The Challenge of Brain Complexity

A Brief Discussion about a Fractal Intermittency-based Approach

Paolo Paradisi

1,2

, Marco Righi

and Umberto Barcaro

Istituto di Scienza e Tecnologie dell’Informazione “A. Faedo” (ISTI-CNR), Via G. Moruzzi 1, 56124 Pisa, Italy

Basque Center of Applied Mathematics (BCAM), Alameda Mazarredo 14, 48009 Bilbao, Bizkaia, Spain

Dipartimento di Informatica, Universit

a di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy

Keywords:

Biomedical Signal Processing, Electroencephalogram, Brain Events, Fractal Intermittency, Threshold

Analysis, Pattern Recognition, Complex Systems.

Abstract:

In the last years, the complexity paradigm is gaining momentum in many research ﬁelds where large multi-

dimensional datasets are made available by the advancements in instrumental technology. A complex system

is a multi-component system with a large number of units characterized by cooperative behavior and, conse-

quently, emergence of well-deﬁned self-organized structures, such as communities in a complex network. The

self-organizing behavior of the brain neural network is probably the most important prototype of complexity

and is studied by means of physiological signals such as the ElectroEncephaloGram (EEG). Physiological

signals are typically intermittent, i.e., display non-smooth rapid variations or crucial events (e.g., cusps or

abrupt jumps) that occur randomly in time, or whose frequency changes randomly. In this work, we introduce

a complexity-based approach to the analysis and modeling of physiological data that is focused on the char-

acterization of intermittent events. Recent ﬁndings about self-similar or fractal intermittency in human EEG

are reviewed. The deﬁnition of brain event is a crucial aspect of this approach that is discussed in the last part

of the paper, where we also propose and discuss a ﬁrst version of a general-purpose event detection algorithm

for EEG signals.

1 INTRODUCTION

A crucial focus in Physiological Computing is the

development of human-computer interaction devices

whose applications span from the monitoring of the

health state of a patient, with an associated decision

support system including diagnosis and prognosis, to

the application of speciﬁc therapies, (e.g., administra-

tion of stimuli).

In this general framework, the so-called complex-

ity approach (Sol

e and Bascompte, 2006; Sornette,

2006), focused on the analysis and modeling of

emerging self-organization in multi-component sys-

tems and complex networks, is nowadays gaining mo-

mentum in the ﬁeld of biomedical signal processing.

In order to extract useful information from large clini-

cal datasets, storing many different physiological data

and signals, algorithms for the reduction of data com-

plexity are needed to derive reliable diagnostic in-

dices. Then, a great interest is focused in deﬁning,

developing and testing statistical indices that can en-

close the minimal information required to character-

ize physiological signals in a both efﬁcient and reli-

able way. This is of relevance in clinical activities in-

volving, e.g., disorders of consciousness (DOC) (De

Biase et al., 2014; Monti et al., 2010; Fingelkurts

et al., 2013; Fingelkurts et al., 2014; Laureys and

Tononi, 2009; Tononi, 2008) or attention deﬁcit hy-

peractivity disorders (ADHD) (Robbie et al., 2016).

However, such indices are useful if they are able

to describe the key features of the signals and if

these features can be exploited by physicians in their

clinical activity, e.g., in the evaluation of a medical

condition or disease (diagnosis); in foretelling the

course of a disease (prognosis); in the consequent

choice of the proper therapy (decision making).

Interestingly, the possibility of applying complex

stimuli with fractal features as innovative thera-

peutic strategies is attracting the attention of many

researchers (Zueva, 2015; H

agerh

all et al., 2015;

Spehar et al., 2015).

In this work we introduce and discuss an approach to

the processing of ElectroEncephaloGram (EEG) sig-

Paradisi, P., Righi, M. and Barcaro, U.

The Challenge of Brain Complexity - A Brief Discussion about a Fractal Intermittency-based Approach.

DOI: 10.5220/0005998601230129

In Proceedings of the 3rd International Conference on Physiological Computing Systems (PhyCS 2016), pages 123-129

ISBN: 978-989-758-197-7

123

nals that is based on the observation that, in many

complex systems, such as the human physiology, the

dynamics trigger crucial events, each one associated

with the emergence or decay of self-organized struc-

tures in the system (Allegrini et al., 2009; Allegrini

et al., 2010a; Allegrini et al., 2010b; Allegrini et al.,

2011; Paradisi et al., 2013; Allegrini et al., 2013; Al-

legrini et al., 2015; Beggs and Plenz, 2003; Plenz

and Thiagarjan, 2007; Fraiman et al., 2009; Chialvo,

2010; Grigolini and Chialvo, 2013).

