Applications of Mathematics, Vol. 68, No. 1, pp. 35-50, 2023
Global bifurcations in a dynamical model of recurrent neural networks
Anita Windisch, Péter L. Simon
Received July 31, 2021. Published online November 18, 2022. OPEN ACCESS
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.
Keywords: saddle-node; Hopf; homoclinic; cycle fold bifurcation; Hopfield model
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Affiliations: Anita Windisch, Department of Applied Analysis and Computational Mathematics, Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary, e-mail: anita.windisch@ttk.elte.hu; Péter L. Simon (corresponding author), Department of Applied Analysis and Computational Mathematics, Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary; Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Széchenyi István sqr. 9, 1051 Budapest, Hungary, e-mail: peter.simon@ttk.elte.hu
