A nearly exact discretization of a two-neuron system with a time delay

Abstract Delay differential equations (DDEs) play a crucial role in modeling dynamical systems where the future state depends on both the present and past values. These equations arise in various scientific fields, including neuroscience, engineering, and economics. However, their numerical discretization is challenging, as standard methods often fail to preserve essential properties such as stability and bifurcation behavior. This study applies the nearly exact discretization scheme (NEDS) to a two-neuron system with time delay, converting it into a (2 $k+2$ k + 2 )-dimensional discrete-time model while maintaining its key dynamical features. Moreover, we introduce a novel transformation to obtain a consistent discretized system corresponding to the time-delay system. Specifically, when the delay parameter is set to zero, the resulting discretized system is reduced to the discretized form of the associated nondelay differential equation. We conduct a detailed theoretical analysis of local stability and Neimark–Sacker bifurcation to gain insights into the system’s behavior. Additionally, we introduce a simplified hybrid control method to stabilize the discretized system, providing an efficient alternative to conventional stability analyses. To support our theoretical findings, we examine a four-dimensional discrete system as a special case and present numerical simulations demonstrating the effectiveness of the proposed approach.