数学と現象： - 宮崎大学 情報基盤センター

研究集会
数学と現象：
Mathematics and Phenomena
in Miyazaki 2016
日時：平成２８年１１月１８日∼１１月１９日
場所：宮崎大学工学部 B棟２階B２１０教室
世話人：辻川亨, 飯田雅人, 今隆助, 梅原守道,
平山浩之, 坂田繁洋, 出原浩史（宮崎大学）
URL: http://www.cc.miyazaki-u.ac.jp/math/mpm/mpm2016/
研究集会 「数学と現象： Mathematics and Phenomena
in Miyazaki 2016 (略称：MPM2016)」
日時：
2016 年 11 月 18 日（金）∼ 11 月 19 日（土）
会場：
宮崎大学工学部 B 棟 2 階 B210 教室
案内：
http://www.cc.miyazaki-u.ac.jp/math/mpm/
プログラム
11 月 18 日（金）
午後の部
14:00-14:55
本多 泰理（NTT）
「On synchronization and brain network」
15:15-16:10
可香谷 隆（東京工業大学）
「界面ダイナミクスに対する接触エネルギーの効果について」
16:30-17:25
櫻井 建成（千葉大学）
「走化性大腸菌における進行パルス波と沈殿」
11 月 19 日（土）
午前の部 ! MPM2016 特別体験講座 "
10:15-12:15 石田 祥子（明治大学）
「折紙の数理と構造設計」
注
宮交バス「橘通り 3 丁目→宮崎大学（木花キャンパス）」の土曜日の朝の時刻表：
8:21 → 8:54，8:41 → 9:18，9:11 → 9:52，9:21 → 9:54，9:41 → 10:18（遅刻！）
午後の部
14:00-14:55
中田 行彦（島根大学）
「Delay equations for epidemic models : instability due to waning immunity」
15:15-16:10
吉川 周二（大分大学）
「Structure-preserving finite difference schemes for the Cahn-Hilliard equation
with dynamic boundary conditions in the one-dimensional case」
16:30-17:25
好村 滋行（首都大学東京）
「細胞中の異常拡散とマイクロレオロジー」
本研究集会は，科学研究費補助金
課題番号
種目
代表者
26400173
15K04963
16K05279
16KT0135
26800071
26800084
基盤 (C)
基盤 (C)
基盤 (C)
基盤 (C)
若手 (B)
若手 (B)
辻川亨
飯田雅人
今隆助
出原浩史
梅原守道
出原浩史
課題名
縮約系を応用した高次元空間にみられる現象の解明と解析的手法の構築
漸近解構築に基づく反応拡散系の解の形と動きの解明
常微分方程式で近似できる構造化生態系モデルの数理的研究
生命現象における階層を超えるミクロとマクロとをつなぐ理論の構築
天文現象における自己重力流体の運動の数学解析
生物の集合形成メカニズムに対する数理モデルからの探求
の援助を受けています．
世話人：
連絡先：
辻川 亨，飯田 雅人，今 隆助，梅原 守道，平山 浩之，坂田 繁洋，出原 浩史（宮崎大学）
出原 浩史 (Hirofumi Izuhara)
〒 889-2192 宮崎市学園木花台西 1-1 宮崎大学工学部工学基礎教育センター
E-mail：[email protected]
TEL：0985-58-7384 ／ 0985-58-7288（事務室）& FAX：0985-58-7289
作成日：2016.11.2
On synchronization and brain network
Hirotada HONDA (NTT Network Technology Laboratories)§
1. Introduction
Theoretical investigations of weakly coupled limit cycle oscillators [6] are intensively
developed over several research fields these days. For example, in network science,
various network models are being taken into account.
It is noteworthy that the mathematical analysis of this field has been promoted
recently. In this talk, we first introduce some of our results concerning the KuramotoSakaguchi equation. Then, as an application, we show the equation of the resting state
network, which is one of the most attractive topics in the brain network analysis these
days.
2. Kuramoto-Sakaguchi equation
The Kuramoto-Sakaguchi equation is a physical model of the behavior of weakly coupled oscillators. It describes the temporal evolution of the probability density of the
phase of each oscillator.
In this section, we first introduce some existing results concerning the KuramotoSakaguchi equation.
8
@%
@ 2%
@%
>
>
°
D
+!
>
>
2
>
@t
@µ
@µ
>
"
#
>
>
Z
Z 2º
>
>
@
>
0
0
0
>
+K
%(µ, !, t)
g(! )d!
sin(¡ ° µ)%(¡, ! , t)d¡ = 0,
>
>
@µ
<
R
0
(1)
(µ, !, t) 2 Ω £ R £ (0, 1),
>
>
>
>
>
>
@ i % ØØ
@ i % ØØ
>
>
=
(i = 0, 1), (!, t) 2 R £ (0, 1),
Ø
Ø
>
>
@µi µ=0 @µi µ=2º
>
>
>
>
: ØØ
% t=0 = %0 (µ, !), (µ, !) 2 (0, 2º) £ R,
where Ω ¥ (0, 2º), g(!) is the probability distribution function of the natural frequency
!, D, the diffusion coefficient, and K is the coupling strength.
3. Resting state network : application of Kuramoto model
In this section, we consider another problem concerning the brain network, derived on
the basis of Cabral’s works [1][2] as an application of the Kuramoto theory.
In the region of the brain analysis, it is reported that a synchronous cooperation
of multiple regions emerges when the individuals are at rest. They are now called as
the resting state network. Recently, Cabral [1][2] derived a system of ordinary equation
as a model of the average neuronal behavior in each region of the brain in the resting
state network. It reads
N
X
°
¢
dµn
= !n + K
cnp sin µp (t ° ønp ) ° µn (t) (n = 1, 2, . . . , N ),
dt
p=1
2000 Mathematics Subject Classification: 45K05, 45M10.
Keywords: Kuramoto-Sakaguchi equation, brain network, resting state network..
§
e-mail: [email protected]
(2)
where the unknown µn (t) (n = 1, 2, . . . , N ) are the average phase of the neuronal firing
in each region numbered n at time t; cnp , the coupling strength between neurons n and
p; K, the global coupling strength which scales all connection strength; ønp , the delay
in the axon between neurons n and p, and !n , the natural frequency of the signal of a
neuron numbered n.
In [4], we derived a Fokker-Planck equation corresponding to the equation (2), and
discussed the mathematical well-posedness, stability and vanishing diffusion limit.
On the other hand, since the resting state network contains the dynamical property,
the dynamics of the model has been left for further study [2].
Here, we consider a dynamical model of the resting state network. On the basis of
existing models of modified versions of Kuramoto-Saguchi equation [5][7], it reads
8
@%
@%
@ 2 % kK(t) @
>
>
+
!
°
D
+
F[%, %] = 0
>
>
>
@t
@¡
@¡2
P (t) @¡
>
>
>
>
t > 0, (¡, k, x, !) 2 Ω £ R+ £ R £ R,
<
(3)
>
@ i % ØØ
@ i % ØØ
>
>
=
(i = 0, 1), t > 0, (k, x, !) 2 R+ £ R £ R
Ø
Ø
>
>
@¡i ¡=0 @¡i ¡=2º
>
>
>
>
: ØØ
% t=0 = %0 (¡, k, x, !) 2 Ω £ R+ £ R £ R,
where
F[%1 , %2 ] ¥ %1 (¡, t; x, !)
Z
R
G(x ° y)dy
Z
0
2º
Z
R
0
g(! )d!
0
Z
k 0 P (k 0 )dk 0
R+
Γ(¡0 ° ¡)%2 (¡0 , æ(t; x, y); k 0 , y, ! 0 ) d¡0 ,
P (t; k) and g(!) are the probability densities of the node degree k and natural frequency
!, respectively; D, the diffusion coefficient; Γ(·), the coupling function; K(t), the
coupling strength; G(·, t), the coupling strength
between each node; æ(t; x, y), the
R
delay between nodes x and y, and P (t) ¥ R+ kP (t; k)dk is the expected value of the
degree.
References
[1] J. Cabral, E. Hugues, O. Sporns and G. Deco, Role of local network oscillations in
resting-state functional connectivity, NeuroImage, 57 (2011), 130–139.
[2] J. Cabral, M. L. Kringelbach and G. Deco, Exploring the network dynamics underlying
brain activity during rest, Progress in Neurobiology, 114 (2014), 102–131.
[3] Ha, S. Y. and Xiao, Q., Remarks on the nonlinear stability of the Kuramoto-Sakaguchi
equation, J. Diff. Eq. 259 (2015), 2430–2457.
[4] H. Honda, On mathematical modeling and analysis of brain network, preprnit.
[5] Ichinomiya, T., Frequency synchronization in a random oscillator network, Phys. Rev. E,
70 026116 (2004).
[6] Kuramoto, Y., in Int. Symp. on Mathematical problems in theoretical physics, edited by
H. Araki (Springer, New York), Lect. N. Phys., 39 (1975), 420–422.
[7] W.S.Lee, E.Odd and T.M.Antonsen, Large Coupled Oscillator Systems with Heterogeneous Interaction Delays, Phys. Rev. Lett., 103 (2009), 044101.
(
)
∗
A⊂Ω
E(A) := Hn−1 (∂A ∩ Ω) + cos θHn−1 (∂A ∩ ∂Ω).
