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Bifurcation structure of two coupled FHN neurons with delay

Farajzadeh Tehrani, N ; Sharif University of Technology | 2015

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  1. Type of Document: Article
  2. DOI: 10.1016/j.mbs.2015.09.008
  3. Publisher: Elsevier Inc , 2015
  4. Abstract:
  5. This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified
  6. Keywords:
  7. Delay differential equation ; Fold of limit cycles ; Hopf-Pitchfork bifurcation ; Asymptotic stability ; Bifurcation (mathematics) ; Delta sigma modulation ; Differential equations ; Dynamical systems ; Neural networks ; Time delay ; Delay differential equations ; Double Hopf bifurcation ; Hopf-pitchfork bifurcations ; Limit-cycle ; Neural modeling ; Torus bifurcation ; Hopf bifurcation ; Bifurcation ; Detection method ; Mathematical analysis ; Numerical model ; Artificial neural network ; FitzHugh nagumo neural model ; Linear system ; Mathematical computing ; mathematical model ; Neimark Sacker bifurcation ; Nonlinear system ; Prediction ; Process development ; Simulation ; Statistical model
  8. Source: Mathematical Biosciences ; Volume 270 , 2015 , Pages 41-56 ; 00255564 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0025556415001959