From invasion to extinction in heterogeneous neural fields

Journal of Mathematical Neuroscience

Volumn 2, Issue 1, 2012, Pages 1-21

  Bressloff, Paul C ^a  

^a UNIVERSITY OF UTAH   (United States)

Author keywords

Hamilton jacobi equation; Heterogeneous media; Invasive pulled fronts; Large fluctuations; Multiplicative noise; Neural fields; Path integrals; Population extinction

Indexed keywords

ARTICLE; CELL HETEROGENEITY; ENVIRONMENTAL HEALTH; HAMILTON SCALE; MOTION; NERVOUS SYSTEM; NOISE; PRIORITY JOURNAL; REINFORCEMENT; SENSITIVITY ANALYSIS; STOCHASTIC MODEL; SYNAPSE; VELOCITY; VISUAL FIELD;

EID: 84859482937     PISSN: None     EISSN: 21908567     Source Type: Journal    
DOI: 10.1186/2190-8567-2-6     Document Type: Article

References (61)

1. Fisher RA: The wave of advance of advantageous genes. Ann Eugenics 1937, 7:353-369.
2. Kolmogorff A, Petrovsky I, Piscounoff N: Etude de l'équation de la diffusion avec croissance de la quantité de matiére et son application àun probléme biologique. Moscow University Bull Math 1937, 1:1-25.
3. Murray JD: Mathematical Biology. Berlin: Springer; 1989.
4. Holmes EE, Lewis MA, Banks JE, Veit RR: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 1994, 75:17-29.
5. Shigesada N, Kawasaki K: Biological Invasions: Theory and Practice. Oxford: Oxford University Press; 1997.
6. Cantrell RS, Cosner C: Spatial Ecology via Reaction-Diffusion Equations. Chichester: Wiley; 2003.
7. Volpert V, Petrovskii S: Reaction-diffusion waves in biology. Physics of Life Reviews 2009, 6:267-310.
8. van Saarloos W: Front propagation into unstable states. Phys Rep 2003, 386:29-222.
9. Brunet E, Derrida B: Shift in the velocity of a front due to a cutoff. Phys Rev E 1997, 56:2597-2604.
10. Mendez V, Fort J, Rotstein HG, Fedotov S: Speed of reaction-diffusion fronts in spatially heterogeneous media. Phys Rev E 2003, 68:041105.
11. Rocco A, Ebert U, van Saarloos W: Subdiffusive fluctuations of "pulled" fronts with multiplicative noise. Phys Rev E 2000, 65:R13.
12. Shigesada N, Kawasaki K, Teramoto E: Traveling periodic waves in heterogeneous environments. Theor Popul Biol 30:143-160.
13. Xin J: Front propagation in heterogeneous media. SIAM Rev 2000, 42:161-230.
14. Weinberger HF: On spreading speeds and traveling waves for growth and migration in a periodic habitat. J Math Biol 2002, 45:511-548.
15. Berestycki H, Hamel F, Roques L: Analysis of the periodically fragmented environment model: II-biological invasions and pulsating travelling fronts. J Math Biol 2005, 51:75-113.
16. Gartner J, Freidlin MI: On the propagation of concentration waves in periodic and random media. Soviet Math Dokl 1979, 20:1282-2128.
17. Freidlin MI: Limit theorems for large deviations and reaction-diffusion equations. Ann Probab 1985, 13:639-675.
18. Freidlin MI: Geometric optics approach to reaction-diffusion equations. SIAM J Appl Math 1986, 46:222-232.
19. Evans LC, Sougandis PE: A PDE approach to geometric optics for certain semilinear parabolic equations. Ind Univ Math J 1989, 38:141-172.
20. Ovaskainen O, Meerson B: Stochastic models of population extinction. Trends Ecol and Evol 2010, 25:643-652.
21. Elgart V, Kamanev A: Rare event statistics in reaction-diffusion systems. Phys Rev E 2004, 70:041106.
22. Kamanev A, Meerson B: Extinction of an infectious disease: a large fluctuation in a nonequilibrium system. Phys Rev E 2008, 77:061107.
23. Assaf M, Kamanev A, Meerson B: Population extinction risk in the aftermath of a catastrophic event. Phys Rev E, 2009 79:011127.
24. Meerson B, Sasorov PV: Extinction rates of established spatial populations. Phys Rev E 2011, 83: 011129.
25. Freidlin MI, Wentzell AD: Random Perturbations of Dynamical Systems. Springer, New York; 1984.
26. Dykman MI, Mori E, Ross J, Hunt PM: Large fluctuations and optimal paths in chemical kinetics. J Chem Phys A 1994, 100:5735-5750.
27. Stein RS, Stein DL: Limiting exit location distribution in the stochastic exit problem. SIAM J Appl Math 1997, 57:752-790.
28. Ermentrout GB: Neural networks as spatio-temporal pattern-forming systems. Rep Prog Phy 1998, 61:353-430.
