Stable and unstable periodic orbits in complex networks of spiking neurons with delays

Discrete and Continuous Dynamical Systems

Volumn 28, Issue 4, 2010, Pages 1555-1588

  Memmesheimer, Raoul Martin ^a,b   Timme, Marc ^c,d,e  

^a HARVARD UNIVERSITY   (United States)

^b RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

^c MAX PLANCK INSTITUTE FOR DYNAMICS AND SELF ORGANIZATION   (Germany)

^d Bernstein Center for Computational Neuroscience   (Germany)

^e UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Attractor neural networks; Hybrid dynamical system; Local cortical circuits; Periodic orbit; Spiking neural network; Stability; Synchronization

Indexed keywords

EID: 77958026979     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2010.28.1555     Document Type: Article

References (46)

1. L. F. Abbott and C. van Vreeswijk, Asynchronous states in networks of pulse-coupled oscillators, Phys. Rev. E, 48 (1993), 1483-1490.
2. M. Abeles, Time is precious, Science, 304 (2004), 523-524.
3. M. Abeles, H. 'Bergman, F. Margalit and E. Vaadia, Spatiotemporal firing patterns in the frontal cortex of behaving monkeys, J. Neurophysiol., 70 (1993), 1629-1638.
4. P. Ashwin and M. Timme, Unstable attractors: Existence and robustness in networks of oscillators with delayed pulse coupling, Nonlinearity, 18 (2005), 2035-2060.
5. D. D. Bainov and P. S. Simeonov, "Systems with Impulse Effect. Stability, Theory and Applications," Horwood, London, 1989.
6. M. S. Branicky, "Introduction to Hybrid Systems," Handbook of Networked and Embedded Control Systems (D. Hristu-Varsakelis and W.S. Levine, eds.), Birkhäuser, 2005.
7. N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, J. Comp. Neurosci., 8 (2000), 183-208.
8. N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Comp., 11 (1999), 1621-1671.
9. M. Denker, M. Timme, M. Diesmann, F. Wolf and T. Geisel, Breaking synchrony by heterogeneity in complex networks, Phys. Rev. Lett., 92 (2004), 074103.
10. U. Ernst, K. Pawelzik and T. Geisel, Synchronization induced by temporal delays in pulsecoupled oscillators, Phys. Rev. Lett., 74 (1995), 1570-1573.
11. -, Deiay-induced multistable synchronization of biological oscillators, Phys. Rev. E, 57 (1998), 2150-2162.
12. K. Gansel and W. Singer, Replay of second-order spike patterns with millisecond precision in the visual cortex, Soc. Neurosci. Abstr. 276.8, 2005.
13. -, Recurring second-order spike patterns concur with increased symmetry in functional neuronal couplings, Soc. Neurosci. Abstr. 634.15, 2006.
14. C. Grebogi, E. Ott and J. A. Yorke, Chaotic attractors in crisis, Phys. Rev. Lett., 48 (1982), 1507-1510.
15. R. H. R. Hahnloser, A. A. Kozhevnikov and M. S. Fee, .An ultra-sparse code underlies the generation of neural sequences in a songbird, Nature, 419 (2002), 65-70.
16. D. Hansel and G. Mato, Existence and stability of persistent states in large neuronal networks, Phys. Rev. Lett., 86 (2001), 4175-4178.
17. S. Jahnke, R.-M. Memmesheimer and M. Timme, Stable irregular dynamics in complex neural networks, Phys. Rev. Lett., 100 (2008), 048102.
18. S. Jahnke, R.-M. Memmesheimer and M. Timme, How chaotic is the balanced state?, Front. Comput. Neurosci., 3 (2009), 13.
19. D. Z. Jin, Fast convergence of spike sequences to periodic patterns in recurrent networks, Phys. Rev. Lett., 89 (2002), 208102.
20. A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.
21. A. Kumar, S. Schrader, A. Aertsen and S. Rotter, The high-conductance state of cortical networks, Neural Comp., 20 (2008), 1-34.
