Biological Cybernetics

, Volume 113, Issue 5–6, pp 561–577 | Cite as

Investigating the role of gap junctions in seizure wave propagation

  • Laura R. González-RamírezEmail author
  • Ava J. Mauro
Original Article


The effect of gap junctions as well as the biological mechanisms behind seizure wave propagation is not completely understood. In this work, we use a simple neural field model to study the possible influence of gap junctions specifically on cortical wave propagation that has been observed in vivo preceding seizure termination. We consider a voltage-based neural field model consisting of an excitatory and an inhibitory population as well as both chemical and gap junction-like synapses. We are able to approximate important properties of cortical wave propagation previously observed in vivo before seizure termination. This model adds support to existing evidence from models and clinical data suggesting a key role of gap junctions in seizure wave propagation. In particular, we found that in this model gap junction-like connectivity determines the propagation of one-bump or two-bump traveling wave solutions with features consistent with the clinical data. For sufficiently increased gap junction connectivity, wave solutions cease to exist. Moreover, gap junction connectivity needs to be sufficiently low or moderate to permit the existence of linearly stable solutions of interest.


Traveling waves Seizure termination Gap Junctions 

Mathematics Subject Classification

35C07 37N25 45K05 92C20 



  1. Amari S (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27:77–87 PubMedGoogle Scholar
  2. Braitenberg V, Schuz A (1998) Cortex: statistics and geometry of neuronal connectivity. Springer, BerlinGoogle Scholar
  3. Bressloff P (2012) Spatiotemporal dynamics of continuum neural fields. J Phys A Math Theor 45(3):033001Google Scholar
  4. Bressloff P (2016) Diffusion in cells with stochastically gated gap junctions. SIAM J Appl Math 76:1658–1682Google Scholar
  5. Bressloff P, Cowan J, Golubitsky M, Thomas P, Wiener M (2001) Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex. Philos Trans R Soc B 356:299–330Google Scholar
  6. Carlen P, Skinner F, Zhang L, Naus C, Kushnir ea M (2000) The role of gap junctions in seizures. Brain Res Rev 32:235–241PubMedGoogle Scholar
  7. Chen L, Meng M (1995) Compact and scattered gap junctions in diffusion mediated cell–cell communication. J Theor Biol 176:39–45PubMedGoogle Scholar
  8. Chow C, Kopell N (2000) Dynamics of spiking neurons with electrical coupling. Neural Comput 12:1643–1678PubMedGoogle Scholar
  9. Coombes S (2005) Waves, bumps, and patterns in neural field theories. Biol Cybern 93:91–108PubMedGoogle Scholar
  10. Coombes S (2008) Neuronal networks with gap junctions: a study of piecewise linear planar neuron models. SIAM J Appl Dyn Syst 7:1101–1129Google Scholar
  11. Coombes S, Zachariou M (2009) Gap junctions and emergent rhythms. In: Josic K, Rubin J, Matias M, Romo R (eds) Coherent behavior in neuronal networks. Springer series in computational neuroscience, vol 3. Springer, New York, NY Google Scholar
  12. Coombes S, Beim Graben P, Potthast R, Wright J (2014) Neural fields: theory and applications. Springer, BerlinGoogle Scholar
  13. Dudek F, Yasamura T, JE R (1998) Non-synaptic mechanisms in seizures and epileptogenesis. Cell Biol Int 22:793–805PubMedGoogle Scholar
  14. Elvin A (2008) Pattern formation in a neural field model. PhD thesis, Massey University, College of SciencesGoogle Scholar
  15. Ermentrout G (1998) Neural networks as spatio-temporal pattern-forming systems. Rep Prog Phys 61:353–430Google Scholar
  16. Ermentrout G (2006) Gap junctions destroy persistent states in excitatory networks. Phys Rev E Stat Nonlinear Soft Matter Phys 74:031918Google Scholar
  17. Ermentrout G, Cowan J (1979) A mathematical theory of visual hallucination patterns. Biol Cybern 34:137–150PubMedGoogle Scholar
  18. Ermentrout G, Terman D (2010) Mathematical foundations of neuroscience. Springer, BerlinGoogle Scholar
  19. Evangelista E, Benar C, Bonini F, Carron R, Colombet B, Regis J, Bartolomei F (2015) Does the thalamo-cortical synchrony play a role in seizure termination? Front Neurol 6:192PubMedPubMedCentralGoogle Scholar
