Final state sensitivity and fractal basin boundaries from coupled Chialvo neurons

TL;DR

This study analyzes how final-state uncertainty and fractal basin boundaries arise in a minimal neural motif: two identically parameterized Chialvo neurons coupled electro-physically with asymmetry. Using basin stability, Sprott–Xiong basin classification, uncertainty exponent, and basin entropy, the authors show a robust two-attractor structure where the chaotic basin dominates the four-dimensional state space while a smaller nonchaotic basin extends via chance synchronization, and the boundary between basins is fractal with dimension about $3.8$ and an uncertainty exponent near $0.182$. The results reveal extreme sensitivity to initial conditions and connect fractal geometry to synchronization phenomena, with potential implications for neural dynamics and neurological disease. The methodologies demonstrated can be applied to other discrete-time neuronal models to quantify final-state uncertainty and basin geometry. Overall, the work provides quantitative tools to understand multistability and fractal basins in neural systems and their possible biological consequences.

Abstract

We investigate and quantify the basin geometry and extreme final state uncertainty of two identical electrically asymmetrically coupled Chialvo neurons. The system's diverse behaviors are presented, along with the mathematical reasoning behind its chaotic and nonchaotic dynamics as determined by the structure of the coupled equations. The system is found to be multistable with two qualitatively different attractors. Although each neuron is individually nonchaotic, the chaotic basin takes up the vast majority of the coupled system's state space, but the nonchaotic basin stretches to infinity due to chance synchronization. The boundary between the basins is found to be fractal, leading to extreme final state sensitivity. This uncertainty and its potential effect on the synchronization of biological neurons may have significant implications for understanding human behavior and neurological disease.

Figures (9)

• Figure 1: Different behaviors of the individual Chialvo neuron
• Figure 2: Fixed points and cobweb plot corresponding to $y=0.5$ for a shift from $I=0.03$ (shown in red) to $I=0.4$ (shown in blue), demonstrating the role of $\mathfrak{C}^{\text{el}}_{1,2}$ in creating a spike in $x$ even when $y<1$.
• Figure 3: Attractors of the coupled Chialvo system
• Figure 4: Behavior of coupling parameters $\mathfrak{C}^{\text{el}}_{1,n}$ and $\mathfrak{C}^{\text{el}}_{2,n}$ with $a=1.0$, $b=2.2$, $c=0.26$, $I=0.04$
• Figure 5: Projections onto the $xy$ and $\alpha\beta$ planes of orbits of the nonchaotic and chaotic attractors of the coupled Chialvo map with $a=1.0$, $b=2.2$, $c=0.26$, $I=0.04$