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8910 Accepted 12 April 201311 Available online xxxx121314151617
Markov process18 Discrete dynamics19 Partition20 State space21
Graph
22of23the actual nature of its time evolution remains unclear.
Computational models propose specic state space24the dynamic
repertoire of the resting brain. Nevertheless, methods devoted
to25organization of brain state space from empirical data still
lack and thus preclude26
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43 1. Introduction
44 Brain activity is a spatiotemporal signal with deterministic
and45
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61none of its temporal nature.62
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NeuroImage xxx (2013) xxxxxx
YNIMG-10353; No. of pages: 15; 4C: 5, 6, 7, 8, 9, 10, 11, 14
Contents lists available at SciVerse ScienceDirect
NeuroIm
.e lUNCOstochastic characteristics. As time evolves the brain
dynamics traces
out a trajectory in a high-dimensional state space. The
objective of allanalysis methods is to provide a characterization
of this trajectory.The most frequently used approaches apply some
sort of informationcompression with the objective that the achieved
reduction in com-plexity is still sufciently informative to allow
for comparisons betweendata sets, let it be different experimental
conditions, different subjectgroups or between model and empirical
data. Such approaches includeentropy, dimension estimates, various
forms of complexity, Lyapunovexponents, and covariances just to
name a few (see e.g. Stam (2005)
We illustrate this thought along a hotly debated example in
neuro-science, that is the resting state dynamics of the brain,
which istypically accessed through functional magnetic resonance
imaging(fMRI), electroencephalographic (EEG) and
magnetoencephalographic(MEG) measurements. The fMRI measures the
blood oxygenationdependent level (BOLD) signals of the hemodynamic
response on highresolution spatial scales (mm) and slow temporal
scales (sec). Here alimited number of 810 robust network patterns
of BOLD activityhave been observed in the primate brain during rest
in the BOLD signals(Biswal et al., 1995; Damoiseau et al., 2006;
Raichle et al., 2001). Theseand Subha et al. (2010), for reviews).
For eachple derivatives exist with various degrees of sapproaches
have in common by constructionof dynamics, that is the actual
notion of tempo
Corresponding author at: Institut de NeuroscienceAix-Marseille
Universit, Inserm, 27 Bd. Jean Moulin,
E-mail address: [email protected] (L. Peza
1053-8119/$ see front matter 2013 Published by
Elhttp://dx.doi.org/10.1016/j.neuroimage.2013.04.041
Please cite this article as: Hadriche, A., et al.,
Mj.neuroimage.2013.04.041R consequence quantitative descriptions of
brain activity are impove-rished and only capture limited aspects
of the brain trajectory andRECTE
DWepropose here an algorithm based on set oriented approach of
dynamical system to extract a coarse-grainedorganization of brain
state space on the basis of EEG signals. We use it for comparing
the organization of thestate space of large-scale simulation of
brain dynamics with actual brain dynamics of resting activity in
healthysubjects.The dynamical skeleton obtained for both simulated
brain dynamics and EEG data depicts similar structures.The skeleton
comprised chains of macro-states that are compatible with current
interpretations of brainfunctioning as series of metastable states.
Moreover, macro-scale dynamics depicts correlation featuresthat
differentiate them from random dynamics.We here propose a procedure
for the extraction and characterization of brain dynamics at a
macro-scale level.It allows for the comparison between models of
brain dynamics and empirical measurements and leads to thedenition
of an effective coarse-grained dynamical skeleton of spatiotemporal
brain dynamics.
2013 Published by Elsevier Inc.Keywords:Resting state comparison
of the hypothetical dynamical repertoire of the brain with the
actual one.organization which denesthe characterization of
theMapping the dynamic repertoire of the re
Abir Hadriche a,b, Laurent Pezard a,, Jean-Louis NandAbdennaceur
Kachouri b, Viktor K. Jirsa a
a Aix Marseille Universit, INSERM, INS UMR S 1106, 27 Bd. Jean
Moulin, 13005 Marseille,b Universit de Sfax, ENIS, LETI, Route de
Soukra Km 2.5, BP. 1173 3038, Sfax, Tunisiac Universit de Lille
Nord-de-France, URECA EA 1059, Domaine Universitaire du Pont de
Bo
a b s t r a c ta r t i c l e i n f o
Article history: The resting state dynamics
j ourna l homepage: wwwof thesemeasures multi-ophistication. All
of thesethat they lose their senseral evolution is lost. As a
des Systmes, UMR S 1106,13005 Marseille, France.rd).
sevier Inc.
apping the dynamic repertoOOF
ing brain
no c, Hamadi Ghariani b,
ce
9653 CEDEX Villeneuve d'Ascq, France
the brain shows robust features of spatiotemporal pattern
formation but
age
sev ie r .com/ locate /yn img72patterns are characterized by
spontaneous intermittent coherentuctu-73ations on an ultraslow
scale of b0.1 Hz. Correlation patterns amongst74the resting state
networks with strong positive and negative correla-75tions have
been observed (Fox et al., 2005). Though this nding pro-76vides a
constraint on the permissible spatiotemporal brain dynamics,77the
details remain still unclear. We still do not know if this
temporal78dynamics translates into an oscillatory coactivation of
selected resting79state networks, or a pattern competition, or even
a hopping process
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
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2 A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxUNCORREC
from one resting state network to another. These important
details willexpress themselves through features of the trajectories
in the statespace of the brain network, including their geometry
and topology, aswell as zones of increased convergence and
divergence of the trajectorydensities. The total loss of temporal
dynamics has been nicelydemonstrated by Wen et al. (2012) who
applied common techniquesof prevailing statistical approaches to
surrogate fMRI data (subjectedto covariant randomization of the
temporal order) and obtained identi-cal results regarding the
resting state networks.
We nd a similar situation in electroencephalographic data.
Firstof all, the BOLD signal and its electrophysiological correlate
show aclear but non-trivial relationship. To investigate this
relationshipbetween the BOLD signal and its underlying neural
activity Logothetiset al. (2001) demonstrated in anesthetized
monkeys that simultaneousfMRI and intracortical electrical
recordings of localeld potentials showpositively correlated
spontaneous uctuations. Laufs et al. (2003)recorded simultaneous
fMRI and EEG in awake human subjects at restand found that the
-band power was positively correlated with theBOLD signals in the
posterior cingulate, the precuneus, the temporo-parietal and
dorsomedial prefrontal areas. These areas are knownfrom one of the
resting state network patterns referred to as the DefaultMode
Network (DMN), which is characterized by a reduced activationduring
task conditions and elevated activation in absence of any
task.Mantini et al. (2007) also showed that different frequency
bands corre-late differentlywith the various resting state network
patterns, whereasa recent study by Scheeringa et al. (2011)
demonstrated that the BOLDsignal in humans performing a cognitive
task is related to synchroniza-tion across different frequency
bands. In the latter, trial-by-trial BOLDuctuations correlated
positively with high power of the EEG andnegativelywith and. Hence,
there is empirical evidence that the spa-tiotemporal dynamics of
the BOLD signal is related to the spatiotempo-ral dynamics of
electroencephalographic signals, but the actual natureof the
spatiotemporal organization, in particular its time evolution isnot
clear.
The situation is similar for the theoretical studies in the
restingstate literature, certainly motivated by the objective to
achieve corre-spondence with empirical data (see Deco et al.
