Speed of synchronization in complex networks of neural oscillators: Analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, wepresent an exact linear stability theory of the synchronous state for networksof arbitrary connectivity. For general neuron rise functions, stability isdetermined 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 stabilityoperators become degenerate and standard eigenvalue analysis becomes a suitabletool. Interestingly, this class is found to consist of networks of leakyintegrate and fire neurons. For random networks of inhibitoryintegrate-and-fire neurons, we then develop an analytical approach, based onthe theory of random matrices, to precisely determine the eigenvaluedistribution. This yields the asymptotic relaxation time for perturbations tothe synchronous state which provides the characteristic time scale on whichneurons can coordinate their activity in such networks. For networks withfinite in-degree, i.e. finite number of presynaptic inputs per neuron, we finda speed limit to coordinating spiking activity: Even with arbitrarily stronginteraction strengths neurons cannot synchronize faster than at a certainmaximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao