Ensuring safety and achieving human-level driving performance remain challenges for autonomous vehicles, especially in safety-critical situations. As a key component of artificial intelligence, reinforcement learning is promising and has shown great potential in many complex tasks; however, its lack of safety guarantees limits its real-world applicability. Hence, further advancing reinforcement learning, especially from the safety perspective, is of great importance for autonomous driving. As revealed by cognitive neuroscientists, the amygdala of the brain can elicit defensive responses against threats or hazards, which is crucial for survival in and adaptation to risky environments. Drawing inspiration from this scientific discovery, we present a fear-neuro-inspired reinforcement learning framework to realize safe autonomous driving through modeling the amygdala functionality. This new technique facilitates an agent to learn defensive behaviors and achieve safe decision making with fewer safety violations. Through experimental tests, we show that the proposed approach enables the autonomous driving agent to attain state-of-the-art performance compared to the baseline agents and perform comparably to 30 certified human drivers, across various safety-critical scenarios. The results demonstrate the feasibility and effectiveness of our framework while also shedding light on the crucial role of simulating the amygdala function in the application of reinforcement learning to safety-critical autonomous driving domains.