(2011), for a recent re-view). Essentially, in a series of
theoretical studies (Deco et al., 2009;Ghosh et al., 2008; Honey et
al., 2007) the authors demonstrate thatthe resting state pattern
formation can be understood as emergingfrom the network
interactions of a brain network with a full brainconnectivity
derived from either the CoCoMac database or usingtractographic data
(DTI/DSI). Convergent evidence based on thesetheoretical models was
provided that multiple states exist duringthe resting state, which
are explored in a stochastic process drivenby noise. Such models
agree on the importance of noise-driven explo-ration of the brain
dynamical repertoire, but do not fully agree on thedeterministic
skeleton of this dynamical repertoire. The dynamicrepertoire is the
set of all themacrostates thatmay be occupied duringthe resting
state dynamics and the deterministic skeleton is the setof
transitions between these macrostates. Taken together they
thusinclude the geometry and topology of the trajectories, as well
astheir convergence properties indicating stability. One can view
the en-semble of dynamic repertoire and skeleton as a graph
composed ofmacrostates on the nodes and the edges indicating the
possible transi-tions. Such a visualization is effectively a
discrete representation of thecontinuous time evolution of the
brain network in its state space.Ghosh et al. (2008) named the
noise driven exploration of such abrain dynamics graph the
exploration of the dynamic repertoire ofthe human brain due to the
similarity between the resting state pat-terns and network
activations known from task-related conditions.
Initial interpretations of neuronal dynamics have mainly
focusedon attractor dynamics (Amit, 1989; Hopeld, 1982), leading to
aclear hypothesis concerning the organization of the brain state
space.Nevertheless, it soon appeared that transients and
metastability play
an important role due to e.g. high dimension or sparse
connectivity
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
ROOF
(Friston, 1997; Tsuda, 2001). This leads to the proposal that
such a com-plex coordinated system as the brain is best understood
as amultistableand metastable system at many levels from
multifunctional neuralcircuits to large-scale neural circuits
(Kelso, 2012). Nevertheless, theseproposals have been
quantitatively compared with real data (fMRI,EEG or MEG) mainly on
the basis of the comparison of time-series ofspectral contents
(Freyer et al., 2011) or the comparisons of covariancematrices
(functional connectivity) (Deco and Jirsa, 2012) betweenexperiment
and model. For example, Deco and Jirsa (2012) recentlycorrelated
the functional connectivity matrices of the simulated brainnetwork
data with the experimental BOLD signals and obtained
highcorrespondence. They also demonstrated the existence of ve
restingstate patterns for an elevated value of coupling strength
but alsofound that the best t of functional connectivity is
obtained at a lowerlevel of coupling, i.e. when the resting state
patterns do not exist asstable states in the state space, but are
close to their creation(a so-called bifurcation). For this reason,
Deco and Jirsa named themghost attractors, which comprise the
dynamic repertoire of the brainnetwork. Here a subtle distinction
plays an important role: Theseghost attractors are assumed to exist
due to their parametric proximityto the bifurcation. This situation
is called criticality. The assumptionbehind this hypothesis is that
below the bifurcation point (subcritical)and above (supercritical),
the density of the trajectories in the statespace changes smoothly.
Though this assumption is reasonable, it stillremains to be shown
that the ghost attractors indeed exist in the sub-critical regime.
Once their existence, and thus the dynamic repertoire,is
established, then this will be the rst step towards the full
character-ization of the trajectories within the state space
connecting the ghostattractors, and hence the full characterization
of the spatiotemporalbrain dynamics.
As we illustrated above, although the presence of a
spatio-temporalorganization in brain activity has clear empirical
support, the under-standing and interpretation of this organization
remains an openproblem. Moreover, the deterministic dynamical
skeleton in the statespace is not easily characterized at the level
of scalar metrics (spectralanalysis or even correlation) and render
the comparison of model andempirical brain activity in its state
space indispensable. From the viewpoint of dynamical system theory
the only unambiguous representa-tion of a dynamic system is given
by its ow in the state space. Saiddifferently, two dynamical
systems are different from each other ifand only if the topology of
their ow is different. If not, then theybelong to the same class of
systems. The ow prescribes the evolutionof the state vector of the
system as a function of its current state and ismost commonly
written as a set of differential equations. In terms ofstate space
analysis, two approaches are typically taken in the litera-ture.
The rst approach is based on nonlinear time series analysis
de-rived from dynamical systems theory (see e.g. Kantz and
Schreiber(2004)) and has mostly focused on the properties of
continuoustrajectories in the brain's state space. The second
approach is basedon the stochastic approach of dynamical systems
(Lasota and Mackey,1994). The main difference between these two
complementaryapproaches lies in the fact that the rst one supposes
a differentialstructure of the trajectory, whereas the second one
does not and thusallows one to link dynamical systems and
stochastic processes. Thestochastic approach of dynamical systems
thus becomes a naturaltheoretical setting for the characterization
of brain dynamics all themore that high dimensional dynamics can be
hardly differentiatedfrom stochastic processes (Lachaux et al.,
1997). Nevertheless, thereare subtleties that go beyond the scope
of this article, but should bekept in mind when interpreting data.
To some extent noise, which isinvariably present in biological
systems, can obscure the underlyingdifference but not always: A
subcritical bifurcation is only smoothedinto an apparent
supercritical bifurcation by additive noise of a suf-cient
amplitude otherwise the discontinuity of the former remains.In the
presence of multiplicative noise, the statistics of the
system211will be also qualitatively different. Similarly, the
symmetry of noise
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3A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxUNCORREC
may play a crucial role in the transition dynamics of
multistable sys-tems (Freyer et al., 2011). In the current article
we focus to uncoverthe regimes in the state space, in which the
brain dynamics spends alonger time as compared to other regimes.
Under these conditionsthese former regimes will correspond to
either stable network patternsor zones of slow and convergent ows,
which Deco and Jirsa (2012)called ghost attractors. The latter play
effectively a similar role as thenetwork states, since in the
presence of noise the difference betweenthem is smeared out.
Thus our goal is twofold: rst propose an approach of
braindynamics which allows obtaining information on the
organizationof brain dynamics in its state space and then to
compare state spaceorganization for large-scale simulation of brain
electrical activity atrest and experimental EEG data in healthy
subjects. Let us considerbrain activity within the framework of
ergodic theory i.e. let B; T be the brain dynamical system with B
its state space and T : BBthe transformation that allows changing
brain state. We have accessto B only through a measurement m : BE
where E is the statespace of the measurement apparatus. In the case
of EEG, the measure-ment space is related to the electrode
placement system and theresolution of the analog to digital
conversion system; thus E is discreteand thus constitutes a
partition of B. This partition is our highestresolution andwewill
call it ourmicro-scale. Nevertheless, due to nitetime observation
this partition is too rened to be studied at themicro-scale level
and we propose a coarse-graining procedure whichmakes a compromise
between description scale and statistical errors.Such an
alternative approach is based on proposals in the context
ofthermodynamics of non-equilibrium systems (Gaveau and
Schulman,1996), set oriented approach of dynamical systems
(Dellnitz andJunge, 2002), approximation of dynamics (Dellnitz and
Junge, 1999;Froyland, 2001), nite resolution dynamics (Luzzatto and
Pilarczyk,2011), symbolic image of dynamics (Osipenko, 2007) and
symbolicapproach of complex systems (Gupta and Ray, 2009).
Regardingnonlinear approach of brain dynamics, this approach, also
proposedby Allefeld et al. (2009), does not emphasize the
estimation of invariantindices but deals with the
qualitative/topological organization of thebrain phase space.
The initial step of this procedure is a subdivision of the
measure-ment space which allows one to obtain an approximate model
ofbrain's dynamics in terms of a meso-scale discrete Markov
process(Dellnitz and Junge, 1999). Once the discrete representation
isobtained one can search for regions of the phase space with
specicdynamical features such as dynamical slowing down or
recurrence.The set of coarse-grained regions and their dynamical
transitionsdenes a coarse-grained dynamical skeleton in measurement
statespace and as such an effective macro-scale model of brain
dynamics.