The paper is organized as follows. In Section 2 we in-

troduce the fractal intermittency description of com-

plex systems and we discuss a measure of complexity

and a reliable method to estimate it. In Section 3 we

review some results about the application of fractal

intermittency to brain dynamics, introducing the ba-

sic concept of crucial event in the brain. In Section 4

we discuss a proposal for a general-purpose algorithm

for event detection in human EEG and in Section 5 we

draw some conclusions.

2 THE PARADIGM OF FRACTAL

INTERMITTENCY IN

COMPLEXITY

Even if a general deﬁnition of complex system is

not universally accepted in the scientiﬁc community,

some features are recognized to be ubiquitous in the

presence of complexity (Paradisi et al., 2015c; Par-

adisi et al., 2015b; Sol

e and Bascompte, 2006; Sor-

nette, 2006). Without demanding completeness, we

give a brief list of the most important features:

• A complex system is composed of many parts,

i.e., degrees of freedom: many individuals, par-

ticles, units or, in general, many sub-systems that

are embedded in a network of strong nonlinear in-

teractions.

• The behavior of a multi-component system can be

considered complex if the nonlinear dynamics is

cooperative, thus giving rise to self-organized or

coherent states.

• Non-reducibility: self-organized states have tem-

poral and spatial scales that are hardly obtained as

a simple (linear) function of an external forcing

or by means of linear coarse graining procedures

(e.g., average or weighted sum over single com-

ponents).

• Self-organized states display long-range space

and/or time correlations (slow power-law decay).

The correlation exponents are an important exam-

ple of emergent properties that can be used as syn-

thetic indicators of the cooperative dynamics in

the complex system.

In order to introduce the approach discussed in this

paper, it is worth noting that the above list of features

is not complete. In fact, the following general obser-

vations are also in order:

• Self-organized states are usually metastable

states. They are in fact characterized by relatively

long life-times and by fast transition events among

two successive coherent states.

• The sequence of fast transition events among self-

organized states is described by a intermittent

birth-death point process of self-organization (i.e.,

coherence): {t

}; t

n+1

> t

= 0; n = 0, 1,2, ...,

being t

the occurrence times of the n-th event.

Even if the states are in a far-from-equilibrium

condition, the overall dynamics are stationary (on

average) if there are no external perturbations.

• The life-times of coherent structures are deﬁned

by the time intervals between two successive tran-

sition events: τ

= t

−t

n−1

; n = 1,2,.... In com-

plex systems these times, here denoted as Waiting

Times (WTs), are random variables whose sta-

tistical distribution has a inverse power-law tail:

ψ(τ) ∼ 1/τ

(Allegrini et al., 2009; Allegrini

et al., 2010a; Allegrini et al., 2010b; Allegrini

et al., 2011; Paradisi et al., 2013; Allegrini et al.,

2013; Allegrini et al., 2015).

• The fast transitions among metastable states are

typically associated with a fast memory drop in

the dynamics, so that each self-organized state is

often independent from each other, as such as the

transition events. This is denoted as renewal con-

dition and the sequence of transition events is a

renewal point process (Cox, 1962; Paradisi and

Allegrini, 2015; Paradisi et al., 2015a; Paradisi

et al., 2012a; Paradisi et al., 2012b; Paradisi et al.,

2009b; Akin et al., 2006; Paradisi et al., 2008;

Akin et al., 2009; Bianco et al., 2007; Paradisi

et al., 2009a) In this case the WTs τ

are mutually

independent random variables. Conversely, in the

time interval (t

n+1

) between two events, i.e., in

correspondence of a self-organized state, the dy-

namics are strongly correlated.

The inverse power-law in the WT distribution is the

manifestation of a self-similar behavior in the coop-

erative dynamics of the complex system. This power-

law behavior in the WT distribution is a crucial emer-

gent property, characterizing the capacity of the com-

plex system to trigger self-organization. This condi-

tion is denoted as fractal intermittency (Paradisi et al.,

2012b; Paradisi et al., 2013; Allegrini et al., 2013;

Paradisi et al., 2015b; Paradisi and Allegrini, 2015).

