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Stability, bifurcation and phase-locking of time-delayed excitatory-inhibitory neural networks

  • Received: 31 August 2020 Accepted: 03 November 2020 Published: 10 November 2020
  • We study a model for a network of synaptically coupled, excitable neurons to identify the role of coupling delays in generating different network behaviors. The network consists of two distinct populations, each of which contains one excitatory-inhibitory neuron pair. The two pairs are coupled via delayed synaptic coupling between the excitatory neurons, while each inhibitory neuron is connected only to the corresponding excitatory neuron in the same population. We show that multiple equilibria can exist depending on the strength of the excitatory coupling between the populations. We conduct linear stability analysis of the equilibria and derive necessary conditions for delay-induced Hopf bifurcation. We show that these can induce two qualitatively different phase-locked behaviors, with the type of behavior determined by the sizes of the coupling delays. Numerical bifurcation analysis and simulations supplement and confirm our analytical results. Our work shows that the resting equilibrium point is unaffected by the coupling, thus the network exhibits bistability between a rest state and an oscillatory state. This may help understand how rhythms spontaneously arise in neuronal networks.

    Citation: Hwayeon Ryu, Sue Ann Campbell. Stability, bifurcation and phase-locking of time-delayed excitatory-inhibitory neural networks[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7931-7957. doi: 10.3934/mbe.2020403

    Related Papers:

  • We study a model for a network of synaptically coupled, excitable neurons to identify the role of coupling delays in generating different network behaviors. The network consists of two distinct populations, each of which contains one excitatory-inhibitory neuron pair. The two pairs are coupled via delayed synaptic coupling between the excitatory neurons, while each inhibitory neuron is connected only to the corresponding excitatory neuron in the same population. We show that multiple equilibria can exist depending on the strength of the excitatory coupling between the populations. We conduct linear stability analysis of the equilibria and derive necessary conditions for delay-induced Hopf bifurcation. We show that these can induce two qualitatively different phase-locked behaviors, with the type of behavior determined by the sizes of the coupling delays. Numerical bifurcation analysis and simulations supplement and confirm our analytical results. Our work shows that the resting equilibrium point is unaffected by the coupling, thus the network exhibits bistability between a rest state and an oscillatory state. This may help understand how rhythms spontaneously arise in neuronal networks.


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    [1] P. Dayan, L. Abbott, Theoretical Neuroscience, MIT Press, Cambridge, MA, 2001.
    [2] G. Ermentrout, D. Terman, Mathematical Foundations of Neuroscience, Springer, New York, NY, 2010.
    [3] E. M. Izhikevich, Dynamical Systems in Neuroscience, MIT Press, Cambridge, MA, 2007.
    [4] N. Brunel, X. J. Wang, What determines the frequency of fast network oscillations with irregular neural discharges? I. synaptic dynamics and excitation-inhibition balance, J. Neurophysiol., 90 (2003), 415-430.
    [5] G. B. Ermentrout, N. Kopell, Fine structure of neural spiking and synchronization in the presence of conduction delays, Proc. Natl. Acad. Sci., 95 (1998), 1259-1264. doi: 10.1073/pnas.95.3.1259
    [6] N. Kopell, G. Ermentrout, M. Whittington, R. Traub, Gamma rhythms and beta rhythms have different synchronization properties, Proc. Natl. Acad. Sci., 97 (2000), 1867-1872. doi: 10.1073/pnas.97.4.1867
    [7] J. E. Rubin, D. Terman, Analysis of clustered firing patterns in synaptically coupled networks of oscillators, J. Math. Biol., 41 (2000), 513-545. doi: 10.1007/s002850000065
    [8] B. Pfeuty, G. Mato, D. Golomb, D. Hansel, The combined effects of inhibitory and electrical synapses in synchrony, Neural Comput., 17 (2005), 633-670. doi: 10.1162/0899766053019917
    [9] M. A. Whittington, R. Traub, N. Kopell, B. Ermentrout, E. Buhl, Inhibition-based rhythms: experimental and mathematical observations on network dynamics, Int. J. Psychophysiol., 38 (2000), 315-336. doi: 10.1016/S0167-8760(00)00173-2
    [10] J. E. Rubin, D. Terman, Geometric analysis of population rhythms in synaptically coupled neuronal networks, Neural Comput., 12 (2000), 597-645. doi: 10.1162/089976600300015727
    [11] J. E. Rubin, D. Terman, Geometric singular perturbation analysis of neuronal dynamics, Handbook Dyn. Syst., 2 (2002), 93-146.
