

= (2−N)TXN−j→Xj

+

N∑m=1m6=j

TXN−mj→Xj

(24a)

∆Xj→XN−j

= (2−N)TXj→XN−j

+

N∑m=1m6=j

TXj→XN−mj

(24b)

∆XN−j

•Xj= (2−N)IXN

−j•Xj

+

N∑m=1m6=j

IXN−mj

•Xj(24c)

quantify the informational character of the directed information transfer from XN−j to Xj , of the directed informa-

tion transfer from Xj to XN−j , and of the instantaneous information shared between Xj and XN

−j , respectively; theinformational character of each term is redundant when the term is positive, and synergistic when the term is negative.

C. Linear parametric formulation

The OIR framework considers a network of Q stationary stochastic processes Y = Y1, . . . , YQ, whose dynamicsare represented by the linear parametric model given by Eq. 5 of the main paper:

Yn =

p∑k=1

AkYn−k + Un. (25)

In our analyses, the original processes are grouped into M blocks X1, . . . , XM , from which the processes Z1 = Xj

and Z2 = XN−mj of dimension R1 and R2 are selected (varying m in the range 0, 1, . . . ,M , all processes appearing

in 21b can be considered). The linear parametric description of the vector process Z = Z1, Z2 is given by the VARsub-model in Eq. 6 of the main paper,

Zn =

∞∑k=1

BkZn−k +Wn, (26)

where Zn = [Zᵀ1,nZ

ᵀ2,n]ᵀ and Wn = [W ᵀ

1,nWᵀ2,n]ᵀ are vectors of dimension R = R1 + R2, and the R × R innovation

covariance ΣW = E[WnWᵀn ] is a block matrix having the R1 × R1 matrix ΣW11

= E[W1,nWᵀ1,n] and the R2 × R2

matrix ΣW22= E[W2,nW

ᵀ2,n] as diagonal blocks. The model (26) that provides a joint description of Z = Z1, Z2 is

denoted as ”full” model; then, the processes Z1 and Z2 can be described individually by the ”restricted” models

Zi,n =

∞∑k=1

Ci,kZi,n−k + Vi,n, i ∈ 1, 2, (27)

for which Ci,k is an Ri × Ri coefficient matrix and Vi is an Ri-dimensional zero-mean white noise process withcovariance matrix ΣVi

= E[Vi,nVᵀi,n] of dimension Ri ×Ri. The covariance matrices of the full and restricted models

(26) and (27) are exploited to define measures of global, causal and instantaneous interdependence between Z1 andZ2 as

IZ1;Z2=

1

2log|ΣV1

||ΣV2|

|ΣW |, (28a)

TZ1→Z2=

1

2log|ΣV2

||ΣW22

|, TZ2→Z1

=1

2log|ΣV1

||ΣW11

|, (28b)

IZ1 •Z2 =1

2log|ΣW11 ||ΣW22 ||ΣW |

, (28c)

which satisfy Eq. 3 of the main paper when the observed processes have a joint Gaussian distribution [44]: IZ1;Z2=

TZ1→Z2+ TZ2→Z1

+ IZ1 •Z2. Note that this decomposition is the same as that defined here in (22), generalized to the

15

generic vector processes Z1 and Z2. The measures (28), defined without the multiplicative factor 12 , were originally

proposed by [19] in the framework of linear prediction. Here, exploiting the equivalence (up to a factor two) of linearprediction and information-theoretic measures valid for Gaussian processes [27, 44], we have provided their directformulation as measures of MIR (28a), transfer entropy (28b), and instantaneous information transfer (28c).

The derivations above show that, for Gaussian systems, the decomposition of MIR and the consequent computationof any OIR measure can be performed using nested linear regression models. For these models, all the partial covariancemeasures (i.e., the determinants of the innovation covariance matrices appearing in (28)) which are needed for thecomputation of MIR and OIR can be obtained from the parameters of the original VAR process (25) after expressingthis process as a state space (SS) process. To do this, we define the pQ-dimensional state process Sn = [Y ᵀ

n−1 · · ·Yᵀn−p]

ᵀ

that, together with Yn, obeys the equations of the SS model defined in Eq. 15 of the main paper,

Sn+1 = ASn + KUn, (29a)

Yn = CSn + Un. (29b)

The parameters of the SS model (29) are (A,C,K,V), of dimension (pQ× pQ,Q× pQ, pQ×Q,Q×Q), where

A =

A1 · · · Ap−1 Ap

IQ · · · 0Q 0Q...

