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References
Atay FM (2003a) Distributed delays facilitate amplitude death of coupled oscillators. Phys. Rev. Lett. 91 (9), 094101
Atay FM (2003b) Total and partial amplitude death in networks of diffusively coupled oscillators. Physica D 183, 1–18
Atay FM (2006) Oscillator death in coupled functional differential equations near Hopf bifurcation. J. Differential Equations 221 (1), 190–209
Bélair J, Campbell SA, van den Driessche P (1996) Frustration, stability and delay-induced oscillations in a neural network model. SIAM J. Applied Mathematics 56 (1), 245–255
Bernard S, Bélair J, Mackey MC (2001) Sufficient conditions for stability of linear differential equations with distributed delay. Discrete and Continuous Dynamical Systems 1B, 233–256
Breakspear M, Jirsa VK (2006) Neuronal dynamics and brain connectivity. In: McIntosh R, Jirsa VK (eds), Handbook of Brain Connectivity. Springer-Verlag, New York
Burić N, Grozdanović I, Vasović N (2005) Type I vs. type II excitable systems with delayed coupling. Chaos, Solitons and Fractals 23, 1221–1233
Burić N, Todorović D (2003) Dynamics of Fitzhugh-Nagumo excitable systems with delayed coupling. Physical Review E 67, 066222
Burić N, Todorović D (2005) Bifurcations due to small time-lag in coupled excitable systems. International Journal of Bifurcations and Chaos 15 (5), 1775–1785
Campbell SA, Edwards R, van den Dreissche P (2004) Delayed coupling between two neural network loops. SIAM J. Applied Mathematics 65 (1), 316–335
Campbell SA, Ncube I (2006) Some effects of gamma distribution on the dynamics of a scalar delay differential equation. Preprint
Campbell SA, Ncube I, Wu J (2006) Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system. Physica D 214 (2), 101–119
Campbell SA, Smith A (2007) Phase models and delayed coupled Fitzhugh-Nagumo oscillators. Preprint
Campbell SA, Yuan Y, Bungay S (2005) Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18, 2827–2846
Carson RG, Byblow WD, Goodman D (1994) The dynamical substructure of bimanual coordination. In: Swinnen S, Heuer H, Massion J, Casaer P (eds), Interlimb coordination: Neural, dynamical and cognitive constraints. Academic Press, San Diego, pp 319–337
Crook SM, Ermentrout GB, Vanier MC, Bower JM (1997) The role of axonal delay in the synchronization of networks of coupled cortical oscillators. J. Computational Neuroscience 4, 161–172
Cushing JM (1977) Integrodifferential Equations and Delay Models in Popluation Dynamics. Vol. 20 of Lecture Notes in Biomathematics. Springer-Verlag, Berlin; New York
Desmedt JE, Cheron G (1980) Central somatosensory conduction in man: Neural generators and interpeak latencies of the far-field components recorded from neck and right or left scalp and earlobes. Electro. Clin. Electroencephalog. 50, 382–403
Dhamala M, Jirsa VK, Ding M (2004) Enhancement of neural synchrony by time delay. Physical Review E 92 (7), 074104
Diekmann O, van Gils SA, Verduyn Lunel SM, Walther H-O (1995) Delay Equations. Springer-Verlag, New York
Engelborghs K, Luzyanina T, Roose D (2002) Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Transactions on Mathematical Software 28 (1), 1–21
Engelborghs K, Luzyanina T, Samaey G (2001) DDE-BIFTOOL v. 2.00: a MATLABpackage for bifurcation analysis of delay differential equations. Tech. Rep. TW-330, Department of Computer Science, K.U. Leuven, Leuven, Belgium
Ermentrout GB (1994) An introduction to neural oscillators. In: Ventriglia F (ed), Neural Modeling and Neural Networks. Pergamon, Oxford, UK, pp 79–110
Ermentrout GB (2002) Similating, Analyzing and Animating Dynamical Systems: A Guide to XPPAUT for Researcher and Students. SIAM, Philadelphia, PA
Ermentrout GB (2005) XPPAUT 5.91 – the differential equations tool. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA
Ermentrout GB, Kopell N (1998) Fine structure of neural spiking and synchronization in the presence of conduction delays. PNAS 95 (3), 1259–64
Fitzhugh R (1960) Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J. General Physiology 43, 867
