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We propose a novel three-layer neural network architecture with threshold activations for tabular data classification problems. The hidden layer units correspond to trainable neurons with arbitrary weights and biases and a step activation. These neurons are logically equivalent to threshold logic functions. The output layer neuron is also a threshold function that implements a conjunction of the hidden layer threshold functions. This neural network architecture can leverage state-of-the-art network training methods to achieve high prediction accuracy, and the network is designed so that minimal human understandable explanations can be readily derived from the model. Further, we employ a sparsity-promoting regularization approach to sparsify the threshold functions to simplify them, and to sparsify the output neuron so that it only depends on a small subset of hidden layer threshold functions. Experimental results show that our approach outperforms other state-of-the-art interpretable decision models in prediction accuracy.