......

0Q · · · IQ 0Q

,C = [A1 · · ·Ap], K = [IQ0Q×Q(p−1)]

ᵀ, and V = E[UnUᵀn ] = ΣU . Then, to represent the R-dimensional process

Z = Z1, Z2 we replace (26) with a reduced SS model which has the same state equation of the overall model and areduced observation equation, i.e. the model

Sn+1 = ASn + KUn, (30a)

Zn = C(r,:)Sn +Wn. (30b)

The parameters of the model (30) are (A,C(r,:),KVKᵀ,V(r,r),KV(:,r)), where the superscripts (r,:), (:,r), and (r,r)

denote selection of the rows and/or columns with indices r in a matrix. Since the model (30) has a different formthan the original SS model, to exploit this model for the Granger-causal analysis of Z it is necessary to lead its formback to that of (29) [22], which gives Eq. 16 of the main paper:

Sn+1 = ASn + KWn, (31a)

Zn = CSn +Wn. (31b)

The parameters of the model (31) are (A, C, K, V), of dimension (pQ × pQ,R × pQ, pQ × R,R × R); two of them

can be retrieved directly from the original SS parameters, i.e. A = A and C = C(r,:), while the gain K and thereduced innovation covariance V = E[WnW

ᵀn ] = ΣW can be obtained from the parameters in (29) and (30) by solving

a discrete algebraic Riccati equation (DARE) (see, e.g., refs. [28, 44] for detailed derivations).The computation of Granger-causal measures requires to formulate two additional reduced models, i.e. those

describing the blocks Z1 and Z2 which compose the reduced process Z. These models, which are defined and identifiedas described above using the subprocesses Z1 or Z2 in place of Z in (30) and (31), constitute the SS representationof the restricted VAR models formulated in (27). Their identification solving the DARE equation returns, among theother parameters, the covariance matrices of the innovations V1 and V2, which can be used in (28) together with thecovariance of the innovations W to derive the terms of the causal decomposition of the MIR between Z1 and Z2. Wenote that this procedure is alternative to the one working in the frequency domain described in the main paper, whichis presented in extended form in the next section of this Supplemental Material. The advantage of the proceduredescribed here is that it does not need to translate the models in the frequency domain and then to provide fullspectral integration to retrieve the time-domain measures, while the disadvantage is that it requires the formulationof the two additional restricted models to derive all the partial variances appearing in (28).

D. Frequency domain expansion

Starting from the subset Z = Z1, Z2 of the observed multivariate process, the joint parametric description (26)can be expressed in the frequency domain taking the FT to obtain Eq. 7 of the main paper:

Z(ω) = [IR −∞∑k=1

Bke−jωk]−1W (ω) = H(ω)W (ω). (32)

16

Eq. (32) allows to obtain the transfer function matrix H(ω) starting from the parameters Bk of the reduced VARmodel (26). However, since the identification of this infinite-order model is impractical, we replace it with theidentification of the reduced model (31); the frequency domain analysis of this model is performed taking the FT of(31a), which reads

S(ω) = AS(ω) + KW (ω)e−jω, (33)

from which it is easy to derive the PSD of the state process, S(ω), and to substitute it in the FT of (31b) to obtainEq. (17) of the main paper:

Z(ω) =(IR + C[IpQ − Ae−jω]−1Ke−jω

)W (ω) = H(ω)W (ω). (34)

Then, given transfer function matrix H(ω), the spectral factorization theorem allows to derive the PSD of theprocess using also the innovation covariance matrix [45]:

SZ(ω) = H(ω)ΣWH∗(ω). (35)

The matrix SZ(ω) can be factorized in blocks to make explicit the power spectral densities of Z1 and Z2, SZ1(ω) and

