Nested mixed-mode oscillations, Part III: Comparison of bifurcation structures between a driven Bonhoeffer-van der Pol oscillator and Nagumo-Sato piecewise-linear discontinuous one-dimensional map

In our previous studies (Inaba and Kosaka, (2020); Inaba and Tsubone, (2020)), we discovered bifurcation structures represented by nested mixed-mode oscillations (MMOs) generated by a driven Bonhoeffer-van der Pol (BVP) oscillator. BVP oscillators are equivalent to FitzHugh Nagumo models and have been a subject of intense research for the last six decades. In this study, we consider the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch. In this case, Poincaré return maps can be rigorously constructed one-dimensionally, which consist of two downward convex branches. We also consider the Poincaré return map that is approximated as a two-segment piecewise-linear discontinuous one-dimensional map. Such a piecewise-linear map was proposed by Nagumo and Sato and generates nested period-adding bifurcations. We show that the possible solutions of un-nested, singly, and doubly nested period-adding bifurcations generated by the Nagumo-Sato map coincide with those of un-nested, singly, and doubly nested MMO-incrementing bifurcations, respectively, generated with the BVP oscillator.


Iterative Image Reconstruction Algorithm with Parameter Estimation by Neural Network for Computed Tomography


Image reconstruction algorithm using weighted mean of ordered-subsets EM and MART for computed tomography


Block-iterative reconstruction from dynamically selected sparse projection views using extended power-divergence measure


Noise-Robust Image Reconstruction Based on Minimizing Extended Class of Power-Divergence Measures


Intensity-modulated radiation therapy optimization for acceptable and remaining-one unacceptable dose-volume and mean-dose constraint planning

We give a novel approach for obtaining an intensity-modulated radiation therapy (IMRT) optimization solution based on the idea of continuous dynamical methods. The proposed method, which is an iterative algorithm derived from the discretization of a continuous-time dynamical system, can handle not only dose-volume but also mean-dose constraints directly in IMRT treatment planning. A theoretical proof for the convergence to an equilibrium corresponding to the desired IMRT planning is given by using the Lyapunov stability theorem. By introducing the concept of `acceptable', which means the existence of a nonempty set of beam weights satisfying the given dose-volume and mean-dose constraints, and by using the proposed method for an acceptable IMRT planning, one can resolve the issue that the objective and evaluation are different in the conventional planning process. Moreover, in the case where the target planning is totally unacceptable and partly acceptable except for one group of dose constraints, we give a procedure that enables us to obtain a nearly optimal solution close to the desired solution for unacceptable planning. The performance of the proposed approach for an acceptable or unacceptable planning is confirmed through numerical experiments simulating a clinical setup.

Computational and Mathematical Methods in Medicine


Hybrid Euler method for discretizing continuous-time tomographic dynamical system


Convergence Study of Iterative Image Reconstruction Algorithm with Power Exponent for Computed Tomography


Continuous-time image reconstruction based on Hellinger distance minimization for medical X-ray CT imaging

Basic & Clinical Pharmacology & Toxicology (Abstracts)


Hybrid of ML-EM and MART algorithms for X-ray CT image reconstruction


Extended ordered-subsets expectation-maximization algorithm with power exponent for noise-robust image reconstruction in computed tomography

The maximum-likelihood expectation-maximization (ML-EM) algorithm is the most popular iterative reconstruction method in emission-computed tomography with a noise model based on the Poisson distribution. The ordered-subsets EM (OS-EM) algorithm is known owing to accelerating the convergence of the ML-EM algorithm with the drawback of slow convergence. In this paper, we propose an extended OS-EM algorithm with a power exponent. We theoretically prove the asymptotic stability of an equilibrium corresponding to the solution of the nonlinear hybrid dynamical system whose numerical discretization based on multiplicative calculus coincides with the extended OS-EM algorithm. We provide a numerical experiment to demonstrate the effectiveness of the proposed system and confirm the acceleration of the proposed method and the robustness against noise. The reconstruction of high-quality images made by the method even when the projection data is noisy allows patient dose reduction in clinical practice.


