EPJ Nonlinear Biomed Phys
Volume 2, Number 1, December 2014
Advances in Neural Population Models and Their Networks
Article Number 3
Number of page(s) 18
Published online 06 March 2014
  1. Hodgkin AL, Huxley AF: A quantitative description of membrane and its application to conduction and excitation in nerve.J Physiol 1952, 117:500–544.139241312991237 [Google Scholar]
  2. Rinzel J: Electrical excitability of cells, theory and experiment: review of the Hodgkin-Huxley foundation and an update.Bull Math Biol 1990, 52:3–23.10.1007/BF02459567 [Google Scholar]
  3. Carpenter G: Traveling wave solutions of nerve impulse equations.PhD thesis. University of Wisconsin, Madison; 1974 [Google Scholar]
  4. Evans J: Nerve axon equations: IV The stable and unstable impulse.Indiana University Math J 1975, 24:1169–1190.10.1512/iumj.1975.24.24096 [Google Scholar]
  5. Miller RN, Rinzel J: The dependence of impulse propagation speed on firing frequency, dispersion, for the Hodgkin-Huxley model.Biophys J 1981, 34:227–259.10.1016/S0006-3495(81)84847-313274697236850 [Google Scholar]
  6. Steriade M, Jones EG, Línas RR: Thalamic Oscillations and Signalling. New York: Wiley; 1990. [Google Scholar]
  7. Connors BW, Amitai Y: Generation of epileptiform discharges by local circuits in neocortex. In Epilepsy: Models, Mechanisms and Concepts. Edited by: Schwartzkroin PA. Cambridge: Cambridge University Press; 1993:388–424. [Google Scholar]
  8. Ermentrout GB, Kleinfeld D: Traveling electrical waves in cortex: Insights from phase dynamics and speculation on a computational role.Neuron 2001, 29:33–44.10.1016/S0896-6273(01)00178-711182079 [Google Scholar]
  9. Coombes S: Waves, bumps, and patterns in neural field theories.Biol Cybernetics 2005, 93:91–108.10.1007/s00422-005-0574-y [Google Scholar]
  10. Bressloff PC: Spatiotemporal dynamics of continuum neural fields.J Phys A 2012, 45:033001.10.1088/1751-8113/45/3/033001 [Google Scholar]
  11. Liley DTJ, Cadusch PJ, Dafilis MP: A spatially continuous mean field theory of electrocortical activity.Network: Comput Neural Syst 2002,13(1):67–113.10.1080/net. [Google Scholar]
  12. Breakspear M, Roberts JA, Terry JR, Rodrigues S, Mahant N, Robinson PA: A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis.Cerebral Cortex 2006, 16:1296–1313.16280462 [Google Scholar]
  13. Goodfellow M, Schindler K, Baier G: Self-organised transients in a neural mass model of epileptogenic tissue dynamics.NeuroImage 2011, 55:920–932.10.1016/j.neuroimage.2010.12.07421195779 [Google Scholar]
  14. Curtu R, Ermentrout B: Pattern formation in a network of excitatory and inhibitory cells with adaptation.SIAM J Appl Dynamical Syst 2004, 3:191–231.10.1137/030600503 [Google Scholar]
  15. Venkov NA, Coombes S, Matthews PC: Dynamic instabilities in scalar neural field equations with space-dependent delays.Physica D 2007, 232:1–15.10.1016/j.physd.2007.04.011 [Google Scholar]
  16. Coombes S: Large-scale neural dynamics: simple and complex.NeuroImage 2010, 52:731–739.10.1016/j.neuroimage.2010.01.04520096791 [Google Scholar]
  17. Nunez PL: The brain wave equation: a model for the EEG.Math Biosci 1974, 21:279–297.10.1016/0025-5564(74)90020-0 [Google Scholar]
  18. Jirsa V K Haken H: Field theory of electromagnetic brain activity.Phys Rev Lett 1996, 77:960–963.10.1103/PhysRevLett.77.96010062950 [Google Scholar]
  19. Benda J, Herz AVM: A universal model for spike-frequency adaptation.Neural Comput 2003, 15:2523–2564.10.1162/08997660332238506314577853 [Google Scholar]
  20. Coombes S, Laing CR: Delays in activity based neural networks.Philos Trans R Soc A 2009, 367:1117–1129.10.1098/rsta.2008.0256 [Google Scholar]