SZ2(ω), as diagonal blocks, and the cross-spectral densities between Z1 and Z2, SZ1Z2

(ω) and SZ2Z1(ω) = S∗Z1Z2

(ω),as off-diagonal blocks. From this factorization, a logarithmic spectral measure of the interdependence between Z1 andZ2 is defined by Eq. 8 of the main paper [19]:

fZ1;Z2(ω) = log|SZ1

(ω)||SZ2(ω)|

|SZ(ω)|; (36)

this measure quantifies the total (symmetric) coupling between the two block processes, and is related to the so-calledblock coherence [46], which extends to vector processes the standard spectral coherence function [47]. Moreover, afterusing (35) to expand the PSD of Zi, i = 1, 2, as

SZi(ω) = Hii(ω)ΣWiH∗ii(ω) + Hij(ω)ΣWjH

∗ij(ω) + Hij(ω)ΣWjiH

∗ii(ω) + Hii(ω)ΣWijH

∗ij(ω), (37)

where Hij describes the transfer from Wj to Zi in the frequency domain and ΣWji = E[Wj,nWᵀi,n], the logarithmic

spectral measure of the causal effect of Zi on Zj can be computed by Eq. 9 of the main paper as [19]

fZj→Zi(ω) = log

|SZi(ω)|

|Hii(ω)ΣWiH∗ii(ω)|

. (38)

To complete the representation of the pairwise interactions between Z1 and Z2, a spectral equivalent of the measureIZ1 •Z2

given in (28c) can be defined as

fZ1 •Z2(ω) = log

|H11(ω)ΣW11H∗11(ω)||H22(ω)ΣW22H

∗22(ω)|

|SZ(ω)|, (39)

so as to satisfy the frequency-domain given in Eq. 10 of the main paper:

fZ1;Z2(ω) = fZ1→Z2(ω) + fZ2→Z1(ω) + fZ1 •Z2(ω). (40)

Note that, since the measure (39) is defined ad-hoc to satisfy the decomposition of the total interaction, its physicalmeaning is not straightforward [20]; as its formulation depends on both the transfer functions H12 and H21, it hasbeen recently interpreted as a spectral measure which reflects the ”mixing effects” between the directed interactionsalong the directions Z1 → Z2 and Z2 → Z1, i.e. the part of the interactions between the two processes which cannotbe disentangled and assigned to one of the two causal directions [48].

Importantly, all the spectral measures appearing in (40) can be linked to the similar measures given in the timedomain in (28). In fact, it can be shown (see, e.g., [20]) that the integration over the whole frequency axis of thespectral measures yields the corresponding time-domain measure:

IZ1;Z2 =1

4π

∫ π

−πfZ1;Z2(ω) dω, (41a)

TZ1→Z2=

1

4π

∫ π

−πfZ1→Z2

(ω) dω, TZ2→Z1=

1

4π

∫ π

−πfZ2→Z1

(ω) dω, (41b)

IZ1 •Z2 =1

4π

∫ π

−πfZ1 •Z2(ω). (41c)

17

The relations in (41), together with the formulation of the time-domain measures (28) in terms of MI rates, give tothe spectral measures an information-theoretic meaning. In particular, the total coupling fZ1;Z2

(ω) is a non-negativequantity measuring the density of information shared between Z1 and Z2 at the angular frequency ω. Similarly, thecausal measures fZ1→Z2

(ω) and fZ2→Z1(ω) quantify the density of the information transferred from one process to

the other as a function of frequency. We note that these two measures can in general take negative values at somefrequencies, although their integration over all frequencies is non-negative; negative values are possible when the off-diagonal blocks of ΣW are non-zero, i.e. when the block process is not strictly causal. In the case of strict causality,i.e. when ΣWij = ΣWji = 0, the non-negativity of fZ1→Z2(ω) and fZ2→Z1(ω) is guaranteed at each frequency [20, 48].On the contrary, the measure fZ1 •Z2(ω) can take negative values even for strictly causal processes, as its integral overall frequencies is IZ1 •Z2 = 0 when ΣWij = ΣWji = 0 (see (28c)) [48].