Hybrid algorithm of maximum-likelihood expectation-maximization and multiplicative algebraic reconstruction technique for iterative tomographic image reconstruction

Proceedings Volume 11049, International Workshop on Advanced Image Technology (IWAIT) 2019


Tomographic Inverse Problem with Estimating Missing Projections


Bifurcation Analysis of Reduced Network Model of Coupled Gaussian Maps for Associative Memory

International Journal of Modern Nonlinear Theory and Application


Tomographic image reconstruction based on minimization of symmetrized Kullback-Leibler divergence


Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

International Journal of Bifurcation and Chaos


Remarkable similarities of two pairs of stable and saddle canards in a van der Pol oscillator under extremely weak periodic perturbation

Progress of Theoretical and Experimental Physics


Multivalued discrete tomography using dynamical system that describes competition

Multivalued discrete tomography involves reconstructing images composed of three or more gray levels from projections. We propose a method based on the continuous-time optimization approach with a nonlinear dynamical system that effectively utilizes competition dynamics to solve the problem of multivalued discrete tomography. We perform theoretical analysis to understand how the system obtains the desired multivalued reconstructed image. Numerical experiments illustrate that the proposed method also works well when the number of pixels is comparatively high even if the exact labels are unknown.

離散値トモグラフィは3つ以上の複数レベルの濃度値を持つ画像を，投影から再構成する問題である．我々はこれら離散値トモグラフィを解く，競争原理を効果的に導入した非線形力学系に基づく連続時間最適化手法を提案する．この系がいかに画像再構成を達成するかに関する理論解析を与えた．また，数値実験では，ラベル数を与えていない高解像度画像であっても適切な結果が得られることを示している．


Continuous analog of accelerated OS-EM algorithm for computed tomography


Discrete-Time Dynamic Image Segmentation Using Oscillators with Adaptive Coupling


Noise reduction in computed tomography using a multiplicative continuous-time image reconstruction method

Progress in Biomedical Optics and Imaging - Proceedings of SPIE


Continuous analog of multiplicative algebraic reconstruction technique for computed tomography


Robustification of a Nonlinear Dynamical System with a Stability Index and a Matrix Inequality

A problem in the field of nonlinear dynamics is considered with a technique of control theory. In particular, for preventing a nonlinear dynamical system from making undesirable bifurcation, it is proposed to control its parameter with a stability index and a matrix inequality. A basic idea is to update the parameter of the system so as to minimize the stability index. Since the stability index is not differentiable as a function of the parameter, its minimization is carried out through equivalent transformation to a smooth minimization problem, where a matrix inequality is used. The proposed method is applied to a simple dynamical system and shown to be effective. Its generalization is also considered for driving a system to avoid chaos.

SICE Journal of Control, Measurement, and System Integration


Dose-volume constrained optimization in intensity-modulated radiation therapy treatment planning

We present a novel optimization method to handle dose-volume constraints (DVCs) directly in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods. Most of the conventional methods are constructed for solving inconsistent inverse problems with, e.g., dose-volume based objective functions, and one expects to obtain a feasible solution that minimally violates the DVCs. We introduce the concept of `acceptable' meaning that there exists a nonempty set of radiation beam weights satisfying the given DVCs and resolve the issue that the objective and evaluation are different in the conventional planning approach. We apply the initial-value problem of the proposed dynamical system to an acceptable and inconsistent inverse problem and prove that the convergence to an equilibrium in the acceptable set of solutions is theoretically guaranteed by using the Lyapunov theorem. Indeed, we confirmed that we can obtain acceptable beam weights through numerical experiments using phantom data simulating a clinical setup for an acceptable and inconsistent IMRT planning system.

Journal of Inequalities and Applications


Robust bifurcation analysis based on optimization of degree of stability

International Journal of Innovative Computing, Information and Control


A Method for Finding Optimal Parameter Values Using Bifurcation-Based Procedure

International Journal of Modern Nonlinear Theory and Application


Continuous-time method and its discretization to inverse problem of intensity-modulated radiation therapy treatment planning

We propose a novel approach for solving box-constrained inverse problems in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods and split-feasibility algorithms. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient-based optimization algorithms. We prove theoretically that a double Kullback Leibler divergence can be used as the Lyapunov function for the IMRT planning system. Moreover, we propose a non-negatively constrained iterative method formulated by discretizing a differential equation in the continuous method. We give proof for the convergence of a desired solution in the discretized system, theoretically. The proposed method not only reduces computational costs but also does not produce a solution with an unphysical negative radiation beam weight in solving IMRT planning inverse problems. The convergence properties of solutions for an ill-posed case are confirmed by numerical experiments using phantom data simulating a clinical setup.