    [12] C. D. Acker, N. Kopell, J. A. White, Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics, J. Comput. Neurosci., 15 (2003), 71-90. doi: 10.1023/A:1024474819512
    [13] P. J. Hellyer, B. Jachs, C. Clopath, R. Leech, Local inhibitory plasticity tunes macroscopic brain dynamics and allows the emergence of functional brain networks, Neuroimage, 24 (2016), 85-95.
    [14] E. M. Izhikevich, G. M. Edelman, Large-scale model of mammalian thalamocortical systems, Proc. Natl. Acad. Sci., 105 (2008), 3593-3598. doi: 10.1073/pnas.0707563105
    [15] M. Bazhenov, N. F. Rulkov, I. Timofeev, Effect of synaptic connectivity on long-range synchronization of fast cortical oscillations, J. Neurophysiol., 100 (2008), 1562-1575. doi: 10.1152/jn.90613.2008
    [16] A. Ghosh, Y. Rho, A. McIntosh, R. Kötter, V. Jirsa, Cortical network dynamics with time delays reveals functional connectivity in the resting brain, Cognitive Neurodyn., 2 (2008), 115. doi: 10.1007/s11571-008-9044-2
    [17] D. Guo, Q. Wang, M. Perc, Complex synchronous behavior in interneuronal networks with delayed inhibitory and fast electrical synapses, Phys. Rev. E, 85 (2012), 061905. doi: 10.1103/PhysRevE.85.061905
    [18] T. Pérez, G. C. Garcia, V. M. Eguiluz, R. Vicente, G. Pipa, C. Mirasso, Effect of the topology and delayed interactions in neuronal networks synchronization, PLoS ONE, 6 (2011), e19900. doi: 10.1371/journal.pone.0019900
    [19] X. Sun, L. Guofang, Fast regular firings induced by intra- and inter-time delays in two clustered neuronal networks, Chaos, 28 (2017), 106310.
    [20] Q. Wang, Q. Lu, G. Chen, Synchronization transition induced by synaptic delay in coupled fastspiking neurons, Int. J. Bifurcat. Chaos, 18 (2008), 1189-1198. doi: 10.1142/S0218127408020914
    [21] S. Crook, G. Ermentrout, M. Vanier, J. Bower, The role of axonal delay in synchronization of networks of coupled cortical oscillators, J. Comput. Neurosci., 4 (1997), 161-172. doi: 10.1023/A:1008843412952
    [22] P. Bressloff, S. Coombes, Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays, Physica D, 126 (1999), 99-122. doi: 10.1016/S0167-2789(98)00264-4
    [23] P. Bressloff, S. Coombes, Travelling waves in chains of pulse-coupled integrate-and-fire oscillators with distributed delays, Physica D, 130 (1999), 232-254. doi: 10.1016/S0167-2789(99)00013-5
    [24] T. W. Ko, G. B. Ermentrout, Effects of axonal time delay on synchronization and wave formation in sparsely coupled neuronal oscillators, Phys. Rev. E, 76 (2007), 056206. doi: 10.1103/PhysRevE.76.056206
    [25] S. A. Campbell, Z. Wang, Phase models and clustering in networks of oscillators with delayed coupling, Physica D, 363 (2018), 44-55. doi: 10.1016/j.physd.2017.09.004
    [26] G. Orosz, Decomposition of nonlinear delayed networks around cluster states with applications to neurodynamics, SIAM J. Appl. Dyn. Syst., 13 (2014), 1353-1386. doi: 10.1137/130915637
    [27] Z. Wang, S. A. Campbell, Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks, Chaos, 27 (2017), 114316. doi: 10.1063/1.5006921
    [28] D. Golomb, G. B. Ermentrout, Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity, Proc. Natl. Acad. Sci., 96 (1999), 13480-13485. doi: 10.1073/pnas.96.23.13480
    [29] D. Golomb, G. B. Ermentrout, Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity, Network: Comput. Neural Syst., 11 (2000), 221-246. doi: 10.1088/0954-898X_11_3_304
    [30] A. Roxin, N. Brunel, D. Hansel, Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks, Phys. Rev. Lett.s, 94 (2005), 238103. doi: 10.1103/PhysRevLett.94.238103
    [31] M. Dahlem, G. Hiller, A. Panchuk, E. Schöll, Dynamics of delay-coupled excitable neural systems, Int. J. Bifurc. Chaos, 29 (2009), 745-753.
