Journal of Mathematical Biology

, Volume 66, Issue 6, pp 1303–1338 | Cite as

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

  • Grégory FayeEmail author
  • James Rankin
  • Pascal Chossat


The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.


Localized state Neural field equation Reversible Hopf-bifurcation Normal form Orbital stability Numerical continuation 

Mathematics Subject Classification

37G05 34D20 37M05 92B20 


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  1. Amari S.-I. (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27(2): 77–87MathSciNetzbMATHCrossRefGoogle Scholar
  2. Brezis N (1983) Analyse fonctionnelle. Théorie et applications. MassonGoogle Scholar
  3. Burke J, Knobloch E (2006) Localized states in the generalized Swift–Hohenberg equation. Phys Rev E 73(5): 056211MathSciNetCrossRefGoogle Scholar
  4. Burke J, Knobloch E (2007) Homoclinic snaking: structure and stability. Chaos 17(3): 7102MathSciNetCrossRefGoogle Scholar
  5. Burke J, Knobloch E (2007) Normal form for spatial dynamics in the Swift–Hohenberg equation. Discret Continuous Dyn Syst Ser S, pp 170–180Google Scholar
  6. Champneys AR (1998) Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Phys D: Nonlinear Phenom 112(1–2): 158–186MathSciNetzbMATHCrossRefGoogle Scholar
  7. Chapman SJ, Kozyreff G (2009) Exponential asymptotics of localized patterns and snaking bifurcation diagrams. Phys D: Nonlinear Phenom 238: 319–354MathSciNetzbMATHCrossRefGoogle Scholar
  8. Chossat P, Faye G, Faugeras O (2011) Bifurcations of hyperbolic planforms. J Nonlinear Sci 21(4): 465–498. doi: 10.1007/s00332-010-9089-3 MathSciNetzbMATHCrossRefGoogle Scholar
  9. Chossat P, Faugeras O (2009) Hyperbolic planforms in relation to visual edges and textures perception. PLoS Comput Biol 5(12): e1000625MathSciNetCrossRefGoogle Scholar
  10. Chossat P, Lauterbach R (2000) Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific Publishing Company, River Edge, NJzbMATHCrossRefGoogle Scholar
  11. Coombes S, Lord GJ, Owen MR (2003) Waves and bumps in neuronal networks with axo-dendritic synaptic interactions. Phys D: Nonlinear Phen 178(3–4): 219–241MathSciNetzbMATHCrossRefGoogle Scholar
  12. Coombes S (2005) Waves, bumps, and patterns in neural fields theories. Biol Cybern 93(2): 91–108MathSciNetzbMATHCrossRefGoogle Scholar
  13. Doedel EJ, Champneys AR, Fairgrieve TF, Kuznetsov YA, Sandstede B, Wang X (1997) AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont)Google Scholar
  14. Elvin AJ, Laing CR, McLachlan RI, Roberts MG (2010) Exploiting the hamiltonian structure of a neural field model. Phys D: Nonlinear Phenom 239(9): 537–546MathSciNetzbMATHCrossRefGoogle Scholar
  15. Faugeras O, Grimbert F, Slotine J-J (2008) Abolute stability and complete synchronization in a class of neural fields models. SIAM J Appl Math 61(1): 205–250MathSciNetCrossRefGoogle Scholar
  16. Faye G, Chossat P (2011) Bifurcation diagrams and heteroclinic networks of octagonal h-planforms. J Nonlinear Sci (accepted for publication)Google Scholar
  17. Faye G, Chossat P, Faugeras O (2011) Analysis of a hyperbolic geometric model for visual texture perception. J Math Neurosci 1(4)Google Scholar
  18. Folias SE, Bressloff PC (2004) Breathing pulses in an excitatory neural network. SIAM J Appl Dyn Syst 3: 378–407MathSciNetzbMATHCrossRefGoogle Scholar
