On Markov Chains, Attractors, and Neural Nets

The work presented here relates specific aspects of the theory of Markov chains with some concepts that arise in the theory of complexity. This is done in order to support and help explain more recent hypotheses on how networks of neurons process information. The reader is referred to [1] where a good treatment of the subject of Markov chains is offered, and to [2] where a good introductory review on complexity is given. In [3,4], a closed form solution was given to some simple time varying Markov chains defined on a binary state space. These stochastic processes were characterized by their convergence to a unique finite stationary cycle of probability distributions which is independent of time and independent of initial conditions. They were also shown to exhibit weak ergodicity in their distribution functions. Any one realization, except for the repetitive nature of the cycle, ultimately becomes independent of time and independent of initial conditions while yielding all relevant information about the long run behavior of the process. In the projected extensions to this work, a particular type of time varying Markov chain was outlined that had the following interesting property: If one considered entropy in its probability context, then the processes could start with the highest levels of entropy possible as initial conditions (in the state probability distribution) and end up at cyclic stationarity with absolute zero entropy. An appeal was made to the work in [5] which was essential to show this. The hypothesis was then suggested that it is possible for neural pathways to process information in this manner. It is the purpose of this work, then, to expand on that notion.