# 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 & Continuous Dynamical Systems - A, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
##### References:

show all references

##### References:
 [1] Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677 [2] Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057 [3] Xu Chen, Jianping Wan. Integro-differential equations for foreign currency option prices in exponential Lévy models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 529-537. doi: 10.3934/dcdsb.2007.8.529 [4] Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [5] Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160 [6] Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541 [7] Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191 [8] Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 [9] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [10] Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693 [11] Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119 [12] Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015 [13] Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053 [14] Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053 [15] Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249 [16] Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367 [17] Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026 [18] Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517 [19] Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069 [20] Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741

2018 Impact Factor: 1.143