This paper introduces a groundbreaking computing paradigm based on neural oscillatory circuits, which harnesses the power of oscillatory dynamics found in the brain to overcome the fundamental limitations of traditional computing architectures. By encoding and processing information in the phase and amplitude dynamics of coupled oscillators, these circuits achieve unprecedented levels of parallelism, energy efficiency, and adaptability. We present a comprehensive mathematical framework for designing and analyzing oscillatory computing systems, drawing upon advanced concepts from dynamical systems theory, stochastic differential equations, and complex network analysis. Through extensive numerical simulations and physical prototypes, we demonstrate that oscillatory circuits can solve complex computational problems orders of magnitude faster and more efficiently than state-of-the-art algorithms, paving the way for transformative applications in domains ranging from neuromorphic computing and artificial intelligence to quantum computing and smart sensors. Our results provide compelling evidence that neural oscillatory circuits represent a paradigm shift in computing, with profound implications for the future of technology and our understanding of the computational principles underlying biological intelligence.