8.4 Modeling cell assemblies  (Page 3/9)

Ion channels

The cell dynamics use a model that is derived from the Hodgkin-Huxley model. The ion currents take the general form of a driving force (which is due to the different concentrations of the ions on the inside and outside of the cell) being multiplied by a conductance term (the reciprocal of resistance) and a product of voltage dependant variables that indicate the fraction of the ion channels open. For example, the sodium current is:

Where $V_{Na}$ is the reversal potential (indicating the concentration of $N{a}^{+}$ in the surrounding fluid). $G_{Na}$ is the conductance of $N{a}^{+}$ . $m$ is called the activation variable of the $N{a}^{+}$ channels and $h$ is the inactivation variable. They are governed by the following differential equations:

Here ${\alpha }_m$ is the opening rate and ${\beta }_m$ is the closing rate. These rates are dependent on the soma potential. A, B and C are fitting parameters that will later be prescribed.

The inactivation variable $h$ is slightly different, and has its own fitting parameters.

The $K^{+}$ channels are similar. There is only one gating variable $n$ .

For more details on the Hodgkin Huxley model visit (External Link) or look at the series of papers this module is designed after.

The change in net potential due the calcium flow is negligible. However, we still need to track it in order to control the Calcium dependent Potassium channels. As cell repeatedly fires calcium will build up leading to stronger repolarizations this will help allow for firing rate adaptation and provide the assembly a mechanism through which it will die out eventually.

The Calcium dependent Potassium channels:

$\left[C{a}_{AP}\right]$ is the intracellular calcium level. Constants ${\rho }_{AP}$ and ${\delta }_{AP}$ are the rates of calcium ion influx and efflux (decay).

The second form of Calcium comes through NMDA channels that are located on the post-synapse. It is important to separate this Calcium source since it enters and leaves the cell with different time constants due to the different method it enters the cell. This contribution will be discussed in more details in the next section. The Calcium pool $\left[C{a}_{NMDA}\right]$ will be added to the total $C{a}^{2+}$ count changing the total current from Calcium dependent Potassium channels to:

Synaptic connections

Cells communicate with synaptic connections where the bouton and post-synapse meet. The firing cell releases neurotransmitter into the synaptic cleft. These chemical open up ion channels in the postsynaptic receptor cell. This allows either a gain in postsynaptic potential (if the presynaptic cell is excitatory) or a depression in potential (for a inhibitory cell). The model for the synaptic current is show below: