Delays in activity-based neural networks

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1891, 2009, Pages 1117-1129

  Coombes, Stephen ^a   Laing, Carlo ^b  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

^b MASSEY UNIVERSITY   (New Zealand)

Author keywords

Chaos; Multiple delays; Phase locking; Wilson Cowan networks

Indexed keywords

BIFURCATION (MATHEMATICS); DYNAMICS; MOTION ESTIMATION;

ACTIVITY-BASED; BURSTING OSCILLATIONS; CHAOS; CHAOTIC BEHAVIOURS; DIRECT SIMULATIONS; DISCRETE DELAYS; FIRING RATES; FIXED POINTS; HEAVISIDE; LINEAR STABILITY THEORIES; MULTIPLE DELAYS; NATURAL EXTENSIONS; NEURONAL POPULATIONS; NON LINEARITIES; NUMERICAL BIFURCATION ANALYSIS; PHASE LOCKING; WILSON-COWAN NETWORKS;

NEURAL NETWORKS;

ANIMAL; ARTICLE; BIOLOGICAL MODEL; HUMAN; NERVE CELL; NERVE CELL NETWORK; OSCILLOMETRY; PHYSIOLOGY; REACTION TIME; SYNAPSE; SYNAPTIC POTENTIAL;

ANIMALS; HUMANS; MODELS, NEUROLOGICAL; NERVE NET; NEURONS; OSCILLOMETRY; REACTION TIME; SYNAPSES; SYNAPTIC POTENTIALS;

EID: 60749120055     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2008.0256     Document Type: Article

References (35)

