Dynamic analysis of the discrete fractional-order Rulkov neuron map.

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Dynamical Systems

Background:

• Human evolution involves genetic systems and nervous system information transmission.
• Mathematical neural models, particularly discrete-time models, are crucial in computational neuroscience for understanding brain function.
• Discrete fractional order neuron models enhance dynamic models by incorporating memory.

Purpose of the Study:

• To introduce and analyze the fractional order discrete Rulkov neuron map.
• To investigate the dynamical properties and synchronization capabilities of the new model.
• To explore the impact of fractional order and model parameters on system behavior.

Main Methods:

• Analysis of the Rulkov neuron map using phase plane, bifurcation diagrams, and Lyapunov exponents.
• Investigation of the fractional order discrete Rulkov neuron map's dynamical behaviors (silence, bursting, chaotic firing).
• Theoretical and numerical determination of stability regions and synchronization analysis of coupled fractional-order models.

Main Results:

• The fractional order discrete Rulkov neuron map replicates biological neuron behaviors like silence, bursting, and chaotic firing.
• Increasing the fractional order decreases the stability regions of the system.
• Complete synchronization is not achievable between two coupled fractional-order Rulkov neuron models.

Conclusions:

• The fractional order discrete Rulkov neuron model offers a novel approach to simulating neuronal dynamics with memory.
• System stability is inversely related to the fractional order, suggesting limitations for complex dynamics.
• The inability to achieve complete synchronization in these fractional-order systems has implications for understanding neural network communication.