Ω ⊂ Rn
n
(1)
(1)
θ ∈ (0, π)
0 < m < |Ω|
1:
Σm := {A ⊂ Ω : |A| = m}
(1)
∂Ω
θ
(i)
A ∈ Σm
Σm
(
1
A
∂A ∩ Ω
)
(1)
Ω = {(x, y) ∈ R2 : y > 0}
A(0) ⊂ Ω
[1]
∂Ω
(1)
Γ := ∂A ∩ Ω
x
l− (Γ), l+ (Γ)
ψ− , ψ+ ∈ (0, π/2)
2:
1
Ẽ(A) := H (Γ) − l+ (Γ) cos ψ+ + l− (Γ) cos ψ−
A(0)
Γ(t) := ∂A(t) ∩ Ω
(Eq) V
(BC) Γ(t)
(IC) Γ(0)
∗ e-mail:
x
κ
2
A(t)
Γ(t)
Γ(t)
x
(
Ẽ
Γ(t)
!
A(t)
Γ(t)
V =κ− !
2
)
κ dH1
Γ(t)
C2
dH1
on Γ(t).
A(t)
ψ− , ψ+ ∈ (0, π/2)
(BC)
[email protected]
1
(ii)
[3]
ε>0
!
!
ε|∇u|2
W (u)
Eε (u) :=
+
dx +
σ(u) dHn−1
2
ε
Ω
∂Ω
u
Ω
W
R
σ
Eε
{uε ≈ −1}
(
3
W (s) = (1 − s2 )2 )
double well potential(
uε
{uε ≈ 1}
Ω
{uε ≈ 0}
)
ε→0
uε → u0 = ±1
∃m s.t. {u0 = 1}
E
Hn -a.e. x ∈ Ω as
for
Σm
3:
uε
ε → 0,
E
(2)
.
θ
σ(1) − σ(−1)
θ = arccos " 1 #
2W (s) ds
−1
(A1) Ω
R
(A2) W
W ≥ 0, W (±1) = 0,
(−1, 1)
R
(A3) σ
C ∈ [0, 1)
(A4) εi → 0(i ∈ N, i → ∞)
(uεi
ν
|σ " (s)| ≤ C
uεi
#
2W (s)
λi
Ẽ
)

"

−εi ∆uε + W (uεi ) = λε
i
i
εi

εi *∇uε , ν+ = −σ " (uε )
i
i
Ω
(A5) ,uεi ,L∞ (Ω) , |λi |, Eεi (uεi )
(A6)
W "" (s) > 0(|s| ≥ γ)
γ ∈ (0, 1)
εi |∇uε |2
W (uεi )
−
2
εi
in Ω,
on ∂Ω.
i
L1 (Ω)
0
varifold
(2)
varifold
[2]
[1] T. Kagaya and M. Shimojo, Exponential stability of a traveling wave for an area preserving curvature
motion with two endpoints moving freely on a line, Asymptotic Analysis 96 (2016), no. 2, pp 109–134.
[2] T. Kagaya and Y. Tonegawa, A fixed contact angle condition for varifolds, submitted, arXiv:1606.00164.
[3] T. Kagaya and Y. Tonegawa, A singular perturbation limit of diffused interface energy with a fixed contact
angle condition, submitted, arXiv:1609.00191.
2
Swarm ring
REFERENCES
[1] E.O. Budrene and H.C. Berg, Nature 349, 630 (1995)
[2] L. Tsimring, H. Levine, I. Aranson, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 79,
1859 (1995)
[3] R. Tyson, S.R. Lubkin, and J.D. Murray, Proc. R. Soc. Lond. 266, 299 (1999)
[4] A. Aotani, M Mimura, and T. Mollee, Japan J. Indust. Appl. Math. 27, 5 (2010)
DELAY EQUATIONS FOR EPIDEMIC MODELS: INSTABILITY
DUE TO WANING IMMUNITY
YUKIHIKO NAKATA
Abstract. The talk will begin with introduction of ”delay equation formulation” of structured population models using the Kermack and McKendrick
epidemic model as an example [1, 2]. We briefly review analytical results (such
as a threshold dynamics and final size relation) and then extend the Kermack
and McKendrick model to describe reinfection due to waning immunity of
recovered individuals. Together with stability analysis, it is suggested that
recovered period is the source of destabilisation of the system. We also discuss
global stability of equilibria and periodicity of a periodic solution for specific
cases. Comparison of two models will illustrate importance of the immune period and its distribution in the periodic oscillation of infectious diseases. The
talk is based on the collaboration studies with R. Omori and G. Röst [3, 4].
References
[1] D. Breda, O. Diekmann, W.F. de Graaf, A. Pugliese, R. Vermiglio, On the formulation of
epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn. 6, Suppl. 2 (2012)
103-117.
[2] W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics.
