Title: Redefining The Self-Normalization Property. Abstract: The approaches that prevent gradient explosion and vanishing have boosted the performance of deep neural networks in recent years. A unique one among them is the self-normalizing neural network (SNN), which is generally more stable than initialization techniques without explicit normalization. The self-normalization property of SNN in previous studies comes from the Scaled Exponential Linear Unit (SELU) activation function. %, which has achieved competitive accuracy on moderate-scale benchmarks.  However, it has been shown that in deeper neural networks, SELU either leads to gradient explosion or loses its self-normalization property. Besides, its accuracy on large-scale benchmarks like ImageNet is less satisfying.  In this paper, we analyze the forward and backward passes of SNN with mean-field theory and block dynamical isometry. A new definition for self-normalization property is proposed that is easier to use both analytically and numerically. A proposition is also proposed which enables us compare the strength of the self-normalization property between different activation functions. We further develop two new activation functions, leaky SELU (lSELU) and scaled SELU (sSELU), that have stronger self-normalization property. The optimal parameters in them can be easily solved with a constrained optimization program. Besides, analysis on the activation's mean in the forward pass reveals that the self-normalization property on mean gets weaker with larger fan-in, which explains the performance degradation on ImageNet. This can be solved with weight centralization, mixup data augmentation, and centralized activation function. On moderate-scale datasets CIFAR-10, CIFAR-100, and Tiny ImageNet, the direct application of lSELU and sSELU achieves up to 2.13% higher accuracy. On Conv MobileNet V1 - ImageNet, sSELU with Mixup, trainable $\lambda$, and centralized activation function reaches 71.95% accuracy that is even better than Batch Normalization.(code in Supplementary Material)