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Abstract In this work, we use a simple multi-agent-based-model (MABM) of a social network, implementing selfish algorithm (SA) agents, to create an adaptive environment and show, using a modified diffusion entropy analysis (DEA), that the mutual-adaptive interaction between the parts of such a network manifests complexity synchronization (CS). CS has been shown to exist by processing simultaneously measured time series from among organ-networks (ONs) of the brain (neurophysiology), lungs (respiration), and heart (cardiovascular reactivity) and to be explained theoretically as a synchronization of the multifractal dimension (MFD) scaling parameters characterizing each time series. Herein, we find the same kind of CS in the emergent intelligence of groups formed in a self-organized social interaction without macroscopic control but with biased self-interest between two groups of agents playing an anti-coordination game. This computational result strongly suggests the existence of the same CS in real-world social phenomena and in human–machine interactions as that found empirically in ONs.

Abstract Herein we address the measurable consequences of the network effect (NE) on time series generated by different parts of the brain, heart, and lung organ-networks (ONs), which are directly related to their inter-network and intra-network interactions. Moreover, these same physiologic ONs have been shown to generate crucial event (CE) time series, and herein are shown, using modified diffusion entropy analysis (MDEA) to have scaling indices with quasiperiodic changes in complexity, as measured by scaling indices, over time. Such time series are generated by different parts of the brain, heart, and lung ONs, and the results do not depend on the underlying coherence properties of the associated time series but demonstrate a generalized synchronization of complexity. This high-order synchrony among the scaling indices of EEG (brain), ECG (heart), and respiratory time series is governed by the quantitative interdependence of the multifractal behavior of the various physiological ONs’ dynamics. This consequence of the NE opens the door for an entirely general characterization of the dynamics of complex networks in terms of complexity synchronization (CS) independently of the scientific, engineering, or technological context. CS is truly a transdisciplinary effect.

About a hundred years ago, the Russian biologist A. Gurwitsch, based on experiments with onion plants by measuring their growth rate, hypothesized that plants emit a weak electromagnetic field that somehow influences cell growth. This interesting observation remained fundamentally ignored by the scientific community; only in the 1950s the electromagnetic emission from some plants was measured using a photomultiplier used in single counting mode. Later, in the 1980s, several groups around the world started extensive work to understand the origin and role of this ultraweak emission, now called biophotons, coming from living organisms. Biophotons are an endogenous very small production of photons in the visible energy range in and from cells and organisms, and this emission is characteristic of living organisms. Today, there is no doubt that biophotons exist, this emission has been measured by many groups and for many different living organisms, from humans to bacteria. However, the origin of biophotons and whether organisms use them to exchange information is not yet well understood; no model proposed to date is capable of reproducing and interpreting the great variety of experimental data coming from the many different living systems measured so far. In this brief review, we present our experimental work on the biophotons coming from germinating seeds, the main experimental results, and some new methods we are using to analyze the data to open the door for interpretative models of this phenomenon clarifying its function in the regulation and communication between cells and living organisms. We also discuss ideas on how to increase the signal-to-noise ratio of the measured signal to open up new experimental possibilities that allow the measurement and the characterization of currently unmeasurable quantities.

Abstract Introduction Detrended fluctuation analysis (DFA) is a well‐established method to evaluate scaling indices of time series, which categorize the dynamics of complex systems. In the literature, DFA has been used to study the fluctuations of reaction time Y(n) time series, where n is the trial number. Methods Herein we propose treating each reaction time as a duration time that changes the representation from operational (trial number) time n to event (temporal) time t, or X(t). The DFA algorithm was then applied to the X(t) time series to evaluate scaling indices. The dataset analyzed is based on a Go–NoGo shooting task that was performed by 30 participants under low and high time‐stress conditions in each of six repeated sessions over a 3‐week period. Results This new perspective leads to quantitatively better results in (1) differentiating scaling indices between low versus high time‐stress conditions and (2) predicting task performance outcomes. Conclusion We show that by changing from operational time to event time, the DFA allows discrimination of time‐stress conditions and predicts performance outcomes.