PhyCS 2016 - 3rd International Conference on Physiological Computing Systems

124

This complex behavior is also known as Temporal

Complexity (Grigolini, 2015; Beig et al., 2015; Tu-

ralska et al., 2011; Grigolini and Chialvo, 2013), a

term that was introduced to underline the difference of

the intermittency-based approach to complexity, fo-

cused on the study of the temporal structure of self-

organization, with the more extensively investigated

approach associated with the estimation of topologi-

cal and spatial indicators of complexity (e.g., the de-

gree distribution in a complex network, the avalanche

size distribution) (Beggs and Plenz, 2003; Plenz and

Thiagarjan, 2007; Fraiman et al., 2009; Chialvo,

2010; Grigolini and Chialvo, 2013).

In summary, when a system is characterized by fractal

intermittency, the essential dynamics are an alterna-

tion of metastable states, with strong coherence and

long life-times, and transition (intermittent) events

that occur randomly in time, develop in very short

time, can be considered instantaneous and are asso-

ciated with a fast memory drop. In this condition, the

experimental observation of long-range (power-law)

correlations is strictly connected to the inverse power-

law decay in the statistical distribution of the random

life-times (Allegrini et al., 2009).

2.1 Intermittency-based Scaling

Analysis as a Signature of

Complexity

The emergence of intermittency per se is not a sig-

nature of complexity. In fact, let us consider a in-

termittent system with many units whose dynamics

is independent from each other. An example of this

condition is given by neural cultures in vitro with

connections that are chemically inhibited. The ab-

sence of interactions involves the absence of coop-

eration and, thus, of self-organized structures. Inde-

pendently from the single neuron intermittent dynam-

ics, the overall behavior of the network is described

by a renewal Poisson process (Cox, 1962), which in-

volves an exponential WT distribution. This simple

observation implies that Poisson intermittency in a

multi-dimensional system without external perturba-

tions is associated with absence of self-organization

and, thus, it can be taken as a non-complexity refer-

ence condition. Then, a necessary condition for com-

plex self-organization in intermittent systems is the

emergence of crucial events with non-Poisson statis-

tics.

The emergence of non-Poisson statistics by itself is

not a signature of complexity but, as observed above,

a inverse power-law tail in the WT distribution is of-

ten observed in complex systems. As a consequence,

the power exponent µ can be exploited as a signature

of complex behavior and it is denoted as complexity

index. This intermittency-based complexity measure

is very powerful and represents a robust way of quan-

tifying the amount of complexity in a system, being

the value of µ typically associated with the ability of

the system of triggering complex self-organized struc-

tures.

2.2 Diffusion Scaling and the EDDiS

Method

The complexity index µ is formally deﬁned by the

asymptotic behavior of the WT distribution. How-

ever, due to secondary pseudo-events associated with

the presence of noise in the signals and/or with the un-

avoidable limitations of the event detection algorithm,

the direct estimation of µ is often not possible as a

clear power-law decay is not seen in the hystogram

(Allegrini et al., 2010a; Paradisi et al., 2012a; Paradisi

et al., 2012b; Paradisi and Allegrini, 2015) (Allegrini

et al., 2010a) proved that, when a intermittent pro-

cess with complexity index µ is superposed to noisy

events, the WT distribution can display an apparent

exponent µ

that is completely different from the un-

derlying real µ. For this reason, (Allegrini et al., 2009)

exploited an algorithm for the indirect estimation of µ.

In this method, denoted as analysis of Event-Driven

Diffusion Scaling (EDDiS), three different diffusion

processes driven by the same sequence of event oc-

currence times {t

} are ﬁrstly built. Then, some

scaling exponents of the event-driven diffusion pro-

cesses are evaluated by applying well-known tools

of statistical data analysis: the Detrended Fluctua-

tion Analysis (DFA) for the second moment scaling

H: hX

(t)i ∼ t

(Peng et al., 1994); and the Dif-

fusion Entropy (DE) for the self-similarity exponent

∆: P(x,t) = F(x/t

∆

)/t

∆

(see Ref. (Paradisi and Alle-

grini, 2015) and references therein).

The EDDis method exploits a combination of some

theoretical results about the relationships between µ

and the exponents H and µ that are well-known in

literature by many years (for a short review see Ref.

(Paradisi and Allegrini, 2015) and references therein).

