Fractional analysis of a coupled dual-capacitance neuronal model with state-wise surrogate neural networks

Neural systems exhibit memory-dependent dynamics that are difficult to capture with classical integer-order models. The current work fulfills this gap by using fractional derivative with nonlocal and nonsingular kernel. A fractional coupled dual-capacitance neuronal model build in with inductive coil (L), driven by Bessel function-modulated external stimuli to obtain intricate dynamics, is investigated. Mathematical analysis includes derivation of Picard stability, uniqueness of solution, and Ulam-Hyres stability via functional analysis's notions. Numerical solutions are derived by the application of Sumudu transform and two stage Newton interpolation polynomial. For fractional order system, synchronization analysis is studied to reduce the error dynamics in the system. The complex dynamics such as phase portraits, time series analysis, fractional neuronal dynamics like spike trains, inter spike intervals (ISI), and ISI distributions, are simulated for different fractional orders. Moreover, we propose a state-wise neural surrogate networks (SNNs) for learning parametric and temporal system dynamics. Instead of training single global model, we train independent neural surrogates for each state to get improved accuracy, stability, and scalability. The temporal behaviors are captured with Fourier feature encoding to express temporal dynamics without recurrent architectures. The accuracy and capability of the proposed network are demonstrated via prediction of phase portraits, time-series analysis, Poincare maps, error metrics, and comparison with long short-term memory model (LSTM).