On the curved surfaces of living and nonliving materials, planar excitable wavefronts frequently exhibit a directional change and subsequently undergo a discontinuous (topological) change; i.e., a series of wavefront dynamics from collision, annihilation to splitting. Theoretical studies have shown that excitable planar stable waves change their topology significantly depending on the initial conditions on flat surfaces, whereas the directional change of the waves occurs based on the geometry of curved surfaces. However, it is not clear if the geometry of curved surfaces induces this topological change. In this study, we first demonstrated that the curved surface geometry induces bending, collision, and splitting of a planar stable wavefront by numerically solving an excitable reaction–diffusion equation on a bell-shaped surface. We determined two necessary conditions for inducing the topological change: the characteristic length of the curved surface (i.e., the height of the bell-shaped structure) should be greater than the width of the wave, and the ratio of the height to the width of the bell shape should be greater than a threshold. As for the geometrical mechanism of the latter, we found that a bifurcation of the geodesics on the curved surface provides the alternative minimal paths of the wavefront, which circumvent the surface region with a high local curvature, thereby resulting in the topological change. These conditions imply that the topological change of the wavefront can be predicted on the basis of the curved surfaces, whose structures are larger than the wave width.

1.
A. T.
Winfree
, The Geometry of Biological Time, Interdisciplinary Applied Mathematics Vol. 12 (Springer, New York, NY, 2001).
2.
D.
Taniguchi
,
S.
Ishihara
,
T.
Oonuki
,
M.
Honda-Kitahara
,
K.
Kaneko
, and
S.
Sawai
, “
Phase geometries of two-dimensional excitable waves govern self-organized morphodynamics of amoeboid cells
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
5016
5021
(
2013
).
3.
G.
Gerisch
,
M.
Ecke
,
B.
Schroth-Diez
,
S.
Gerwig
,
U.
Engel
,
L.
Maddera
, and
M.
Clarke
, “
Self-organizing actin waves as planar phagocytic cup structures
,”
Cell Adhes. Migr.
3
,
373
382
(
2009
).
4.
G.
Gerisch
,
B.
Schroth-Diez
,
A.
Müller-Taubenberger
, and
M.
Ecke
, “
PIP3 waves and PTEN dynamics in the emergence of cell polarity
,”
Biophys. J.
103
,
1170
1178
(
2012
).
5.
M.
Gerhardt
,
M.
Ecke
,
M.
Walz
,
A.
Stengl
,
C.
Beta
, and
G.
Gerisch
, “
Actin and PIP3 waves in giant cells reveal the inherent length scale of an excited state
,”
J. Cell Sci.
127
,
4507
4517
(
2014
).
6.
Y.
Arai
,
T.
Shibata
,
S.
Matsuoka
,
M. J.
Sato
,
T.
Yanagida
, and
M.
Ueda
, “
Self-organization of the phosphatidylinositol lipids signaling system for random cell migration
,”
Proc. Natl. Acad. Sci. U.S.A.
107
,
12399
12404
(
2010
).
7.
H. G.
Döbereiner
,
B. J.
Dubin-Thaler
,
J. M.
Hofman
,
H. S.
Xenias
,
T. N.
Sims
,
G.
Giannone
,
M. L.
Dustin
,
C. H.
Wiggins
, and
M. P.
Sheetz
, “
Lateral membrane waves constitute a universal dynamic pattern of motile cells
,”
Phys. Rev. Lett.
97
,
038102
(
2006
).
8.
M.
Hörning
and
T.
Shibata
, “
Three-dimensional cell geometry controls excitable membrane signaling in dictyostelium cells
,”
Biophys. J.
116
,
372
382
(
2019
).
9.
S.
Alonso
,
M.
Bar
, and
B.
Echebarria
, “
Nonlinear physics of electrical wave propagation in the heart: A review
,”
Rep. Prog. Phys.
79
,
96601
(
2016
).
10.
S.
Chun
, “
A mathematical model of the unidirectional block caused by the pulmonary veins for anatomically induced atrial reentry
,”
J. Biol. Phys.
40
,
219
258
(
2014
).
11.
A.
Neic
,
F. O.
Campos
,
A. J.
Prassl
,
S. A.
Niederer
,
M. J.
Bishop
,
E. J.
Vigmond
, and
G.
