Neural complexity through a nonextensive statistical–mechanical approach of human electroencephalograms

The brain is a complex system whose understanding enables potentially deeper approaches to mental phenomena. Dynamics of wide classes of complex systems have been satisfactorily described within q-statistics, a current generalization of Boltzmann-Gibbs (BG) statistics. Here, we study human electroencephalograms of typical human adults (EEG), very specifically their inter-occurrence times across an arbitrarily chosen threshold of the signal (observed, for instance, at the midparietal location in scalp). The distributions of these inter-occurrence times differ from those usually emerging within BG statistical mechanics. They are instead well approached within the q-statistical theory, based on non-additive entropies characterized by the index q. The present method points towards a suitable tool for quantitatively accessing brain complexity, thus potentially opening useful studies of the properties of both typical and altered brain physiology.


Introduction
The brain is widely recognized as a complex system since it is composed by billions of cells (neurons) which express individual behaviors and, at same time, they build a fully interconnected network with emergent, self-organized collective behaviors [1].Thus, traditional reductionist scientific methodology from mechanistic rationality appears to fail for deeply understanding the brain and its associated mind inside a multidimensional environment [2].On one hand, a humanity's great unresolved problem is to establish a suitable mental medicine, from epistemology [3] to the biomedical perspective.The problem begins in differentiating normality from typicality, illness from neurodiversity.And, upon this basis, to establish a taxonomy about mental typology for a more realistic nosography.On the other hand, several studies have explored brain complexity through entropic measures within the electroencephalogram (EEG), and found relationships between brain complexity and different mind conditions [4].However, this issue yet is incipient.
The pioneering works of Boltzmann [5] and Gibbs [6] (BG) established a magnificent theory which is structurally associated with the BG entropic functional and consistent expressions for continuous or quantum variables; k is a conventional positive constant adopted once forever (in physics, k is chosen to be the Boltzmann constant k B ; in information theory and computational sciences, k = 1 is frequently adopted).
In the simple case of equal probabilities, this functional becomes S BG = k ln W . Eq. ( 1) is generically additive [7].Indeed, if A and B are two probabilistically independent systems (i.e., p A+B ij = p A i p B j ), we straightforwardly verify that S BG (A + B) = S BG (A) + S BG (B).The celebrated entropic functional (1) is consistent with thermodynamics for all systems whose N elements are either independent or weakly interacting in the sense that only basically local (in space/time) correlations are involved.For example, if we have equal probabilities and the system is such that the number of accessible microscopic configurations is given by W (N ) ∝ µ N (µ > 1; N → ∞), then S BG (N ) is extensive as required by thermodynamics.Indeed S BG (N ) = k ln W (N ) ∼ k(ln µ)N .
However, complex systems are typically composed of many elements which essentially are non-locally correlated, building an intricate network of interdependencies from where collective states can emerge [8].BG statistical mechanics appears to be generically inadequate for such systems because this theory assumes (quasi) independent components with short-range (stochastic or deterministic) interactions.Indeed, if the correlations are nonlocal in space/time, S BG may become thermodynamically inadmissible.Such is the case of equal probabilities with say W (N ) ∝ N ν (ν > 0; N → ∞): it immediately follows S BG (N ) ∝ ln N , which violates thermodynamical extensivity [8].To satisfactorily approach cases such as this one, it was proposed in 1988 [9] to build a more general statistical mechanics based on the nonadditive entropic functional with the q-logarithmic function ; (e z 1 = e z ; [z] + = z if z > 0 and vanishes otherwise); for q < 0, it is necessary to exclude from the sum the terms with vanishing p i .We easily verify that equal probabilities yield S q = k ln q W . Also, we generically have the following functional nonadditivity Consequently, in the (1 − q)/k → 0 limit, we recover the S BG additivity.For the anomalous class of systems mentioned above, namely if W (N ) ∝ N ν , we obtain, ∀ν, the extensive entropy S 1−1/ν (N ) = k ln 1−1/ν W (N ) ∝ N , as required by the Legendre structure of thermodynamics [10,11].Finally, the optimization of S q under simple constraints yields q-exponential distributions for the (quasi)stationary states, instead of the usual BG exponentials.