    [32] N. Burić, I. Grozdanović, N. Vasović, Type I vs type II excitable systems with delayed coupling, Chaos Solitons Fractals, 23 (2005), 1221-1233. doi: 10.1016/j.chaos.2004.06.033
    [33] N. Burić, D. Todorović, Dynamics of Fitzhugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066-222.
    [34] N. Burić, D. Todorović, Bifurcations due to small time-lag in coupled excitable systems, Int. J. Bifurc. Chaos, 15 (2005), 1775-1785. doi: 10.1142/S0218127405012831
    [35] S. A. Campbell, R. Edwards, P. van den Dreissche, Delayed coupling between two neural network loops, SIAM J. Appl. Math., 65 (2004), 316-335.
    [36] E. Schöll, G. Hiller, P. Hövel, M. A. Dahlem, Time-delay feedback in neurosystems, Phil. Trans. Roy. Soc. A, 367 (2009), 1079-10956. doi: 10.1098/rsta.2008.0258
    [37] C. U. Choe, T. Dahms, P. Hovel, E. Schöll, Controlling synchrony by delay coupling in networks: From in-phase to splay and cluster states, Phys. Rev. E, 81 (2010), 025205. doi: 10.1103/PhysRevE.81.025205
    [38] T. Dahms, J. Lehnert, E. Scholl, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E, 86 (2012), 016202. doi: 10.1103/PhysRevE.86.016202
    [39] J. Lehnert, T. Dahms, P. Hovel, E. Schöll, Loss of synchronization in complex neuronal networks with delay, EPL (Europhys. Lett.), 96 (2011), 60013. doi: 10.1209/0295-5075/96/60013
    [40] Y. Song, J. Xu, Inphase and antiphase synchronization in a delay-coupled system with applications to a delay-coupled Fitzhugh-Nagumo system, IEEE Trans. Neural. Netw. Learn. Syst., 23 (2012), 1659-1670. doi: 10.1109/TNNLS.2012.2209459
    [41] A. Panchuk, D. Rosin, P. Hovel, E. Schöll, Synchronization of coupled neural oscillators with heterogeneous delays, Int. J. Bifurc. Chaos, 23.
    [42] S. R. Campbell, D. Wang, Relaxation oscillators with time delay coupling, Physica D, 111 (1998), 151-178. doi: 10.1016/S0167-2789(97)80010-3
    [43] J. J. Fox, C. Jayaprakash, D. Wang, S. R. Campbell, Synchronization in relaxation oscillator networks with conduction delays, Neural Comput., 13 (2001), 1003-1021. doi: 10.1162/08997660151134307
    [44] A. Bose, S. Kunec, Synchrony and frequency regulation by synaptic delay in networks of selfinhibiting neurons, Neurocomputing, 38 (2001), 505-513.
