As cyber-physical systems become increasingly virtualized, Digital Twins are emerging as critical components for real-time system monitoring, simulation, and control. However, their growing complexity and exposure to dynamic network environments render them susceptible to sophisticated cyber threats. Traditional cybersecurity models, often based on rules or based on machine learning, struggle to adapt in real time to evolving attack patterns, especially within decentralized and resourceconstrained settings. In this conceptual paper, we introduce the Neuromorphic Cyber-Twin (NCT), a brain-inspired architectural framework that leverages spiking neural networks (SNNs) and event-driven cognition to endow digital twins with adaptive, self-evolving cyber defense capabilities. The NCT framework is grounded in neuromorphic principles such as sparse coding, temporal encoding, and synaptic plasticity, e.g., Spike-Timing-Dependent Plasticity (STDP), enabling it to process telemetry data from the digital twin layer as spike-based sensory input. We present a layered architecture wherein the cognitive layer continuously monitors behavioral deviations, performs anomaly inference, and autonomously adapts its defense responses in alignment with system dynamics. This biologically inspired paradigm offers low-latency detection, contextual awareness, and energy efficiency, essential for scalable security in virtualized ecosystems. We discuss theoretical underpinnings, architectural components, and application scenarios across smart infrastructure, autonomous transport, and industrial control systems. The paper concludes with key research challenges and a roadmap for future implementation of neuromorphic cybersecurity within digital twin environments.