  21. Sandstede B: Evans functions and nonlinear stability of travelling waves in neuronal network models.Int J Bifurcation and Chaos 2007, 17:2693–2704.10.1142/S0218127407018695 [Google Scholar]
  22. Coombes S, Lord GJ, Owen MR: Waves and bumps in neuronal networks with axo-dendritic synaptic interactions.Physica D 2003, 178:219–241.10.1016/S0167-2789(03)00002-2 [Google Scholar]
  23. Coombes S, Owen MR: Evans functions for integral neural field equations with Heaviside firing rate function.SIAM J Appl Dynamical Syst 2004, 3:574–600.10.1137/040605953 [Google Scholar]
  24. Bressloff PC, Folias SE: Front bifurcations in an excitatory neural network.SIAM J Appl Dynamical Syst 2004, 65:131–151. [Google Scholar]
  25. Laing CR, Troy WC: PDE methods for nonlocal models.SIAM J Appl Dyn Syst 2003, 2:487–516.10.1137/030600040 [Google Scholar]
  26. Laing C: Spiral waves in nonlocal equations.SIAM J Appl Dynamical Syst 2005,4(3):588–606.10.1137/040612890 [Google Scholar]
  27. Shusterman V, Troy WC: From baseline to epileptiform activity: a path to synchronized rhythmicity in large-scale neural networks.Phys Rev E 2008, 77:061911. [Google Scholar]
  28. Steyn-Ross ML, Steyn-Ross DA, Sleigh JW: Interacting Turing-Hopf instabilities drive symmetry-breaking transitions in a mean-field model of the cortex: a mechanism for the slow oscillation.Phys Rev X 2013, 3:021005. [Google Scholar]
  29. Jirsa VK, Haken H: A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics.Physica D 1997, 99:503–526.10.1016/S0167-2789(96)00166-2 [Google Scholar]
  30. Dhooge A, Govaerts W, Kuznetsov YA, Meijer HGE, Sautois B: New features of the software MatCont for bifurcation analysis of dynamical systems.Math Comput Modell Dynamical Syst 2008,14(2):147–175.10.1080/13873950701742754 [Google Scholar]
  31. Meijer HGE, Coombes S: Travelling waves in a neural field model with refractoriness.J Math Biol 2014,68(5):1249–1268. online first, DOI:10.1007/s00285–013–0670-x10.1007/s00285-013-0670-x394861623546637 [Google Scholar]
  32. Rankin J, Avitabile D, Baladron J, Faye G, Lloyd DJ: Continuation of localised coherent structures in nonlocal neural field equations.SIAM J Sci Comput36–1(2014):B70-B93, arXiv:1304.7206. [Google Scholar]
  33. Kuznetsov YA: Impulses of a complicated form in models of nerve conduction.Selecta Mathematica (formerly Sovietica) 1994, 13:127–142. [Google Scholar]
  34. Champneys AR, Kirk V, Knobloch E, Oldeman BE, Sneyd J: When Shilnikov meets Hopf in excitable systems.SIAM J Appl Dynamical Syst 2007, 6:663–693.10.1137/070682654 [Google Scholar]
  35. Röder G, Bordyugov G, Engel H, Falcke M: Wave trains in an excitable FitzHugh-Nagumo model: bistable dispersion relation and formation of isolas.Phys Rev E 2007, 75:036202. [Google Scholar]
  36. Guckenheimer J, Kuehn C: Homoclinic orbits of the Fitz Hugh-Nagumo equation: bifurcations in the full system.SIAM J Appl Dynamical Syst 2010,9(1):138–153.10.1137/090758404 [Google Scholar]
  37. Keener J, Sneyd J: Mathematical Physiology. New York: Springer; 1998. [Google Scholar]
  38. Kuznetsov YA: Elements of Applied Bifurcation Theory, 3rd edition. New York: Springer; 2004. [Google Scholar]
  39. Homburg AJ, Sandstede B: Homoclinic and heteroclinic bifurcations in vector fields. In Handbook of Dynamical Systems. Volume III, Chap. 8. Edited by: Broer H, Takens F, Hasselblatt B. Amsterdam: Elevier; 2010:379–524. [Google Scholar]
  40. Marten F, Rodrigues S, Benjamin O, Richardson MP, Terry JR: Onset of poly-spike complexes in a mean-field model of human EEG and its application to absence epilepsy.Philos Trans R Soc A 2009, 367:1145–1161.10.1098/rsta.2008.0255 [Google Scholar]