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Waves in random neural media
1. | School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD |
2. | Institute of Information and Mathematical Sciences, Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand |
3. | Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, S-412 96 Göteborg, Sweden |
4. | Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P. O. Box 5003, NO-1432 Ås, Norway |
References:
[1] |
S Coombes, Waves, bumps, and patterns in neural field theories, Biological Cybernetics, 93 (2005), 91-108.
doi: 10.1007/s00422-005-0574-y. |
[2] |
H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12 (1972), 1-24.
doi: 10.1016/S0006-3495(72)86068-5. |
[3] |
H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 (1973), 55-80.
doi: 10.1007/BF00288786. |
[4] |
S. Amari, Homogeneous nets of neuron-like elements, Biological Cybernetics, 17 (1975), 211-220.
doi: 10.1007/BF00339367. |
[5] |
S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biological Cybernetics, 27 (1977), 77-87.
doi: 10.1007/BF00337259. |
[6] |
G. B. Ermentrout and D. Kleinfeld, Traveling electrical waves in cortex: Insights from phase dynamics and speculation on a computational role, Neuron, 29 (2001), 33-44.
doi: 10.1016/S0896-6273(01)00178-7. |
[7] |
B. W. Connors and Y. Amitai, Generation of epileptiform discharges by local circuits in neocortex, in "Epilepsy: Models, Mechanisms and Concepts" (ed. P A Schwartzkroin), Cambridge University Press, (1993), 388-424.
doi: 10.1017/CBO9780511663314.016. |
[8] |
O. Faugeras, F. Grimbert and J.-J. Slotine, Absolute stability and complete synchronization in a class of neural fields models, SIAM Journal on Applied Mathematics, 69 (2008), 205-250.
doi: 10.1137/070694077. |
[9] |
P. C. Bressloff, Traveling fronts and wave propagation failure in an inhomogeneous neural network, Physica D, 155 (2001), 83-100.
doi: 10.1016/S0167-2789(01)00266-4. |
[10] |
H. Schmidt, A. Hutt and L. Schimansky-Geier, Wave fronts in inhomogeneous neural field models, Physica D, 238 (2009), 1101-1112.
doi: 10.1016/j.physd.2009.02.017. |
[11] |
S. Coombes and C. R. Laing, Pulsating fronts in periodically modulated neural field models, Physical Review E, 83 (2011), 011912.
doi: 10.1103/PhysRevE.83.011912. |
[12] |
C. A. Brackley and M. S. Turner, Persistent fluctuations of activity in undriven continuum neural field modles with power-law connections, Physical Review E, 79 (2009), 011918.
doi: 10.1103/PhysRevE.79.011918. |
[13] |
C. A. Brackley and M. S. Turner, Random fluctuations of the firing rate function in a continuum neural field model, Physical Review E, 75 (2007), 041913.
doi: 10.1103/PhysRevE.75.041913. |
[14] |
J. Keener, Homogenization and propagation in the bistable equation, Physica D, 136 (2000), 1-17.
doi: 10.1016/S0167-2789(99)00151-7. |
[15] |
G. Nguetseng, A general convergence result of a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.
doi: 10.1137/0520043. |
[16] |
M. R. Owen, C. R. Laing and S. Coombes, Bumps and rings in a two-dimensional neural field: Splitting and rotational instabilities, New Journal of Physics, 9 (2007), 378.
doi: 10.1088/1367-2630/9/10/378. |
[17] |
P. C. Bressloff and S. E. Folias, Front bifurcations in an excitatory neural network, SIAM Journal on Applied Mathematics, 65 (2004), 131-151.
doi: 10.1137/S0036139903434481. |
[18] |
D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses, SIAM Journal on Applied Mathematics, 62 (2001), 206-225.
doi: 10.1137/S0036139900346453. |
[19] |
C. W. Rowley, I. G. Kevrekidis, J. E. Marsden and K. Lust, Reduction and reconstruction for self-similar dynamical systems, Nonlinearity, 16 (2003), 1257-1275.
doi: 10.1088/0951-7715/16/4/304. |
[20] |
W.-J. Beyn and V. Thümmler, Freezing solutions of equivariant evolution equations, SIAM Journal on Applied Dynamical Systems, 3 (2004), 85-116.
doi: 10.1137/030600515. |
[21] |
G. J. Lord and V. Thümmler, Freezing stochastic travelling waves, arXiv:1006.0428, 2010. |
[22] |
S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM Journal on Applied Dynamical Systems, 34 (2004), 574-600.
doi: 10.1137/040605953. |
[23] |
L. Fronzoni, R. Mannella, P. V. E. McClintock and F. Moss, Postponement of Hopf bifurcations by multiplicative colored noise, Physical Review A, 36 (1987), 834.
doi: 10.1103/PhysRevA.36.834. |
[24] |
C. R. Laing and A. Longtin, Noise-induced stabilization of bumps in systems with long-range spatial coupling, Physica D, 160 (2001), 149-172.
