Biological Basis of the M Current Model

The provided code models the M-current ((I_M)), a potassium ion ((K^+)) current that plays a crucial role in the regulation of neuronal excitability, particularly in cortical pyramidal cells. The M-current is essential for the adaptation of the neuronal firing rate and contributes to the afterhyperpolarization (AHP) that follows an action potential.

Key Biological Aspects Modeled:

1. Ion Channel Specificity:

   • The code models the (M)-current as a potassium ((K^+)) channel. The channel is voltage-dependent and does not inactivate, meaning it remains open when depolarization is present.

2. Membrane Voltage ((v)) Dependency:

   • The activation of the M-current depends on the membrane voltage. The function evaluate_fct computes the steady-state activation variable ((m_{\text{inf}})) and the time constant ((\tau_m)) of the M-current, both of which are functions of the membrane potential ((v)).

3. Steady-State Activation ((m_{\text{inf}})):

   • This represents the fraction of channels that are open at a given membrane potential when the system has reached equilibrium. It is modeled using a sigmoidal function reflecting the probabilistic nature of channel gating. This is influenced by the voltage via the equation: [ m_{\text{inf}} = \frac{1}{1 + e^{-\frac{(v+35)}{10}}} ]

4. Time Constant ((\tau_m)):

   • This describes the time it takes for the channel to reach approximately 63% of its final value following a change in membrane potential. It is reflective of the speed of the channel's response to changes in voltage and is described by: [ \tau_m = \frac{\tau_{\text{peak}}}{3.3 \cdot \text{exp}\left(\frac{(v+35)}{20}\right) + \text{exp}\left(-\frac{(v+35)}{20}\right)} ]

5. Temperature Dependence:

   • Physiological processes are temperature-dependent. The model accounts for this using a Q10 factor, which is a measure of how much a physiological process rate increases with a 10°C increase in temperature. Here, a Q10 value of 2.3 is used to adjust the peak time constant ((\tau_{\text{peak}})).

6. Hodgkin-Huxley Formalism:

   • The model uses a Hodgkin-Huxley type formalism, which is common in modeling ionic currents. This involves the use of differential equations to represent the change in the state variable (m) over time, have it approach its steady state value ((m' = \frac{m_{\text{inf}} - m}{\tau_m})).

This code snippet provides a biophysically plausible representation of the M-current, essential for understanding the dynamics of spike frequency adaptation and overall excitability in cortical pyramidal neurons.