AI 中文总结
研究具有分布式延迟的θ神经元网络，通过渡边-斯特罗加茨约化等方法推导出复序参量的延迟微分方程，得到平衡点稳定性准则和霍普夫分岔条件，数值模拟验证结果，揭示延迟对网络稳定性的多种影响。
AI 中文摘要
我们研究了一个由分布式时间延迟介导突触相互作用的全连接相同θ神经元网络。利用渡边-斯特罗加茨约化，并在运动常数均匀分布的假设下取热力学极限，我们推导出了复序参量的单个延迟微分方程。延迟由具有规定平均延迟的一族延迟核建模，使得离散和分布式延迟能在统一框架中处理。约化系统的平衡点可分为两个几何上不同的族：单位圆上的1型平衡点和实轴上的2型平衡点。对于这两个族，局部稳定性问题归结为涉及延迟核拉普拉斯-斯蒂尔杰斯变换的标量特征方程。我们得到了可允许核的稳定性准则以及狄拉克核的显式霍普夫分岔条件，并与弱和强伽马核进行了额外比较。结果表明，延迟可能保持稳定性、使平衡点不稳定或产生稳定性切换，这取决于平衡分支、参数区域和核的选择。离散延迟情况的数值模拟支持了分析结果，并展示了相应的相图、吸引盆、吸引子共存和延迟诱导的周期动力学。
英文摘要
We study an all-to-all coupled network of identical theta neurons with synaptic interaction mediated by a distributed time delay. Using the Watanabe-Strogatz reduction and passing to the thermodynamic limit under the assumption of uniformly distributed constants of motion, we derive a single delay differential equation for the complex order parameter. The delay is modeled by a family of delay kernels with prescribed mean delay, allowing discrete and distributed delays to be treated in a unified framework. The equilibria of the reduced system can be classified into two geometrically distinct families: type 1 equilibria on the unit circle and type 2 equilibria on the real axis. For both families, the local stability problem reduces to scalar characteristic equations involving the Laplace-Stieltjes transform of the delay kernel. We obtain stability criteria for admissible kernels and explicit Hopf bifurcation conditions for the Dirac kernel, with additional comparison to weak and strong Gamma kernels. The results show that the delay may either preserve stability, destabilize equilibria, or produce stability switching, depending on the equilibrium branch, parameter regime, and choice of kernel. Numerical simulations for the discrete-delay case support the analytical results and illustrate the corresponding phase portraits, basins of attraction, coexistence of attractors, and delay-induced periodic dynamics.
Comments32 pages, 10 figures
