# American Institute of Mathematical Sciences

August  2012, 32(8): 2951-2970. doi: 10.3934/dcds.2012.32.2951

## 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

Received  April 2011 Revised  July 2011 Published  March 2012

Translationally invariant integro-differential equations are a common choice of model in neuroscience for describing the coarse-grained dynamics of cortical tissue. Here we analyse the propagation of travelling wavefronts in models of neural media that incorporate some form of modulation or randomness such that translational invariance is broken. We begin with a study of neural architectures in which there is a periodic modulation of the neuronal connections. Recent techniques from two-scale convergence analysis are used to construct a homogenized model in the limit that the spatial modulation is rapid compared with the scale of the long range connections. For the special case that the neuronal firing rate is a Heaviside we calculate the speed of a travelling homogenized front exactly and find how the wave speed changes as a function of the amplitude of the modulation. For this special case we further show how to obtain more accurate results about wave speed and the conditions for propagation failure by using an interface dynamics approach that circumvents the requirement of fast modulation. Next we turn our attention to forms of disorder that arise via the variation of firing rate properties across the tissue. To model this we draw parameters of the firing rate function from a distribution with prescribed spatial correlations and analyse the corresponding fluctuations in the wave speed. Finally we consider generalisations of the model to incorporate adaptation and stochastic forcing and show how recent numerical techniques developed for stochastic partial differential equations can be used to determine the wave speed by minimising the $L^2$ norm of a travelling disordered activity profile against a fixed test function.
Citation: Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
##### 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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##### 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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