unified approach to catastrophic events: from the normal …...namics fracture has been shown to be...

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Unified approach to catastrophic events: from thenormal state to geological or biological shock in terms of

spectral fractal and nonlinear analysisK. A. Eftaxias, P. G. Kapiris, G. T. Balasis, A. Peratzakis, K. Karamanos, J.

Kopanas, G. Antonopoulos, K. D. Nomicos

To cite this version:K. A. Eftaxias, P. G. Kapiris, G. T. Balasis, A. Peratzakis, K. Karamanos, et al.. Unified approachto catastrophic events: from the normal state to geological or biological shock in terms of spectralfractal and nonlinear analysis. Natural Hazards and Earth System Science, Copernicus Publicationson behalf of the European Geosciences Union, 2006, 6 (2), pp.205-228. �hal-00299278�

https://hal.archives-ouvertes.fr/hal-00299278https://hal.archives-ouvertes.fr

Nat. Hazards Earth Syst. Sci., 6, 205–228, 2006www.nat-hazards-earth-syst-sci.net/6/205/2006/© Author(s) 2006. This work is licensedunder a Creative Commons License.

Natural Hazardsand Earth

System Sciences

Unified approach to catastrophic events: from the normal state togeological or biological shock in terms of spectral fractal andnonlinear analysis

K. A. Eftaxias1, P. G. Kapiris1, G. T. Balasis2, A. Peratzakis1, K. Karamanos3, J. Kopanas1, G. Antonopoulos1, andK. D. Nomicos4

1Solid State Section, Physics Department, University of Athens, Panepistimiopolis, 157-84, Zografos, Athens, Greece2Department of Earth’s Magnetic Field, GeoForschungsZentrum Potsdam, Telegrafenberg, 14473, Potsdam, Germany3Centre for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Campus Plaine, C.P. 231, Boulevarddu Triomphe, B-1050, Brussels, Belgium4Department of Electronics, Technological Education Institute of Athens, Egaleo, 12210, Greece

Received: 3 August 2005 – Revised: 27 January 2006 – Accepted: 27 January 2006 – Published: 3 April 2006

Part of Special Issue “Seismic hazard evaluation, precursory phenomena and seismo electromagnetics”

Abstract. An important question in geophysics is whetherearthquakes (EQs) can be anticipated prior to their occur-rence. Pre-seismic electromagnetic (EM) emissions providea promising window through which the dynamics of EQpreparation can be investigated. However, the existence ofprecursory features in pre-seismic EM emissions is still de-batable: in principle, it is difficult to prove associations be-tween events separated in time, such as EQs and their EMprecursors. The scope of this paper is the investigation ofthe pre-seismic EM activity in terms of complexity. A basicreason for our interest in complexity is the striking similar-ity in behavior close to irreversible phase transitions amongsystems that are otherwise quite different in nature. Inter-estingly, theoretical studies (Hopfield, 1994; Herz and Hop-field 1995; Rundle et al., 1995; Corral et al., 1997) suggestthat the EQ dynamics at the final stage and neural seizuredynamics should have many similar features and can be an-alyzed within similar mathematical frameworks. Motivatedby this hypothesis, we evaluate the capability of linear andnon-linear techniques to extract common features from brainelectrical activities and pre-seismic EM emissions predictiveof epileptic seizures and EQs respectively. The results sug-gest that a unified theory may exist for the ways in whichfiring neurons and opening cracks organize themselves toproduce a large crisis, while the preparation of an epilepticshock or a large EQ can be studied in terms of “IntermittentCriticality”.

Correspondence to:K. Eftaxias([email protected])

1 Introduction

Fracture in disordered media is a complex problem whichstill lacks a definite physical and theoretical treatment. De-spite the large amount of experimental data and the consid-erable effort that has been undertaken by the material scien-tists, many questions about the fracture remain. Earthquakes(EQs) are large scale fracture phenomena in the Earth’s het-erogeneous crust. A vital problem in material science and ingeophysics is to identify precursors of macroscopic defectsor shocks. In physics, the degree to which we can predict aphenomenon is often measured by how well we understandthis. Moreover, it is difficult to prove associations betweenevents separated in time, such as EQs and their precursors.Herein, we hope to uncover more information hidden in thepre-seismic (EM) time series, and thus, to achieve a deeperunderstanding of the physics of the EQ nucleation process.

Both acoustic as well as EM emissions, in a wide fre-quency spectrum ranging from very low frequencies (VLF)to very high frequencies (VHF), are produced by openingcracks, which can be considered as the so-called precursorsof general fracture. We focus on the geophysical view ofthis problem. Our main tool is the monitoring of the micro-fractures, which occur before the final break-up in the focalarea, by recording their VLF-VHF EM emissions.

In the last 15 years, the study of complex systems hasemerged as a recognized field in its own right, although, agood definition of what a complex system has proven elu-sive. When one considers a phenomenon or a thing that is“complex”, one generally associates it with something that is“hard to separate, analyze or to solve”. Instead, we refer to

Published by Copernicus GmbH on behalf of the European Geosciences Union.

206 K. A. Eftaxias et al.: Catastrophic phenomena

“a complex system” as one whose phenomenological laws,which describe the global behavior of the system, are notnecessarily directly related to the “microscopic” laws thatregulate the evolution of its elementary parts. In other words,“complexity” is the emergence of a non-trivial behavior dueto the interactions of the subunits that form the system it-self. The statistical features of complex systems are gener-ally not dependent on the details of the interactions of thesubunits that form the system. Another relevant ingredient ofa complex system is that topological disorder within the sys-tem will generally introduce new, surprising effects, differentthan those one would expect from the simple “microscopic”evolution rules.

A basic reason for our interest in complexity is the strik-ing similarity in behavior close to irreversible phase tran-sitions among systems that are otherwise quite different innature (Stanley, 1999, 2000; Sornette, 2002; Vicsek, 2001,2002). One universal footprint seen in many complex sys-tems is self-affinity. The fractional power-law relationshipis a classic definition of a self-affine structure. Thus, muchwork on complexity focuses on statistical power laws thatdescribe the scaling properties of fractal processes and struc-tures. Recent studies have demonstrated that a large vari-ety of complex processes, including EQs (Bak, 1997), forestfires (Malamud et al., 1998), heartbeats (Peng et al., 1995),human coordination (Gilden et al., 1995; Chen et al., 1997),neuronal dynamics (Worrell et al., 2002), financial markets(Mantegna and Stanley, 1995; Lux and Marchesi, 1999), ex-hibit statistical similarities, most commonly power-law scal-ing behavior of a particular observable.

A few years ago, Bak et al. (1987, 1989) coined the termself-organized criticality (SOC) to describe the phenomenonobserved in a particular automaton model, nowadays knownas the sandpile model. This system is critical in analogywith classical equilibrium critical phenomena, where neithercharacteristic time nor length scales exist. The strong analo-gies between the dynamics of the SOC model for EQs andthat of neurobiology have been realized by numerous au-thors (Hopfield, 1994; Herz and Hopfield, 1995; Corral etal., 1997; Usher et al., 1995; Zhao and Chen, 2002; Plenz,2003). In general, authors have suggested that EQ dynamicsand neurodynamics can be analyzed within similar mathe-matical frameworks (Herz and Hopfield, 1995; Rundle et al.,2002). Characteristically, driven systems of interconnectedblocks with stick-slip friction capture the main features ofEQ process. These models, in addition to simulating the as-pects of EQs and frictional sliding, can also represent the dy-namics of neurological networks (Herz and Hopfield, 1995,and references therein). Hopfield (1994) proposed a modelfor a network ofN integrate-and-fire neurons. In this model,the dynamical equation ofkth neuron Eq. (28) in Hopfield(1994) is based on the Hodgekin-Huxley model for neurody-namics and represents the same kind of mean field limit thathas been examined in Rundle et al. (2002) in connection withEQs.

We have been motivated by the suggestion that EQ dynam-ics and neurodynamics can be analyzed within similar math-ematical frameworks. We concentrate here on the questionwhether common precursory symptoms emerge as epilepticseizures and EQs approach.

