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Spiking Neural Networks (SNNs) are extensively utilized in brain-inspired computing and neuroscience research. To enhance the speed and energy efficiency of SNNs, several many-core accelerators have been developed. However, maintaining the accuracy of SNNs often necessitates frequent explicit synchronization among all cores, which presents a challenge to overall efficiency. In this paper, we propose an asynchronous architecture for Spiking Neural Networks (SNNs) that eliminates the need for inter-core synchronization, thus enhancing speed and energy efficiency. This approach leverages the pre-determined dependencies of neuromorphic cores established during compilation. Each core is equipped with a scheduler that monitors the status of its dependencies, allowing it to safely advance to the next timestep without waiting for other cores. This eliminates the necessity for global synchronization and minimizes core waiting time despite inherent workload imbalances. Comprehensive evaluations using five different SNN workloads show that our architecture achieves a 1.86x speedup and a 1.55x increase in energy efficiency compared to state-of-the-art synchronization architectures.