Exploiting the analogy between the decompositions resulting in the time domain from (28) and in the frequencydomain from (36-39), we can achieve a causal decomposition of the OIR formulated for spectral functions. Forinstance, considering N stochastic processes X1, . . . , XN and setting Z1 = XN

−j and Z2 = Xj , the OIR incrementcan be expanded in frequency as

∆XN−j ;Xj

=1

4π

∫ π

−πδXN

−j ;Xj(ω) dω, (42)

where the frequency-specific OIR increment is defined in analogy to (21b) by Eq. 12 of the main paper,

δXN−j ;Xj

(ω) = (2−N)fXj ;XN−j

(ω) +

N−1∑m=1m 6=j

fXj ;XN−mj

(ω), (43)

and can be expanded through a causal decomposition similar to (23) as

δXN−j ;Xj

(ω) = δXN−j→Xj

(ω) + δXj→XN−j

(ω) + δXN−j

•Xj(ω), (44)

where the three terms on the r.h.s. of (44) are obtained expanding fXj ;XN−j

(ω) and fXj ;XN−mj

(ω) in (43) according to

(40). Moreover, the spectral OIR increment (43) can be used to compute recursively a frequency-domain version ofthe OIR, in analogy to (21), as

νXN (ω) = νXN−j

(ω) + δXN−j ;Xj

(ω). (45)

Considering (44) and (45), and given (42), it is easy to show that the spectral OIR and all terms of the causaldecomposition of the spectral OIR increment satisfy individually the spectral integration property, i.e. the average overall frequencies of each of these spectral functions yields the corresponding information-theoretic function. Therefore,the spectral versions of the high-order interaction measures defined in this section can be meaningfully interpreted asdensities of the synergistic/redundant character of the information shared between multiple stochastic processes.

THEORETICAL EXAMPLES

A. Simulation 1

The first simulation considers three scalar Gaussian processes X1, X2, X3 whose dynamics and interactions aredefined by the trivariate VAR model:

X1,n =

4∑k=1

a11,kX1,n−k + U1,n (46a)

X2,n =

q+1∑k=1

a21,kX1,n−k + a22,1X2,n−1 + a22,2X2,n−2 + U2,n (46b)

X3,n =

q+1∑k=1

a31,kX1,n−k + a32X2,n−1 + U3,n (46c)

In (46), U1, U2, and U3 are white uncorrelated Gaussian noise processes with zero mean and variance set respectivelyto σ2

U1= 2, σ2

U2= 0.5, and σ2

U3= 2. The coefficients determining self-dependencies (i.e., aii,k, i = 1, 2) are set

18

placing two pairs of complex-conjugate poles in the complex plane (with modulus ρ11 = 0.85 and ρ12 = 0.85 andphases φ11 = 2π0.1 rad and φ12 = 2π0.35 rad) to generate autonomous oscillations at the frequencies ∼ 0.1 Hz and∼ 0.35 Hz for X1, and placing one pair of complex conjugate poles in the complex plane (with modulus ρ2 = 0.7and phase φ2 = 2π0.1 rad) to generate an autonomous oscillation at the frequency ∼ 0.1 Hz for X2 (here, we denotefrequencies in Hz assuming sampling frequency fs = 1). The coefficients determining causal dependencies (i.e.,aij,k, i, j = 1, 2, 3, i 6= j) are set as follows: a21,k = 0.4b1,k, where b1,k are the coefficients of a low-pass FIR filter oforder q = 20, with cutoff frequency at 0.2 Hz; a31,k = 0.6b2,k, where b2,k are the coefficients of a high-pass FIR filterof order q = 20, with cutoff frequency at 0.2 Hz; a32 = 1 to realize an all-pass filter.

This simulation is implemented by the script test oir simu1.m of the OIR Matlab toolbox, which produces theresults shown in Fig. 1 of the main paper.

