8.4 Modeling cell assemblies  (Page 4/9)

Here $s$ is an activation level variable. It will only take value 0 or 1. After a presynaptic cell fires the $s$ variable will be held at 1 for fixed number of milliseconds, then reset to 0. The excitatory receptor area will consist of AMPA receptor and NMDA receptors. The AMPA receptor takes the form above and $V_{syn}$ would typically be around 0 mV for an excitatory connection. An inhibitory connection will have a reversal potential, $V_{syn}$ , below or near rest (around -85 mV).

The second type of receptor Channel is the NMDA channel. These receptor channels are blocked by $M{g}^{2+}$ when the post-synaptic cell in near resting potential. Only come out when the receptive cell has been depolarized will the $M{g}^{2+}$ pop out allowing $N{a}^{+}$ and $C{a}^{+}$ to pass through. This allows so that cells can continue firing once they have been activated.These channels are dependent on neurotransmitter from the sending cell and the post-synaptic cell firing. This feature helps for LTP (long term potentiation). The $M{g}^{2+}$ is modeled by a voltage dependant ODE between 0 and 1, $p$ . The form for this current and it's gating variable $p$ is the following:

The NMDA channels are also permeable to $C{a}^{2+}$ so we need to track another pool of calcium. This calcium is modeled by a differential equation. The influx rate is dependent on the $s$ and $p$ variable, as well as a characteristic influx conductance term, ${\rho }_{NMDA}$ . This influx term varies from each synapse and is proportional to the synapse strength (something which is determined in later a section when connection weighting is addressed). The efflux is a constant ${\delta }_{NMDA}$ . The $s$ variable here is the sum of all the binary s variables which connection to the specific cells whose Calcium pool is being calculated.

This term actually looks a little more complicated. There are generally many synapses where $C{a}^{2+}$ can enter a neuron and the entry rate at each synapse is a product of the activation variable $s_{kj}$ (which is 1 if the pre-synaptic neuron j which synapses onto k fired recently, 0 if not) and the connection weight ${\rho }_{NMDA}$ , which is proportional to the conductance wieghting (i.e. ${\rho }_{NMDA}=NMD{A}_{scalar}{G}_{kj}$ ). We must write this term as a summation of all the $C{a}^{2+}$ coming in from different synapses. There are now subscripts and superscripts $j$ and $k$ being used to indicate which cells are be tracked. In the below equation is the differential equation for cell $k$ . The $j$ index is used for other cells that synapse onto $k$ . The compartment here is assumed to be the soma as we will not bother to track the calcium flow from the end compartment to the soma.

Computational methods for solving differential equations

There are several differential equations that need to be solved in order to advance the system. A time-stepping method is used and updates each variable given the selected step size (typically .01 milliseconds). A hybrid Euler scheme is used. In a differential equation the derivative is replaced with the approximation $(v_{i+1}-v_i)/dt$ . The right hand side $v_{i+1}$ is plugged in for $v$ when the equation can be explicitly solved for, this is backward Euler which has better stability properties. However, when we cannot solve for $v_{i+1}$ we use $v_i$ instead, this is a forward Euler stepping scheme. In our main differential equation for cell voltage some terms we can use the backward method. The $m,h,n,q,\left[CaAP\right]$ and $\left[CaNMDA\right]$ variables have to updated using a forward Euler update first. Then they are plugging into their ion current components of the main cell voltage differential equation. This is all then solved for $V_{i+1}$ . The set up is below. Remember the ion currents are only at the soma, below we imagine we are updating the soma compartment. Other compartments are done the same way except the ion currents are not there and some parameters such as $C_m$ take different values. We start with an example of how the $m$ gating variable would be updated.