Tuesday, December 27, 2011

78: And You Thought Those Mice Were Smart

    Last week, you may recall that I discussed an area of probability
theory where mice instinctively calculate a better answer than humans
in a typical experiment.  No sooner had I written that script than my
wife emailed me a link to top it:  an article cirulating the net over
the past few weeks talking about how certain roundworms are able to do
calculus!  Could this be true?  Are worms about to replace freshman in
our university mathematics hallways?
    If I understand the articles right, here's the basics of what the
worms can do.  The worms have two major neurons that control their
motion, a left neuron and a right neuron.  When the worm finds the
concentration of salt nearby increasing, this causes the left neuron
to fire, and the worm continues crawling in the same direction.  When
the worm finds the concentration of salt decreasing, the right neuron
fires, and the worm then turns in another direction.  The claim is
that this instinctive calculation of a difference in salt
concentrations is the equivalent to taking a derivative, or an
instantaneous measurement of a rate of change, in calculus.
    Now, let's take a moment to review what a derivative is in
calculus.  In high school or earlier math classes, you are sure to
have learned about the slope of a line, a measurement of how steeply
the line is pointing up or down.  If you take any two samples of a
value in nature, you can find out the difference, and calculate a
slope.  In more general terms, a slope measures how fast some value is
increasing or decreasing:  the speed of a racing car, for example, is
just a measurement of the slope of the line showing its distance from
the starting point. 
    But what if the car is accelerating, or in general you're
measuring some quantity that is not varying like a smooth line?  For
example, what if the racecar is accelerating rather than moving at a
constant speed?  That's where calculus comes in.  To find the speed of
the car at any given moment, known as the "derivative" of its distance
graph, you figure out what the slope would look like of samples of
that speed at any given time.  Basically you draw a graph of the car's
distance versus time, which will look like a curved line turning
rapidly upwards.  Then to find its speed at, say, 10 seconds after the
start, you draw a line on the graph that is tangent to the curve, or
just barely touches the curve, at the 10-second point.  If you
calculate this in a very precise way, the slope of that line will
match the reading on the car's speedometer at that moment.
    So now let's get back to the worm.  Is it doing this?  Is it
calculating the exact slope of a continuously varying quantity at an
infinitesmal instant in time?  Somehow I don't think so.  From the
description of the worm's talents, it sounds to me like it is taking
samples of the enviroment, calculating differences by looking at
concrete values from time to time.  At best, it sounds to me like the
worms are measuring relative slopes, a neat trick but quite different
from doing calculus.  Saying these worms are doing calculus because
they react to changes in salt concentration is kind of like saying my
22-month-old daughter is doing architecture, since she can stack lego
blocks on top of each other.  So don't worry, while your future
mathematical career may be endangered by competition from many
directions, you are not in danger of being upstaged by a roundworm.
    And this has been your math mutation for today.

Science Daily article on worms doing calculus

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