Exploring Geometrical Properties of Chaotic Systems Through an Analysis of the Rulkov Neuron Maps

While extensive research has been conducted on chaos emerging from a dynamical system's temporal dynamics, our research examines extreme sensitivity to initial conditions in discrete-time dynamical systems from a geometrical perspective. Specifically, we develop methods of detecting, classifying, and quantifying geometric structures that lead to chaotic behavior in maps, including certain bifurcations, fractal geometry, strange attractors, multistability, fractal basin boundaries, and Wada basins of attraction. We also develop slow-fast dynamical systems theory for discrete-time systems, with a specific application to modeling the spiking and bursting behavior emerging from the electrophysiology of biological neurons. Our research mainly focuses on two simple low-dimensional slow-fast Rulkov maps, which model both non-chaotic and chaotic spiking-bursting neuronal behavior. We begin by exploring the maps' individual dynamics and parameter spaces, performing bifurcation analyses, describing and quantifying their chaotic dynamics, and modeling an injection of current into them. Then, by putting these neurons into different physical arrangements and coupling them with a flow of current, we find that complex dynamics and geometries emerge from the existence of multistability and final state sensitivity in higher-dimensional state space. We then analyze the complexity and fractalization of these coupled neuron systems' attractors and basin boundaries using our mathematical and computational methods. This paper begins with a conversational introduction to the geometry of chaos, then integrates mathematics, physics, neurobiology, computational modeling, and electrochemistry to present original research that provides a novel perspective on how types of geometrical sensitivity to initial conditions appear in discrete-time neuron systems.