doi: 10.1016/S0167-2789(01)00351-7. |
[25] |
C. R. Laing, T. A. Frewen and I. G. Kevrekidis, Coarse-grained dynamics of an activity bump in a neural field model, Nonlinearity, 20 (2007), 2127-2146.
doi: 10.1088/0951-7715/20/9/007. |
[26] |
B. Ermentrout and D. Saunders, Phase resetting and coupling of noisy neural oscillators, Journal of Computational Neuroscience, 20 (2006), 179-190.
doi: 10.1007/s10827-005-5427-0. |
[27] |
J. Xin, "An Introduction to Fronts in Random Media," Surveys and Tutorials in the Applied Mathematical Sciences, 5, Springer, New York, 2009. |
[28] |
C. R. Laing, Spiral waves in nonlocal equations, SIAM Journal on Applied Dynamical Systems, 4 (2005), 588-606.
doi: 10.1137/040612890. |
[29] |
S. Hermann and G. A. Gottwald, The large core limit of spiral waves in excitable media: A numerical approach, SIAM Journal on Applied Dynamical Systems, 9 (2010), 536-567.
doi: 10.1137/090780055. |
[30] |
N. Svanstedt and J. Wyller, A one population Wilson-Cowan model with periodic microstructure, in preparation, 2011. |
[31] |
R. Potthast and P. B. Graben, Existence and properties of solutions for neural field equations, Mathematical Methods in the Applied Sciences, 33 (2010), 935-949. |
[32] |
G. Faye and O. Faugeras, Some theoretical and numerical results for delayed neural field equations, Physica D, 239 (2010), 561-578.
doi: 10.1016/j.physd.2010.01.010. |
[33] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. |
[34] |
L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy estimates, Journal de Mathématiques Pures et Appliquées, 92 (2009), 163-187. |
[35] |
A. F. Pazoto and J. D. Rossi, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Analysis, 52 (2007), 143-155. |
[36] |
A. Holmbom, J. Silfver, N. Svanstedt and N. Wellander, On two-scale convergence and related sequential compactness topics, Applications of Mathematics, 51 (2006), 247-262.
doi: 10.1007/s10492-006-0014-x. |
[37] |
A. Visintin, Towards a two-scale calculus, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 371-397.
doi: 10.1051/cocv:2006012. |
[38] |
A. Visintin, Two-scale convergence of some integral functionals, Calculus of Variations and Partial Differential Equations, 29 (2007), 239-265. |
show all references
References:
[1] |
S Coombes, Waves, bumps, and patterns in neural field theories, Biological Cybernetics, 93 (2005), 91-108.
doi: 10.1007/s00422-005-0574-y. |
[2] |
H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12 (1972), 1-24.
doi: 10.1016/S0006-3495(72)86068-5. |
[3] |
H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 (1973), 55-80.
doi: 10.1007/BF00288786. |
[4] |
S. Amari, Homogeneous nets of neuron-like elements, Biological Cybernetics, 17 (1975), 211-220.
doi: 10.1007/BF00339367. |
[5] |
S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biological Cybernetics, 27 (1977), 77-87.
doi: 10.1007/BF00337259. |
[6] |
G. B. Ermentrout and D. Kleinfeld, Traveling electrical waves in cortex: Insights from phase dynamics and speculation on a computational role, Neuron, 29 (2001), 33-44.
doi: 10.1016/S0896-6273(01)00178-7. |
[7] |
B. W. Connors and Y. Amitai, Generation of epileptiform discharges by local circuits in neocortex, in "Epilepsy: Models, Mechanisms and Concepts" (ed. P A Schwartzkroin), Cambridge University Press, (1993), 388-424.
doi: 10.1017/CBO9780511663314.016. |
[8] |
O. Faugeras, F. Grimbert and J.-J. Slotine, Absolute stability and complete synchronization in a class of neural fields models, SIAM Journal on Applied Mathematics, 69 (2008), 205-250.
doi: 10.1137/070694077. |
[9] |
P. C. Bressloff, Traveling fronts and wave propagation failure in an inhomogeneous neural network, Physica D, 155 (2001), 83-100.
doi: 10.1016/S0167-2789(01)00266-4. |
[10] |
H. Schmidt, A. Hutt and L. Schimansky-Geier, Wave fronts in inhomogeneous neural field models, Physica D, 238 (2009), 1101-1112.
doi: 10.1016/j.physd.2009.02.017. |
[11] |
S. Coombes and C. R. Laing, Pulsating fronts in periodically modulated neural field models, Physical Review E, 83 (2011), 011912.
doi: 10.1103/PhysRevE.83.011912. |
[12] |
C. A. Brackley and M. S. Turner, Persistent fluctuations of activity in undriven continuum neural field modles with power-law connections, Physical Review E, 79 (2009), 011918.
