The provided code is a computational model of neuronal activity based on the Hodgkin-Huxley (HH) framework, with an extension to include fractional calculus elements. Such models are crucial for understanding how neurons process information and generate electrical signals, known as action potentials. ### Biological Basis #### Neuronal Membrane Potential At the core of this model is the simulation of the neuron's membrane potential, which is the voltage difference across the neuron's membrane. The model captures how this potential changes over time due to various ionic currents that flow through ion channels. #### Ion Channels and Gating Variables The model uses gating variables (`m`, `h`, and `n`) to describe the dynamics of ion channels that control the flow of sodium (Na⁺) and potassium (K⁺) ions: - **Sodium Channels**: These channels have an activation (`m`) and inactivation (`h`) gate. The model accounts for their dynamics using differential equations that describe the probability of the gates being open, directly influencing the Na⁺ current. - **Potassium Channels**: These channels are primarily governed by the `n` gating variable. This variable controls the activation of K⁺ channels, which are crucial for repolarizing the membrane after an action potential. The equations governing these gating variables are representative of the voltage-dependent opening and closing of the ion channels, following the classical Hodgkin-Huxley descriptions. #### Ionic Currents The model computes the ionic currents flowing through these channels: - **Sodium Current (INa)**: Influenced by `m`, `h`, and the membrane potential. The rapid opening of Na⁺ channels is responsible for the depolarization phase of the action potential. - **Potassium Current (IK)**: Governed by `n`, this current is responsible for repolarizing the cell back to its resting state after depolarization. - **Leak Current (IL)**: A smaller, constant current representing the passive flow of ions through non-specific channels or membrane leaks. #### Fractional Calculus The model incorporates fractional calculus, which introduces non-integer order derivatives to capture memory effects and anomalous diffusion processes in neuronal dynamics. This inclusion reflects a more complex, realistic temporal integration of ion channel state changes, which can be crucial for modeling subthreshold activities and adapting to different time scales. #### Noise and Stochasticity The model incorporates Gaussian noise, representing the inherent stochasticity and variability in biological systems due to synaptic input fluctuations and channel noise. ### Summary Overall, this model extends the classical Hodgkin-Huxley framework by introducing fractional order dynamics and noise, aiming to provide a more nuanced simulation of neuronal behavior that accommodates for the complex, memory-dependent processes observed in real neurons. This approach is valuable for understanding the biophysical mechanisms underlying neuronal excitability and how these mechanisms may vary across different contexts or disease states.