Decoding Turing instability in the convergent and divergent system

Abstract Turing instability plays a central role in the emergence of complex patterns in biological and physical systems. In this work, we present a novel framework that reveals how both convergent and divergent network dynamics can trigger previously undetectable Turing instabilities. By analyzing the eigenvalue distribution of the Laplacian matrix and its pseudo-inverse, we derive general conditions under which such instabilities arise. Applying this to the Hindmarsh–Rose neuronal model, we uncover how subtle changes in network topology, diffusion, and divergent velocity can lead to transitions between stable, periodic, and chaotic states-corresponding to seizure-free and epileptic dynamics. Our findings offer a deeper understanding of the mechanisms underlying pattern formation and seizure occurrence, with potential implications for neurological modeling and intervention strategies.