Correlation measures the simultaneous coupling between two time series, while coherency can be thought of as a spectral analog of correlation and is defined as
where 〈
f x(
ν)〉 and 〈
f y(
ν)〉 are the means of the power spectra of segments of time series
x and
y at frequency
ν (see
Coherency calculations section below), and 〈
f xy(
ν)〉 is the mean of the cross spectrum of segments of
x and
y (Sun et al.,
2004).
Rxy(
ν) is a complex quantity (
a +
ib). In polar coordinates, the absolute value (modulus) is the length of the vector
Rxy(
ν),
, and this corresponds to the magnitude of coupling between
x and
y (taking a value between 0 and 1). The complex argument, or polar angle, of
Rxy(
ν) defines the phase spectrum. Dividing this phase spectrum by a given temporal frequency yields the delay between
x and
y in units of time (taking positive or negative values, depending on the direction of the delay between the time series). Thus, coherency analysis allows the measurement of both the strength of functional coupling between two time series (magnitude) as well as the direction of signal transmission (phase). A more detailed description of coherency analysis and its application to neural signals can be found in Rosenberg, Amjad, Breeze, Brillinger, and Halliday (
1989).
Coherency analysis overcomes a significant problem in interpreting functional connectivity patterns based on correlation-type measurements in the time domain. The hemodynamic response function (HRF) is the transfer function that relates neural activity to the blood oxygenation level-dependent (BOLD) signals measured with fMRI, and the shape of the HRF varies significantly across cortical areas of individual subjects (Handwerker, Ollinger, & D'Esposito,
2004). Because of this, two areas with very similar temporal patterns of neural activity can have substantially different fMRI time courses, resulting in artifactually low levels of correlation. Because coherency magnitudes are computed in the frequency domain and not in the time domain, they are largely unaffected by differences in HRF shape across cortical areas (Sun et al.,
2004).