Epileptic seizures can be anticipated by geometric-topological entropy analysis

Epilepsy is a complex brain disorder characterized by an hypersynchronous activity of neural ensemble in the brain. Nowadays electroencephalography (EEG) is the golden stan- dard for studying, monitoring and diagnosing epilepsy. Signals (time series), recorded by EEG, represent a description of the dynamics of the brain. Epilepsy is an emergent behavior given by a phase transition between a non-epileptic state (pre-ictal state) and an epileptic one (ictal state) of the neural hypergraph [1-2]. Traditional linear techniques applied to EEG show some limitation to identify these transitions while the non-linear ones seem to be more promising. The understanding of the underlying mechanisms of ictogenesis and propagation requires a suitable formal method to compute the model that supports the anticipation of ictal states. Recently, Topological Data Analysis and topological entropy [3-4], the so-called persistent entropy, are proven to be encouraging for distinguishing healthy from unhealthy patients by showing numerical evidence of the occurrence of phase transitions. We extend the previous work by providing a theoretical justification, based on statistical indexes (skewness and kurtosis), persistent entropy and topological invariants (Betti numbers), of the preliminary numerical results which describe the occurrence of a phase transition; moreover, we also intend to investigate the role of geometric entropy in quantifying the complexity of the networks since a change of complexity is also an indicator of a phase transition [5]. References 1. Varela F.J.; Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science Edited by Jean, Petitot, Francisco J. Varela, Bernard Pachoud abd Jean-Michel Roy Stanford University Press, Stanford Chapter 9, pp.266-329 2. Piangerelli M.; Merelli E.; RNN-based Model for Self-adaptive Systems - The Emer- gence of Epilepsy in the Human Brain. IJCCI (NCTA).2014: 356-361 3. Merelli E.; Piangerelli M.; Rucco M.; Toller D.; A topological approach for multivariate time series characterization: the epileptic brain.2015 4. Rucco M.; Castiglione F.; Merelli E.; Pettini M.; Characterization of idiotypic immune network through Persistent Entropy. In Proc. Complex2015 5. Franzosi R.; Felice D.; Mancini M.; Pettini M.; A geometric entropy detecting the Erdös-Rényi phase transition. EPL.201