Theoretically, it should possible to get 6 different es-

timates of µ, but sometimes not all the estimates are

reliable due to the presence of noise (Allegrini et al.,

2010a; Paradisi and Allegrini, 2015). In any case, at

least 2 values of µ can be usually obtained. Being the

relationships H = H(µ) and ∆ = ∆(µ) based on the re-

newal assumption, the self-consistency of the µ values

is used as a test for the renewal assumption and, in the

positive case, this also gives a robust estimation of µ

(Paradisi et al., 2012b; Paradisi and Allegrini, 2015).

The EDDiS method is used to estimate the complex-

The Challenge of Brain Complexity - A Brief Discussion about a Fractal Intermittency-based Approach

125

ity index µ in the presence of noisy pseudo-events.

However, the scaling exponents H and/or ∆ can be

also used directly to characterize the complexity of

the system. Actually, when the renewal condition is

satisﬁed, using the diffusion scaling or the power-law

exponent µ is totally equivalent.

3 COMPLEXITY AND FRACTAL

INTERMITTENCY IN THE

BRAIN

Brain dynamics is probably the most important exam-

ple of complex self-organization with a great variety

of quantitatively and qualitatively different behaviors,

also spanning over several time scales. The efﬁciency

of brain information processing is strictly associated

with the metastability of the highly non-stationary

neural assemblies continuously forming and decay-

ing, usually in response to external and internal stim-

uli in order to maintain the homeostatic equilibrium.

In this section we introduce the fractal intermittency

approach in brain research and we brieﬂy discuss

some interesting ﬁndings about brain complexity.

3.1 Transition Eevents in Brain

Dynamics

Fingelkurts and Fingelkurts proposed a conceptual

model of the brain, the Brain Operation Architec-

tonics, that could explain the self-organizing and

multi-scale behavior of brain dynamics (Fingelkurts

and Fingelkurts, 2001; Fingelkurts et al., 2008; Fin-

gelkurts et al., 2010; Fingelkurts and Fingelkurts,

2015; Fingelkurts et al., 2013). According to this sce-

nario, a neural assembly is a group of neurons for

which coordinated activity persists over substantial

time intervals and underlies basic operations of infor-

mation processing during brain functioning. Thus, the

neural assemblies are the metastable, self-organized,

states introduced in Section 2.

From this brain theory, a general approach to the sig-

nal processing of EEG data was derived (Kaplan et al.,

2005). In this paper EEG signals are recognized to

be strongly non-stationary, reﬂecting the switching

among different neural assemblies. During the life-

time of neural assemblies, the EEG signals should dis-

play periods of quasi-stationarity, while rapid changes

can be seen in the EEG traces in correspondence

of the emergence and decay of the neural assem-

blies themselves. These rapid changes are called

Rapid Transition Processes (RTPs) and mark the pas-

sages between two quasi-stationary periods and, as

said above, are interpreted as the birth or death of

metastable neural assemblies (Kaplan et al., 2005).

3.2 Deep Sleep, Wakefulness and REM:

Towards an Index for Consciousness

RTPs are a prototype of complex transition events in

the brain and RTP deﬁnition was exploited to apply

the EDDiS method introduced above. This allowed

to discover the presence of fractal intermittency in

the brain functioning (Allegrini et al., 2009; Allegrini

et al., 2010a; Allegrini et al., 2010b; Allegrini et al.,

2011; Paradisi et al., 2013; Allegrini et al., 2013; Al-

legrini et al., 2015). The interesting range of the com-

plexity index is: 1 < µ < 3. (Allegrini et al., 2009)

found an approximate value µ = 2.1 for the brain basal

condition (wakefulness with closed eyes). The in-

teresting result is that this value remain quite stable

among different subjects. More interestingly, µ was

proven to be associated with the degree of integra-

tion/segregation of the brain dynamics and, thus, with

consciousness (Paradisi et al., 2013; Allegrini et al.,

2013; Allegrini et al., 2015). This result was obtained

by applying the EDDiS method to EEG data recorded

during sleep condition, including about 30 minutes

before the sleep onset. This allowed to estimate the

diffusion scaling H, whose values was found to be in

agreement with the integrated states of consciousness

and the segregated states of unconscious deep sleep.

These results indicate that intermittency-based com-

plexity measures could be good candidates as indica-

tors of the consciousness state of a patient with brain

injury.