Communications in Nonlinear Science and Numerical Simulation


Bifurcation scenarios for a 3D torus and torus-doubling

Bifurcation transitions between a 1D invariant closed curve (ICC), corresponding to a 2D torus in vector fields, and a 2D invariant torus (IT), corresponding to a 3D torus in vector fields, have been the subjects of intensive research in recent years. An existing hypothesis involves the bifurcation boundary between a region generating an ICC and a region generating an IT. It asserts that an IT would be generated from a stable fixed point as a consequence of two Hopf (or two Neimark Sacker) bifurcations. We assume that this hypothesis may puzzle many researchers because it is difficult to assess its validity, although it seems to be a reasonable bifurcation scenario at first glance. To verify this hypothesis, we conduct a detailed Lyapunov analysis for a coupled delayed logistic map that can generate an IT, and indicate that this hypothesis does not hold according to numerical results. Furthermore, we show that a saddle-node bifurcation of unstable periodic points does not coincide with the bifurcation boundary between an ICC and an IT. In addition, the bifurcation boundaries of torus doubling do not coincide with a period-doubling bifurcation of unstable periodic points. To conclude, torus bifurcations have no relation with the bifurcations of unstable periodic points. Additionally, we exactly derive a quasi-periodic Hopf bifurcation boundary introducing a double Poincaré map.

Progress of Theoretical and Experimental Physics


Behavior of MMOs in the extended BVP oscillator under weak periodic perturbation

拡張BVP発振器における微小周期外乱下でのMMOsの挙動

Mixed-mode oscillatons (MMOs) are complicated oscillations found in chemical experiments. In this study, to investigate the fragility of MMOs in the presence of noise, the extended BVP oscillator under weak periodic perturbation is investigated. MMOs are sensitive to extremely weak perturbation. This is clarified by calculating the Lyapunov exponents, observing the attractors, and drawing a two-parameter bifurcation diagram. Furthermore, by choosing one inductor to be large, the extended BVP oscillator be- comes one of three time-scale systems. In this case, the physical mechanisms of the generation of MMOs can be explained by the construction of the oscillator.

mixed-mode oscillations (MMOs) は化学実験において見出された現象である．MMOsはcanardと深い関係があるが，canardの崩壊によるMMOsを発生する最も簡素な回路ダイナミックスとして拡張BVP発振器と，その微小周期外力応答を数値解析によって調べる．MMOsは超微小周期外力印加の下でカオスへと崩壊することを，分岐集合を作成し，リアプノフ指数を計算することにより明らかにする．このような微小外乱に鋭敏な現象が実際の化学実験で見られるということは極めて興味深い．次に本論文では，拡張BVP発振器では，1つのパラメータを変化させるだけでcanard-induced MMOsと3 time-scale系のMMOsとの対応が得られることを明らかにする．特に，パラメータ値を3 time-scale系となるように選んだ場合には，その回路構成によりMMOsの発生機構の原理に関する物理的な解釈が得られた．

The Transactions of the Institute of Electronics, Information and Communication Engineers A

電子情報通信学会論文誌(A)


A coupling model of light propagation based on the radiative transfer and diffusion equations


Continuous-Time Image Reconstruction for Binary Tomography

Binary tomography is the process of reconstructing a binary image from a finite number of projections. We present a novel method for solving binary tomographic inverse problems using a continuous-time image reconstruction (CIR) system described by nonlinear differ- ential equations based on the minimization of a double Kullback Leibler divergence. We prove theoretically that the divergence measure monotonically decreases in time. More- over, we demonstrate numerically that the quality of the reconstructed images of the non- linear CIR system is better than those from an iterative reconstruction method.


