Two-dimensional bumps in piecewise smooth neural fields with synaptic depression*

SIAM Journal on Applied Mathematics

Volumn 71, Issue 2, 2011, Pages 379-408

  Bressloff, Paul C ^a,b   Kilpatrick, Zachary P ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b UNIVERSITY OF UTAH   (United States)

Author keywords

Neural fields; Piecewise smooth dynamics; Synaptic depression

Indexed keywords

CONTINUUM DYNAMICS; HEAVISIDE FUNCTIONS; INTEGRAL KERNEL; LOCAL ACTIVITY; LOCAL STABILITY; MEAN FIRING RATE; NEURAL FIELD MODELS; NEURAL FIELDS; NONLOCAL INTEGRO-DIFFERENTIAL EQUATION; PIECEWISE SMOOTH; PIECEWISE-SMOOTH DYNAMICS; SYNAPTIC DEPRESSION; SYNAPTIC WEIGHT; WEIGHT DISTRIBUTIONS;

INTEGRODIFFERENTIAL EQUATIONS; NEURAL NETWORKS; TWO DIMENSIONAL;

DISEASES;

EID: 79957500220     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/100799423     Document Type: Article

References (45)

1. L. F. Abbott, J. A. Varela, K. Sen, and S. B. Nelson, Synaptic depression and cortical gain control, Science, 275 (1997), pp. 220-224. (Pubitemid 27034764)
2. S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybernet., 27 (1977), pp. 77-87. (Pubitemid 8146632)
3. P. C. Bressloff, Spontaneous symmetry breaking in self-organizing neural fields, Biol. Cybernet., 93 (2005), pp. 256-274. (Pubitemid 41500508)
4. P. C. Bressloff, Weakly interacting pulses in synaptically coupled excitable neural media, SIAM J. Appl. Math., 66 (2005), pp. 57-81.
5. S Coombes, Waves, bumps and patterns in neural field theories, Biol. Cybernet., 93 (2005), pp. 91-108. (Pubitemid 41205590)
6. R. D. Chervin, P. A. Pierce, and B. W. Connors, Periodicity and directionality in the propagation of epileptiform discharges across neocortex, J. Neurophysiol., 60 (1988), pp. 1695-1713.
7. 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. (Pubitemid 40578317)
8. S. Coombes and M. R. Owen, Bumps, breathers, and waves in a neural network with spike frequency adaptation, Phys. Rev. Lett., 94 (2005), 148102.
9. S. Coombes and M. R. Owen, Exotic dynamics in a firing rate model of neural tissue with threshold accommodation, in Fluids and Waves, Contemp. Math. 440, AMS, Providence, RI, 2007, pp. 123-144.
10. M. Di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. O. Tost, and P. T. Piiroinens, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50 (2008), pp. 629-701.
11. G. B. Ermentrout, Neural networks as spatio-temporal pattern forming systems, Rep. Prog. Phys., 61 (1998), pp. 353-430.
12. O. Faugeras, R. Veltz, and F. Grimbert, Persistent neural states: Stationary localized activity patterns in nonlinear continuous n-population, q-dimensional neural networks, Neural Comput., 21 (2009), pp. 147-187.
13. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), pp. 445-466.
14. S. E. Folias and P. C. Bressloff, Breathing pulses in an excitatory neural network, SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 378-407.
15. S. E. Folias and P. C. Bressloff, Breathers in two-dimensional neural media, Phys. Rev. Lett., 95 (2005), 208107.
16. 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. (Pubitemid 43385175)
17. J. P. Keener, Waves in excitable media, SIAM J. Appl. Math., 39 (1980), pp. 528-548.
18. Z. P. Kilpatrick, S. E. Folias, and P. C. Bressloff, Traveling pulses and wave propagation failure in an inhomogeneous neural media, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 161- 185.
19. Z. P. Kilpatrick and P. C. Bressloff, Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network, Phys. D, 239 (2010), pp. 547- 560.
20. Z. P. Kilpatrick and P. C. Bressloff, Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression, J. Comput. Neurosci., 28 (2010), pp. 193-209.
