Brain Dynamics Explained by Means of Spectral-Structural Neuronal Networks

Abstract

Starting from the morphological-functional assumption of the fractal brain, a mathematical model is given by activating brain's non-differentiable dynamics through the determinism-nondeterminism inference of the responsible mechanisms. The postulation of a scale covariance principle in Schrödinger's type representation of the brain geodesics implies the spectral functionality of the brain dynamics through mechanisms of tunelling, percolation etc., while in the hydrodynamical type representation, it implies their structural functionality through mechanisms of wave schock, solitons type etc. For external constraints proportional with the states density, the fluctuations of the brain stationary dynamics activate both the spectral neuronal networks and the structural ones through a mapping principle of two distinct classes of cnoidal oscillation modes. The spectral-structural compatibility of the neuronal networks generates the communication codes of algebraic type, while the same compatibility on the solitonic component induces a strange topology (the direct product of the spectral topology and the structural one) that is responsible of the quadruple law (for instance, the nucleotide base from the human DNA structure). Implications in the elucidation of some neuropsychological mechanisms (memory's location and functioning, dementia etc.) are also presented.

Toward Interactions through Information in a Multifractal Paradigm

“…In what follows, we admit that the motions of the entities of any complex system are described by continuous and non-differentiable curves (multifractal curves). Such a “non–differentiable” procedure to approach these motions has important consequences [ 1 , 2 , 17 , 18 , 19 , 20 , 21 ]: The lengths of multifractal curves tend to infinity when the scale resolution tends to zero, according to the Lebesgue theorem [ 3 ]. In these conditions, the space becomes a Mandelbrot’s multifractal.…”

“…are operational. Thus, let us select for the average of the subsequent functionality: with To describe the dynamics of complex systems the operator (4) needs to function as a scale covariant derivative [ 17 , 18 , 19 ]: where …”

“…To describe the dynamics of complex systems the operator (4) needs to function as a scale covariant derivative [ 17 , 18 , 19 ]: where …”

“…For irrotational dynamics (5) become: where In (19) and (20), is the states function (on the significances of , see [ 1 , 2 ]), while is the scalar potential of (5). By substituting (19) in (16) through specific mathematical procedures [ 17 , 18 , 19 ], the geodesics equation (16) can be written as a multifractal Schrödinger-type equation: …”

“…This is due to the fact that motion curves belonging to the entities of any complex structure are multifractal curves (i.e., they are continuous and non–differentiable) simultaneously characterized by several fractal dimensions and explicated by the singularity spectrum (with being the singularity index [ 3 ]). To this end, we present in Section 2 , the multifractalization procedure as being part of a “general methodology” of SRT, in the form of the multifractal theory of motion [ 17 , 18 , 19 , 20 , 21 ]. For such a conjecture, in Section 3 , we analyze the dynamics of complex systems in the form of Schrödinger- and hydrodynamic-type “regimens” at various scale resolutions.…”