The provided code models the electrical activity of cardiac cells using the Bueno-Orovio-Cherry-Fenton (BOCF) model. This mathematical representation is aimed at simulating the dynamics of cardiac action potentials, focusing on the deterministic features of individual heart cells in different layers of the heart tissue: epicardium (epi), endocardium (endo), and midmyocardial (midmyo) areas. ### Biological Basis #### Action Potential Modeling The central goal of the code is to simulate cardiac action potentials, which are the changes in membrane potential that occur as heart cells transmit electrical signals. These action potentials are crucial for initiating muscle contractions that enable the heart to pump blood. #### Membrane Potential (`Vm`) - **Resting Potential (`V_0`)**: The baseline voltage of the cell membrane when it is not excited. - **Depolarization and Repolarization**: Changes in the membrane potential are modeled through the variable `u`, which integrates contributions from various current components representing ionic flows. #### Ionic Currents The code breaks down the transmembrane ionic currents into several components: - **Fast Inward Current (`J_fi`)**: This represents the rapid influx of ions during the initial phase of an action potential, simulating sodium ion channels' activity responsible for depolarization. - **Slow Outward Current (`J_so`)**: Simulates the outward flow of ions, likely representing potassium ions' role in repolarizing the membrane. - **Inward Calcium Current (`J_si`)**: Corresponds to the calcium ion exchange, pivotal for heart muscle contraction and prolonged depolarization. #### Gating Variables Gating variables such as `m`, `p`, `q`, `r`, `v`, `w`, and `s` are used to model the opening and closing of ion channels, controlling ionic currents dynamically: - **`m`, `p`, `q`, `r`**: Fast activation and inactivation parameters of sodium channels. - **`v`, `w`**: Represent ion channel gating kinetics regulating the timing and duration of phases in action potential. - **`s`**: Calcium-mediated inactivation variable, affecting calcium ion's role in the action potential. #### Temporal and Spatial Dynamics The code includes elements that govern the time course of action potentials (e.g., `tau_fi`, `tau_so`, etc.), with parameters (`tau_`) representing the time constants that control the various transitions and durations of phases like depolarization, plateau, and repolarization in cardiac cells. #### Stimulation and External Influence (`J_stim`) The model includes external stimulation to simulate the effects of periodic electrical impulses on cardiac cells, depicted by the variable `J_stim`. This mimics experimental conditions where heart cells might be paced in a controlled manner. ### Tissue-Specific Parameters The model addresses tissue heterogeneity by allowing different parameter values (`case`) for epicardial, endocardial, and midmyocardial cells, which reflect regional variations in action potential duration and morphology seen in these cell types. In summary, the code intricately combines the complexities of ionic currents, channel gating, and other physiological processes involved in cardiac excitation to provide a detailed simulation of heart cell electrical activity. This can be a significant tool for understanding pathological conditions or testing pharmacological interventions affecting cardiac electrophysiology.