I can see where a weak battery could cause problems. The alternator, unlike the old generators, puts out an alternating current, which is changed to pulsating direct current via the diodes. Part of the battery'* job is to smooth out those ripples into nice clean D.C. If it doesn't, some of those pulses could find their way into the serial data cables and confuse the computer.
I'm still waiting for my daughter'* boyfriend to find his paperwork. He'* moved a couple of times since his problem, so they're packed away....somewhere. The other side of the battery issue is that his car (and thus the battery) was virtually brand new when he had the problem.
Black 2000 SSEi, Silverstar Headlights, Angel Eyes fogs, 3rd brake light overlay, hi-flo cat, drilled 180 degree thermostat, Intense FWI, PCM, shift kit, and 3.4 Pulley. Solid front mount.