Biological Basis of the Cardiac Sodium Current Model

The provided code is part of a computational model that simulates the cardiac sodium current, which is crucial for the initiation and propagation of action potentials in cardiac cells. The core biological concepts behind this model relate to the behavior of sodium ion channels in cardiac myocytes, particularly as described in the foundational work by Beeler and Reuter (1977).

Key Biological Components

Sodium Channels

• Function: Sodium channels are membrane proteins that allow for the rapid influx of Na⁺ ions into the cell. This influx is critical for depolarizing the cell membrane, triggering the upstroke of the cardiac action potential.
• Role in Cardiac Cells: These channels are essential for the propagation of electrical signals across the heart, facilitating coordinated contractions necessary for effective pumping of blood.

Gating Variables

• m, h, n: The model includes three state variables, m, h, and n, which represent the gating of these sodium channels:
  • m (Activation Gate): Represents the opening of the channel in response to membrane depolarization. It is a fast process contributing to the rapid onset of the sodium current.
  • h & n (Inactivation Gates): These represent the closing of the channel, a slower process relative to activation which ensures the current is transient. The h gate corresponds to "fast" inactivation, and n serves as an additional inactivation variable.

Hodgkin-Huxley Model

• The code uses a framework inspired by the Hodgkin-Huxley model, which describes how ion channel conductance changes with voltage due to gating variables.
• Equations for Dynamics: The differential equations in the DERIVATIVE states block simulate how m, h, and n change over time, affecting the overall sodium current (ina).

Parameters and Functions

Conductance Parameters

• gnabar: Represents the maximum conductance of the sodium channels, directly influencing the size of the sodium current.
• gnac: Represents a constant background sodium conductance, modeling a small, persistent sodium current.

Rate Constants (alp and bet)

• These functions (alp and bet) provide the voltage-dependent rate constants for transitions between different states of the channel (e.g., open vs. closed, inactivated vs. activated). These rates determine how quickly the gates open or close in response to changes in membrane potential.

Biological Processes

Action Potential

• The main output of the model, ina, is the fast sodium current that initiates the rapid depolarization phase of the cardiac action potential.
• Depolarization and Repolarization: The opening of sodium channels (characterized by the gating variable m) leads to the depolarization of the cell, while the inactivation (characterized by h and n) allows the cell membrane to repolarize, resetting for the next action potential.

Temperature Dependence

• The parameter celsius reflects the temperature dependency typical in biological systems, modulating the rates of the gating processes.

This model encapsulates the essential properties of sodium ion channel behavior in cardiac cells, providing a mathematical representation of how these channels contribute to electrical signaling in the heart. It is a foundational component for more complex cardiac electrophysiological models.