Extended Continuous-Time Image Reconstruction System for Binary and Continuous Tomography


GPU Implementation of Oscillator Network System with Plastic Coupling for Dynamic Image Segmentation


Identification of target image regions based on bifurcations of a fixed point in a discrete-time oscillator network

International Journal of Innovative Computing, Information and Control


Convergence of Iterative Method as Discretization of Continuous-Time Image Reconstruction System for Computed Tomography


Bifurcation analyses of nonlinear dynamical systems: From theory to numerical computations

In this paper, we explain how to compute bifurcation parameter values of periodic solutions for non-autonomous nonlinear differential equations. Although various approaches and tools are available for this problem nowadays, we introduce the simplest method composed by computational algorithms appeared in textbooks for beginner's, i.e., the Newton's method and Runge-Kutta method. We formulate the given bifurcation problem as a boundary value problem and use the Newton's method as a solver consistently. All derivatives required in each iteration are obtained by solving variational equations about the state and the parameter. Thank to quadratic convergence ability of the Newton's method, accurate results can be obtained without using any sophisticated mathematical library or software. If a discontinuous periodic force is applied to the system, we can take the same strategy to solve the bifurcation problem. The key point of this method is deriving a differentiable composite map from consideration of the various information about the problem such as the location of sections, the periodicity, the Poincaré mapping, and so on.

非線形非自律系を例に周期解の分岐の計算方法について解説する．現在まで多くの分岐解析手法・計算方法が提案されているが，初学者にとって取っ付き易いと思われるニュートン法とルンゲクッタ法のみを用いた簡単でかつパワフルな手法を紹介する．分岐問題をまず2点境界値問題として定式化し，ニュートン法を適用する．その際必要な写像の微分値などは，変分方程式の数値積分によって与えられる．これの手法は特別な数値解析ツールを一切使用しない．

Nonlinear Theory and Its Applications, IEICE


Sudden change from chaos to oscillation death in the Bonhoeffer-van der Pol oscillator under weak periodic perturbation

Physical Review E, Statistical, Nonlinear, and Soft Matter Physics


Discrete-Time Dynamic Image Segmentation Based on Oscillations by Destabilizing a Fixed Point

IEEJ Transactions on Electrical and Electronic Engineering (TEEE)


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.


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


Implementation of continuous-time image reconstruction system in analog electronic circuit

Far East Journal of Electronics and Communications


FPGA Implementation of Discrete-Time Neuronal Network for Dynamic Image Segmentation

We have developed a discrete-time dynamical system for dynamic image segmentation. It consists of a global inhibitor and modified chaotic neurons that can generate oscillatory responses. Dynamic image segmentation is performed using its oscillatory responses. This letter presents an implementation of our system in a field programmable gate array (FPGA) device and a successful result of dynamic image segmentation.

IEEJ Transactions on Electronics, Information and Systems


Bifurcations of Oscillatory Responses Observed in Discrete-Time Coupled Neuronal System for Dynamic Image Segmentation


Block continuous-time image reconstruction for computed tomography


Period-Doubling Cascades of Canards from the Extended Bonhoeffer-van der Pol Oscillator

This Letter investigates the period-doubling cascades of canards, generated in the extended Bonhoeffer–van der Pol oscillator. Canards appear by Andronov–Hopf bifurcations (AHBs) and it is confirmed that these AHBs are always supercritical in our system. The cascades of period-doubling bifurcation are followed by mixed-mode oscillations. The detailed two-parameter bifurcation diagrams are derived, and it is clarified that the period-doubling bifurcations arise from a narrow parameter value range at which the original canard in the non-extended equation is observed.


Iterative Method Based on a Discretization of Continuous-Time Image Reconstruction for Computed Tomography


極めて簡素な区分定数強制発振器にみられる同期現象

極めて簡素な区分定数強制発振器にみられる同期現象

本論文では,極めて簡素な区分定数強制発振器を解析し,この回路に見られる同期現象を写像法によって調べた.この方程式においては,ポアンカレ写像は区分線形サークル写像として陽に導出できる.このポアンカレ写像を解析することにより,基本調波引込領域から奇数次の分数調波引込領域の境界が陽に求められることを明らかにした.

