There is the idea that mental work is tiring. Does the brain as a whole need to consume more glucose and oxygen when we are "mentally active"?
I think Chris Dennet's answer is a bit misleading, and I would like to see quoted sections for such a complex question rather than some lonely links. In my opinion, there is no direct link between fMRI signal and neuronal activity.
I looked for research on the changes of brain activity (sleep, resting, cognitive tasks...) and overall energy consumption of the brain. I found two very good articles putting your question into a bigger picture, quite academically written, but they focus on the statements concerning energy consumption. The articles also discuss what actually can be derived from brain imaging data (fMRI, PET) and how higher conscious brain functions relate to physiological changes measured by these techniques.
The basic conclusion is that the brain doesn't vary its energy consumption much, whether resting, tasked. In fact, it needs high average activity (high metabolism, energy consumption) to make specific functionality possible at all. So it's not like a computer where you start a program (analogous to higher conscious brain function, e.g. playing chess) and then the processor and memory consumption rises. Instead, the energy consumption is already and constantly on a high average level, otherwise the operating system (brain) couldn't run distinct software (function) at all.**
I've quoted the most important parts, but both articles give a pretty good overview and draw the bigger picture around your question.
brain represents about 2% of the body weight. Remarkably, despite its relatively small size, the brain accounts for about 20% of the oxygen and, hence, calories consumed by the body. This high rate of metabolism is remarkably constant despite widely varying mental and motoric activity
showing that the maximum values of oxygen consumption and spike frequency achieved during stimulation were approximately the same from both baselines (i.e., both levels of anesthesia). The authors assert that an overall level of ongoing activity must be achieved for a particular function to occur
This high metabolic activity is present when we are completely passive and resting as well as when we are observably doing something. Two lines of investigation have recently converged in their analysis on how this energy is being used. Both have focused on the metabolic requirements associated with glutamate signaling in the brain. This focus would seem reasonable, considering that greater than 80% of neurons are excitatory and greater than 90% of synapses release glutamate (6, 7). Attwell and Laughlin (8) have taken a bottom up modeling approach using extant data on the blowfly retina and the mammalian cerebral cortex. Estimates from their approach indicate that most of the energy used in the brain is required for the propagation of action potentials and for restoring postsynaptic ion fluxes after receptors have been stimulated by the neurotransmitter. In contrast, maintenance of the resting potential in neurons and glial cells accounts for less than 15% of the total energy consumption. Shulman and his colleagues (9, 10) in a very different approach using MRS in anesthetized rats have shown remarkably converging evidence that a very large fraction (≈80%) of the energy use in the brain is correlated with glutamate cycling and, hence, active signaling processes
An intriguing hypothesis has emerged that the responsiveness of neurons to changes in their input depends on a continuous, high-level but balanced input of both excitatory and inhibitory activity (for review, see ref. 29). Importantly, it is the balance between this continuous excitatory and inhibitory input that determines the gain or responsiveness of the neurons to correlations in their input. In this formulation, spontaneous ongoing activity becomes a critical enabling factor in the creation of functional connections within circuits responsible for specific behaviors. Furthermore, this correlation-induced functional connectivity can be modified without causing variations in the mean firing rates of the involved cells. As Salinas and Sejnowski have pointed out in their review (29), balanced neurons have rich dynamics and can react to external stimuli on effective timescales that are much smaller than the membrane time constant of a single neuron.
So, how might this relate to our analysis of the energy budget of the brain? It should be noted that most of the neurophysiology discussed above concerns synaptic activity at the input to neurons. Because the highest energy-demanding processes in the brain are centered at these sites (27, 28), it suggests that much of the ongoing or baseline metabolism is devoted to processes occurring there. We might therefore posit that, in the brain, a large majority of its metabolic activity is devoted to ongoing synaptic processes associated with maintaining a proper balance between excitatory and inhibitory activity. Maintenance of this balance allows neurons to respond appropriately to correlational changes in their input and establish the functional connectivity as required for a particular task.
Thus, we may entertain the possibility that the very high baseline or ongoing metabolic activity of the brain not only supports processes necessary for the maintenance of the proper responsiveness of neurons for the transient and ever changing functions of the brain but also instantiates a sustained functionality.
Indeed, relative to the high rate of ongoing or “basal” brain metabolism,6 the amount dedicated to task-evoked regional imaging signals is remarkably small (estimated to be less than 5%). The brain continuously expends a considerable amount of energy, even in the absence of a particular task (i.e., when a subject is awake and at rest). A significant fraction of the energy consumed by the brain (quite possibly the majority) has been shown to be a result of functionally significant spontaneous neuronal activity.7 From this cost-based analysis of brain functional activity, it seems reasonable to conclude that intrinsic activity may be as significant, if not more so, than evoked activity in terms of overall brain function.
fMRI scanner images show increased areas of blood flow in parts of the brain during mental work. Increased neural activity causes an increased demand for oxygen, and the vascular system actually overcompensates for this, increasing the amount of oxygenated hemoglobin relative to deoxygenated hemoglobin1.
Because deoxygenated hemoglobin attenuates the MR signal, the vascular response leads to a signal increase that is related to the neural activity2.
However, it is not known if this correlates precisely with neural activity, and it is an ongoing area of research3.
Edit: I've learned that this article should not be trusted, due to multiple weaknesses.
Perhaps Gailliot et al.'s1 work on blood glucose and self-control is a more direct answer than the fMRI results (I'm also a bit skeptical like footnote 3). I think the link with blood glucose is not specific to self-control.
The present work suggests that self-control relies on glucose as a limited energy source. Laboratory tests of self-control (i.e., the Stroop task, thought suppression, emotion regulation, attention control) and of social behaviors (i.e., helping behavior, coping with thoughts of death, stifling prejudice during an interracial interaction) showed that (a) acts of self-control reduced blood glucose levels, (b) low levels of blood glucose after an initial self-control task predicted poor performance on a subsequent self-control task, and (c) initial acts of self-control impaired performance on subsequent self-control tasks, but consuming a glucose drink eliminated these impairments. Self-control requires a certain amount of glucose to operate unimpaired. A single act of self-control causes glucose to drop below optimal levels, thereby impairing subsequent attempts at self-control.
1 Gailliot, Matthew T.; Baumeister, Roy F.; DeWall, C. Nathan; Maner, Jon K.; Plant, E. Ashby; Tice, Dianne M.; Brewer, Lauren E.; Schmeichel, Brandon J. Journal of Personality and Social Psychology, Vol 92(2), Feb 2007, 325-336. doi: 10.1037/0022-3522.214.171.1245