Bottom-up approach to torus bifurcation in neuron models

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Dynamical Systems Theory

Background:

• Neuronal activity exhibits diverse firing patterns, including tonic spiking and bursting.
• Understanding the transitions between these patterns is crucial for comprehending neural dynamics.
• Biologically plausible models are essential for studying these phenomena.

Purpose of the Study:

• To investigate quasi-periodicity at the transition between tonic spiking and bursting.
• To analyze these transitions in both Hodgkin-Huxley and reduced phenomenological models.
• To elucidate the underlying bifurcation mechanisms driving these dynamic changes.

Main Methods:

• Geometric slow-fast dissection to analyze model dynamics.
• Parameter continuation approach to trace bifurcations.
• Examination of bifurcations on the 2D slow-motion manifold near a fold.

Main Results:

• Identified torus bifurcation and period-doubling bifurcation as key transition mechanisms.
• Characterized various torus bifurcations: stable/saddle torus-canards, resonant tori.
• Observed co-existence of nested tori and torus breakdown leading to complex/bistable dynamics.

Conclusions:

• Quasi-periodicity and complex dynamics arise from specific bifurcations at the spiking-bursting transition.
• The study provides a unified framework for understanding these transitions in different neural models.
• Findings contribute to the theoretical understanding of neural excitability and dynamical complexity.