Entropy and Chaos in Self-Organizing Systems

Self-organizing systems arise in complex biomechanical structures, human locomotion, and neural control hierarchies, yet quantitative methods for describing order formation and loss of stability remain limited. This study develops a mathematical framework for analyzing self-organization using entropy-based measures, indicators of chaotic dynamics, and network-theoretic structure. The approach (the LET framework) combines Lyapunov exponents with entropy families and graph metrics (algebraic connectivity, Load-Path Heterogeneity Index) to: (i) examine transitions between ordered and disordered states, (ii) assess sensitivity to perturbations, and (iii) characterize structural coherence in evolving cervical spine kinematics. Analytical models and computational validations are presented for cervical stability and post-operative Adjacent Segment Disease (ASD) using the Branney–Breen dataset. The findings indicate that entropy and chaos measures identify regime shifts and the emergence of a “stability corridor” more clearly than task-oriented indices, and provide finer resolution of dynamical variability within self-organizing processes. Network metrics complement these results by linking local segmental interactions to global structural fragility transfer. The study shows that entropy, chaos indicators, and network structure together form a consistent basis for describing self-organization in biomechanical systems, enabling quantitative comparison of dynamical regimes and improved interpretation of emergent pathological behavior. The approach utilizes a hybrid kinematic surrogate model to resolve passive and active components, bypassing direct force measurements by employing viscoelastic mechanotransduction principles.