I am looking for a closed form, an estimation, or a bound for the sum of $M$ out of $N$ terms of a Gaussian sequence expressed as $$ \sum_{x=0}^{M}\exp\left[-a \frac{x^2}{N^2}\right] $$
where $a$ is a constant positive integer and $M<N$ is the number of arbitrarily chosen terms of the sequence. I believe that the problem here is that the sum will depend on the indices of the $M$ chosen terms. However, we can ignore that to solve the problem initially.
There is an answered question titled Sum of Gaussian Sequence. However, that question addressed only the $M=N-1$ case.