This paper is organized as follows: in the next section, we
describeour approach in a general manner. It proceeds in several
transforma-tions with the goal of replacing the micro-scale of the
EEG signal by agraph of transitions between macro-scale brain
states. The steps ofthis procedure are then applied to a
multistable stochastic system inorder to test this procedure on a
simple benchmark example. Thewhole procedure is then applied to
resting brain dynamics in bothlarge-scale simulation and in real
EEG data from healthy subjects. Itallows one to compare the
macroscopic organization of the realbrain dynamics with simulated
one and to characterize resting activ-ity by a set of
coarse-grained patterns.
2. Materials and methods
2.1. Procedure
We provide here a general description of the procedure used for
thecoarse-grained approach of brain dynamics. It allows us to
introducethe notions used throughout the remainder of this article.
Each step
of the procedure and specic notations are detailed in Appendix
A.
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
ROOF
Our approach is based on transformations from the
measurementspace (i.e. micro-scale) to a coarse-grained
representation (i.e. macro-scale) through an intermediate level
(i.e. meso-scale).
2.1.1. From micro-scale to meso-scaleThe rst step of the
procedure is an iterative discretization of the
measurement space which leads to the meso-scale representation.
Ourapproach is based on a subdivision's algorithm (Dellnitz and
Hohmann,1997) inspired by Ulam's discretization process (Ulam,
1960). It isthe basis of set oriented numerical methods for
dynamical systems(Dellnitz and Junge, 2002). The entropy rate of
the discrete process isused to choose the last discretization step
as described in Appendix A.When the last discretization step is
found, the system's dynamics is rep-resented by a sequence of
discrete meso-states. Each meso-state corre-sponds to a set of
micro-states, i.e. the set of spatial EEG patterns in thepartition
element.We can represent ameso-state as amean EEG patternaveraged
over all the EEG patterns in one partition.
As a working hypothesis, the sequence of meso-states is then
con-sidered as the result of a rst-order Markov process. This
hypothesisprovides a simple approximation of the dynamics at the
meso-scalewhich is characterized by its stationary distribution and
its transitionmatrix. This working hypothesis is difcult to check
experimentallyand should be further investigated in subsequent
studies.
2.1.2. From meso-scale to macro-scaleThe transformation from
meso-scale to macro-scale is based on
the network representation of master equation systems, Markov
pro-cesses or symbolic dynamical systems (Kitchens, 1998;
Osipenko,2007; Schnakenberg, 1976). In fact, it has been shown that
dynamicalproperties of complex systems can be related to the
properties of theassociated network representation (Donner et al.,
2010; Marwanet al., 2009). The network representation of a Markov
process is agraph, whose nodes are the states of the process and
the edges theobserved transitions between the states (see Appendix
A).
On the basis of this graph, our next step needed to identify
regions ofthe meso-scale state space with specic dynamical
characteristics. First,regions of the state space where the
dynamics slows down or withhigh recurrence are identied as sets
ofmesostateswith high probability.Second, parts of the graph with
complete connectivity (i.e. cliques, seeAppendix A) can be
associated with regions depicting a kind of dynami-cal invariance
perturbed by random uctuations. Moreover, since theMarkov process
is built from experimental data, some properties (suchas connected
or attracting sets) trivially correspond to the whole graphand the
clique organization is an intermediate level. On the basis ofthese
constraints, our algorithm uses the probabilistic properties of
thestationary distribution and the clique organization of the
network repre-sentation of the meso-scale Markov process for
coarse-graining themeso-states intomacro-states. A representation
of the EEGpattern corre-sponding to a macro-state can be obtained
as the weighted average ofEEG patterns associated with the
meso-states in the macro-state. Thedynamics of the system is thus a
sequence of macro-states, which isstudied to characterize the
macro-scale dynamics.
2.1.3. Adaptation of the procedure to brain dynamicsThe
procedure explained above should be adapted to the practical
case of brain dynamics using a preprocessing step. In fact, in
the caseof brain dynamics one starts with a measurement space where
eachdimension has a physical meaning related to the electrode
placement.This arbitrary order can lead to misleading results in
the subdivisionstep. In order to avoid such a problem, a principal
component analysis(PCA) is used as a preprocessing step. PCA
induces an order in thedimensions since it orders the dimensions
according to decreasingvariance. PCA is used here only for rotating
axis and not for dimensionreduction; the latter would necessitate a
truncation of the PCA expan-sion, which we do not perform, and we
leave the dimension reduction337to the discretization process.
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oImage xxx (2013) xxxxxxUNCORREC
2.1.4. Statistical validationEach step of the procedure was
statistically validated against
adapted surrogate data (Schreiber and Schmitz, 2000; Theiler et
al.,1992) depending on the null hypothesis to be tested.
Multivariate phase-randomized surrogate data (Prichard
andTheiler, 1994) were used to test the rst step of our procedure
i.e. theresult of the subdivision procedure. Since these surrogate
data keeppower spectra of single signals and cross-correlation
between signals,they allow us to test the null hypothesis that the
subdivision processleads to the same process for raw data as for
multivariate randomprocesses. The entropy rate of the effective
Markov process was usedas a statistics for this test. The null
hypothesis was rejected using anon-parametric procedure with a
signicant threshold = 0.05 for aset of 50 multi-variate
phase-randomized surrogate data for each rawdata set. When the null
hypothesis can be rejected, we can considerthe meso-scale Markov
process as a validated discrete approximationof the raw
dynamics.
The determination of the macro-states for the raw dynamics
wastested against macro-states obtained for random sequences.
Namely,macro-states obtained using the meso-scale Markov
approximationof the dynamics can be characterized by their size
(i.e. the number ofmeso-states in the macro-state) and their
probability (i.e. the sum ofprobability of the meso-states in the
macro-state). The coarse-graining was thus computed for the raw
meso-scale Markov processand for shufed versions of this process.
The shufed surrogate datakeep the same stationary distribution as
the raw meso-scale Markovprocess, but randomize the transitions.
For each raw meso-scalesequence, a set of 50 shufed surrogate data
was generated. Themacro-states obtained from the raw meso-scale
process were thencompared with those obtained from the shufed
surrogate data. Foreach size, only macro-states with higher
probability than that of allthe surrogate macro-states of the same
size were selected for furtherinvestigation. The macro-states that
were not selected are then consid-ered as a random component of the
macro-scale dynamics and theselected macro-states as a non-random
component.
2.1.5. Characterization of the macro-scale dynamicsOnce
non-random macro-states were selected, we reduced our
investigation to these states by discarding the macro-states of
therandom component from the sequence of macro-states. The
dynamicsof these states was then characterized for both their
typical residencetimes (i.e. their correlation) and for their
transitions.
Firstly, the typical residence time was computed for each
macro-state as the mean length of runs and compared with the
typical resi-dence time obtained for shufed random sequences. The
same statisti-cal procedure as for the statistical validation
(Section 2.1.4) was usedhere i.e. 50 shufed surrogate sequences and
a statistical thresholdp = 0.05.
Secondly, the transitions between non-random macro-states
wereestimated using sequences where successive repetitions of the
samemacro-state were reduced to one single occurrence of the
macro-state. The transitions were then considered signicant if they
werehigher than those observed for 50 shufed surrogate versions of
thereduced sequence. These validated transitions observed for the
non-random component allow one to infer a macro-scale dynamical
skele-ton between the macro-states.
2.2. Data
2.2.1. Multistable stochastic systemIn order to extract an
effective macro-scale dynamical skeleton
frommultidimensional data, we studied a two dimensional
stochasticsystem (x1,x2) in discrete time as in Allefeld et al.