The Transactions of the Institute of Electronics, Information and Communication Engineers A

電子情報通信学会論文誌(A)


Continuous-Time Image Reconstruction Using Differential Equations for Computed Tomography

Communications in Nonlinear Science and Numerical Simulation


Reduced Model of Discrete-time Dynamic Image Segmentation System and its Bifurcation Analysis

International Journal of Imaging Systems and Technology


Bifurcation Phenomena of Periodic Points With High Order of Period Observed in Discrete-Time Two-Coupled Chaotic Neurons


Bifurcations in an Iterative Optimization Process for Intensity Modulated Radiation Therapy

International Journal of Bifurcation and Chaos


Discrete-time dynamic image segmentation system


Bifurcation Analysis of Iterative Image Reconstruction Method for Computed Tomography

Of the iterative image reconstruction algorithms for computed tomography (CT), the power multiplicative algebraic reconstruction technique (PMART) is known to have good properties for speeding convergence and maximizing entropy. We analyze here bifurcations of fixed and periodic points that correspond to reconstructed images observed using PMART with an image made of multiple pixels and we investigate an extended PMART, which is a dynamical class for accelerating convergence. The convergence process for the state in the neighborhood of the true reconstructed image can be reduced to the property of a fixed point observed in the dynamical system. To investigate the speed of convergence, we present a computational method of obtaining parameter sets in which the given real or absolute values of the characteristic multiplier are equal. The advantage of the extended PMART is verified by comparing it with the standard multiplicative algebraic reconstruction technique (MART) using numerical experiments.

コンピュータトモグラフィーにおける反復画像再構成アルゴリズムとしてPMARTが広く知られている．本論文ではPMARTの反復過程を分岐問題と捉え，観測され固定点や周期点について解析する．また，加速アルゴリズムを付与したPMARTについても検討する．固定点問題を解き，分岐図を示すとともに，収束速度などの性能について言及する．MARTとの比較を行い，数値結果とともにPMARTの優位性を述べる．


Bifurcations in an Oscillator Network with Synaptic Coupling for Improving Image Segmentation

画像領域分割の改良を目的としたシナプス結合振動子ネットワークに見られる分岐

振動子ネットワーク系に見られる時空ダイナミックスを画像領域分割に応用した処理系としてLEGION (Locally-Excitatory Globally-Inhibitory Oscillator Network)が知られている.本論文では,まず,LEGION力学系に生じる現象の本質的な発生機構を解明するために低次元系(低画素数の画像)を解析対象とし,画像領域分割を実現する現象が入力画像に応じた多様な対称・非対称パターンの周期解に対応していることを示す.次に,非連結画像領域に対応した複数個の振動子の挙動が,初期値によっては同相で同期することを明らかにし,同相周期解が画像領域分割にとって不適切な挙動であることを示す.また,周期解の分岐解析を通じて,同相周期解が比較的広いパラメータ範囲で存在していることも明らかにする.更に,同相周期解の発生を回避する方法としてシナプス結合を有するLEGION処理系を提案し,定性的分岐理論に基づいた解析を通じ,同相周期解の不安定化にとってシナプス結合が有効であることを例証する.

電子情報通信学会論文誌(A)


Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.


Bifurcation structure of successive torus doubling


Bifurcations in Morris-Lecar neuron model


Bifurcation of synchronization phenomena observed in a coupled neuron model for circadian clock rhythms


Collapse of duck solution in a circuit driven by an extremely small periodic force

Electronics and Communications in Japan, Part 3 : Fundamental Electronic Science


A method to accelerate the convergence of the PMART for computed tomography


Bifurcation of Synchronization Phenomena Observed in a Coupled Neuron Model for Circadian Clock Rhythms

時計遺伝子リズムのニューロンモデルに見られる同調現象の分岐

本研究では,2個のMorris-Lecarニューロンモデルが電気シナプス(gap-junction)と化学シナプス(a関数)により相互結合し,かつ,周期の異なる正弦波外力が個々のニューロンに印加された系を解析する.このモデルは時計遺伝子リズムに起因したニューロン相互の同調を説明するために提案した力学系であり,実際,平均発火率リズム(発火/非発火応答の持続時間変動)の相互同調及び周期外力への同調現象が観察された.系に見られる分岐現象の解析により,生理実験で報告されている同調/脱同調の発生メカニズムが説明できることを明らかにする.