Plank
, “
Efficient computation of electrograms and ECGs in human whole heart simulations using a reaction-eikonal model
,”
J. Comput. Phys.
346
,
191
211
(
2017
).
12.
E. V.
Lubenov
and
A. G.
Siapas
, “
Hippocampal theta oscillations are travelling waves
,”
Nature
459
,
534
539
(
2009
).
13.
G.
Agarwal
,
I. H.
Stevenson
,
A.
Berenyi
,
K.
Mizuseki
,
G.
Buzsaki
, and
F. T.
Sommer
, “
Spatially distributed local fields in the hippocampus encode rat position
,”
Science
344
,
626
630
(
2014
).
14.
D.
Rubino
,
K. A.
Robbins
, and
N. G.
Hatsopoulos
, “
Propagating waves mediate information transfer in the motor cortex
,”
Nat. Neurosci.
9
,
1549
1557
(
2006
).
15.
T. P.
Zanos
,
P. J.
Mineault
,
K. T.
Nasiotis
,
D.
Guitton
, and
C. C.
Pack
, “
A sensorimotor role for traveling waves in primate visual cortex
,”
Neuron
85
,
615
627
(
2015
).
16.
L. E.
Martinet
,
G.
Fiddyment
,
J. R.
Madsen
,
E. N.
Eskandar
,
W.
Truccolo
,
U. T.
Eden
,
S. S.
Cash
, and
M. A.
Kramer
, “
Human seizures couple across spatial scales through travelling wave dynamics
,”
Nat. Commun.
8
,
14896
(
2017
).
17.
E.
Santos
,
M.
Schöll
,
R.
Sánchez-Porras
,
M. a.
Dahlem
,
H.
Silos
,
A.
Unterberg
,
H.
Dickhaus
, and
O. W.
Sakowitz
, “
Radial, spiral and reverberating waves of spreading depolarization occur in the gyrencephalic brain
,”
NeuroImage
99
,
244
255
(
2014
).
18.
V. M.
Verkhlyutov
and
V. V.
Balaev
, “
A novel methodology for simulation of EEG traveling waves on the folding surface of the human cerebral cortex
,” in
International Conference on Neuroinformatics
(
Springer
,
2018
), pp.
51
63
.
19.
J. A.
Roberts
,
L. L.
Gollo
,
R. G.
Abeysuriya
,
G.
Roberts
,
P. B.
Mitchell
,
M. W.
Woolrich
, and
M.
Breakspear
, “
Metastable brain waves
,”
Nat. Commun.
10
,
1056
(
2019
).
20.
S.
Heitmann
,
T.
Boonstra
, and
M.
Breakspear
, “
A dendritic mechanism for decoding traveling waves: Principles and applications to motor cortex
,”
PLoS Comput. Biol.
9
,
e1003260
(
2013
).
21.
M.
Mimura
and
M.
Nagayama
, “
Nonannihilation dynamics in an exothermic reaction-diffusion system with mono-stable excitability
,”
Chaos
7
,
817
826
(
1997
).
22.
S.
Ei
,
M.
Mimura
, and
M.
Nagayama
, “
Pulse–pulse interaction in reaction–diffusion systems
,”
Physica D
165
,
176
198
(
2002
).
23.
C. K. R. T.
Jones
, “
Stability of the travelling wave solution of the FitzHugh-Nagumo system
,”
Trans. Am. Math. Soc.
286
,
431
469
(
1984
).
24.
T.
Tsujikawa
,
T.
Nagai
,
M.
Mimura
,
R.
Kobayashi
, and
H.
Ikeda
, “
Stability properties of traveling pulse solutions of the higher dimensional FitzHugh-Nagumo equations
,”
Jpn. J. Appl. Math.
6
,
341
366
(
1989
).
25.
Y.
Biton
,
A.
Rabinovitch
,
I.
Aviram
, and
D.
Braunstein
, “
Two mechanisms of spiral-pair-source creation in excitable media
,”
Phys. Lett. A
373
,
1762
1767
(
2009
).
26.
P. K.
Brazhnik
,
V. A.
Davydov
, and
A. S.
Mikhailov
, “
Kinematic approach to the description of autowave processes in active media
,”
Theor. Mat. Phys.
74
,
440
447
(
1988
).
27.
V.
Davydov
and
V.
