Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback

Area of Science:

• Nonlinear Dynamics
• Computational Neuroscience
• Mathematical Biology

Background:

• Excitable systems, like the Morris-Lecar model, exhibit complex dynamics crucial for understanding biological processes.
• Time-delayed feedback introduces intricate behaviors, including the formation and stability of temporal dissipative solitons.
• Understanding soliton dynamics is key to modeling phenomena ranging from neural signaling to pattern formation.

Purpose of the Study:

• To analyze the dynamics and bifurcations of temporal dissipative solitons in an excitable system (Morris-Lecar model) subjected to time-delayed feedback.
• To elucidate the role of homoclinic and heteroclinic bifurcations in shaping soliton behavior and stability.
• To explain the emergence of stable pulse packages from destabilized multi-pulse solutions.

Main Methods:

• Utilized the Morris-Lecar model as a prototypical excitable system.
• Employed concepts from classical homoclinic bifurcation theory, treating large-delay soliton solutions as homoclinic solutions.
• Analyzed delay-differential equations and their relationship to advanced argument equations.

Main Results:

• Demonstrated that a homoclinic orbit flip of a single-pulse soliton destabilizes equidistant multi-pulse solutions.
• Showed the emergence of stable pulse packages due to this destabilization.
• Identified that this transition is triggered by a heteroclinic orbit flip in the feedback-free system, linked to the model's excitability.

Conclusions:

• Time-delayed feedback in excitable systems can lead to complex soliton dynamics and the formation of stable pulse packages.
• Homoclinic and heteroclinic bifurcations are critical mechanisms governing these transitions.
• The study provides a theoretical framework for understanding pulse behavior in excitable media with delayed feedback.