This work explores distinct complex dynamics of simplified two nodes of coupled gene regulatory networks with multiple delays in two self-inhibitory and mutually activated genes. We have identified the emergence of extreme events within a specific range of system parameter values. A detailed analysis of the time delay-induced emergence of extreme events is illustrated using bifurcation analysis, two-parameter phase diagrams, return maps, temporal plots, and probability density functions. The reasons behind the advent of extreme events are discussed in detail, with possible analogies to simplified two nodes of gene regulatory networks. The occasional large-amplitude bursting originated in the system via interior crisis-induced intermittency, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic intermittency routes. Additionally, we have used various recurrence quantification statistical measures, such as mean recurrence time, determinism, and recurrence time entropy, to describe the transition from periodic or chaotic to unforeseen large deviations. Our approach shows that the sudden surge of variance and mean recurrence time at the transition points can be used as a new metric to detect the critical transitions of distinct extreme bursting events. The comprehensive overview of the interaction between gene regulatory networks, with insights into the formation of unusual dynamics, is beneficial to grasping different neuronal diseases. [ABSTRACT FROM AUTHOR]

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