I've had this issue before but there can be many causes.
Note your power supply seems to have more than 1 rail, make sure you are not overloading a rail's maximum amps somewhere.
Check your northbridge temperatures as this is directly related to hardware, a northbridge fault/overheat can cause random issues.
May be worth checking your RAM out also.
Other than that if there hasn't been any other time when it shuts down randomly then it sounds like it may be something to do with the GPU, whether this be drivers or an overload on PSU rail somewhere.
Double check your physical PCI-E connectors are not causing the issue, for instance use the provided molex to 6-pin to see if a simply faulty cable is causing power loss to the card
EDIT: Also your power supply is bronze rated, so I wouldn't let the maximum power go above 344w as to not overload it, but you don't seem to be near that

HD 3870 is quite a hungry card it draws quite some amps