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Complexity Matching and Requisite Variety bioRxiv preprint doi: https://doi.org/10.1101/414755; this version posted September 11, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available for use under a CC0 license. Complexity Matching and Requisite Variety Korosh Mahmoodi 1;∗, Bruce J. West 2, Paolo Grigolini 1 1 Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA and 2 Information Science Directorate, Army Research Office, Research Triangle Park, NC 27708 Abstract Complexity matching emphasizes the condition necessary to efficiently transport information from one complex system to another and the mechanism can be traced back to the 1957 Introduction to Cybernetics by Ross Ashby. Unlike this earlier work we argue that complexity can be expressed in terms of crucial events, which are generated by the processes of spontaneous self-organization. Complex processes, ranging from biological to sociological, must satisfy the homeodynamic condition and host crucial events that have recently been shown to drive the information transport between complex systems. We adopt a phenomenological approach, based on the subordination to periodicity that makes it possible to combine homeodynamics and self-organization induced crucial events. The complexity of crucial events is defined by the waiting- time probability density function (PDF) of the intervals between consecutive crucial events, which have an inverse power law (IPL) PDF (τ) / 1=(τ)µ with 1 < µ < 3. We establish the coupling between two temporally complex systems using a phenomenological approach inspired by models of swarm cognition and prove that complexity matching, namely sharing the same IPL index µ, facilitates the transport of information, generating perfect synchronization, reminiscent of, but distinct from chaos synchronization. This advanced form of complexity matching is expected to contribute a significant progress in understanding and improving the bio-feedback therapies. Author Summary This paper is devoted to the control of complex dynamical systems, inspired to real processes of biological and sociological interest. The concept of complexity we adopt focuses on the assumption that the processes of self-organization generate intermittent fluctuations and that the time distance between two consecutive fluctuations is described by a distribution density with an inverse power law structure making the second moment of these time distances diverge. These fluctuations are called crucial events and are responsible for the ergodicity breaking that is widely revealed by the experimental observation of biological dynamics. We argue that the information transport from one to another complex system is ruled by these crucial events and we propose an efficient theoretical prescription leading to qualitative agreement with experimental results, shedding light into the processes of social learning. The theory of this paper is expected to have important medical applications, such as an improvement of the biofeedback techniques, the heart-brain communication and a significant progress on cognition and the contribution of emotions to cognition. I. INTRODUCTION It is important to stress that there exists further re- search directed toward the foundation of social learning It is slightly over a half century since Ross Ashby, in his [9{12] that is even more closely connected to the ambi- masterful book [1], alerted scientists to be aware of the tious challenge made by Ashby. In fact, this research difficulty of regulating biological systems, and that \the aims at evaluating the transfer of information from the main cause of difficulty is the variety in the disturbances brain of one player to that of another, by way of the inter- that must be regulated against". This insightful observa- action the two players established through their avatars tion need not lead to the conclusion that complex systems [10]. The results are exciting in that the trajectories of cannot be regulated. It is possible to regulate them if the the two players turn out to be significantly synchronized. regulators share the same high intelligence (complexity) But even more important than synchronization is the fact as the systems being regulated. Herein we refer to the that the trajectories of the two avatars have a universal Ashby's requisite variety with the modern term complex- structure based on the shared EEGs of the human brain. ity matching [2]. The term complexity matching has been This paper provides a theoretical understanding of the widely used in the recent past [3{8] including the syn- universal structure representing the brain of the two in- chronization between the finger tapping and a complex teracting individuals. In addition, the theory can be metronome interpreted to be a system as complex as the adapted to the communication, or information transfer, human brain. These synchronization studies are today's between the heart and the brain [13] of a single individ- realizations of the regulation of the brain, in conformity ual. with the observations of Ashby. The transfer of information between interacting sys- bioRxiv preprint doi: https://doi.org/10.1101/414755; this version posted September 11, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available for use under a CC0 license. 2 tems has been addressed using different theoretical tools, ceive the mean field of the driving system. At criticality examples of which include: chaos syncronization [14], the choice made by these units is interpreted as swarm self-organization [15], and resonance [16]. On the other intelligence [31], and, in the case of the Decision Mak- hand, in a system as complex as the brain [17] there is ing Model (DMM) adopted in [30] is associated to the experimental evidence for the existence of crucial events. index µ = 1:5. In [30] this synchronization is observed These crucial events, for our purposes here, can be inter- when both systems are in the supercritical condition and preted as organization rearrangements, or renewal fail- it is destroyed if one system is in the subcritical regime ures. The interval between consecutive crucial events and the other in the supercritical regime, or viceversa. is described by a waiting-time IPL PDF (t) / 1=tµ, This suggests that the maximal synchronization is real- with 1 < µ < 3. The crucial events generate ergodicity ized when both systems are at criticality, namely, they breaking and are widely studied to reveal fundamental share the same power law index µ. biological statistical properties [18]. This ideal condition of complexity matching is stud- Another important property of biological processes ied in this paper supplemented by homeodynamics, not is homeodynamics [19], which seems to be in conflict considered in the earlier work. The theory that we use with homeostasis as understood and advocated by Ashby. herein should not be confused with the unrelated phe- Lloyd et al [19] invoke the existence of bifurcation points nomenon of chaos synchronization. In fact, the intent of to explain the transition from homeostasis to homeody- the present approach is to establish the proper theoretical namics. This transition, moving away from Ashby's em- framework to explain, for instance, brain-heart commu- phasis on the fundamental role of homeostasis, has been nication. Here the heart is considered to be a complex, studied by Ikegami and Suzuki [20] and by Oka et al. but not chaotic, system, in accordance with a growing [21] who coined the term dynamic homeostasis. They consensus that the heart dynamics are not chaotic [32]. used Ashby's cybernetics to deepen the concept of self and to establish if the behavior of the Internet is similar to that of the human brain. II. METHOD Experimental results exist for the correlation between the dynamics of two distinct physiological systems [22], To deal with the ambitious challenge of Ashby [1] we but they are not explained using any of the earlier men- adopt the perspective of subordination theory [33]. This tioned theoretical approaches [14{18]. Herein we relate theoretical perspective is closely connected to the Con- this correlation to the occurrence of crucial events. These tinuous Time Random Walk (CTRW) [34, 35], which is crucial events are responsible for the generation of 1=f known to generate anomalous diffusion. We use this noise, S(f) / 1=f 3−µ [23] and the results of the psycho- viewpoint to establish an approach to explaining the logical experiment of Correll [24]. The experimental data experimental results showing the remarkable oscillatory imply that activating cognition has the effect of making synchronization between different areas of the brain [22]. the IPL index µ < 3 cross the barrier between the L´evy Consider a clock, whose discrete hand motion is punc- and Gauss basins of attraction, namely making µ > 3 tuated by ticks and the time interval between consecutive [25]. This is in line with Heidegger's phenomenology [26]. ticks is, ∆t = 1, by assumption. At any tick the angle The crossing of a basin's boundary is a manifestation θ of the clock hand increases by 2π=T , where T is the of the devastating effect of violating the linear response number of ticks necessary to make a complete rotation condition, according to which a perturbation should be of 2π. We implement subordination theory by selecting sufficiently weak as to not affect a system's dynamic com- for the time interval between consecutive ticks a value plexity [27]. The experimental observation obliged us τ= < τ > where τ is picked from an IPL waiting -time to go beyond the linear response theory adopted in ear- PDF (τ) with a complexity index µ > 2.
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