dorsal/arxiv
View SchemaSpeed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory
| Authors | Marc Timme, Theo Geisel, Fred Wolf |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0510046 |
| URL | https://arxiv.org/abs/q-bio/0510046 |
| DOI | 10.1063/1.2150775 |
| Journal | Chaos 16, 015108 (2006) |
Abstract
We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general non-standard solution to the multi-operator problem. Subsequently, we derive a class of rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate and fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distribution. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity: Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.
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"abstract": "We analyze the dynamics of networks of spiking neural oscillators. First, we\npresent an exact linear stability theory of the synchronous state for networks\nof arbitrary connectivity. For general neuron rise functions, stability is\ndetermined by multiple operators, for which standard analysis is not suitable.\nWe describe a general non-standard solution to the multi-operator problem.\nSubsequently, we derive a class of rise functions for which all stability\noperators become degenerate and standard eigenvalue analysis becomes a suitable\ntool. Interestingly, this class is found to consist of networks of leaky\nintegrate and fire neurons. For random networks of inhibitory\nintegrate-and-fire neurons, we then develop an analytical approach, based on\nthe theory of random matrices, to precisely determine the eigenvalue\ndistribution. This yields the asymptotic relaxation time for perturbations to\nthe synchronous state which provides the characteristic time scale on which\nneurons can coordinate their activity in such networks. For networks with\nfinite in-degree, i.e. finite number of presynaptic inputs per neuron, we find\na speed limit to coordinating spiking activity: Even with arbitrarily strong\ninteraction strengths neurons cannot synchronize faster than at a certain\nmaximal speed determined by the typical in-degree.",
"arxiv_id": "q-bio/0510046",
"authors": [
"Marc Timme",
"Theo Geisel",
"Fred Wolf"
],
"categories": [
"q-bio.NC",
"cond-mat.dis-nn"
],
"doi": "10.1063/1.2150775",
"journal_ref": "Chaos 16, 015108 (2006)",
"title": "Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory",
"url": "https://arxiv.org/abs/q-bio/0510046"
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