2,741 research outputs found
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
Synchronization of coupled neural oscillators with heterogeneous delays
We investigate the effects of heterogeneous delays in the coupling of two
excitable neural systems. Depending upon the coupling strengths and the time
delays in the mutual and self-coupling, the compound system exhibits different
types of synchronized oscillations of variable period. We analyze this
synchronization based on the interplay of the different time delays and support
the numerical results by analytical findings. In addition, we elaborate on
bursting-like dynamics with two competing timescales on the basis of the
autocorrelation function.Comment: 18 pages, 14 figure
Deterministic Brownian motion generated from differential delay equations
This paper addresses the question of how Brownian-like motion can arise from
the solution of a deterministic differential delay equation. To study this we
analytically study the bifurcation properties of an apparently simple
differential delay equation and then numerically investigate the probabilistic
properties of chaotic solutions of the same equation. Our results show that
solutions of the deterministic equation with randomly selected initial
conditions display a Gaussian-like density for long time, but the densities are
supported on an interval of finite measure. Using these chaotic solutions as
velocities, we are able to produce Brownian-like motions, which show
statistical properties akin to those of a classical Brownian motion over both
short and long time scales. Several conjectures are formulated for the
probabilistic properties of the solution of the differential delay equation.
Numerical studies suggest that these conjectures could be "universal" for
similar types of "chaotic" dynamics, but we have been unable to prove this.Comment: 15 pages, 13 figure
Fractional Calculus - Theory and Applications
In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications
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