8.4 Modeling cell assemblies  (Page 6/9)

From conditional probability rule

arises Bayes Rule, which can then be iterated.

Taking the logrithm of this term gives us something in the form we wanted:

Thus we have that:

The method for computing the probabilities is now explicated. For single probabilities (such as $p\left(q\right)$ ), we simply look at each pattern and compute the proportion in which cell q is are active. If either $p\left(q\right)$ or $p\left(h\right)$ be zero set $w_{hq}$ to zero. For joint probabilities (such as $p(h\&q)$ ), compute the proportion in which both are active. If both $p\left(q\right)$ and $p\left(h\right)$ are not zero but $p(h\&q)$ is zero, set $w_{hq}$ to $log(1/\#patterns)$ .

There exists a more complicated version in which different events/patterns carry more significance than others and are then weighted accordingly by ${\kappa }^{\left(\alpha \right)}$ , this capability is not implemented in our simulations but the code can handle this option. The code for building the weight matrix (where entry $w_{ij}$ is connection strength from $i$ to $j$ ) is carried out by the code make $\_$ W.m.

The method that produces the weight matrix $W$ will produce a symmetric matrix. This should not be that limiting on the network model, but is not exactly representative of physiology.

Hooking up the network

The output from the weight matrix will yield positive and negative entries. If the entry is positive we have an excitatory connection. If an entry is negative there is an inhibitory connection. If the absolute value is less than a chosen tolerance then there is no connection.

Now comes an important step that is not entirely biologically correct but serves to make the over network model simpler. The patterns that we used in constructing the weight matrix in fact will consist of all excitatory cells. For each of these excitatory cells we will now designate a inhibitory companion cell through that only synapses onto the excitatory cell. All inhibition of the excitatory cells is achieved by other excitatory cells that synapse onto the inhibitory companion. Thus, there are an equal number of E and I cells. (as mentioned earlier there will be 50 E and 50 I cells in the simulations produced ahead).

Simulations

The bulk of the theory is completed. Now, the specifications for the simulations are presented. The parameters that are used in the biological model presented earlier are given values. Then there is a few examples of how to use the matlab code. The code used can be downloaded at (External Link) .