×

zbMATH — the first resource for mathematics

Two-parameter bifurcations in a network of two neurons with multiple delays. (English) Zbl 1136.34058
The following system of delay-differential equations is considered
\[ \dot x_1(t) = -x_1(t) + \beta f(x_1(t-\tau)) + a_{12} f(x_2(t-\tau_1)), \]
\[ \dot x_2(t) = -x_2(t) + \beta f(x_2(t-\tau)) + a_{21} f(x_1(t-\tau_2)). \] Here \(\tau\), \(\tau_1\) and \(\tau_2\) are positive time delays, which satisfy \(\tau_1+\tau_2=2\tau\), and \(f:\mathbb R\to\mathbb R\) ia a \(C^1\)-smooth function with \(f(0)=0\).
Considering the corresponding characteristic equation for the equilibrium \(x_1=x_2=0\), the authors obtain conditions for various codimension-1 and codimension-2 bifurcations, give formulas for the normal form coefficients, and give information about the bifurcating solutions.

MSC:
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Algaba, A.; Merino, M.; Freire, E.; Gamero, E.; Rodriguez-Luis, A.J., Some results on Chua’s equation near a triple-zero linear degeneracy, Internat. J. bifur. chaos appl. sci. engrg., 13, 58-608, (2003) · Zbl 1083.93020
[2] Buono, P.L.; Bélair, J., Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. differential equations, 189, 234-266, (2003) · Zbl 1032.34068
[3] Bélair, J.; Campbell, S.A., Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. appl. math., 54, 1402-1424, (1994) · Zbl 0809.34077
[4] Bélair, J.; Campbell, S.A.; van den Driessche, P., Frustration, stability, and delay-induced oscillations in a neural network model, SIAM J. appl. math., 56, 245-255, (1996) · Zbl 0840.92003
[5] Broer, H.W.; Vegter, G., Subordinate šil’nikov bifurcations near some singularities of vector fields having low codimension, Ergodic theory dynam. systems, 4, 509-525, (1984) · Zbl 0553.58024
[6] Chen, Y.; Wu, J., Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential integral equations, 14, 1181-1236, (2001) · Zbl 1023.34065
[7] Diekmann, O.; van Gils, S.A.; Verduyn Lunel, S.M.; Walther, H.-O., Delay equations, functional-, complex-, and nonlinear analysis, (1995), Springer-Verlag New York · Zbl 0826.34002
[8] Dumortier, F.; Ibáñez, S., Singularities of vector fields on \(\mathbb{R}^3\), Nonlinearity, 11, 1037-1047, (1998) · Zbl 0907.58051
[9] Elphick, C.; Tirapegui, E.; Brachet, M.E.; Coullet, P.; Iooss, G., A simple global characterization for normal forms of singular vector fields, Phys. D, 29, 95-127, (1987) · Zbl 0633.58020
[10] Faria, T., On a planar system modelling a neuron network with memory, J. differential equations, 168, 129-149, (2000) · Zbl 0961.92002
[11] Faria, T.; Magalhães, L.T., Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities, J. dynam. differential equations, 8, 35-70, (1996) · Zbl 0853.34064
[12] Freire, E.; Gamero, E.; Rodríguez-Luis, A.J.; Algaba, A., A note on the triplezero linear degeneracy: normal forms, dynamical and bifurcation behaviors of an unfolding, Internat. J. bifur. chaos appl. sci. engrg., 12, 2799-2820, (2002) · Zbl 1043.37042
[13] Gavrilov, N., On some bifurcations of an equilibrium with one zero and a pair of pure imaginary roots, () · Zbl 0456.34029
[14] Guckenheimer, J., On a codimension two bifurcation, (), 99-142
[15] Guckenheimer, J., Multiple bifurcation problems of codimension two, SIAM J. math. anal., 15, 1-49, (1984) · Zbl 0543.34034
[16] Guckenheimer, J.; Holmes, P.J., Nonlinear oscillations: dynamical system and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001
