We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
55/2006
Cortical network modeling: analytical methods for firing rates and some properties of networks of LIF neurons
Henry Tuckwell
Abstract
The circuitry of cortical networks involves interacting populations of excitatory (E) and inhibitory (I) neurons whose relationships are now known to a large extent. Inputs to E-cells and I-cells may have their origins in remote or local cortical areas. We consider a rudimentary model involving E- and I-cells.
One of our goals is to test an analytic approach to finding firing rates in neural networks without using a diffusion approximation and to this end we consider in detail networks of excitatory neurons with leaky integrate-and-fire (LIF) dynamics. A simple measure of synchronization, denoted by $S_q$, where $q$ is between 0 and 100 is introduced. Fully connected E-networks have a large tendency to become dominated by synchronously firing groups of cells, except when inputs are relatively weak.
We observed random or asynchronous firing in such networks with diverse sets of parameter values. When such firing patterns were found, the analytical approach was often able to accurately predict average neuronal firing rates. We also considered several properties of E-E networks, distinguishing several kinds of firing pattern. Included were those with silences before or after periods of intense activity or with periodic synchronization. We investigated the occurrence of synchronized firing with respect to changes in the internal excitatory postsynaptic potential (EPSP) magnitude in a network of 100 neurons with fixed values of the remaining parameters. When the internal EPSP size was less than a certain value, synchronization was absent. The amount of synchronization then increased slowly as the EPSP amplitude increased until at a particular EPSP size the amount of synchronization abruptly increased, with $S_5$ attaining the maximum value of 100%.
We also found network frequency transfer characteristics for various network sizes and found a linear dependence of firing frequency over wide ranges of the external afferent frequency, with nonlinear effects at lower input mnfrequencies. The theory may also be applied to sparsely connected networks whose firing behaviour was found to change abruptly as the probability of a connection passed through a critical value. The analytical method was also found to be useful for a feed-forward excitatory network and a network of excitatory and inhibitory neurons.
