Real-time analysis of neuronal firing patterns via Hawkes processes

Quick and reliable real-time detection of the onset of epileptic seizures is a key computational component for the development of high quality closed loop neurostimulators. The objective of this paper is the construction of a low-cost algorithm for real-time statistical analysis of EEG signals of epileptic patients. The mathematical methodology is the theory of self-exiting point processes or Hawkes processes, see Truccolo (2016) or Lambert et al. (2018). The main advance is the development of a computationally feasible recursive maximum likelihood method for fitting a Hawkes process the impulse response function of which is a sum of exponential functions. The viability of the method is proven by strong mathematical heuristics and numerical tests on simulated data, based on a priori analysis of human data recorded during both inter-ictal and ictal epochs.