(2009), dened by:
xi a xi2x3i
bii 1
4 A. Hadriche et al. / NeurPlease cite this article as:
Hadriche, A., et al., Mapping the dynamic
repertoj.neuroimage.2013.04.041were excluded from the analysis.EEG
signals were digitized on 16 bits of precision using a 250 Hz
sampling frequency and ltered using a band-pass lter between0.5
Hz and 70 Hz. EEG recordings were obtained from each participantin
eyes closed and resting condition. About 2 min (i.e.: 117 s or
3.104
time samples) of artifact-free EEG recordings was visually
selectedfrom each subject. Effective coarse-grained dynamical
skeleton wasthus obtained for 12 subjects in the 17-dimensional
measurement
4ED P
ROterms of the neuron model used. The connectivity matrix for
the full
brain network is an extrapolation of the human anatomy from
thema-caque monkey brain's connectivity (Kotter, 2004; Kotter and
Wanke,2005). The interaction between the vertices of the network
wasimplemented using delayed interaction terms. The network
modelwas solved using the EulerMaruyama method as implemented inthe
Virtual Brain platform (see Jirsa et al. (2010), for more
details).The equilibrium state of the network is a function of
transmissiondelay and coupling strength (Deco et al., 2011; Ghosh
et al., 2008;Jirsa and Ghosh, 2010). All parameters were identical
as used in thesubcritical regime by Ghosh et al. (2008). The delay
and strengthparameters were chosen in the neighborhood of the
instability ofequilibrium state of the deterministic system. Thus,
when the noisedrives a system out of equilibrium state, the system
explores theneighborhood of the equilibriumpoint and the subsequent
relaxationsestablish the resting-state dynamics. For these
parameter settings, noother stable equilibrium points exist.
Two sets of data of length T = 3.104 time points were
simulatedusing this model. Since FitzhughNagumo model has two state
vari-ables, these data sets correspond to brain dynamics in a 38 2
dimen-sional space. Only the voltage state variable is taken into
account as ourmeasurement and the measurement space is thus
38-dimensional. Thesampling rate was 250 Hz. Those data sets were
studied, as in the caseof real EEG data, after a PCA preprocessing
step, using discretizationand coarse-graining procedure.
2.2.3. Resting EEG in healthy subjectsA set of EEG sample was
used to compare coarse-grained dynamical
skeleton of large-scale simulated brain dynamics to real EEG
data. TheEEG recordings were performed on 12 healthy subjects in
the Depart-ment of Clinical Neurophysiology (University Hospital of
St. Philibert,Lomme, France) using 17 Ag/AgCl electrodes (C3, C4,
Cz, F3, F4, F7,F8, Fz, O1, O2, P3, P4, T3, T4, T5, T6) placed on
the scalp according tothe 10/20 international electrode placement
system. The referenceelectrode was placed on the nose. Prefrontal
electrodes (Fp1, Fp2,OF
where 1; 2 are a standard normal two dimensional white noise
anda = 0.01, b1 = 0.03 and b2 = 0.05 and T = 5.105. The
deterministicpart of this system has four stable xed points 1=
2
p;1=
2
p
which dene the multistable dynamical organization in the
phasespace. The Markov transition matrix of the model system has
foureigenvalues close to unity associated with four distributions,
whichcorrespond to almost-invariant sets and have been referred to
asmetastable with regard to the Markov operator (Allefeld et al.,
2009).
2.2.2. Large-scale simulated systemResting state brain dynamics
was simulated using the large-scale
model of Ghosh et al. (2008). It is based on a network of 38
vertices,where the local node dynamics is implemented using noisy
StefanescuJirsa populationmodels (Stefanescu and Jirsa, 2008) based
on coupledFitzHughNagumo neurons (see e.g. Fitzhugh and Izhikevich
(2006)).This population model takes into account dispersion of the
neuronalparameters, in this case a distribution of threshold
values. The virtueof this population model is that it is
analytically reducible to a mathe-460space E for T = 3.10 data
points after PCA.
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
TOOF
461
462
463
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465
466
467
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472
473474475476477478479480481
482
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2.0
1.5
.5
2
1
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
0.0544
0.0436
0.0327
0.0218
0.0109
Y
Fig. 1. Illustration of the subdivision process and of the
coarse-graining step for the multistalammac) foransited a
5A. Hadriche et al. / NeuroImage xxx (2013) xxxxxx3. Results
3.1. Technical results
(right). In both cases, the result of the subdivision algorithm
(after 11 steps) based on Usoscale (left) microstates are colored
according to the mesostate they belong to. For theThe second line
depicts the complete transition graph (before surrogate data
selectioncumulative occurrence probability of each macro-state and
the edges represent the trmacro-states with a size larger than 3
have been depicted. The spheres have been loca1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0-1.5 -1.0 -0.5 0.0 0.5 1.0 1
2
1UNCORRECNumerical simulations were used to check the validity
of the rststep of the procedure on systems with known properties.
Multivari-
ate Gaussian white noise process and deterministic chaotic
systemwere used to demonstrate nite-size effects in the subdivision
pro-cess. These technical validations are reported in Appendix
B.
3.2. Multistable stochastic system
An illustration of the results of the successive steps of the
proce-dure is given in Fig. 1.
The application of the discretization step stops for k = 11
leadingto a Markov representation of Mj j 2008 actual meso-states.1
Theentropy rate obtained is h = 3.66 bits per sampling unit. In the
caseof the phase-randomized multivariate surrogate data, the number
ofmeso-states varies from 3217 to 3698 and entropy rates vary
from3.80 to 3.92 bits per sampling unit. Thus, the meso-scale
dynamicsobtained for raw dynamics is signicantly different from
thatobtained for surrogate data. Themeso-scale approximation of the
sys-tem thus keeps the information of the raw data and does not
confuseit with the random system.
The network representation of the meso-scale dynamics was
thencoarse-grained to dene the macro-states on the basis of
cliquesearch in the graph. The same procedure was applied to a set
of 50shufed meso-scale surrogate data. The macro-states for both
rawand shufed data ordered according to their occurrence
probabilityand size are depicted in Fig. 2. The comparison between
raw and
1 Although 211 = 2048 all the subdivision elements do not
contain a microstate sothat a lower number of meso-states is
actually visited by the system's dynamics.
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
R
surrogate macro-states allows one to select 4 non-random
macro-states. The averaged coordinates of the 4 macrostates are
(0.69,0.70), (0.67, 0.68), (0.79, 0.71) and (0.71, 0.70) which
are
3.40e-05Z X
ble stochastic system. The rst line represents the mesoscale
(left) and the macroscale's procedure is depicted as black lines.
The dots correspond to microstates. For the me-roscale (right) the
microstates are colored according to the macrostate they belong
to.the coarse-grained multistable stochastic system. The size of
the spheres denotes thetions between the macrostates (colored
according to the transition probability). Onlyt their average
coordinates.490correct approximations of the 4 stable states of the
system.491Macro-scale dynamics was then studied for both residence
time492and transition probabilities. For each macro-state we found
residence493times of 2.97, 2.84, 3.26 and 3.23 (sampling unit),
which were all494signicantly longer than their respective residence
time in the shufed495surrogate sequence. The study of the
transition probabilities allows one496to select signicant
transitions between macro-states that can be com-497pared with the
dynamics in the system's state space (see Fig. 3).
The498organization of the macro-scale space reproduces the main
features
0
Fig. 2. Surrogate data testing of the macrostates for the
multistable stochastic system.Macro-states for raw (in red) and
shufed surrogate data (in blue) represented in thespace of their
size and occurrence probability. Only raw macro-states, for which
theoccurrence probability is higher than that of all the surrogate
data of the same size,are selected for further analysis. Here, four
macro-states are selected.