4 TOWARDS A

GENERAL-PURPOSE EVENT

DETECTION ALGORITHM

In our opinion, even if the global RTP events extracted

from EEG signals are good candidates to characterize

the state of consciousness of a subject, the applica-

tion of the fractal intermittency approach for general

purposes would require an extension in the deﬁnition

of brain events. In fact, EEG signals are strongly

non-stationary and display large variability at several

time scales and this variability is not only quantitative

but also qualitative with very different waveforms and

grapho-elements, also depending on the experimental

condition. It is then desiderable to extend the RTP

deﬁnition of brain event to a multi-modal deﬁnition,

thus allowing to have different kind of events that can

be used individually or jointly depending on the par-

PhyCS 2016 - 3rd International Conference on Physiological Computing Systems

126

ticular application and scope. For example, we expect

that a kind of brain events is suitable to establish some

pathological condition, while another kind is not.

The starting point here proposed is based on the stan-

dard approach based on splitting the EEG signal into

the classical frequency bands. For each band, some

basic events are extracted, so that we can deﬁne at

least one event typology for each band. In our opin-

ion, this would allow to implement a ﬁrst kernel of

event detection algorithms that can be improved and

upgraded in future developments. Further, in brain re-

search each frequency band is often the starting point

for the building of neural correlates. This could al-

low for a easier link between the technical aspects

of the signal processing procedure used to extract

the events and the neuro-physiological interpretation

of these same events. Then, the intermittency-based

complexity index can be also extended to a vector of

complexity measures, whose elements can be selected

and/or coupled in different ways depending of the par-

ticular application and/or pathology in order to get a

reliable statistical index.

The approach here introduced is essentially an exten-

sion of an already existing algorithm that was orig-

inally developed to detect the high activity epochs

within each frequency band by means of a threshold-

ing technique (see, e.g., (Navona et al., 2002; Barcaro

et al., 2004; Magrini et al., 2015; Righi et al., 2008))

In the intermittency-based complexity approach the

focus is on the transition events from high (+) to low

(−) states (epochs) and vice versa. The event detec-

tion algorithm is applied to EEG signals after a pre-

processing, essentially consisting of the usual notch

ﬁlter at 50 Hz and a successive artifact removal. In

the following we assume that the EEG signal was al-

ready pre-processed and artifacts were removed.

Then, the event detection algorithm works as follows:

(1) The frequency bands of the EEG signal are com-

puted. The following band ranges are usually con-

sidered: (a) δ band (0.5 −4 Hz); (b) θ band (4 −8

Hz); (c) α band (8 − 12 Hz); (d) σ band (12 − 16

Hz); (e) β band (16 − 35 Hz); (f) γ band (35 − 64

Hz).

(2) For each frequency band, the absolute value, i.e.,

the component amplitude is considered. Then,

two moving-window time averages are computed

at different time scales, short and long, that are

typically chosen as 2 and 64 sec., respectively.

(3) Calculation of non-dimensional descriptors for

each frequency band: (short-time average - long-

time average)/long-time average [denoted as A =

A(t) in Fig. 1].

(4) Identiﬁcation of epochs and transition events be-

Figure 1: Identiﬁcation of epochs and transition events. The

curve A = A(t) is the descriptor deﬁned in item (3) for a

given frequency band, L/H is the low/high threshold.

tween epochs for a given band. In Fig. 1

we sketch an example with 6 different kinds of

events, all obtained by considering the crossings

through two different thresholds: L (Low) and H

(High). Typical values are L= 0 and H= 1.

In the example of Fig. 1, the epoch typologies are:

(i) L

: epochs with descriptor above (+) or below

(−) the threshold L; (ii) H

: same as before for

the threshold H; (iii) C

: epochs L

containing

at least one epoch H

; (iv) C

−

(complementary

to the epochs C

. Similarly, the crossing times

corresponding to these epochs deﬁne the events

of the kind L

, H

and C

(5) Storing in a database (spatio-temporal event

maps) and feature extraction (e.g., number of

events per time unit for each band and EEG trace).

5 CONCLUSIONS

In this work we have discussed the complexity

paradigm in the context of the analysis and modeling

of physiological signals and, in particular, the fractal

intermittency approach applied to the signal process-

ing of EEG recordings. We argued that the opera-

tional deﬁnition of the events and, consequently, the

speciﬁc algorithm of event detection can play a cru-

cial role in the evaluation of EEG complexity asso-

ciated with fractal intermittency. For this reason, we

discussed a proposal to extend the deﬁnition of RTP

events in order to have a set of different kind of brain

events that could be used for different applications

and to build a general-purpose algorithm for event

detection and, thus, for the estimation intermittency-

based complexity measures.

The Challenge of Brain Complexity - A Brief Discussion about a Fractal Intermittency-based Approach

127

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