Variational calculation of the limit cycle and its frequency in a two-neuron model with delay

Abstract

We consider a model system of two coupled Hopfield neurons, which is described by delay differential equations taking into account the finite signal propagation and processing times. When the delay exceeds a critical value, a limit cycle emerges via a supercritical Hopf bifurcation. First, we calculate its frequency and trajectory perturbatively by applying the Poincaré-Lindstedt method. Then, the perturbation series are resummed by means of the Shohat expansion in good agreement with numerical values. However, with increasing delay, the accuracy of the results from the Shohat expansion worsens. We thus apply variational perturbation theory (VPT) to the perturbation expansions to obtain more accurate results, which moreover hold even in the limit of large delays.

Distributed delays stabilize neural feedback systems

“…We are thus especially interested in the case where the coupling strengths a 1 and a 2 are of opposite sign. For a 1 a 2 ≤ −1, the fixed point at the origin is asymptotically stable as long as the mean of the time delays (τ 1 + τ 2 )/2 does not exceed a critical value τ 0 (Babcock and Westervelt 1987;Wei and Ruan 1999;Brandt et al 2006a):…”

“…To interpret the potential impact of the measured distribution of delays on the dynamics of neural feedback systems, we investigated a model system of two coupled Hopfield neurons (Hopfield 1984;Babcock and Westervelt 1987;Marcus and Westervelt 1989;Brandt et al 2006a, described by the first-order delay differential equations…”

“…In response to visual stimulation, the Ipc neurons undergo a transition from quiescence to rhythmic firing (Marin et al 2005(Marin et al , 2007. Delays can drive a neural feedback loop over a stability boundary resulting in oscillatory behavior (Babcock and Westervelt 1987;Marcus and Westervelt 1989;Laing and Longtin 2003;Brandt et al 2006a. To elucidate the impact of a delay distribution on the system dynamics, we investigated, through numerical simulations and mathematical analysis, a model of reciprocally coupled neurons with distributed delays.…”

Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations

“…Lindstedt's method, which ignores transient dynamics, is used in [34,35]. Center manifold reductions, used in [12,22,23,36], project the infinite dimensional system onto a plane and study the dynamics thereon.…”

Bifurcation analysis on a turning system with large and state-dependent time delay

“…A number of bifurcation analyses of the chatter vibration have been attempted by many researchers using the center manifold reduction method [31][32][33][34], Lindstedt's method [35,36], the MMS [37][38][39], etc. These studies mostly used the lower dimensional systems that reduced the original delayed (infinite-dimensional) systems to the finite dimensional ones by assuming the chatter vibration to be of a mono-frequency (i.e., chatter frequency).…”