This investigation highlights an application of one-dimensional dynamic systems theory to information processing. We describe and analyze a method invented by Andreyev et al. that allows symbolic strings (e.g. phone numbers like 867-5309) to be encoded in a piecewise 1D map via limit cycles whose stability can be tweaked deliberately via the Linear Stability Theorem. When prompted by a stimuli that matches a previously encoded string, the corresponding cycle stabilizes and the resulting periodic attractor is interpreted as "recognition" of the stimuli. Advantages of this encoding system include the ability to recognize partial subsets of stored strings, storage capacity for multiple strings, random access to stored information, and some robustness to stimuli error.
To fulfill my final mathematics requirement at Olin, I took a class on Nonlinear Dynamics and Chaos with Prof. John Geddes. The class focused on interactively exploring basic concepts in the field, covering discrete and continuous theory. The course's only deliverables were four-page research reports in the style of Physical Review Letters highlighting results of a multi-week investigation on a topic of the student's choosing (related to the current topic, of course). This page highlights my first such report, which focused on the topic of 1D maps. I conducted a literature review to find an application that used 1D maps and produced Matlab simulations to explore ideas of my own.
Several neurological investigations suggest that information processing within the brain may be modeled by chaotic systems. Memory and recognition tasks in particular have been observed to exhibit dynamic properties. For example, when studying the neurological response of rabbits to various odor stimuli, Freeman observed that known smells gave rise to activity of the olfactory potential that was spatially coherent and nearly periodic . Alternatively, the presentation of a novel, unknown smell results in a state of low-dimensional chaos, as if the rabbit's olfactory processing indicated an "I don't know" response (as cited in ). This research has spawned much academic interest in the role of chaotic systems for information processing.
Although much of this research remains neurological, some mathematical study has also been undertaken.
Complex neural dynamics such as membrane potential have been successfully modeled with a one-dimensional circle map . More excitingly, however, Andreyev et al. have invented a stimulus recognition process (similar to Freeman's experiments) using dynamically-constructed one-dimensional maps . The result is a virtual memory system that can be initialized to remember certain symbolic strings (e.g. character sequences like A-B-C-D-E or numeric ones like 1-3-3-7). Later, when prompted with known stimuli A-B-C the system dynamics will be pushed toward a stable periodic orbit that passes through all symbols of the initially programmed string ...-A-B-C-D-E-A-B-C-D-E-... . Overall, Andreyev et al. demonstrate a "memory system" that can successfully store many strings simultaneously and differentiate known stimuli from novel ones via the resulting non-transient dynamics. Their procedure can successfully recognize fragments of stored strings, allow random access to stored strings, and even correct some stimuli error (e.g. C-D-E-F may still collapse to the A-B-C-D-E sequence under certain conditions.
The theoretical investigation reported here seeks to elucidate the principles behind Andreyev et al.'s novel memory system. Although the original work covers many concepts repeated here, this work embarks on a more in-depth analysis of the transition between periodic and chaotic states and studies the recognition problem with much larger strings.
 C. A. Skarda and W. J. Freeman, "How brains make chaos in order to make sense of the word." Behavioral Brain Sci., 10, pp. 161-165, (1987).
 Y. V. Andreyev, A. S. Dmitriev, and S. O. Starkov. "Information Processing in 1-D Systems with Chaos." IEEE
Trans. on Circuits and Systems., 44, 1, (1997).
 M. Zeller, M. Bauer, and W. Martienssen. "Neural dynamics modelled by one-dimensional circle maps." Chaos, Solitons & Fractals. 5, 6, pp. 885-893, (1995).
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