To survive, the brain has evolved to solve problems it faces at the best it can, from the problems of escaping a stalking tiger, finding food, competing for the fit mate in the ancient time to solving physics/math problems, trying to make ends meet, and planning for a secure future in the present time.
But the ability for a particular brain to solve any problem depends on both the hard-wiring of the involved neural circuits of the cognitive areas and the information, both declarative (the memory that you’re conscious of) and non-declarative (the skills, the experiences you’ve learned), that the brain has. The hard-wiring is mostly genetically-determined, which is different among individuals, but is and can be modified by practicing, aging, and diseases. The information the brain has is determined by what the brain has accrued from previous encounters, which are obviously vastly different among individuals. So, the ability to solve problems among different brain are different.
That’s why some people are genius, some are average, and some are handicapped. But no matter of what type they are, in the normal conditions, their brains always try to do it best to solve problems because it is inherent in the nature of the brain to do so. Of course, in some situations such as when the brain is bored from repeated stimuli, fatigued from prolonged working, in a sleepy stage, or in a pathological stage (i.e., having a cerebral concussion, an encephalopathy, or an encephalitis), it might not try to solve the problems as usual. But those are abnormal conditions.
So, when a student who can’t come up with any ideas to write longer essays is trying to come up with sentences to write about, the brain is still working and does not refuse to fire neurons to come up with ideas. And the ability to solve problems is not about firing electricity in the brain at a high enough amplitude but is about firing signals in the correct ways in the capable and knowledgeable neural circuits, which depend on the hard-wiring and the information as mentioned above.
Accordingly, to be better at solving problems, one has to practice solving problems (this will modify the hard-wiring of the involved neural circuits) and learn more and more (this will help increase the information that may be useful in solving problems).
You may find more information about the brain function on the net. The fairly good and not-very to understand one is Neuroscience
(Purves D, Augustine GJ, David Fitzpatrick D, Hall WC, Lamantia AS, McNamara JO, Williams SM, editors. Neuroscience. 3rd ed. Sunderland, Massachusetts: Sinauer Associates Inc; 2004. ISBN-13: 9780878937257 ISBN-10: 0878937250.) in Unit IV The Changing Brain and Unit V Complex Brain Functions.