B. Simulation 2

The second simulation considers ten scalar Gaussian processes whose dynamics and interactions are defined by the10-variate VAR model:

Y1,n = 2ρ1 cos(2πf1)Y1,n−1 − ρ21Y1,n−2 + U1,n (47a)

Y2,n = 0.5Y1,n−1 + U2,n (47b)

Y3,n = 0.5Y2,n−1 + U3,n (47c)

Y4,n = −0.5Y1,n−2 + 0.2Y3,n−1 + 0.5Y10,n−1 + U4,n (47d)

Y5,n = 2ρ1 cos(2πf1)Y5,n−1 − ρ21Y5,n−2 + U5,n (47e)

Y6,n = 0.3Y7,n−2 + U6,n (47f)

Y7,n = 2ρ1 cos(2πf1)Y7,n−1 − ρ21Y7,n−2 + 0.3Y6,n−1 + U7,n (47g)

Y8,n = 2ρ2 cos(2πf2)Y8,n−1 − ρ22Y8,n−2 + 0.4Y2,n−2 + 0.3Y3,n−1 − 0.4Y5,n−1 + 0.3Y7,n−1 + U8,n (47h)

Y9,n = 0.7Y8,n−1 − 0.2Y10,n−2 + U9,n (47i)

Y10,n = 0.4Y9,n−1 + U10,n (47j)

In (47), the innovation processes U1, . . . U10 are white uncorrelated Gaussian noise processes with zero mean andunit variance. The coefficients determining self-dependencies (i.e., aii,k, i = 1, 2) are set placing one pair of complexconjugate poles in the complex plane to generate an autonomous oscillation at the frequency ∼ 10 Hz for Y1, Y5 andY7 (pole modulus ρ1 = 0.9, pole frequency f1 = 10/fs = 0.1 Hz), and at the frequency ∼ 25 Hz for Y8 (pole modulusρ2 = 0.8, pole frequency f1 = 25/fs = 0.1 Hz), with simulated sampling frequency fs = 100 Hz. The coefficientsdetermining causal dependencies are set with the values indicated in (47), to obtain directed dependencies betweenpairs of processes as depicted in Fig. 3a of the main paper. The analysis of HOIs is then performed after grouping theoriginal 10 processes into 5 blocks organized as follows: X1 = Y1, Y2, Y3, Y4; X2 = Y5; X3 = Y6, Y7; X4 = Y8;X5 = Y9, Y10.

This simulation is implemented by the script test oir simu2.m of the OIR Matlab toolbox, which produces theresults shown in Fig. 3 of the main paper.

CARDIOVASCULAR AND RESPIRATORY INTERACTIONS DURING PACED BREATHING

A. Data acquisition and experimental protocol

The analyzed data belong to an historical database of cardiovascular and respiratory time series measured duringa protocol of paced breathing [9, 31]. The signals were recorded with a sampling frequency of fs = 300 Hz on acontrol group composed of 19 healthy subjects (11 females, 8 males, age: 27-35, median: 31 years). Data consisted inelectrocardiographic (ECG, lead II), noninvasive arterial blood pressure (BP) (Finapres 2300, Ohmeda, Englewood,CO), and respiratory flow (RF) (nasal thermistor by Marazza, Monza, Italy) signals. The experimental protocolincluded four different sessions, with subjects laying in the supine position normally breathing (spontaneous breathing,SB) or following a controlled metronome breathing at 10, 15 and 20 breaths/min (CB10, CB15 and CB20, respectively).The period of spontaneous respiration was always carried out first, followed by the controlled breathing sessions insteadperformed in a random order. The data of one subject were excluded from the analyses, since the SAP time series,extracted from the corresponding BP signal, was not readable during the condition CB20. The total number ofanalysed subjects was then reduced to 18.

19

B. Data pre-processing

Three time series were extracted from ECG, AP and RF signals on a beat-to-beat basis as follows: (i) the heartperiod (HP) series was extracted as the sequence of the temporal distances between consecutive R peaks (R-R intervals)of the ECG signal; (ii) the systolic arterial pressure (SAP) series was obtained as the sequences of the maximum valuesof the BP signal measured within each detected R-R interval; and (iii) the respiration (RESP) series was extractedas the sequence of RF values sampled at the onset of each detected R-R interval. Further information about signalacquisition and series extraction can be found in [9, 31]. For each subject and experimental condition, stationarysegments of N = 256 points were selected through a visual inspection. Before the analysis, the series were linearlydetrended using a zero-phase autoregressive (AR) high-pass filter with a cut-off frequency equal to 0.0156 fs, andthen normalized to zero mean [49].

