Does the brain consume more calories when we think harder?

When your brain is more active, compared to a rest/slow wave sleep state, this means that more of your neurons are firing, and they are firing more often. When firing they release neurotransmitter (along with several other types of messengers) which then need to be replenished by the neuron. Creating more neurotransmitters takes energy. So the more neurotransmitters that are being released, and therefore the more that need to be created, the more calories you are burning.