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

Mathematica Bohemica (2010)

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

top

Abstract

top
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

top

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

top
1. Wang, G. Q., Cheng, S. S., 10.1016/j.apm.2007.11.013, Appl. Math. Modelling 33 (2009), 499-506. (2009) Zbl1167.39304MR2458519DOI10.1016/j.apm.2007.11.013
2. Mawhin, J., Willem, M., Critical Point Theory and Hamiltonian Systems, Springer, New York (1989). (1989) Zbl0676.58017MR0982267
3. Roberts, J. S., Artificial Neural Networks, McGraw-Hill, Singapore (1997). (1997)
4. Haykin, S., Neural Networks: a Comprehensive Foundation, Englewood Cliffs, Macmillan Company, NJ (1994). (1994) Zbl0828.68103
5. Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, AMS, number 65 (1986). (1986) Zbl0609.58002MR0845785
6. Zhou, Z., Yu, J. S., Guo, Z. M., Periodic solutions of higher-dimensional discrete systems, Proc. Royal Soc. Edinburgh 134A (2004), 1013-1022. (2004) Zbl1073.39010MR2099576
7. Wang, G. Q., Cheng, S. S., 10.4171/PM/1772, Portugaliae Math. 64 (2007), 3-10. (2007) Zbl1141.39011MR2298108DOI10.4171/PM/1772
8. Guo, Z. M., Yu, J. S., Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Science in China (Series A) 46 (2003), 506-515. (2003) Zbl1215.39001MR2014482
9. Guo, Z. M., Yu, J. S., 10.1112/S0024610703004563, J. London Math. Soc. 68 (2003), 419-430. (2003) MR1994691DOI10.1112/S0024610703004563
10. Zhou, Z., 10.1016/S0898-1221(03)00075-0, Comput. Math. Appl. 45 (2003), 1155-1161. (2003) Zbl1052.39016MR2000585DOI10.1016/S0898-1221(03)00075-0
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
13. Cheng, S. S., Yen, H. T., 10.1016/S0024-3795(00)00133-6, Linear Alg. Appl. 312 (2000), 193-201. (2000) MR1770367DOI10.1016/S0024-3795(00)00133-6
14. Cheng, S. S., Partial Difference Equations, Taylor and Francis (2003). (2003) Zbl1016.39001MR2193620

NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.