Tetsuya Yoshinaga

Bifurcations in Morris-Lecar neuron model

The Morris-Lecar (M-L) equations are an important neuron model that exhibits classes I and II excitabilities when system parameters are set appropriately. Although many papers have clarified characteristic behaviors of the model, the detailed transition between two classes is unclear from the viewpoint of bifurcation analyses. In this paper, we investigate bifurcations of invariant sets in a five-dimensional parameter space, and identify an essential parameter of the half-activated potential of the potassium activation curve that contributes to the alternation of the membrane properties of the M-L neuron. We also show that the membrane property can be controlled by varying the value of the single parameter.

A METHOD TO CALCULATE BIFURCATIONS IN SYNAPTICALLY COUPLED HODGKIN–HUXLEY EQUATIONS

We propose a numerical method for calculating bifurcations of periodic solutions observed in a model equation of Hodgkin–Huxley neurons coupled by excitatory synapses with a time delay. To illustrate the validity of the method, bifurcations in two-coupled Hodgkin–Huxley equations with variation of a coupling coefficient and time delay are studied.

CHAOTIC BURSTS AND BIFURCATION IN CHAOTIC NEURAL NETWORKS WITH RING STRUCTURE

We investigate a noninvertible map describing burst firing in a chaotic neural network model with ring structure. Since each neuron interacts with many other neurons in biological neural systems, it is important to consider global dynamics of networks composed of nonlinear neurons in order to clarify not only mechanisms of emergence of the burst firing but also its possible functional roles. We analyze parameter regions in which burst firing can be observed, and show that dynamics of strange attractors with burst firing is related to the generation of a homoclinic-like situation and vanishing of an invariant closed curve of the map.

BIFURCATIONS AND CHAOTIC STATES IN FORCED OSCILLATORY CIRCUITS CONTAINING SATURABLE INDUCTORS

Bifurcations of periodic, quasi-periodic and chaotic oscillations are most frequently observed in forced nonlinear circuits containing saturable inductors, whose dynamics are described by nonautonomous di erential equations with periodic external forcing terms. A Poincar e mapping is commonly applied to the study of the bifurcation phenomena. We present some elementary discussion of the codimension one and two bifurcations of periodic solution and give a practical method to obtain various bifurcation diagrams. As illustrated examples, we show numerical results for typical circuits.

Bifurcations in synaptically coupled BVP neurons

We investigate bifurcations of periodic solutions in model equations of two and three Bonhoffer–van der Pol (BVP) neurons coupled through the characteristics of synaptic transmissions with a time delay. Bifurcations of the coupled BVP neurons are compared with bifurcations of synaptically coupled Hodgkin–Huxley neurons. We obtained a parameter set of the BVP system, such that the two systems are qualitatively very similar from a bifurcational point of view. This study is a base for the analysis of synaptically coupled neurons with a large number of coupling strategies.

Common Lyapunov function based on Kullback-Leibler divergence for a switched nonlinear system.

Many problems with control theory have led to investigations into switched systems. One of the most urgent problems related to the analysis of the dynamics of switched systems is the stability problem. The stability of a switched system can be ensured by a common Lyapunov function for all switching modes under an arbitrary switching law. Finding a common Lyapunov function is still an interesting and challenging problem. The purpose of the present paper is to prove the stability of equilibrium in a certain class of nonlinear switched systems by introducing a common Lyapunov function; the Lyapunov function is based on generalized Kullback–Leibler divergence or Csiszars I-divergence between the state and equilibrium. The switched system is useful for finding positive solutions to linear algebraic equations, which minimize the I-divergence measure under arbitrary switching. One application of the stability of a given switched system is in developing a new approach to reconstructing tomographic images, but nonetheless, the presented results can be used in numerous other areas.

Codimension Two Bifurcation and Its Computational Algorithm

In this paper we discuss a method for calculating the bifurcation value of parameters of periodic solution in nonlinear ordinary differential equations. The generic bifurcations of the periodic solution are known as codimension one bifurcations: tangent bifurcation, period doubling bifurcation and the Hopf bifurcation. At the parameters for which bifurcation occurs, if a periodic solution satisfies two bifurcation conditions, then the bifurcation is referred to as a codimension two bifurcation. Our method enables us to obtain directly both codimension one and two bifurcation values from the original equations without special coordinate transformation. Hence we can easily trace out various bifurcation sets in an appropriate parameter plane, which correspond to nonlinear phenomena such as jump and hysteresis phenomena, frequency entrainment, etc. Some electrical circuit examples are analyzed and shown to illustrate the validity of our method.

Modeling Light Adaptation in Circadian Clock: Prediction of the Response That Stabilizes Entrainment

Periods of biological clocks are close to but often different from the rotation period of the earth. Thus, the clocks of organisms must be adjusted to synchronize with day-night cycles. The primary signal that adjusts the clocks is light. In Neurospora, light transiently up-regulates the expression of specific clock genes. This molecular response to light is called light adaptation. Does light adaptation occur in other organisms? Using published experimental data, we first estimated the time course of the up-regulation rate of gene expression by light. Intriguingly, the estimated up-regulation rate was transient during light period in mice as well as Neurospora. Next, we constructed a computational model to consider how light adaptation had an effect on the entrainment of circadian oscillation to 24-h light-dark cycles. We found that cellular oscillations are more likely to be destabilized without light adaption especially when light intensity is very high. From the present results, we predict that the instability of circadian oscillations under 24-h light-dark cycles can be experimentally observed if light adaptation is altered. We conclude that the functional consequence of light adaptation is to increase the adjustability to 24-h light-dark cycles and then adapt to fluctuating environments in nature.

A numerical algorithm for computing Neimark-Sacker bifurcation parameters

We propose an efficient method for computing parameter values of the Neimark-Sacker bifurcation in this paper. To handle the complex eigenvalue problem, we expand the characteristic equation into a sum of determinants of the involving matrices. We solve for the location of the fixed point, period, parameter, and arguments of the complex eigenvalues for the bifurcation simultaneously, by using Newtons method and solutions of variational equations. This algorithm does not rely on the analytic structure of the underlying system.

A computation of bifurcation parameter values for limit cycles

This paper describes a new computational method to obtain the bifurcation parameter value of limit cycles in nonlinear autonomous systems. Local bifurcations, e.g., tangent, period-doubling and Neimark-Sacker bifurcations, can be calculated by using the characteristic equation for a fixed point of the Poincare mapping. Conventionally a period of the limit cycle is not used explicitly since the Poincare mapping is needed whether the orbit reaches a cross-section or not. In our method, we regard the period as an independent variable for Newtons method and obtain location of a fixed point, the bifurcation parameter and the period simultaneously. Although the number of variables increases, the Jacobi matrix becomes simple and the method converges rapidly against the conventional method.