# Epileptiform Activity and Seizure Risk Follow Long‐Term Non‐Linear Attractor Dynamics

## TL;DR

This study uses a mathematical framework to model seizure risk patterns in epilepsy patients, enabling better prediction and personalized treatment.

## Contribution

The novel use of HAVOK analysis to model nonlinear seizure dynamics and improve multi-day seizure risk forecasting in epilepsy.

## Key findings

- Seizures occur in regions of strongly nonlinear dynamics within multi-day attractor cycles.
- HAVOK analysis accurately predicts slower multi-day rhythms using short-period forcings.
- The framework provides a pathway for personalized, data-driven epilepsy interventions.

## Abstract

Many biological systems display circadian and slow multi‐day rhythms, such as hormonal and cardiac cycles. In patients with epilepsy, these cycles also manifest as slow cyclical fluctuations in seizure propensity. However, such fluctuations in symptoms are consequences of the complex interactions between the underlying physiological, pathophysiological, and external causes. Therefore, identifying an accurate model of the underlying system that governs the multi‐day rhythms allows for a more reliable seizure risk forecast and targeted interventions. The primary aim is to develop a personalized strategy for inferring long‐term trajectories of epileptiform activity and, consequently, seizure risk for individual patients undergoing long‐term ECoG sampling via implantable neurostimulation devices. To achieve this goal, the Hankel alternative view of Koopman (HAVOK) analysis is adopted to approximate a linear representation of nonlinear seizure propensity dynamics. The HAVOK framework leverages Koopman theory and delay‐embedding to decompose chaotic dynamics into a linear system of leading delay‐embedded coordinates driven by the low‐energy coordinate (i.e., forcing). The findings reveal the topology of attractors underlying multi‐day seizure cycles, showing that seizures tend to occur in regions of the manifold with strongly nonlinear dynamics. Moreover, it is demonstrated that the identified system driven by forcings with short periods up to a few days accurately predicts patients' slower multi‐day rhythms, which improves seizure risk forecasting.

This study leverages the HAVOK framework to model long‐term, nonlinear attractor dynamics underlying epileptiform activity and seizure risk in epilepsy patients. By identifying key forcing mechanisms driving chaotic transitions, the findings improve seizure risk forecasting over multi‐day cycles and provide a pathway for personalized, data‐driven interventions in epilepsy management.