Effective nonlocal kernels on reaction-diffusion networks

JOURNAL OF THEORETICAL BIOLOGY

卷:509

Article

Ei, Shin-Ichiro¹  Ishii, Hiroshi¹  Kondo, Shigeru²  Miura, Takashi³  Tanaka, Yoshitaro⁴ 

[1] Hokkaido Univ, Fac Sci, Dept Math, Sapporo, Hokkaido, Japan

[2] Osaka Univ, Grad Sch Frontier Biosci, Osaka, Japan

[3] Kyushu Univ, Grad Sch Med Sci, Fukuoka, Japan

[4] Future Univ Hakodate, Sch Syst Informat Sci, Hakodate, Hokkaido, Japan

关键词: Non-local convolution;    Pattern formation;    Network;    Reaction-diffusion;    Turing pattern;   

DOI  :  10.1016/j.jtbi.2020.110496

来源: Elsevier

【 摘 要 】

A new method to derive an essential integral kernel from any given reaction-diffusion network is proposed. Any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations called effective equation including the reduced integral kernel (called effective kernel) in the convolution type. As one typical example, the Mexican hat shaped kernel is theoretically derived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentially the same spatial and temporal patterns. Thus, we can identify two different systems with the understanding in unified concept through the reduced effective kernels. Other two applications of this method are also given: Applications to pigment patterns on skins (two factors network with long range interaction) and waves of differentiation (called proneural waves) in visual systems on brains (four factors network with long range interaction). In the applications, we observe the reproduction of the same spatial and temporal patterns as those appearing in pre-existing models through the numerical simulations of the effective equations. (C) 2020 The Authors. Published by Elsevier Ltd.

【 授权许可】

Free