Methodology and results
We analized the EEG signal of ten typical adult humans from a match-to-sample task experiment with neutral affective interference for access working memory and attention, such in Yang and Zhen's study [12].This work was approved by our ethical board for human research, under CAAE 50137721.4.0000.5269.Each EEG signal has 5-10 minutes length recorded with open eyes at 1000Hz sampling rate, through 20 channels disposed at 10-20 montage with eyes open.The high, low and band-pass filters were respectively 0.5, 150Hz and 60Hz.We did not apply any other filter to minimize signal manipulation.
We accessed signal recorded at the midparietal (P z ) site (see FIG. 3), where classical cognitive event-related potentials, as P300 [13], manifest during attention tasks.A threshold was set at -1.0 standard deviation from P z signal average (FIG.1, from subject B006).Taking negative voltages we are minimizing the effect of blink artifacts, which are positive waves, amplier in frontal places.
The constant a is determined by imposing normalization, i.e., ∞ 0 dx y(x) = 1.Consequently, for q > 1 and 1 q−1 − 1+cq ηq > 0. In the q → 1 limit, we obtain It is observed that EEGs at P z position from all subjects express very similar distributions of distances.The EEG regularity was modelled by the q-statistics function instead BG one (FIG.3).

Discussion
This preliminary study exhibits that q-statistics can somehow reveal the brain complexity, at least for typical people on scalp regions where cognitive-related potential P300 emerges (a site almost free of blink artifacts).Consistently, we have verified here that the brain phenomenology is not properly described within BG statistics (i.e., q=1).This is by no means surprising since BG statistics generically disregards inter-component long-range correlations and their collective behavior, which is well known in neural systems [1].In contrast, plethoric evidence exists that q-statistics satisfactorily models vast classes of complex systems [11] even involving c q = 0, from basic chemical reactions through quantum tunneling [17] to financial market behavior [18], COVID-19 spreading [19], commercial air traffic networks [20] (see [22,23,24,25,26] for illustrations with c q = 0).We are led to believe that we are dealing with universality classes of complexity, thus revealing, in what concerns information processing and energy dynamics, far more integrative networks than one might a priori expect from neural structures [27].By generalizing the BG theory, q-statistics shows that it could be a most suitable and promising path to explore brain complexity.Our expectancy is that the q parameter can be sensitive to different brain/mental states, to brain/mind development, and to neural diversity, perhaps clarifying the boundaries between the normal and the ill mind.Consistently, a key outcome of emergence of self-organized new states in complex systems is an adaptive behavior facing environmental constraints [1].Indeed, the concept of disease has also been related to reduced adaptive capabilities, and to the alteration of complexity [4,28].Along the lines of the seminal philosophical work of G. Canguilhem [3], normality should be related to the ability to create new rules (i.e., adaptation) instead of living by the same old norms.We intend to further explore, in the future, the neural diversity through the most remarkable paradigm of complexity.Superimposed signal recorded on the Pz location of ten subjects performing a work memory task.Amplitude threshold = 1.0 standard deviation.Fitting within Boltzmann-Gibbs statistical mechanics for non-complex systems (i.e., q = 1, dashed red curve).Fitting within nonextensive statistical mechanics for complex systems (i.e., q = 1, black continuous curve).See Methodology for details. http://arxiv.org/ps/2303.03128v1

Figure 1 :
Figure 1: Segment of ongoing EEG from one subject (B006), recorded on the mid-parietal (Pz) location of the head.Red dots: time values when ddp (signal amplitude) crosses downwards the bottom threshold (1.0 standard deviation; red line).EEG sampling rate was 1000 Hz.

Figure 2 :
Figure 2: Sequence of inter-event time intervals from EEG signal, as detected in FIG. 1.

Figure 3 :
Figure 3: Probability distributions of EEG inter-occurrence times (500 equal logarithmic bins) and fittings with statistical models.