Weakly connected quasi-periodic oscillators, FM interactions, and multiplexing in the brain

SIAM Journal on Applied Mathematics

Volumn 59, Issue 6, 1999, Pages 2193-2223

  Izhikevich, Eugene M ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FREQUENCY MODULATION; MATHEMATICAL MODELS; VECTORS;

QUASI-PERIODIC OSCILLATORS;

OSCILLATORS (ELECTRONIC);

EID: 0033348899     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139997330623     Document Type: Article

References (48)

1. M. ABELES, Y. PRUT, H. EERGMAN, AND E. VAADIA, Synchronization in neuronal transmission and its importance for information processing, in Temporal Coding in the Brain, G. Buzsáki, R. Llinás, W. Singer, A. Berthoz, and Y. Christen eds., Springer-Verlag, New York 1994.
2. M. A. ARBIB, P. ÉRDI, AND J. SZENTÁGOTHAI. Neural Organization, MIT Press, Cambridge, MA, 1998.
3. C. BAESENS, J. GUCKENHEIMER, S. KIM, AND R. S. MACKAY, Three coupled oscillators: Modelocking, global bifurcations and toroidal chaos, Phys. D, 49 (1991), pp. 387-475.
4. W. BAIR, C. KOCH, W. NEWSOME, K. BRITTEN, AND E. NIEBUR, Power spectrum analysis of bursting cells in area MT in the behaving monkey, J. Neurosci., 14 (1994), pp. 2870-2892.
5. N. YU. BIBIKOV, Multi-Frequency Nonlinear Oscillations and Their Bifurcations, Leningrad University Press, Leningrad. 1991 (in Russian).
6. G. N. BORISYUK, R. M. BORISYUK, A. I. KHIBNIK, AND D. ROOSE, Dynamics and bifurcations of two coupled neural oscillators with different connection types, Bull. Math. Biol., 57 (1995), pp. 809-840.
7. H. W. BROER, G. B. HUITEMA, AND M. B. SEVRYUK, Quasi-Periodic Motions in Families of Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, 1996.
8. G. BUZSÁKI, Z. HORVÁTH, R. URIOSTE, J. HETKE, AND K. WISE, High-frequency network oscillation in the hippocampus, Science, 256 (1992), pp. 1025-1027.
9. G. B. ERMENTROUT, n:m phase-locking of weakly coupled oscillators, J. Math. Biol., 12 (1981), pp. 327-342.
10. G. B. ERMENTROUT AND N. KOPELL, Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol., 29 (1991), pp. 195-217.
11. N. FENICHEL, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), pp. 193-226.
12. Y. FRÉGNAC, V. BRINGUIER, AND A. BARANYI, Oscillatory neuronal activity in visual cortex: A critical re-evaluation, in Temporal Coding in the Brain, G. Buzsáki, R. Llinás, W. Singer, A. Berthoz, and Y. Christen eds., Springer-Verlag, Berlin, New York, 1994.
13. C. M. GRAY, Synchronous oscillations in neuronal systems: Mechanism and functions, J. Comput. Neurosci., 1 (1994), pp. 11-38.
14. C. M. GRAY, P. KONIG, A. K. ENGEL, AND W. SINGER, Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties, Nature, 338 (1989), pp. 334-337.
15. W. HEILIGENBERG, The coding and processing of temporal information in the electro sensory system of fish, in Temporal Coding in the Brain, G. Buzsáki, R. Llinás, W. Singer, A. Berthoz, and Y. Christen, eds., Springer-Verlag, Berlin, New York, 1994.
16. M. W. HIRSCH, C. C. PUGH, AND M. SHUB, Invariant Manifolds, Springer-Verlag, New York, 1977.
17. M. W. HIRSCH, Differential Topology, Springer-Verlag, New York, 1976.
18. F. C. HOPPENSTEADT AND E. M. IZHIKEVICH, Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: I. Analysis of Canonical Model, Biological Cybernetics, 75 (1996), pp. 117-127.
19. F. C. HOPPENSTEADT AND E. M. IZHIKEVICH, Weakly Connected Neural Networks, Springer-Verlag, New York, 1997.
20. E. M. IZHIKEVICH, Weakly pulse-coupled oscillators. FM interactions, synchronization, and oscillatory associative memory, IEEE Trans. Neural Networks, 10 (1999), pp. 508-526.
21. YA. B. KAZANOVICH AND R. M. BORISYUK. Synchronization in a neural network of phase oscillators with the central element, Biological Cybernetics, 71 (1994), pp. 177-185.
22. N. KOPELL, Chains of coupled oscillators, in Brain Theory and Neural Networks, M.A. Arbib, ed., MIT Press, Cambridge, MA, 1995, pp. 178-183.
