SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1177-1208, January 2021.
Threshold-linear networks (TLNs) are recurrent networks where the dynamics are threshold-linear (linearly rectified at zero). Mathematically, they consist of coupled nonsmooth ordinary differential equations. When the nodes in the network are assumed to be neurons or neuronal populations, TLNs represent firing rate models. We investigate the dynamics of a subclass of TLNs referred to as competitive TLNs where all the connections between different nodes are inhibitory. We prove the existence of periodic solutions in competitive TLNs with three nodes using a combination of mathematical analysis and numerical simulations. We calculate the analytical expressions of the periodic solutions, then we consider a reduced system of transcendental equations and apply a Kantorovich's convergence result to demonstrate the existence of these solutions. We then analyze the attributes (frequency and amplitude) of these periodic solutions as the model parameters vary. Finally, we study the entrainment properties of competitive TLNs in the oscillatory regime, by examining their response to external periodic inputs to one of the nodes in the network. We numerically determine the ranges of input amplitudes and frequencies for which competitive TLNs are able to follow the periodic input for three-node networks and larger networks with cyclic symmetry.

中文翻译：

---

阈值线性网络中的周期解及其夹带

SIAM Journal on Applied Dynamical Systems，第 20 卷，第 3 期，第 1177-1208 页，2021 年 1 月。
阈值线性网络 (TLN) 是循环网络，其中动态是阈值线性的（线性校正为零）。在数学上，它们由耦合的非光滑常微分方程组成。当网络中的节点被假定为神经元或神经元群时，TLN 代表放电率模型。我们研究了被称为竞争性 TLN 的 TLN 子类的动态，其中不同节点之间的所有连接都是抑制性的。我们使用数学分析和数值模拟的组合证明了具有三个节点的竞争性 TLN 中周期性解的存在。我们计算周期解的解析表达式，然后我们考虑一个简化的超越方程组并应用Kantorovich 的收敛结果来证明这些解的存在。然后，随着模型参数的变化，我们分析这些周期解的属性（频率和幅度）。最后，我们通过检查它们对网络中一个节点的外部周期性输入的响应来研究振荡状态下竞争性 TLN 的夹带特性。我们在数值上确定了输入幅度和频率的范围，在这些范围内，竞争性 TLN 能够遵循三节点网络和具有循环对称性的更大网络的周期性输入。