Foss J, Longtin A, Mensour B, Milton JG (1996) Multistability and delayed recurrent loops. Phys. Rev. Letters 76, 708–711
Foss J, Milton JG (2000) Multistability in recurrent loops arising from delay. J. Neurophysiol. 84, 975–985
Foss J, Milton JG (2002) Multistability in delayed recurrent loops. In: Milton J, Jung P (eds), Epilepsy as a Dynamic Disease. Springer-Verlag, New York, pp 283–295
Foss J, Moss F, Milton JG (1997) Noise, multistability and delayed recurrent loops. Phys. Rev. E 55, 4536–4543
Fox JJ, Jayaprakash C, Wang DL, Campbell SR (2001) Synchronization in relaxation oscillator networks with conduction delays. Neural Computation 13, 1003–1021
Golomb D, Ermentrout GB (1999) Continuous and lurching travelling pulses in neuronal networks with delay and spatially decaying connectivity. PNAS 96, 13480–13485
Golomb D, Ermentrout GB (2000) Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity. Network: Comput. Neural Syst. 11, 221–246
Gopalsamy K, Leung I (1996) Delay induced periodicity in a neural netlet of excitation and inhibition. Physica D 89, 395–426
Guckenheimer J, Holmes PJ (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York
Haken H, Kelso JAS, Bunz H (1985) A theoretical model of phase transitions in human hand movements. Biological Cybernetics 51, 347–356
Hale JK, Verduyn Lunel SM (1993) Introduction to Functional Differential Equations. Springer-Verlag, New York
Hoppensteadt FC, Izhikevich EM (1997) Weakly connected neural networks. Springer-Verlag, New York
Ikeda I, Matsumoto K (1987) High dimensional chaotic behaviour in systems with time-delayed feedback. Physica D 29, 223–235
Izhikevich EM (1998) Phase models with explicit time delays. Physical Review E 58 (1), 905–908
Jantzen KJ, Kelso JAS (2006) Neural coordination dynamics of human sensorimotor behaviour: A review. In: McIntosh R, Jirsa VK (eds), Handbook of Brain Connectivity. Springer-Verlag, New York
Jirsa VK, Ding M (2004) Will a large complex system with delays be stable? Physical Review Letters 93, 070602
Karbowski J, Kopell N (2000) Multispikes and synchronization in a large neural network with temporal delays. Neural Computation 12, 1573–1606
Keener J, Sneyd J (1998) Mathematical Physiology. Springer-Velag, New York
Kelso JAS (1984) Phase transitions and critical behaviour in human bimanual coordination. American J. Physiology: Regulatory, Integrative and Comparative Physiology 15, R1000–R1004
Kelso JAS, Holt KG, Rubin P, Kugler PN (1981) Patterns of human interlimb coordination emerge from nonlinear limit cycle oscillatory processes: theory and data. J. Motor Behaviour 13, 226–261
Kleinfeld D, Raccuia-Behling F, Chiel HJ (1990) Circuits constructed from identified Aplysia neurons exhibit multiple patterns of activity. Biophysical J. 57 (4), 697–715
Koch C (1999) Biophysics of Computation. Oxford University Press, New York
Kolmanovskii VB, Nosov VR (1986) Stability of functional differential equations. Vol. 180 of Mathematics in Science and Engineering. Academic Press
Kopell N, Ermentrout GB (2002) Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. In: Fiedler B (ed), Handbook of Dynamical Systems, vol 2: Toward Applications. Elsevier, Amsterdam
Kopell N, Ermentrout GB, Whittington MA, Traub R (2000) Gamma rhythms and beta rhythms have different synchronization properties. PNAS 97 (4), 1867–1872
Kuznetsov YA (1995) Elements of Applied Bifurcation Theory. Vol. 112 of Applied Mathematical Sciences. Springer-Verlag, Berlin; New York
MacDonald N (1978) Time lags in biological models. Vol. 27 of Lecture notes in biomathematics. Springer-Verlag, Berlin; New York
Mackey MC, Van der Heiden U (1984) The dynamics of recurrent inhibition. J. Mathematical Biology 19, 211–225
Maex R, De Schutter E (2003) Resonant synchronization in heterogeneous networks of inhibitory neurons. J. Neuroscience 23 (33), 10503–10514
Milton JG, Foss J (1997) Oscillations and multistability in delayed feedback control. In: Othmer HG, Adler FR, Lewis MA, Dallon JC (eds), The Art of Mathematical Modeling: Case Studies in Ecology, Physiology and Cell Biology. Prentice Hall, New York, pp 179–198
Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070
Olgac N, Sipahi R (2002) An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Transactions on Automatic Control 47 (5), 793–797
Olgac N, Sipahi R (2005) Complete stability robustness of third-order LTI multiple time-delay systems. Automatica 41, 1413–1422
Plant RE (1981) A Fitzhugh differential-difference equation modeling recurrent neural feedback. SIAM J. Applied Mathematics 40 (1), 150–162
Ramana Reddy DV, Sen A, Johnston GL (1998) Time delay induced death in coupled limit cycle oscillators. Physical Review Letters 80, 5109–5112
Ramana Reddy DV, Sen A, Johnston GL (1999) Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D 129, 15–34
Ramana Reddy DV, Sen A, Johnston GL (2000) Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators. Physical Review Letters 85 (16), 3381–3384
Sen AK, Rand R (2003) A numerical investigation of the dynamics of a system of two time-delayed coupled relaxation oscillators. Communications on Pure and Applied Mathematics 2 (4), 567–577
Shampine LF, Thompson S (2001) Solving DDEs in MATLAB. Applied Numerical Mathematics 37, 441–458
Shayer LP, Campbell SA (2000) Stability, bifurcation and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Applied Mathematics 61 (2), 673–700
Shepherd G (1994) Neurobiology. Oxford University Press, New York
Skinner FK, Bazzazi H, Campbell SA (2005a) Two-cell to n-cell heterogeneous, inhibitory networks: precise linking of multistable and coherent properties. J. Computational Neuroscience 18 (3), 343–352
Skinner FK, Chung JYJ, Ncube I, Murray PA, Campbell SA (2005b) Using heterogeneity to predict inhibitory model characteristics. J. Neurophysiology 93, 1898–1907
Stépán G (1989) Retarded Dynamical Systems. Vol. 210 of Pitman Research Notes in Mathematics. Longman Group, Essex
Strogatz SH (1998) Death by delay. Nature 394, 316–317
Terman D, Wang DL (1995) Global competition and local cooperation in a network of neural oscillators. Physica D 81, 148–176
Thiel A, Schwegler H, Eurich CW (2003) Complex dynamics is abolished in delayed recurrent systems with distributed feedback times. Complexity 8 (4), 102–108
van Vreeswijk C, Abbott LF, Ermentrout GB (1994) When inhibition not excitation synchronizes neural firing. J. Computational Neuroscience 1, 313–321
Wang X-J, Buzsáki G (1998) Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J. Neuroscience 16 (20), 6–16
Wang X-J, Rinzel J (1992) Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Computation 4, 84–97
Wang X-J, Rinzel J (1993) Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons. Neuroscience 53, 899–904
White JA, Chow CC, Ritt J, Soto-Trevino C, Kopell N (1998) Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. J. Computational Neuroscience 5, 5–16
Wirkus S, Rand R (2002) The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dynamics 30, 205–221
Wischert W, Wunderlin A, Pelster A, Olivier M, Groslambert J (1994) Delay-induced instabilities in nonlinear feedback systems. Physical Review E 49 (1), 203–219
Wu J, Faria T, Huang YS (1999) Synchronization and stable phase-locking in a network of neurons with memory. Math. Comp. Modelling 30 (1-2), 117–138
Yuan Y, Campbell SA (2004) Stability and synchronization of a ring of identical cells with delayed coupling. J. Dynamics and Differential Equations 16 (1), 709–744
Zhou J, Chen T, Lan Z (2004a) Robust synchronization of coupled delayed recurrent neural networks. In: Advances in Neural Networks - ISNN 2004. Vol. 3173 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, pp 144–149
Zhou S, Liao Z, Yu J, Wong K (2004b) Chaos and its synchronization in two-neuron systems with discrete delays. Chaos, Solitons and Fractals 21, 133–142
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Campbell, S.A. (2007). Time Delays in Neural Systems. In: Jirsa, V.K., McIntosh, A. (eds) Handbook of Brain Connectivity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71512-2_2
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