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Edited by: Srdjan Kesić, University of Belgrade, Serbia

Reviewed by: Andrew A. Fingelkurts, BM-Science, Finland; Sebastian Wallot, Max-Planck-Institut für Empirische Ästhetik, Germany

This article was submitted to Fractal Physiology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Earlier research work on the dynamics of the brain, disclosing the existence of crucial events, is revisited for the purpose of making the action of crucial events, responsible for the 1/

Following the dynamics of the brain is a challenging issue that has forced researchers to go beyond applying the conventional forms of non-equilibrium statistical physics (Papo,

A parallel line of inquiry has recently been developed that focuses on the connection between the dynamics of the brain and the phenomenon of criticality (Aburn et al., _{C} where this transition occurs, a long-range correlation between the thermally disordered spins occurs. It is widely thought that an analogous condition is fulfilled by brain dynamics with the consequence of strongly correlating the functionality of different physical regions of the brain. This connection between brain dynamics and phase transition processes at criticality led the present investigators to focus on the concept of crucial events.

For context, let us briefly consider how the concept of crucial events was introduced in neurophysiology. Contoyiannis and Diakonos (Contoyiannis and Diakonos,

Returning to neurophysiology, we can make the conjecture that crucial events in the EEGs are signaled by abrupt transitions from regular to a fast irregular behavior, Rapid Transition Processes (RTPs). Allegrini et al. (

On the basis of earlier remarks, crucial events are defined in terms of their statistical properties as follows. The time intervals between consecutive crucial events are described by the waiting-time PDF ψ(τ) having an IPL structure

with the IPL index μ in the interval

From the earlier arguments, furthermore, it is clear that the crucial events are renewal and consequently the times τ_{i} should not be correlated. If a sequence of crucial events are defined by the time intervals τ_{1},τ_{2},τ_{3},… then the time-average correlation function is a Kronecker delta function where the time average is indicated by an overbar

This correlation function is properly normalized, thereby yielding _{i} and τ_{j}, Π(τ_{i}, τ_{j}), when

where _{i}) and _{j}) are the probability of occurrence of τ_{i} and τ_{j}, respectively.

Allegrini et al. (

It is not yet clear what kind of criticality generates crucial events, either that determined by externally tuning a control parameter (Ising-like), or that achieved spontaneously through the internal system dynamics, i.e., self-organized criticality (SOC), is expected to afford a sufficient theoretical picture. We discuss the open issue of the proper form of criticality to use to increase our understanding of the brain dynamics in sections 5 and 6.

Here we stress that the research lines of this paper are determined by the recent form of self-organization called self-organized temporal criticality (SOTC) (Mahmoodi et al.,

The second problem left unsettled by the results of Allegrini et al. (

The main result of the present paper is establishing an approach that simultaneously detects the statistical properties of crucial events and a connection with the wave-like nature of brain dynamics. The adoption of the RTP method is very attractive but, as shown in section 2, its adoption does not make it possible for us to measure the complexity of brain dynamics directly and in addition requires a filtering process. Herein we propose a technique of analysis of EEG time series data not requiring the detection of RTPs, and that leads to the detection of scaling directly from raw data. We show that the resulting scaling is identical to that obtained in earlier work using RTP's. More importantly, the present technique helps establish a bridge between EEG waves (di Santo et al.,

In section 2 we review the procedure adopted to detect RTP events. We devote section 3 to an intuitive introduction to the process of self-organization combining periodicity and crucial events and in section 4 we analyze the spectrum of one EEG to point out the interesting qualitative agreement with the predictions of section 3 . In section 5 we illustrate a technique of detection of crucial events that facilitates the analysis of EEG time series. Finally, in section 6 we draw some conclusions and present plans for future work.

As mentioned earlier, the efficacy of the RTP method in the study of brain dynamics has been established by the brothers Fingelkurts and co-workers (Fingelkurts,

These EEG time series data have been filtered between 0.15-28 Hz and the sampling rate (_{s}) is 2048 Hz. We select one healthy subject, from the dataset, and the top panel of Figure

Illustration of the RTP procedure of Kaplan et al. (

The third panel from the top of Figure

Note that this procedure for finding the crucial events is not sufficiently accurate to be restricted to detecting only renewal events. It is known that the events revealed by this analysis are a mixture of crucial events and ordinary Poisson events (Allegrini et al.,

Note that the latter scaling dominates asymptotically in the time due to Equation (5) resulting in δ > 0.5 when the condition 2 < μ < 3 applies (Grigolini et al.,

To be explicit, since in this paper as far as the scaling detection is concerned, we adopt the same procedure as that proposed by Grigolini et al. (

Using a moving window of size

The PDF constructed from the diffusion process has the scaling form

Then inserting Equation (8) into Equation (7), after some algebra yields

where

To make this treatment compatible with the subsequently discussed arguments about intermediate asymptotics, we rewrite Equation (9) in the following way

where

It is important to stress that a significant advance of the theoretical justification of Equation (11) based on an extension of the theory of SOC, incorporating complexity in the time domain, is called SOTC (Mahmoodi et al.,

Detection of the scaling δ applying DEA to the diffusion process generated by a random walker making a jump ahead when a crucial event occurs.

