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Periodic oscillations arising and death in delay-coupled neural loops. (English) Zbl 1168.34044

The paper considers a system of two identical rings of \(n\) neurons which are coupled via a single neuron of each loop with a discrete time delay. Within each loop the single neurons are described by scalar differential equations of the form \(\dot x_j=-x_j+f(b\,x_{j-1})\) (\(n=(0,\ldots,n) \mod n\)). The function \(f\) is a sigmoidal (\(\tanh\) like) coupling function and \(b\) determines if the coupling is excitatory (\(b>0\)) or inhibitory (\(b<0\)). At neuron \(n\) of each loop a term \(r\,f(b\,x^o_n(t-\tau))\) is added where \(x^o_n\) is neuron \(n\) of the other loop, and \(r\) is the coupling strength of the cross-loop coupling relative to the intra-loop coupling.
The paper then studies linear stability, Hopf and Pitchfork bifurcations (and their criticality) of the ground state (all \(x_j=0\)) in the \((r,\tau)\)-parameter space.

MSC:

34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
92C20 Neural biology

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[1] DOI: 10.1016/j.jde.2005.01.007 · Zbl 1099.34066 · doi:10.1016/j.jde.2005.01.007
[2] DOI: 10.1137/S0036139993248853 · Zbl 0809.34077 · doi:10.1137/S0036139993248853
[3] DOI: 10.1016/S0166-2236(97)01151-X · doi:10.1016/S0166-2236(97)01151-X
[4] Campbell S. A., Fields Inst. Commun. 21 pp 65–
[5] DOI: 10.1137/S0036139903434833 · Zbl 1072.92003 · doi:10.1137/S0036139903434833
[6] DOI: 10.1016/j.physd.2005.12.008 · Zbl 1100.34054 · doi:10.1016/j.physd.2005.12.008
[7] Edwards R., Dyn. Contin. Discr. Impuls. Syst. Ser. B Appl. Algorith. 10 pp 287–
[8] Ermentrout B., Softw. Environ. Tools 14 (2002)
[9] DOI: 10.1006/jdeq.1995.1144 · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144
[10] DOI: 10.1006/jdeq.1995.1145 · Zbl 0836.34069 · doi:10.1006/jdeq.1995.1145
[11] DOI: 10.1007/BF02547797 · Zbl 0391.92001 · doi:10.1007/BF02547797
[12] DOI: 10.1016/S0022-5193(05)80127-4 · doi:10.1016/S0022-5193(05)80127-4
[13] DOI: 10.1016/0167-2789(95)00203-0 · Zbl 0883.68108 · doi:10.1016/0167-2789(95)00203-0
[14] DOI: 10.1016/S0959-4388(05)80043-1 · doi:10.1016/S0959-4388(05)80043-1
[15] DOI: 10.1007/978-1-4612-1140-2 · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[16] DOI: 10.1007/978-1-4615-9968-5 · doi:10.1007/978-1-4615-9968-5
[17] DOI: 10.1007/978-1-4612-4342-7 · doi:10.1007/978-1-4612-4342-7
[18] Kim A., Time-Delay System Toolbox, User’s Guide (2001)
[19] DOI: 10.1137/0150062 · Zbl 0711.34029 · doi:10.1137/0150062
[20] DOI: 10.1016/S0167-2789(96)00215-1 · Zbl 0887.34069 · doi:10.1016/S0167-2789(96)00215-1
[21] DOI: 10.1103/PhysRevA.39.347 · doi:10.1103/PhysRevA.39.347
[22] Ncube I., Fields Inst. Commun. 36 pp 179–
[23] DOI: 10.1137/S0036139998344015 · Zbl 0992.92013 · doi:10.1137/S0036139998344015
[24] DOI: 10.1016/S0167-2789(00)00216-5 · Zbl 1007.34072 · doi:10.1016/S0167-2789(00)00216-5
[25] DOI: 10.1016/S0895-7177(99)00120-X · Zbl 1043.92500 · doi:10.1016/S0895-7177(99)00120-X
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