Electroencephalogram (EEG) time series provide a win-dow through which the dynamics of epileptic seizure prepa-ration can be investigated under well-controlled conditions.Consequently, the analysis of a pure pre-epileptic time se-ries would help in establishing a reliable collection of crite-ria to indicate the approach to the biological shock. In termsof complexity, such a collection of criteria indicates the ap-proach to global instability in the pre-focal area as well.

The results of the present analysis imply that a unified the-ory may exist for the ways in which firing neurons/openingcracks organize themselves to produce a significant epilepticseizure/EQ: we show that many similar distinctive symptoms(including common alteration in associated scaling parame-ters) emerge as epileptic seizures and EQs are approaching.The present state of research requires a refined definition ofa possible pre-seismic EM anomaly, as well as the develop-ment of more objective methods of distinguishing seismo-genic emissions from non-seismic EM events. The observedsimilarities support the consideration that the detected EManomaly is originated during the micro-fracturing in the pre-focal area of the impending EQ.

2 Pre-seismic electromagnetic activity

When a heterogeneous material is strained, its evolution to-wards breaking is characterized by local nucleation and co-alescence of micro-cracks. Crack propagation is the basicmechanism of material’s failure. The motion of a crack in dy-namics fracture has been shown to be governed by a dynami-cal instability causing oscillations in its velocity and struc-ture on the fracture surface. Experimental evidence indi-cates that the instability mechanism is that of local branching(Sharon and Fineberg, 1996). A multi-crack state is formedby repetitive, frustrated micro-fracturing events (Sharon andFineberg, 1999).

In many materials, emission of photons, electrons, ionsand neutral particles are observed during the formation ofnew surface features in fracturing, deformation, wearing,peeling, and so on. Collectively, we refer to these emissionsas fractoemission (Langford et al., 1987; Dickinson et al.,1988; Gonzalez and Pantano, 1990; Miura and Nakayama,2000; Takeuchi and Nagahama, 2004; Mavromatou et al.,2004; Wu et al., 2004). It is worth mentioning that labora-tory experiments show that more intense fractoemissions areobserved during the unstable crack growth (Gonzalez andPantano, 1990). The rupture of inter-atomic (ionic) bondsalso leads to intense charge separation, the origin of the elec-tric charge between the micro-crack faces. On the faces ofa newly created micro-crack the electric charges constitute

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Fig. 1. (a)10 kHz magnetic field variations associated with the Athens EQ. The consecutive green, red, and green epochs show the normalstate, the pre-seismic phase, and the aftershock state (new normal state), correspondingly. The sampling rate was 1 Hz.(b) Rat EEG timeseries. The sampling rate was 200 Hz. Bicuculline i.p. injection was used to induce the rat epileptic seizures. The green, red and ochreepochs show the normal state, the pre-epileptic phase, and the stage including the epileptic seizure, correspondingly.

an electric dipole or a more complicated system. Due tothe crack strong wall vibration in the stage of the micro-branching instability, it behaves as an efficient EM emitter.These EM precursors are detectable both at a laboratory andgeological scale. Our main tool is the monitoring of themicro-fractures, which possibly occur in the pre-focal areabefore the final break-up, by recording their EM emissions.A multidisciplinary analysis in terms of fault modeling (Ef-taxias et al., 2001), laboratory experiments (Eftaxias et al.,2002), criticality (Kapiris et al., 2004a; Contoyiannis et al.,2004, 2005), scaling similarities of multiple fracturing ofsolid material (Kapiris et al., 2004b), fractal electrodynamics(Eftaxias et al., 2004) and complexity (Kapiris et al., 2005a),seems to validate their association with the fracturing processin the pre-focal area.

Figure 1a shows the time series of the 10 kHz magneticfield variations associated with the Athens EQ. We concen-trate on this kHz EM activity observed before the 7 Septem-ber 1999 Athens EQ (38.2◦ N, 23.6◦ E) with a magnitude Ms(AT H)=5.9 at Zante station (Eftaxias et al., 2000, 2001,2003, 2004; Kapiris et al., 2004). In terms of literature, thisprecursory emission has a strange, long duration (approxi-mately a few days), thus it provides sufficient data for a sta-tistical analysis. We note that the data were sampled at 1 Hz.

3 Electroencephalogram

The brain is composed of very large numbers of non-identicalnon-linear neurons that are embedded in a vast and com-

plex network. Electroencephalogram (EEG) provides a win-dow, through which the dynamics of epilepsy preparationcan be investigated. Epileptic seizures are abnormal, tempo-rary manifestations of dramatically increased neuronal syn-chrony, either occurring regionally (partial seizures) or bilat-erally (generalized seizures) in the brain. There is presentlyno standard mathematical model of EEG activity. There-fore investigators have been using various methods of signalsanalysis to describe stochastic and deterministic features ofthese signals. The prediction of epilepsy seizures plays animportant role in epileptology (Litt and Javier, 2002), be-cause we can propose a new diagnostic tool and a novel ap-proach to seizure control based on this prediction (Iasemidis,2003). Figure 1b shows a rat’s epileptic seizure.

Adult Sprague-Dawley rats were used to study the epilep-tic seizures in EEG recordings. The rats were anaesthetizedwith an i.p. injection of Nembutal (sodium pentobarbital,65 mg/kg of body weight), and mounted in a stereotaxicapparatus. An electrode was placed in epidural space torecord the EEG signals from temporal lobe. The animalswere housed separately postoperatively with free access tofood and water, allow 2–3 days to recover, and handled gen-tly to familiarize them with the recording procedure. Eachrat was initially anaesthetized with a dose of pentobarbi-tal (60 mg/kg, i.p.), while constant body temperature wasmaintained (36.5–37.5◦C) with a piece of blanket. The de-gree of anaesthesia was assessed by continuously monitor-ing the EEG, and additional doses of anesthetic were ad-ministered at the slightest change towards an awake pattern

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Fig. 2. Upper part. Number of electric pulses of the pre-ictal (redepoch in Fig. 1b) with energies higher than that given by the corre-sponding abscissa. The continuous line is the least squares fit of thepower-lawN(>E)=E−b. Lower part. Number of electromagneticpulses of the pre-seismic phase (red epoch in Fig. 1a) with energieshigher than that given by the corresponding abscissa. The continu-ous line is the least squares fit of the power-lawN(>E)=E−b. Thepower-laws observed here is the footprint of a scale-free activity; anabsence of characteristic lengths in the system.

(i.e., an increase in the frequency and reduction in the am-plitude of the EEG waves). Then, bicuculline i.p. injectionwas used to induce the rat epileptic seizures. EEG signalswere recorded using an amplifier with band-pass filter set-ting of 0.5–100 Hz. The sampling rate was 200 Hz, and theanalogue-to-digital conversion was performed at 12-bit reso-lution. The seizure onset time is determined by visual identi-fication of a clear electrographic discharge, and then look-ing backwards in the record for the earliest EEG changesfrom baseline associated with the seizure. The earliest EEG

change is selected as the seizure onset time. The intervalbetween the seizure onset time and injection time are con-sidered as the maximum prediction duration or extended pre-ictal phase.

4 Fingerprints of scale free time and length behavior inthe pre-seismic/pre-epileptic phase

Complexity manifests itself in linkages between space andtime, generally producing patterns on many scales and theemergence of fractal structure. A lot of work on complexityfocuses on statistical power-laws, which describe the scal-ing properties of fractal processes and structures that arecommon among systems that at least qualitatively are con-sidered complex. Scaling behavior or “scale-free” behaviormeans that no characteristic scales dominate the dynamicsof the underlying process. Scale-free behavior reflects a ten-dency of complex systems to develop correlations that de-cay more slowly and extend over larger distances in time andspace than the mechanisms of the underlying process wouldsuggest (Bassingthwaihte et al., 1994; Barabasi and Albert,1999; Bak, 1997). The emergence of a scale-free behav-ior is generally named “criticality” (Kadanoff et al., 1967).Thus, firstly we focus on the fundamental question whetherpower-law scaling behaviors in associated time series emergeas epileptic seizures or EQs approach.

4.1 Focus on pre-seismic signals

We have already studied this question in relation to pre-seismic EM emissions (Kapiris et al., 2004b) and, so far,find that the sequence of precursory EM pulses associatedwith the Athens EQ follow: (i) power-law distribution ofthe quiescent (waiting) times between successive EM pulses,i.e. (1t)−1.8 ; (ii) power-law distribution of the durations ofthe EM events,(1t)−1.9; (iii) a “Gutenberg-Richter type”distribution, namely, the cumulative numberN(>A) of EMpulses having amplitudes larger thanA follows the power-law N(>A)∼A−0.62.

