Jordan-chevalley decomposition of recorded electroencephalography signals during epileptic seizures

Abstract

Epilepsy is a disease that can be identified by its main features that are the recurrent and unpredictable electrical discharges of the cerebral cortex that trigger disturbances in brain functions. It can be diagnosed through a noninvasive tool called electroencephalography (EEG), which records the electrical signals emanated by the brain. The recorded signals provide important information about brain functions in terms of observed electrical potentials. This information is critical especially for diagnosing brain disorders, particularly epilepsy. In fact, EEG signals during epileptic seizures can form a semigroup of upper triangular matrices under matrix multiplication. This semigroup can be represented as the product of its elementary components through the Krohn-Rhodes decomposition technique. The representation of EEG signals in terms of the product of its elementary components is somehow connected to how any positive integer can be written as a product of prime numbers. In this study, the elementary EEG signals are shown to act as the building blocks of EEG signals, similar to the prime numbers being the building blocks of the positive integers. The primary goal of this research is to describe and view the elementary components of EEG signals during epileptic seizures as prime numbers. Firstly, the well-ordered property of prime numbers hints at a similar attribute to be found in the case of elementary EEG signals. Therefore, a new way of ordering matrices, namely the precede operator, is introduced to achieve this goal, and several theorems are developed in the process. Secondly, the elementary EEG signals are decomposed via the Jordan-Chevalley decomposition technique, which produces the summation of its simpler parts and reveals that it resonates with one of the main properties of prime numbers, particularly the Twin-Prime Conjecture. Finally, the decomposition method is implemented on the real EEG data of three patients. In this way, the theoretical framework of the EEG signals’ building blocks is developed. In short, it is obtained that the ordered property of the elementary EEG signals and its representation through the Jordan-Chevalley decomposition exhibits similar properties with certain results in prime numbers as anticipated by the Krohn-Rhodes semigroup theory. The results hint that, due to their similar properties, the EEG signals could have a similar pattern to that of prime numbers.