2013/11/12 by Simona Olmi, Antonio Politi, Alessandro Torcini
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #Artificial intelligence #Artificial neural network #Asynchronous communication #Computer science #Dynamical systems theory #Field (mathematics) #Focus (optics) #Instability #Linear stability #Mathematical analysis #Mathematics #Neural dynamics and brain function #Nonlinear Dynamics and Pattern Formation #Optics #Phase (matter) #Phase response #Physics #Pulse (music) #Pure mathematics #Quantum mechanics #Series (stratigraphy) #Smoothness #Stability (learning theory) #Statistical physics #Topology (electrical circuits) #cond-mat.dis-nn #physics.bio-ph #q-bio.NC #stochastic dynamics and bifurcation
paper · pdf · doi:10.3389/fncom.2014.00008
published as Front. Comput. Neurosci. 8:8 (2014) · 13 pages, 7 figures, submitted to Frontiers in Computational Neuroscience
arxiv created 2013/11/12 · openalex publication_date 2014/01/01 · arxiv updated 2014/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In a first step toward the comprehension of neural activity, one should focus on the stability of the possible dynamical states. Even the characterization of an idealized regime, such as that of a perfectly periodic spiking activity, reveals unexpected difficulties. In this paper we discuss a general approach to linear stability of pulse-coupled neural networks for generic phase-response curves and post-synaptic response functions. In particular, we present: (1) a mean-field approach developed under the hypothesis of an infinite network and small synaptic conductances; (2) a "microscopic" approach which applies to finite but large networks. As a result, we find that there exist two classes of perturbations: those which are perfectly described by the mean-field approach and those which are subject to finite-size corrections, irrespective of the network size. The analysis of perfectly regular, asynchronous, states reveals that their stability depends crucially on the smoothness of both the phase-response curve and the transmitted post-synaptic pulse. Numerical simulations suggest that this scenario extends to systems that are not covered by the perturbative approach. Altogether, we have described a series of tools for the stability analysis of various dynamical regimes of generic pulse-coupled oscillators, going beyond those that are currently invoked in the literature.