The Transactions of the Institute of Electronics, Information and Communication Engineers D-II

電子情報通信学会論文誌(D-II)


Bifurcation structure of fractional harmonic entrainments in the forced Rayleigh oscillator


超微小な周期外力の印加によるあひる解の崩壊とカオス

超微小な周期外力の印加によるあひる解の崩壊とカオス

本論文では,あひる解を発生するvan der Po1発振器に超微小な周期的外力を印加した場合にどのような分岐現象が生じるのかを調べた.分岐ダイアグラムを作成し,基本調波引込領域,1/2分数調波引込領域,1/3分数調波引込領域の分岐の構造を明らかにした.各々の領域において,周期外力が超微小な領域において周期倍分岐の連鎖的発生が観察され,カオスが発生することが分かった.また,超微小周期外力の印加のもとで二つのあひる解が共存するという興味深い現象も見られた.

The Transactions of the Institute of Electronics, Information and Communication Engineers A

電子情報通信学会論文誌(A)


Itinerant memory dynamics and global bifurcations in chaotic neural networks


Bifurcations in synaptically coupled Hodgkin-Huxley neurons with a periodic input

International Journal of Bifurcation and Chaos


強制レイリー発振器の分数調波引き込み領域にみられる分岐の構造

強制レイリー発振器の分数調波引き込み領域にみられる分岐の構造

本論文では,強制レイリー方程式の分数調波引込領域の分岐の構造を系の対称性と関連づけて解析する.系の対称性により,状態空間上において原点について対称な形状をしたアトラクタが発生する場合と,原点について非対称な形状をした二つのアトラクタが互いに原点対称な位置に発生する場合の2種類の分数調波周期解の引込領域が存在する.それらの分岐集合のパラメータ平面上での形状を調べたところ,前者と後者の分数調波引込領域は基本調波周明解のネイマルク・サッカー分岐曲線の近傍で異なった構造をもっていることが明らかになった.

電子情報通信学会論文誌(A)


Bifurcations of synchronized responses in synaptically coupled Bonhoeffer-van der Pol neuronss

The Bonhöffer-van der Pol (BvdP) equation is considered as an important model for studying dynamics in a single neuron. In this paper, we investigate bifurcations of periodic solutions in model equations of four and five BvdP neurons coupled through the characteristics of synaptic transmissions with a time delay. The model can be considered as a dynamical system whose solution includes jumps depending on a condition related to the behavior of the trajectory. Although the solution is discontinuous, we can define the Poincaré map as a synthesis of successive submaps, and give its derivatives for obtaining periodic points and their bifurcations. Using our proposed numerical method, we clarify mechanisms of bifurcations among synchronized oscillations with phase-locking patterns by analyzing periodic solutions observed in the coupling system and its subsystems. Moreover, we show that a global behavior of chaotic itinerancy or a phenomenon of chaotic transitions among several quasiattracting states can be observed in higher-dimensional systems of the synaptically four and five coupled neurons.

Physical Review E, Statistical, Nonlinear, and Soft Matter Physics


Investigation of not-in-phase synchronization on coupled system of four oscillators


Chaotic wandering and bifurcations in coupled chaotic neurons

Nonlinear Analysis: Theory, Methods & Applications


Burst firing and bifurcation in chaotic neural networks with ring structure


Bifurcations in synaptically coupled BVP neurons


4個の発振器の結合系に生じる非同期同相同期現象の考察

4個の発振器の結合系に生じる非同期同相同期現象の考察

電子情報通信学会論文誌(A)


A Method to Calculate Neimark-Sacker Bifurcation in Autonomous Systems

高次元自律系におけるNeimark-Sacker分岐の一計算法

自律系にみられるリミットサイクルにおいて， パラメータの変化により特性乗数が複素単位円を横切る場合， Neimark-Sacker 分岐を生じる．本論文ではこの分岐パラメータ値を 求める新しい計算方法を提案する． 固定点条件，ポアンカレ 断面条件に加えて， 共役複素数の特性乗数を代入した特性方程式を 実部と虚部に分離し，それぞれを独立した条件として採用する． これら条件式を連立させ，ニュートン法により解く． 新たな独立変数として，共役複素な特性乗数の偏角を用いた． ニュートン法のヤコビ行列の各要素は，複素数演算， 数値微分を用いることなく，変分方程式の解と行列操作によって 求めることができるため，ニュートン法の収束性もよい． 例として8次元自律系のリミットサイクルのNeimark-Sacker分岐の 計算結果を挙げる．

