Papers - Ei Shin-Ichiro
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Bifurcation of co-existing traveling wave solutions in a three-component competition–diffusion system Reviewed
Shin-Ichiro Ei, Hideo Ikeda, Toshiyuki Ogawa
Physica D: Nonlinear Phenomena 448 133703 - 133703 2023.06
Language：English Publishing type：Research paper (scientific journal) Publisher：Elsevier BV
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Alien invasion into the buffer zone between two competing species Reviewed
Shin-Ichiro Ei, Hideo Ikeda, Toshiyuki Ogawa
Discrete and Continuous Dynamical Systems - B 2023
Language：English Publishing type：Research paper (scientific journal) Publisher：American Institute of Mathematical Sciences (AIMS)
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Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains Reviewed
Shin-Ichiro Ei, Hiroyuki Ochiai, Yoshitaro Tanaka
Journal of Computational and Applied Mathematics 402 113795 - 113795 2022.03
Language：English Publishing type：Research paper (scientific journal) Publisher：Elsevier BV
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Oscillations and bifurcation structure of reaction–diffusion model for cell polarity formation Reviewed
Masataka Kuwamura, HirofumiIzuhara, Shin-ichiro Ei
Journal of Mathematical Biology 84 ( 4 ) 2022.02
Language：English Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC
DOI： 10.1007/s00285-022-01723-5
Other Link： https://link.springer.com/article/10.1007/s00285-022-01723-5/fulltext.html
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Intracellular trafficking of Notch orchestrates temporal dynamics of Notch activity in the fly brain Reviewed
Miaoxing Wang, Xujun Han, Chuyan Liu, Rie Takayama, Tetsuo Yasugi, Shin-Ichiro Ei, Masaharu Nagayama, Yoshitaro Tanaka, Makoto Sato
Nature Communications 12 ( 1 ) 2021.12
Language：English Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC
<title>Abstract</title>While Delta non-autonomously activates Notch in neighboring cells, it autonomously inactivates Notch through <italic>cis</italic>-inhibition, the molecular mechanism and biological roles of which remain elusive. The wave of differentiation in the <italic>Drosophila</italic> brain, the ‘proneural wave’, is an excellent model for studying Notch signaling in vivo. Here, we show that strong nonlinearity in <italic>cis</italic>-inhibition reproduces the second peak of Notch activity behind the proneural wave in silico. Based on this, we demonstrate that Delta expression induces a quick degradation of Notch in late endosomes and the formation of the twin peaks of Notch activity in vivo. Indeed, the amount of Notch is upregulated and the twin peaks are fused forming a single peak when the function of Delta or late endosomes is compromised. Additionally, we show that the second Notch peak behind the wavefront controls neurogenesis. Thus, intracellular trafficking of Notch orchestrates the temporal dynamics of Notch activity and the temporal patterning of neurogenesis.
DOI： 10.1038/s41467-021-22442-3
Other Link： http://www.nature.com/articles/s41467-021-22442-3
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Corrigendum to “Interaction of non-radially symmetric camphor particles” [Physica D 366 (2018) 10–26] (Physica D: Nonlinear Phenomena (2018) 366 (10–26), (S0167278917303603), (10.1016/j.physd.2017.11.004)) Reviewed
Shin Ichiro Ei, Hiroyuki Kitahata, Yuki Koyano, Masaharu Nagayama
Physica D: Nonlinear Phenomena 422 2021.08
Language：English Publishing type：Research paper (scientific journal)
The authors regret that the experimental condition was wrongly described in Section 4 in page 20 and the caption of Fig. 5 in page 21. The concentration of camphor methanol solution was 3 mol/L, not 0.3 mol/L. The authors would like to apologise for any inconvenience caused.
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Correction to: A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices Reviewed International journal
Shin-Ichiro Ei, Hiroshi Ishii, Makoto Sato, Yoshitaro Tanaka, Miaoxing Wang, Tetsuo Yasugi
Journal of Mathematical Biology 82 ( 6 ) 57 2021.05
Language：English Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC
In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta-Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model.
