Stability and synchronization of random brain networks with a distribution of connection strengths

The impact of stability and synchronization of electrical activity on the structure of random brain networks with a distribution of connection strengths is investigated using a physiological model of brain activity. Connection strength is measured by the gain of the connection, which describes the effect of changes in the firing rate of neurons in one component on the neurons of another component. The stability of a network is calculated from the eigenvalue spectrum of the network's matrix of gains. Using random matrix theory, we predict and numerically verify the eigenvalue spectrum of randomly connected networks with gain values determined by a probability distribution. From the eigenvalue spectrum, the probability that a network is stable is calculated and shown to constrain the structural and physiological parameters of the network. In particular, stability constrains the variance of the gains. The complex vector of component amplitudes, or mode, corresponding to each dispersion root is an eigenvector of the network's gain matrix and is used to calculate the synchronization of each component's electrical activity. Synchronization is shown to decrease as the variance of the connection gain increases and inhibitory connections become more likely. Brain networks with large gain variance are shown to have multiple eigenvalues close to the stability boundary and to be partially synchronized. Such a network would have multiple partially synchronized modes strongly excited by a stimulus.