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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Bifurcations and Dynamics in Modified Two Population Neuronal Network Models

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 237--257 | DOI:10.5890/DNC.2021.06.006

S. Roy Choudhury , Gizem S. Oztepe

Department of Mathematics, University of Central Florida, Orlando, FL32816, USA Department of Mathematics, Ankara University, Ankara, 06100, Turkey

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Abstract

A canonical modified two population neuronal network model of Laplace convolution type is considered via the 'linear chain trick'. Linear stability analysis of this system and conditions for Hopf bifurcation initiating spatiotemporal oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. For more steeply negative firing-rate functions, the Hopf bifurcations occur at larger values of both the delay and the inhibitory time constant. Other bifurcations such as double Hopf or generalized Hopf modes occurring from the homogeneous background state are also shown to be impossible for our model. In this first model, the Hopf-generated limit cycles turn out to be remarkably stable under very large variations of all four system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. Numerical simulations reveal strong distortion of the limit cycle shapes in phase space as the parameters are pushed far into the post-Hopf regime, and also reveal other features, such as the increase of the oscillation amplitudes of the physical variables on the limit cycle attractor, as well as decrease of their time periods, as both the delay and the inhibitory time constant are increased. The final section considers alternative Fourier convolution models with general functional forms for the synaptic connectivity functions. In particular, we develop an approach to derive the large variable or asymptotic behaviors in both space and time for arbitrary functional forms of the connectivity functions.

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