We introduce a low-dimensional but explicitly nonlinear spatiotemporal model for the encoding of hand position by primary motor cortical (MI) neurons during a random pursuit tracking task. We find that the probability of a spike in a small time bin of size $dt$ is well approximated by the following formula: $P(spike|x) = exp(Ax+b)) dt$, where $b$ is a scalar and $A$ is a linear functional of the time-varying, two-dimensional hand position signal $x$. We give an algorithm for efficiently fitting the parameters $A$ and $b$ to data. The model is compact and yet appears to account for all previously described spatiotemporal hand position tuning properties of MI cells (Fellows et al, SFN `01). Given the above model and an additional (maximum-entropy) assumption on the population encoding properties of MI, we can define the mean-square optimal estimator of the hand position signal given population neural activity; we give an algorithm to compute this estimator. Although the encoding properties of the observed MI...