23. Y. KURAMOTO, Collective synchronization of pulse-coupled oscillators and excitable units, Phys. D, 50 (1991), pp. 15-30.
24. Y. KURAMOTO, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, New York, 1984.
25. R. R. LLINÁS, The intrinsic electrophysiological properties of mammalian neurons: A new insight into CNS function, Science, 242 (1988), pp. 1654-1664.
26. R. R. LLINÁS, A. A. GRACE, AND Y. YAROM, In vitro neurons in mammalian cortical layer 4 exhibit intrinsic oscillatory activity in the 10-to 50-Hz frequency range, Proc. Natl. Acad. Sci., 88 (1991), pp. 897-901.
27. C. VON DER MALSBURG, Dynamic link architecture, in Brain Theory and Neural Networks, M.A. Arbib, ed., MIT Press, Cambridge. MA, 1995, pp. 329-331.
28. A. MASON, A. NICOLL, AND K. STRATFORD, Synaptic transmission between individual pyramidal neurons of the rat visual cortex in vitro, J. Neurosci., 11 (1991), pp. 72-84.
29. B. L. MCNAUGHTON, C. A. BARNES, AND P. ANDERSEN, Synaptic efficacy and EPSP summation in granule cells of rat fascia dentata studied in vitro. J. Neurophysiol., 46 (1981), pp. 952-966.
30. R. MILES AND R. K. S. WONG, Excitatory synaptic interactions between CA3 neurons in the guinea-pig hippocampus, J. Physiol., 373 (1986), pp. 397-418.
31. R. E. MIROLLO AND S. H. STROGATZ, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), pp. 1645-1662.
32. m, m ≥ 3, Comm. Math. Phys., 64 (1978), pp. 35-40.
33. P. L. NUNEZ, Neocortical Dynamics and Human EEG Rhythms, Oxford University Press, New York, London, 1995.
34. C. S. PESKIN, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975.
35. R. H. RAND, A. H. COHEN, AND P. J. HOLMES, Systems of coupled oscillators as models of central pattern generators, in Neural Control of Rhythmic Movements in Vertebrates, A. H Cohen, S. Grillner, and S. Rossignol, eds., Wiley, New York, 1987, pp. 369-413.
36. D. RUELLE AND F. TAKENS, On the nature of turbulence, Comm. Math. Phys., 20 (1971), pp. 167-192.
37. A. M. SAMOILENKO, Elements of the Mathematical Theory of Multi-Frequency Oscillations, Mathematics and Its Applications (Soviet Series) 71, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1991.
38. R. J. SAYER, M. J. FRIEDLANDER, AND S. J. REDMAN, The time course and amplitude of EPSPs evoked at synapses between pairs of CA3/CA1 neurons in the hippocampal slice, J. Neurosci., 10 (1990) pp. 826-836.
39. J. A. SIMMONS, M. FERRAGAMO, C. F. MOSS, S. B. STEVENSON, AND R. A. ALTES, Discrimination of jittered sonar echoes by the echolocating bat, Eptesicus fuscus: The shape of target images in echolocation, J. Comp. Physiol., A167 (1990), pp. 589-616.
40. W. SINGER AND C. M. GRAY, Visual feature integration and the temporal correlation hypothesis, Ann. Rev. Neurosci., 18 (1995), pp. 555-586.
41. D. TERMAN AND D. L. WANG, Global competition and local cooperation in a network of neural oscillators, Phys. D, 81 (1995), pp. 148-176.
42. M. J. TOVEE AND E. T. ROLLS, Oscillatory activity is not evident in the primate temporal visual cortex with static stimuli, NeuroReport, 3 (1992), pp. 369-372.
43. D. L. WANG, Emergent synchrony in locally coupled neural oscillators, IEEE Trans. Neural Networks, 6 (1995), pp 941-948.
44. S. WIGGINS, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer-Verlag, New York, 1994.
45. H. R. WILSON AND J. D. COWAN, Excitatory and inhibitory interaction in localized populations of model neurons, Biophys. J., 12 (1972), pp. 1-24.
46. H. R. WILSON AND J. D. COWAN, A mathematical theory of the functional dynamics of cortical and thalamic nervous issue, Kybernetik, 13 (1973), pp. 55-80.
47. A. WINFREE, The Geometry of Biological Time, Springer-Verlag, New York, 1980.
48. M. P YOUNG, K. TANAKA, AND S. YAMANE. On oscillating neuronal responses in the visual cortex of the monkey, J. Neurophysiol., 67 (1992), pp. 1464-1474.