Inhomogeneous coherent states in small-world networks: application to the functional brain networks

TL;DR

This paper models small-world networks with power-law degree distributions, deriving equations for their dynamics, and applies these findings to understand brain processes and the emergence of mind.

Contribution

It introduces an analytical framework using fractional differential equations to describe inhomogeneous coherent states in small-world networks, with applications to brain dynamics.

Findings

Derived an equation of motion for the order parameter in small-world networks.

Obtained analytical solutions for various stable phases of the system.

Applied the model to describe processes in the brain and emergence of mind.

Abstract

We study the dynamics of the processes in the small-world networks with a power-law degree distribution where every node is considered to be in one of the two available statuses. We present an algorithm for generation of such network and determine analytically a temporal dependence of the network nodes degrees and using the maximum entropy principle we define a degree distribution of the network. We discuss the results of the Ising discrete model for small-world networks and in the framework of the continuous approach using the principle of least action, we derive an equation of motion for the order parameter in these networks in the form of a fractional differential equation. The obtained equation enables the description of the problem of a spontaneous symmetry breaking in the system and determination of the spatio-temporal dependencies of the order parameter in varies stable phases of the system. In the cases of one and two component order parameters with taken into account major and secondary order parameters we obtain analytical solutions of the equation of motion for the order parameters and determine solutions for various regimes of the system functioning. We apply the obtained results to the description of the processes in the brain and discuss the problems of emergence of mind.