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Traveling wave solutions to a neural field model with oscillatory synaptic coupling types

Department of Mathematics, Lehigh University, 14 East Packer Ave., Bethlehem, PA 18015, USA

In this paper, we investigate the existence, uniqueness, and spectral stability of traveling waves arising from a single threshold neural field model with one spatial dimension, a Heaviside firing rate function, axonal propagation delay, and biologically motivated oscillatory coupling types. Neuronal tracing studies show that long-ranged excitatory connections form stripe-like patterns throughout the mammalian cortex; thus, we aim to generalize the notions of pure excitation, lateral inhibition, and lateral excitation by allowing coupling types to spatially oscillate between excitation and inhibition. With fronts as our main focus, we exploit Heaviside firing rate functions in order to establish existence and utilize speed index functions with at most one critical point as a tool for showing uniqueness of wave speed. We are able to construct Evans functions, the so-called stability index functions, in order to provide positive spectral stability results. Finally, we show that by incorporating slow linear feedback, we can compute fast pulses numerically with phase space dynamics that are similar to their corresponding singular homoclinical orbits.
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Keywords integral differential equations; traveling wave solutions; existence; stability; Evans function

Citation: Alan Dyson. Traveling wave solutions to a neural field model with oscillatory synaptic coupling types. Mathematical Biosciences and Engineering, 2019, 16(2): 727-758. doi: 10.3934/mbe.2019035

