The provided code implements a computational model of the **Hodgkin-Huxley (HH) equations** with an extension to include the effects of an external electrical field on neuronal activity. The Hodgkin-Huxley model is a foundational mathematical representation that describes how action potentials in neurons are initiated and propagated. This model accounts for the dynamics of ion channels in the neuronal membrane, which are key to generating electrical signals in neurons. ### Key Biological Concepts 1. **Ion Channels and Gating Variables:** - The model specifies channels for sodium (Na\(^+\)) and potassium (K\(^+\)) ions, which are critical for action potential generation. - The variables `m`, `h`, and `n` represent gating variables associated with these ion channels. These variables correspond to the probabilities of the ion channels being in open or closed states. - **`m`** is the activation variable for sodium channels. - **`h`** is the inactivation variable for sodium channels. - **`n`** is the activation variable for potassium channels. - The equations governing these variables determine the direction and rate of change of the gating dynamics. 2. **Membrane Potential (V):** - The model tracks the membrane potential, `V`, which is influenced by the flow of ions through their respective channels. In this model, the membrane potential is initially set to 0 mV. 3. **Ionic Currents:** - Three types of ionic currents are represented in the model: - **Sodium current (I\(_{Na}\))**: Governed by the variables `m` and `h`, it is key to the rapid depolarization phase of the action potential. - **Potassium current (I\(_{K}\))**: Governed by the variable `n`, it facilitates repolarization. - **Leakage current (I\(_{L}\))**: Represents a constant baseline current due to non-specific ion permeability. 4. **External Electrical Field:** - The extension of the HH model in this code is the inclusion of an external electrical field, represented by `vex(t)`. This term adds an oscillatory component that simulates the effects of an external sinusoidal electric field on the neuron. The amplitude and frequency of this field can be adjusted via parameters, allowing the study of its effect on neuronal behavior. 5. **Parameters:** - Various parameters set biological constants like conductance levels (`gNa`, `gK`, `gL`), reversal potentials (`VNa`, `VK`, `VL`), and capacitance (`cm`) of the neuronal membrane. These constants are derived from experimental data on squid giant axon, which was the basis for the original HH model. Overall, this model aims to simulate neuronal excitation and electrical activity, accounting for the dynamic changes and interactions between different ion channels. The introduction of an external electric field suggests a focus on understanding how such fields influence neuronal excitability, possibly mimicking experimental or physiological conditions where neurons are exposed to external electromagnetic stimuli.