DOI： 10.1007/s00285-021-01610-5
Other Link： https://link.springer.com/article/10.1007/s00285-021-01610-5/fulltext.html
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Effective nonlocal kernels on reaction–diffusion networks Reviewed
Shin-Ichiro Ei, Hiroshi Ishii, Shigeru Kondo, Takashi Miura, Yoshitaro Tanaka
Journal of Theoretical Biology 509 110496 - 110496 2021.01
Language：English Publishing type：Research paper (scientific journal) Publisher：Elsevier BV
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Noise-induced scaling in skull suture interdigitation Reviewed
Yuto Naroda, Yoshie Endo, Kenji Yoshimura, Hiroshi Ishii, Shin-Ichiro Ei, Takashi Miura
PLOS ONE 15 ( 12 ) e0235802 - e0235802 2020.12
Language：English Publishing type：Research paper (scientific journal) Publisher：Public Library of Science (PLoS)
Sutures, the thin, soft tissue between skull bones, serve as the major craniofacial growth centers during postnatal development. In a newborn skull, the sutures are straight; however, as the skull develops, the sutures wind dynamically to form an interdigitation pattern. Moreover, the final winding pattern had been shown to have fractal characteristics. Although various molecules involved in suture development have been identified, the mechanism underlying the pattern formation remains unknown. In a previous study, we reproduced the formation of the interdigitation pattern in a mathematical model combining an interface equation and a convolution kernel. However, the generated pattern had a specific characteristic length, and the model was unable to produce a fractal structure with the model. In the present study, we focused on the anterior part of the sagittal suture and formulated a new mathematical model with time–space-dependent noise that was able to generate the fractal structure. We reduced our previous model to represent the linear dynamics of the centerline of the suture tissue and included a time–space-dependent noise term. We showed theoretically that the final pattern from the model follows a scaling law due to the scaling of the dispersion relation in the full model, which we confirmed numerically. Furthermore, we observed experimentally that stochastic fluctuation of the osteogenic signal exists in the developing skull, and found that actual suture patterns followed a scaling law similar to that of the theoretical prediction.
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Shin-Ichiro Ei, Hiroshi Ishii, Makoto Sato, Yoshitaro Tanaka, Miaoxing Wang, Tetsuo Yasugi
Journal of Mathematical Biology 81 ( 4-5 ) 981 - 1028 2020.11
Language：English Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC
Abstract
In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta–Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model.DOI： 10.1007/s00285-020-01534-6
Other Link： https://link.springer.com/article/10.1007/s00285-020-01534-6/fulltext.html
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Existence of traveling wave solutions to a nonlocal scalar equation with sign-changing kernel Reviewed
Shin-Ichiro Ei, Jong-Shenq Guo, Hiroshi Ishii, Chin-Chin Wu
Journal of Mathematical Analysis and Applications 487 ( 2 ) 124007 - 124007 2020.07
Language：English Publishing type：Research paper (scientific journal) Publisher：Elsevier BV
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Center Manifold Theory for the Motions of Camphor Boats with Delta Function Reviewed
Kota Ikeda, Shin-Ichiro Ei
Journal of Dynamics and Differential Equations - 37 2020.01
Language：English Publishing type：Research paper (scientific journal)
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JAK/STAT guarantees robust neural stem cell differentiation by shutting off biological noise. Reviewed
Tanaka Y, Yasugi T, Nagayama M, Sato M, Ei SI
Scientific reports 8 ( 1 ) 12484 2018.08
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Heterogeneity-induced effects for pulse dynamics Reviewed
Chao-Nien Chen, Shin-Ichiro Ei, Shyuh-yaur Tzeng
PHYSICA D-NONLINEAR PHENOMENA 2018.07
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Interaction of non-radially symmetric camphor particles Reviewed
Shin-Ichiro Ei, Hiroyuki Kitahata, Yuki Koyano, Masaharu Nagayama
Physica D: Nonlinear Phenomena 366 10 - 26 2018.03
Language：English Publishing type：Research paper (scientific journal) Publisher：Elsevier B.V.
