Temporally Smoothed Wavelet Coherence for Multivariate Point-Processes and Neuron-Firing

In neuroscience, it is of key importance to assess how neurons interact with each other as evidenced via their firing patterns and rates. We here introduce a method of smoothing the wavelet periodogram (scalogram) in order to reduce variance in spectral estimates and allow analysis of tune-varying dependency between neurons at different scale levels. Previously such smoothing methods have only received analysis in the setting of regular real-valued (Gaussian) time series. However, in the context of neuron-firing, observations may be modelled as a point-process which when binned, or aggregated, gives rise to an integer-valued time-series. In this paper we propose an analytical asymptotic distribution for the smoothed wavelet spectra, and then contrast this, via synthetic experiments, with the finite sample behaviour of the spectral estimator. We generally find good alignment with the asymptotic distribution, however, this may break down if the level of smoothing, or the scale under analysis is very small. To conclude, we demonstrate how the spectral estimator can be used to characterize real neuron-firing dependency, and how such relationships vary over time and scale.