Abstract
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.
Highlights
The origin of cognition is a subject of spirited debate among a number of research camps having significantly different views of the underlying mechanisms
To understand the importance of this bridge we review the infinite memory of fractional Brownian motion, a theory without crucial events, and the continuous time random walk [8] (CTRW), a theory with crucial events and very well described by a fractional diffusion equation
This article has been devoted to the joint action of fractional Brownian motion (fBm) and the Caputo fractional derivative, interpreted as the joint action of quantum-like coherence and crucial events
Summary
The origin of cognition is a subject of spirited debate among a number of research camps having significantly different views of the underlying mechanisms. In this article we refer to criticality along the lines of the earlier work in [7] We refer to this form of criticality as self-organized temporal criticality (SOTC), meaning that the process of transition to criticality is spontaneous and does not require fine tuning of an external control parameter. This criticality is called temporal because, adopting the view of turbulence, it generates short time regions of chaotic behavior punctuating large time regions of regular behavior, called laminar regions. We interpret the theoretical procedure as a combination of criticality generating crucial events along with coherence of quantum origin. We stress the phenomenological attempt done in the recent past to establish a theoretical bridge between the theories of fBm and crucial events
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