An Investigation into the Mathematical Nature of Electrophysiological Signals with Applications

—In this work we have proposed, possibly for the first time, a rigorous mathematical definition for the one dimensional time domain electrophysiological signals and established its relationship with two of the three Dirichlet's conditions. We have argued that any such signal can be represented as the trajectory of a particle moving in a force field with one degree of freedom. At each point on the trajectory, that is, on the signal, the kinetic energy dissipated by the particle embeds semantic information into the trajectory or the signal in terms of giving its shape. We have shown that the rate of kinetic energy dissipation operator or the power operator P is of importance in shape analysis of the signal by considering its sign changes. Operating the P-operator on digital signals we have mathematically proved that its sign change can induce 13 different shapes to a three successive point configuration. We have shown that the entropy of distribution of these 13 different shapes or configurations or features across focal intracranial electroencephalogram (iEEG) signals from patients with epilepsy can distinguish the epoch before an epileptic seizure from the epoch during the seizure in a statistically significant way. It has also been shown that these 13 features can clearly distinguish between raw signals, their shifted surrogates, the power spectrum preserved shifted surrogates and Gaussian white noise signals, provided signals of a minimum length are available.