Activation functions are fundamental for enabling nonlinear representations in deep neural networks. However, the standard rectified linear unit (ReLU) often suffers from inactive or "dead" neurons caused by its hard zero cutoff. To address this issue, we introduce N-ReLU (Noise-ReLU), a zero-mean stochastic extension of ReLU that replaces negative activations with Gaussian noise while preserving the same expected output. This expectation-aligned formulation maintains gradient flow in inactive regions and acts as an annealing-style regularizer during training. Experiments on the MNIST dataset using both multilayer perceptron (MLP) and convolutional neural network (CNN) architectures show that N-ReLU achieves accuracy comparable to or slightly exceeding that of ReLU, LeakyReLU, PReLU, GELU, and RReLU at moderate noise levels (sigma = 0.05-0.10), with stable convergence and no dead neurons observed. These results demonstrate that lightweight Gaussian noise injection offers a simple yet effective mechanism to enhance optimization robustness without modifying network structures or introducing additional parameters.