Heteroclinic networks in an ensemble of generalized Lotka-Volterra elements

Area of Science:

• Dynamical Systems Theory
• Computational Neuroscience
• Mathematical Biology

Background:

• The Lotka-Volterra model is a foundational ecological model.
• Neuronal models are crucial for understanding brain function.
• Heteroclinic cycles represent complex dynamic behaviors in systems.

Purpose of the Study:

• To analyze the generalized four-dimensional Lotka-Volterra model.
• To identify and characterize heteroclinic networks within the model's phase space.
• To explore the implications of these networks for neuronal activity and system stability.

Main Methods:

• Phase space analysis of the four-dimensional Lotka-Volterra model.
• Construction of parameter space partitions to delineate regions of heteroclinic network existence.
• Investigation of the conditions leading to stable heteroclinic cycles and multistability.

Main Results:

• Existence of heteroclinic networks, defined as unions of heteroclinic cycles, was demonstrated.
• A comprehensive partition of the coupling parameter plane identified regions supporting various heteroclinic networks.
• The model was shown to exhibit multistability, indicating multiple coexisting stable states.

Conclusions:

• The identified heteroclinic networks in the Lotka-Volterra model provide insights into complex dynamics.
• Stable heteroclinic cycles can be interpreted as implementing neuronal activity switching.
• The model's capacity for multistability has significant implications for understanding neuronal network behavior.