Multichannel evoked potentials as voltage space trajectories

Multichannel brain evoked potentials can be represented as trajectories V(t) in n-dimensional voltage space, analogous to three-channel Lissajous trajectories (3-CLT). Equations of motion are developed based on an arbitrary number of dipole generators at arbitrary locations within the brain, and the properties of 3-CLT are generalized to the higher dimensional case. The trajectory is shown to be limited to k < n dimensions, and k channels are found to be sufficient for predicting the potential difference between any two points on the head, provided that an empirically determined set of linear functionals is known. A method for choosing the "best" m-channel montage (m < or = k) is described, by associating with each montage an alternating m-tensor on Rk. Planar analysis of the voltage trajectories is generalized to the k-dimensional case, in which m-planes are compared using a mapping between the Grassman manifold and real projective space.