22. L. Lapicque, Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarisation, J. Physiol. Pathol. Gen., 9 (1907), 620-635.
23. A. K. Lee and M. A. Wilson, Memory of sequential experience in the hippocampus during slow wave sleep, Neuron, 36 (2002), 1183-1194.
24. I. J. Matus Bloch and C. Romero Z., Firing sequence storage using inhibitory synapses in networks of pulsatil nonhomogeneous integrate-and-fire neural oscillators, Phys. Rev. E, 66 (2002), 036127.
25. M. L. Mehta, "Matrix Theory: Selected Topics and Useful Results," Hindustan Publishing Corporation, Delhi, 1989.
26. R.-M. Memmesheimer, "Precise Spike Timing in Complex Neural Networks," Doctoral thesis, Department of Physics, Georg-August University of Göttingen, 2008.
27. R.-M. Memmesheimer and M. Timme, Designing complex networks, Physica D, 224 (2006), 182-201.
28. -, Designing the dynamics of spiking neural networks, Phys. Rev. Lett., 97 (2006), 188101.
29. R. E. Mirollo and S. H. Strogatz, Synchronization of pulse- coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662.
30. Z. Nádasdy, H. Hirase, A. Czurkó, J. Csicsvari and G. Buzsáki, Replay and time compression of recurring spike sequences in the hippocampus, J. Neurosci., 19 (1999), 9497-9507.
31. G. Pipa, A. Riehle and S. Grün, Validation of task-related excess of spike coincidences based on NeuroXidence, Neurocomputing, 70 (2007), 2064-2068.
32. W. Singer, Neuronal synchrony: A versatile code for the definition of relations?, Neuron, 24 (1999), 49-65.
33. S. H. Strogatz and R. E. Mirollo, Splay states in globally coupled Josephson arrays: Analytical prediction of floquet multipliers, Phys. Rev. E, 47 (1993), 220-227.
34. M. Timme, Does dynamics reflect topology in directed networks?, Europhys. Lett., 76 (2006), 367-373.
35. M. Timme and F. Wolf, The simplest problem in the collective dynamics of neural networks: Is synchrony stable?, Nonlinearity, 21 (2008), 1579-1599.
36. M. Timme, F. Wolf and T. Geisel, Prevalence of unstable attractors in networks of pulsecoupled oscillators, Phys. Rev. Lett., 89 (2002), 154105.
37. -, Unstable attractors induce perpetual synchronization and desynchronization, Chaos, 13 (2003), 377-387.
38. M. Tsodyks, I. Mitkov and H. Sompolinsky, Pattern of synchrony in inhomogeneous networks of oscillators with pulse interactions, Phys. Rev. Lett., 71 (1993), 1280-1283.
39. C. van Vreeswijk, Partial synchronization in populations of pulse-coupled oscillators, Phys. Rev. E, 54 (1996), 5522-5537.
40. -, Analysis of the asynchronous state in networks of strongly coupled oscillators, Phys. Rev. Lett., 84 (2000), 5110-5113.
41. C. van Vreeswijk, L. F. Abbott and G. B. Ermentrout, When inhibition not excitation synchronizes neural firing, J. Comp. Neurosci., 1 (1994), 313-321.
42. D. Werner, "Funktionalanalysis," Springer, Berlin, 1997.
43. J. Zhang, K. H. Johansson, J. Lygeros and S. Sastry, "Dynamical Systems Revisited: Hybrid Systems with Zeno Executions," Hybrid Systems: Computation and Control (N. Lynch and B. Krogh, eds.), Springer, 2000.
44. R. Zillmer, N. Brunel and D. Hansel, Very long transients, irregular firing, and chaotic dynamics in networks of randomly connected inhibitory integrate-and-fire neurons, Phys. Rev. E, 79 (2009), 031909.
45. R. Zillmer, R. Livi, A. Politi and A. Torcini, Desynchronization in diluted neural networks, Phys. Rev. E, 74 (2006), 036203.
46. A. Zumdieck, M. Timme, T. Geisel and F. Wolf, Long chaotic transients in complex networks, Phys. Rev. Lett., 93 (2004), 244103.