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Asymmetric coupling of nonchaotic Rulkov neurons: Fractal attractors, quasimultistability, and final state sensitivity

Brandon B. Le

TL;DR

The paper investigates the geometrical properties of basins of attraction and their boundaries in a minimal neuronal system: two asymmetrically electrically coupled nonchaotic Rulkov neurons. It combines fractal analysis, Lyapunov dynamics, and uncertainty-exponent methods to reveal a short-lived chaotic spiking-bursting pseudo-attractor coexisting with a nonchaotic spiking attractor (quasimultistability), and to classify basin structures in both 2D slices and full 4D space. The main findings show fractal chaotic attractors with $d\approx1.84$ and highly dimensional basin boundaries in 4D with $d_4\approx3.963$, accompanied by extreme final-state sensitivity (requiring initial-uncertainty reductions of about $10^{27}$ to reduce final-state uncertainty by a factor of 10). These results illuminate how multistability and fractal geometry can arise in discrete-time neuronal maps and may have implications for understanding sensitivity in brain dynamics, with future directions including basin-entropy analyses and experimental validation in neural circuits.

Abstract

Although neuron models have been well studied for their rich dynamics and biological properties, limited research has been done on the complex geometries that emerge from the basins of attraction and basin boundaries of multistable neuron systems. In this paper, we investigate the geometrical properties of the strange attractors, four-dimensional basins, and fractal basin boundaries of an asymmetrically electrically coupled system of two identical nonchaotic Rulkov neurons. We discover a quasimultistability in the system emerging from the existence of a chaotic spiking-bursting pseudo-attractor, and we classify and quantify the system's basins of attraction, which are found to have complex fractal geometries. Using the method of uncertainty exponents, we also find that the system exhibits extreme final state sensitivity, which results in a dynamical uncertainty that could have important applications in neurobiology.

Asymmetric coupling of nonchaotic Rulkov neurons: Fractal attractors, quasimultistability, and final state sensitivity

TL;DR

The paper investigates the geometrical properties of basins of attraction and their boundaries in a minimal neuronal system: two asymmetrically electrically coupled nonchaotic Rulkov neurons. It combines fractal analysis, Lyapunov dynamics, and uncertainty-exponent methods to reveal a short-lived chaotic spiking-bursting pseudo-attractor coexisting with a nonchaotic spiking attractor (quasimultistability), and to classify basin structures in both 2D slices and full 4D space. The main findings show fractal chaotic attractors with and highly dimensional basin boundaries in 4D with , accompanied by extreme final-state sensitivity (requiring initial-uncertainty reductions of about to reduce final-state uncertainty by a factor of 10). These results illuminate how multistability and fractal geometry can arise in discrete-time neuronal maps and may have implications for understanding sensitivity in brain dynamics, with future directions including basin-entropy analyses and experimental validation in neural circuits.

Abstract

Although neuron models have been well studied for their rich dynamics and biological properties, limited research has been done on the complex geometries that emerge from the basins of attraction and basin boundaries of multistable neuron systems. In this paper, we investigate the geometrical properties of the strange attractors, four-dimensional basins, and fractal basin boundaries of an asymmetrically electrically coupled system of two identical nonchaotic Rulkov neurons. We discover a quasimultistability in the system emerging from the existence of a chaotic spiking-bursting pseudo-attractor, and we classify and quantify the system's basins of attraction, which are found to have complex fractal geometries. Using the method of uncertainty exponents, we also find that the system exhibits extreme final state sensitivity, which results in a dynamical uncertainty that could have important applications in neurobiology.

Paper Structure

This paper contains 9 sections, 47 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The model and dynamics
  3. Geometry of the chaotic pseudo-attractor
  4. Basin classification
  5. Two-dimensional basin slices
  6. Four-dimensional basins
  7. Fractal basin boundary analysis
  8. Conclusions
  9. Jacobian Matrix Partition

Figures (9)

  • Figure 1: Graphs of the fast variable orbits $x_{1,k}$ and $x_{2,k}$ of two asymmetrically electrically coupled nonchaotic Rulkov neurons with parameters $\sigma_1=\sigma_2=-0.5$, $\alpha_1=\alpha_2=4.5$, and electrical coupling strengths $g^e_1=0.05$, $g^e_2=0.25$
  • Figure 2: Graphs showing the attraction of two chaotic orbits to the system's nonchaotic spiking attractor
  • Figure 3: Projections of the four-dimensional nonchaotic spiking attractor of the asymmetrically coupled Rulkov neuron system
  • Figure 4: Projections of the four-dimensional chaotic spiking-bursting pseudo-attractor of the asymmetrically coupled Rulkov neuron system
  • Figure 5: Plots of the first three Lyapunov exponents at each point in the orbit of $\mathbf{X}_{0} = (-0.56, -3.25, -1, -3.25)$
  • ...and 4 more figures