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
TOOF
499 of the state-space structure: four macrostates and preferred
transitions500
501
502
503
504
505
506
507
508
509 subdivision iteration number 7 leading to Mj j 126 and Mj j
123 ac-510 tual meso-states. The entropy rates of the associated
effective Markov511512513514515516517
518the simulated brain dynamics preserves the information of the
raw data519520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
a
0
2
1
3
b
Fig. 3. Comparison of the dynamics of the multistable stochastic
system in its phase space and in the macro-scale space. (a)
Representation of the phase space of the system: arrowsdepict the
orientation of the gradient; blue dots correspond to micro-states
visited by the numerical simulation. (b) Macrostate network for the
multistable stochastic system:nodes represent the selected
macro-states and arrows the signicant transitions.
6 A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxRRECprocesses
were respectively h = 0.74103 and h = 0.69.103 bits per
second. The subdivision process of the phase-randomized
surrogatedata sets leads to a number of 256 meso-states for the
surrogate data.The entropy rate of the effective Markov process
associated with surro-gate data vary from 0.96.103 to 1.00.103 bits
per second. Thus, themeso-scale dynamics obtained for rawdynamics
is signicantly differentfrom that obtained for surrogate data. The
meso-scale approximation ofin the y-direction where the noise level
is the highest.Our procedure using a discrete approximation of the
system's
dynamics and on a coarse-graining based on clique search in
thegraph representation of this approximation allows revealing
themain structure of the dynamics of the metastable system in
itsphase space. This procedure is now applied to simulated and
recordedbrain dynamics.
3.3. Large-scale simulated system
For the two simulateddata sets, the discretization procedure
stops forUNCO
a
Fig. 4.Macro-states for raw (in red) and shufed surrogate data
(in blue) represented in theset. Only raw macro-states for which
the occurrence probability is higher than that of all theframed
lower left corner of the main plot.
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
Rand does not confuse it with the random system.The network
representation of the meso-scale dynamics was thencoarse-grained to
dene the macro-states on the basis of cliquesearch in the graph.
The same procedure was applied to a set of 50shufed meso-scale
surrogate data for each simulated data set. Themacro-states are
depicted for both raw and shufed data accordingto their occurrence
probability and size in Fig. 4. The comparisonbetween raw and
surrogate data results allows selecting 19 non-random macro-states
for the rst simulated data set and 16 non-random macro-states for
the second simulated data set.
Macro-scale dynamics was then studied for both residence timeand
non-random transitions. Mean residence times were comparedfor each
macro-state with those obtained for random dynamics (seeFig. 5). In
each case, raw mean residence times are signicantly longerthan
those of random dynamics demonstrating that macro-statedynamics
depict signicant correlations that are not present in randomwalk
dynamics. The study of the transition probabilities allowsselecting
the transitions between macro-states which are signicantlymore
probable than those of shufed surrogate data. When the transi-tions
are bidirectional (which happens in almost all of the cases)
thehighest transition indicates the direction of the dynamics as
depictedb
space of their size and occurrence probability for rst (a) and
second (b) simulated datasurrogate data of the same size are
selected for further analysis. The inset enlarges the
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
TROOF
540 in Fig. 6. This graph reveals the actual noise driven
dynamics of the sys-541 tem. In both cases, one can notice that the
graphs are not totally542 connected demonstrating that preferred
transitions exist within the543 phase space of the large scale
simulated brain dynamics. Moreover,544
545
546
547
548
549
550
551
552
553
554
555
556
557varies from 0.76.103 to 0.97.103 bits per second (mean:
0.86.103, sd.:5580.06.103). The subdivision process of the
phase-shufed surrogate559data stops for k = 8 in 6 data sets and
for k = 9 for the other5606 data sets leading to a number of
meso-states |M| = 256 or |M| =561512. For all the data sets, the
entropy rate of the effective Markov pro-562cess associated with
surrogate data is signicantly higher than that of563the raw data.
The subdivision process thus keeps the structure of the564raw
dynamics that is destroyed by phase shufing. The
difference565between the entropy rates obtained for real and
simulated brain566dynamics was tested using a bootstrap procedure
for the difference of567mean. Entropy rate for simulated data is
signicantly lower than that568of real data (p-value: 0.006 for 1000
resampling).569The coarse-graining procedure allows obtaining
macro-states for570real electrical brain dynamics. The occurrence
probability as a func-571tion of the macro-states' size is depicted
in Fig. 7 for all the subjects.572The selection procedure allows to
extract from 7 to 12 macro-states573for each EEG data set leading
to a total of 105 macro-states selected574for the whole set of 12
EEG samples.575For each EEG sample, the macro-scale dynamics was
studied576for both residence time and preferred transitions. The
signicant577macro-scale transition graphs are given in Fig. 8 for
the highest prob-578abilities of transitions as in Fig. 6. As in
the case of simulated data,579the macro-scale graphs clearly depict
preferred transitions between580macro-states of the non-random
component. The organization of581these transitions depicts one
macrostate whose role in the graph is582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
Fig. 5. The mean residence times for simulated data sets as a
function of the macro-statenumber. The continuous lines depict the
results for raw data whereas the dotted lines de-pict the maximum
value obtained for shufed surrogate data.
7A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxRREC
the structure of the graphs depicts an attracting node which
corre-sponds to the equilibrium point, cycles and chains of
macro-states.Such chains maybe realized via heteroclinic cycles
(Rabinovich et al.,2012) together with possible cycles of variate
lengths as suggested byprevious work on smooth dynamical systems
(Ashwin and Chossat,1998; Ashwin and Field, 1999; Dellnitz et al.,
1995) or may correspondto ghost attractors (Deco and Jirsa, 2012).
The latter differ from the for-mer that the chains are
probabilistic (as opposed to deterministic) andhence may provide
varying sequences for each realization as a functionof the
transition probability.
3.4. Application to resting EEG in healthy subjects
For the 12 data sets, we obtained a meso-scale Markov
representa-tion with Mj j 28 after the subdivision procedure. The
entropy rate
18
2UNCO
0
1
3
17
5
4
16
13
11
9
615
14
10
127
8
a) First datasetFig. 6. Macrostate network for simulated data.
The nodes are the non-random macro-stshufed surrogate data. When
transitions are bidirectional (which happens in almost
alldynamics.
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED Psimilar to the
equilibrium point in the case of simulated data andchains and
cycles of macro-states as in the simulated case.
The mean residence times observed for the macro-scale dynamicsof
the EEG data are depicted in Fig. 9. They are signicantly
longerthan those of the shufed surrogate data demonstrating that
the EEGdynamics at the macro-scale level have a correlation
structure, whichis not present in a mere random walk dynamics.
Moreover, the timescale of these correlations is similar with that
observed for the simulat-ed data.
The 105 patterns of activity associated with macro-states are
hardlyvisually comparable between subjects. We used a clustering
techniquebased on afnity propagation (Frey and Dueck, 2007) to
extract com-mon features between subjects. This procedure, based on
Euclideandistance between patterns identies the number of clusters
in anunsupervised way and thus extracts common features without a
priori
15
5
12
0
11
14
13
7
10
6
4
9
18
3
2
b) Second datasetates. The edges represent the transitions that
are signicantly higher than those ofthe cases) the highest
transition probability is depicted to dene the direction of theire
of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
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597Q3598
599
600
601
8 A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxUNCORRECT
assumption on the number of clusters. The non-random
macro-stateEEG patterns for all the subjects were taken into
account. Fig. 10depicts the average pattern of the macro-state EEG
identied in eachcluster. Thus the most important patterns at the
macro-scale can besummarized to 3 archetypes which can be
considered as coarse-
(a) (b
(d) (e
(g) (h
(j) (kFig. 7. Macro-states for raw (in red) and shufed surrogate
data (in blue) represented inmacro-states for which the occurrence
probability is higher than that of all the surrogatelower left
corner of the main plot.
