Hopfield Neural Lattice Models
Type of DegreePhD Dissertation
Mathematics and Statistics
Restriction TypeAuburn University Users
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Hopfield neural network model is a continuous deterministic model proposed by John J. Hopfield in the early 1980’s. The model was proposed in an attempt to produced an artificial neural network architecture that mimic the activity of the brain neurons. The continuous deterministic model was developed so that it incorporate one of the basic properties of biological neurons which is a continuous input-output relations. The model captured the attention of many researchers and since then, the model has been extensively studied, modified and extended. As of the time of writing this dissertation, the paper where the model was first introduced has received more than eight thousand citations. In this dissertation, we start by looking at the very first motivational part of the model, which is the brain. An average human brain has about 86 billion neurons, with this fact in mind, we have created a Hopfield neural network model where the number of neurons is increasingly large, that is a Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of square summable bi-infinite sequences. Then the existence and uniqueness of solutions, as well as long term dynamics of solutions are investigated. In the second part of the dissertation, we study infinite dimensional extension of the classical Hopfield model and its corresponding finite dimensional approximations. The existence of global attractors is established for both the lattice system and its finite dimensional approximations. Moreover, the global attractors for the finite dimensional approximations are shown to converge to the attractor for the infinite dimensional lattice system upper semi-continuously. In the final part of the dissertation we present the ongoing works, which are the stochastic Hopfiled neural lattice model and the non autonomous Hopfield lattice model.