21. Z. P. Kilpatrick and P. C. Bressloff, Stability of bumps in piecewise smooth neural fields with nonlinear adaptation, Phys. D, 239 (2010), pp. 1048-1060.
22. Z. P. Kilpatrick and P. C. Bressloff, Binocular rivalry in a competitive neural network with synaptic depression, SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 1303-1347.
23. K. Kishimoto and S. Amari, Existence and stability of local excitations in homogeneous neural fields, J. Math. Biol., 7 (1979), pp. 303-318. (Pubitemid 9255725)
24. C. R. Laing, W. C. Troy, B. Gutkin, and G. B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J. Appl. Math., 63 (2002), pp. 62-97.
25. C. R. Laing and W. C. Troy, PDE methods for nonlocal models, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 487-516. (Pubitemid 38818581)
26. C. R. Laing, Spiral waves in nonlocal equations, SIAM J Appl. Dyn. Syst., 4 (2005), pp. 588-606. (Pubitemid 43385167)
27. J. S. Lund, A. Angelucci, and P. C. Bressloff, Anatomical substrates for functional columns in macaque monkey primary visual cortex, Cerebral Cortex, 12 (2003), pp. 15-24. (Pubitemid 35462888)
28. O. Melamed, O. Barak, G. Silberberg, H. Markram, and M. Tsodyks, Slow oscillations in neural networks with facilitating synapses, J. Comput. Neurosci., 25 (2008), pp. 308-316.
29. M. R. Owen, C. R. Laing, and S. Coombes, Bumps and rings in a two-dimensional neural field: Splitting and rotational instabilities, New J. Phys., 9 (2007), 378. Available online at http://iopscience.iop.org/1367-2630/9/ 10/378.
30. 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. (Pubitemid 33779905)
31. 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. (Pubitemid 33779906)
32. B. Sandstede, Evans functions and nonlinear stability of traveling waves in neuronal network models, Internat. J. Bifur. Chaos, in Appl. Sci. Engrg., 17 (2007), pp. 2693-2704. (Pubitemid 47479149)
33. V. Shusterman and W. C. Troy, From baseline to epileptiform activity: A path to synchronized rhythmicity in large-scale neural networks, Phys. Rev. E, 77 (2008), 061911.
34. J. Tabak, W. Senn, M. J. O'Donovan, and J. Rinzel, Modeling of spontaneous activity in developing spinal cord using activity-dependent depression in an excitatory network, J. Neurosci., 20 (2000), pp. 3041-3056. (Pubitemid 30220329)
35. W. C. Troy and V. Shusterman, Patterns and features of families of traveling waves in large-scale neuronal networks, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 263-292.
36. M. V. Tsodyks and H. Markram, The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability, Proc. Natl. Acad. Sci. USA, 94 (1997), pp. 719-723. (Pubitemid 27053700)
37. M. Tsodyks, K. Pawelzik, and H. Markram, Neural networks with dynamic synapses, Neural Comput., 10 (1998), pp. 821-835. (Pubitemid 128463179)
38. X.-J.Wang, Synaptic reverberation underlying mnemonic persistent activity, Trends Neurosci., 24 (2001), pp. 455-463. (Pubitemid 32695357)
39. H. Werner and T. Richter, Circular stationary solutions in two-dimensional neural fields, Biol. Cybern., 85 (2001), pp. 211-217. (Pubitemid 33605215)
40. 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.
41. 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. (Pubitemid 29270135)
42. J.-Y. Wu, Propagating waves of activity in the neocortex: What they are, what they do, Neuroscientist, 14 (2008), pp. 487-502.
43. L. Zhang, Traveling waves of singularly perturbed system of integral-differential equations arising from neuronal networks, J. Dynam. Differential Equations, 17 (2005), pp. 489-522. (Pubitemid 41261184)
44. Y. Zou, T. Kupper, and W. J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Sci., 16 (2006), pp. 159-177.
45. R. S. Zucker and W. G. Regehr, Short-term synaptic plasticity, Ann. Rev. Physiol., 64 (2002), pp. 355-405. (Pubitemid 34259234)