Dynamical theory of spike train transitions in networks of integrate-and-fire oscillators

SIAM Journal on Applied Mathematics

Volumn 60, Issue 3, 2000, Pages 820-841

  Bressloff, P C ^a   Coombes, S ^a  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); DIFFERENTIAL EQUATIONS; LINEARIZATION; NEURAL NETWORKS; NEUROPHYSIOLOGY; SPECTRUM ANALYSIS;

HOPF BIFURCATIONS; INTEGRATE AND FIRE NEURAL OSCILLATORS; PHASE EQUATIONS; PULSE COUPLED OSCILLATORS; RETICULAR THALAMIC NUCLEUS; SPIKE TRAIN TRANSITIONS;

MATHEMATICAL MODELS;

EID: 0033717490     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139998339643     Document Type: Article

References (36)

1. D. J. AMIT AND M. V. TSODYKS, Quantitative study of attractor neural networks retrieving at low spike rates I: Substrate - spikes, rates and neuronal gain, Networks, 2 (1991), pp. 259-274.
2. P. ASHWIN AND J. W. SWIFT, The dynamics of n weakly coupled identical oscillators, J. Nonlinear Sci., 2 (1992), pp. 69-108.
3. A. ATIYA AND P. BALDI, Oscillations and synchronization in neural networks: An exploration of the labelling hypothesis, Int. J. Neural Syst., 1 (1989), pp. 103-124.
4. P. C. BRBSSLOFF, Resonant-like synchronization and bursting in a model of pulse-coupled neurons with active dendrites, J. Coniput. Neurosci., 6 (1999), pp. 237-249.
5. P. C. BRESSLOFF AND S. COOMBES, Desynchronization, mode-locking and bursting in strongly coupled integrate-and-fire oscillators, Phys. Rev. Lett., 81 (1998), pp. 2168-2171.
6. P. C. BRESSLOFF AND S. COOMBES, Spike train dynamics underlying pattern formation in networks of integrate-and-fire oscillators, Phys. Rev. Lett., 81 (1998), pp. 2384-2387.
7. P. C. BRESSLOFF AND S. COOMBES, Travelling waves in a chain of pulse-coupled oscillators, Phys. Rev. Lett.. 81 (1998), pp. 4815-4818.
8. P. C. BRESSLOFF AND S. COOMBES, Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays, Phys. D, 126 (1999), pp. 99-122.
9. P. C. BRESSLOFF AND S. COOMBES, Travelling waves in a chain of pulse-coup led integrate-and-fire oscillators with distributed delays, Phys. D, 130 (1999), pp. 232-254.
10. P. C. BRESSLOFF, S. COOMBES, AND B. D. SOUZA, Dynamics of a ring of pulse-coupled oscillators: A group theoretic approach. Phys. Rev. Lett., 79 (1997), pp. 2791-2794.
11. S. COOMBES AND G. J. LORD, Desynchronisation of pulse-coup led integrate-and-fire neurons, Phys. Rev. E, 55 (1997), pp. 2104R-2107R.
12. G. B. ERMENTROUT, Reduction of conductance-based models with slow synapses to neural nets, Neural Comput., 6 (1994). pp. 679-695.
13. G. B. ERMENTROUT, Neural networks as spatio-temporal pattern-forming systems, Rep. Progr. Phys., 61 (1998), pp. 353-430.
14. G. B. ERMENTROUT AND N. KOPELL, Frequency plateaus in a chain of weakly coupled oscillators. I, SIAM J. Math. Anal., 15 (1984), pp. 215-237.
15. U. ERNST, K. PAWELZIK, AND T. GIESEL, Synchronization induced by temporal delays in pulse-coupled oscillators, Phys. Rev. Lett., 74 (1995), pp. 1570-1573.
16. W. GERSTNER, Time structure of the activity in neural-network models, Phys. Rev. E. 51 (1995), pp. 738-758.
17. W. GERSTNER, J. L. VAN HEMMEN, AND J. D. COWAN, What matters in neuronal locking, Neural Comput., 8 (1996), pp. 1653-1676.
18. D. GOLOMB AND J. RINZEL, Clustering in globally coupled inhibitory neurons, Phys. D, 72 (1994), pp. 259-282.
19. M. GOLUBITSKY, I. N. STEWART, AND D. G. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. 2, Appl. Math. Sci. 69, Springer-Verlag, New York, 1988.
20. C. M. GRAY, Synchronous oscillations in neuronal systems: Mechanisms and functions, J. Comput. Neurosci., 1 (1994), pp. 11-38.
21. D. HANSEL, G. MATO, AND C. MEUNIER, Synchrony in excitatory neural networks, Neural Comput., 7 (1995), pp. 307-337.
22. A. L. HODGKIN AND A. F. HUXLEY, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (Lond.), 117 (1952), pp. 500-544.
23. J. J. B. JACK, D. NOBLE, AND R. W. TSIEN, Electric Current Flow in Excitable Cells, Clarendon Press, Oxford, 1975.
24. W. M. KISTLER, R. SEITZ, AND J. L. VAN HEMMEN, Modeling collective excitations in cortical tissue, Phys. D, 114 (1998), pp. 273-295.
25. N. KOPELL, Chains of coupled oscillators, in The Handbook of Brain Theory and Neural Networks, M. A. Arbib, ed., MIT Press, Cambridge, MA, 1995, pp. 178-183.
26. R. E. MIROLLO AND S. H. STROGATZ, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), pp. 1645-1662.
27. G. P. K. P. ASHWIN AND J. W. SWIFT, Three identical oscillators with symmetric coupling, Nonlinearity, 3 (1990), pp. 585-601.
28. P. F. PINSKY AND J. RINZEL, Synchrony measures for biological neural networks, Biol. Cybern., 73 (1995), pp. 129-137.
29. R. H. RAND AND P. J. HOLMES, Bifurcation of periodic motions in two weakly coupled van der pol oscillators, Internat. J. Non-Linear Mech., 15 (1980), pp. 387-399.
30. H. C. TUCKWELL, Introduction to Theoretical Neurobiology, Vol. I, Cambridge University Press, Cambridge, UK, 1988.
31. C. VAN VREESWIJK, Partial synchronization in populations of pulse-coupled oscillators, Phys. Rev. E, 54 (1996), pp. 5522-5537.
32. C. VAN VREESWIJK, L. F. ABBOTT, AND G. B. ERMENTROUT, When inhibition not excitation synchronizes neural firing, J. Comput. Neurosci., 1 (1994), pp. 313-321.
33. X.-J. WANG AND J. RINZEL, Alternating and synchronous rhythms in reciprocally inhibitory model neurons, Neural Comput., 4 (1992), pp. 89-97.
34. J. A. WHITE, C. C. CHOW, J. RITT, C. SOTO-TREVINO, AND N. KOPELL, Synchronization and oscillatory dynamics in heterogeneous mutually inhibited neurons, J. Comput. Neurosci., 5 (1998), pp. 5-16.
35. H. R. WILSON AND J. D. COWAN, Excitatory and inhibitory interactions in localized populations of model neurons, J. Biophys., 12 (1972), pp. 1-24.
36. M. A. WILSON AND J. M. BOWER, Cortical oscillations and temporal interactions in a computer simulation of piriform cortex, J. Neurophysiol., 67 (1992), pp. 981-995.