Chih-Wen Shih

Multistability in Recurrent Neural Networks

Stable stationary solutions correspond to memory capacity in the application of associative memory for neural networks. In this presentation, existence of multiple stable stationary solutions for Hopfield-type neural networks with delay and without delay is investigated. Basins of attraction for these stationary solutions are also estimated. Such a scenario of dynamics is established through formulating parameter conditions based on a geometrical setting. The present theory is demonstrated by two numerical simulations on the Hopfield neural networks with delays.

Complete stability in multistable delayed neural networks

We investigate the complete stability for multistable delayed neural networks. A new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property. An iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity. The existence of 3n equilibria and 2n positively invariant sets for the n-neuron system remains valid under the new formulation. The theory is demonstrated by a numerical illustration.

Convergent dynamics for multistable delayed neural networks

This investigation aims at developing a methodology to establish convergence of dynamics for delayed neural network systems with multiple stable equilibria. The present approach is general and can be applied to several network models. We take the Hopfield-type neural networks with both instantaneous and delayed feedbacks to illustrate the idea. We shall construct the complete dynamical scenario which comprises exactly 2n stable equilibria and exactly (3n − 2n) unstable equilibria for the n-neuron network. In addition, it is shown that every solution of the system converges to one of the equilibria as time tends to infinity. The approach is based on employing the geometrical structure of the network system. Positively invariant sets and componentwise dynamical properties are derived under the geometrical configuration. An iteration scheme is subsequently designed to confirm the convergence of dynamics for the system. Two examples with numerical simulations are arranged to illustrate the present theory.

Multiple almost periodic solutions in nonautonomous delayed neural networks

A general methodology that involves geometric configuration of the network structure for studying multistability and multiperiodicity is developed. We consider a general class of nonautonomous neural networks with delays and various activation functions. A geometrical formulation that leads to a decomposition of the phase space into invariant regions is employed. We further derive criteria under which the n-neuron network admits 2n exponentially stable sets. In addition, we establish the existence of 2n exponentially stable almost periodic solutions for the system, when the connection strengths, time lags, and external bias are almost periodic functions of time, through applying the contraction mapping principle. Finally, three numerical simulations are presented to illustrate our theory.

Multistability for Delayed Neural Networks via Sequential Contracting

In this paper, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise-linear activation functions. Several numerical examples illustrate the present analytic theory.

Cycle-symmetric matrices and convergent neural networks

Abstract This work investigates a class of neural networks with cycle-symmetric connection strength. We shall show that, by changing the coordinates, the convergence of dynamics by Fiedler and Gedeon [Physica D 111 (1998) 288] is equivalent to the classical results. This presentation also addresses the extension of the convergence theorem to other classes of signal functions with saturations. In particular, the result of Cohen and Grossberg [IEEE Trans. Syst. Man Cybernet. SMC-13 (1983) 815] is recast and extended with a more concise verification.

GLOBAL SYNCHRONIZATION AND ASYMPTOTIC PHASES FOR A RING OF IDENTICAL CELLS WITH DELAYED COUPLING

We consider a neural network which consists of a ring of identical neurons coupled with their nearest neighbors and is subject to self-feedback delay and transmission delay. We present an iteration scheme to analyze the synchronization and asymptotic phases for the system. Delay-independent, delay-dependent, and scale-dependent criteria are formulated for the global synchronization and global convergence. Under this setting, the possible asymptotic dynamics include convergence to single equilibrium, multiple equilibria, and synchronous oscillation. The study aims at elucidating the effects from the scale of network, self-decay, self-feedback strength, coupling strength, and delay magnitudes upon synchrony, convergent dynamics, and oscillation of the network. The disparity between the contents of synchrony induced by distinct factors is investigated. Two different types of multistable dynamics are distinguished. Moreover, oscillation and desynchronization induced by delays are addressed. We answer two conj...

A General Approach to Synchronization of Coupled Cells

This investigation presents a general framework to establish synchronization of coupled cells and coupled systems. Each individual subsystem is represented by nonlinear differential equations with or without internal or intracellular delay. A general coupling function is employed to depict the communication or interaction between subsystems or cells. Under this framework, the problem of establishing the synchronization for delayed coupled nonlinear systems is transformed to solving a corresponding linear system of algebraic equations. We start by considering a cell-to-cell system under symmetric coupling to present the main idea of the approach. The framework is then extended to the N -cell system under circulant coupling. Delay-dependent, delay-independent, and network- scale-dependent criteria for global synchronization will be established, respectively. The developed scheme can accommodate a wide range of coupled systems. We demonstrate the applications of the present approach to establish synchronization for a gene regulation model, a neuronal model, and some neural networks.

Synchronized oscillations in a mathematical model of segmentation in zebrafish

Somitogenesis is a process for the development of somites which are transient, segmental structures that lie along the anterior–posterior axis of vertebrate embryos. The pattern of somites is governed by the segmentation clock and its timing is controlled by the clock genes which undergo synchronous oscillation over adjacent cells in the posterior presomitic mesoderm (PSM). In this paper, we analyze a mathematical model which depicts the kinetics of the zebrafish segmentation clock genes subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta–Notch signalling. Our goal is to elucidate how synchronous oscillations are generated for the cells in the posterior PSM, and how oscillations are arrested for the cells in the anterior PSM. For this system of delayed equations, an iteration technique is employed to derive the global convergence to the synchronous equilibrium, which corresponds to the oscillation-arrested. By applying the delay Hopf bifurcation theory and the center manifold theorem, we derive the criteria for the existence of stable synchronous oscillations for the cells at the tail bud of the PSM. Our analysis provides the basic parameter ranges and delay magnitudes for stable synchronous, asynchronous oscillation and oscillation-arrested. We exhibit how synchronous oscillations are affected by the degradation rates and delays. Extended from the analytic theory, further numerical findings linked to the segmentation process are presented.

The kinetics in mathematical models on segmentation clock genes in zebrafish

Somitogenesis is the process for the development of somites in vertebrate embryos. This process is timely regulated by synchronous oscillatory expression of the segmentation clock genes. Mathematical models expressed by delay equations or ODEs have been proposed to depict the kinetics of these genes in interacting cells. Through mathematical analysis, we investigate the parameter regimes for synchronous oscillations and oscillation-arrested in an ODE model and a model with transcriptional and translational delays, both with Michaelis–Menten type degradations. Comparisons between these regimes for the two models are made. The delay model has larger capacity to accommodate synchronous oscillations. Based on the analysis and numerical computations extended from the analysis, we explore how the periods and amplitudes of the oscillations vary with the degradation rates, synthesis rates, and coupling strength. For typical parameter values, the period and amplitude increase as some synthesis rate or the coupling strength increases in the ODE model. Such variational properties of oscillations depend also on the magnitudes of time delays in delay model. We also illustrate the difference between the dynamics in systems modeled with linear degradation and the ones in systems with Michaelis–Menten type reactions for the degradation. The chief concerns are the connections between the dynamics in these models and the mechanism for the segmentation clocks, and the pertinence of mathematical modeling on somitogenesis in zebrafish.