We analyzed the number of cumulative positive cases of COVID-19 as a function of time in countries around the World. We tracked the increase in cases from the onset of the pandemic in each region for up to 150 days. We found that in 81 out of 146 regions the trajectory was described with a power-law function for up to 30 days. We also detected scale-free properties in the majority of sub-regions in Australia, Canada, China, and the United States (US). We developed an allometric model that was capable of fitting the initial phase of the pandemic and was the best predictor for the propagation of the illness for up to 100 days. We then determined that the power-law COVID-19 exponent correlated with measurements of human mobility. The COVID-19 exponent correlated with the magnitude of air passengers per country. This correlation persisted when we analyzed the number of air passengers per US states, and even per US metropolitan areas. Furthermore, the COVID-19 exponent correlated with the number of vehicle miles traveled in the US. Together, air and vehicular travel explained 70% of the variability of the COVID-19 exponent. Taken together, our results suggest that the scale-free propagation of the virus is present at multiple geographical scales and is correlated with human mobility. We conclude that models of disease transmission should integrate scale-free dynamics as part of the modeling strategy and not only as an emergent phenomenological property.

Abstract The reproduction number of an infectious disease, such as CoViD-19, can be described through a modified version of the susceptible-infected-recovered (SIR) model with time-dependent contact rate, where mobility data are used as proxy of average movement trends and interpersonal distances. We introduce a theoretical framework to explain and predict changes in the reproduction number of SARS-CoV-2 in terms of aggregated individual mobility and interpersonal proximity (alongside other epidemiological and environmental variables) during and after the lockdown period. We use an infection-age structured model described by a renewal equation. The model predicts the evolution of the reproduction number up to a week ahead of well-established estimates used in the literature. We show how lockdown policies, via reduction of proximity and mobility, reduce the impact of CoViD-19 and mitigate the risk of disease resurgence. We validate our theoretical framework using data from Google, Voxel51, Unacast, The CoViD-19 Mobility Data Network, and Analisi Distribuzione Aiuti.

Biophotons are an ultra-weak emission of photons in the visible energy range from living matter. In this work, we study the emission from germinating seeds using an experimental technique designed to detect light of extremely small intensity. The emission from lentil seeds and single bean was analyzed during the whole germination process in terms of the different spectral components through low pass filters and the different count distributions in the various stages of the germination process. Although the shape of the emission spectrum appears to be very similar in the two samples used in our experiment, our analysis can highlight the differences present in the two cases. In this way, it was possible to correlate the various types of emissions to the degree of development of the seed during germination.

The transdisciplinary nature of science as a whole became evident as the necessity for the complex nature of phenomena to explain social and life science, along with the physical sciences, blossomed into complexity theory and most recently into complexitysynchronization. This science motif is based on the scaling arising from the 1/f-variability in complex dynamic networks and the need for a network of networks to exchange information internally during intra-network dynamics and externally during inter-network dynamics. The measure of complexity adopted herein is the multifractal dimension of the crucial event time series generated by an organ network, and the difference in the multifractal dimensions of two organ networks quantifies the relative complexity between interacting complex networks. Information flows from dynamic networks at a higher level of complexity to those at lower levels of complexity, as summarized in the ‘complexity matching effect’, and the flow is maximally efficient when the complexities are equal. Herein, we use the scaling of empirical datasets from the brain, cardiovascular and respiratory networks to support the hypothesis that complexity synchronization occurs between scaling indices or equivalently with the matching of the time dependencies of the networks’ multifractal dimensions.

In this paper, we study the spreading of epidemic in a network of individuals who may either contract a disease through contact with infected nearest neighbors or be vaccinated under the influence of neighbors who are already vaccinated. We show that both interaction between susceptible S and infected individuals I and the imitation of vaccination, a form of sociological interaction between susceptible S and vaccinated V individuals, may lead to a phase transition. If the spreading of epidemic is in the supercritical condition, corresponding to an unlimited growth of infection, the interaction between S and V must reach the supercritical condition to generate control of the spreading of infection, and bring the system to criticality. By adopting a theoretical perspective like that of multilayer complex networks, we study the case where the epidemiological network is under the influence of a sociological debate on whether to be vaccinated. We show that at criticality this debate generates clusters of individuals in favor of vaccination and clusters of individuals opposing it.3,4 We study the influence of this debate on the spread of infection. We show that because of this debate in the epidemic network, a pattern emerges mirroring the structures of the sociological network. Finally, we introduce feedback of the epidemic network on the sociological network, and we prove that because of this feedback the sociological system undergoes a process of self-organization maintaining it at criticality. This system exhibits temporal complexity and critical slowing down.1 We hope that these results may have an important effect of giving interesting suggestions to behavioral psychologists and information scientists actively involved in the analysis of the social debate on the moral issues connected to sexual activities.

