Your electrician could add a SquareD sub panel that would have the four SquareD AFCI breakers, fed from the GE panel with a standard two pole breaker.
AFCI breakers can be sensitive to certain loads. I've experienced problems with florescent lighting, power tools, and vacuum cleaners tripping the breaker.
Some surge protectors work by "redirecting" surges to the equipment ground, which is pretty much the definition of a ground fault. In this case, it's definitely possible that surge suppressors could be tripping GFCI receptacles/breakers (given a large enough surge).
As for the AFCIs... When electrical contacts separate, they often create an arc between the contacts. This arc spans the gap until the contacts are too far apart, and the arc can no longer be sustained. It's possible (in theory) that the GFCI tripping could create a large enough arc, for long enough to cause the AFCI to trip. It's not likely that a properly functioning GFCI would do this, but if the GFCI was old and worn it could be opening the contacts slow enough that a detectable arc is being produced.
Determining if the GFCI tripping is causing the AFCI to trip, is a simple test. Simply press the test button on the GFCI. If the AFCI trips, you'll have figured out half the problem. If the ACFI does not trip, you could repeat the test using a receptacle tester with a GFCI test function.
If it's determined that the GFCI is causing the AFCI breaker to trip, replacing the GFCI receptacle may resolve that part of the issue.
Figuring out if the surge suppressor is tripping the GFCI, is a more difficult proposition. It will likely require you to monitor the supply voltage over some time, to identify and document possible surges.
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So Pat, the OP has not been seen since the day they posted, in Oct 2013. Hopefully this will be of help to new viewers
I had to comment that I highly doubt they have 14 GFCI breakers in their panel. I would all but guarantee that they are referring to AFCI breakers, and they are experiencing nuisance trips. Could even be that the breakers are the early SqD AFCI breakers that were recalled. This was very common of early AFCI breakers which is why you see so many subsequent incarnations.
I'll also add that grounding and bonding, or the absence thereof, would have NOTHING to do with nuisance tripping of AFCI or GFCI breakers.