References

  • 1. S. i. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybernet., 27 (1977), 77–87.
  • 2. D. Avitabile and H. Schmidt, Snakes and ladders in an inhomogeneous neural field model, Phys. D, 294 (2015), 24–36.
  • 3. A. Benucci, R. A. Frazor and M. Carandini, Standing waves and traveling waves distinguish two circuits in visual cortex, Neuron, 55 (2007), 103–117.
  • 4. F. Botelho, J. Jamison and A. Murdock, Single-pulse solutions for oscillatory coupling functions in neural networks, J. Dynam. Differential Equations, 20 (2008), 165–199.
  • 5. P. C. Bressloff, Traveling fronts and wave propagation failure in an inhomogeneous neural network, Phys. D, 155 (2001), 83–100.
  • 6. P. C. Bressloff, Waves in Neural Media, Springer Science & Business Media, New York, 2014. 18–19.
  • 7. H. J. Chisum, F. Mooser and D. Fitzpatrick, Emergent properties of layer 2/3 neurons reflect the collinear arrangement of horizontal connections in tree shrew visual cortex., J. Neurosci., 23 (2003), 2947–2960.
  • 8. B.W. Connors and Y. Amitai, Generation of epileptiform discharges by local circuits in neocortex, Epilepsy: Models, Mechanisms and Concepts, 388–424.
  • 9. S. Coombes, G. J. Lord and M. R. Owen, Waves and bumps in neuronal networks with axodendritic synaptic interactions, Phys. D, 178 (2003), 219–241.
  • 10. S. Coombes and M. R. Owen, Evans Functions for Integral Neural Field Equations with Heaviside Firing Rate Function, SIAM J. Appl. Dyn. Syst., 3 (2004), 574–600.
  • 11. S. Coombes and C. Laing, Pulsating fronts in periodically modulated neural field models, Phys. Rev. E, 83 (2011), 011912.
  • 12. S. Coombes and H. Schmidt, Neural fields with sigmoidal firing rates: Approximate solutions, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 1369–1379.
  • 13. R. J. Douglas and K. A. Martin, Neuronal circuits of the neocortex, Annu. Rev. Neurosci., 27 (2004), 419–451.
  • 14. A. J. Elvin, C. R. Laing, R. I. McLachlan and M. G. Roberts, Exploiting the Hamiltonian structure of a neural field model, Phys. D, 239 (2010), 537–546.
  • 15. B. Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Rep. Prog. Phys., 61 (1998), 353–430.
  • 16. G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123A (1993), 461–478.
  • 17. G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, vol. 35, Springer Science & Business Media, 2010.
  • 18. J. W. Evans, Nerve axon equations. I. Linear approximations, Indiana Univ. Math. J., 21 (1972), 877–885.
  • 19. J. W. Evans, Nerve axon equations. II. Stability at rest, Indiana Univ. Math. J., 22 (1972), 75–90.
  • 20. J. W. Evans, Nerve axon equations. III Stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972), 577–593.
  • 21. J. W. Evans, Nerve axon equations. IV The stable and the unstable impulse, Indiana Univ. Math. J., 24 (1975), 1169–1190.
  • 22. G. Faye, Existence and stability of traveling pulses in a neural field equation with synaptic depression, SIAM J. Appl. Dyn. Syst., 12 (2013), 2032–2067.
  • 23. G. Faye and A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Adv. Math., 270 (2015), 400–456.
  • 24. W. Gerstner, Time structure of the activity in neural network models, Phys. Rev. E, 51 (1995), 738–758.
  • 25. C. D. Gilbert and T. N. Wiesel, Columnar specificity of intrinsic horizontal and corticocortical connections in cat visual cortex, J. Neurosci., 9 (1989), 2432–2442.
  • 26. D. Golomb and Y. Amitai, Propagating neuronal discharges in neocortical slices: computational and experimental study, J. Neurophysiol., 78 (1997), 1199–1211.
  • 27. B. S. Gutkin, G. B. Ermentrout and J. O'Sullivan, Layer 3 patchy recurrent excitatory connections may determine the spatial organization of sustained activity in the primate prefrontal cortex, Neurocomputing, 32-33 (2000), 391–400.
  • 28. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500–544.
  • 29. E. M. Izhikevich and R. FitzHugh, Fitzhugh-nagumo model, Scholarpedia, 1 (2006), 1349.
  • 30. C. K. Jones, Stability of the travelling wave solution of the fitzhugh-nagumo system, Trans. Amer. Math. Soc., 286 (1984), 431–469.
  • 31. C. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), 64–88.
  • 32. Z. P. Kilpatrick, S. E. Folias and P. C. Bressloff, Traveling pulses and wave propagation failure in inhomogeneous neural media, SIAM J. Appl. Dyn. Syst., 7 (2008), 161–185.
  • 33. C. R. Laing, W. C. Troy, B. Gutkin and G. B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J. Appl. Math., 63 (2002), 62–97.
  • 34. C. R. Laing andW. C. Troy, PDE methods for nonlocal models, SIAM J. Appl. Dyn. Syst., 2 (2003), 487–516.
  • 35. C. R. Laing and W. C. Troy, Two-bump solutions of Amari-type models of neuronal pattern formation, Phys. D, 178 (2003), 190–218.
  • 36. J. W. Lance, Current concepts of migraine pathogenesis., Neurology, 43 (1993), S11–5.
  • 37. S.-H. Lee, R. Blake and D. J. Heeger, Traveling waves of activity in primary visual cortex during binocular rivalry, Nat. Neurosci., 8 (2005), 22–23.
  • 38. J. B. Levitt, D. A. Lewis, T. Yoshioka and J. S. Lund, Topography of pyramidal neuron intrinsic connections in macaque monkey prefrontal cortex (areas 9 and 46), J. Comp. Neurol., 338 (1993), 360–376.
  • 39. S. Lowel and W. Singer, Selection of intrinsic horizontal connections in the visual cortex by correlated neuronal activity, Science, 255 (1992), 209–212.
  • 40. J. S. Lund, T. Yoshioka and J. B. Levitt, Comparison of intrinsic connectivity in different areas of macaque monkey cerebral cortex, Cereb. Cortex, 3 (1993), 148–162.
  • 41. G. Lv and M.Wang, Traveling waves of some integral-differential equations arising from neuronal networks with oscillatory kernels, J. Math. Anal. Appl., 370 (2010), 82–100.
  • 42. F. M. G. Magpantay and X. Zou,Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections, Math. Biosci. Eng., 7 (2010), 421–442.
  • 43. K. A. C. Martin, S. Roth and E. S. Rusch, A biological blueprint for the axons of superficial layer pyramidal cells in cat primary visual cortex, Brain Struct. Funct., 0 (2017), 1–24.
  • 44. D. S. Melchitzky, S. R. Sesack, M. L. Pucak and D. A. Lewis, Synaptic targets of pyramidal neurons providing intrinsic horizontal connections in monkey prefrontal cortex, J. Comp. Neurol., 390 (1998), 211–224.
  • 45. B. Pakkenberg, D. Pelvig, L. Marner, M. J. Bundgaard, H. J. G. Gundersen, J. R. Nyengaard and L. Regeur, Aging and the human neocortex, Exp. Geront., 38 (2003), 95–99.
  • 46. D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: I. traveling fronts and pulses, SIAM J. Appl. Math., 62 (2001), 206–225.
  • 47. D. J. Pinto, R. K. Jackson and C. E. Wayne, Existence and stability of traveling pulses in a continuous neuronal network, SIAM J. Appl. Dyn. Syst., 4 (2005), 954–984.
  • 48. B. Sandstede, Evans functions and nonlinear stability of traveling waves in neuronal network models, Int. J. Bifurc. Chaos, 17 (2007), 2693–2704.
  • 49. T. K. Sato, I. Nauhaus and M. Carandini, Traveling waves in visual cortex, Neuron, 75 (2012), 218–229.
  • 50. H. Schmidt, A. Hutt and L. Schimansky-Geier,Wave fronts in inhomogeneous neural field models, Phys. D, 238 (2009), 1101–1112.
  • 51. R. Traub, J. Jefferys and R. Miles, Analysis of the propagation of disinhibition-induced afterdischarges along the guinea-pig hippocampal slice in vitro., J. Physiol., 472 (1993), 267–287.
  • 52. H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons., Biophys. J., 12 (1972), 1–24.
  • 53. L. Zhang, Existence and uniqueness of wave fronts in neuronal network with nonlocal postsynaptic axonal and delayed nonlocal feedback connections, Adv. Difference Equ., 2013 (2013), 243.
  • 54. L. Zhang, L. Zhang, J. Yuan and C. Khalique, Existence of wave front solutions of an integral differential equation in nonlinear nonlocal neuronal network, in Abstr. Appl. Anal., Hindawi Publishing Corporation, 2014.
  • 55. L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differ. Integral Equ., 16 (2003), 513–536.
  • 56. L. Zhang, How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks?, SIAM J. Appl. Dyn. Syst., 6 (2007), 597–644.
  • 57. L. Zhang and A. Hutt, Traveling wave solutions of nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks, J. Appl. Anal. Comp., 4 (2014), 1–68.

 

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