Neural systems, both Leleubiological and artificial, exhibit rich dynamical properties that govern stability, adaptationAdaptation, and optimizationOptimization. In this chapter, we explore two interconnected perspectives: (1) the spatio-temporal propagationPropagation of neural burstsBurst in cortical circuitsCircuit and (2) gradient-based optimizationOptimization methods inspired by neural dynamics. Both approaches rely on simplified artificial neural networkNeural network models, which, despite their constraints, offer rigorous theoretical insights into the fundamental principles of brain dynamics. Notably, chaotic and complex activity patterns in the cortex, often perceived as noise, may instead serve as an intrinsic optimizationOptimization mechanism—enabling efficient explorationExploration of solution spaces. This idea finds a parallel in chaotic amplitude control, a strategy for accelerating optimizationOptimization processes. By examining these perspectives together, we highlight how neural circuitsCircuit and mathematical optimizationOptimization techniques converge in their use of structured yet adaptable dynamics, shedding light on the deep computational principles underlying both natural and artificial intelligence.