In this paper, we present a model for the problem of separating active neurons from their superposition at an array of electrodes in a neural recording setting. In this model, we allow unknown continuous delays between firing neurons and attenuations introduced by the pulses traveling through the medium. Additionally, we show how to utilize this model in the compressed sensing and the finite rate of innovation framework and then present a total of four solutions for the extraction of the pulse shape, number, and activity patterns of active neurons. To support our claims, we present extensive numerical simulations comparing our proposed algorithms with the state of the art.