Chaotic Computation is the exploitation of chaotic systems to perform computational tasks. The abundance of uncountable distinct behaviours by chaotic systems, along with their embedded determinism, position such systems as perfect candidates for developing a new computational environment. The present dissertation focuses on algorithms developed over the past decade within the realm of Chaotic Computation. After a brief exposition of general Chaos Theory, we proceed to give detailed instructions for performing such algorithms, as well as specific examples of implementations. We begin with multiple methods for number representation and basic arithmetic manipulations, providing from the start evidence of the flexibility of Chaotic Computation. The compatibility with Turing machines is subsequently shown through an algorithm for logic operations whose general form is a recurrent theme. We soon though, proceed further than Turing machines and present a solution to the Deutsch-Jozsa problem, of arbitrary binary functions. Even more, a practical issue is also handled by showing how chaotic systems have a natural way for selecting matches of a searched item from within an unsorted database. Finally we present our latest results in handling “prolonged” evolution of chaotic systems. Specifically we demonstrate the dominance of selecting the appropriate behaviour, for a computational task, over being exact with specific state values, or even confined to specific physical quantities.

Pulse coupled oscillators (PCOs) represent an ubiquitous model for a number of physical and biological systems. Phase response curves (PRCs) provide a general mathematical framework to analyze patterns of synchrony generated within these models. A general theoretical approach to account for the nonlinear contributions from higher-order PRCs in the generation of synchronous patterns by the PCOs is still lacking. Here, by considering a prototypical example of a PCO network, i.e., two synaptically coupled neurons, we present a general theory that extends beyond the weak-coupling approximation, to account for higher-order PRC corrections in the derivation of an approximate discrete map, the stable fixed point of which can predict the domain of 1:1 phase locked synchronous states generated by the PCO network.

We propose and characterize an iterated map whose nonlinearity has a simple (i.e., minimal) electronic implementation. We then demonstrate explicitly how all the different fundamental logic gates can be implemented and morphed using this nonlinearity. These gates provide the full set of gates necessary to construct a general-purpose, reconfigurable computing device. As an example of how such chaotic computing devices can be exploited, we use an array of these maps to encode data and to process information. Each map can store one of M items, where M is variable and can be large. This nonlinear hardware stores data naturally in different bases or alphabets. We also show how this method of storing information can serve as a preprocessing tool for exact or inexact pattern-matching searches.

J Crayton Pruitt Department of Biomedical Engineering,University of Florida, Gainesville, FL 32611(Dated: January 30, 2009)Spike time response curves (STRC’s) are used to study the influence of synaptic stimuli on thefiring times of a neuron oscillator without the assumption of weak coupling. They allow us toapproximate the dynamics of synchronous state in networks of neurons through a discrete map,which can then be used to predict the stability of patterns of synchrony in the network. Generaltheory for taking into account the contribution from higher order STRC terms, resulting fromperturbations caused by synaptic stimuli lasting for more than one firing cycle of a neuron, in theapproximation of the discrete map is still lacking. Here we present a general theory to account forhigher order STRC corrections in the approximation of discrete map to determine the domain of 1:1phase locked state in a network of two interacting neurons. We test the ability of this discrete mapto predict the domain of 1:1 phase locked state in a network of two neurons interacting through ashunting synapse.

We propose a direct and flexible implementation of logic operations using the dynamical evolution of a nonlinear system. The concept involves the observation of the state of the system at different times to obtain different logic outputs. We explicitly implement the basic NAND, AND, NOR, OR and XOR logic gates, as well as multiple-input XOR and XNOR logic gates. Further we demonstrate how the single dynamical system can do more complex operations such as bit-by-bit addition in just a few iterations.

By applying nonlinear dynamics to the dense storage of information, we demonstrate how a single nonlinear dynamical element can store M items, where M is variable and can be large. This provides the capability for naturally storing data in different bases or in different alphabets and can be used to implement multilevel logic. Further we show how this method of storing information can serve as a preprocessing tool for (exact or inexact) pattern matching searches. Since our scheme involves just a single procedural step, it is naturally set up for parallel implementation and can be realized with hardware currently employed for chaos-based computing architectures.

J Crayton Pruitt Department of Biomedical Engineering,University of Florida, Gainesville, FL 32611(Dated: June 3, 2008)We present two novel methods for performing logic operations. Our methods are based on usingthe time dimension for programming and data representation. The first method is based on varyingthe sampling moment in time of a neuronal action potential, and the second method is based on aneural delay system, where the generation of the action potential is delayed by specific time lengths,to be sampled at a fixed moment in time. Both methods are supported by explicit examples.

We present two novel methods for performing logic operations. Our methods are based on using the time dimension for programming and data representation. The first method is based on varying the sampling moment in time of a neuronal action potential, and the second method is based on a neural delay system, where the generation of the action potential is delayed by specific time lengths, to be sampled at a fixed moment in time. Both methods are supported by explicit examples.

We propose a method that uses nonlinear dynamics to address the task of determining the existence of a specific item in an unsorted arbitrarily large database. The scheme involves a single global operation applied simultaneously to all elements of the database and a single global monitoring threshold for verification of existence. We also show how this global threshold can be altered for identification of the existence of items with characteristics close to the searched one