In the last years of the 20th century, the attention of physicists working in statistical physics moved from equilibrium processes characterized by stationary correlation functions and Poisson dynamics to biological processes exhibiting ergodicity breaking. The discovery of these processes raised a debate on whether basic properties such as the Onsager principle had to be abandoned or properly revisited.1 The discovery of Levy processes led many researchers to replace the conventional central limit theorem with the generalized central limit theorem, responsible for a striking departure from the ordinary Gaussian statistics. The discovery of the processes of self-organization made the study of avalanches become very popular.2 The observation of turbulent processes led to the discovery of new waiting time distribution densities, characterized by inverse power laws3 and a new stochastic central limit theorem was invented to explain the emergence of Mittag-Leffler function, which is now widely used for the foundation of fractional derivatives.4 The traditional Linear Response Theory of Kubo was replaced by a new form of linear response, compatible with the ergodic breakdown of complex systems, and this new form of linear response was used for the foundation of Complexity Matching (CM).5 I plan to prove that crucial events are responsible for ergodicity breaking and that the CM phenomenon is a manifestation of crucial events. One problem still open in this field of research is the origin of [Formula: see text] noise that is traditionally interpreted as a manifestation of the Mandelbrot Fractional Brownian Motion (FBM).6 I plan to show that the [Formula: see text] noise proposed in5 for the foundation of the CM phenomenon has a completely different nature, involving crucial events rather than the FBM infinite memory. A recent result of my research group7 proved that the progress of autonomic neuropathy makes the heartbeats of healthy individuals, dominated by crucial events, turn into FBM. Quite surprisingly, the same phenomenon of transition from the crucial event to the FBM regime was observed in the germination process of lentils8 in the absence of light. The transition from Levy to Gauss statistics is supposed to be generated by environmental fluctuations and I will illustrate the experimental and theoretical research works that will shed light into their nature.

Several studies have used the logistic equation to model the growth of cancer cell populations1 as seen in Eq. (1). This has included correlated multiplicative, [Formula: see text] and additive, [Formula: see text], noise terms. These noise terms can affect the growth rate, [Formula: see text], and death rate, [Formula: see text], of tumor cells and can be induced from factors such as radiotherapy or other cancer treatments. Depending on the intensity of the noise the terms, the fluctuations can induce a phase transition. Noise-induced transitions of nonlinear stochastic systems have applications in the fields of physics, chemistry and biology. (1)dxdt=ax−bx2+xϵ(x)−Γ(t). We study the logistic differential equation with a multiplicative noise term before and at phase transition. Computational methods used to investigate this cancer cell model include a Diffusion Entropy Analysis method and a waiting time distribution method.2,3,4 DEA will establish the scaling of a simulated series without altering the data through detrending. We hypothesize the treatment that causes a phase transition in the logistic model will induce tumor extinction and management. Understanding how to better evaluate and study cancer cell growth models will assist in assessing the efficacy of cancer treatments. Future work will include running simulations with a modified DEA method that includes the use of stripes.2 For better statistics, the code will be adopted to run ensembles of simulated data instead of a single series. Generating and analyzing these large datasets can be computationally expensive. Through multiprocessing and the use of a supercomputer, we believe these computational limitations can be overcome.

We review the literature to argue the importance of the occurrence of crucial events in the dynamics of physiological processes. Crucial events are interpreted as short time intervals of turbulence, and the time distance between two consecutive crucial events is a waiting time distribution density with an inverse power law (IPL) index μ, with μ < 3 generating non-stationary behavior. The non-stationary condition is characterized by two regimes of the IPL index: (a) perennial non-stationarity, with 1 < μ < 2 and (b) slow evolution toward the stationary regime, with 2 < μ < 3. Human heartbeats and brain dynamics belong to the latter regime, with healthy physiological processes tending to be closer to the border with the perennial non-stationary regime with μ = 2. The complexity of cognitive tasks is associated with the mental effort required to address a difficult task, which leads to an increase of μ with increasing task difficulty. On this basis we explore the conjecture that disease evolution leads the IPL index μ moving from the healthy condition μ = 2 toward the border with Gaussian statistics with μ = 3, as the disease progresses. Examining heart rate time series of patients affected by diabetes-induced autonomic neuropathy of varying severity, we find that the progression of cardiac autonomic neuropathy (CAN) indeed shifts μ from the border with perennial variability, μ = 2, to the border with Gaussian statistics, μ = 3 and provides a novel, sensitive index for assessing disease progression. We find that at the Gaussian border, the dynamical complexity of crucial events is replaced by Gaussian fluctuation with long-time memory.