In this study, the interaction between two non-radially symmetric camphor particles is theoretically investigated and the equation describing the motion is derived as an ordinary differential system for the locations and the rotations. In particular, slightly modified non-radially symmetric cases from radial symmetry are extensively investigated and explicit motions are obtained. For example, it is theoretically shown that elliptically deformed camphor particles interact so as to be parallel with major axes. Such predicted motions are also checked by real experiments and numerical simulations.
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Masataka Kuwamura, Sungrim Seirin-Lee, Shin-ichiro Ei
SIAM Journal on Applied Mathematics 78 ( 6 ) 3238 - 3257 2018.01
Publishing type：Research paper (scientific journal) Publisher：Society for Industrial & Applied Mathematics (SIAM)
DOI： 10.1137/18m1163749
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JAK/STAT guarantees robust differentiation of neural stem cells by shutting off biological noises in the developing fly brain Reviewed
Makoto Sato, Tetsuo Yasugi, Yoshitaro Tanaka, Masaharu Nagayama, Shin-Ichiro Ei
CYTOKINE 100 127 - 127 2017.12
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Mathematical modeling for meshwork formation of endothelial cells in fibrin gels Reviewed
Daiki Sasaki, Hitomi Nakajima, Yoshimi Yamaguchi, Ryuji Yokokawa, Shin-Ichiro Ei, Takashi Miura
JOURNAL OF THEORETICAL BIOLOGY 429 95 - 104 2017.09
Language：English Publishing type：Research paper (scientific journal) Publisher：ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Vasculogenesis is the earliest process in development for spontaneous formation of a primitive capillary network from endothelial progenitor cells. When human umbilical vein endothelial cells (HUVECs) are cultured on Matrigel, they spontaneously form a network structure which is widely used as an in vitro model of vasculogenesis. Previous studies indicated that chemotaxis or gel deformation was involved in spontaneous pattern formation. In our study, we analyzed the mechanism of vascular pattern formation using a different system, meshwork formation by HUVECs embedded in fibrin gels. Unlike the others, this experimental system resulted in a perfusable endothelial network in vitro. We quantitatively observed the dynamics of endothelial cell protrusion and developed a mathematical model for one-dimensional dynamics. We then extended the one-dimensional model to two-dimensions. The model showed that random searching by endothelial cells was sufficient to generate the observed network structure in fibrin gels. (C) 2017 Elsevier Ltd. All rights reserved.
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ANNIHILATION OF TWO INTERFACES IN A HYBRID SYSTEM Reviewed
Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura, Takashi Teramoto
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S 8 ( 5 ) 857 - 869 2015.10
Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES-AIMS
We consider the mixed ODE-PDE system called a hybrid system, in which the two interfaces interact with each other through a continuous medium and their equations of motion are derived in a weak interaction framework. We study the bifurcation property of the resulting hybrid system and construct an unstable standing pulse solution, which plays the role of a separator for dynamic transition from standing breather to annihilation behavior between two interfaces.
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INSTABILITY OF MULTI-SPOT PATTERNS IN SHADOW SYSTEMS OF REACTION-DIFFUSION EQUATIONS Reviewed
Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 14 ( 2 ) 717 - 736 2015.03
Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES
Our aim in this paper is to prove the instability of multi-spot patterns in a shadow system, which is obtained as a limiting system of a reaction-diffusion model as one of the diffusion coefficients goes to infinity. Instead of investigating each eigenfunction for a linearized operator, we characterize the eigenspace spanned by unstable eigenfunctions.
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APPLICATION OF A CENTER MANIFOLD THEORY TO A REACTION-DIFFUSION SYSTEM OF COLLECTIVE MOTION OF CAMPHOR DISKS AND BOATS Reviewed
S.-I. Ei, K. Ikeda, M. Nagayama, A. Tomoeda
MATHEMATICA BOHEMICA 139 ( 2 ) 363 - 371 2014
Language：English Publishing type：Research paper (scientific journal)
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Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems Reviewed
Shin-Ichiro Ei, Toshio Ishimoto
Networks and Heterogeneous Media 8 ( 1 ) 191 - 209 2013
Language：English Publishing type：Research paper (scientific journal)
We consider pulse-like localized solutions for reaction-diffusion sys- tems on a half line and impose various boundary conditions at one end of it. It is shown that the movement of a pulse solution with the homogeneous Neumann boundary condition is completely opposite from that with the Dirichlet boundary condition. As general cases, Robin type boundary conditions are also considered. Introducing one parameter connecting the Neumann and the Dirichlet boundary conditions, we clarify the transition of motions of solutions with respect to boundary conditions. © American Institute of Mathematical Sciences.
