Activity in a brain region and additional blood requirement: What is the causal relation? May I know causal dependence of the blood flow in the brain?

How it is determined by the brain that particular brain region requires additional blood?
The ion gradient that neurons use to transmit electrical signals is not free and costs energy to maintain, therefore needs oxygen and blood. 

A brain region is considered "Active" if the neurons in said region are firing (or at least some subset of them). If a neuron fires then it releases chemicals known as neurotransmitters that stimulate (or suppress) activity in the down stream neuron it is connected to. 

The cool thing is that the entire process of creating and sending neurotransmitters produces Nitric Oxide (NO) and in fact released as co-transmitters along with the main neurotransmitter. 

Nitric Oxide is a small gas and as such diffuses through the brain relatively fast and causes the nearby blood vessels to relax( know as vasodilatation). This causes the blood cells to open and increase blood flow to the brain region.

However their are a few caveats as the relationship is not perfect one to one. What I mean by this is a group of neurons could be releasing a inhibitory neurotransmitter (and thus Nitric Oxide) onto a brain region and thus producing less activation, but still increases blood flow. 

Here is a source on the internet if your interested in more. Sections 14.8 and 14.13 are of particular interest for NO and vasodilation in general.

Hope this helps. :)