Sadatoshi Kumagai

Complex nonlinear dynamics of the Hodgkin–Huxley equations induced by time scale changes

Abstract. The Hodgkin–Huxley equations with a slight modification are investigated, in which the inactivation process (h) of sodium channels or the activation process of potassium channels (n) is slowed down. We show that the equations produce a variety of action potential waveforms ranging from a plateau potential, such as in heart muscle cells, to chaotic bursting firings. When h is slowed down – differently from the case of n variable being slow – chaotic bursting oscillations are observed for a wide range of parameter values although both variables cause a decrease in the membrane potential. The underlying nonlinear dynamics of various action potentials are analyzed using bifurcation theory and a so-called slow–fast decomposition analysis. It is shown that a simple topological property of the equilibrium curves of slow and fast subsystems is essential to the production of chaotic oscillations, and this is the cause of the large difference in global firing characteristics between the h-slow and n-slow cases.

Generation of Very Slow Neuronal Rhythms and Chaos Near the Hopf Bifurcation in Single Neuron Models

We have presented a new generation mechanism of slow spiking or repetitive discharges with extraordinarily long inter-spike intervals using the modified Hodgkin-Huxley equations (Doi and Kumagai, 2001). This generation process of slow firing is completely different from that of the well-known potassium A-current in that the steady-state current-voltage relation of the neuronal model is monotonic rather than the N-shaped one of the A-current. In this paper, we extend the previous results and show that the very slow spiking generically appears in both the three-dimensional Hodgkin-Huxley equations and the three dimensional Bonhoeffer-van der Pol (or FitzHugh-Nagumo) equations. The generation of repetitive discharges or the destabilization of the unique equilibrium point (resting potential) is a simple Hopf bifurcation. We also show that the generation of slow spiking does not depend on the stability of the Hopf bifurcation: supercritical or subcritical. The dynamics of slow spiking is investigated in detail and we demonstrate that the phenomenology of slow spiking can be categorized into two types according to the type of the corresponding bifurcation of a fast subsystem: Hopf or saddle-node bifurcation.

Reachability of LSFC nets

Free choice nets are a class of Petri nets that can represent the substantial features of systems by modeling both choice and concurrency. A reachability problem of live and safe free choice (LSFC) nets is considered. A necessary and sufficient condition for the reachability of a class of LSFC nets is presented. A structural analysis of LSFC nets is presented.<<ETX>>

Spectral Analysis of Stochastic Phase Lockings and Stochastic Bifurcations in the Sinusoidally Forced van der Pol Oscillator with Additive Noise

Noise effects on the phase lockings and bifurcations in the sinusoidally forced van der Pol relaxation oscillator are investigated. Deterministic (noise-free) one-dimensional Poincaré mapping is extended to the iteration of the operator defined by a stochastic kernel function. Stochastic phase lockings and bifurcations are analyzed in terms of the density evolution by the operator. In particular, a new method which uses spectra (eigenvalues and eigenfunctions) of the operator to analyze stochastic bifurcations intensively is proposed.

Reduction of inductance and current rating of the coil and enhancement of fault current limiting capability of a rectifier type superconducting fault current limiter

A rectifier type superconducting fault current limiter is mainly composed of a rectifier using diodes and/or thyristors, and a superconducting coil. It has advantages in the following aspects. It can control the fault current level by controlling the coil current. AC losses can be reduced because the current through the superconducting coil is rectified DC current. But the longer the fault current limiting time is, the larger the superconducting coil current becomes and the fault current level increases due to the increase of the coil current. This paper shows how to reduce the inductance and current rating of the coil and how to increase the fault current limiting capability. The idea is to put a resistor in series with the superconducting coil for absorbing the incoming energy into the coil and turn on and off the switching device connected in parallel with the resistor in order to control the coil current in a pre-set region. Simulation results assuming the fault current limiter as an inter-connecting device between a generator of IPP (independent power producers) and utility grids, and some experimental results with a laboratory model will be shown.

Magnet power supply with power fluctuation compensating function using SMES for high intensity synchrotron

The power supply for high intensity synchrotron magnet draws large amount of power from utility network. The JHF 50-GeV main ring will require 104 MW of total active power and 28.8 MW of dissipation power by estimation. Moreover, the charging and discharging cycle is repeated with 3.64 s of the cycle time at the initial operation, and the repeating frequency will be raised up by twice in future. Taking this situation into consideration, energy storage system using adjustable speed type flywheel and IGBT power converter are studied in the JHF project. In this paper, the power supply using SMES is proposed. The power supply can absorb the fluctuation of active and reactive power caused by charging and discharging the synchrotron magnet. The system is composed of current source ac/dc converter, chopper circuits and superconducting magnets. The chopper circuits for superconducting magnets and synchrotron magnets can be connected to the same dc bus of the power supply and this feature can reduce the power rating of ac/dc converters.

Experiment on voltage sag compensation with minimum energy injection by use of a micro-SMES

In an attempt to moderate the cost of a micro-SMES voltage sag compensating system the minimum energy injection algorithm has been adopted. To protect the load voltage from sudden phase angle shift when the source voltage phase jump occurs simultaneously with the voltage sag, a phase-invariant voltage injection scheme is applied. A method for fast detection of the fundamental positive sequence components and a minimum energy injection algorithm are proposed and described respectively. Experiments of voltage sag compensation were carried out to confirm the validity of the proposed methods.

A Cooperative Algorithm for Autonomous Distributed Vehicle Systems with Finite Buffer Capacity

Recently, there are so many researches on Autonomous Distributed Manufacturing Systems (ADMSs), where cooperation among agents is used to solve problems, such as the scheduling problem and the vehicle routing problem. We target ADMSs where an ADMS consists of two sub-systems: a Production System (PS) and an Autonomous Transportation System (ATS). This paper discusses an on-line Tasks Assignment and Routing Problem (TARP) for ATSs under conditions of given production schedule and finite buffer capacity. The TARP results in a constrained version of the Pickup and Delivery Problem with Time Windows (PDPTW), and this paper gives a mathematical formulation of the problem. This paper, also, proposes a cooperative algorithm to obtain suboptimal solutions in which no deadlocks and buffer overflows occur. By computational experiments, we will examine the effectiveness of the proposed algorithm. Computational experiments show that the proposed algorithm is able to obtain efficient and deadlock-free routes even though the buffer capacity is less.

An agent net approach to autonomous distributed systems

When we design a large scale system, it will be more effective to consider the system as a set of plural objects, and to design each object and relations among objects. This concept is well known for object-oriented methodologies. To consider the control of such system, it will be hard to apply the traditional centralized control because of the deficiencies in robustness, flexibility, and set-up speed of the control. To overcome this problem it is necessary to implement an autonomous distributed system. In this system, each object acts on its own judgment in order to accomplish the purpose of the whole system. Petri nets are well known for an efficient mathematical model for concurrent systems. Thus an integration of the concept of Petri nets and object-oriented methodologies will bring us a good model to describe autonomous distributed systems. In this paper, we discuss a multi-agent net model for autonomous distributed systems, which is an integration of the concept of Petri nets and object oriented modeling techniques.

Parameter estimation of various Hodgkin-Huxley-type neuronal models using a gradient-descent learning method

The automatic parameter identification method proposed by Doya et al. (1994) of the Hodgkin-Huxley-type equations (1952) is investigated in detail. The Hodgkin-Huxley-type equations describe membrane currents and conduction and excitation in nerves. An improved estimation method is proposed and it is shown that our method resolves the difficulties in estimating parameters of such equations with complicated membrane potential waveforms such as a chaotic bursting and also much improves the parameter estimation (learning) speed.