  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.
