Multivariate Hawkes processes are past-dependant point processes originally introduced to model excitation effects, later extended to a nonlinear framework to account for the opposite effect, known as inhibition. Motivated by applications in neuroscience, where the memory of a neuron may reset upon firing, we introduce a new class of nonlinear Hawkes processes with variable length memory. Our model generalises classical Hawkes processes, with or without inhibition, describing the situation where the probability of an event occurring within a given subprocess may depend differently on the history before and after its last event. In particular, if the subprocess does not depend on the history before its last event, it is said to have a variable length memory. Our main contributions are to prove existence of such processes, and to derive a workable likelihood maximisation method, capable of identifying both classical and variable memory dynamics. We demonstrate the effectiveness of our approach both on synthetic data, and on a neuronal activity dataset.

翻译：多元霍克斯过程是依赖于过去历史的点过程，最初用于建模激发效应，后扩展至非线性框架以描述相反效应（即抑制）。受神经科学应用启发——其中神经元的记忆可能在放电后重置——我们引入了一类具有可变长度记忆的新型非线性霍克斯过程。我们的模型概括了经典霍克斯过程（含或不含抑制），描述了子过程中事件发生概率可能对最后一次事件前后的历史产生不同依赖的情况。特别地，若子过程不依赖于最后一次事件之前的记忆，则称其具有可变长度记忆。我们的主要贡献在于证明此类过程的存在性，并提出一种可行的似然最大化方法，能够同时识别经典记忆动态与可变记忆动态。我们在合成数据及神经元活动数据集上验证了该方法的有效性。