  Granger causality is well established within the neurosciences for inference
of directed functional connectivity from neurophysiological data. These data
usually consist of time series which subsample a continuous-time
biophysiological process. While it is well-known that subsampling can lead to
imputation of spurious causal connections where none exist, here we address the
equally important issue of the effects of subsampling on the ability to
reliably detect causal connections which do exist.
  Neurophysiological processes typically feature signal propagation delays on
multiple time scales; accordingly, we base our analysis on a distributed-lag,
continuous-time stochastic model, and consider Granger causality in continuous
time at finite prediction horizons. Via exact analytical solutions, we identify
relationships among sampling frequency, underlying causal time scales and
detectability of causalities. Our analysis reveals complex interactions between
the time scale(s) of neural signal propagation and sampling frequency: we
demonstrate that Granger causality decays exponentially as the sample time
interval increases beyond causal delay times, identify detectability "black
spots" and "sweet spots", and show that subsampling may sometimes improve
detectability. We also demonstrate that the invariance of Granger causality
under causal, invertible filtering fails at finite prediction horizons. We
discuss the implications of our results for inference of Granger causality at
the neural level from various neurophysiological recording modes, and emphasise
that sampling rates for causal analysis of neurophysiological time series
should be informed by domain-specific time scales.