To explain using an intuitive interpretation the intermediate asymptotics, we notice that the short-time region corresponds to the time scale where the self-organization is not yet perceived by the interacting units. According to SOTC (Mahmoodi et al.,

The events generated by SOTC are renewal, which explains adopting Equation (5) for the connection between δ and μ, which is based in fact on the renewal assumption (Grigolini et al.,

It is convenient to stress the fact that the choice of the RTP method, illustrated in Figure

Establishing the statistics of the crucial events manifest in EEG time series by means of the detection of RTP, unfortunately, does not help us to build a bridge between the wavelike nature of EEG time series and crucial events. To establish the theoretical connection between the periodicity of EEG time series and crucial events, we adopt the SOTC model of units with an individual periodicity, for instance the SOTC model (Mahmoodi et al.,

The research work done in the recent past on the brain with the help of the RTP method led Allegrini et al. (

The derivation of this spectrum was done by other researchers (Margolin and Barkai,

To establish a bridge between crucial events and periodicity, as done by Ascolani et al. (_{RR} clicks with the time interval Δt between one click and the next. Thus,

The crucial events, some of which have been detected by Allegrini et al. (

corresponding to the survival probability

The parameter

This procedure of infusing the original perfect coherence of the clock with complex randomness establishes a bridge between waves and crucial events. This has the effect of turning the frequency Ω into an effective frequency Ω_{eff}, thereby modeling a process of self-organization of interacting oscillators, each of which is characterized by its own frequency, into a collective homeodynamic process.

According to the theoretical treatment of Lambert et al. (to be submitted), the effective frequency is, valid for μ > 2,

This theoretical prediction suggests, in agreement with Figure _{eff} = Ω (Lambert et al., to be submitted).

Power Spectra obtained averaging over 300 trajectories with numerical parameters

This illustration of subordination makes it evident that crucial events are not only at the border between consecutive pieces of the sausage but the oscillatory-like behavior within a sausage hosts crucial events. It is surprising that the same conclusion has been achieved by the Fingelkurts brothers (Fingelkurts and Fingelkurts,

Figure _{eff} that shifts to the right upon decreasing μ and disappears for μ < 2. At the left of the Ω_{eff} peak the slope of the spectrum β is determined to be

We see that the spectrum becomes flat at μ = 3 and remains flat for higher values of μ, as clearly shown in Figure

Note that due to the average of many realizations, which is not possible with real EEG time series, the region of low frequency is regular and is not affected by the fluctuations that would appear when evaluating the spectrum with only one time series. For this reason, the adoption of surrogate time series makes it possible for us to prove that, as expected, subordination is compatible with the emergence of 1/

Let us now discuss the spectra depicted in Figure

Power Spectra derived with permission from Aburn et al. (

When μ < 3 there exists a close connection between periodicity and complexity, as indicated in Figure

In this section we discuss the spectrum generated by real EEG time series fluctuations as shown in Figure

Power Spectrum obtained from raw EEG time series data.

To stress the multi-frequency nature of the real spectrum, again using the HHT method (Huang and Wu,

Power Spectra of the same subject with different HHT frequency components.

In Figure

Power Spectra generated using the data recorded in Figure

The stripe method was originally adopted to detect the scaling of crucial events hosted by heartbeats (Allegrini et al.,

In section 3 we used an intuitive illustration of the process of self-organization, based on subordination that affords theoretical support for the adoption of the method of stripes. The central idea is that the RTP method detects only a small fraction of crucial events, whereas real EEG time series and subordination theory with them, host a much larger number of crucial events, even if they remain invisible.

Figure

Illustration of the method of the stripes. The size of stripes is Δ

The change from one firing rate to another is an event. Of course this event is not necessarily a crucial event. As a consequence, the time interval between consecutive events cannot be used to define the important parameter μ. This lack of precision in determining the occurrence of crucial events applies also to the RTP method. Let us call _{T} the total number of events detected, _{c} the total number of (unknown) crucial events and _{nc} the total number of non-crucial, possibly Poisson events. The intermediate asymptotics, revealing the complex scaling δ of Equation (5), begins earlier upon increase of the ratio

In both cases, the adoption of the DEA method is essential. In fact, after recording events with the method of stripes, as done with the method of RTP, we adopt the prescription of Grigolini et al. (

The result illustrated in Figure _{c} of Equation (18) significantly larger than the RTP method.

DEA applied to the diffusion process generated by the stripe-crossing events.

The adoption of the RTP method makes it easy to establish the non-local nature of the brain criticality (Allegrini et al.,

The sausage-like structure of the model studied by Bologna et al. (

The SOTC is a new form of self-organization studied by Mahmoodi et al. (

In the case of SOTC the time necessary to evolve toward the condition of temporal criticality is finite (Mahmoodi et al.,

The processes of phase transitions are characterized by IPL PDFs with indices expressing the universality of criticality. The construction of renormalization group theory made it possible to determine without a detailed knowledge of the micro-interactions of the system, the scaling nature of phase transitions. In the case of the brain the micro-units, whose dynamics depart from the erratic behavior of independent units to collective behavior at criticality, are neurons. However, in spite of the frequent use of the term SOC these models rest on tuning a control parameter to a critical value that establishes global properties making the micro-dynamics unimportant in favor the macro-dynamics of criticality. If a neuron fires all the neurons linked to it makes a step ahead toward the firing level. Criticality is a condition generated by a suitable value of the control parameter that establishes a complex dynamics characterized by temporal complexity, namely, the crucial events defined in section 1. An interesting example of “Self-organized criticality” is given by Levina et al. (

The results of this paper suggest promising directions to establish homeodynamics as a form of genuinely spontaneous organization. SOTC (Mahmoodi et al.,

These remarks lead us to conclude that the subordination theory used in this paper is an appropriate way to mimic the self-organization of units characterized by periodicity, as manifest in their spectra.

The comments we make in section 3 on the surprising agreement between the physical meaning of SOTC and the architectonic structures of Fingelkurts and Fingelkurts (

We base our analysis on data derived from Matran-Fernandez and Polo (

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Welch and ARO for financial support through Grant No. B-1577 and W911NF- 15-1-0245, respectively.