These experimental results witness that the heterogeneoussystem in the focal area evolves to its global instability with-out characteristic time and length scales. This behavior im-plies the existence of an underlying critical dynamics.

Remark: Rabinovitch et al. (2001) have recently stud-ied the fractal nature of EM radiation induced by rock frac-ture. The analysis of the pre-fracture EM time series revealsthat the cumulative distribution function of the amplitudesfollows the powerN(>A)∼A−0.62. The accord of expo-nents in phenomena involving remarkably different scalesshould be considered as a further hint that the same fracto-electromagnetic critical dynamics holds from the geophysi-cal scale down to the microscopic scale.



4.2 Focus on EEG time series

We investigate the potential presence of power-law distri-butions in rat pre-epileptic phase (red epoch in Fig. 1b).The statistical analysis shows that the distribution of elec-tric pulses of energy larger thanE, N(>E), is well fitted bythe power-lawN(>E)∼E−b, whereb=1.22 (Fig. 2 upperpart). The energy distribution of the binned data will followthe power-lawN(E)∼E−δ, whereδ=(1+ b)=2.22 (Kapiriset al., 2004b). For comparison reasons, Fig. 2b (lower part)exhibits the number of EM events with energyE higher thanthat given by the corresponding abscissa. The continuousline is the least square fit of the power lawN(>E)∼E−b,whereb=0.31.

The distribution of quiescent time between successivefluctuations,l, in EEG time series, is well fitted by the power-law N(l)∼l−η, whereη=1.15 (Fig. 3 upper part). For com-parison, in Fig. 3 (lower part), the same distribution is alsopresented for the pre-seismic EM time series (red epoch ofFig. 1a).

In summary, in the preparation stage of EQ (or epilepticseizure) the population of activated cracks (or firing neurons)in the focal area (or brain), seems to evolve through a nonlinear feedback mechanism triggering transitions betweendifferent metastable states, while, these transitions take theform of intermittent avalanche like eventsdistributed with-out characteristic time and length scales.

5 Emergence of common distinctive alternations in as-sociated scaling dynamical parameters as epilepticseizures and EQs approach

The main feature of collective behavior is that an individualunit’s (opening-crack or firing neuron) action is dominatedby the influence of its neighbors; the unit behaves differentlyfrom the way it would behave on its own. Finally, orderingphenomena are emerged as the units simultaneously changetheir behavior to a common pattern (Vicsek, 2001, 2002).The emphasis in the structure formation in complex systemsis bridging the gap between what one element does and whatmany of them do when they function cooperatively. Con-sequently, it follows that the science of complexity is aboutrevealing the principles that govern the ways in which thesenew properties appear. Based on this background, we attemptto investigate not only the presence of power-law behavior inthe pre-seismic (or pre-ictal) period, but mainly the temporalevolution of the associated scaling parameters as the mainshock approaches. The visually apparent “patchiness” andevolution of the pre-seismic/pre-epileptic signals (Fig. 1) im-plies that different parts of time series have different scalingproperties.

First, we focus on the statistics of the recorded series offluctuations with respect to their amplitude, let’s sayA(ti).Any time series can exhibit a variety of autocorrelation struc-

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Fig. 3. Upper part. The distribution of the waiting times betweentwo pre-epileptic electric pulses (red epoch in Fig. 1b). The straightline is the respective power-law. Lower part. The distribution ofthe waiting times between two pre-seismic electromagnetic pulses(red epoch in Fig. 1a). The strait line is the respective power-law.The occurrence of power-laws indicate that both systems developcritical correlations without characteristic time scales.

tures; successive terms can show strong (brown noise), mod-erate (pink noise) or no (white noise) correlation with previ-ous terms. The strength of these correlations provides use-ful information about the inherent “memory” of the system.The power spectrum,S(f ), is probably the most commonlyused technique to detect structure in time series. The quantityS(f )df is understood as the contribution to the total powerfrom those components of the time series, whose frequen-cies lie in the interval betweenf and f +df . If the timeseriesA(ti) is a self-affine time series that series cannot havea characteristic time scale. But a fractal time series cannothave any characteristic frequency either. The only possibility



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is then that the power spectrumS(f ) has a scaling form:

S(f ) ∼ f −β (1)

where the power spectrumS(f ) quantifies the correlationsat the time scaleτ∼1/f andf is the frequency of the Fouriertransform. In a lnS(f )− ln f representation the power spec-trum is a line with spectral slopeβ. The linear correlationcoefficient,r, is a measure of the goodness of fit to the power-law (1).

Our approach is to calculate the fractal parameterβ andthe linear correlation coefficientr of the power-law fit di-viding the signal into successive segments of 1024 sampleseach, in order to study not only the presence of a power-law

S(f )∼f −β but, as it was mentioned, the temporal evolutionof these parameters.

The Continuous Wavelet Transform (CWT), using Morletwavelet, is applied to compute the power spectrum, since be-ing superior to the Fourier spectral analysis providing excel-lent decompositions of even transient, non-stationary signals(Kaiser, 1994). It has the ability of providing a representa-tion of the signal in both the time and frequency domains. Incontrast to the Fourier transform, which provides the descrip-tion of the overall regularity of signals, the wavelet transformidentifies the temporal evolution of various frequencies. Thisproperty suits the signals under investigation, because they



are not stationary by their nature, and have a time varyingfrequency content.

5.1 Information hidden in the temporal evolution of coeffi-cient correlationr

Figures 4a and b present the temporal evolution ofr as themain geological/biological event is approached. We focuson the precursory epochs 2 and 3. We observe a significantincrease of the correlation values, while, the percentage ofr that exceeds a certain threshold increases. At the tail ofthe precursory activity the fit to the power-law is excellent: aregion withr close to 1 is approached during the last stageof the pre-seismic/pre-seizure activity. The fact that the dataare well fitted by the power-law (1) suggests that the pre-seizure (or pre-seismic) activity can be ascribed to a multi-time-scale cooperative activity of numerous activated firing-neurons/emitting-cracks. Here, an individual unit’s behavioris dominated by its neighbors so that all units simultaneouslyalter their behavior to a common large scale fractal pattern.The gradual increase ofr indicates that the clustering of fir-ing neurons/opening cracks in more “compact” fractal struc-tures in brain/pre-focal area is strengthened with time.

We concentrate on the normal (quiescent) EM period (firstepoch), preceding the emergence of the possible seismogenicEM anomaly (second and third epochs). We observe thatthe majority of segments (∼95%) do not follow the powerlaw (1). This means the EM background of the record doesnot behave as a temporal fractal. Thus the appearance offractal structures seems to foretell on the launch of a possibleseismogenic emission.

5.2 Information hidden in the temporal evolution of scalingspectral exponentβ

Two classes of signal have been widely used to modelstochastic fractal time series: fractional Gaussian noise (fGn)and fractional Brownian motion (fBm). These are, respec-tively, generations of white Gaussian noise and Brownianmotion. The main difference between fBm and regularBrownian motion is that while the increments in Brownianmotion are independent they are dependent in fBm. The na-ture of fractal behavior (i.e., fGn versus fBm) provides in-sight into the physical mechanism that generates the correla-tions: The fGn random fields describe fluctuations (additivenoise) around mean state, while the fBm random model isderived from growth, deposition or random-walk processes.

The scaling exponentβ associated with the fBm modellies between 1 and 3 (Heneghan and McDarby, 2000). Therange ofβ from 0 to 1 indicates the fGn class. Figures 4aand b reveal that during the pre-ictal (or pre-seismic) phase(epochs 2 and 3) theβ-values are distributed in the regionfrom 1 to 3. This observation implies that the associated partof both time series follow the fBm model.

Remark: We refer to the pre-seismic EM time series. Itwas mentioned that a very small number of segments (∼5%)of the epoch 1 follows the power law (1). Theirβ-values aresmaller than 1, consequently, these segments belongs to thefGn class. This alteration further reveals the launch of theseismogenic emission from the background.

