Neuromorphic computing, inspired by the human brain's neural architecture, is revolutionizing artificial intelligence and edge computing with its low-power, adaptive, and event-driven designs. However, these unique characteristics introduce novel cybersecurity risks. This paper proposes Neuromorphic Mimicry Attacks (NMAs), a groundbreaking class of threats that exploit the probabilistic and non-deterministic nature of neuromorphic chips to execute covert intrusions. By mimicking legitimate neural activity through techniques such as synaptic weight tampering and sensory input poisoning, NMAs evade traditional intrusion detection systems, posing risks to applications such as autonomous vehicles, smart medical implants, and IoT networks. This research develops a theoretical framework for NMAs, evaluates their impact using a simulated neuromorphic chip dataset, and proposes countermeasures, including neural-specific anomaly detection and secure synaptic learning protocols. The findings underscore the critical need for tailored cybersecurity measures to protect brain-inspired computing, offering a pioneering exploration of this emerging threat landscape.