29. Coombes S: Waves, bumps and patterns in neural field theories. Biol Cybern 2005, 93:91-108.
30. Bressloff PC: Spatiotemporal dynamics of continuum neural fields. Topical review. J Phys A: Math Theor 2012, 45:033001.
31. Pinto D, Ermentrout GB: Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses. SIAM J Appl Math 2001, 62:206-225.
32. Richardson KA, Schiff SJ, Gluckman BJ: Control of traveling waves in the mammalian cortex. Phys Rev Lett 2005, 94:028103.
33. Huang X, Troy WC, Yang Q, Ma H, Laing CR, Schiff SJ, Wu J: Spiral waves in disinhibited mammalian neocortex. J Neurosci 2004, 24:9897-9902.
34. Pinto D, Patrick SL, Huang WC, Connors BW: Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms. J Neurosci 2005, 25:8131-8140.
35. Wilson HR, Blake R, Lee SH: Dynamics of traveling waves in visual perception. Nature 2001, 412:907-910.
36. Lee SH, Blake R, Heeger DJ: Traveling waves of activity in primary visual cortex during binocular rivalry. Nat Neurosci 2005, 8:22-23.
37. Kang M, Heeger DJ, Blake R: Periodic perturbations producing phase-locked fluctuations in visual perception. J Vision 2009, 9:1-12.
38. Bressloff PC, Webber MA: Neural field model of binocular rivalry waves. J Comput Neurosci, in press.
39. Amari S: Dynamics of pattern formation in lateral inhibition type neural fields. Biol Cybern 1977, 27:77-87.
40. Ermentrout GB, McLeod JB: Existence and uniqueness of travelling waves for a neural network. Proc Roy Soc Edin A 1993, 123:461-478.
41. Bressloff PC: Traveling fronts and wave propagation failure in an inhomogeneous neural network. Physica D, 2001 155:83-100.
42. Coombes S, Laing CR: Pulsating fronts in periodically modulated neural field models. Phys Rev E 2011, 83:011912.
43. Coombes S, Laing CR, Schmidt H, Svanstedt N, Wyller JA: Waves in random neural media. Disc Cont Dyn Syst A, in press.
44. Folias SE, Bressloff PC: Stimulus-locked traveling pulses and breathers in an excitatory neural network. SIAM J Appl Math 2005, 65:2067-2092.
45. Ermentrout GB, Jalics JZ, Rubin JE: Stimulus-driven traveling solutions in continuum neuronal models with a general smooth firing rate functions. SIAM J Appl Math 2010, 70:3039-3054.
46. Buice M, Cowan JD: Field-theoretic approach to fluctuation effects in neural networks. Phys Rev E 2007, 75:051919.
47. Beggs JM, Plenz D: Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures. J Neurosci 2004, 24:5216.
48. Plenz D, Thiagarajan TC: The organizing principles of neuronal avalanches: cell assemblies in the cortex? Trends Neurosci 2007, 30:101.
49. Bressloff PC, Webber MA: Front propagation in stochastic neural fields. SIAM J Appl Dyn Syst, in press.
50. Gourley SA: Travelling front solutions of a nonlocal Fisher equation. J Math Biol 2000, 41:272-284.
51. Kilpatrick ZP, Folias SE, Bressloff PC: Traveling pulses and wave propagation failure in an inhomogeneous neural network. SIAM J Appl Dyn Syst 2008, 7:161-185.
52. Xu W, Huang X, Takagaki K, Wu J-Y: Compression and reflection of visually evoked cortical waves. Neuron 2007, 55:119-129.
53. Ebert U, van Saarloos W: Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Physica D 2000, 146:1-99.
54. Buice M, Cowan JD, Chow CC: Systematic fluctuation expansion for neural network activity equations. Neural Comp 2010, 22:377-426.
55. Bressloff PC: Stochastic neural field theory and the system-size expansion. SIAM J Appl Math 2009, 70:1488-1521.
56. Bressloff PC: Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics. Phys Rev E 2010, 85:051903.
57. Gardiner CW: Stochastic Methods: A Handbook for the Natural and Social Sciences (Fourth Edition). Berlin: Springer; 2009.
58. Justin JZ: Quantum Field Theory and Critical Phenomena (Fourth Edition. Oxford: Oxford University Press; 2002.
59. Tauber UC: Field-theory approaches to nonequilibrium dynamics. Lecture Notes in Physics 2007, 716:295-348.
60. Chow CC, Buice M: Path integral methods for stochastic differential equations. 2011 arXiv:1009.5966.
61. Bedard C, Destexhe A: Macroscopic models of local field potentials the apparent 1/f noise in brain activity. Biophys J 2009, 96:2589-2603.