Co-evolutionary dynamics for two adaptively coupled Theta neurons

Abstract

Natural and technological networks exhibit dynamics that can lead to complex cooperative behaviors, such as synchronization in coupled oscillators and rhythmic activity in neuronal networks. Understanding these collective dynamics is crucial for deciphering a range of phenomena from brain activity to power grid stability. Recent interest in co-evolutionary networks has highlighted the intricate interplay between dynamics on and of the network with mixed time scales. Here, we explore the collective behavior of excitable oscillators in a simple networks of two Theta neurons with adaptive coupling without self-interaction. Through a combination of bifurcation analysis and numerical simulations, we seek to understand how the level of adaptivity in the coupling strength, $a$, influences the dynamics. We first investigate the dynamics possible in the non-adaptive limit; our bifurcation analysis reveals stability regions of quiescence and spiking behaviors, where the spiking frequencies mode-lock in a variety of configurations. Second, as we increase the adaptivity $a$, we observe a widening of the associated Arnol'd tongues, which may overlap and give room for multi-stable configurations. For larger adaptivity, the mode-locked regions may further undergo a period-doubling cascade into chaos. Our findings contribute to the mathematical theory of adaptive networks and offer insights into the potential mechanisms underlying neuronal communication and synchronization.

References in corpus (16)

• The complex network of global cargo ship movements
• Adaptive Coevolutionary Networks: A Review
• The physics of brain network structure, function, and control
• Macroscopic description for networks of spiking neurons
• Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review
• Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks
• Relevance of Metric-Free Interactions in Flocking Phenomena
• Plasticity and learning in a network of coupled phase oscillators
• Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators
• Desynchronization transitions in adaptive networks
• Hierarchical frequency clusters in adaptive networks of phase oscillators
• Topological phenotypes constitute a new dimension in the phenotypic space of leaf venation networks
• Exact finite-dimensional description for networks of globally coupled spiking neurons
• Transitions from trees to cycles in adaptive flow networks
• Complex dynamics in adaptive phase oscillator networks
• Integrability of a globally coupled complex Riccati array: quadratic integrate-and-fire neurons, phase oscillators and all in between