Shigeki Tsuji

BIFURCATIONS IN TWO-DIMENSIONAL HINDMARSH–ROSE TYPE MODEL

We analyze a two-dimensional Hindmarsh–Rose type model exhibiting properties of both Class 1 and Class 2 neurons. Although the system is two-dimensional and contains only four parameters, the obtained bifurcation diagrams show that the bifurcation structure satisfies conditions for emergence of both features with constant stimuli.

BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

The Bonhoffer-van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions are disappeared simultaneously by the homoclinic bifurcation on the Poincare map and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it is disappeared by the heteroclinic bifurcation on the Poincare map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

An advanced design method of bursting in Fitzhugh-Nagumo model

Spiking and bursting observed in nerve membranes seem to be important when we investigate the information representation model of the brain. Many topologically different bursting responses are observed in the mathematical models and their related bifurcation mechanisms have been clarified. In this paper, we propose an advanced design method to generate bursting responses in FitzHugh-Nagumo (FHN) model with a simple periodic external force based on bifurcation analysis. Some effective parameter perturbations for the amplitude of the external input are given from the 2-parameter bifurcation diagram. A forced BVP oscillator as an analogy of FHN model indicates an expected type of bursting. We also present laboratory experiments.

Partial delayed feedback control and its DSP implementation

This paper proposes the partial delayed feedback control for chaos circuits with interrupt characteristics. The control input is added during the certain period defined by the switching policy. The total consumed energy may be saved compared with the conventional delayed feedback control system. This method is easily realized by a digital signal processor (DSP) since the delayed information can be generated by storing the state of the system into the memory in the DSP with a sufficient high-speed sampling rate. As an illustrative example, we apply the control method to a chaotic piecewise smooth circuit.

A DESIGN METHOD OF BURSTING USING TWO-PARAMETER BIFURCATION DIAGRAMS IN FITZHUGH–NAGUMO MODEL

Spiking and bursting observed in nerve membranes seem to be important when we investigate information representation model in the brain. Many topologically different bursting responses are observed in the mathematical models and their related bifurcation mechanisms have been clarified. In this paper, we propose a design method to generate bursting responses in FitzHugh–Nagumo model with a simple periodic external force based on bifurcation analysis. Some effective parameter perturbations for the amplitude of the external input are given from the two-parameter bifurcation diagram.

Calculation of the isocline for the fixed point with a specified argument of complex multipliers

This paper proposes a calculation method to obtain the location of the fixed point in given nonlinear discrete system when the argument of complex multipliers is specified. We can calculate an isocline corresponding to the specified argument, then the stability and instantaneous phase around the fixed point can be discussed. As an application, we also propose a new calculation method of NS bifurcation parameter values with the specified argument. In addition, cross points of a cusp in a periodic entrainment region and the NS bifurcation curve is identified by this application.

Occasional Delayed Feedback Control for Switched Autonomous Systems

We propose the occasional delayed feedback control for chaotic circuits with interrupt characteristics. The control input generated by the difference between the current state and the delayed state is added to the system occasionally, i.e., during a certain interval defined by the switching policy. This control is reasonable for chaotic systems including some switches since they are not always feasible to admit the control input, e.g., a fixed control input terminal may be separated according to the switching action. We apply the control method to a chaotic switched circuit and demonstrate availability by some numerical experiments.

Synchronization and Bifurcation Phenomena in Inhibitory Neurons with Gap-junction

Electrical coupling has been discovered extensively between certain neurons in some regions of the brain. They are also coupled by bi-directional or uni-directional inhibitory synapses. However, the relationship between synchronous firing of them and different functional roles of each coupling have not been revealed perfectly. To clarify the above problem, we investigate bifurcation phenomena in coupled neuron models interconnected by both electrical and inhibitory synapses. As a result, we show that coupled neurons show in-phase, anti-phase solutions and complex behavior by the interaction between electrical and inhibitory synapses and that large scale network can show irregular switching between in-phase and anti-phase solutions

Bifurcations in modified BVP neurons connected by inhibitory and electrical coupling

Electrical coupling among inhibitory interneurons has been discovered in various regions of the brain. Since these local networks indicate a sophisticated activity by the interplay between electrical and inhibitory synapses, the emergence of synchronous phenomena have been investigated by many researchers. We propose coupled neuronal models connected by both inhibitory and electrical synapses and investigate the emergence of various synchronous phenomena and its bifurcational structures by bifurcation analysis. Although it is generally known that a strong gap junction synchronizes coupled neurons, we show in-phase and anti-phase phenomena in coupled neurons connected by both inhibitory and electrical synapses. They change to chaotic solutions via period-doubling and pitchfork bifurcations.