The provided code models the Hodgkin-Huxley (HH) neuron dynamics using both conventional and fractional calculus approaches. The Hodgkin-Huxley model is a foundational mathematical model in computational neuroscience used to describe how action potentials in neurons are initiated and propagated. This model is based on the ionic mechanisms underlying the initiation and propagation of action potentials, focusing on the key ions and their movement through the neuron membrane.

Key Biological Components Modeled in the Code

1. Membrane Potential (v):

   • The variable v represents the membrane potential of the neuron. This potential is crucial for the generation of action potentials.

2. Ion Channels:

   • Sodium Channels (gNa): Sodium ions (Na+) are handled by the gating variable m, which controls the activation of these channels. alpham and betam define the rate of transition between open and closed states.
   • Potassium Channels (gK): Potassium ions (K+) are managed by the gating variable n, reflecting the slower dynamics of these channels compared to Na+.
   • Leak Channels (gL): The leak channel manages the passive flow of ions, contributing to the neuron's resting membrane potential.

3. Gating Variables:

   • The code includes gating variables m, h, and n, which represent the probabilities of ion channel states (open or closed). These variables are crucial for simulating ionic currents through the Na+ and K+ channels.
   • The m and n variables are associated with Na+ activation and K+ activation, respectively, while h corresponds to Na+ inactivation.

4. Equilibrium Potentials:

   • Equilibrium potentials for K+ (EK), Na+ (ENa), and leak (EL) are specified. These are critical for determining the direction and magnitude of ion flow across the membrane.

5. Fractional Derivatives:

   • The model incorporates fractional calculus (denoted by the parameter alpha) to capture memory effects and variations in the neuron’s response that may be characteristic of biological neurons beyond the classic HH model. This approach can represent long-term correlations or complex dynamic behaviors not captured by integer-order models.

6. Differential Equations and Dynamics:

   • At its core, the model relies on differential equations derived from the dynamic changes in membrane voltage and gating variables. The Runge-Kutta methods (HH_RK and HH_RKfull) are employed to numerically solve these equations.

Biological Basis and Purpose

The code attempts to model a neuron's electrical behavior based on the classic Hodgkin-Huxley model, integrating both traditional dynamics and fractional dynamics to possibly achieve a more realistic representation. Key ions involved are sodium and potassium, which are fundamental in action potential generation. The fractional aspect may suggest an interest in exploring complex biological behaviors such as adaptation, neuronal plasticity, or other long-term phenomena that the traditional HH model may not fully capture on its own.

This type of modeling is crucial in computational neuroscience for better understanding neural excitability, synaptic integration, and the overall limitations and capabilities of neurons under diverse conditions.