The distribution ofβ-exponents is also shifted to highervalues (Figs. 4a and b) during the precursory periods. Thisshift reveals several important features of the underlyingmechanism. The fractal-laws observed corroborate to theexistence of memory. The current value of the biologi-cal/geophysical signal co-varies not only with its most recentvalue but also with its long-term history in a scale-invariant,fractal manner, namely, the system refers to its history inorder to define its future (Hausdorff and Peng, 1996). Asthe β-exponent increases the spatial correlation in the timeseries also increases. Consequently, the observed increaseof β-values with time indicates the gradual increase of thememory, and thus the gradual reduction of complexity in theunderlying dynamics.

Concerning epilepsy, the driving force behind a large num-ber of studies on dimension analysis of EEG is that epilep-tic seizures are regarded as emergent states with reducedcomplexity compared to non-epileptic activity (Lai, 2002,and references therein): seizure onset represents a transitionfrom relatively less orderly state to one of more orderly state(Iasemidis and Sackellares, 1996). Figure 4b also revealsthat the epoch later in the epileptic seizure is of greater com-plexity than the early epoch. In summary, as the seizure ap-proaches there is a transition from higher to lower complexityand then back to higher complexity prior to seizure termina-tion.

The 1/f observed power relationship implies that pertur-bations occurring at low frequencies can cause a cascade ofenergy dissipation at higher frequencies and that widespreadslow oscillation modulate faster local events (Buzsaki andDraguhn, 2004). Indeed, Figs. 5a and b show that coherentfluctuations in a higher range of frequencies occur as the crit-ical point is approached.

The observed color-type behavior of the power spectrumdensity (β>0) means that the spectrum manifests morepower at low frequencies than at high frequencies. If allfrequencies are equally important, namely white noise, wehaveβ=0. The increase, however, in the spectrum slopeβ witnesses the gradual enhancement of lower frequencyfluctuations, namely, the system selects to transmit morepower at lower frequencies.These observations are con-sistent with the following physical picture: the activatedmicro-cracks/neurons interact and coalesce to form largerfractal structures, i.e., the events are initiated at the lowestlevel of the hierarchy, with the smallest elements merging inturn to form larger and larger ones. Concerning brain ac-tivity, higher frequency oscillations are confined to a smallneuronal space, whereas very large networks are recruitedduring slow oscillations (Buzsaki and Draguhn, 2004, and



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Fig. 5. The wavelet power spectrum of the time series associated with the Athens EQ (Fig. 1a) and the EEG time series including a ratepileptic seizure (Fig. 1b), correspondingly. The intensity scale on the top shows colour corresponding to the values of the square spectralamplitudes in arbitrary units.

references therein). There is only limited knowledge aboutseizure-generating mechanisms in humans. However, modelsimulations and animal experiments have led to the theorythat seizure activity will be induced when a “critical mass” ofneurons is progressively involved in synchronized discharg-ing (Lehnertz and Elger, 1998).

Nature seems to paint the following picture: first, singleisolated opening-cracks/firing-neurons emerge which, sub-sequently, grow and multiply. This leads to cooperative ef-fects. Long-range correlations build up through local inter-actions until they extend throughout the entire system; fi-nally, the main shock forms.The challenge is to determinethe “critical epoch” during which the “short-range” corre-lations evolve into “long-range”ones.Figure 4 reveals thatthe closer the global instability, the larger the percentagesof segments withr close to 1 and the larger the shift ofβ-exponent to higher values; theβ-values are maximal at the

tail of the pre-seizure/pre-seismic state. This behavior re-veals the emergence of a “critical epoch”, namely, the tran-sition to the epoch of synchronization of oscillating neuronaldischarges (or fracto-EM oscillators).

6 Monitoring the transition from anti-persistent to per-sistent behavior

We attempt to shed more light on the physical significanceof the systematic increase of theβ-exponent during the pre-seismic/pre-ictal period. Theβ-exponent is related to theHurst-exponent,H , by the formula (Turcotte, 1992)

β = 2H + 1 with 0 < H < 1, , 1 < β < 3 (2)

for the fractional Brownian motion (fBM) random fieldmodel (Henegham and McDardy, 2000). The exponentH



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Fig. 6. Decomposition of time series into subsets, each characterized by a different local Hurst exponentH . We pay attention to the findingthat the behavior of both the(a) pre-seismic signal and(b) pre-ictal signal become persistent in the tail of the precursory phase.

characterizes the persistent/anti-persistent properties of thesignal according to the following scheme. At the limitH=0, the signal does not grow at all and is stationary. Therange 0


positive feedback process. This implies that the processacquires to a great degree the property of irreversibility,namely, the system is close to irreversible out of equilibriumphase transition. Consequently, the emergence of the persis-tent activity gives a significant hint of a considerable proba-bility for a forthcoming global instability, namely, a signifi-cant seismic or epileptic event.

It is worth mentioning that laboratory experiments bymeans of acoustic EM emission verify that the main rup-ture occurs after the appearance of strong persistent behavior(Ponomarev et al., 1997; Lei et al., 2000, 2004).

7 A key parameter: the order of heterogeneity

In natural systems the units are not identical (Golomb andRinzel, 1993). Most natural dynamical systems evolve inthe presence of one or another type of disorder. Qualita-tively, the presence of strong intermittency in the time se-ries under study itself implies the effect of a highly hetero-geneous regime, in which the heterogeneity occurs at manytime-scales. Herz and Hopfield (1995) conclude that inho-mogeneities are needed to generate a potentially rich con-nection between EQ cycles and neural reverberations. Inthe following, effects of heterogeneity on the dynamical sys-tems under investigation are studied, in order to understandthe cause of the transition from a purely anti-persistent to apurely persistent behavior.

Focus on fracture. In a highly disordered medium thereare long-range anti-correlations, in the sense that a high valueof threshold for breaking is followed by a low value, andvice versa (Sahimi and Arbadi, 1996). The anti-persistentcharacter of the detected precursory EM emission reflectsthe fact that areas with a lowthreshold for breakingalter-nate with much stronger volumes. Crack growth contin-ues until a much stronger region is encountered. When thishappens, crack growth stops and another crack opens in aweaker region, and so on. In addition, the observed de-crease of the anti-persistency can be understood if we acceptthat the micro-heterogeneity of the system becomes less anti-correlated with time.

On the other hand, in the case of a homogeneous medium,once a crack nucleates in the rock, stress enhancement atits tip is larger than at any other point of the medium, andtherefore the next micro-crack almost surely develops atthe tip (Sahimi and Arbadi, 1996). Thus, one expects tosee long-range positive correlations, i.e., persistent behavior,0.5


al. (2003) have reported characteristic spatial and temporalshifts in synchronization that appear to be strongly relatedto pathological epileptic activity. On the other hand, authorshave reported in-phase changes of the temporal and spatialHurst exponents during sample deformation in laboratoryacoustic/EM emission experiments (Ponomarev et al., 1997,and references therein). If we accept this consideration, thedecrease of the fractal dimension reflects the appearance ofa clear preferred direction of elementary activities just be-fore the main shock. In general, the role of anisotropy inself-organized criticality has been highlighted (Pruessner andJensen, 2003).

Focus on fracture. The terminal phase of fracture pro-cess is accompanied by a significant increase in localizationand directionality (Eftaxias et al., 2004; Contoyiannis et al.,2005). The network of activated cracks becomes less rami-fied just before the main shock (Kapiris et al., 2004a).

Theoretical (Kolesnikov and Chelidze, 1985) aspects asindicate the following two major points: (i) elementary rup-tures interact due to overlapping of their (scalar) stress field,giving rise to correlation effects, and (ii) at the same timethe action of anisotropy inherent in the material leads to theexistence of preferred directions in the arrangement of ele-mentary ruptures.

On the laboratory scale, it is found that the spatial distri-bution of the hypocenters of the opening cracks (“laboratoryEQs”) shows fractal structure (Hirata et al., 1987). However,the fractal dimension significantly decreases with the evo-lution of the micro-fracturing process. Recent experimentsfurther verify this behaviour (Lei et al., 2000, 2004).

As it was mentioned, Ponomarev et al. (1997) have re-ported in-phase changes of the temporal and spatial Hurstexponents during sample deformation in laboratory acous-tic/EM emission experiments. Thus both correlation andanisotropy should be taken into account simultaneously(Chelidze, 1994).

