Mathematical Investigation of Chaotic Behaviour in Coupled Oscillator Networks Modelling Neuronal Activity

This study conducts a mathematical investigation of chaotic behaviour in coupled oscillator networks modelled on the connectivity structure of mammalian cortical neuronal circuits, contributing to the intersection of dynamical systems theory and computational neuroscience at a level of mathematical rigour appropriate to the Nigerian postgraduate mathematics research environment. Understanding whether and under what coupling conditions neuronal network models exhibit chaotic dynamics has significant implications for theories of information processing in the brain and for the mathematical modelling of neurological disorders characterised by abnormal synchrony and desynchrony. A network of Hodgkin-Huxley conductance-based neural oscillators is coupled through diffusive gap junction connections arranged in random, small-world, and scale-free topologies, with network sizes ranging from 50 to 500 nodes. The largest Lyapunov exponents of the coupled system are estimated numerically using the Benettin algorithm, and bifurcation diagrams are constructed as coupling strength is varied across each network topology. Synchronisation transitions are characterised through order parameters, and the Kuramoto order parameter is tracked through the coupling-strength parameter space. The impact of network topology on the onset and extent of synchronisation-induced chaos suppression is analysed. Results demonstrate that small-world networks exhibit a broader parameter regime of synchronised non-chaotic behaviour relative to random and scale-free topologies, with implications for understanding the functional significance of small-world connectivity in biological neural circuits. Keywords: coupled oscillators, chaos theory, Lyapunov exponents, neuronal networks, synchronisation