電子情報通信学会論文誌(A)


Period doubling bifurcation and chaos observed in regions of higher-harmonic entrainment in the forced Rayleigh oscillator

Electronics and Communications in Japan (Part II: Electronics)


A fast convergence method with simultaneous iterative reconstruction technique for computerized tomography

International Journal of Imaging Systems and Technology


Bifurcation of a modified BVP circuit model for neurons generating rectangular waves

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences


A method to calculate bifurcations in synaptically coupled Hodgkin-Huxley equations

International Journal of Bifurcation and Chaos


強制レイリー発振器の高調波引込領域にみられる周期倍分岐とカオス

強制レイリー発振器の高調波引込領域にみられる周期倍分岐とカオス

本論文は，強制レイリー発振器の高調波引込領域にみられる周期倍分岐とカオスについて分岐理論を用いて解析したものである．

The Transactions of the Institute of Electronics, Information and Communication Engineers A

電子情報通信学会論文誌(A)


睡眠覚醒リズムの結合発振器モデルにみられる分岐現象

睡眠覚醒リズムの結合発振器モデルにみられる分岐現象

本論文は，睡眠覚醒と体温のリズムに関する結合発振器モデルにみられる分岐現象を解析したものである．


A Method to Calculate Homoclinic Points of a Two Dimensional Noninvertible Map

本論文は，非可逆写像のホモクリニック点を求めるアルゴリズムを提案したものである．具体的な計算例もあげられている．


ウェーブレット変換を用いたMRIの体動アーチファクト削減方法

ウェーブレット変換を用いたMRIの体動アーチファクト削減方法

Japanese Journal of Medical Electronics and Biological Engineering


ヒト気道トリプターゼによるフィブリノーンの分解

ヒト気道トリプターゼによるフィブリノーンの分解

The Japanese Journal of Thoracic Diseases


Codimension two bifurcation observed in a phase converter circuit


ヒト喀痰から分離した気道トリプターゼと肥満細胞トリプターゼの性質の比較

ヒト喀痰から分離した気道トリプターゼと肥満細胞トリプターゼの性質の比較


Bifurcations in a coupled Roessler system


Bifurcations of quasi-periodic responses in coupled van der Pol oscillators with external force


Synchronized quasi-periodic oscillations in a ring of coupled oscillators with hard characteristics

Electronics and Communications in Japan, Part 3 : Fundamental Electronic Science


環状に結合した硬い発振器に見られる同期した準周期振動

環状に結合した硬い発振器に見られる同期した準周期振動

電子情報通信学会論文誌(A)


New strategy of informational processing: utilization of nonlinear dynamics to chemical sensing

Physica A: Statistical Mechanics and its Applications


Mode bifurcations in diffusively coupled van der Pol equations

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences


Codimension two bifurcation problems in forced nonlinear circuits


周期的外力を加えた非線形回路にみられる余次元2の分岐

周期的外力を加えた非線形回路にみられる余次元2の分岐

電子情報通信学会論文誌(A)


水·油界面に生じる化学的非線形振動の回路モデル

水·油界面に生じる化学的非線形振動の回路モデル

電子情報通信学会論文誌(A)


多変数系の測定出力と達成可能な外部特性のクラス

多変数系の測定出力と達成可能な外部特性のクラス

Transactions of the Society of Instrument and Control Engineers


EDB/CMSと汎用LMSを用いたe-ラーニングシステムの構築

EDB/CMSと汎用LMSを用いたe-ラーニングシステムの構築

大学教育研究ジャーナル(別冊)

大学教育研究ジャーナル(別冊)


A research of subjective complaints in expectoration of sputum in patients with chronic airway diseases through a questionnaire