Growing evidence suggests that rupture in strongly dis-ordered systems can be viewed as a type of critical phe-nomenon (Gluzman and Sornette, 2001, and referencestherein). Experiments (Anifrani et al., 1995; Lamaignere etal., 1996; Garcimartin et al., 1997; Johansen and Sornette,2000), numerical simulations (Sornette and Vaneste, 1992;Sahimi and Arbadi, 1996a; Sornette and Andersen, 1998)and theoretical aspects (Andersen et al., 1997) confirm thisconcept. Recently, we have shown that the anti-persistentphase of the precursory EM activity can be described as anal-ogous to a second-order phase transition: from the phase ofa sparse almost symmetrical random cracking distributionto a localized cracking zone that includes the backbone ofstrong asperities (Contoyiannis et al., 2005). A characteristicsignature of the onset of a continuous (second-order) phasetransition is symmetry breaking (S-B). This signature is hid-den in the anti-persistent pre-seismic time series (Contoyian-nis et al., 2005). The S-B here lies in the transition fromnon-directional uncorrelated to directional correlated crack

growth. The completion of the S-B in the anti-persistentepoch signals that micro-fractures in the heterogeneous com-ponent of the pre-focal area, that surrounds the backboneof strong asperities on the fault plane, have finished: the“siege” of the backbone of asperities begins (Contoyianniset al., 2005).

Consequently, the observed decrease of fractal dimensionwith time implies that the fracto-EM events have been re-stricted in the narrow zone including the backbone of stronglarge asperities that sustain the system (Eftaxias, 2004;Kapiris et al., 2004a). The emergence of strong persistentEM emission in the tail of the precursory emission revealsthat the fracture of asperities has been started (Contoyianniset al., 2005).

Focus on epileptic seizure. It has been reported thatthe characteristic abnormalities associated with epileptic-seizures include simplification of the dendrite tree (Sackel-lares et al., 2000), which indicates that the network of firingneurons will be less ramified just before the next new seizure.This justifies a significant decrease ofd-values with time inEEG time series just before the main event.

Usher et al. (1995) have presented a generic model thatgenerates long-range (power-law) temporal correlations, 1/fnoise, and fractal signals in the activity of neural populations.Strong correlations are induced in the model by spontaneoussymmetry breakingof the spatial pattern of activity across thenetwork, leading to a preferred axis. This pattern is, however,metastable in the presence of noise: clusters of high activitydiffuse throughout the system while undergoing strong in-ternal fluctuations. The spatial pattern’s persistent dynamicsresult in fractal (power-law) intervened interval distributionsand 1/f noise in the temporal dynamics. These features arein agreement with the results of the present study.

9 Information hidden in the accelerating persistentstage of precursory activity

Figures 5 and 7 reveal a significant accelerating energy re-lease as the main geophysical or biological shock is ap-proached. The observed acceleration implies that the twosystems are not only near the “critical point” in the sense ofhaving power-law correlations, but also in terms of exhibit-ing high susceptibility to perturbations.Due to the high levelof the clustering of opening micro-cracks (or firing neurons),even a new small-activated cluster is able to connect otherlarge clusters, and thus can generate a significant crisis. Theemergence of persistency in the tail of the precursory emis-sions strongly encourages this hypothesis.

Concerning the EEG time series, the appearance of per-sistency just before the main shock indicates both: (i) theincrease of the degree of excitability of the neural network(due to the increase of the average value of excitatory con-nections), and (ii) the increase of connectivity strength. Asimulation study performed by Bondarenko and Ree Chay



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Fig. 7. The cumulative energy release, in arbitrary units, as a function of time:(a) for the pre-seismic EM time series (see Fig. 1a) and(b)for the EEG time series (see Fig. 1b).

(1998) suggests that the combination of these two actionsleads to the synchronization of neurons outputs. Figure 5bshows that new higher frequencies gradually emerge in thespectrum during the pre-ictal phase. The authors propose thatthe appearance of a high-frequency component allows a bet-ter synchronization and, as a consequence, an increase of theaverage activity amplitude. These suggestions are in agree-ment with the observed acceleration of energy release duringthe pre-ictal period. Bondarenko and Ree Chay (1998) con-clude that these properties can be considered as the basis ofthe appearance of epilepsy.

The sensitivity of energy release and rich anisotropy pro-vides two cross checking precursors for the impending catas-trophic seismic or biological event.

10 Footprints of an underlying self-synchronization

Synchronization is one of the most fascinating non-linearphenomena appearing in a wide range of fields (Marodi etal., 2002, and references therein). The reduction of com-plexity with time in combination with the acceleration of en-ergy release, the gradual reduction of anti-persistency andthe emergence of persistent properties in the tail of the pre-cursory epoch, are features consistent with the concept offorced synchronization (self-organization), which can be de-scribed qualitatively as follows. The population of openingcracks in the stage of the micro-branching instability (seesection 2) can be considered as a source of radiated EMwaves. Due to the interaction of these nonlinear oscilla-tors, the energy exchange occurs. Gradually, due to internal

relations, the energy interchange can result in synchroniza-tion of phases of oscillators. Finally, the radiation graduallybecomes more coherent. During this process, the resultingEM energy grows considerably. Indeed, in the case of non-coherent radiation, namely if the phase distribution is chaoticand independent, the resulting energy W is proportional tothe number of oscillators-n. On the contrary, in the case offull coherent radiation, the energy W is proportional to thesquare number of interacting oscillators-n2 (Nicolis, 1986).We stress that if we accept the emergence of synchroniza-tion, then, the magnitudes of piezo-stimulated EM bursts canbe influenced not only by the population of activated cracksbut also can depend on the degree of coherency. Theoreticalaspects as well as laboratory experience imply the appear-ance of synchronization in the final stage of EQ preparationprocess.

Aksenov and Lokajicek (1997), based on the comparisonof laboratory and field observations, have shown the exis-tence of symptoms of forced synchronization (spectral sim-plification and transformation into a narrow spectral band)and coherency of radiation in seismic foci.

Chelidze and Lursmanashvili (2003), based on experi-ments on the spring-slider system subjected to a constant pulland superimposed with weak mechanical or electromagneticperiodic forces, have recently shown that at definite con-ditions the systems manifests synchronization of micro-slipevents with weak excitation. The regimes of slip vary fromthe perfect synchronization of slip events (acoustic emission)with the perturbing periodic mechanical or EM impact, to thecomplete desynchronization of micro-slip events and pertur-bations.



Gil and Sornette (1996) have shown a general correspon-dence between SOC and synchronization of threshold os-cillators. Recently, theoretical studies imply that the phe-nomenon of synchronized chaos also appears in fundamen-tal models for EQs (de Sousa Vieira, 1999; Akishin et al.,2000; Tsukamoto et al., 2003). Characteristically, Akishinet al. (2000) have presented an extension of the Burridge-Knopoff (BK) model of EQ dynamics, one of the basic mod-els of theoretical seismicity. The extension is based on theintroduction of non-linear terms for the inter-block springs ofthe BK model. Their analysis reveals synchronization of thecollective motion and produces stronger seismic events. Wepay attention to the fact that for the BK system the Hurst ex-ponent is much greater than 0.5. It lies between 0.6 and 0.8,stabilizing about 0.7, when time increases. This value, whichindicates persistency, is very close to the values of Hurst ex-ponents obtained in the tail of the pre-seismic EM activityunder investigation, as well as for real EQs that is also about0.7 (Lomnitz, 1994).

We turn our attention to epileptic seizures. In general, toproduce a voltage change large networks of neurons have tobe active synchronously (Buzsaki and Draguhn, 2004).

The epileptic seizure models exhibit distinct collectiveproperties that range from systemwide synchronization toself-organized criticality: phase locking without global syn-chronization has been identified as an important borderlinephenomenon between both cases and has been observed un-der various conditions (Herz and Hopfield, 1995, and refer-ences therein).

Numerous of studies suggest that neuronal hyper-synchrony underlies seizures (Lehnertz and Elger, 1995;Bondarenco and Ree Chay, 1998; Gong et al., 2003; Buzsakiand Draguhn, 2004). Non-linear analyses have suggestedthat the seizure onset represents a transition from the inter-ictal period (the period between seizures) to one of increasedsynchronous activity, and that a more orderly state character-izes an epileptic seizure (Iasemidis and Sackellares, 1996).

