Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays

Area of Science:

• Computational Neuroscience
• Dynamical Systems Theory
• Nonlinear Dynamics

Background:

• Understanding complex neural system dynamics is crucial for neuroscience.
• Time delays significantly influence the behavior of neural networks.
• Bifurcation theory provides a framework for analyzing transitions in dynamical systems.

Purpose of the Study:

• To construct and analyze an inertial two-neuron system with multiple time delays.
• To investigate the coexistence of multiple dynamical behaviors including equilibria, periodic, and quasi-periodic orbits.
• To explore the impact of bifurcations and time delays on system stability and dynamics.

Main Methods:

• Utilized three first-order delayed differential equations to model the neural system.
• Employed center manifold reduction and normal form method to analyze pitchfork bifurcations.
• Analyzed equilibrium stability using the characteristic equation and investigated Hopf bifurcations.

Main Results:

• Demonstrated pitchfork bifurcations leading to multiple stable equilibria.
• Identified Hopf bifurcations causing periodic orbits and Hopf-Hopf bifurcations leading to coexistence of two periodic orbits.
• Observed quasi-periodic behavior arising from multiple frequencies and stable coexistence of equilibria, periodic, and quasi-periodic states.

Conclusions:

• The constructed two-neuron system exhibits rich dynamical behaviors, including stable coexistence of various states.
• Time delays play a critical role in stability switching and the emergence of complex dynamics.
• The study highlights the intricate interplay of bifurcations and delays in neural system function.