Continual Test-Time Adaptation (CTTA), which aims to adapt the pre-trained model to ever-evolving target domains, emerges as an important task for vision models. As current vision models appear to be heavily biased towards texture, continuously adapting the model from one domain distribution to another can result in serious catastrophic forgetting. Drawing inspiration from the the encoding characteristics of neuron activation in neural networks, we propose the Mixture-of-Activation-Sparsity-Experts (MoASE) for the CTTA task. Given the distinct reaction of neurons with low and high activation to domain-specific and agnostic features, MoASE decomposes the neural activation into high-activation and low-activation components in each expert with a Spatial Differentiable Dropout (SDD). Based on the decomposition, we devise a Domain-Aware Router (DAR) that utilizes domain information to adaptively weight experts that process the post-SDD sparse activations, and the Activation Sparsity Gate (ASG) that adaptively assigns feature selection thresholds of the SDD for different experts for more precise feature decomposition. Finally, we introduce a Homeostatic-Proximal (HP) loss to maintain update consistency between the teacher and student experts to prevent error accumulation. Extensive experiments substantiate that our methodology achieves state-of-the-art performance in both classification and segmentation tasks.