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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..

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    $\begingroup$ Tens of thousands of papers have been written about the stability of nonlinear ODE systems. You provide too little detail here about what exactly your system is, but in any case I suspect that this is more of a mathematical than a Computational Science question. $\endgroup$ Commented Apr 22, 2015 at 19:44
  • $\begingroup$ It is a mathematical question out of a computational neuroscience project. I included a link of the synaptic matrix and a clarification the continuity of the transfer function. Do you require more details to answer the question, or are you suggesting that this forum is not suitable at all for it? $\endgroup$
    – kalfasyan
    Commented Apr 22, 2015 at 19:56
  • $\begingroup$ The latter. But who knows, maybe someone with the right knowledge is around here :-) $\endgroup$ Commented Apr 23, 2015 at 11:28

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Some brief comments, since this is an old question.

  • Some results on theoretical examination of eigenspectra of random matrices for neural networks: (Rajan & Abbott 2006, Muir & Mrsic-Flogel 2015). Much of random matrix theory doesn't apply however, since graphs of realistic networks are non-Hermitian and can be quite constrained.
  • In general you can certainly use results from linear systems analysis to examine stability of the network. However, the results will only hold while the system is operating in a linear way — while the activity of any neuron in the active partition is far from saturation, and while the active partition does not change (i.e. no active neuron drops below threshold).
  • However, you can analyse each active partition separately (see Hahnloser 1998).
  • Instability will lead to saturation of firing rates, since you are using a saturating transfer function.
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