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.

The theory of medicine and its compliment systems biology are intended to explain the workings of the large number of mutually interdependent complex physiologic networks in the human body, and to apply that understanding to maintaining the functions for which nature designed them. Therefore, when what had originally been made as a simplifying assumption or a working hypothesis becomes foundational to understanding the operation of physiologic networks, it is in the best interests of science to replace or at least update that assumption. The replacement process requires, among other things, an evaluation of how the new hypothesis affects modern day understanding of medical science. This paper identifies linear dynamics and Normal statistics as being such arcane assumptions and explores some implications of their retirement. Specifically, we explore replacing Normal with fractal statistics, and examine how the latter are related to nonlinear dynamics and chaos theory. The observed ubiquity of inverse power laws in physiology entails the need for a new calculus, one that describes the dynamics of fractional phenomena and captures the fractal properties of the statistics of physiological time series. We identify these properties as a necessary consequence of the complexity resulting from the network dynamics, and refer to them collectively as The Network Effect.

The scaling of respiratory metabolism with body size in animals is considered by many to be a fundamental law of nature. An apparent corollary of this law is the scaling of physiologic time with body size, implying that physiologic time is separate and distinct from clock time. However, these are only two of the many allometry relations that emerge from empirical studies in the physical, social and life sciences. Herein, we present a theory of allometry that provides a foundation for the allometry relation between a network function and the size that is entailed by the hypothesis that the fluctuations in the two measures are described by a scaling of the joint probability density. The dynamics of such networks are described by the fractional calculus, whose scaling solutions entail the empirically observed allometry relations.

Herein we study the effects on Anderson localization of correlations in the energy distribution of the sites of a tight-binding Hamiltonian. The lattice correlations are introduced by means of classical maps generating anomalous diffusion, that have recently been found to account for the correlated disorder of ‘‘biological’’ lattices. We show that the enhancement of localization length takes place on a much wider band of energies than in the case of the random-dimer model if the random walk on the site energies of the tight-binding Hamiltonian is determined by the joint action of short- and long-range correlations. © 1996 The American Physical Society.

The Army Research Laboratory program on the Network Science of Human Decision Making brings together researchers from a variety of disciplines to work on a complex research problem that defies confinement within any single discipline. Consequently, new and rewarding solutions have been obtained for a problem of importance to society and the Army, that being, the human dimension of complex networks. This program investigates the basic research foundation of a science of networks supporting the linkage between the cognitive and social domains as they relate to human decision making. The research strategy extends recent methods of non-equilibrium statistical physics to non-stationary, renewal stochastic processes characteristic of the interactions among nodes in complex networks. The theoretical analyses of complex networks, although mathematically rigorous, often elude analytic solutions and require simulation and computation to analyze the underlying dynamic process. The information transfer between two complex networks is calculated using the principle of complexity management as well as direct numerical calculation of the decision making model developed within the project.