  Complex numbers appear naturally in biology whenever a system can be analyzed
in the frequency domain, such as physiological data from magnetoencephalography
(MEG). For example, the MEG steady state response to a modulated auditory
stimulus generates a complex magnetic field for each MEG channel, equal to the
Fourier transform at the stimulus modulation frequency. The complex nature of
these data sets, often not taken advantage of, is fully exploited here with new
methods. Whole-head, complex magnetic data can be used to estimate complex
neural current sources, and standard methods of source estimation naturally
generalize for complex sources. We show that a general complex neural vector
source is described by its location, magnitude, and direction, but also by a
phase and by an additional perpendicular component. We give natural
interpretations of all the parameters for the complex equivalent-current dipole
by linking them to the underlying neurophysiology. We demonstrate complex
magnetic fields, and their equivalent fully complex current sources, with both
simulations and experimental data.
