Patterns and features of families of traveling waves in large-scale neuronal networks

SIAM Journal on Applied Dynamical Systems

Volumn 6, Issue 1, 2007, Pages 263-292

  Troy, William C ^a   Shusterman, Vladimir ^a  

^a UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

Excitatory; Integro differential equation; Nonlocal; Waves

Indexed keywords

A STABLES; ACTIVITY LEVELS; COUPLING FUNCTIONS; CRITICAL VALUES; DO-MAINS; EIGENVALUES; ENTIRE DOMAINS; EXCITATORY; EXCITATORY NEURONS; EXTERNAL INPUTS; GAUSSIAN; INDEPENDENT VARIABLES; INTEGRO-DIFFERENTIAL EQUATION; LONG RANGES; MULTIRING; NEURONAL NETWORKS; NONLOCAL; NUMERICAL EXPERIMENTS; PULSE WAVES; SPIRAL WAVES; STABLE SOLUTIONS; TIME DEPENDENTS; TRAVELING WAVES; TWO SPACE DIMENSIONS; WAVE FRONTS;

COMPUTER NETWORKS; DIFFERENTIAL EQUATIONS; INTEGRODIFFERENTIAL EQUATIONS; NEURAL NETWORKS; NEURONS; POPULATION STATISTICS;

WAVE EQUATIONS;

EID: 45749147923     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/06066638X     Document Type: Article

References (56)