Proc. R. Soc. Lond. B Biol. Sci. 115 (1927) 700-721.
[3] R. Omori, Y. Nakata, H. L. Tessmer, S. Suzuki, K. Shibayama, The determinant of periodicity
in Mycoplasma pneumoniae incidence: an insight from mathematical modelling. Scientific
Reports 5, 14473; doi: 10.1038/srep14473 (2015)
[4] Y. Nakata, G. Rost, Global stability of an SIS epidemic model with a finite infectious period,
submitted
Department of Mathematics, Shimane University, Nishikawatsu-cho 1060,
Matsue, 690-8504
1
Structure-preserving finite difference schemes for the Cahn-Hilliard equation
with dynamic boundary conditions in the one-dimensional case
Shuji Yoshikawa
Oita University, Oita, Japan
email: [email protected]
Abstract.
(CH)
We study the following one-dimensional Cahn–Hilliard equation:
!
∂t u − ∂x2 p = 0,
p = −γ∂x2 u + F ! (u),
(x, t) ∈ (0, L) × (0, T ],
under two kinds of dynamic boundary condition. The unknowns u := u(x, t) and p := p(x, t)
are the order parameter and the chemical potential, respectively, F ! is some derivative of the
potential F , and γ is a positive constant. The first problem, under the standard dynamic
boundary condition, is of the following form:
!
∂t u(0, t) + ∂x u(0, t) = ∂t u(L, t) − ∂x u(L, t) = 0,
(DBC1)
∂x p(0, t) = ∂x p(L, t) = 0,
t ∈ (0, T ].
The second problem, recently proposed by Goldstein–Miranville–Schimperna [4], is


∂t u(0, t) + ∂x p(0, t) = ∂t u(L, t) − ∂x p(L, t) = 0,
(DBC2)
p(0, t) = γ∂x u(0, t) + F ! (u(0, t)),


p(L, t) = −γ∂x u(L, t) + F ! (u(L, t)) = 0,
t ∈ (0, T ].
We propose structure-preserving finite difference schemes (see [2] and [3]) for problems (CH)
with (DBC1) and (CH) with (DBC2), and give mathematical results such as an error estimate
and unique existence of solution for the scheme, by using the energy method introduced in [5].
This result is based on a collaborated research with Prof. Takeshi Fukao (Kyoto University of
Education) and Ms. Saori Wada (Ehime University), which has been submitted as [1].
Bibliography
[1] T. Fukao, S. Yoshikawa and S. Wada, Structure-preserving finite difference schemes for the Cahn-Hilliard
equation with dynamic boundary conditions in the one-dimensional case, submitted.
[2] D. Furihata, A stable and conservative finite difference scheme for the Cahn–Hilliard equation, Numer.
Math., 87 (2001), 675–699.
[3] D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Numerical Analysis and Scientific Computing series, CRC Press/Taylor & Francis, 2010.
[4] G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn–Hilliard model in a domain with non-permeable
walls, Phys. D, 240 (2011), 754–766.
[5] S. Yoshikawa, Energy method for structure-preserving finite difference schemes and some properties of difference quotient, Journal of Computational and Applied Mathematics 311, 394-413 (2017).
Anomalous diffusion and microrheology in cells
(Tokyo Metropolitan University) S. Komura, K. Yasuda, and R. Okamoto
(Fritz-Haber-Institut der Max-Planck-Gesellschaft) A. S. Mikhailov
With the use of the “two-fluid model”, we discuss anomalous diffusion induced by active
protein molecules in viscoelastic media. Active proteins in living cells generate non-thermal
fluctuating flows that lead to a substantial increment of the diffusion in the cytoplasm [1].
Using the Green's function of the two-fluid model, we first obtain passive (thermal) two-point
correlation functions including the displacement cross-correlation function between the two
point particles separated by a finite distance. We then calculate active (non-thermal) one-point
and two-point correlation functions due to active force dipoles representing proteins. The time
correlation of the force dipole is assumed to decay exponentially with a characteristic time
scale. We show that the active component of the displacement cross-correlation function
exhibits various crossovers from super-diffusive to sub-diffusive behaviors depending on the
characteristic time scales and the particle separation. Our theoretical results are intimately
connected to the microrheology technique, and also reproduce the experimental result [2] by
adding both passive and active contributions to the mean squared displacement.
µ
η
d
!
Figure 1: Active proteins in a viscoelastic media
References
[1] A. S. Mikhailov and R. Kapral, Proc. Nat. Acad. Sci. USA 112, E3639 (2015).
[2] M. Guo, A. J. Ehrlicher, M. H. Jensen, M. Renz, J. R. Moore, R. D. Goldman, J.
Lippincott-Schwartz, F. C. MacKintosh, and D. A. Weitz, Cell 158, 822 (2014).