    [45] S. Kunec, A. Bose, Role of synaptic delay in organizing the behavior of networks of self-inhibiting neurons, Phys. Rev. E, 63 (2001), 021908. doi: 10.1103/PhysRevE.63.021908
    [46] S. Kunec, A. Bose, High-frequency, depressing inhibition facilitates synchronization in globally inhibitory networks, Network: Comput. Neural Syst., 14 (2003), 647-672. doi: 10.1088/0954-898X_14_4_303
    [47] H. Ryu, S. A. Campbell, Geometric analysis of synchronization in neuronal networks with global inhibition and coupling delays, Phil. Trans. R. Soc. A, 337 (2019), 20180129.
    [48] X. Ji, J. Lu, J. Lou, J. Qiu, K. Shi, A unified criterion for global exponential stability of quaternionvalued neural networks with hybrid impulses, Int. J. Robust Nonlinear Control, 2020.
    [49] B. Sun, Y. Cao, Z. Guo, Z. Yan, S. Wen, Synchronization of discrete-time recurrent neural networks with time-varying delays via quantized sliding mode control, Appl. Math. Comput., 375 (2020), 125093.
    [50] Y. Tian, Z. Wang, Stability analysis for delayed neural networks based on the augmented Lyapunov-Krasovskii functional with delay-product-type and multiple integral terms, Neurocomputing, 410 (2020), 295-303. doi: 10.1016/j.neucom.2020.05.045
    [51] J. Xiao, Z. Zeng, A. Wu, S. Wen, Fixed-time synchronization of delayed Cohen-Grossberg neural networks based on a novel sliding mode, Neural Networks, 128 (2020), 1-12. doi: 10.1016/j.neunet.2020.04.020
    [52] E. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: a Guide to XPPAUT for Researchers and Students, SIAM, 2002.
    [53] K. Engelborghs, T. Luzyanina, G. Samaey, DDE-BIFTOOL v. 2.00: a Matlab Package for Bifurcation Analysis of Delay Differential Equations, Technical Report TW-330, Department of Computer Science, K.U. Leuven, Leuven, Belgium, 2001.
    [54] C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle fibre, Biophysical J., 35 (1981), 193-213. doi: 10.1016/S0006-3495(81)84782-0
    [55] J. Rinzel, Excitation dynamics: insights from simplified membrane models, Fed. Proc, 44 (1985), 2944-2946.
    [56] D. Terman, D. Wang, Global competition and local cooperation in a network of neural oscillators, Physica D, 81 (1995), 148-176. doi: 10.1016/0167-2789(94)00205-5
    [57] J. Hale, S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993.
    [58] Z. Wang, Clustering of Networks with Time Delayed All-to-all Coupling, Ph.D thesis, University of Waterloo, 2017,
    [59] R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6
    [60] J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceeding IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235
    [61] M. Golubitsky, I. Stewart, D. G. Scherffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York, 1988.
    [62] J. Wu, Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2
    [63] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
    [64] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer Science & Business Media, 2013.
    [65] J. Bélair, S. A. Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math., 54 (1994), 1402-1424. doi: 10.1137/S0036139993248853
    [66] Z. Kilpatrick, B. Ermentrout, Sparse gamma rhythms arising through clustering in adapting neuronal networks, PLoS Comput. Biol., 7 (2011), e1002281. doi: 10.1371/journal.pcbi.1002281
    [67] S. A. Campbell, M. Waite, Multistability in coupled Fitzhugh-Nagumo oscillators, Nonlinear Anal., 47 (2001), 1093-1104. doi: 10.1016/S0362-546X(01)00249-8
    [68] F. K. Skinner, H. Bazzazi, S. A. Campbell, Two-cell to N-cell heterogeneous, inhibitory networks: precise linking of multistable and coherent properties, J. Comput. Neurosci., 18 (2005), 343-352. doi: 10.1007/s10827-005-0331-1
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