Brain oscillations have traditionally been studied by time-frequency analysis of the electrophysiological signals. In this work we demonstrated the usefulness of two nonlinear combinations of differential operators on intracranial EEG (iEEG) recordings to study abnormal oscillations in human brain during intractable focal epileptic seizures. Each one dimensional time domain signal was visualized as the trajectory of a particle moving in a force field with one degree of freedom. Modeling of the temporal difference operators to be applied on the signals was inspired by the principles of classical Newtonian mechanics. Efficiency of one of the nonlinear combinations of the operators in distinguishing the seizure part from the background signal and the artifacts was established, particularly when the seizure duration was long. The resultant automatic detection algorithm is linear time executable and detects a seizure with an average delay of 5.02s after the electrographic onset, with a mean 0.05/h false positive rate and 94% detection accuracy. The area under the ROC curve was 0.959. Another nonlinear combination of differential operators detects spikes (peaks) and inverted spikes (troughs) in a signal irrespective of their shape and size. It was shown that in a majority of the cases simultaneous occurrence of all the spikes and inverted spikes across the focal channels was more after the seizure offset than during the seizure, where the duration after the offset was taken equal to the duration of the seizure. It has been explained in terms of GABAergic inhibition of seizure termination. 
Analysis of cortical rhythms in intracranial EEG by temporal difference operators during epileptic seizures