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Abstract
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Applied Mathematics

Kernel Reconstruction for Delayed Neural Field Equations

Profile image of Jehan AlswaihliJehan Alswaihli

2018, Journal of mathematical neuroscience

https://doi.org/10.1186/S13408-018-0058-8
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24 pages

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Abstract

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for t...

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References (41)

  1. Coombes S, beim Graben P, Potthast R, Wright J. Neural fields: theory and applications. Berlin: Springer; 2014.
  2. Wilson HR, Cowan JD. Excitatory and inhibitatory interactions in localized populations of model neurons. Biophys J. 1972;12:1-24.
  3. Wilson HR, Cowan JD. A mathematical theory of the functional dynamics of cortical and thelamic nervous tissue. Kybernetik. 1973;13:55-80.
  4. Amari S. Dynamics of patterns formation in lateral-inhibition type neural fields. Biol Cybern. 1977;27:77-87.
  5. Boeree CG. The neurons. General Psychology. 2015. p. 1-6.
  6. Bressloff PC, Coombes S. Physics of the extended neuron. Int J Mod Phys B. 1997;11:2343-92.
  7. Nogaret A, Meliza CD, Margoliash D, Abarbanel HDI. Automatic construction of predictive neuron models through large scale assimilation of electrophysiological data. Sci Rep. 2016;6:32749.
  8. Gils S, Janssens SG, Kuznetsov Y, Visser S. On local bifurcations in neural field models with trans- mission delays. J Math Biol. 2013;66:837-87.
  9. Venkov NA. Dynamics of neural field models [PhD thesis]. 2008.
  10. Faye G, Faugeras O. Some theoretical and numerical results for delayed neural field equations. Phys- ica D. 2010;239:561-78.
  11. Atay FM, Hutt A. Stability and bifurcations in neural fields with finite propagation speed and general connectivity. SIAM J Appl Math. 2005;65(2):644-66.
  12. Potthast R, beim Graben P. Existence and properties of solutions for neural field equations. Math Methods Appl Sci. 2010;33:935-49.
  13. Venkov NA, Coombes S, Matthews PC. Dynamic instabilities in scalar neural field equations with space-dependent delays. Physica D. 2007;232:1-15.
  14. Veltz R, Faugeras O. Stability of the stationary solutions of neural field equations with propagation delays. J Math Neurosci. 2011;1:1.
  15. Veltz R, Faugeras O. A center manifold result for delayed neural fields equations. SIAM J Math Anal. 2013;45(3):1527-62.
  16. beim
  17. Graben P, Potthast R. Inverse problems in dynamic cognitive modeling. Chaos, Interdiscip J Nonlinear Sci. 2009;19:015103.
  18. Freitag MA, Potthast RWE. Synergy of inverse problems and data assimilation techniques. In: Large scale inverse problems. Radon series on computational and applied mathematics. 2013. p. 1-54.
  19. Potthast R. Inverse problems and data assimilation for brain equations-state and current challenges. 2015.
  20. Potthast R. Inverse problems in neural population models. In: Encyclopedia of computational neuro- science. 2013.
  21. Potthast R, beim Graben P. Dimensional reduction for the inverse problem of neural field theory. Front Comput Neurosci. 2009;3:17.
  22. Nakamura G, Potthast R. Inverse modeling: an introduction to the theory and methods of inverse problems and data assimilation. Bristol: IOP Publishing; 2015.
  23. Hutt A. Generalization of the reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equa- tions and effects of finite propagation speeds. Phys Rev E. 2007;75:026214.
  24. Coombes S, Venkov N, Shiau L, Bojak I, Liley D, Laing C. Modeling elactrocortical activity through improved local approximations of integral neural field equations. Phys Rev E. 2007;76:051901.
  25. Dijkstra K, van Gils SA, Janssens SG. Pitchfork-Hopf bifurcations in 1D neural field models with transmission delays. Physica D. 2015;297:88-101.
  26. Engl HW, Hankle M, Neubauer A. Regularization of inverse problems. Mathematics and its applica- tions. Dordrecht: Springer; 2000.
  27. Groetsch CW. Inverse problems in the mathematical sciences. Theory and practice of applied geo- physics series. Wiesbaden: Vieweg; 1993.
  28. Kress R. Linear integral equations. Applied mathematical sciences. vol. 82. New York: Springer; 1999.
  29. Coombes S, beim Graben P, Potthast R. Tutorial on neural field theory. In: Neural fields: theory and applications. 2014.
  30. James MP, Coombes S, Bressloff PC. Effects of quasioctive membrane on multiply periodic travelling waves in integrate-and-fire systems. 2003.
  31. Laing CR, Coombes S. The importance of different timings of excitatory and inhibitory pathways in neural field models. 2005.
  32. Bojak I, Liley DT. Axonal velocity distributions in neural field equations. PLoS Comput Biol. 2010;6(1):e1000653.
  33. Coombes S, Schmidt H. Neural fields with sigmoidal firing rates: approximate solutions. Nottingham e Prints. 2010.
  34. Rankin J, Avitabil D, Baladron J, Faye G, Lloyd DJ. Continuation of localised coherent structures in nonlocal neural field equations. 2013. arXiv:1304.7206.
  35. Bressloff PC, Kilpatrick ZP. Two-dimensional bumps in piecewise smooth neural fields with synaptic depression. SIAM J Appl Math. 2011;71:379-408.
  36. Diekmann O. Delay equations: functional-, complex-, and nonlinear analysis. Berlin: Springer; 1995.
  37. Hutt A, Buhry L. Study of gabaergic extra-synaptic tonic inhibition in single neurons and neural pop- ulations by traversing neural scales: application to propofol-induced anaesthesia. J Comput Neurosci. 2014;37(3):417-37.
  38. Kirsch A. An introduction to the mathematical theory of inverse problems. Applied mathematical sciences. New York: Springer; 2011.
  39. Hutt A, Bestehorn M, Wennekers T. Pattern formation in intracortical neural fields. Netw Comput Neural Syst. 2003;14:351-68.
  40. Wennekers T. Orientation tuning properties of simple cells in area V1 derived from an approximate analysis of nonlinear neural field models. Neural Comput. 2001;13:1721-47.
  41. Potthast R, Graben P. Inverse problems in neural field theory. SIAM J Appl Dyn Syst. 2009;8(4):1405- 33.

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