In heaven the soul will not be required or have need to "think" on earthy terms... all that will be irrelevent perhaps. Using your analogy of the computer turned off : Even after my PC is turned off for the night, even after being unplugged... it continues to function, the time and date, for example are still being processed. Also you can think of the fact that data is still in storage on the hard drive to be brought back to use when turned on again. Perhaps the memory of some relevant information is not only implanted on the brain, but on the soul as well.
It is not totally turned off. There is a battery inside that keeps the clock running. Also you can think of the fact that data is still in storage on the hard drive to be brought back to use when turned on again.
Although I don't think it is completely worked out as to how memory is stored, in some cases the body is completely destroyed, e.g. by incineration. Thus, there is no permananent storage medium.
Perhaps the memory of some relevant information is not only implanted on the brain, but on the soul as well.
Perhaps there is some distributed information theoretic object pervading the universe (God?) such that when a localized information theoretic object (soul?) becomes sufficiently complex, the localized object can continue to exist after its physical representation ceases to be. But this seems pretty far from orthodox Christianity.