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
ROOF
602grained dynamical modes of the resting state network. These
modes603comprise a low amplitude pattern, which can be related to
an average604level of activity and twomodes of higher amplitude
with occipital acti-605vation. These patterns are compatible with
the -rhythm of the resting606condition recorded here (see Fig.
10).
) (c)
) (f)
) (i)
) (l)the space of their size and occurrence probability for all
the EEG data sets. Only rawdata of the same size are selected for
further analysis. The inset enlarges the framed
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
ORRECTE
D P
ROOF
607 4. Discussion
608 EEG brain dynamics is only accessed through a nite
resolution609 and nite time window dened by the electrode placement
system610 and the duration of the recording session. Due to this
limited informa-611 tion, one can only model brain dynamics at a
nite scale. We pro-612 posed here a procedure which denes an
effective meso-scale for613 brain dynamics as the result of a
subdivision of the measurement614 space. In order to obtain
geometrical information on the brain615 dynamics, we dened
macro-states as regions of the meso-scale616 space with specic
dynamical characteristics such as slowing down,617 recurrence or
clique-invariance on the basis of the graph associated618 to the
Markov dynamics in the meso-scale. In that sense, the macro-619
states emerge from the properties of meso-scale dynamics. This
proce-620 dure was inspired by several trends in non-linear
dynamics which621 can be qualied as the set oriented approach of
dynamical systems622 (Dellnitz and Junge, 2002). Using this
unsupervised procedure, we623 showed that we were able to recover
the main qualitative properties
624of a multistable stochastic system on the sole basis of the
time series.625We then studied both the simulated and real brain
dynamics.626When applied to the brain's resting state dynamics, our
coarse-627grained approach allows reconstructing the dynamic
repertoire of a628brain network model based on large scale
connectivity, the so-called629connectome. In this particular case
we focused on the model of630Ghosh et al. (2008), for which the
existence of multiple macro-631states has not been known yet. Our
approach successfully recovers632several macro-states in the state
space of the network model, of633which one is highly dominant and
the others signicantly weaker.634This property was observed both in
the geometrical representation635of the transition network at the
macro-scale and in the residence636time. What has been known so far
is that in the subcritical regime637the spatial network activations
are dominated by two networks638showing an oscillatory dynamics in
the 10 Hz range. Our analysis639suggests that ghost attractors may
also exist in the model of Ghosh640et al. (2008). Effectively a
ghost attractor will correspond here to an641unstable attractive
regime in the state space, which predicts for the
a) FRA b) ANG c) LAH
OR
s. Th
9A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxUNC
d) DCH e) FFig. 8. Macrostate network for EEG data. The nodes
are the non-random macro-state
surrogate data. When transitions are bidirectional (which
happens in almost all the cases) the h
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041f) aLIe edges represent the
transitions that are signicantly higher than those of shufed
ighest transition probability is depicted to dene the preferred
direction of the dynamics.
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
642
643
644
645
646
647
648
649
10 A. Hadriche et al. / NeuroImage xxx (2013)
xxxxxxUNCORRECT
stochastic system an increased transition probability from
onemacro-state to another. Allowed transitions between
macro-statesare illustrated in the topology of the transition
network as edges.The absence of an edge in this graph indicates
that the transition can-not be differentiated from a pure random
walk transition. In thissense the transition network captures
nicely the spatiotemporal dy-namics of the brain network by
quantifying the preferred transitionsbetween macro-states. Given
the stochastic nature of this process,
a) LAE b) BE
d) CAN e) Fig. 8 (cont
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
ROOF
650we have expressed the network's time evolution through
transition651probabilities. The relative sizes of the macro-states
are dependent652on the degree of the coarse graining, which in our
method is deter-653mined automatically through the algorithm by
construction. The654dominant macro-state expresses the sampling of
the mostly linear re-655gime around the stable xed point. In case
of a smaller coarse656graining, more smaller macro-states might
have been identied in657its immediate neighborhood. In the
particular case of the brain
A c) LUDO
HEI f) FRFinued).
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
T PROOF
658 network model though, the algorithm identied a coarser
scale,659 which emphasizes the strong asymmetry in the topology of
its transi-660 tion graph: the dominant macro-state is located at
the periphery of661 the transition network and the associated
transitions to the other662 macro-states occur in to a particular
direction in the state space.663 This phenomenon is due to the
characteristics of the network shaping664 the nonlinear ow in the
state space, in particular the nonlinearities,665 the connectome
and the signal transmission delays.666
667
668
669
670Q4671
672
673the simulated one. This result can be both interpreted as a
higher674level of noise in real brain dynamics or as a higher
dimensional dy-675namics for the real brain than for the simulated
large-scale brain dy-676namics. Second, the main macro-states
differed less in their size than677in the simulated brain dynamics.
Nevertheless, the transition net-678works show similar structure
between real and simulated data: the679presence of dominating
macro-state related to the equilibrium state680in the simulated
data and the presence of chains of macrostates. In681order to
extract some regularity from all the macro-states found in682the
empirical data set, clusters of macro-states were studied.
These683clusters allow dening three archetypical averaged patterns
of activ-684ity that can be regarded as dynamical modes of the
resting brain net-685work comprising its dynamic repertoire.686The
representation of coarse-grained regions of the meso-scale as687an
average pattern does not address satisfactorily the dynamical
na-688ture of those ghost-attractors. A specic study of the
dynamics689within these macro-states would complement our
coarse-grained ap-690proach. Nevertheless, several theoretical
proposals about cognitive691architectures and how they relate to
brain dynamics and cognition692(Rabinovich et al., 2012; Tsuda,
2001; Woodman et al., 2011) suppose693the presence of metastable
states as randomly visited organizing cen-694ters of the dynamics.
Our results are one of the rsts to give empirical695support to this
model of resting state brain dynamics using EEG data696from human
subjects.697The approach developed here is a completely
unsupervised method698with no a priori hypothesis about what a
brain macro-state should be.699Brain states are dened here on
dynamical properties and in a data700driven manner so that
macro-states appear as regions discovered by701the clique search in
the meso-scale transition network. Nevertheless,702
703
704
705
706
707
708
Fig. 9. Residence time for all the EEG data sets. The continuous
lines depict the resultsfor the real data sets whereas the dotted
lines depict the maximum result for the shuf-ed surrogate data. In
all the cases, the residence times are longer for the real data
thanfor the random walk dynamics.
11A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxRREC
In the case of the empirical EEG data, the meso-scale and
macro-scale properties were similar between subjects although
variabilityincreases when the EEG patterns were taken into account.
Themeso-scale and macro-scale properties were qualitatively
compara-ble with those found for the simulated brain network
dynamics.Nevertheless, noticeable differences were found. First the
entropyrate of the real brain dynamics was signicantly higher than
that ofUNCO
Fig. 10. Averaged modes of the brain dynamics obtained from the
clustering procedure for alpattern is a low amplitude pattern
whereas the two other patterns depict higher amplitude wthe resting
state condition as depicted on the averaged power spectrum density
of electrod
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041EDit supposes several
simplifying assumptions and methodological
choices that certainly need further attention.Our main
assumptions are the following. First, although we do not
suppose reversibility, we considered that the resting state can
be con-sidered as a stationary homogeneous process at the
observation scaleused here. Moreover, the Markov representation of
brain dynamicswas limited to a rst-order Markov process which
represents a simple
l the EEG patterns of the non-random component of the
macro-scale dynamics. The rstith occipital activation. These
patterns are compatible with the -rhythm activation of
es Fz, Cz, O1 and O2.
