The geometry of rest-spike bistability

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Dynamical Systems Theory

Background:

• The Morris-Lecar model is a foundational conductance-based model for neuronal excitability.
• Existing models often lack a balance between physiological realism and mathematical simplicity.
• Understanding bistability is crucial for modeling complex neural behaviors like bursting.

Purpose of the Study:

• To augment the Morris-Lecar model with a slow inward current.
• To achieve bistability between resting and spiking states in a simplified dynamical system.
• To provide a model with both physiological interpretation and mathematical tractability.

Main Methods:

• Augmentation of the standard Morris-Lecar model by introducing a single slow inward ionic current.
• Analysis of the resulting dynamical system to identify conditions for bistability.
• Comparison of the proposed model's properties with existing alternative models in the literature.

Main Results:

• The augmented model successfully demonstrates bistability between a stable resting state and a spiking limit cycle for a range of input currents.
• The model captures key dynamical phenomena, including slow spiking and bursting patterns.
• The proposed model retains physiological interpretability while enhancing mathematical tractability.

Conclusions:

• The augmented Morris-Lecar model provides a robust and simplified framework for studying neuronal excitability and complex firing patterns.
• This approach offers a valuable balance between physiological relevance and mathematical analysis.
• The model serves as a core structure for understanding diverse dynamical behaviors in neural systems.