Thursday, January 13, 2011

Connecting Fibonacci polynomials to eyeball muscles

This post will require a day or so, but here is the summary.

I will show that neurons can easily be used to construct nth order Fibonacci generating trees.  Then I take the leaves of the trees for each polynomial and connect them to the muscles around the eye circumference. Then I show that this is the minimum variational method to trace a bounded function in the eye viewing plane. The eye will trace out bounds using spirals of varying resolution.

I want to create this eye movement, and a finite combination of these movements. These movement, approximated, are movements from corner to corner of the increasingly sized tilings.  Movements are proportional to spike trains, so right away we want to say that Y in Y(X), is proportional to firing rate, and X is proportional to some base rate, we have converted to channel form.

Spacially, the distance the eye moves is proportional to the firing rate, so the goal here is to dirsribute the fibonacci numbers around the eye every 90 degrees.



And this neural net will generate the fib numbers.
  Consider the input to the net to be the base rate, the Fibonacci numbers are the number of nodes along any horizontal rank.  When I start firing this network rom the root, the electro potential of the neurons build up over time and each rank level is activated in sequence as the electro potential builds up.  The numbers are taken as the direct firing sum occurring at each rank, and distributed to the eye muscl at the next 90 degree increment.  One can see the eye will follow the approximate spiral at some rate.

To get variable precision I simply duplicate this network multiple times, and each separate network can be activated as needed causing the curvature to be controlled within a finite time and space precision.  I can trace paths through the vision plane with combinations of curvature.

Think of the sequence of activation of these various fib nets as a digital series, 0001, 0101, 0110, occurring in time.  Such a small sequence could, fo example, define a bounded path through my house.

If we make these generators part of the generic bounded function definition, then we are not limited to the eye, we can connect these to the legs, the hands, and the fingers.  Useful for walking around the house or typing blog posts.
  The remaining problem to be solves is phase locking these sequences to specific guideposts in the environment.

Dunno if I will go beyond this point.

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