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Keywords:
saddle-node; Hopf; homoclinic; cycle fold bifurcation; Hopfield model
saddle-node; Hopf; homoclinic; cycle fold bifurcation; Hopfield model
Summary:
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.
References:
[1] Beer, R. D.: On the dynamics of small continuous-time recurrent neural networks. Adapt. Behav. 3 (1995), 469-509 \99999DOI99999 10.1177/1059712395003004 . DOI 10.1177/105971239500300405
[2] Beer, R. D.: Parameter space structure of continuous-time recurrent neural networks. Neural Comput. 18 (2006), 3009-3051 \99999DOI99999 10.1162/neco.2006.18.12.3009 . MR 2265210 | Zbl 1107.68075
[3] Breakspear, M.: Dynamic models of large-scale brain activity. Nature Neurosci. 20 (2017), 340-352 \99999DOI99999 10.1038/nn.4497 .
[4] Brunel, N.: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci. 8 (2000), 183-208 \99999DOI99999 10.1023/A:1008925309027 . Zbl 1036.92008
[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 \99999DOI99999 10.7554/eLife.71850 .
[6] Ermentrout, B.: Neural networks as spatio-temporal pattern-forming systems. Rep. Progr. Phys. 61 (1998), Article ID 353, 78 pages. DOI 10.1088/0034-4885/61/4/002
[7] Ermentrout, B., Terman, D. H.: Mathematical Foundations of Neuroscience. Interdisciplinary Applied Mathematics 35. Springer, New York (2010),\99999DOI99999 10.1007/978-0-387-87708-2 . MR 2674516 | Zbl 1320.92002
[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 \99999DOI99999 10.1371/journal.pcbi.1004992 .
[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 \99999DOI99999 10.1007/978-3-319-68297-6_5 . MR 3793028 | Zbl 1401.92026
[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 \99999DOI99999 10.1016/0893-6080(88)90021-4 .
[12] Hopfield, J. J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79 (1982), 2554-2558 \99999DOI99999 10.1073/pnas.79.8.255 . MR 0652033 | Zbl 1369.92007
[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 \99999DOI99999 10.1073/pnas.81.10.3088 . Zbl 1371.92015
[14] Kuznetsov, Y. A.: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences 112. Springer, New York (2004),\99999DOI99999 10.1007/978-1-4757-3978-7 . MR 2071006 | Zbl 1082.37002
[15] Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics 7. Springer, New York (2001),\99999DOI99999 10.1007/978-1-4613-0003-8 . MR 1801796 | Zbl 0973.34001
[16] Trappenberg, T.: Fundamentals of Computational Neuroscience. Oxford University Press, Oxford (2010),\99999MR99999 2583115 . MR 2583115 | Zbl 1179.92012
[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. MR 4612555 | Zbl 07541738
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