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Dynamics and interactions of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D Reviewed
Shin-Ichiro Ei, Toshio Ishimoto
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 30 ( 1 ) 69 - 90 2013
Language：English Publishing type：Research paper (scientific journal) Publisher：SPRINGER JAPAN KK
It is known that for special types of reaction-diffusion Systems, such as the Gierer-Meinhardt model and the Gray-Scott model, stable stationary spike solutions exist on boundary points with maximal curvature. In this paper, we rigorously give the equation describing the motion of spike solutions along boundaries for general types of reaction-diffusion systems in R-2. We also apply the general results to the Gierer-Meinhardt model and show that a single spike solution moves toward a boundary point with locally maximal curvature. Moreover, by showing the repulsive interaction of spikes along boundaries for solutions of the Gierer-Meinhardt model, we have stable multispike stationary solutions in the neighborhood of a boundary point with locally maximal curvature.
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INFINITE DIMENSIONAL RELAXATION OSCILLATION IN AGGREGATION-GROWTH SYSTEMS Reviewed
Shin-Ichiro Ei, Hirofumi Izuhara, Masayasu Mimura
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B 17 ( 6 ) 1859 - 1887 2012.09
Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES
Two types of aggregation systems with Fisher-KPP growth are proposed. One is described by a normal reaction-diffusion system, and the other is described by a cross-diffusion system. If the growth effect is dominant, a spatially constant equilibrium solution is stable. When the growth effect becomes weaker and the aggregation effect become dominant, the solution is destabilized so that spatially non-constant equilibrium solutions, which exhibit Turing's patterns, appear. When the growth effect weakens further, the spatially non-constant equilibrium solutions are destabilized through Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when the growth effect is extremely weak, there appear spatio-temporal periodic solutions exhibiting infinite dimensional relaxation oscillation.
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DYNAMICS OF A BOUNDARY SPIKE FOR THE SHADOW GIERER-MEINHARDT SYSTEM Reviewed
Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 11 ( 1 ) 115 - 145 2012.01
Language：English Publishing type：Research paper (scientific journal) Publisher：AMER INST MATHEMATICAL SCIENCES-AIMS
The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. The authors of [3] showed that if an initial value is close to a spiky pattern and its peak is far away from the boundary, the solution of the shadow Gierer-Meinhardt system, called a interior spike solution, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a boundary spike solution moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.
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Dynamics of pulses on a thin strip-like domain in R² Reviewed
EI Shin-Ichiro
RIMS Kokyuroku Bessatsu 31 ( B31 ) 195 - 210 2012
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Infinite dimensional relaxation oscillation in reaction-diffusion systems Reviewed
S.-I. Ei, H. Izuhara, M. Mimura
RIMS Kokyuroku Bessatsu 35 ( B35 ) 31 - 40 2012
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A NEW TREATMENT FOR PERIODIC SOLUTIONS AND COUPLED OSCILLATORS Reviewed
Shin-Ichiro Ei, Kunishige Ohgane
KYUSHU JOURNAL OF MATHEMATICS 65 ( 2 ) 197 - 217 2011.09
Language：English Publishing type：Research paper (scientific journal) Publisher：KYUSHU UNIV, FAC MATHEMATICS
We develop a systematic method for deriving the phase dynamics of perturbed periodic solutions. The method is to regard periodic solutions as slowly modulated traveling solutions on the circle. There, problems are reduced to the perturbed problems from stationary solutions on the circle. This makes the treatment of periodic solutions far easier and systematic. We also give the rigorous proofs for this method.
