Stability of synchronous states in sparse neuronal networks
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ORIGINAL PAPER
Stability of synchronous states in sparse neuronal networks Afifurrahman · Ekkehard Ullner
· Antonio Politi
Received: 3 July 2020 / Accepted: 3 August 2020 © The Author(s) 2020
Abstract The stability of synchronous states is analyzed in the context of two populations of inhibitory and excitatory neurons, characterized by two different pulse-widths. The problem is reduced to that of determining the eigenvalues of a suitable class of sparse random matrices, randomness being a consequence of the network structure. A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the overall stability depends crucially on the relative pulsewidth. This has implications for the overall property of the asynchronous (balanced) regime. Keywords Stability analysis · Synchronization · Neuronal network · Sparse network
1 Introduction Networks of oscillators are widely studied in many fields: mechanical engineering [1,2], power grids [3], arrays of Josephson junctions [4], cold atoms [5], neuAfifurrahman · E. Ullner (B)· A. Politi Institute for Pure and Applied Mathematics and Department of Physics (SUPA), Old Aberdeen, Aberdeen AB24 3UE, UK e-mail: [email protected] Afifurrahman e-mail: [email protected] A.Politi e-mail: [email protected]
ral networks [6], and so on. Such networks can be classified according to the single-unit dynamics, the coupling mechanism, the presence of heterogeneity, and network topology. Since phases are the most sensitive variables to any kind of perturbation [7], most of the attention is devoted to setups composed of phase oscillators [8], i.e., one-dimensional dynamical systems. However, even the study of such relatively simple models is not as straightforward as it might appear. In fact, a wide variety of dynamical regimes can emerge even in mean field models of identical oscillators, ranging from full synchrony to splay states, and including hybrids, such as partial synchronization [9], chimera [10], and cluster states [11]. General theory of synchronization is, therefore, a much investigated field. In this paper, we focus on synchronous states by referring to a rather popular class of neural networks, but the whole formalism can be easily extended to more general systems so long as the coupling is mediated by the emission of pulses. In neuroscience, the neuron dynamics is often described by a single variable, the membrane potential, which evolves according to a suitable velocity field. The resulting model is equivalent to a phase oscillator, where the variable of the bar system increases linearly in time and the complexity of the evolution rule is encoded in the phase response curve (PRC), which accounts for the mutual coupling [12]. Under the additional approximation of a weak coupling strength, the model can be further simplified and cast into a Kuramoto-Daido form, where the coupling depends on phase differences between pairs of
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oscillators [13,14]. Here, however, w
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