Variational calculation of the limit cycle and its frequency in a two-neuron model with delay.
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.
Brandt
Sebastian F
SF
Department of Physics, Campus Box 1105, Washington University in St. Louis, Missouri 63130-4899, USA. sbrandt@physics.wustl.edu
Pelster
Axel
A
Wessel
Ralf
R
D000818:Animals; D001703:Biophysics / Q000379:methods; D003198:Computer Simulation; D025461:Feedback, Physiological; D008433:Mathematics; D008959:Models, Neurological; D015233:Models, Statistical; D008962:Models, Theoretical; D016571:Neural Networks, Computer; D009434:Neural Pathways; D009474:Neurons / Q000378:metabolism*; D009716:Numerical Analysis, Computer-Assisted; D009435:Synaptic Transmission; D013997:Time Factors