ODE file for simulation in XPPAUT from Geometric analysis of synchronization in neuronal networks with global inhibition and coupling delays

We study synaptically coupled neuronal networks to identify the role of coupling delays in network synchronized behaviour. We consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory are coupled with time-delayed synapses. The excitatory population is uncoupled, while the inhibitory population is tightly coupled without time delay. A geometric singular perturbation analysis yields existence and stability conditions for periodic solutions where the excitatory cells are synchronized and different phase relationships between the excitatory and inhibitory populations can occur, along with formulae for the periods of such solutions. In particular, we show that if there are no delays in the coupling oscillations where the excitatory population is synchronized cannot occur. Numerical simulations are conducted to supplement and validate the analytical results. The analysis helps to explain how coupling delays in either excitatory or inhibitory synapses contribute to producing synchronized rhythms.This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

本研究围绕突触耦合神经元网络展开，旨在厘清耦合延迟对网络同步行为的调控机制。本研究考虑一类由可兴奋弛豫振荡器神经元构成的网络，其中包含两类功能迥异的神经元群：兴奋性群与抑制性群，二者通过带时延的突触实现连接。其中兴奋性神经元群无耦合连接，而抑制性神经元群则实现了无时延的紧密耦合。通过几何奇异摄动分析，本研究推导得到了兴奋性细胞同步的周期解的存在性与稳定性条件，明确了兴奋性与抑制性群之间可存在的多种相位关系，并给出了此类周期解的周期计算公式。特别地，本研究证明：当耦合无时延时，兴奋性群的同步振荡将无法出现。本研究通过数值模拟对解析结果进行补充与验证。本分析有助于阐释兴奋性或抑制性突触中的耦合延迟如何助力同步节律的产生。本文为「延迟系统的非线性动力学」专题特刊的组成部分。