Positive fixed point theorems arising from seeking steady states of neural networks

Gen Qiang Wang; Sui-Sun Cheng

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 1, page 99-112
  • ISSN: 0862-7959

Abstract

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Biological systems are able to switch their neural systems into inhibitory states and it is therefore important to build mathematical models that can explain such phenomena. If we interpret such inhibitory modes as `positive' or `negative' steady states of neural networks, then we will need to find the corresponding fixed points. This paper shows positive fixed point theorems for a particular class of cellular neural networks whose neuron units are placed at the vertices of a regular polygon. The derivation is based on elementary analysis. However, it is hoped that our easy fixed point theorems have potential applications in exploring stationary states of similar biological network models.

How to cite

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Wang, Gen Qiang, and Cheng, Sui-Sun. "Positive fixed point theorems arising from seeking steady states of neural networks." Mathematica Bohemica 135.1 (2010): 99-112. <http://eudml.org/doc/38114>.

@article{Wang2010,
abstract = {Biological systems are able to switch their neural systems into inhibitory states and it is therefore important to build mathematical models that can explain such phenomena. If we interpret such inhibitory modes as `positive' or `negative' steady states of neural networks, then we will need to find the corresponding fixed points. This paper shows positive fixed point theorems for a particular class of cellular neural networks whose neuron units are placed at the vertices of a regular polygon. The derivation is based on elementary analysis. However, it is hoped that our easy fixed point theorems have potential applications in exploring stationary states of similar biological network models.},
author = {Wang, Gen Qiang, Cheng, Sui-Sun},
journal = {Mathematica Bohemica},
keywords = {positive fixed point; neural network; periodic solution; difference equation; discrete boundary condition; critical point theory; periodic solutions; difference equations; discrete boundary condition; critical point theory},
language = {eng},
number = {1},
pages = {99-112},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Positive fixed point theorems arising from seeking steady states of neural networks},
url = {http://eudml.org/doc/38114},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Wang, Gen Qiang
AU - Cheng, Sui-Sun
TI - Positive fixed point theorems arising from seeking steady states of neural networks
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 1
SP - 99
EP - 112
AB - Biological systems are able to switch their neural systems into inhibitory states and it is therefore important to build mathematical models that can explain such phenomena. If we interpret such inhibitory modes as `positive' or `negative' steady states of neural networks, then we will need to find the corresponding fixed points. This paper shows positive fixed point theorems for a particular class of cellular neural networks whose neuron units are placed at the vertices of a regular polygon. The derivation is based on elementary analysis. However, it is hoped that our easy fixed point theorems have potential applications in exploring stationary states of similar biological network models.
LA - eng
KW - positive fixed point; neural network; periodic solution; difference equation; discrete boundary condition; critical point theory; periodic solutions; difference equations; discrete boundary condition; critical point theory
UR - http://eudml.org/doc/38114
ER -

References

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  11. Wang, G. Q., Cheng, S. S., Positive periodic solutions for nonlinear difference equations via a continuation theorem, Advance in Difference Equations 4 (2004), 311-320. (2004) Zbl1095.39011MR2129756
  12. Cheng, S. S., Lin, S. S., Existence and uniqueness theorems for nonlinear difference boundary value problems, Utilitas Math. 39 (1991), 167-186. (1991) Zbl0729.39002MR1119771
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  14. Cheng, S. S., Partial Difference Equations, Taylor and Francis (2003). (2003) Zbl1016.39001MR2193620

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