Dynamical Analysis Of Multistable Neural Networks

The brain is a system, which has the most complex structure, the most mysterious mechanism and the most perfect function. The high-ranking intelligent activities of human, such as the feeling, thinking, learning, and memory, are the results of neural networks. Artificial neural networks are proposed to simulate structure and mechamism of biological neural networks. Since the 1980s, artificial neural networks have attracted much interest of many scientists and have made significant process in both theory and application.The dynamical properties of neural networks play crucial roles in their applications. Generally speaking, the stable mode of neural networks can be divided into two classes: monostability and multistability. A multistable neural network can possess multiple stable equilibria. It characterizes properties of biological neural network in essence and depicts internal mechanism of neural biology in deep.Neurons in the brain cortex are activitied when the brain is affected by an external stimulus. A response is triggered by a certain stimulus in the neural system composed of huge neurons. Neural activity evoked by the stimulus has been changed subtly. However, little is known about the complex and subtle change. In 2003, the proposed background neural netowrk has provided a new theoretical model. However, since the model is a coupled and nonlinear dynamical system, the form of division in the equation brought many difficulties to us. Moreover, there still exist a lot of unresolved problems about the original model. Therefore, in the dissertation, we propose several classes of improved background neural network model and discuss its dynamical behavior.The main contributions of the dissertation are as follows:(1) In Chapter 2, a class of improved background neural network model with uniform firing is proposed. Dynamical behavior of the proposed model is studied. Conditions for boundness and invariant set of networks are estabilished. By constructing a new energy function, complete convergence of networks is proved.(2) Chapter 3 focuses on a class of improved background neural networks with two subnetworks. Convergence of the networks is investigated. Global attractive set of the network is obtained. By using Jacobian matrix, a local stable condition for equilibrium point of the network is derived. Complete convergence of the network is rigorous proved by constructing a new energy function.(3) Chapter 4 presents a class of improved n-dimensional background neural networks. Four basic dynamical properties are discussed in detail: boundness, invarianty, global attractivity, and complete convergence. An invariant set is obtained. Moreover, the expressions of invariant set and global attractive set are given respectively. By using a new energy function again, complete convergence of the networks is proved.(4) In Chapter 5, based on the equivalent model of original background neural networks, a class of improved background neural network model with a relatively large number of neurons is proposed. In two cases, i.e, the background input is zero or nonzero, the conditions for continuous attractors are derived and the representations of them are obtained.(5) In Chapter 6, multistability for one-dimensional and two-dimensional improved background neural network models with arbitrary exponents are studied. For one-dimensional case, conditions for the existence and stability of equilibrium points are derived and complete convergence is investigated. For two-dimensional case, local stable condition for equilibrium point is also derived.(6) Finally, in Chapter 7, the dynamical behavior of a class of two-dimensional neural networks is discussed. An invariant set and boundedness of networks are given. By using the winder number of the vector field and constructing a closed curve, we obtain a condition under which there at least exists an equilibrium point in the network.