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 brain connectivity.

翻译：现代技术的进步使得同时记录神经脉冲活动成为可能，这些活动在统计上可表示为多元点过程。我们通过谱密度矩阵（协方差矩阵的频域等价形式）来刻画该过程的二阶结构。在神经科学分析中，基于谱密度矩阵的统计量可用于推断单个神经元之间脑网络的连接性。然而，脉冲序列数据的高维特性往往导致这些统计量的计算极为困难，甚至在某些情况下无法实现。本文探讨了基于正则化方法的谱估计的重要性，并针对点过程场景提出了新型方法论。我们建立了所提出估计量的渐近性质，并在由多元霍克斯过程模拟的合成数据上评估其性能。最后，将所提出的方法应用于神经科学脉冲序列数据，以证明其在推断脑连接性方面的能力。