The goal of this thesis is to explore some variations of neural networks. The thesis is mainly split into two parts: a variation of the shaping functions in neural networks and a variation of learning rules in neural networks. In the first part, we mainly investigate polynomial perceptrons - a perceptron with a polynomial shaping function instead of a linear one. We prove the polynomial perceptron convergence theorem and illustrate the notion by showing that a higher order perceptron can learn the XOR function through empirical experiments with implementation. In the second part, we propose three models (SMLP, SA, SA2) for stream learning and anomaly detection in streams. The main technique allowing these models to perform at a level comparable to the state-of-the-art algorithms in stream learning is the learning rule used. We employ mini-batch gradient descent algorithm and stochastic gradient descent algorithm to speed up the models. In addition, the use of parallel processing with multi-threads makes the proposed methods highly efficient in dealing with streaming data. Our analysis shows that all models have linear runtime and constant memory requirement. We also demonstrate empirically that the proposed methods feature high detection rate, low false alarm rate, and fast response. The paper on the first two models (SMLP, SA) is published in the 29th Canadian AI Conference and won the best paper award. The invited journal paper on the third model (SA2) for Computational Intelligence is under peer review.