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doi: 10.3934/era.2021022
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## Complexity in time-delay networks of multiple interacting neural groups

 Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China

* Corresponding author: Xiaochen Mao

Received  December 2020 Revised  January 2021 Early access March 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China under Grant Nos. 11872169 and 11472097, Fundamental Research Funds for the Central Universities of China under Grant No. B200202114, and Natural Science Foundation of Jiangsu Province of China under Grant No. BK20191295

Coupled networks are common in diverse real-world systems and the dynamical properties are crucial for their function and application. This paper focuses on the behaviors of a network consisting of mutually coupled neural groups and time-delayed interactions. These interacting groups can include different sets of nodes and topological architecture, respectively. The local and global stability of the system are analyzed and the stable regions and bifurcation curves in parameter planes are obtained. Different patterns of bifurcated solutions arising from trivial and non-trivial equilibrium points are given, such as the coexistence of non-trivial equilibrium points and periodic responses and multiple coexisting periodic orbits. The bifurcation diagrams are shown and plenty of complex dynamic phenomena are observed, such as multi-period oscillations and multiple coexisting attractors.

Citation: Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, doi: 10.3934/era.2021022
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##### References:
Stability region of three delay-coupled networks with different nodes
Stability region of four delay-coupled networks, where the solid curves are $1-\alpha-2\alpha^2-\beta_1+\alpha\beta_1 = 0$ and $1-\alpha-2\alpha^2+\beta_1-\alpha\beta_1 = 0$
Amplitudes of the periodic oscillations when the sum of coupling time delays varies. (a) $c_1 = -1.4$, $c_2 = 0.4$, $c_3 = 2.5$, $\tau_1 = 0.3$, and $\tau_2 = 0.5$; (b) $c_1 = 1.5$, $c_2 = 0.8$, $c_3 = 1.4$, $\tau_1 = 0.1$, and $\tau_2 = 0.1$
Stability region of the trivial equilibrium of three coupled networks with different topologies. (a) $a = 0.4$; (b) $c = -1.3$
Responses of interacting three-node networks with different topologies. (a) $\tau_s = 1.5$; (b) $\tau_s = 1.7$; (c) $\tau_s = 2.1$
Phase trajectories of the network when $\tau_s = 0.17$. (a) Two coexisting period-6 oscillations under initial conditions IC1 (0.1, 0.2, 0.5, 0.8, 0.3, 0.7, 0.9, 0.4, 0.6) and IC2 (-0.1, -0.2, -0.5, -0.8, -0.3, -0.7, -0.9, -0.4, -0.6), respectively; (b) Two chaotic motions under initial conditions IC3 (-0.1, 0.2, 0.5, 0.8, 0.3, 0.7, -0.9, 0.4, 0.6) and IC4 (0.1, -0.2, -0.5, -0.8, -0.3, -0.7, 0.9, -0.4, -0.6), respectively. The blue, red, green, and purple curves represent phase trajectories under initial conditions IC1, IC2, IC3, and IC4, respectively
Phase trajectories of the network when $\tau_s = 0.25$. (a) Two period-3 oscillations under initial conditions IC1 and IC2, respectively; (b) Two chaotic motions under initial conditions IC3 and IC4, respectively
Phase trajectories of the network when $\tau_s = 0.43$. (a) Two separated chaotic attractors under initial conditions IC1 and IC2, respectively; (b) Two coexisting period-4 orbits under initial conditions IC3 and IC4, respectively
Phase trajectories of the network when $\tau_s = 0.47$. (a) Two chaotic responses under initial conditions IC1 and IC2, respectively; (b) Two symmetric period-2 oscillations under initial conditions IC3 and IC4, respectively
Phase trajectories of the network when $\tau_s = 1.73$. (a) A pair of period-4 responses under initial conditions IC1 and IC2, respectively; (b) Another pair of period-4 oscillations under initial conditions IC3 and IC4, respectively
Bifurcation diagrams on the Poincaré section when the time delay varies. The blue, red, green, and purple dots are obtained under the initial conditions IC1, IC2, IC3, and IC4, respectively
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