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Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network. (English) Zbl 1196.37088
This paper considers a two-cell inhibitory neural network with adaptation modeled by a four-dimensional system of ordinary differential equations. It was found that the system will admit the mixed-mode oscillations (MMOs), and the authors proved that the MMOs are due to a singular Hopf bifurcation point situated in close distance to the transition point to the winner-take-all case.

37G35Attractors and their bifurcations
92C20Neural biology
Full Text: DOI
[1] Hudson, J.; Hart, M.; Marinko, D.: An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov--zhabotinskii reaction, J. chem. Phys. 71, 1601-1606 (1979)
[2] Petrov, V.; Scott, S. K.; Showalter, K.: Mixed-mode oscillations in chemical systems, J. chem. Phys. 97, No. 9, 6191-6198 (1992)
[3] Krischer, K.; Lübke, M.; Eiswirth, M.; Wolf, W.; Hudson, J. L.; Ertl, G.: A hierarchy of transitions to mixed mode oscillations in an electrochemical system, Physica D 62, 123-133 (1993) · Zbl 0800.92015 · doi:10.1016/0167-2789(93)90277-8
[4] Epstein, I. R.; Showalter, K.: Nonlinear chemical dynamics: oscillations, patterns, and chaos, J. phys. Chem. 100, 13132-13147 (1996)
[5] Vanag, V. K.; Yang, L.; Dolnik, M.; Zhabotinsky, A. M.; Epstein, I. R.: Oscillatory cluster patterns in a homogeneous chemical system with global feedback, Nature 406, 389-391 (2000)
[6] Vanag, V. K.; Zhabotinsky, A. M.; Epstein, I. R.: Pattern formation in the Belousov--Zhabotinsky reaction with photochemical global feedback, J. phys. Chem. A 104, 11566-11577 (2000)
[7] Kovacs, K.; Leda, M.; Vanag, V. K.; Epstein, I. R.: Small-amplitude and mixed-mode ph oscillations in the bromate-sulfite-ferrocyanide-$aluminum(III)$ system, J. phys. Chem. A 113, 146-156 (2009)
[8] Hayashi, T.: Mixed-mode oscillations and chaos in a glow discharge, Phys. rev. Lett. 84, 3334-3337 (2000)
[9] Mikikian, M.; Cavarroc, M.; Couedel, L.; Tessier, Y.; Boufendi, L.: Mixed-mode oscillations in complex-plasma instabilities, Phys. rev. Lett. 100, 1-4 (2008)
[10] Alonso, A. A.; Llinás, R. R.: Subthreshold na+-dependent theta like rhythmicity in stellate cells of entorhinal cortex layer II, Nature 342, 175-177 (1989)
[11] Del Negro, C. A.; Wilson, C. G.; Butera, R. J.; Rigatto, H.; Smith, J. C.: Periodicity, mixed-mode oscillations, and quasiperiodicity in a rhythm-generating neural network, Biophys. J. 82, 206-214 (2002)
[12] Medvedev, G.; Cisternas, J. E.: Multimodal regimes in a compartmental model of the dopamine neuron, Physica D 194, No. 3--4, 333-356 (2004) · Zbl 1055.92012 · doi:10.1016/j.physd.2004.02.006
[13] Kuznetsov, A.; Kopell, N.; Wilson, C.: Transient high-frequency firing in a coupled-oscillator model of the mesencephalic dopaminergic neuron, J. neurophysiol. 95, 932-947 (2006)
[14] Rubin, J.; Wechselberger, M.: The selection of mixed-mode oscillations in a Hodgkin--Huxley model with multiple timescales, Chaos 18, 1-12 (2008) · Zbl 1306.34060
[15] Ermentrout, B.; Wechselberger, M.: Canards, clusters and synchronization in a weakly coupled interneuron model, SIAM J. Appl. dyn. Syst. 8, No. 1, 253-278 (2009) · Zbl 1167.34352 · doi:10.1137/080724010
[16] Rotstein, H.; Wechselberger, M.; Kopell, N.: Canard induced mixed-mode oscillations in a medial entorhinal cortex layer II stellate cell model, SIAM J. Appl. dyn. Syst. 7, 1582-1611 (2008) · Zbl 1167.34353 · doi:10.1137/070699093
[17] Shilnikov, L.; Shilnikov, A.: Shilnikov bifurcation, Scholarpedia 2, No. 8, 1981 (2007) · Zbl 1300.00035