The reduction of complexity with time in EEG time se-ries, the acceleration of energy release, and the emergence ofpersistent properties in the tail of the pre-ictal epoch are fea-tures consistent with the concept that the epileptic seizuresare episodic events resulting from abnormal synchronous dis-charges from cerebral neuronal networks.

Buzsaki and Draguhn (2004) concluded that “An impor-tant function of the brain is the prediction of future probabil-ities. Feedforward and feedback networks predict well whathappens next. Oscillators are very good predicting when”.

11 A possible common scenario for the development ofsevere epileptic and seismic shock

By monitoring the temporal evolution of the fractal spec-tral characteristics in electroencephalograms and pre-seismicEM emissions we find that common distinctive alterations in

the associated scaling parameters indicate a transition fromthe low-activity (normal) state to an abnormal (high-activity)state (major EQ/epileptic seizure), as follows: (i) Appear-ance of memory effects, namely, emergence of long-rangecorrelations. This implies a multi-time-scale cooperative ac-tivity of numerous activated sub-units. (ii) Increase of thespatial correlationin the time serieswith time. This indicatesa gradual transition from a less orderly state to a more or-derly state. (iii) Gradual increase of the number of time inter-vals with “critical fractal characteristics”. It indicates that theclustering of activated units in more “compact” fractal struc-tures strengthened with time; the self-organized complexitybecomes more obvious. (iv) Predominance of large events asthe main biological or geophysical shock is approached. Thisobservation signals that the precursory events are initiated atthe lowest level of the hierarchy, with the smallest elementsmerging in turn to form larger and larger ones. (v) Appear-ance of strong anti-persistent behavior in the first epoch ofthe pre-ictal and pre-seismic activity. It implies the existenceof a non-linear negative feedback mechanism that “kicks”the system away from extremes. (vi) Decrease of the anti-persistence behavior with time. It implies that the mentionednon-linear negative feedback mechanism gradually loses itsability to “kick” the system away from extremes. (vii) Ap-pearance of persistency in the “tail” of the precursory activ-ities. This gives a hint that the system acquires to a greatdegree the property of irreversibility. (viii) Significant accel-eration of the energy release as the main shock approaches.It shows that the system exhibits high susceptibility even to asmall perturbation. (ix) Decrease of the fractal dimension ofthe time series with time. This suggests the establishment ofpreferred directions in elementary activities. (x) Gradual ap-pearance of higher frequencies in the spectrum with a simul-taneously increase of the amplitudes at each emission rate asthe shock approaches, mainly characterizing lower emissionrates. The coherent fluctuations at all scales further indicatethat the critical point is approached.

The aforementioned crucial precursory footprints (includ-ing temporal alterations in associated scaling parameters)may indicate the following common scenario for the devel-opment of a severe epileptic or seismic shock. During thefirst epoch of the transition from the normal state to the ab-normal one, the neuronal (or fracture) mechanism of patient(or strained focal area) is in self-organized complexity. Theenergy is released spasmodically in avalanches with a self-similar size distribution, however, with a restricted and sys-tematically fluctuating correlation length. The system is ina sub-critical anti-persistent state, in which case the externalinput signal can only access a small, limited part of the sys-tem. Long-range correlations gradually build up through lo-cal interactions until instabilities extend throughout the entiresystem.The scale over which the interacting units are cor-related sets the size of the largest event that can be expectedat that time. Such instabilities can channel the brain/focalarea into a globally nonequilibrium super-critical state. Weak



events can be the agents by which longer correlations are es-tablished. A population of weak events will advance the cor-relation length by an amount depending on the brain/focalarea state, triggering intense shocks only if the condition isright. In a sub-critical anti-persistent regime, a population ofsmall activated events leads to a quickly decaying activity.In the super-critical persistent state is just able to continue“indefinitely”. This naturally explains why while brain/focalarea not being close to criticality can exhibit only a small re-sponse to a small perturbation, its response near criticalityto this same small perturbation may become explosive dueto the very high “susceptibility”. An intense shock destroyslong correlations, creating a new normal period during whichthe process repeats by rebuilding correlation lengths towardscriticality and the next large event. Thus a large epileptic orseismic shock can act as a sort of “critical point” dividing theepileptic/seismic cycle into a period of growing correlationsbefore the great event and a relatively uncorrelated phase af-ter.

Concerning the brain, this view is consistent with Bak’sview (1997). Bak argued that the brain can be neither sub-critical nor super-critical but self-organized critical. In thefirst case, the external input signal can only access a smalllimited part of the information. In the second case, any in-put would cause an explosive branching process within thebrain, and connect the input with essentially everything thatis stored in the brain. Except in last case, the brain has an ap-propriate sensitivity to small shocks (Zhao and Chen, 2002).

Remark: The precursory symptoms indicate that the pro-posed method of analysis is interesting for readers belongingto different scientific fields. Major magnetospheric distur-bances are undoubtedly among the most important phenom-ena in space physics and also a core subject of space weather.They are relatively rare events: as in the case of atmosphericstorms, EQs, solar flares, etc., the occurrence of geomagneticstorms rapidly decreases as their magnitude grows. Mag-netic storms occur when the accumulated input power fromthe solar wind exceeds a certain threshold. MS intensity isusually represented by an average of the geomagnetic per-turbations measured at four mid-latitude magnetometer sta-tions (http://swdcwww.kugi.kyoto-u.ac.jp/). The resultinggeomagnetic index Dst is a widely used measure of the sym-metric ring current during MSs. During MSs the complexsystem of the Earth’s magnetosphere, which corresponds toan open (input – output) spatially extended nonequilibriumsystem, manifests itself in linkages between space and time,producing characteristic fractal structures. We have shown(Balasis et al., 2005) that the above mentioned distinctive al-terations in scaling parameters of Dst index time series occuras a strong magnetic storm approaches. The increase of thesusceptibility coupled with the transition from anti-persistentto persistent behavior indicates that the onset of a severemagnetic storm is imminent. This evidence enhances the uni-versal character of the precursory symptoms under study.

11.1 Footprints of an underlying intermittent criticality

Pure SOC models imply a system perpetually near global in-stability, hence reducing the degree of predictability of indi-vidual events (Bak et al., 1987; Bak and Tang, 1989). How-ever, natural systems very rarely break down without present-ing time-dependent effects. Thus, these simplified modelsfail to reproduce some important properties of the precursoryspatiotemporal clustering in real systems (Grasso and Sor-nette, 1998; Main and Al-Kindy, 2002). Characteristically,in a recent Letter, Yang et al. (2004) conclude that “EQs areunlikely phenomena of SOC; therefore further research onEQ prediction should not be discouraged or opposed by theargument that EQs are SOC and as a result unpredictable”.Authors, in order to bridge both, the hypothesis of an un-derlying self-organized critical state and the occurrence ofprecursory phenomena analyze more complex and realisticmodels (Sornette and Sammis, 1995; Sammis et al., 1996;Heimpel, 1997; Huang et al., 1998, Hainzl et al., 2000).In this direction, the school of “intermittent criticality” pre-dicts a time-dependant variation in the activity as the “criti-cal point” is approached, implying a degree of predictability.The aforementioned evolution incorporates a time dynamicswith memory effects, which is overall characterized as inter-mittent criticality.

Particularly, this hypothesis predicts two different precur-sory phenomena in space and time, namely, the accelerationenergy release and the growth of the correlation length. Interms of intermittent criticality the acceleration of energy re-lease is the consequence of the growth of the spatial corre-lation length. Thus, a large shock which is not immediatelypreceded by a period of accelerating energy release repre-sents a system which had previously reached a critical statebut has not yet had a large event to perturb the system awayfrom the critical state. On the other hand, an accelerating ac-tivity which is not followed by a large shock mirrors a systemwhich has achieved criticality but in which a large event hasnot nucleated.

The present study suggests that it can be important to dis-tinguish between SOC and intermittent criticality in the studyof the seismic/epileptic cycle. A proper recognition and un-derstanding of tuning parameters leads to the developmentof improved models having higher performance reliability.One of the main features in the complexity is the role that thetopological disorder plays in such systems. The range of sizescales characterizing heterogeneities of the thresholds mightbe acted as a tuning parameter of the underlying final dynam-ics.

