Real world datasets commonly contain nonlinearly separable classes, requiring nonlinear classifiers. However, these classifiers are less computationally efficient than their linear counterparts. This inefficiency wastes energy, resources and time. We were inspired by the efficiency of the brain to create a novel type of computationally efficient Artificial Neural Network (ANN) called Rank Similarity Filters. They can be used to both transform and classify nonlinearly separable datasets with many datapoints and dimensions. The weights of the filters are set using the rank orders of features in a datapoint, or optionally the 'confusion' adjusted ranks between features (determined from their distributions in the dataset). The activation strength of a filter determines its similarity to other points in the dataset, a measure based on cosine similarity. The activation of many Rank Similarity Filters transforms samples into a new nonlinear space suitable for linear classification (Rank Similarity Transform (RST)). We additionally used this method to create the nonlinear Rank Similarity Classifier (RSC), which is a fast and accurate multiclass classifier, and the nonlinear Rank Similarity Probabilistic Classifier (RSPC), which is an extension to the multilabel case. We evaluated the classifiers on multiple datasets and RSC is competitive with existing classifiers but with superior computational efficiency. Code for RST, RSC and RSPC is open source and was written in Python using the popular scikit-learn framework to make it easily accessible (https://github.com/KatharineShapcott/rank-similarity). In future extensions the algorithm can be applied to hardware suitable for the parallelization of an ANN (GPU) and a Spiking Neural Network (neuromorphic computing) with corresponding performance gains. This makes Rank Similarity Filters a promising biologically inspired solution to the problem of efficient analysis of nonlinearly separable data.