The code provided models the electrical behavior of a neuronal system based on the Hodgkin-Huxley (HH) model, with alterations to account for coupling between two HH neurons and the effects of sodium channel availability. The key biological aspects being modeled are as follows: ### Neuronal Membrane Dynamics 1. **Hodgkin-Huxley Framework**: - This well-established model describes how action potentials in neurons are initiated and propagated, focusing on the dynamics of the membrane potential. - It uses differential equations to model ionic currents through voltage-gated sodium (Na⁺) and potassium (K⁺) channels which are critical for generating action potentials. 2. **Ion Channels**: - **Sodium Channels (Na⁺):** The code accounts for fast activation and slow inactivation dynamics. The availability of these channels is modulated by state variables `m` and `h`, which represent the activation and inactivation states of sodium channels, respectively. - **Potassium Channels (K⁺):** Modeled by state variable `n`, representing the gating variable for potassium channels that are responsible for repolarizing the neuron after an action potential. 3. **Ion Current Contributions**: - Ionic currents are calculated based on conductance values (`gNa`, `gK`, and `gL` for leak), reversal potentials (`VNa`, `VK`, `VL`), and the membrane potential `V`. ### Stimulation and Response 1. **Stimulation Current**: - The model simulates the effect of a fixed-length current pulse. The strength of the pulse (`I0`) is varied to assess the neuron's response. 2. **Action Potentials (APs)**: - The condition for an action potential is defined when the membrane potential crosses a threshold (e.g., -10 mV). The code computes the probability of responses (spikes) under different stimulation conditions. ### Sodium Channel Inactivation 1. **Sodium Channel Availability**: - Slow inactivation of sodium channels is modeled through the variables `s1` and `s2`. This reflects the biological reality that sodium channels experience both fast and slow inactivation, influencing neuronal excitability over time. 2. **Inactivation Kinetics**: - The parameters for slow sodium inactivation and its recovery (e.g., `phi_s1`, `amp`, `sigma`, `Vhalf`) are based on empirical findings. These parameters determine the rate of inactivation and recovery, influencing the neuron's response to successive stimuli. ### Purpose of the Model The primary biological goal of this model is to investigate how variations in sodium channel availability and different current pulse amplitudes affect the excitability of a neuron. Such models provide insights into the role of ion channel kinetics in neuronal signaling, particularly the reliability and temporal precision of action potential generation under varying conditions. By simulating these dynamics computationally, researchers can better understand mechanisms like adaptation and neuronal plasticity, which are fundamental in processes ranging from sensory perception to learning and memory.