An innovative mathematical architecture for modelling neuronal electrical activity is presented, called the cognitive rhythm generator (CRG), wherein the proposed architecture is a hybrid model comprised of three interconnected stages, namely: (1) a bank of neuronal modes; (2) a ring device (limit-cycle oscillator); and (3) a static output nonlinearity (mapper). Coupled CRG networks are employed to emulate and elucidate the dynamics of biological neural networks, including the recurrent networks in the hippocampus. Several species of ring devices are described and investigated, including the clock, labile clock, hourglass and multistable ring systems, and their applications to neuronal modelling explored. Complexity measures such as the maximum Lyapunov exponent, correlation dimension and detrended fluctuation analysis are applied to compare model and biological records and validate the CRG methodology. The basis of neural coding is also examined in mathematical detail, with particular regard to its description by Volterra-Wiener kernel formalism, from which the neuronal modes are derived. Applications to theta-gamma coding are discussed. Further on in the thesis, a CRG epileptiform network model of spontaneous seizure-like events (SLEs) is developed and used as a platform to test neuromodulation approaches for seizure abatement. (Neuromodulation mentioned here refers to methods involving electrical stimulation of neural tissue for therapeutic benefit). Spontaneous SLE transitions in the epileptiform network are shown to be related to the mechanism of intermittency, as determined by examining the state space dynamics of the model. The onset of SLEs is associated with increased network excitability and decreased stability, consistent with experimental results from the low-magnesium/high-potassium in vitro model of epilepsy. Lastly, a novel strategy for therapeutic neuromodulation is presented wherein a coupled CRG network (called the “therapeutic network”) is interfaced with the epileptiform network model, forming a closed loop for responsive, biomimetic neuromodulation of the epileptiform network. Relevance to clinical applications and future work is discussed.

An innovative mathematical architecture for modelling neuronal electrical activity is presented, called the cognitive rhythm generator (CRG), wherein the proposed architecture is a hybrid model comprised of three interconnected stages, namely: (1) a bank of neuronal modes; (2) a ring device (limit-cycle oscillator); and (3) a static output nonlinearity (mapper). Coupled CRG networks are employed to emulate and elucidate the dynamics of biological neural networks, including the recurrent networks in the hippocampus. Several species of ring devices are described and investigated, including the clock, labile clock, hourglass and multistable ring systems, and their applications to neuronal modelling explored. Complexity measures such as the maximum Lyapunov exponent, correlation dimension and detrended fluctuation analysis are applied to compare model and biological records and validate the CRG methodology. The basis of neural coding is also examined in mathematical detail, with particular regard to its description by Volterra-Wiener kernel formalism, from which the neuronal modes are derived. Applications to theta-gamma coding are discussed. Further on in the thesis, a CRG epileptiform network model of spontaneous seizure-like events (SLEs) is developed and used as a platform to test neuromodulation approaches for seizure abatement. (Neuromodulation mentioned here refers to methods involving electrical stimulation of neural tissue for therapeutic benefit). Spontaneous SLE transitions in the epileptiform network are shown to be related to the mechanism of intermittency, as determined by examining the state space dynamics of the model. The onset of SLEs is associated with increased network excitability and decreased stability, consistent with experimental results from the low-magnesium/high-potassium in vitro model of epilepsy. Lastly, a novel strategy for therapeutic neuromodulation is presented wherein a coupled CRG network (called the “therapeutic network”) is interfaced with the epileptiform network model, forming a closed loop for responsive, biomimetic neuromodulation of the epileptiform network. Relevance to clinical applications and future work is discussed.