Global bifurcations in a dynamical model of recurrent neural networks

Anita Windisch; Péter L. Simon

Applications of Mathematics (2023)

  • Volume: 68, Issue: 1, page 35-50
  • ISSN: 0862-7940

Abstract

top
The dynamical behaviour of a continuous time recurrent neural network model with a special weight matrix is studied. The network contains several identical excitatory neurons and a single inhibitory one. This special construction enables us to reduce the dimension of the system and then fully characterize the local and global codimension-one bifurcations. It is shown that besides saddle-node and Andronov-Hopf bifurcations, homoclinic and cycle fold bifurcations may occur. These bifurcation curves divide the plane of weight parameters into nine domains. The phase portraits belonging to these domains are also characterized.

How to cite

top

Windisch, Anita, and Simon, Péter L.. "Global bifurcations in a dynamical model of recurrent neural networks." Applications of Mathematics 68.1 (2023): 35-50. <http://eudml.org/doc/299353>.

@article{Windisch2023,
abstract = {The dynamical behaviour of a continuous time recurrent neural network model with a special weight matrix is studied. The network contains several identical excitatory neurons and a single inhibitory one. This special construction enables us to reduce the dimension of the system and then fully characterize the local and global codimension-one bifurcations. It is shown that besides saddle-node and Andronov-Hopf bifurcations, homoclinic and cycle fold bifurcations may occur. These bifurcation curves divide the plane of weight parameters into nine domains. The phase portraits belonging to these domains are also characterized.},
author = {Windisch, Anita, Simon, Péter L.},
journal = {Applications of Mathematics},
keywords = {saddle-node; Hopf; homoclinic; cycle fold bifurcation; Hopfield model},
language = {eng},
number = {1},
pages = {35-50},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global bifurcations in a dynamical model of recurrent neural networks},
url = {http://eudml.org/doc/299353},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Windisch, Anita
AU - Simon, Péter L.
TI - Global bifurcations in a dynamical model of recurrent neural networks
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 1
SP - 35
EP - 50
AB - The dynamical behaviour of a continuous time recurrent neural network model with a special weight matrix is studied. The network contains several identical excitatory neurons and a single inhibitory one. This special construction enables us to reduce the dimension of the system and then fully characterize the local and global codimension-one bifurcations. It is shown that besides saddle-node and Andronov-Hopf bifurcations, homoclinic and cycle fold bifurcations may occur. These bifurcation curves divide the plane of weight parameters into nine domains. The phase portraits belonging to these domains are also characterized.
LA - eng
KW - saddle-node; Hopf; homoclinic; cycle fold bifurcation; Hopfield model
UR - http://eudml.org/doc/299353
ER -

References

top
  1. Beer, R. D., 10.1177/105971239500300405, Adapt. Behav. 3 (1995), 469-509 9999DOI99999 10.1177/1059712395003004 . (1995) DOI10.1177/105971239500300405
  2. Beer, R. D., Parameter space structure of continuous-time recurrent neural networks, Neural Comput. 18 (2006), 3009-3051 9999DOI99999 10.1162/neco.2006.18.12.3009 . (2006) Zbl1107.68075MR2265210
  3. Breakspear, M., Dynamic models of large-scale brain activity, Nature Neurosci. 20 (2017), 340-352 9999DOI99999 10.1038/nn.4497 . (2017) 
  4. Brunel, N., Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, J. Comput. Neurosci. 8 (2000), 183-208 9999DOI99999 10.1023/A:1008925309027 . (2000) Zbl1036.92008
  5. A. Ecker, B. Bagi, E. Vértes, O. Steinbach-Németh, M. R. Karlócai, O. I. Papp, I. Mik-lós, N. Hájos, T. F. Freund, A. I. Gulyás, S. Káli, Hippocampal sharp wave-ripples and the associated sequence replay emerge from structured synaptic interactions in a network model of area CA3, eLife 11 (2022), Article ID e71850, 29 pages 9999DOI99999 10.7554/eLife.71850 . (2022) 
  6. Ermentrout, B., 10.1088/0034-4885/61/4/002, Rep. Progr. Phys. 61 (1998), Article ID 353, 78 pages. (1998) DOI10.1088/0034-4885/61/4/002
  7. Ermentrout, B., Terman, D. H., Mathematical Foundations of Neuroscience, Interdisciplinary Applied Mathematics 35. Springer, New York (2010),9999DOI99999 10.1007/978-0-387-87708-2 . (2010) Zbl1320.92002MR2674516
  8. Fasoli, D., Cattani, A., Panzeri, S., The complexity of dynamics in small neural circuits, PLoS Comput. Biology 12 (2016), Article ID e1004992, 35 pages 9999DOI99999 10.1371/journal.pcbi.1004992 . (2016) 
  9. Fasoli, D., Cattani, A., Panzeri, S., Bifurcation analysis of a sparse neural network with cubic topology, Mathematical and Theoretical Neuroscience Springer INdAM Series 24. Springer, Cham (2017), 87-98 9999DOI99999 10.1007/978-3-319-68297-65 . (2017) Zbl1401.92026MR3793028
  10. Govaerts, W., Kuznetsov, Y. A., DeWitte, V., Dhooge, A., Meijer, H. G. E., Mestrom, W., Rietand, A. M., Sautois, B., MATCONT and CL_MATCONT: Continuation Toolboxes in Matlab, Gent University and Utrecht University, Gent and Utrecht (2011) . 
  11. Grossberg, S., Nonlinear neural networks: Principles, mechanisms, and architectures, Neural Netw. 1 (1988), 17-61 9999DOI99999 10.1016/0893-6080(88)90021-4 . (1988) 
  12. Hopfield, J. J., Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA 79 (1982), 2554-2558 9999DOI99999 10.1073/pnas.79.8.255 . (1982) Zbl1369.92007MR0652033
  13. Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA 81 (1984), 3088-3092 9999DOI99999 10.1073/pnas.81.10.3088 . (1984) Zbl1371.92015
  14. Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, Applied Mathematical Sciences 112. Springer, New York (2004),9999DOI99999 10.1007/978-1-4757-3978-7 . (2004) Zbl1082.37002MR2071006
  15. Perko, L., Differential Equations and Dynamical Systems, Texts in Applied Mathematics 7. Springer, New York (2001),9999DOI99999 10.1007/978-1-4613-0003-8 . (2001) Zbl0973.34001MR1801796
  16. Trappenberg, T., Fundamentals of Computational Neuroscience, Oxford University Press, Oxford (2010),9999MR99999 2583115 . (2010) Zbl1179.92012MR2583115
  17. Windisch, A., Simon, P. L., The dynamics of the Hopfield model for homogeneous weight matrix, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 64 (2021), 235-247. (2021) Zbl07541738MR4612555

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.