Advances in modern technology have enabled the simultaneous recording of neural spiking activity, which statistically can be represented by a multivariate point process. We characterise the second order structure of this process via the spectral density matrix, a frequency domain equivalent of the covariance matrix. In the context of neuronal analysis, statistics based on the spectral density matrix can be used to infer connectivity in the brain network between individual neurons. However, the high-dimensional nature of spike train data mean that it is often difficult, or at times impossible, to compute these statistics. In this work, we discuss the importance of regularisation-based methods for spectral estimation, and propose novel methodology for use in the point process setting. We establish asymptotic properties for our proposed estimators and evaluate their performance on synthetic data simulated from multivariate Hawkes processes. Finally, we apply our methodology to neuroscience spike train data in order to illustrate its ability to infer connectivity in the brain network.

翻译：现代技术的进步使得同时记录神经尖峰活动成为可能，这在统计学上可以用多元点过程来表示。我们通过谱密度矩阵（协方差矩阵在频域中的对应量）来刻画该过程的二阶结构。在神经分析中，基于谱密度矩阵的统计量可用于推断脑网络中单个神经元之间的连接性。然而，尖峰序列数据的高维特性往往导致这些统计量的计算困难，甚至有时无法实现。本文讨论了基于正则化方法进行谱估计的重要性，并提出了适用于点过程场景的新方法。我们建立了所提估计量的渐近性质，并在多元霍克斯过程模拟的合成数据上评估其性能。最后，将所提方法应用于神经科学尖峰序列数据，以展示其推断脑网络连接性的能力。