[17] Guo, S.; Huang, L., Periodic solutions in an inhibitory two-neuron network, J. comput. appl. math., 161, 217-229, (2003) · Zbl 1044.34034
[18] Guo, S.; Huang, L.; Wang, L., Linear stability and Hopf bifurcation in a two-neuron network with three delays, Internat. J. bifur. chaos appl. sci. engrg., 14, 2799-2810, (2004) · Zbl 1062.34078
[19] Guo, S.; Huang, L.; Wu, J., Global attractivity of a synchronized periodic orbit in a delayed network, J. math. anal. appl., 281, 633-646, (2003)
[20] Guo, S.; Huang, L.; Wu, J., Convergence and periodicity in a delayed network of neurons with threshold nonlinearity, Electron. J. differential equations, 2003, 61, 1-14, (2003) · Zbl 1054.34112
[21] Guo, S.; Huang, L.; Wu, J., Regular dynamics in a delayed network of two neurons with all-or-none activation functions, Phys. D, 206, 32-48, (2005) · Zbl 1081.34069
[22] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
[23] Holmes, P.J., Unfolding a degenerate nonlinear oscillators: A codimension two bifurcation, (), 473-488
[24] Hopfield, J.J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. natl. acad. sci. USA, 81, 3088-3092, (1984) · Zbl 1371.92015
[25] Iooss, G.; Adelmeyer, M., Topics in bifurcation theory and applications, (1992), World Scientific Singapore · Zbl 0968.34027
[26] Iooss, G.; Langford, W.F., Conjectures on the routes to turbulence via bifurcation, (), 489-505 · Zbl 0469.58011
[27] Kuznetsov, Y.A., Elements of applied bifurcation theory, (1998), Springer-Verlag New York · Zbl 0914.58025
[28] Li, S.; Liao, X.; Li, C.; Wong, K.-W., Hopf bifurcation of a two-neuron network with different discrete time delays, Internat. J. bifur. chaos appl. sci. engrg., 15, 1589-1601, (2005) · Zbl 1092.34563
[29] Milton, J., Dynamics of small neural populations, (1996), Amer. Math. Soc. Providence, RI · Zbl 0879.92005
[30] Olien, L.; Bélair, J., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Phys. D, 102, 349-363, (1997) · Zbl 0887.34069
[31] Ruan, S.; Wei, J., Periodic solutions of planar systems with two delays, Proc. roy. soc. Edinburgh sect. A, 129, 1017-1032, (1999) · Zbl 0946.34062
[32] Shayer, L.P.; Campbell, S.A., Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays, SIAM J. appl. math., 61, 673-700, (2000) · Zbl 0992.92013
[33] Skinner, F.K.; Bazzazi, H.; Campbell, S.A., Two-cell to N-cell heterogeneous, inhibitory networks: precise linking of multistable and coherent properties, J. comput. neurosci., 18, 343-352, (2005)
[34] Takens, F., A nonstabilizable jet of a singularity of a vector field, (), 583-597
[35] Takens, F., Normal forms for certain singularities of vector fields, Ann. inst. Fourier (Grenoble), 23, 163-195, (1973) · Zbl 0266.34046
[36] Takens, F., Singularities of vector fields, Publ. math. inst. hautes études sci., 43, 47-100, (1974) · Zbl 0279.58009
[37] Tu, F.; Liao, X.; Zhang, W., Delay-dependent asymptotic stability of a two-neuron system with different time delays, Chaos solitons fractals, 28, 437-447, (2006) · Zbl 1084.68109
[38] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Phys. D, 130, 255-272, (1999) · Zbl 1066.34511
[39] Wei, J.J.; Velarde, M.G., Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos, 14, 940-953, (2004) · Zbl 1080.34064
[40] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (2003), Springer-Verlag New York · Zbl 1027.37002
[41] Wu, J., Introduction to neural dynamics and signal transmission delay, (2001), Walter de Gruyter New York · Zbl 0977.34069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.