C. Data analysis

The analysis was performed computing, in both time and spectral domains, the OIR increments obtained addingthe HP series (X3) to the bivariate network formed by the RESP (X1) and SAP (X2) series. Specifically, we identifieda VAR model fitting the vector process comprising the three time series of interest (X1, X2, X3) by using leastsquares identification. Model order selection was first performed using the Akaike criterion, setting the maximumlag to 14. Then, model orders were manually adjusted by visual inspection in the range 3-8. The OIR toolbox wasthen used to compute, in the frequency domain, the OIR increment relevant to the addition of HP to SAP, RESP(δX3;X1,X2

(ω)), the causal OIR increment from HP to SAP, RESP (∆X3→X1,X2(ω)), the causal OIR increment from

SAP, RESP to HP (∆X1,X2→X3(ω)) and the instantaneous OIR increment (∆X3·X1,X2(ω)). The spectral profileswere integrated within the Low Frequency (LF) and High Frequency (HF) bands defined as follows: regarding the LFband, we considered the range [0.04-0.12 Hz]; regarding the HF band, we first identified the respiratory peak withinthe range [0.12-0.4] Hz for each subject, and then selected the range around the peak with a width of ±0.04 Hz.

D. Statistical analysis

The statistical significance of the distributions obtained for the measures was performed using non-parametrictests, given the small sample size and since the assumption of normality was rejected for most distributions using theAnderson-Darling test. The non-parametric one-way Friedman test was employed to assess the statistical significanceof the differences of the median of the distributions, followed by a post-hoc Wilcoxon test with Bonferroni-Holmcorrection for multiple comparison (n = 3) to assess the difference between each distribution in a controlled breathingcondition versus the spontaneous breathing reference (i.e. C10 vs SB, C15 vs SB, C20 vs SB). All the statistical testswere carried out with 5% significance level.

E. Results

The OIR decomposition analysis of cardiovascular time series is implemented by the script test oir HPRESPSAP.mof the OIR Matlab toolbox, which produces for one representative subject the spectral profiles shown in Fig. 4a ofthe main paper.

Fig. 6 reports the distributions across subjects (boxplots and individual values) of the OIR increments obtainedadding HP to SAP, RESP, as well as of the two causal terms and the instantaneous term of the decomposition ofsuch increments, computed in the four experimental conditions and integrated over all frequencies as well as withinthe selected LF and HF bands.

NEURAL INTERACTIONS FROM ECOG SIGNALS IN THE ANESTHETIZED MACAQUE MONKEY

A. Data acquisition and experimental protocol

The dataset used in this study can be downloaded from http://neurotycho.org/expdatalist/listview?task=45; theanalyzed data are relevant to the macaque monkey named Su. During the experiment analyzed in our work, themonkey was seated in a primate chair with both arms and head movement restricted and eyes covered to avoid

20

FIG. 6. OIR decomposition of cardiovascular and cardiorespiratory interactions computed during spontaneous breathing (SB)and controlled breathing (CB) at 10 (CB10), 15 (CB15) and 20 (CB20) breaths/min. (a), Time-domain, (b) LF (0.04-0.12Hz) and (c) HF (fRESP ± 0.04 Hz) values of the total OIR increment (∆X3;X1,X2) obtained with the addition of HP to SAP,RESP, of the causal OIR increments from SAP, RESP to HP (∆X1,X2→X3) and from HP to SAP, RESP (∆X3→X1,X2), andthe instantaneous OIR increment (∆X3·X1,X2). Asterisks denote statistically significant difference between the CB conditioncompared with SB (Wilcoxon signed-rank test: black, uncorrected; red, Bonferroni-Holm correction for multiple comparisons).

evoking visual response during the experimental period. We considered two experimental conditions, relevant toa resting state before (REST) and after (ANES) the injection of a cocktail of anesthetics consisting of ketaminehydrochloride (1.15 ml (8.8mg/Kg)) and Medetomidine (0.35 ml (0.05 mg/Kg)). All the experimental and surgicalprocedures were performed in a previous study according with the experimental protocol (No. H24-2-203(4)) approvedby the RIKEN ethics committee [23].