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
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12 A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxUNCORREC
approximation of brain dynamics. This hypothesis has been used
bothat the meso- and macro-scales. In the case of the meso-scale
this ismainly an intermediate step and other models might be used
here toapproximate the meso-scale dynamics. In the case of the
macro-scaledynamics, more general models, such as variate length
Markov chains(Buhlmann and Wyner, 1999) or -machines (Crutcheld,
2012)might be used to obtain a more precise description of the
macro-scale brain dynamics. Our approach is also restricted to
theobservation/electrode space and does not allow studying the
problemin the source space. Moreover, the number of recording site
is limitedhere and might weaken the study of brain dynamics in the
observationspace. An extension of this study should take into
account higher den-sity recordings of brain electrical activity in
order to obtain a largeramount of information on the brain dynamics
and might also beextended in the directions of the source space
after inverse problemresolution. Finally, simulated data were
obtained in the state space ofthe model but they were compared to
the data in the measurementspace. Another extension of this work
should compare the data in thesame space such as after the forward
solution. Nevertheless, the pre-processing step using a PCA may
reduce this limitation.
In our methodological strategy the algorithm used to
coarse-grainthe meso-scale state space is based on heuristics
choices. They wereadapted to the purpose of the identication of
regions with specicdynamical characteristics. The correct
identication of the character-istics of the multistable stochastic
system somehow validates empir-ically these choices. Moreover, the
surrogate data testing used toselect the macro-states ensures a
statistical validity. Nevertheless,the denition of the macro-states
depends on the denition of thecliques which can be biased rst by
the fact that transitions are notobserved during the recording
period and second by the choice ofthe simple criteria of high
probability. In the rst case, a softer crite-rion might ensure a
more robust procedure against the inevitableerrors in estimating
transition probabilities. In the second case morecomplicated
criteria such as combination of local transitions andlocal graph
structure could be explored.
The role of noise on real and simulated brain dynamics can also
berelated to its inuence on the interpretation of our results.
First, noisehas been introduced in the data generating model
systems as additiveGaussian white noise. This type of noise is the
most common in neu-roscience modeling; we still wish to point out
that other variations ofnoise such as multiplicative noise, skewed
noise (with an asymmetricdistribution), and colored noise may
display qualitative effects (seefor instance Freyer et al. (2011)).
These situations need to be furtherconsidered on a case by case
basis. Then, metastability on the macro-scopic scale is to be
understood in terms of the Markov operatorresulting in a dynamics
towards a node, followed by its escape to an-other macro-state.
This dynamics needs to be complemented by theview of the dynamics
on the meso-scale, in which the neural networkdynamics follows a
deterministic ow perturbed by random noise.
The comparison between models of brain dynamics and real
braindynamics need criteria for their evaluation. We proposed here
a com-parison at a macro-scale oriented toward a coarse-grained
dynamicalskeleton. Once the simulated and real data are represented
in thesame space i.e. either the source space or the observation
space,the large scale dynamical skeletons and their topological or
metricproperties obtained in both models and real data could be
compared.This might be a criterion for deciding between alternative
models. In-deed, our approach does not deal with the signal but
with an effectivelarge-scale skeleton whichmight be a very simple
but robust criterionfor the evaluation of models.
Acknowledgments
We wish to thank two anonymous reviewers for their advices
thatallowed us to improve our manuscript. This research was
initiated in
the CODYSEP project supported by the Neuroinformatics Program
of
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
ROOF
the CNRS (p.i.: LP). AH received nancial support from the
InstitutFranais de Coopration and The Virtual Brain Project (see
www.thevirtualbrain.org). VKJ was supported by the Brain Network
Recov-ery Group through the James S. McDonnell Foundation and
theFP7-ICT BrainScales. We wish to thank B. Lenne for his help with
dataacquisition. Most of the computation of this article was done
usingfree software and we are indebted to the developers and Debian
main-tainers of the following packages: TEXLive, vim, R, python,
python-networkx, python-numpy, python-mvpa, python-sklearn,
mayavi2,graphviz, to mention only a few.
Appendix A. Procedure
1. We denote the measurement space E. For each electrode, the
ana-log to digital conversion limits the available measurement to
an in-terval ofN equivalent to I 0;2b1 where b is the resolution
ofthe analog to digital converter in bits. In the case of a
multichannelrecording system with N electrodes, E can be considered
to be: E IN with card E Ej j 2b
N.
In this space a recording session of Ts seconds leads to a
sequenceof row vectors e(t) = (e1(t), ,eN(t)) with t = 0, ,T 1 with
T =Ts/t the number of recording samples in a recording session
(with tthe recording period in seconds).
We thus used here a spatial embedding of the dynamics instead
ofthe time-delay embedding (Kantz and Schreiber, 2004). This
choicewas motivated rstly, by our interest in the whole brain state
spaceand, secondly, by the fact that time-delay embedding of single
chan-nel recordings is not adapted in the case of spatiotemporal
dynamicssuch as brain dynamics (Lachaux et al., 1997).
2. The rst step of the procedure is an iterative discretization
of themeasurement space E which leads to the meso-scale space
M.Since the discretization proceeds by iterative bisections of
inter-vals until a nite k step, Mj j 2k . The criteria to choose k
inthe k iterations will be described below. At this meso-scale,
braindynamics can be approximated by an effective Markov
processwith Mj j states. This Markov process is characterized by
its sta-tionary probability distribution and its associated graph
represen-tation. The meso-states are then grouped into macro-states
usinga graph-based algorithm using both probabilistic properties
ofthe stationary distribution of the Markov process and clique
orga-nization of the graph related to the transition matrix.
3. The macro-scale or coarse-grained scale of this study is thus
C Cif gi1;; Cj j which corresponds to the states that belong to
thesame clique ordered by occurrence probability. This scale is
de-ned in order to obtain geometrical information on the
topologicalorganization in the measurement space.
A.1. Discretization step
The rst step consists in a subdivision or partition of the
measure-ment space i.e. in the denition of non-overlapping regions
Ei so thatiEi E and iEi . There is obviously no unique way for the
par-tition of E and each partition procedure has its own goal. For
example,in the dynamical systems literature, one typically searches
for a gen-erating partition (see e.g. Badii and Politi (1999))
which has strongtheoretical settings. Nevertheless, a generating
partition is hardlydetermined on the basis of noisy experimental
data.
Our approach is based on a subdivision's algorithm (Dellnitz
andHohmann, 1997) leading to the generation of a sequence E 0 ; E 1
;of nite collection E k E k j : j 0;1;;2k1
n oof 2k rectangles
such as 2kj1E k j E. The rst collection E 0 E emin1 ; emax1
min max
min max min max830e2 ; e2 eN ; eN with ei and ei is respectively
the
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
T831 minimum and maximum values of the i-th coordinate of eQ6 .
E k1 is832 constructed by the bisection of each rectangle E k on
the k modN th833834
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13A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxUNCORREC
j -
dimension.This is an unsupervised method which leads to an
equal-width
discretization process (Kotsiantis and Kanellopoulos, 2006). In
Allefeldet al. (2009) another unsupervised technique was used based
on anequal-frequency discretization process. Both techniques lead
to anapproximation of the probability density E e by a simple
functiondened as
k e i k i I k i e A:1
where IE k i is the characteristic function of Ek i i.e. IE k i
e 1 if eE
k i
and IE k i e 0 otherwise. In the case of Allefeld et al. (2009)
i, i =T/2k and the measure of E k i vary whereas it is the reverse
case for thepresent subdivision procedure.
