A hybrid neural-computational paradigm for complex firing patterns and excitability transitions in fractional Hindmarsh-Rose neuronal models

The Hindmarsh-Rose (HMR) model, widely regarded as a cornerstone in the field of computational neuroscience, distills complex neuronal dynamics into a tractable framework capable of reproducing diverse firing patterns – ranging from tonic spiking to bursting and chaotic dynamics – while maintaining an essential balance of biological plausibility and mathematical simplicity for the exploration of neuronal excitability, synaptic interactions, synchronization patterns and emergent network-level phenomena. This paper presents the fractional-order extension of the HMR neuronal model, demonstrating a diverse range of firing behaviors, including slow spiking, chaotic bursting, fast spiking, Type I and Type II bursting, and chaotic spiking. The fractional HMR neuronal models' spatiotemporal dynamics are numerically simulated using an efficient Caputo Fractional Adams-Bashforth-Moulton Predictor-Corrector (FABM-PECE) solver. A novel Nonlinear AutoRegressive eXogenous Neural Network enhanced with hybrid second-order Levenberg Marquardt optimization algorithm (LMNARXNNs) is designed to delineate, analyze and simulate the fractional HMR neuronal models. A comprehensive experimental investigation is conducted to compare the proposed intelligent computing technique with reference HMR solutions. This proposed neural network strategy undergoes extensive analysis using iterative performance curves (MSE) for training, testing and validation, along with error autocorrelation, error histograms, regression analysis and correlation examinations between exogenous inputs and errors. Through comparative graphical illustrations and absolute error evaluations between the LMNARXNN and FAMB-PECE solutions, it is observed that LMNARXNN encapsulates each fractional HMR neuronal model impeccably with error evaluations in the ranges of 10−02 to 10−05. Further scrutiny on 1-Step and multi-step (5-Step) predictions, with errors on the order of 10−07 to 10−09, validates the robustness and precision of the LMNARXNN approach in accurately delineating the intricate fractional HMR firing patterns. These findings underscore that the LMNARXNN strategy constitutes a highly accurate methodological framework for modeling and forecasting neuronal dynamics, firing patterns, excitability transitions and complex temporal structures.