RESUMEN
We study the noisy FitzHugh-Nagumo model, representative of the dynamics of excitable neural elements, and derive a Fokker-Planck equation for both a single element and for a network of globally coupled elements. We introduce an efficient way to numerically solve this Fokker-Planck equation, especially for large noise levels. We show that, contrary to the single element, the network can undergo a Hopf bifurcation as the coupling strength is increased. Furthermore, we show that an external sinusoidal driving force leads to a classical resonance when its frequency matches the underlying system frequency. This resonance is also investigated analytically by exploiting the different time scales in the problem.
RESUMEN
We study a system of globally coupled two-dimensional nonlinear oscillators [using the two-junction superconducting quantum interference device (SQUID) as a prototype for a single element] each of which can undergo a saddle-node bifurcation characterized by the disappearance of the stable minima in its potential energy function. This transition from fixed point solutions to spontaneous oscillations is controlled by external bias parameters, including the coupling coefficient. For the deterministic case, an extension of a center-manifold reduction, carried out earlier for the single oscillator, yields an oscillation frequency that depends on the coupling; the frequency decreases with coupling strength and/or the number of oscillators. In the presence of noise, a mean-field description leads to a nonlinear Fokker-Planck equation for the system which is investigated for experimentally realistic noise levels. Furthermore, we apply a weak external time-sinusoidal probe signal to each oscillator and use the resulting (classical) resonance to determine the underlying frequency of the noisy system. This leads to an explanation of earlier experimental results as well as the possibility of designing a more sensitive SQUID-based detection system.
RESUMEN
We investigate a discrete model consisting of self-propelled particles that obey simple interaction rules. We show that this model can self-organize and exhibit coherent localized solutions in one- and in two-dimensions. In one-dimension, the self-organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges. In two-dimensions, we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions, even in the absence of a confining boundary. Furthermore, we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent.
Asunto(s)
Conducta Animal , Movimiento Celular , Modelos Biológicos , Animales , Aves , Simulación por Computador , Peces , Vuelo Animal , Cómputos MatemáticosRESUMEN
We investigate dendritic sidebranching during crystal growth in an undercooled melt by simulation of a phase-field model which incorporates thermal noise of microscopic origin. As a nontrivial quantitative test of this model, we first show that the simulated fluctuation spectrum of a one-dimensional interface in thermal equilibrium agrees with the exact sharp-interface spectrum up to an irrelevant short-wavelength cutoff comparable to the interface thickness. Simulations of dendritic growth are then carried out in two dimensions to compute sidebranching characteristics (root-mean-square amplitude and sidebranch spacing) as a function of distance behind the tip. These quantities are compared quantitatively to the predictions of the existing linear WKB theory of noise amplification. The extension of this study to three dimensions remains needed to determine the origin of noise in experiments.