For each iteration step k, one hypothesizes a Markov process
basedon the discretized trajectory i.e. the brain micro-states e(t)
are re-placed by meso-states m(k)(t) with m k t M k 0;2k1
wherem(k)(t) corresponds to the index j of the region E k j where
e(t) ispresent. We consider the sequence of m(k)(t) as the
realization ofa rst-order Markov process M(k) with stationary
distribution k k 0 ;;
k 2k1
and 2k 2k transition matrix (k) = (i,j(k)) =
(Pr(m(k)(t) = j | m(k)(t 1) = i). This Markov process can be
charac-terized by its entropy rate h(M(k)) given by:
h M k
i;j k i
k i;j log
k i;j A:2
where the (k) and (k) are both maximum likelihood estimate
of,respectively, the stationary distribution and probability of
transitionand log is taken as natural logarithm.2
The iteration of the subdivision process lead to ner and ner
par-titions and thus the entropy rate h(M(k)) increases for k = 1,
2, toits limit of the entropy rate of the underlying
transformation(Petersen, 1989). Nevertheless, in the case of
experimental data thiscannot be repeated for an arbitrary number of
times since data arelimited to the available T data points. For
large k, the number ofmeso-states becomes large and the number of
data point needed toobtain a valid estimation of the entropy rate
increases. Thus in practi-cal situations, one enters the domain of
bad statistics (Lesne et al.,2009) where block entropy saturates
and thus entropy rate tends tozero. One should enter a compromise
between statistical error andprecision of the partition (Holstein
and Kantz, 2009). We choosehere a pragmatic and simple criteria
based on the behavior of theentropy rate of the Markov process and
we thus dened k as: k =argmax h(M(k)).
This rst step can be summarized as follows: for k 1, 2, ,kmax1.
Split the data according to the dimension k modN;2. Estimate the
stationary distribution and transition matrix (k) and
(k);3. Compute h(M(k)).
Then, choose k 1,kmax where h(M(k)) is maximal (practicallykmax
10).
At the end of the iterative process we consider the discretized
tra-jectory as a Markov process with 2k
states and thus we keep the 2k
vector of stationary probability and 2k 2k transition matrix. In
the
remaining parts of this article, we will drop the (k) in our
notationsand will for example use M for M k and m(t) for m k t and
so onfor all the quantities that are obtained at the nal k step
such as ,and .
2 Entropy rates are converted to bits per sampling unit when the
time scale is arbi-
trary and to bits per second when time scale is explicit.
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
ROOF
The meso-scale Markov representation of brain dynamics is
nowstudied using the graph representation of the process.
A.2. Coarse-graining
Let us associate to the transition matrix a weighted
directedgraph G with Mj j vertices vi (each vertex corresponds to
an elementmi of M) and edges li,j from vertex i to vertex j if i,j
> 0 (i,j is theweight of edge li,j). The set of vertices is also
associated with the sta-tionary distribution in the meso-scale
space.
Within this graph G one can dene cliques Ci which are the
largestcomplete subgraphsCiGwith at least three vertices and
characterizedby: vk; vjCi; kj > 0 and jk > 0 which means that
there are tran-sitions from nj to ni and reciprocally. Thus, if
time evolution starts witha meso-state in a clique Ci then with
positive probability the nextmeso-states will stay in Ci but there
is also a positive probability thatit will leave Ci. Since Pr(m(t)
Ci) can be roughly estimated toCi =j jMj j (Barber and Long, 2012),
process that starts in larger cliqueshave a higher probability to
stay in these cliques. Then cliques withthe largest size and
containing the meso-states with the higheststationary probability
have the highest probabilistic invariance.
Macro-scale is then dened using both cliques in G and the
sta-tionary distribution in the meso-scale spaceM. Cliques are
searchedin G according to decreasing probabilities of meso-states
as follows:
1. Select the most probable meso-state mj associated with vertex
vjwhich has not yet been attributed to a clique.
2. Find the largest clique to which the vertex vj belongs to.3.
Gather all the vertices vj of this clique into one coarse-grained
state
and exclude them from next step.4. Iterate this process.
In fact this procedure is empirically equivalent to an
algorithmwhich rst identies all the cliques in the meso-scale graph
and thenselects the clique with the highest probability since
probability andsize criteria are most of the time equivalent. Step
3 ensures thatCiCj for i j. For each clique Ci one can thus
associate a macro-statewhich gathers the meso-states mj associated
to the vertices vj such asvjCi. These macro-states are
characterized by probability viCk iand a size |Ck| i.e. the number
of vertices (meso-states) in the clique.
A.3. Adaptation of the procedure to brain dynamics
In the case of nite observation length T, the subdivision
processstops for small k. If the dimension of the measurement space
is great-er than k, this implies that information is mainly taken
into accountfrom the rst dimensions and ignored for the latest
ones. Since theorder of the electrodes is arbitrary, this would
emphasize arbitrarilyrst electrodes. In order to avoid such a
problem, a principal compo-nent analysis (PCA) was used as a
preprocessing step. After PCA, e isreplaced by linear combinations
p and the discretization process isthen applied to pE.
In the case of real physiological data, once the coarse-grained
(ormacro) states are dened, one can test for physiological
relevance ofthe procedure and project back to E the average
activity of a state.Each macro-state Ci is a set of vertices vj
associated to meso-statesmj. In turn, eachmj is a set of pl or el
and one can thus dene an aver-age pattern of brain electrical
activity eh ij lLj el=jmjj with Lj l : elmj
associated to a meso-state mj. The macro-state patternCi is
then
Ci jj eh ij A:3
where j is the occurrence probability of mj. This macro-state
pattern
943can be projected to the scalp for inspection of physiological
realism.
ire of the resting brain, NeuroImage (2013),
http://dx.doi.org/10.1016/
-
T944
945
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965966967968969970
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978
97998098198298398498598698798898999099199299399499599699799899910001001
iviste Gbeh
14 A. Hadriche et al. / NeuroImage xxx (2013) xxxxxxREC
Appendix B. Technical results
B.1. Multivariate Gaussian white noise
Five independent random processes with standard Gaussian
dis-tribution were used to simulate multivariate independent and
identi-cally distributed processes of dimension 5 for an increasing
length T.The discretization step was applied to these data sets and
the entropyrate of the meso-scale Markov processesM(k) was computed
for eachstep k in each dataset. In that case, for iteration step k
the theoreticalentropy rate is h = log(2k) since the partition has
2k independentand equiprobable states. The results of these
simulations are depictedin Fig. 11(a).
For initial iteration steps, the entropy rate follows the
theoreticalvalue h = log(2k). The value of k estimated for each
length T variesfrom 5 for T = 104 to 6 for T = 3.104 and T = 105.
For k > k the es-timated entropy rate decreases as a consequence
of nite size effects.We thus check here the valid initial evolution
of the estimation of en-tropy rate for k b k and the expected
nite-size decrease for k > k.
a
Fig. 11. Entropy rate of the meso-scale Markov process (h(M(k)))
as a function of subddashed lines represent the theoretical value
(h) of the entropy rate for (a) Multivariathe initial increase of
the entropy rate (for k b k) follows the theoretically
predictedlead to a decrease of the entropy rate.UNCORB.2. Dynamical
system with known dynamics
Lorenz attractor (Lorenz, 1963) was used as a model for low
di-mensional dynamics with known properties. The system of
nonlineardifferential equations which dene Lorenz attractor is:
x: yx y: Rxyxzz: bz xy
8 k.
Please cite this article as: Hadriche, A., et al., Mapping the
dynamic repertoj.neuroimage.2013.04.041ED P
ROOF
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15A. Hadriche et al. / NeuroImage xxx (2013)
xxxxxxUNCORRECPlease cite this article as: Hadriche, A., et al.,
Mapping the dynamic repertoj.neuroimage.2013.04.041ire of the
resting brain, NeuroImage (2013), http://dx.doi.org/10.1016/
Mapping the dynamic repertoire of the resting brain1.
Introduction2. Materials and methods2.1. Procedure2.1.1. From
mic
