The provided code models the dynamics of calcium ion (Ca++) concentration just beneath the neuronal membrane, specifically focusing on the rapid changes in submembrane calcium concentrations that are crucial for various cellular processes. Here's the biological basis of the model:

Biological Context

1. Calcium Signaling: Calcium ions play an essential role as secondary messengers in numerous cellular processes within neurons, including neurotransmitter release, gene expression, and neuronal excitability. The precise regulation of intracellular calcium concentration is vital for these functions.

2. Submembranal Calcium Dynamics: The model explicitly targets the calcium concentration in the small shell of cytoplasm just beneath the membrane. This submembranal region is critical for rapid signaling events because it closely interacts with voltage-gated calcium channels and calcium-dependent proteins.

3. Calcium Influx and Extrusion:

   • Calcium Influx: The calcium current (ica) through voltage-gated calcium channels increases submembranal calcium concentration. This is a rapid process occurring during neuronal activity when these channels open in response to depolarization.
   • Calcium Extrusion: The model incorporates a mechanism for calcium removal from the submembranal region to maintain homeostasis. Calcium extrusion is modeled as a first-order process with a time constant (taur), representing mechanisms like calcium pumps and exchangers that expel calcium from the cell or sequester it into intracellular stores.

4. Equilibrium Concentration: The parameter cainf represents the equilibrium or baseline calcium concentration, reflecting the resting state where calcium influx and extrusion are balanced.

5. Shell Depth: The depth parameter indicates the thickness of the submembranal shell, allowing for the calculation of how much the calcium concentration changes due to calcium fluxes.

6. Biophysical Principles:

   • Faraday's Constant: The model uses Faraday's constant to relate the ionic current (ica) to molar fluxes, as it converts electric current into a measure of ion flow.
   • Current Contribution to Calcium Increase: The calculation of drive_channel represents how the influx of calcium ions through the membrane contributes to changes in calcium concentration in the submembranal region. This influx is temporarily set to zero if it would lead to an inward pump, reflecting the unidirectional nature of typical calcium extrusion mechanisms.

Biological Implications

The model provides a framework for understanding how calcium ions are rapidly managed in neurons, which is crucial for cellular signaling and homeostasis. By simulating calcium dynamics with respect to membrane currents and extrusion rates, this model assists in exploring how neurons encode information and respond to stimuli, and how disruptions in these processes could contribute to pathophysiological conditions.