The provided code models the electrical activities in cardiac tissue, specifically focusing on the generation and dynamics of spiral waves. These spiral waves are critical to understanding cardiac arrhythmias, such as reentrant tachycardia, a condition that leads to rapid and irregular heartbeats. ### Biological Basis #### Model Framework The code implements a monodomain reaction-diffusion model, a mathematical framework often used in cardiac electrophysiology. It is based on a variant of the Fitzhugh-Nagumo equations, which are simplified models of action potentials. These equations help simulate the cardiac action potential by modeling the voltage across the cardiac cell membrane and its recovery dynamics. #### State Variables - **Membrane Voltage (v):** Represents the electrical potential difference across the cardiac cell membrane. It is initialized to a hyperpolarized state (-80 mV) across the simulated domain. - **Recovery Variable (r):** Not explicitly defined in the code, but typically represents processes like the reactivation of ion channels after an action potential. #### Ionic Currents and Conductances - **External Current (Iex):** Used to simulate stimuli applied to the cardiac tissue, initiating the action potential and wave propagation. - **Calcium Current (\(I_{ca}\)):** Driven by the difference between membrane potential and calcium equilibrium potential (\(E_{ca}\)), and modulated by gating variables, modeling the flow of calcium ions which are crucial for action potential propagation. - **Potassium Current (\(I_{k}\)):** Represents the outflow of potassium ions, which are vital for the repolarization phase of the action potential. The code introduces heterogeneity in potassium conductance, which can reflect regional variations in cardiac tissue properties. #### Gating Variables - **D and F gates:** These are dynamic and state-dependent variables that govern the opening and closing of ion channels, specifically calcium channels in this model. They resemble the behavior of real cardiac ion channels that transition between open, closed, and inactivated states. - **X gate:** This gating variable modulates potassium conductance, impacting repolarization. #### Stimulation Protocols The model tests how different stimulation protocols can initiate spiral waves: 1. **Two-Point Stimulation:** Mimics a scenario where localized stimuli create partial wave reentries. 2. **Cross-Field Stimulation:** Initiates initial plane wavefronts, followed by stimuli perpendicular to initial wavefronts, potentially leading to spiral wave formations. #### Boundary Conditions and Spatial Dynamics - **Neumann Boundary Conditions:** Ensure that there are no net fluxes across boundaries, simulating a closed system. - **Finite Difference Method:** Solves spatial derivatives, facilitating simulations of wave propagation across the cardiac tissue grid. ### Conclusion The code serves as a platform to simulate and study the complex wave dynamics in cardiac tissue, emphasizing the role of ionic currents, transmembrane potentials, and channel gating variables. Understanding these dynamics is crucial in exploring the mechanisms leading to cardiac arrhythmias and potential therapeutic interventions.