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Keywords:

  • Bivariate time series;
  • cross-coherence;
  • dual frequency coherence;
  • coherence between amplitudes;
  • Fourier transform;
  • harmonizable processes;
  • lagged coherence;
  • Loève spectrum;
  • spectral analysis

Our goal is to investigate dependence between brain wave oscillations in the nucleus accumbens (NAc) and the hippocampus (Hc) regions of a macaque monkey during a learning association task. The classical approach to studying dependence in the spectral domain is via cross-coherence. It is computed for each frequency (or band) and identifies the frequency bands that drive the linear association between the components in a multi-variate time series. However, cross-coherence may not fully capture the complex dependence structure in brain signals such as local field potentials. In this article, we develop new tools for discovering associations between the theta (4–8 Hz) and gamma (32–50 Hz) activities at both contemporaneous blocks and lagged time blocks. We propose a class of piecewise harmonizable processes under which we give a precise definition of these dependence measures and develop simple estimators in the case where the time-series data are recorded over several replicated trials. Our analysis clearly demonstrates strong dependence between the theta and gamma oscillations in the NAc and the Hc regions of a macaque monkey during learning. Moreover, we determined the lagged dependence that differentiate the ‘correct’ responses (i.e., the monkey was able to identify the correct association) from the ‘incorrect’ responses.