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Multivariate point processes are increasingly being used to model neuronal response properties in the cortex. Estimating the conditional intensity functions underlying these processes is important to characterize and decode the firing patterns of cortical neurons. This paper proposes a new approach for estimating these intensity functions directly from a compressed representation of the neurons' extracellular recordings. The approach is based on exploiting a sparse representation of the extracellular spike waveforms, previously demonstrated to yield near-optimal denoising and compression properties. We show that by restricting this sparse representation to a subset of projections that simultaneously preserve features of the spike waveforms in addition to the temporal characteristics of the underlying intensity functions, we can reasonably approximate the instantaneous firing rates of the recorded neurons with variable tuning characteristics across a multitude of time scales. Such feature is highly desirable to detect subtle temporal differences in neuronal firing characteristics from single-trial data. An added advantage of this approach is that it eliminates multiple steps from the typical processing path of neural signals that are customarily performed for instantaneous neural decoding. We demonstrate the decoding performance of the approach using a stochastic cosine tuning model of motor cortical activity during a natural, nongoal-directed 2-D arm movement.