Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Theoretical Neuroscience

Background:

• Integrate-and-fire models are fundamental in computational neuroscience for simulating neuronal activity.
• Existing one-dimensional models capture basic neuronal firing but lack complex dynamics.
• FitzHugh-Nagumo and Morris-Lecar models are established systems for studying neuronal excitability.

Purpose of the Study:

• To develop and analyze two-dimensional generalizations of integrate-and-fire models.
• To explore novel dynamical properties arising from these higher-dimensional models.
• To investigate the relationship between these models and existing neuronal modeling paradigms.

Main Methods:

• Derivation of two-dimensional integrate-and-fire models via piecewise linear idealization.
• Analytical investigation of model properties, including stability and transitions.
• Comparison with established models like FitzHugh-Nagumo, Morris-Lecar, and resonate-and-fire.

Main Results:

• The piecewise linear FitzHugh-Nagumo model exhibits bistability between stationary and oscillatory states (class-II behavior).
• The piecewise linear Morris-Lecar model demonstrates a noncanonical class-I transition to oscillations with logarithmic dependence.
• A connection to the resonate-and-fire model is established, showing potential for multiple spikes from short input pulses.

Conclusions:

• Two-dimensional integrate-and-fire models offer richer dynamics than their one-dimensional counterparts.
• These models provide new insights into neuronal excitability and firing patterns.
• The derived models offer a simplified yet powerful framework for studying complex neural behaviors.