Instability modes in network-organized system with delay

Pattern-forming instabilities are fundamental to spatiotemporal dynamics in complex biological systems, yet the role of instability modes in the manifestation of pattern formation remains poorly understood. We develop and analyze a generalized Hindmarsh-Rose network model that integrates (1) normal and abnormal diffusion via Laplacian and pseudo-inverse coupling and (2) both synaptic delay and distributed memory kernels. Linear stability analysis yields explicit criteria for Turing and delay-induced instabilities, highlighting how the network spectrum-through the eigenvalues of the coupling operator-governs stability boundaries and transversality. We show that mean delay reshapes which spectral modes set the critical synaptic delay threshold, switching control from the most negative to the most positive Laplacian eigenmodes as memory increases. Direct simulations on random networks reveal that the number of spatial instability modes scales with the number of neuronal spikes via a robust power law, providing a quantitative marker of seizure intensity. Abnormal diffusion and strong coupling promote pathological pattern selection, while increased connectivity enhances robustness and suppresses abnormal discharges. These results establish instability modes as a mechanistic bridge between network spectra, delay mechanisms, and epileptiform activity, and suggest tunable control parameters for prediction and mitigation in delayed, network-organized dynamical systems.