Synchronization of coupled neutral-type neural networks with jumping-mode-dependent discrete and unbounded distributed delays

This is the post-print version of the Article. The official published version can be accessed from the links below - Copyright @ 2013 IEEE.In this paper, the synchronization problem is studied for an array of N identical delayed neutral-type neural networks with Markovian jumping parameters. The coupled networks involve both the mode-dependent discrete-time delays and the mode-dependent unbounded distributed time delays. All the network parameters including the coupling matrix are also dependent on the Markovian jumping mode. By introducing novel Lyapunov-Krasovskii functionals and using some analytical techniques, sufficient conditions are derived to guarantee that the coupled networks are asymptotically synchronized in mean square. The derived sufficient conditions are closely related with the discrete-time delays, the distributed time delays, the mode transition probability, and the coupling structure of the networks. The obtained criteria are given in terms of matrix inequalities that can be efficiently solved by employing the semidefinite program method. Numerical simulations are presented to further demonstrate the effectiveness of the proposed approach.This work was supported in part by the Royal Society of the U.K., the National Natural Science Foundation of China under Grants 61074129, 61174136 and 61134009, and the Natural Science Foundation of Jiangsu Province of China under Grants BK2010313 and BK2011598

Linear stability in networks of pulse-coupled neurons

In a first step towards the comprehension of neural activity, one should focus on the stability of the various dynamical states. Even the characterization of idealized regimes, such as 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: (i) a mean-field approach developed under the hypothesis of an infinite network and small synaptic conductances; (ii) a "microscopic" approach which applies to finite but large networks. As a result, we find that no matter how large is a neural network, its response to most of the perturbations depends on the system size. There exists, however, also a second class of perturbations, whose evolution typically covers an increasingly wide range of time scales. 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. The general validity of this scenarion is confirmed by numerical simulations of systems that are not amenable to a perturbative approach.Comment: 13 pages, 7 figures, submitted to Frontiers in Computational Neuroscienc

Frequency and phase locking of laser cavity solitons

Self-localized states or dissipative solitons have the freedom of translation in systems with a homogeneous background. When compared to cavity solitons in coherently driven nonlinear optical systems, laser cavity solitons have the additional freedom of the optical phase. We explore the consequences of this additional Goldstone mode and analyse experimentally and numerically frequency and phase locking of laser cavity solitons in a vertical-cavity surface-emitting laser with frequency-selective feedback. Due to growth-related variations of the cavity resonance, the translational symmetry is usually broken in real devices. Pinning to different defects means that separate laser cavity solitons have different frequencies and are mutually incoherent. If two solitons are close to each other, however, their interaction leads to synchronization due to phase and frequency locking with strong similarities to the Adler-scenario of coupled oscillators

Wavelength selection of rippling patterns in myxobacteria

Rippling patterns of myxobacteria appear in starving colonies before they aggregate to form fruiting bodies. These periodic traveling cell density waves arise from the coordination of individual cell reversals, resulting from an internal clock regulating them, and from contact signaling during bacterial collisions. Here we revisit a mathematical model of rippling in myxobacteria due to Igoshin et al.\ [Proc. Natl. Acad. Sci. USA {\bf 98}, 14913 (2001) and Phys. Rev. E {\bf 70}, 041911 (2004)]. Bacteria in this model are phase oscillators with an extra internal phase through which they are coupled to a mean-field of oppositely moving bacteria. Previously, patterns for this model were obtained only by numerical methods and it was not possible to find their wavenumber analytically. We derive an evolution equation for the reversal point density that selects the pattern wavenumber in the weak signaling limit, show the validity of the selection rule by solving numerically the model equations and describe other stable patterns in the strong signaling limit. The nonlocal mean-field coupling tends to decohere and confine patterns. Under appropriate circumstances, it can annihilate the patterns leaving a constant density state via a nonequilibrium phase transition reminiscent of destruction of synchronization in the Kuramoto model.Comment: Revtex 26 pages, 7 figure

Coupled Oscillators on a Circle

We consider a continuum of diffusively coupled oscillators on a circle. When each oscillator is of Lienard type, very little is known about the corresponding hyperbolic POE. When each oscillator is represented by a lossless transmission line, we obtain a partial neutral delay differential equation and give the beginnings of a qualitative theory for the dynamics. In particular, we discuss the properties of the solution map, the existence of the global attractor, behavior near an equilibrium point including the Hopf bifurcation, and some elementary properties near a periodic orbit