A recent experiment proves the therapeutic effect of arm-in-arm walking, showing that if an aged participant walks in close synchrony with a young companion, the complexity matching effect results in the restoration of complexity in the former. A clear manifestation of complexity restoration is a perfect synchronization. The authors of this interesting experiment leave open two important problems. The first is the measure of complexity that is interpreted as a degree of multifractality. The second problem is the lack of a theoretical derivation of synchronization, which is experimentally observed with no theoretical derivation. The main goal of this paper is to establish a physiological foundation of these important results based on the recent advances on the dynamics of the brain, interpreted as a system at criticality. Criticality is a phenomenon requiring the cooperative interaction of units, the neurons of the brain, and is hypothesized as the main source of cognition. Using the criticality-induced intelligence, we define complexity as a property of crucial events, a form of temporal complexity, and we prove that the perfect synchronization is due to the interaction between the two systems, with the more complex system restoring the temporal complexity of the less complex system. The phenomenon of temporal complexity is characterized by ergodicity breaking that has made it difficult in the past to derive the perfect synchronization generated by complexity matching. For this reason, we supplement the main result of this paper with a comparison between complexity matching and complexity management.

In the fourth paper of this Special Issue, we bridge the theoretical debate on the role of memory and criticality discussed in the three earlier manuscripts, with a review of key concepts in biology and focus on cell-to-cell communication in organismal development. While all living organisms are dynamic complex networks of organization and disorder, most studies in biology have used energy and biochemical exchange to explain cell differentiation without considering the importance of information (entropy) transfer. While all complex networks are mixtures of patterns of complexity (non-crucial and crucial events), it is the crucial events that determine the efficiency of information transfer, especially during key transitions, such as in embryogenesis. With increasing multicellularity, emergent relationships from cell-to-cell communication create reaction–diffusion exchanges of different concentrations of biochemicals or morphogenetic gradients resulting in differential gene expression. We suggest that in conjunction with morphogenetic gradients, there exist gradients of information transfer creating cybernetic loops of stability and disorder, setting the stage for adaptive capability. We specifically reference results from the second paper in this Special Issue, which correlated biophotons with lentil seed germination to show that phase transitions accompany changes in complexity patterns during development. Criticality, therefore, appears to be an important factor in the transmission, transfer and coding of information for complex adaptive system development.

The bounded rationality mainstream is based on interesting experiments showing human behaviors violating classical probability (CP) laws. Quantum probability (QP) has been shown to successfully figure out such issues, supporting the hypothesis that quantum mechanics is the central fundamental pillar for brain function and cognition emergence. We discuss the decision-making model (DMM), a paradigmatic instance of criticality, which deals with bounded rationality issues in a similar way as QP, generating choices that cannot be accounted by CP. We define this approach as criticality-induced bounded rationality (CIBR). For some aspects, CIBR is even more satisfactory than QP. Our work may contribute to considering criticality as another possible fundamental pillar in order to improve the understanding of cognition and of quantum mechanics as well.

We study the emission of photons from germinating seeds using an experimental technique designed to detect light of extremely small intensity. We analyze the dark count signal without germinating seeds as well as the photon emission during the germination process. The technique of analysis adopted here, called diffusion entropy analysis (DEA) and originally designed to measure the temporal complexity of astrophysical, sociological and physiological processes, rests on Kolmogorov complexity. The updated version of DEA used in this paper is designed to determine if the signal complexity is generated either by non-ergodic crucial events with a non-stationary correlation function or by the infinite memory of a stationary but non-integrable correlation function or by a mixture of both processes. We find that dark count yields the ordinary scaling, thereby showing that no complexity of either kinds may occur without any seeds in the chamber. In the presence of seeds in the chamber anomalous scaling emerges, reminiscent of that found in neuro-physiological processes. However, this is a mixture of both processes and with the progress of germination the non-ergodic component tends to vanish and complexity becomes dominated by the stationary infinite memory. We illustrate some conjectures ranging from stress induced annihilation of crucial events to the emergence of quantum coherence.

