Scroll Waves and Filaments in excitable Media of higher spatial Dimension

TL;DR

This paper extends the concept of scroll waves and filaments to higher-dimensional excitable media, using numerical simulations and mathematical analysis to explore their properties and potential biological relevance.

Contribution

It introduces a framework for understanding scroll waves in four or more dimensions and derives evolution equations for superfilaments as minimal surfaces in N-dimensional space.

Findings

Vortices rotate around a superfilament in four-dimensional simulations.

Superfilaments are shown to be minimal surfaces in N-dimensional space.

Biological systems may be modeled as multidimensional excitable media.

Abstract

Excitable media are ubiquitous in nature, and in such systems the local excitation tends to self-organize in traveling waves, or in rotating spiral-shaped patterns in two or three spatial dimensions. Examples include waves during a pandemic or electrical scroll waves in the heart. Here we show that such phenomena can be extended to a space of four or more dimensions and propose that connections of excitable elements in a network setting can be regarded as additional spatial dimensions. Numerical simulations are performed in four dimensions using the FitzHugh-Nagumo model, showing that the vortices rotate around a two-dimensional surface which we define as the superfilament. Evolution equations are derived for general superfilaments of co-dimension two in an N -dimensional space and their equilibrium configurations are proven to be minimal surfaces. We suggest that biological excitable systems, such as the heart or brain which have non-local connections can be regarded, at least partially, as multidimensional excitable media and discuss further possible studies in this direction.