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

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 nonstandard solution to the multioperator problem. Subsequently, we derive a class of neuronal 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 distributions of the stability operators. 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.The individual units of many physical systems, from the planets of our solar system to the atoms in a solid, typically interact continuously in time and without significant delay. Thus at every instant of time such a unit is influenced by the current state of its interaction partners. Moreover, particles of many-body systems are often considered to have very simple lattice topology (as in a crystal) or no prescribed topology at all (as in an ideal gas). Many important biological systems are drastically different: their units are interacting by sending and receiving pulses at discrete instances of time. Furthermore, biological systems often exhibit significant delays in the couplings and very complicated topologies of their interaction networks. Examples of such systems include neurons, which interact by stereotyped electrical pulses called action potentials or spikes; crickets, which chirp to communicate acoustically; populations of fireflies that interact by short light pulses. The combination of pulse-coupling, delays, and complicated network topology formally makes the dynamical system to be investigated a high dimensional, heterogeneous nonlinear hybrid system with delays. Here we present an exact analysis of aspects of the dynamics of such networks in the case of simple one-dimensional nonlinear interacting units. These systems are simple models for the collective dynamics of recurrent networks of spiking neurons. After briefly presenting stability results for the synchronous state, we show how to use the theory of random matrices to analytically predict the eigenvalue distribution of stability matrices and thus derive the speed of synchronization in terms of dynamical and network parameters. We find that networks of neural oscillators typically exhibit speed limits and cannot synchronize faster than a certain bound defined by the network topology.