1. Atay, F. M. & Hutt, A. 2006 Neural fields with distributed transmission speeds and long-range feedback delays. SIAM J. Appl. Dyn. Syst. 5, 670-698. (doi:10.1137/050629367)
2. Battaglia, D., Brunei, N. & Hansel, D. 2007 Temporal decorrelation of collective oscillations in neural networks with local inhibition and long-range excitation. Phys. Rev. Lett. 99, 238106(1-4). (doi:10.1103/PhysRevLett.99.238106)
3. Bressloff, P. C. 2000 Traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons. J. Math. Biol. 40, 169-183. (doi:10.1007/s002850050008)
4. Campbell, S. A. 2007 Time delays in neural systems. In Handbook of brain connectivity (eds A. R. McIntosh & V. K. Jirsa), pp. 65-90. Berlin, Germany: Springer.
5. Chen, Y. & Wu, J. 1999 Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network. Physica D 134, 185-199. (doi:10.1016/S0167-2789(99)00111-6)
6. Chen, Y. M., Wu, J. H. & Krisztin, T. 2000 Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system. J. Differ. Equat. 163, 130-173. (doi:10.1006/jdeq.1999.3724)
7. Coombes, S. 2005 Waves, bumps, and patterns in neural field theories. Biol. Cybern. 93, 91-108. (doi:10.1007/s00422-005-0574-y)
8. Coombcs, S. & Bressloff, P. C. (eds) 2005 Bursting: the genesis of rhythm, in the nervous system. Republic of Singapore: World Scientific Press.
9. Coombcs, S. & Laing, C. R. In press. Instabilities in threshold-diffusion equations with delay. Physica D. (doi:10.101G/j. physd.2008.10.014)
10. Coombes, S. & Lord, G. J. 1997 Intrinsic modulation of pulse-coupled integrate-and-fire neurons. Phys. Rev. E 56, 5809-5818. (doi:10.1103/PhysRevE.56.5809)
11. Coombes, S., Venkov, N. A., Shiau, L., Bojak, I., Liley, D. T. J. & Laing, C. R. 2007 Modeling electrocortical activity through improved local approximations of integral neural field equations. Phys. Rev. E 76, 051901. (doi:10.1103/PhysRevE.76.051901)
12. Destexlie, A. & Gaspard, P. 1993 Bursting oscillations from a homoclinic tangency in a time-delay system, Phys. Lett. A 173, 386-391. (doi:10.1016/0375-9601(93)90256-Y)
13. Engelborghs, K., Luzyanina, T. & Samaey, G. 2001 DDE-BIFTOOL v. 2.00: a MATLAB package for bifurcation analysis of delay differential equations. Technical report TW-330, Department of Computer Science, KU Leuven, Belgium.
14. Engelborghs, K., Luzyanina, T. & Roose, D. 2002 Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28, 1-21. (doi:10.1145/513001.513002)
15. Ermentrout, B. 1998 Neural networks as spatio-temporal pattern-forming systems. Rep. Prog. Phys. 61, 353-430. (doi:10.1088/0034-4885/61/ 4/002)
16. Giannakapoulos, F. & Zapp, A. 2001 Bifurcations in a planar system of differential delay equations modeling neural activity. Physica D 159, 215-232. (doi:10.1016/S0167-2789(01)00337-2)
17. Glass, L., Beuter, A. & Larocque, D. 1988 Time delays, oscillations, and chaos in physiological control systems. Math. Biosci. 90, 111-125. (doi:10.1016/0025-5564(88)90060-0)
18. Golomb, D. & Ermentrout, G. B. 1999 Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity. Proc. Natl Acad. Sci. USA 96, 13 480-13 485. (doi:10.1073/pnas.96. 23.13480)
19. Guo, S., Huang, L. & Wu, J. 2005 Regular dynamics in a delayed network of two neurons with allor-none activation functions. Physica D 206, 32-48. (doi:10.1016/j.physd.2003.09.049)
20. Hirschberg, P. & Laing, C. R. 1995 Successive homoclinic tangencies to a limit cycle. Physica D 89, 1-14. (doi:10.1016/0167-2789(95) 00211-1)
21. Hoppensteadt, F. C. & Izhikevich, E. M. 1997 Weakly connected neural networks. New York, NY: Springer.
22. Laing, C. R. & Coombes, S. 2006 The importance of different timings of excitatory and inhibitory pathways in neural field models. Network 17, 151-172. (doi: 10.1080/09548980500533461)
23. Laing, C. R. & Longtin, A. 2003 Dynamics of deterministic and stochastic paired excitatoryinhibitory delayed feedback. Neural Comput. 15, 2779-2822. (doi: 10.1162/089976603322518740)
24. Longtin, A. & Milton, J. G. 1998 Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Math. Biosci. 90, 183-199. (doi: 10.1016/0025-5564(88)90064-8)
25. MacDonald, N. 1989 Biological delay systems: linear stability theory. Cambridge, UK: Cambridge University Press.
26. Marcus, C. M. & Westervelt, R. M. 1989 Stability of analog neural networks with delay. Phys. Rev. A 39, 347-359. (doi:10.1103/PhysRevA.39.347)
27. Murdoch, W. W., Nisbet, R. M., Blyth, S. P., Gurney, W. S. C. & Reeve, J. D. 1987 An invulnerable age class and stability in delay-differential parasitoid-host models. Am. Nat. 129, 263-282. (doi:10.1086/284634)
28. Olien, L. & Bélair, J. 1997 Bifurcations, stability, and monotonicity properties of a delayed neural network model. Physica D 102, 349-363. (doi:10.1016/S0167-2789(96)00215-1)
29. Plant, R. E. 1981 A FitzHugh differential-difference equation modeling recurrent neural feedback. SIAM J. Appl. Math. 40, 150-162. (doi:10.1137/0140012)
30. Reddy, R. D. V., Sen, A. & Johnston, G. L. 1998 Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett. 80, 5109-5112. (doi:10.1103/PhysRevLett.80.5109)
31. Roxin, A., Brunei, N. & Hansel, D. 2005 Role of delays in shaping spatiotcmporal dynamics of neuronal activity in large networks. Phys. Rev. Lett. 94, 238103(1-4). (doi:10.1103/PhysRevLett.94.238103)
32. Shayer, L. P. & Campbell, S. A. 2000 Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Appl Math. 61, 673-700. (doi:10.1137/S0036139998344015)
33. Stepan, G. 1989 Retarded dynamical systems. Harlow, UK: Longman Scientific & Technical.
34. Wei, J. J. & Ruan, S. R. 1999 Stability and bifurcation in a neural network model with two delays. Physica D 130, 255-272. (doi: 10.1016/S0167-2789(99)00009-3)
35. Wilson, H. R. & Cowan, J. D. 1972 Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1-24. (doi:10.1016/S0006-3495(72)86068-5)