The complete description of the surgical implantation of the ECoG electrodes can be found in [50]. The acquireddata consisted of 128 ECoG signals recorded with a sampling frequency of 1 kHz with electrodes placed in pairs withan inter-electrode distance of 5 mm to cover the entire left hemisphere of the brain.

B. Data pre-processing

We considered two five-minutes recording sessions during the REST and ANES conditions. The ten bipolar ECoGsignals selected for the analysis (Figure 5a of the main paper) were band-pass filtered between 0.5 and 200 Hz toremove slow and fast components in the power spectrum (zero-phase Butterworth filter; notch filter 49-51 Hz with aslope of 48 dB/oct in the transition band), downsampled to fs = 250 Hz, epoched to extract ∼ 160 trials lasting 2sec for each condition, and finally normalized to zero mean and unit variance within each trial.

C. Data analysis

The analysis of high-order interactions was performed starting from the VAR parameters of the model fitting each2-sec trial. Considering the M = 5 blocks of time series acquired from the 10 pairs of bipolar electrodes analyzed, thespectral OIR was computed for all multiplets of order N = 3, 4, 5. Time-domain OIR values (Ω) were then obtainedintegrating the spectral measures ν(f) within the δ (0.2-3 Hz), θ (4-7 Hz), α (8-12 Hz), β (12-30 Hz) and γ (31-70Hz) frequency bands, as well as cumulatively between 0 and 70 Hz (T).

21

c)

T δ θ α β γ

-2

0

2

4

6

8

10

12x 10-3

ΩX1,X2,X3,X4,X5

*

*

**

**

ΩX1,X2,X4

**

T δ θ α β γ

ΩX1,X2,X3,X5

** * *

*

ΩX2,X3,X4

**

ΩX1,X2,X5

**

T δ θ α β γ

ΩX1,X2,X4,X5

*** * *

ΩX2,X3,X5

* ** *

ΩX1,X3,X4

*

T δ θ α β γ

ΩX1,X3,X4,X5

** *

ΩX2,X4,X5

* *

ΩX1,X3,X5

*

T δ θ α β γ

ΩX2,X3,X4,X5

* *

**

ΩX3,X4,X5

* * *

-2

0

2

4

x 10-3ΩX1,X2,X3

** *

T δ θ α β γ

-202468x 10-3ΩX1,X2,X3,X4

*

* *

-2

0

2

4

ΩX1,X4,X5

*

x 10-3

a)

b)

FIG. 7. OIR analysis of neurophysiological interactions in the anesthetized monkey named Su. The plots depict the values(distribution across ∼ 160 trials) of the time-domain measure of HOIs obtained by integrating the spectral OIR over the wholefrequency axis (T) or within the δ,θ,α,β and γ bands. Results are reported for all multiplets of order 3 (a), all multiplets oforder 4 (a), and the multiplet of order 5 including all time series (c). Asterisks denote statistically significant difference betweenREST (red dots) and ANES (blue dots) obtained after Wilcoxon singed-rank test with Bonferroni correction (p < αc/6)).

D. Statistical analysis

Since we were only interested in the difference between the two experimental conditions (REST vs ANES), irre-spective of the analyzed multiplet of time series, we performed, for each multiplet and for each interval of integration,a Wilcoxon signed rank test with significance level (α) equal to 0.05, followed by Bonferroni correction for multiplecomparisons (six multiple comparison were considered, i.e. those between REST and ANES relevant to the OIRmeasure obtained within the δ, θ, α, β and γ bands, plus the whole-band time domain measure).

E. Results

The OIR decomposition analysis for the ECoG time series is implemented by the script test oir ECoG.m of theOIR Matlab toolbox, which produces for one representative 2-sec trial the spectral profiles shown in Fig. 5b of themain paper.

Fig. 7 reports the distributions across subjects (boxplots and individual values) of the OIR, computed in the twoexperimental conditions (REST, ANES) and integrated over all frequencies (T) as well as within the five selectedfrequency bands ( δ, θ, α, β and γ).