doi: 10.1103/PhysRevE.79.011918. |
[13] |
C. A. Brackley and M. S. Turner, Random fluctuations of the firing rate function in a continuum neural field model, Physical Review E, 75 (2007), 041913.
doi: 10.1103/PhysRevE.75.041913. |
[14] |
J. Keener, Homogenization and propagation in the bistable equation, Physica D, 136 (2000), 1-17.
doi: 10.1016/S0167-2789(99)00151-7. |
[15] |
G. Nguetseng, A general convergence result of a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.
doi: 10.1137/0520043. |
[16] |
M. R. Owen, C. R. Laing and S. Coombes, Bumps and rings in a two-dimensional neural field: Splitting and rotational instabilities, New Journal of Physics, 9 (2007), 378.
doi: 10.1088/1367-2630/9/10/378. |
[17] |
P. C. Bressloff and S. E. Folias, Front bifurcations in an excitatory neural network, SIAM Journal on Applied Mathematics, 65 (2004), 131-151.
doi: 10.1137/S0036139903434481. |
[18] |
D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses, SIAM Journal on Applied Mathematics, 62 (2001), 206-225.
doi: 10.1137/S0036139900346453. |
[19] |
C. W. Rowley, I. G. Kevrekidis, J. E. Marsden and K. Lust, Reduction and reconstruction for self-similar dynamical systems, Nonlinearity, 16 (2003), 1257-1275.
doi: 10.1088/0951-7715/16/4/304. |
[20] |
W.-J. Beyn and V. Thümmler, Freezing solutions of equivariant evolution equations, SIAM Journal on Applied Dynamical Systems, 3 (2004), 85-116.
doi: 10.1137/030600515. |
[21] |
G. J. Lord and V. Thümmler, Freezing stochastic travelling waves, arXiv:1006.0428, 2010. |
[22] |
S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM Journal on Applied Dynamical Systems, 34 (2004), 574-600.
doi: 10.1137/040605953. |
[23] |
L. Fronzoni, R. Mannella, P. V. E. McClintock and F. Moss, Postponement of Hopf bifurcations by multiplicative colored noise, Physical Review A, 36 (1987), 834.
doi: 10.1103/PhysRevA.36.834. |
[24] |
C. R. Laing and A. Longtin, Noise-induced stabilization of bumps in systems with long-range spatial coupling, Physica D, 160 (2001), 149-172.
doi: 10.1016/S0167-2789(01)00351-7. |
[25] |
C. R. Laing, T. A. Frewen and I. G. Kevrekidis, Coarse-grained dynamics of an activity bump in a neural field model, Nonlinearity, 20 (2007), 2127-2146.
doi: 10.1088/0951-7715/20/9/007. |
[26] |
B. Ermentrout and D. Saunders, Phase resetting and coupling of noisy neural oscillators, Journal of Computational Neuroscience, 20 (2006), 179-190.
doi: 10.1007/s10827-005-5427-0. |
[27] |
J. Xin, "An Introduction to Fronts in Random Media," Surveys and Tutorials in the Applied Mathematical Sciences, 5, Springer, New York, 2009. |
[28] |
C. R. Laing, Spiral waves in nonlocal equations, SIAM Journal on Applied Dynamical Systems, 4 (2005), 588-606.
doi: 10.1137/040612890. |
[29] |
S. Hermann and G. A. Gottwald, The large core limit of spiral waves in excitable media: A numerical approach, SIAM Journal on Applied Dynamical Systems, 9 (2010), 536-567.
doi: 10.1137/090780055. |
[30] |
N. Svanstedt and J. Wyller, A one population Wilson-Cowan model with periodic microstructure, in preparation, 2011. |
[31] |
R. Potthast and P. B. Graben, Existence and properties of solutions for neural field equations, Mathematical Methods in the Applied Sciences, 33 (2010), 935-949. |
[32] |
G. Faye and O. Faugeras, Some theoretical and numerical results for delayed neural field equations, Physica D, 239 (2010), 561-578.
doi: 10.1016/j.physd.2010.01.010. |
[33] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. |
[34] |
L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy estimates, Journal de Mathématiques Pures et Appliquées, 92 (2009), 163-187. |
[35] |
A. F. Pazoto and J. D. Rossi, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Analysis, 52 (2007), 143-155. |
[36] |
A. Holmbom, J. Silfver, N. Svanstedt and N. Wellander, On two-scale convergence and related sequential compactness topics, Applications of Mathematics, 51 (2006), 247-262.
doi: 10.1007/s10492-006-0014-x. |
[37] |
A. Visintin, Towards a two-scale calculus, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 371-397.
doi: 10.1051/cocv:2006012. |
[38] |
A. Visintin, Two-scale convergence of some integral functionals, Calculus of Variations and Partial Differential Equations, 29 (2007), 239-265. |
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