[18] Koper, M. T. M.: Bifurcations of mixed-mode oscillations in a three-variable autonomous van der Pol--Duffing model with a cross-shaped phase diagram, Physica D 80, 72-94 (1995) · Zbl 0889.34034 · doi:10.1016/0167-2789(95)90061-6
[19] Larter, R.; Steinmetz, C.: Chaos via mixed-mode oscillations, Phil. trans. R. soc. London A 337, 291-298 (1991) · Zbl 0744.92017 · doi:10.1098/rsta.1991.0124
[20] Guckenheimer, J.; Willms, A.: Asymptotic analysis of subcritical Hopf-homoclinic bifurcation, Physica D 139, 195-216 (2000) · Zbl 0954.34032 · doi:10.1016/S0167-2789(99)00225-0
[21] Szmolyan, P.; Wechselberger, M.: Canards in R3, J. differential equations 177, 419-453 (2001)
[22] Brons, M.; Krupa, M.; Wechselberger, M.: Mixed mode oscillations due to the generalized canard phenomenon, Fields inst. Commun. 49, 39-63 (2006) · Zbl 1228.34063
[23] Rotstein, H.; Kopell, N.; Zhabotinsky, A. M.; Epstein, I. R.: Canard phenomenon and localization of oscillations in the Belousov--Zhabotinsky reaction with global feedback, J. chem. Phys. 119, 8824-8832 (2003)
[24] Drover, J.; Rubin, J.; Su, J.; Ermentrout, B.: Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies, SIAM J. Appl. math. 62, No. 1, 69-92 (2004) · Zbl 1121.34059 · doi:10.1137/S0036139903431233
[25] Guckenheimer, J.: Singular Hopf bifurcation in systems with two slow variables, SIAM J. Appl. dyn. Syst. 7, No. 4, 1355-1377 (2008) · Zbl 1168.37025 · doi:10.1137/080718528
[26] Wechselberger, M.: Existence and bifurcation of canards in R3 in the case of a folded node, SIAM J. Applied dynamical systems 4, 101-139 (2005) · Zbl 1090.34047 · doi:10.1137/030601995
[27] R. Curtu, J. Rubin, Canards and folded saddle-node singularities of type II in a reduced firing rate inhibitory network, Chaos (2010) (submitted for publication)
[28] M. Krupa, M. Wechselberger, Local analysis near a folded saddle-node singularity (2008), preprint · Zbl 1198.34103
[29] Laing, C. R.; Chow, C. C.: A spiking neuron model for binocular rivalry, J. comput. Neurosci. 12, 39-53 (2002)
[30] Curtu, R.; Shpiro, A.; Rubin, N.; Rinzel, J.: Mechanisms for frequency control in neuronal competition models, SIAM J. Appl. dyn. Syst. 7, No. 2, 609-649 (2008) · Zbl 1167.34351 · doi:10.1137/070705842
[31] Shpiro, A.; Curtu, R.; Rinzel, J.; Rubin, N.: Dynamical characteristics common to neuronal competition models, J. neurophysiol. 97, 462-473 (2007)
[32] Calabrese, R. L.: Half-center oscillators underlying rhythmic movements, The handbook of brain theory and neural networks, 444-447 (1995)
[33] Skinner, F. K.; Kopell, N.; Marder, E.: Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks, J. comput. Neurosci. 1, 69-87 (1994) · Zbl 0838.92002 · doi:10.1007/BF00962719
[34] Taylor, A. L.; Cottrell, G. W.; Kristan, W. B.: Analysis of oscillations in a reciprocally inhibitory network with synaptic depression, Neural comput. 14, 561-581 (2002) · Zbl 0989.92010 · doi:10.1162/089976602317250906
[35] Ermentrout, B.: Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students, Software environ. Tools 14 (2002) · Zbl 1003.68738 · doi:10.1137/1.9780898718195
[36] Ermentrout, B.:
[37] Dhooge, A.; Govaerts, W.; Kuznetsov, Yu.A.: MATCONT: A Matlab package for numerical bifurcation analysis of odes, ACM trans. Math. software 29, 141-164 (2003) · Zbl 1070.65574 · doi:10.1145/779359.779362
[38] Braaksma, B.: Singular Hopf bifurcation in systems with fast and slow variables, J. nonlinear sci. 8, 457-490 (1998) · Zbl 0907.34026 · doi:10.1007/s003329900058