We note that the preparation of a major magnetic stormcan be also studied in terms of “Intermittent Criticality”.


http://swdcwww.kugi.kyoto-u.ac.jp/


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Fig. 9. (a) Rat EEG time series. The sampling rate was 200 Hz. Bicuculline i.p. injection was used to induce the rat epileptic seizures.The green, red and ochre epochs show the normal state, the pre-epileptic phase, and the stage of the epileptic seizure, correspondingly. (b)Decomposition of time series into subsets, each characterized by a different local Hurst exponentH. The behavior of the pre-ictal signalbecomes persistence in the tail of the precursory phase.

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Fig. 8. The grey sector shows the percentage of the time in each epoch where the time series does not follow a power law of the typeS(f) ∼ f−β . The green and red sectors demonstrate the percentage of the time in each epoch where the time series follow the fBm modelexhibiting anti-persistent and persistent behavior respectively. The light blue sectors reveal the percentage of the time in each epoch wherethe time series follow the fGn model. This figure suggests that a reduced complexity of brain activity (lower part) or pre-focal area activity(upper part) coupled with the appearance of fBm persistent behavior, as soon as it is of sufficient duration, can be regarded as a specificfeature defining states which proceed to an epileptic seizure or EQ.

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Fig. 9. (a) Rat EEG time series. The sampling rate was 200 Hz. Bicuculline i.p. injection was used to induce the rat epileptic seizures.The green, red and ochre epochs show the normal state, the pre-epileptic phase, and the stage of the epileptic seizure, correspondingly. (b)Decomposition of time series into subsets, each characterized by a different local Hurst exponentH. The behavior of the pre-ictal signalbecomes persistence in the tail of the precursory phase.

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Fig. 8. The grey sector shows the percentage of the time in each epoch where the time series does not follow a power law of the typeS(f )∼f −β . The green and red sectors demonstrate the percentage of the time in each epoch where the time series follow the fBm modelexhibiting anti-persistent and persistent behavior, respectively. The light blue sectors reveal the percentage of the time in each epoch wherethe time series follow the fGn model. This figure suggests that a reduced complexity of brain activity (lower part) or pre-focal area activity(upper part) coupled with the appearance of fBm persistent behavior, as soon as it is of sufficient duration, can be regarded as a specificfeature defining states which proceed to an epileptic seizure or EQ.

12 Is the evolution towards global instability inevitableafter the appearance of distinctive symptoms?

A question that arises is whether the evolution towards globalinstability of the system is inevitable after the appearance ofdistinctive footprints in the pre-catastrophic time series. Thetransition from anti-persistency to persistency and the signif-icant acceleration of the energy release, namely, the increaseof the susceptibility of the system, the extension of fluctu-ations to new frequency scales with simultaneous predomi-nance of large events and the emergence of strong anisotropy,i.e. the appearance of preferential direction, indicates that thegeneration of a very intense seismic/epileptic event becomesunavoidable.

Our analysis reveals an interesting transition from the anti-persistent to the persistent regime. The anti-persistent behav-ior characterizes the brain/pre-focal area during weak events,

while the catastrophic events are in reasonable agreementwith persistent models. It is important to bear in mind thatthe Democratic Fiber Bundle Model (DFBM) for fracture ex-hibits an interesting transition as a function of the amplitudeof the disorder. As the disorder decreases, there exists a tri-critical transition (Andersen et al., 1997) from a progressivedamage (second-order) regime to a Griffith-type abrupt rup-ture (first-order) regime, where rupture occurs without signif-icant precursors. The corresponding pictures in each regimeare not in contradiction.

Contoyiannis et al. (2002) have introduced the intermittentdynamics of critical fluctuation (IDCF) method of analysisfor the critical fluctuations in systems which undergo a con-tinuous phase transition at equilibrium. This method is basedon a previous work (Contoyiannis et al., 2000) according towhich the fluctuations of the order parameter at the criticalpoint of a continuous phase transition obey an intermittent



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Fig. 9. (a) Rat EEG time series. The sampling rate was 200 Hz. Bicuculline i.p. injection was used to induce the rat epileptic seizures.The green, red and ochre epochs show the normal state, the pre-epileptic phase, and the stage of the epileptic seizure, correspondingly.(b)Decomposition of time series into subsets, each characterized by a different local Hurst exponentH . The behavior of the pre-ictal signalbecomes persistent in the tail of the precursory phase.

dynamics which can be analytically expressed by a one di-mensional map of intermittency type I. Recently, the focalarea has been modelled by (i) the backbone of strong and al-most homogeneous large asperities that sustains the system,and (ii) the strongly heterogeneous medium that surroundsthe family of strong asperities (Contoyiannis et al., 2005).The analysis of pre-seismic EM signals in terms of the IDCF-method (Contoyiannis et al., 2005) suggests that the initialanti-persistent part of the EM precursor is triggered by frac-tures in the highly disordered heterogeneous component thatsurrounds the essentially homogeneous backbone of asperi-ties, and described in analogy with a thermal phase transitionof second order. On the contrary, the persistency in the tail ofthe emission is thought to be due to the fracture of the highstrength asperities if and when the local stress exceeds theirfracture stress, and it is a purely out of equilibrium process.The abrupt emergence of strong VLF emission in the tail ofthe precursory EM radiation, showing persistent behavior, isregarded as a fingerprint of the local dynamic fracture of as-perities in the focal zone, namely, faulting nucleation (Con-toyiannis et al., 2005). This concept is strongly supported byseismological observations, information obtained from radarinterferometry, and space-time Thermal Infrared Radiation(TIR)-signals (Kapiris et al., 2005b). In addition, laboratorystudies indicate the appearance of persistent properties in theacoustic emission during the fracture of asperities (Lei et al.,2000, 2004).

Figures 8a and b verify that a reduced complexity of brain(or pre-focal area) activity coupled with the appearance offBm persistent behavior, as soon as it is of sufficient duration,

can be regarded as a specific feature defining states, whichproceed to an epileptic seizure or EQ.

Figure 9 refers to a new example of rat epilepsy. The anal-ysis further supports the concept that the transition to persis-tent regime can be considered as a candidate precursor. Werefer to the first event included in preparation (red) epoch ofthe seizure (Fig. 9). This abnormal event constitutes a kind of“foreshock” of the ensuing generalized epileptic seizure. Wethink that the emergence of the main (generalized) seizureresults from the further increase of the correlation length andthe appearance of persistent properties of sufficient durationin the EEG time series.

13 Extracting precursory symptoms in terms of non-linear techniques

The performed linear fractal spectral analysis suggests asmain result that the last phase of EQ/epileptic seizure prepa-ration process mischaracterized by a clear transition fromhigher to lower complexity. In the sequel, we verify this re-sult in terms of non-linear technique: before the EQ/seizuresymptom appeared, the complexity had begun declining.

13.1 Non-linear analysis in terms of T-complexity

A computable complexity measure, such as T-complexity,gives evidence of state changes leading to the point of globalfracture (Karamanos et al., 2005).

The T-complexity, which has been described in detailselsewhere (Tichener, 1998, 1999) is defined along similar



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Fig. 10. (a)We distinguish six epochs (W1-W6) in the evolution of the EM signal associated with the Athens EQ.(b) Each point denotes thenormalized T-entropy calculated by considering consecutive time windows of 1024 points for the six epochs defined in (a). We observe animportant decrease of the T-entropy values during the emergence of the two strong impulsive emissions at the tail of the precursory activity(windows W3 and W5).(c) Histograms of the probability distribution of the normalized T-entropy, calculated for the six epochs. We alsoobserve a significant reduction of the complexity as the main event is approaching, in particular during the epochs W3 and W5.

lines to the Lempel Ziv production complexity (Lempel andZiv, 1976), which measures the generation rate of new pat-terns along a digital sequence, but a recursive hierarchicalpattern copying algorithm (Tichener, 2000) is used. A char-acteristic example of an actual calculation for a finite stringis given in Ebeling et al. (2001). T-complexity serves as a

measure of complexity of the signal: the lower the value ofT-complexity, the more “ordered” it is. Dividing the time se-ries into sub-series, we study how the T-entropy evolves aswe go from one sub-series to another. The aim is to discovera clear difference of dynamical characteristics as the catas-trophic event is approaching.



