Spectral methods for neural field models with synaptic depression and spike frequency adaptation

Neural field models are widely used to describe spatiotemporal dynamic behaviors in neural networks due to their ability to macroscopically model the activity of neuronal populations. In this study, we propose an efficient numerical method for neural field models with synaptic depression and spike frequency adaptation mechanisms: a combination of the second-order Adams–Bashforth (AB2) method and spectral methods, accompanied by a rigorous error analysis. Numerical experiments show that the computational errors are in excellent agreement with theoretical predictions, fully validating the convergence and reliability of the proposed method. Building upon this foundation, we conducted numerical simulations of the model. Under Gaussian initial conditions and the influence of a piecewise-linear firing function, the system spontaneously evolves a pair of counter-propagating traveling pulses, thereby forming spatially self-sustaining oscillatory structures.