  19. Folias SE, Bressloff PC (2005) Breathers in two-dimensional excitable neural media. Phys Rev Lett 95: 208107CrossRefGoogle Scholar
  20. Guo Y, Chow CC (2005) Existence and stability of standing pulses in neural networks: Ii stability. SIAM J Appl Dyn Syst 4: 249–281MathSciNetzbMATHCrossRefGoogle Scholar
  21. Guo Y, Chow CC (2005) Existence and stability of standing pulses in neural networks: I. existence. SIAM J Appl Dyn Syst 4(2): 217–248MathSciNetzbMATHCrossRefGoogle Scholar
  22. Haragus M, Iooss G (2010) Local bifurcations, center manifolds, and normal forms in infinite dimensional systems. EDP Sci. Springer Verlag UTX seriesGoogle Scholar
  23. Iooss G, Peroueme MC (1993) Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. J Differ Equ 102(1): 62–88MathSciNetzbMATHCrossRefGoogle Scholar
  24. Jirsa V, Haken H (1996) Field theory of electromagnetic brain activity. Phys Rev Lett 77: 960–963CrossRefGoogle Scholar
  25. Kilpatrick ZP, Bressloff PC (2010) Stability of bumps in piecewise smooth neural fields with nonlinear adaptation. Phys D Nonlinear Phenom 239: 1048–1060MathSciNetzbMATHCrossRefGoogle Scholar
  26. Kozyreff G, Chapman SJ (2006) Asymptotics of large bound states of localized structures. Phys Rev Lett 97:044502, 1–4Google Scholar
  27. Laing CR, Troy WC (2003) PDE methods for nonlocal models. SIAM J Appl Dyn Syst 2(3): 487–516MathSciNetzbMATHCrossRefGoogle Scholar
  28. Laing CL, Troy WC, Gutkin B, Ermentrout GB (2002) Multiple bumps in a neuronal model of working memory. SIAM J Appl Math 63(1): 62–97MathSciNetzbMATHCrossRefGoogle Scholar
  29. Laing CR, Troy WC (2003) Two-bump solutions of Amari-type models of neuronal pattern formation. Phys D 178(3): 190–218MathSciNetzbMATHCrossRefGoogle Scholar
  30. Lloyd D, Sandstede B (2009) Localized radial solutions of the Swift–Hohenberg equation. Nonlinearity 22: 485MathSciNetzbMATHCrossRefGoogle Scholar
  31. McCalla S, Sandstede B (2010) Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: a numerical study. Phys D: Nonlinear Phenom 239(16): 1581–1592MathSciNetzbMATHCrossRefGoogle Scholar
  32. Melbourne I (1998) Derivation of the time-dependent Ginzburg–Landau equation on the line. J Nonlinear Sci 8: 1–15MathSciNetzbMATHCrossRefGoogle Scholar
  33. Owen MR, Laing CR, Coombes S (2007) Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities. New J Phys 9(10): 378–401CrossRefGoogle Scholar
  34. Pinto DJ, Ermentrout GB (2001) Spatially structured activity in synaptically coupled neuronal networks: 2. standing pulses. SIAM J Appl Math 62: 226–243MathSciNetzbMATHCrossRefGoogle Scholar
  35. Rubin JE, Troy WC (2001) Sustained spatial patterns of activity in neuronal populations without recurrent excitation. SIAM J Appl Math 64: 1609–1635MathSciNetCrossRefGoogle Scholar
  36. Veltz R, Faugeras O (2010) Illusions in the ring model of visual orientation selectivity. Technical report, arXiv (submitted to PLoS Comp Biol)Google Scholar
  37. Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol Cybern 13(2): 55–80zbMATHGoogle Scholar
  38. Woods PD, Champneys AR (1999) Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation. Phys D: Nonlinear Phenom 129: 147–170MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.NeuroMathComp LaboratoryINRIA, ENS ParisSophia-AntipolisFrance
  2. 2.J-A Dieudonné LaboratoryCNRS and University of Nice Sophia-AntipolisNice Cedex 02France

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