# 8.4 Modeling cell assemblies  (Page 5/9)

 Page 5 / 9
$\begin{array}{cc}\hfill \frac{\left({m}_{i+1}-{m}_{i}\right)}{dt}& ={\alpha }_{m}\left[{V}_{i}\right]\left(1-{m}_{i+1}\right)-{\beta }_{m}\left[{V}_{i}\right]{m}_{i+1}\hfill \\ & =\frac{{A}_{{\alpha }_{m}}\left({V}_{i}-{B}_{{\alpha }_{m}}\right)}{1-{e}^{\left({B}_{{\alpha }_{m}}-{V}_{i}\right)/{C}_{{\alpha }_{m}}}}\left(1-{m}_{i+1}\right)-\frac{{A}_{{\beta }_{m}}\left({B}_{{\beta }_{m}}-{V}_{i}\right)}{1-{e}^{\left({V}_{i}-{B}_{{\beta }_{m}}\right)/{C}_{{\beta }_{m}}}}{m}_{i+1}\hfill \end{array}$

Solving for ${m}_{i+1}$ gives:

${m}_{i+1}=\frac{{m}_{i}+dt{\alpha }_{m}\left[{V}_{i}\right]}{1+dt\left({\alpha }_{m},\left[{V}_{i}\right],+,{\beta }_{m},\left[{V}_{i}\right]\right)}$

Update $n,h,q,\left[C{a}_{AP}\right],\left[C{a}_{NMDA}\right]$ likewise solving for the $i+1$ step. Notice ${\alpha }_{m}$ and ${\beta }_{m}$ are functions of voltage. Plug these results into below equation which is then solved for ${V}_{i+1}$

$\frac{{V}_{i+1}-{V}_{i}}{dt}=\frac{\left({V}_{leak},-,{V}_{i+1}\right){G}_{m}+\sum \left({V}_{comp},-,{V}_{i+1}\right){G}_{core}+{I}_{Na}+{I}_{K}+{I}_{Ca}+{I}_{K\left(Ca\right)}+{I}_{syn}}{{C}_{m}}$

The ion channels have the ${V}_{i+1}$ term in them. For example the $N{a}^{+}$ channel looks like this:

${I}_{Na}=\left({V}_{Na}-{V}_{i+1}\right){G}_{Na}{m}_{i+1}^{3}{h}_{i+1}$

The final result term after solving for ${V}_{i+1}$ :

${V}_{i+1}=\frac{{C}_{m}{V}_{i}+dt\left({G}_{m}{V}_{leak}+{G}_{Na}{m}_{i+1}^{3}{h}_{i+1}{V}_{Na}+{G}_{K}{n}_{i+1}^{4}{V}_{K}+{G}_{K\left(Ca\right)}{V}_{K}\left({\left[C{a}_{AP}\right]}_{i+1}+{\left[C{a}_{NMDA}\right]}_{i+1}\right)+{I}_{Syn}+{I}_{Stim}}{{C}_{m}+dt\left({G}_{m}+{G}_{Na}{m}_{i+1}^{3}{h}_{i+1}+{G}_{K}{n}_{i+1}^{4}+{G}_{K\left(Ca\right)}\left({\left[C{a}_{AP}\right]}_{i+1}+{\left[C{a}_{NMDA}\right]}_{i+1}\right)}$

## Building networks

After constructing models for the individual neurons it is necessary to connect the cells. The mission is to have a small network (50 excitatory and 50 inhibitory neurons will be used in the simulations, more details on why in a bit) and have the cells trained with pre-selected assemblies/patterns. Given different patterns oreventswe want to construct a weighting scheme that will assign a proper weighting. Proper in the sense that cells within a pattern will have strong excitatory connections and cells that are ever in the same pattern will have inhibitory connections. The goal is to have the network able to activate patterns. This means that when a sufficient number of cells belonging a pattern are stimulated it will fully resolve.

The weighting algorithm will output weights that are effective in proportion to another, but will require proper scaling. Should the overall weighting magnitudes be too high the network will seizure (meaning all the cells start firing). If the weighting magnitudes are too low, the network will go dormant once stimulation is removed.

## Picking weights

The term weight is used loosely; it signifies the choice of the measure of conductance at synapse where two neurons meet. This parameter will suffice as way of quantifying the strength of a synapse which is physiologically determined by the amount of neurotransmitter that would be released into the cleft and the number of receptor channels. Look back at the form for a synaptic current to see the conductance term we now will give more detail on:

${I}_{syn}=\left({V}_{syn}-V\right){G}_{syn}s.$

The ${G}_{syn}$ is the conductance term that reflect the weighting. It is expressed as the product of a weighting term that is an output of the weighting algorithm and an overall weighting scale constant that will determined later. Thus, ${G}_{ij}={w}_{ij}*weigh{t}_{syn}$ . Note: there will be several different $weigh{t}_{syn}$ constants for the several synaptic currents used in the model.

The scheme used to produce weights will use a probabilistic approach. Using a simple intuitive approach we want to characterize the level ofsupporta cell gets by the term:

${s}_{q}={b}_{q}+\sum _{h}{w}_{hq}{\pi }_{h}$

Here ${s}_{q}$ is the support level of the cell. The term ${b}_{q}$ will represent a bias of a cell (this would be a way of characterizing a cells firing property independent from other cell activity, since some cells may appear simply more activate in general). ${w}_{hq}$ is the connection weight from $h$ to $q$ . Variable ${\pi }_{h}$ is the activity level of cell $h$ . Assume either 0 or 1 then we reduce this system to a summation over the active cells in all the different patterns, set A.

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research.net
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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