Application to the Athens EQ. We first divide the EMsignal in six consecutive epochs depicted in (Fig. 10a). Foreach of these six epochs, we calculate the T-entropy as-sociated with successive segments of 1024 samples each(Fig. 10b). Thereafter we study the distributions of T-entropyvalues in these windows (Fig. 10c).

We underline the similarity of the distributions of the T-entropy values in the first, fourth and sixth time intervals.This almost common distribution characterizes the order ofcomplexity in the background noise of the EM time series.The associated high T-entropy values indicate a strong com-plexity. We turn our attention to the second and third timeintervals, namely during the emergence of the precursoryemission. We observe a significant decrease of the T-entropyvalues. This evidence reveals a strong reduction of complex-ity in the underlying fracto-electromagnetic mechanism dur-ing the launching of the two strong EM bursts. This findingmight be indicated by the appearance of a new phase at thetail of the EQ preparation process, which is characterized bya higher order of organization. Sufficient experimental evi-dence seems to support the association of the aforementionedtwo EM bursts with the nucleation phase of the impendingEQ (Eftaxias et al., 2000; Kapiris et al., 2005b; Contoyianniset al., 2005).

Application to the EEG signal. We first divide the signalin six consecutive epochs depicted in Fig. 11a. For each ofthese six epochs, we calculate the T-entropy associated withsuccessive segments of 1024 samples each (Fig. 11b).

Thereafter, we study the distributions of T-entropy valuesin these windows (Fig. 11c). As the seizure approaches thereis a transition from higher to lower T-entropy values: we ob-serve a significant decrease of the T-entropy values prior toshock. Higher degrees of synchronization are reflected inlower signal complexity. After seizure initiation, but still rel-atively early in the seizure, the signal is one of relatively lowcomplexity. As the seizure evolves, the complexity of thesignal gradually increases. The observed increasing signalcomplexity during seizure evolution is consistent with pro-gressive de-synchronization of the seizure activity.

In summary, the analysis in terms of T-complexity revealsa significant reduction of complexity prior to an importantEQ/epileptic seizure. We pay attention to the fact that theepoch of high complexity corresponds to that of the anti-persistent behavior, while the epoch of low complexity corre-sponds to that of the persistent behavior. The correspondingpictures in each regime are not in contradiction.

13.2 Non-linear analysis in terms of Correlation Dimen-sion and Approximate Entropy

Other computable complexity measures, namely, correlationdimensionD2 and approximate entropy ApEn, have been re-cently used in pre-seismic EM time series (Nikolopoulos etal., 2004) and epileptic seizure time series data (Radhakrish-nan and Gangadhar, 1998; Lehnertz and Elger, 1998)

The correlation dimensionD2 was first established byGrassberg and Procaccia (1983) and is based on the Tak-ens theorem (Takens, 1981). Generally, the correlation di-mension represents the independent degrees of freedom thatare required for the proper description of a system or for theconstruction of its model. A time series that results from acomplex non-linear dynamic system yields a larger value forthe correlation dimension; a time series, which results from aregular and linear dynamic system, shows lower correlationdimension values.

Entropies indicate the variety of the patterns which areincluded in an ensemble or a certain time interval. In par-ticular, ApEn is a new statistical parameter derived fromthe Kolmogorov-Sinai entropy formula which quantifies theamount of regularity in data. Intuitively, one reason that thepresence of repetitive patterns of fluctuation in a time seriesrender it more predictable than a time series in which suchpatterns are absent. ApEn reflects the likelihood that “simi-lar” patterns of observations will not followed by additional“similar” observations. A time series containing many repet-itive patterns has a relatively small ApEn; a less predictable,namely more complex, process has a higher ApEn. In a per-fectly regular data series the knowledge of the previous val-ues enables the prediction of the subsequent value. The ap-proximate entropy value would be zero. For example, in aperfectly regular data series 0, 0, 1, 0, 0, 1, ..., knowing thatthe two previous values were 0 and 0 enables the predictionthat the subsequent value will be 1. With increasing irreg-ularity, even knowing the previous values, the prediction ofthe subsequent value will get worse. The approximate valuewill increase.

Focus on fracture. The main results obtained from ourstudy of the EM time series associated with the Athens EQin terms of correlation dimension and approximate entropy(Nikolopoulos et al., 2004) are summarized in Fig. 12. Weunderline the almost common distributions of theD2-valuesand ApEn-values in the first, second and fourth time inter-vals (Fig. 12). This almost common distribution refers tothe background noise of the EM time series. The associ-ated highD2-values and ApEn-values indicate a strong com-plexity. We observe a significant abrupt decrease of theD2-values and ApEn-values as we move to the third time win-dow, namely, during the emergence of two impulsive signals.This reveals a strong loss of complexity in the underlyingfracto-EM activity.

Focus on epileptic seizures. Lehnertz and Elger (1998)have evaluated the capability of non-linear series analysis interms of effective correlation dimension to extract featuresfrom brain electrical activity predictive epileptic seizures.Time-resolved analysis of human EEG time series indicatesa significant loss of complexity prior to and during seizures(the related effective correlation dimension decreases up to2), while, far away from any seizure, the associated effectivecorrelation dimension varies around 9.



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Fig. 11.First row: We distinguish six epochs (W1-W6) in the evolution of the EEG signal associated with a rat epileptic seizure. Second row:Each point denotes the normalized T-entropy calculated by considering consecutive time windows of 1024 points. We observe an importantdecrease of the T-entropy values during epochs 2 and 3 at the tail of the pre-epileptic epoch. Third-fourth row: Histograms of the probabilitydistribution of the normalized T-entropy values, calculated for the six epochs. We also observe a reduction of the complexity as the epilepticseizure is approaching, in particular during the epochs W2 and W3.

Radhakrishnan and Gangadhar (1998) have used the ApEnto quantify the regularity embedded in the seizure time seriesdata. They found that the seizure becomes more regular andcoherent in the middle part.

We conclude that the analysis in terms of the correlationdimension and approximate entropy also corroborates the ap-pearance of a new phase in the tail of the EQ/epileptic seizure

preparation process, which is characterized by a higher orderof organization.

The convergence between non-linear and linear analy-sis provides a more reliable detection concerning the emer-gence of the last phase of the EQ preparation process. Moreprecisely, we claim that our results suggest an importantprinciple: significant complexity decrease, and accession of



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persistency in time series can be confirmed at the tail of thepre-catastrophic epoch, which can be used as diagnostic toolsfor the impending shock.

14 Conclusions

Recently, mathematical techniques have been developed thatare useful for quantifying and describing the complex andchaotic signals. Whereas much research in this area has beenconducted in relation to the physiologic systems (cardiac sys-tem, brain activity), we show that the same methods alsopromise in relation to the pre-fracture EM activity.

The existence of precursory features in pre-seismic EMemissions is still in discussion. We bear in mind that, in prin-ciple, it is difficult to prove associations between events sep-arated in time, such as EQs and their EM precursors. Com-plexity suggests a striking similarity in behavior close to irre-

versible phase transitions among systems that are otherwisequite different in nature. Interestingly, theoretical studiessuggest that the EQ dynamics at the final stage and neuralseizure dynamics should have many similar features and canbe analyzed within similar mathematical frameworks.

EEG time series provide a window through which the dy-namics of shock preparation can be investigated under well-controlled conditions, as well as, in the absence of noise.Consequently, the analysis of a pure pre-epileptic time serieswill help in establishing a reliable collection of criteria to in-dicate the approach to the biological shock. In this study wehighlight the precursory symptoms emerging in approachingepileptic seizures, and their resemblance to those precursorysymptoms occurring prior to an EQ.

We observe that both kinds of catastrophic events followcommon behavior in their pre-catastrophic stage. This evi-dence supports the hypothesis that the detected EM anomalyis originated during the micro-fracturing in the pre-focal



area. More precisely, in terms of non-linear analysis, i.e. T-complexity, correlation dimension and approximate entropy,a transition from a regime of high complexity to a regimeof low complexity indicates the approach to bot

unified approach to catastrophic events: from the normal …...namics fracture has been shown to be...

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