Detecting causality between different frequencies

Biological systems are usually non-linear and, as a result, the driving signal frequency (say, M Hz) is in general not identical with the output frequency (say, N Hz). Coherence and causality analysis have been well-developed to measure the (directional) correlation between input and output signals with identical frequencies (N=M), but they are not applicable to the cases with different frequencies (N≠M). In this paper, we propose a novel method called frequency-modified causality (coherence) analysis to resolve the issue. The input or output signal is first modulated by up-sampling or down-sampling, coherence and causality analysis are then applied to the frequency modulated and filtered signals. An optimal coherence and causality is found, revealing the true input-output relationship between signals. The method is successfully tested on data generated from a toy model, the van der Pol oscillator and then employed to analyze data recorded from Parkinson's disease (PD) patients.