From a physiological/anatomical basis it seems to be true that a person will not go into syncope (loss of consciousness) for quite a few seconds even if the heart effectively ceases to act as a viable pump and cerebral blood flow is abruptly stopped. My specialty is not cardiology but I seem to remember from my days in medical school and early residency that given an arrhythmia that stops effective cardiac output, such as with a 3rd degree heart block or with Ventricular Fibrillation for example, a person will go into syncope in around 6-8 seconds.
With a combatant in a violent encounter you would have to also take into account their activated sympathetic nervous system (fight or flight, plus or minus the effects of a simpathomimetic agent such as cocaine, meth-amphetamine, or even epinephrine (apparently some terrorists medicate themselves with epinephrine before fire fights)) and the resultant vasoconstriction that would sustain their blood pressure for quite a few more seconds beyond that.
I think it is hard to predict exactly how many seconds an aggressor will continue to function effectively after having his cardiovascular pump destroyed, in some part due to some of the variables listed above, but interestingly, I think back in the civil war they used to call it "the dead man's 10 seconds".
Furthermore, even a hit to the spinal cord will not ensure that an aggressor will be incapacitated enough not to return fire. It depends on where the spinal cord is hit. In simple terms a hit below the cervico-thoracic junction (lowest part of the neck, where the thorax begins) will still enable upper extremity function, and while the aggressor will fall down in a paraplegic state, they will still be able to use their upper extremities and therefore will still retain the potential to activate a firearm/weapon.
This is of course assuming that bullet placement and penetration even allows for hitting these vital structures.
Oh and I forgot: As far as "knock down" power, here's
The Myth Busters.