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Front dynamics in heterogeneous diffusive media Reviewed
Hideo Ikeda, Shin-Ichiro Ei
PHYSICA D-NONLINEAR PHENOMENA 239 ( 17 ) 1637 - 1649 2010.09
Language：English Publishing type：Research paper (scientific journal) Publisher：ELSEVIER SCIENCE BV
We herein consider two-component reaction-diffusion systems with a specific bistable and odd symmetric nonlinearity, which have the bifurcation structure of pitchfork type traveling front solutions with opposite velocities. We introduce a spatial heterogeneity, for example, a Heaviside-like abrupt change at the origin in the space, into diffusion coefficients. Numerically, the responses of traveling fronts via the heterogeneity can be classified into four types of behavior depending on the strength of the heterogeneity, which, in the present paper, is represented by the height of the jump: passage, stoppage, and two types of reflection. The goal of the present paper is to reduce the PDE dynamics to finite-dimensional ODE systems on a center manifold and show the mathematical mechanism for producing the four types of response in the PDE systems using finite-dimensional ODE systems. The reduced ODE systems include the terms (referred to as heterogeneous perturbations) originating from the interaction between traveling front solutions and the heterogeneity, which is very important for determining the dynamics of the ODE systems. In the present paper, we succeed in calculating these heterogeneous perturbations exactly and explicitly. (C) 2010 Elsevier B.V. All rights reserved.
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Neuron phase shift adaptive to time delay in locomotor control Reviewed
Kunishige Ohgane, Shin-Ichiro Ei, Hitoshi Mahara
APPLIED MATHEMATICAL MODELLING 33 ( 2 ) 797 - 811 2009.02
Language：English Publishing type：Research paper (scientific journal) Publisher：ELSEVIER SCIENCE INC
Based oil neurophysiological evidence, theoretical studies have shown that walking can be generated by mutual entrainment of oscillations of a central pattern generator (CPG) and a body. However, it has also been shown that the time delay in the sensorimotor loop destabilizes mutual entrainment, and results in the failure to walk. Recently, it has been reported that if (a) the neuron model used to construct the CPG is replaced by physiologically faithful neuron model (Bonhoeffer-Van der Pol type) and (b) the mechanical impedance of the body (muscle viscoelasticity) is controlled depending oil the angle between two legs, the phase relationship between CPG activity and body motion could be flexibly locked according to the loop delay and, therefore, mutual entrainment can be stabilized. That is, locomotor control adaptive to the loop delay can emerge from the coupling between CPG and body. Here, we call this mechanism flexible-phase locking. In this paper, we construct a system of coupled oscillators as a simplified model of a walking system to theoretically investigate the mechanism of flexible-phase locking, and to analyze the simplified model. The analysis suggests that the following are required as the essential mechanism: (i) an asymptotically stable limit cycle of the coupling system of CPG and body and (ii) a sign difference between afferent and efferent coupling coefficients. (C) 2007 Elsevier hic. All rights reserved.
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Eigenfunctions of the adjoint operator associated with a pulse solution of some reaction-diffusion systems Reviewed
S.-I. Ei, H. Ikeda, K. Ikeda, E. Yanagida
Bull. Inst. Math. Academia Sinica 3 603 - 666 2008
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'Initial state' coordinations reproduce the instant flexibility for human walking Reviewed
A Ohgane, K Ohgane, S Ei, H Mahara, T Ohtsuki
BIOLOGICAL CYBERNETICS 93 ( 6 ) 426 - 435 2005.12
Language：English Publishing type：Research paper (scientific journal) Publisher：SPRINGER
An important feature of human locomotor control is the instant adaptability to unpredictable changes of conditions surrounding the locomotion. Humans, for example, can seamlessly adapt their walking gait following a sudden ankle impairment (e.g., as a result of an injury). In this paper, we propose a theoretical study of the mechanisms underlying flexible locomotor control. We hypothesize that flexibility is achieved by modulating the posture at the beginning of the stance phase-the initial state. Using a walking model, we validate our hypothesis through computer simulations.
