In order to do this, I will need a good understanding of how the code runs.
Is there a simple way of tracking the order of instantiations when OO code is run?
I don't have a suggestion for the question you asked, but I do have a suggestion for your ultimate goal. You have legacy code (by the first of the modern interpretations on that Wikipedia page, "source code inherited from someone else") but it also sounds like the code may satisfy the definition from Michael Feathers, "code without tests". If that's the case, I recommend starting by writing tests for the methods of those objects using the testing framework included in MATLAB. This may sound intimidating, but I think the script-based infrastructure shouldn't be too difficult to start with. Try looking at the Write Script-Based Unit Tests topic on the Script-Based Unit Tests page linked from the page above for an example of how to create and execute such a test.
Once you have a good collection of tests, running them as soon as you modify the class methods will increase the chances of detecting problems introduced by your modifications right away.
Function-based and class-based tests are more complicated than script-based tests but give increased functionality. The three different methods of testing can't coexist in the same test file, but as you learn more about MATLAB programming you could add function- and/or class-based test files to your suite alongside your script-based test files.