Zykov
, “
Kinematics of spiral waves on nonuniformly curved surfaces
,”
Physica D
49
,
71
74
(
1991
).
28.
V.
Davydov
,
V.
Morozov
, and
N.
Davydov
, “
Ring-shaped autowaves on curved surfaces
,”
Phys. Lett. A
267
,
326
330
(
2000
).
29.
V. A.
Davydov
,
N.
Manz
,
O.
Steinbock
, and
S. C.
Müller
, “
Critical properties of excitation waves on curved surfaces: Curvature-dependent loss of excitability
,”
Europhys. Lett.
59
,
344
350
(
2002
).
30.
V. A.
Davydov
,
N.
Manz
,
O.
Steinbock
,
V. S.
Zykov
, and
S. C.
Müller
, “
Excitation fronts on a periodically modulated curved surface
,”
Phys. Rev. Lett.
85
,
868
871
(
2000
).
31.
V. A.
Davydov
,
V. G.
Morozov
, and
N. V.
Davydov
, “
Critical properties of autowaves propagating on deformed cylindrical surfaces
,”
Phys. Lett. A
307
,
265
268
(
2003
).
32.
F.
Kneer
,
E.
Schöll
, and
M. A.
Dahlem
, “
Nucleation of reaction-diffusion waves on curved surfaces
,”
New J. Phys.
16
,
053010
(
2014
).
33.
C. B.
Macdonald
,
B.
Merriman
, and
S. J.
Ruuth
, “
Simple computation of reaction–diffusion processes on point clouds
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
9209
9214
(
2013
).
34.
Nonlinear Wave Processes in Excitable Media, edited by A. V. Holden, M. Markus, and H. G. Othmer, NATO ASI Series Vol. 244 (Springer, Boston, MA, 1991).
35.
J.
Nagumo
,
S.
Arimoto
, and
S.
Yoshizawa
, “
An active pulse transmission line simulating nerve axon
,”
Proc. IRE
50
,
2061
2070
(
1962
).
36.
R.
FitzHugh
, “
Impulses and physiological states in theoretical models of nerve membrane
,”
Biophys. J.
1
,
445
466
(
1961
).
37.
M.
Kazhdan
,
M.
Bolitho
, and
H.
Hoppe
, “Poisson surface reconstruction,” in
Proceedings of the Symposium on Geometry Processing
(
The Eurographics Assoc.
,
2006
), pp.
61
70
.
38.
J. H.
Ferziger
and
M.
Perić
,
Computational Methods for Fluid Dynamics
(
Springer Science & Business Media
,
2002
).
39.
D.
Geometry
,
F.
de Goes
,
K.
Crane
,
M.
Desbrun
, and
P.
Schröder
, “Digital geometry processing with discrete exterior calculus,” in ACM SIGGRAPH 2013 Courses on SIGGRAPH ’13 (ACM Press, New York, 2013), Vol. 31, pp. 1–126.
40.
A.
de Padua
,
F.
Parisio-Filho
, and
F.
Moraes
, “
Geodesics around line defects in elastic solids
,”
Phys. Lett. A
238
,
153
158
(
1998
).
41.
J. C.
Butcher
, “
Implicit Runge-Kutta processes
,”
Math. Comput.
18
,
50
64
(
1964
).
42.
R.
Martin
,
D. J.
Chappell
,
N.
Chuzhanova
, and
J. J.
Crofts
, “
A numerical simulation of neural fields on curved geometries
,”
J. Comput. Neurosci.
45
,
133
145
(
2018
).
43.
V.
Zykov
and
O.
Morozova
, “
Speed of spread of excitation in two-dimensional excitable medium
,”
Biofizika
24
,
739
744
(
1979
).
44.
Y.
Kuramoto
, “
Instability and turbulence of wavefronts in reaction-diffusion systems
,”
Prog. Theor. Phys.
63
,
1885
1903
(
1980
).
45.
V.
Zykov
, “
Analytical evaluation of the dependence of the speed of an excitation wave in a two-dimensional excitable medium on the curvature of its front
,”
Biofizika
25
,
888
892
(
1980
).
46.
V. A.
Davydov
,
V. S.
Zykov
, and
A. S.
Mikhailov
, “
Kinematics of autowave structures in excitable media
,”
Sov. Phys. Usp.
34
,
665
684
(
1991
).
You do not currently have access to this content.