I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular network. This matrix is close to a random one and Girko's circular law applies partly, in the sense that the bulk of the eigenvalues lies in a circle when you plot their Imag/Real parts; some eigenvalues will be outside the bulk because of the network's modularity.Moreover, my system is nonlinear since it is composed of neurons, whose population activity's transfer function is close to a sigmoid (2). That is Frequency/Input in continuous time. Can I derive some conclusions about my system's stability (locally at least) based on the eigenvalues? For instance, if I have a large real eigenvalue, my system could prove to be unstable, with exponentially increased activity over time..
Locally evaluate nonlinear dynamic system's stability using eigenvalues
linear algebra;eigenvalues;stability;neuroscience