自覚症状からみた慢性気道疾患患者の痰の喀出状態

Bulletin of School of Medical Sciences, the University of Tokushima

徳島大学医療技術短期大学部紀要


Comparison of biochemical characteristics of human airway tryptase and that of lung mast cell tryptase

ヒト喀痰から分離した気道トリプターゼと肥満細胞トリプターゼの性質の比較

徳島大学医療技術短期大学部紀要


MRIの位相エンコード方向に生じた体動アーチファクトの削減II

MRIの位相エンコード方向に生じた体動アーチファクトの削減II

徳島大学医療技術短期大学部紀要


慢性気道感染症患者の臨床像と喀痰の特徴

慢性気道感染症患者の臨床像と喀痰の特徴

徳島大学医療技術短期大学部紀要


MRIの位相エンコード方向に生じた体動アーチファクトの削減

MRIの位相エンコード方向に生じた体動アーチファクトの削減

徳島大学医療技術短期大学部紀要


MRIにおけるスピン密度変化の検出 --- ウェーブレット変換によるMR信号の解析

徳島大学医療技術短期大学部紀要


近年, 微小高調波外力の印加によるTaming Chaosと呼ばれるnon-feedbackカオス制御方式が大きな注目を集め, 活発な議論が行われている. これに対し, 筆者らはTaming Chaos現象は微小周期外力の印加により, 分岐曲線がわずかにシフトして生じる単なる分岐現象であることを主張している. 本論文では, その仮説が正しいことを強制BvP方程式を例にとり検証する.

The Transactions of the Institute of Electronics, Information and Communication Engineers A

電子情報通信学会論文誌(A)


A Calculation Method of the Fixed Point with a Specified Argument and Its Application

指定した偏角を有する固定点の計算とその応用

非線形力学系において， 系に含まれるパラメータが変化し，固定点の特性乗数が複素単位円外に 出ることにより各種の局所的分岐現象を生じる． 本報告では，複素共役な特性乗数をもつ固定点に対して， 偏角を指定してその位置，動径，パラメータ値を数値計算する 方法を述べる．また，その応用として，Neimark-Sacker 分岐 パラメータ値を求める計算方法を提案する． 本稿では，連続力学系のポアンカレ写像または離散力学系を対象とし， 指定した偏角をもつ固定点(周期点)の等高線を パラメータ平面で計算する方法を提案する． 固定点条件，共役複素数の特性乗数を代入した特性方程式を 実部と虚部に分離し，それぞれを独立した条件として連立させ， ニュートン法で固定点位置，パラメータ値について解く． また，この手法の応用として，指定した偏角をもつNS分岐を求める 手法について述べる．これらのツールにより， 固定点とNS分岐，周期引き込み領域との関係を より直接的に調べることができる．

電子情報通信学会論文誌(A)


A computation of bifurcation parameter values for limit cycles

This letter describes a new computational method to obtain the bifurcation parameter value of a limit cycle in nonlinear autonomous systems. The method can calculate a parameter value at which local bifurcations; tangent, period-doubling and Neimark-Sacker bifurcations are occurred by using properties of 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 needs only whether the orbit reaches a cross-section or not. In our method, the period is treated as an independent variable for Newton's method, so an accurate location of the fixed point, its period and the bifurcation parameter value can be calculated simultaneously. Although the number of variables increases, the Jacobian matrix becomes simple and the recurrence procedure converges rapidly compared with conventional methods.

本論文では，非線形自律系としてモデル化された力学系について，リミットサイクルの分岐パラメータ値を計算する新たな計算アルゴリズムを提案している．リミットサイクルの周期は，パラメータ値に依存して変化する値であるため本来変数と扱うべきである．従来法ではポアンカレ写像を取るための数値積分時間を得るため，定数と扱っていた．しかし，この扱いにより，むしろ状態やパラメータに関する方程式は周期に対する依存度が高くなり，ヤコビは複雑となり，分岐パラメータ計算アルゴリズムにおいては収束性に影響を与えていた．本アルゴリズムでは周期を変数とし，方程式の独立性も高くなって，収束性能も高くなったことを示している．

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences


周期倍分岐列に生じる樹状パターン

周期倍分岐列に生じる樹状パターン

電子情報通信学会論文誌(A)