Thanks for the clarification,

Depending on the system under consideration this can be a daunting task. My first approach would be to find a threshold value for the membrane voltage variable that detects spikes, both in spiking and bursting. The interspike intervals should show well-separate intervals between bursts from those in between spikes, which means bursting activity for the chosen parameters. This qualifier can already be used to perform a bi-parametric sweep (with fixed threshold).

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It's really hard to say because you haven't given enough information. Are you trying to solve this analytically or with simulations? If analytically, it obviously depends on the specifics of your system of equations which you have not given, but I would suggest to solve for the fixed points and nullclines. Likely the trajectories of the bursting and spiking modes fall on different sides of those. For simulations, just run the simulation with the ranges of parameters that you are looking at and for each simulation, decide whether it is spiking or not with some metric, like those mentioned by Justus.

you can use xppaut software. There is also a book describing the usage of the AUTO to draw bifurcation diagrams. One important thing is that, you should start from a stable point; from thereon, the program finishes the bifurcation diagrams.