Does Interpolation/Extrapolation is the crucial thing happening in our brain while driving a motor vehichle? What I'd like to know is the mathematics happeing while we are behind the wheel.
PS : I am not quite sure if its the right place to ask, but would surely appreciate your comments/suggestions whether positive/negative.
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A human brain apparently has a lot to do with a neural network, see http://en.wikipedia.org/wiki/Biological_neural_network and http://en.wikipedia.org/wiki/Artificial_neural_network. What a neural network does it essentially performs a multidimensional nonlinear interpolation or extrapolation. Thus, one can argue that all human brain activities (including driving a car) is essentially a set of interpolations or extrapolations of some kind.
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While driving you need to perform decisions. How fast you go, how are you going to go from place A to place B, how much care you need to take, etcetera. The amount of decision making present in the brain is as of yet an unknown quantity, but suffice to say we aren't always thinking about stuff.
Decision making is, however, a branch of operational research, that has already been studied in mathematical terms to some expanse. Assuming a brain can calculate some of the likely probabilities of events (resorting with some extent to past experience), the brain will choose the actions performed by the driver in order to obtain the biggest advantage it might receive (i.e. faster travel, getting out of jams or preventing an accident).
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No "algorithm" used in the brain for any purpose has been identified or mathematically described. For processes like hearing and vision where there is a strong analogy to computer signal processing, nothing like a disassembly of the mental software has been done, it is a level of experimentation that is beyond current methods.
It is known that when you perceive an object in relative motion to be at a certain place now, that position is not where your eyes saw the object a moment earlier, but the brain's forecast of the position based on the visual and motion data at the earlier moment (and the idea of the input happening at one time instant is an oversimplification of a more complicated sensory process). The prediction is to increase the accuracy of perception, in the cases that this ability must have evolved to handle such as predator and prey trajectories, by compensating for the time delay between observation and perception.
From evolutionary considerations one would imagine this kind of process has to be a built from a combination of simple components, such as linear approximation on short time scales, that can be layered on each other to satisfy some "learned" constraints, but I don't know if there is any understanding of how it is really done. On a computer it could be done by an interpolation algorithm but that is only an analogy to how the brain might work.
One of the most well-known examples of competing cognitive theories is that of people catching a ball. In broad strokes, they might follow an algorithm along the lines of "use a subconscious mental model the physics of the ball and use that to predict where the ball lands, updating the prediction in real time as the evidence changes". But they might very well be doing something simpler: something like, "moving so as if to make the ball look like it's moving linearly in your field of vision".
Arguing that that strategy works involves some mathematics which can be found in "How Baseball Outfielders Determine Where to Run to Catch Fly Balls" by McBeath, et al and seeing exactly how that model would make the outfielder move is explored in "A Mathematician Catches a Baseball" by Edward Aboufadel. If you don't want to get deep into the mathematics, you can read an overview of the basic philosophy of this stuff in this blog post from psychsciencenotes.
Along those same lines, maybe driving properly involves "catching" the line in the middle of the road in a similar fashion, or perhaps not.
This whole flavor of explanation of behavior is the subject of Embodied cognitive science, and you can find a little bulleted list of other examples like the baseball one in this other blog post from psychsciencenotes.