We study how the connectivity within a recurrent neural network determines and is determined by the multistable solutions of network activity. To gain analytic tractability we let neural activation be a non-smooth Heaviside step function. This nonlinearity partitions the phase space into regions with different, yet linear dynamics. In each region either a stable equilibrium state exists, or network activity flows to outside of the region. The stable states are identified by their semipositivity constraints on the synaptic weight matrix. The restrictions can be separated by their effects on the signs or the strengths of the connections. Exact results on network topology, sign stability, weight matrix factorization, pattern completion and pattern coupling are derived and proven. Our work may lay the foundation for multistability in more complex recurrent neural networks.

翻译：我们研究循环神经网络内的连接性如何决定网络活动的多稳态解，并受其决定。为获得解析可处理性，我们令神经激活函数为非光滑的亥维赛阶跃函数。这种非线性将相空间划分为具有不同但线性动力学的区域。在每个区域中，要么存在稳定的平衡态，要么网络活动流向区域外部。稳定状态通过其对突触权重矩阵的半正定性约束来识别。这些限制可根据其对连接符号或强度的影响进行区分。我们推导并证明了关于网络拓扑、符号稳定性、权重矩阵分解、模式完成和模式耦合的精确结果。我们的工作可为更复杂循环神经网络中的多稳态奠定基础。