1. S. AMARI, Dynamics of pattern formation in lateral inhibition type neural fields, Biolog. Cybernet., 27 (1977), pp. 77-87.
2. P. C. BRESSLOFF AND S. E. FOLIAS, Front bifurcations in an excitatory neural network, SIAM J. Appl. Math., 65 (2004), pp. 131-151.
3. E. BROWN, J. GAO, P. HOLMES, R. BOGACZ, M. GILZENRAT, AND J. D. COHEN, Simple neural networks that optimize decisions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), pp. 803-826.
4. B. D. BURNS, Some properties of the cat's isolated cerebral cortex, J. Physiol., 111 (1950), pp. 50-68.
5. B. D. BURNS, Some properties of the isolated cerebral cortex in the unanaesthetized cat, J. Physiol., 112 (1951), pp. 156-175.
6. B. D. BURNS AND B. GRAFSTEIN, The function and structure of some neurones in the cat's cerebral mcortex, J. Physiol., 118 (1952), pp. 412-433.
7. G. BUZSAKI AND A. DRAGUHN, Neuronal oscillations in cortical networks, Science, 304 (2004), pp. 1926- 1929.
8. R. D. CHERVIN, P. A. PIERCE, AND B. W. CONNORS, Periodicity and directionality in the propagation of epileptiform discharges across neocortex, J, Neurophys., 60 (1988), pp. 1695-1713.
9. P. H. CHU, J. MLLTON, AND J. D. COWAN, Connectivity and the dynamics of integrate and fire neural networks, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), pp. 237-243.
10. B. W. CONNORS AND Y. AMATI, Generation of epileptiform discharge by local circuits of neocortex, in Epilepsy: Models, Mechanisms and Concepts, P. A. Schwartkroin, ed., Cambridge University Press, Cambridge, UK, 1993, pp. 388-423.
11. S. COOMBES, Waves and bumps in neural field theories, Biolog. Cybernet., 93 (2005), pp. 91-108.
12. S. COOMBES AND M. R. OWEN, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 574-600.
13. S. COOMBES AND M. R. OWEN, Bumps, breathers, and waves in a neural network with spike frequency daptation, Phys. Rev. Lett., 94 (2005), paper 148102.
14. J. D. DROVER AND B. ERMENTROUT, Nonlinear coupling near a degenerate Hopf (Bautin) bifurcation, SIAM J. Appl. Math., 63 (2003), pp. 1627-1647.
15. J. S. EBERSOL AND J. MLLTON, The electroencephalogram (EEG): A measure of neural synchrony, in Epilepsy as a Dynamic Disease, Biol. Med. Phys. Ser., Springer, New York, 2003, Chapter 5.
16. ,7. C. ECCLES, The Understanding of the Brain, McGraw-Hill, New York, 1973.
17. G. B. ERMENTROUT AND J. B. MCLEOD, Existence and, uniqueness of traveling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp. 461-478.
18. I. FEREZOU, S. BOLEA, AND C. PETERSEN, Visualizing the cortical representation of whisker touch: Voltage sensitive dye imaging in freely moving mice, Neuron, 50 (2006), pp. 617-629.
19. S. FOLIAS AND P. BRESSLOFF, Breathers in two dimensional excitable neural media, Phys. Rev. Lett., 95 (2004), paper 208107.
20. S. E. FOLIAS AND P. C. BRESSLOFF, Breathing pulses in an excitatory neural, network, SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 378-407.
21. J. GLANZ, Mastering the nonlinear brain, Science, 277 (1997), pp. 1758-1760.
22. D. GOLOMB, Models of neuronal transient synchrony during propagation of activity through neocortical circuitry, J. Neurophys., 79 (1998), pp. 1-12.
23. D. GOLOMB AND Y. AMITAI, Propagating neuronal discharges in neocortical slices: Computational and experimental study, J. Neurophys., 78 (1997), pp. 1199-1211.
24. Y. GUO AND C. C. CHOW, Existence and stability of standing pulses in neural networks: I. Existence, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 217-248.
25. Y. GUO AND C. C. CHOW, Existence and stability of standing pulses in neural networks: II. Stability, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 249-281.
26. B. GUTKIN, D. PINTO, AND B. ERMENTROUT, Mathematical neuroscience: From neurons to circuits to systems, J. Physiol. Paris, 97 (2003), pp. 209-219.
27. S. P. HASTINGS, Single and multiple pulse waves for the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 42 (1982), pp. 247-260.
28. J. A. HOBSON, Dreaming: An Introduction to the Science of Sleep, Oxford University Press, London, 2004.
29. J. A. HOBSON AND R. W. MCCARLEY, The brain as a dream state generator. An activation-synthesis of the dream process, Amer. J. Psychiatry, 134 (1977), pp. 1335-1348.
30. X. HUANG, W. C. TROY, Q. YANG, H. MA, C. LAING, S. SGHIFF, AND J. Y. WU, Spiral waves in disinhibited mammalian cortex, J. Neurosci., 24 (2004), pp. 9897-9902.
31. M. A. P. IDIART AND L. F. ABBOTT, Propagation of excitation in neural network models, Network, 4 (1993), pp. 285-294.
32. D. KLEINFIELD, K. R. DELANEY, M. S. FEE, J. A. FLORES, D. W. TANK, AND A. GALPERIN, Dynamics of propagating waves in the olfactory network of a terrestrial mollusk: An electrical and optical study, J. Neurophys., 72 (1994), pp. 1402-1419.
33. N. KOPELL AND B. ERMENTROUT, Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks, Proc. Natl. Acad. Sci. USA, 101 (2004), pp. 15482-15487.
34. C. R. LAING AND W. C. TROY, PDE methods in nonlocal models, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 487-516.
35. Y. W. LAM, L. B. COHEN, M. WACHOWIAK, AND M. R. ZOCHOWSKI, Odors elicit three different oscillations in the turtle olfactory bulb, J. Neurosci., 20 (2000), pp. 749-762.
36. R. MILES, R. D. TRAUB, AND R. K. WONG, Spread of synchronous firing in longitudinal slices from the CA3 region of the hippocampus, J. Neurophys., 60 (1988), pp. 1481-1496.
37. J. MILTON, T. MUNDEL, U. AN DER HEIDEN, J. SPRIRE, AND J. COWAN, Traveling activity waves, in Handbook of Brain Theory and Neural Networks, MIT Press, Cambridge, MA, 1994, pp. 994-996.
38. J. MILTON AND P. JUNG, EDS., Epilepsy as a Dynamic Disease, Biol. Med. Phys. Ser., Springer, New York, 2003.
39. C. PESKIN, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences Publication, New York, 1975.
40. D. J. PINTO AND G. B. ERMENTROUT, Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses, SIAM J. Appl. Math., 62 (2001), pp. 206-225.
41. D. J. PINTO AND G. B. ERMENTROUT, Spatially structured activity in synaptically coupled neuronal networks: II. Lateral inhibition and standing pulses, SIAM J. Appl. Math., 62 (2001), pp. 226-243.
42. D. J. PINTO, R. K. JACKSON, AND C. E. WAYNE, Existence and stability of traveling pulses in a continuous neuronal network, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 954-984.
43. D. J. PINTO, S. A. PATRICK, H. W. HUANG, AND B. CONNORS, Initiation, propagation and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms, J. Neurosci., 25 (2005), pp. 8131-8140.
44. D. PINTO AND W. C. TROY, The Effects of Asymmetric Coupling on Traveling Waves in Neocortex, in preparation.
45. J. C. PRECHTL, L. B. COHEN, B. PASARAM, P. P. MITRA, AND D. KLEINFELD, Visual stimuli induce waves of electrical activity in turtle cortex, Proc. Natl. Acad. Sci. USA, 94 (1997), pp. 7621-7626.
46. K. RICHARDSON, S. J. SCHIFF, AND B. J. GLUCKMAN, Control of traveling waves in mammalian cortex, Phys. Rev. Lett., 94 (2005), paper 028103.
47. A. ROSENBLUTH AND W. B. CANNON, Cortical responses to electrical stimulation, A. J. Physiol., 135 (1942), pp. 690-741.
48. B. SANDSTEDE, Evans functions and nonlinear stability of travelling waves in neuronal network models, Internat. J. Bifur. Chaos Appl. Sci. Engrg., to appear.
49. B. SCHECHTER, How the brain gets its rhythm, Science, 274 (1996), p. 339-340.
50. J. SCHOFFLEN, R. OOSTENVELD, AND P. FRIES, Neuronal coherence as a mechanism of effective corticospinal interaction, Science, 308 (2003), pp. 111-113.
51. I. A. SHEVLEV, E. N. TSICALOV, A. M. GORBACH, K. P. BUDKO, AND G. A. SHARAEV, Temperature tomography of the brain cortex: Thermoencephalography, J. Neurosci. Methods, 46 (1992), pp. 49-57.
52. V, SHUSTERMAN, P. J. JANNETTA, B. AYSIN, A. BEIGEL, M. GLUKHOVSKOY, AND I. USIENE, Direct mechanical stimulation of brainstem modulates cardiac rhythm and repolarization in humans, J, Electrocardiol., 35 (2002), pp. 247-256.
53. S. STROGATZ, SYNC, Hyperion Books, New York, 2003.
54. H. R. WILSON AND J. D. COWAN, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 (1973), pp. 55-80.
55. J. Y. WU, L. GUAN, AND Y. TSAU, Propagating activation during oscillations and evoked responses in neocortical slices, J. Neurosci., 19 (1999), pp. 5005-5015.
56. L. ZHANG, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential Integral Equations, 16 (2003), pp. 513-536.