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A variational approach to singular perturbation problems in reaction-diffusion systems Reviewed
SI Ei, M Kuwamura, Y Morita
PHYSICA D-NONLINEAR PHENOMENA 207 ( 3-4 ) 171 - 219 2005.08
Language：English Publishing type：Research paper (scientific journal) Publisher：ELSEVIER SCIENCE BV
In this paper singular perturbation problems in reaction-diffusion systems are studied from a viewpoint of variational principle. The goal of the study is to provide an unified and transparent framework to understand existence, stability and dynamics of solutions with transition layers in contrast to previous works in many literatures on singular perturbation theory. (c) 2005 Elsevier B.V. All rights reserved.
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Emergence of adaptability to time delay in bipedal locomotion Reviewed
K Ohgane, S Ei, K Kazutoshi, T Ohtsuki
BIOLOGICAL CYBERNETICS 90 ( 2 ) 125 - 132 2004.02
Language：English Publishing type：Research paper (scientific journal) Publisher：SPRINGER-VERLAG
Based on neurophysiological evidence, theoretical studies have shown that locomotion is generated by mutual entrainment between the oscillatory activities of central pattern generators (CPGs) and body motion. However, it has also been shown that the time delay in the sensorimotor loop can destabilize mutual entrainment and result in the failure to walk. In this study, a new mechanism called flexible-phase locking is proposed to overcome the time delay. It is realized by employing the Bonhoeffer-Van der Pol formalism - well known as a physiologically faithful neuronal model - for neurons in the CPG. The formalism states that neurons modulate their phase according to the delay so that mutual entrainment is stabilized. Flexible-phase locking derives from the phase dynamics related to an asymptotically stable limit cycle of the neuron. The effectiveness of the mechanism is verified by computer simulations of a bipedal locomotion model.
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Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions Reviewed
SI Ei, JC Wei
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 19 ( 2 ) 181 - 226 2002.06
Language：English Publishing type：Research paper (scientific journal) Publisher：SPRINGER JAPAN KK
In this paper, the Gierer-Meinhardt model systems with finite diffusion constants in the whole space R-2 is considered. We give a regorous proof on the existence and the stability of a single spike solution, and by using such informations, the repulsive dynamics of the interacting multi single-spike solutions is also shown when distances between spike solutions are sufficiently large. This clarifies some part of the mechanism of the evolutional process of localized patterns appearing in the Gierer-Meinhardt model equations.
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2n-Splitting or Edge-Splitting? A Manner of Splitting in Dissipative Systems Reviewed
Shin-Ichiro Ei, Yasumasa Nishiura, Kei-Ichi Ueda
Japan Journal of Industrial and Applied Mathematics 18 ( 2 ) 181 - 205 2001
Language：English Publishing type：Research paper (scientific journal) Publisher：Kinokuniya Co. Ltd
Since early 90's, much attention has been paid to dynamic dissipative patterns in laboratories, especially, self-replicating pattern (SRP) is one of the most exotic phenomena. Employing model system such as the Gray-Scott model, it is confirmed also by numerics that SRP can be obtained via destabilization of standing or traveling spots. SRP is a typical example of transient dynamics, and hence it is not a priori clear that what kind of mathematical framework is appropriate to describe the dynamics. A framework in this direction is proposed by Nishiura-Ueyama [16], i.e., hierarchy structure of saddle-node points, which gives a basis for rigorous analysis. One of the interesting observation is that when there occurs self-replication, then only spots (or pulses) located at the boundary (or edge) are able to split. Internal ones do not duplicate at all. For 1D-case, this means that the number of newly born pulses increases like 2k after k-th splitting, not 2n-splitting where all pulses split simultaneously. The main objective in this article is two-fold: One is to construct a local invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other is to study the manner of splitting by analysing the resulting ODE, and answer the question "2n-splitting or edge-splitting?" starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natural consequence from a physical point of view, because the pulses at edge are easier to access fresh chemical resources than internal ones.
DOI： 10.1007/BF03168570
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Pattern formation in heterogeneous reaction-diffusion- advection systems with an application to population dynamics Reviewed
S.-I. Ei, M. Mimura
SIAM J. Math. Anal. 21 ( 346 ) 361 1990