  Understanding causal relationships within a system is crucial for uncovering
its underlying mechanisms. Causal discovery methods, which facilitate the
construction of such models from time-series data, hold the potential to
significantly advance scientific and engineering fields.
  This study introduces the Cross-Mapping Coherence (CMC) method, designed to
reveal causal connections in the frequency domain between time series. CMC
builds upon nonlinear state-space reconstruction and extends the Convergent
Cross-Mapping algorithm to the frequency domain by utilizing coherence metrics
for evaluation. We tested the Cross-Mapping Coherence method using simulations
of logistic maps, Lorenz systems, Kuramoto oscillators, and the Wilson-Cowan
model of the visual cortex. CMC accurately identified the direction of causal
connections in all simulated scenarios. When applied to the Wilson-Cowan model,
CMC yielded consistent results similar to spectral Granger causality.
  Furthermore, CMC exhibits high sensitivity in detecting weak connections,
demonstrates sample efficiency, and maintains robustness in the presence of
noise.
  In conclusion, the capability to determine directed causal influences across
different frequency bands allows CMC to provide valuable insights into the
dynamics of complex, nonlinear systems.
