A differential equation model of retinal processing for understanding lightness optical illusions

  • Takamichi SushidaEmail author
  • Shintaro Kondo
  • Kokichi Sugihara
  • Masayasu Mimura
Original Paper Area 1


This paper proposes a differential equation model of the human vision system. Our model has a hierarchical structure in the known retinal cell neural network, and hence, enable to explain what aspects of behaviors of the vision system is affected by which parameters. As the result, our model possesses the following two characteristics: First, it enable the derivation of an integral equation model with a Mexican-hat shape kernel as a necessary result of mechanisms (reduction of the retinal image resolution and a self-control mechanism formed by non-local interaction), in contrast to previous models that assumed various types of Mexican-hat shapes a priori. Second, it can explain two mutually contradicting phenomena called lightness contrast and assimilation. Moreover, our model explains the reason why lightness optical illusions do or do not occur via the magnitude of a control parameter.


Retinal information processing Lightness optical illusion Mathematical modeling Convolution Differential equation 

Mathematics Subject Classification

93A30 92C20 44A35 35K57 



The authors would like to thank the reviewers for helpful comments and fruitful suggestions. This work is partly supported by Grant-in-Aid for Basic Scientific Research (A) No. 16H01728 of MEXT. MM is partially supported by JSPS KAKENHI Grant No. 15K13462. Additionally, in this work, we used the computer of the MEXT Joint Usage / Research Center “Center for Mathematical Modeling and Applications”, Meiji University, Meiji Institute for Advanced Study of Mathematical Sciences (MIMS).


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  • Takamichi Sushida
    • 1
    Email author
  • Shintaro Kondo
    • 2
  • Kokichi Sugihara
    • 3
  • Masayasu Mimura
    • 3
  1. 1.Research Institute for Electronic ScienceHokkaido UniversitySapporoJapan
  2. 2.Department of Electrical, Electronic and Computer EngineeringGifu UniversityGifuJapan
  3. 3.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityTokyoJapan

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