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

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/f −noise in the brain, compatible with the wave-like nature of the brain processes. We review the relevant neurophysiological literature to make clear that crucial events are generated by criticality. We also show that although criticality generates a strong deviation from the regular wave-like behavior, under the form of Rapid Transition Processes, the brain dynamics also host crucial events in regions of nearly coherent oscillations, thereby making many crucial events virtually invisible. Furthermore, the anomalous scaling generated by the crucial events can be established with high accuracy by means of direct analysis of raw data, suggested by a theoretical perspective not requiring the crucial events to yield a visible physical effect. The latter follows from the fact that periodicity, waves and crucial events are the consequences of a spontaneous process of self-organization. We obtain three main results: (a) the important role of crucial events is confirmed and established with greater accuracy than previously; (b) we demonstrate the theoretical tools necessary to understand the joint action of crucial events and periodicity; (c) we argue that the results of this paper can be used to shed light on the nature of this important process of self-organization, thereby contributing to the understanding of cognition.

In this paper we emphasize that 1/f noise has two different origins, one compatible with Laplace determinism and one determined by unpredictable crucial events. The dynamics of heartbeats, manifest as heart rate variability (HRV) time series, are determined by the joint action of these different memory sources with meditation turning the Laplace memory into a strongly coherent process while exerting an action on the crucial events favoring the transition from the condition of ideal 1/f noise to the Gaussian basin of attraction. This theoretical development affords a method of statistical analysis that establishes a quantitative approach to the evaluation of the stress reduction realized by the practice of Chi meditation and Kundalini Yoga.

We propose a social model of spontaneous self-organization generating criticality and resilience, called Self-Organized Temporal Criticality (SOTC). The criticality-induced long-range correlation favors the societal benefit and can be interpreted as the social system becoming cognizant of the fact that altruism generates societal benefit. We show that when the spontaneous bottom-up emergence of altruism is replaced by a top-down process, mimicking the leadership of an elite, the crucial events favoring the system’s resilience are turned into collapses, corresponding to the falls of the leading elites. We also show with numerical simulation that the top-down SOTC lacks the resilience of the bottom-up SOTC. We propose this theoretical model to contribute to the mathematical foundation of theoretical sociology illustrated in 1901 by Pareto to explain the rise and fall of elites.

We introduce a form of Self-Organized Criticality (SOC) inspired by the new generation of evolutionary game theory, which ranges from physiology to sociology. The single individuals are the nodes of a composite network, equivalent to two interacting subnetworks, one leading to strategy choices made by the individuals under the influence of the choices of their nearest neighbors and the other measuring the Prisoner's Dilemma Game payoffs of these choices. The interaction between the two networks is established by making the imitation strength K increase or decrease according to whether the last two payoffs increase or decrease upon increasing or decreasing K. Although each of these imitation strengths is selected selfishly, and independently of the others as well, the social system spontaneously evolves toward the state of cooperation. Criticality is signaled by temporal complexity, namely the occurrence of non-Poisson renewal events, the time intervals between two consecutive crucial events being given by an inverse power law index μ = 1.3 rather than by avalanches with an inverse power law distribution as in the original form of SOC. This new phenomenon is herein labeled self-organized temporal criticality (SOTC). We compare this bottom-up self-organization process to the adoption of a global choice rule based on assigning to all the units the same value K, with the time evolution of common K being determined by consciousness of the social benefit, a top-down process implying the action of a leader. In this case self-organization is impeded by large intensity fluctuations and the global social benefit turns out to be much weaker. We conclude that the SOTC model fits the requests of a manifesto recently proposed by a number of European social scientists.

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and 1 / f noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes.

We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.

A two-state, master equation-based decision-making model has been shown to generate phase transitions, to be topologically complex, and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision-making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.

In the wide literature on the brain and neural network dynamics the notion of criticality is being adopted by an increasing number of researchers, with no general agreement on its theoretical definition, but with consensus that criticality makes the brain very sensitive to external stimuli. We adopt the complexity matching principle that the maximal efficiency of communication between two complex networks is realized when both of them are at criticality. We use this principle to establish the value of the neuronal interaction strength at which criticality occurs, yielding a perfect agreement with the adoption of temporal complexity as criticality indicator. The emergence of a scale-free distribution of avalanche size is proved to occur in a supercritical regime. We use an integrate-and-fire model where the randomness of each neuron is only due to the random choice of a new initial condition after firing. The new model shares with that proposed by Izikevich the property of generating excessive periodicity, and with it the annihilation of temporal complexity at supercritical values of the interaction strength. We find that the concentration of inhibitory links can be used as a control parameter and that for a sufficiently large concentration of inhibitory links criticality is recovered again. Finally, we show that the response of a neural network at criticality to a harmonic stimulus is very weak, in accordance with the complexity matching principle.