What connection does Edgar Allan Poe's narrator make between madness and rational thinking in "Tell Tale Heart"?

In the opening paragraph, the narrator of Poe's "The
Tell-Tale Heart" declares that he is nervous, but not mad.  "The disease," he
asserts, has sharpened his senses rather than having damaged them.  However, the senses
of the narrator seem to have become so acute that they become torturous, much like the
senses of Roderick Usher in Poe's "The Fall of the House of
Usher."

With his hypersensitivity, the narrator cannot keep
from focusing on the old man's pale, cloudy eye, an eye that incites coldness in his
blood.  Then the narrator decides to "take the life of the old man, and rid myself of
the eye forever."  In his plan, the narrator insists that he is not mad.  For, he
carefully "dissimulates," deceiving the old man with his kindness.  However, it is also
this nervous sensitivity that causes the narrator to believe that he hears the beating
of the old man's heart after he has buried him beneath the floor boards of the house. 
And, although he declares that he is not mad, the narrator describes himself when the
police are there:

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I foamed--I raved--I swore!...But anything was
better than this agony!  Anything was more tolerable than this derision!  I felt that I
must scream or die!

And,
then, the narrator in his madness uncovers the old man's body, crying "Here, here!  It
is the beating of his hideous heart!"