  20. Evans L (2010) Partial differential equations, 2nd edn. American Mathematical Society, ProvidenceGoogle Scholar
  21. Evans W, Martin P (2002) Gap junctions: structure and function. Mol Membr Biol 19:121–136PubMedGoogle Scholar
  22. Foster B, Boja I, Liley D (2011) Understanding the effects of anesthetic agents on the EEG through neural field theory. In: Conference proceedings of IEEE engineering in medicine and biology society, vol 652Google Scholar
  23. Frascoli F, Van Veen L, Bojak I, Liley D (2011) Metabifurcation analysis of a mean field model of the cortex. Physica D 240:949–962Google Scholar
  24. Fujii Y, Maekawa S, Morita M (2017) Astrocyte calcium waves propagate proximally by gap junction and distally by extracellular diffusion of ATP released from volume-regulated anion channels. Sci Rep 7:13115PubMedPubMedCentralGoogle Scholar
  25. Fukuda T, Kosaka T, Singer W, Galuske R (2006) Gap junctions among dendrites of cortical gabaergic neurons establish a dense and widespread intercolumnar network. J Neurosci 26(13):3434–3443PubMedPubMedCentralGoogle Scholar
  26. González-Ramírez L, Kramer M (2018) The effect of inhibition on the existence of traveling wave solutions for a neural field model of human seizure termination. J Comput Neurosci 44(3):393–409PubMedGoogle Scholar
  27. González-Ramírez L, Ahmed O, Cash S, Wayne C, Kramer M (2015) A biologically constrained, mathematical model of cortical wave propagation preceding seizure termination. PLoS Comput Biol 11:e1004065PubMedPubMedCentralGoogle Scholar
  28. Goodenough D, Paul D (2009) Gap junctions. Cold Spring Harb Perspect Biol 1:a002576PubMedPubMedCentralGoogle Scholar
  29. Jin M, Chen Z (2011) Role of gap junctions in epilepsy. Neurosci Bull 27(6):389–406PubMedPubMedCentralGoogle Scholar
  30. Jirsa VK, Stacey WC, Quilichini PP, Ivanov I, Bernard C (2014) On the nature of seizure dynamics. Brain 137:2210–2230PubMedPubMedCentralGoogle Scholar
  31. Kapitula T, Kutz N, Sandstede B (2004) The evans function for nonlocal equations. Indiana Univ Math J 53:1095–1126Google Scholar
  32. Keener J, Sneyd J (1998) Mathematical physiology. Springer, New YorkGoogle Scholar
  33. Kepler T, Marder E, Abbott L (1990) The effect of electrical coupling on the frequency of model neuronal oscillators. Science 248:83–85PubMedGoogle Scholar
  34. Kopell N, Ermentrout B (2004) Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. Proc Natl Acad Sci USA 101:15482–15487PubMedGoogle Scholar
  35. Lacar B, Young S, Platel J, Bordey A (2011) Gap junction-mediated calcium waves define communication networks among murine postnatal neural progenitor cells. Eur J Neurosci 34(12):1895–1905PubMedPubMedCentralGoogle Scholar
  36. Laing C (2015) Exact neural fields incorporating gap junctions. SIAM J Appl Dyn Syst 14(4):1899–1929Google Scholar
  37. Lewis T, Rinzel J (2003) Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J Comput Neurosci 14:283–309PubMedGoogle Scholar
  38. Liley D, Cadusch P, Dafilis M (2002) A spatially continuous mean field theory of electrocortical activity. Network 13:67–113PubMedGoogle Scholar
  39. Martinet LE, Fiddyment G, Madsen JR, Eskandar EN, Truccolo W, Eden UT, Cash SS, Kramer MA (2017) Human seizures couple across spatial scales through travelling wave dynamics. Nat Commun 8:14896PubMedPubMedCentralGoogle Scholar
  40. Mylvaganam S, Ramani M, Krawczyk M, Carlen P (2014) Roles of gap junctions, connexins, and pannexins in epilepsy. Front Physiol 5:172PubMedPubMedCentralGoogle Scholar
  41. Perucca P, Dubeau F, Gotman J (2013) Intracranial electroencephalographic seizure-onset patterns: effect of underlying pathology. Brain 137:183–96PubMedGoogle Scholar
  42. Peyrache A, Dehghani N, Eskandar E et al (2012) Spatiotemporal dynamics of neocortical excitation and inhibition during human sleep. PNAS 109(5):1731–1736PubMedGoogle Scholar
  43. Proix T, Jirsa VK, Bartolomei F, Guye M, Wilson T (2018) Predicting the spatiotemporal diversity of seizure propagation and termination in human focal epilepsy. Nat Commun 9:1088PubMedPubMedCentralGoogle Scholar
  44. Reimann M, Anastassiou C, Perin R, Hill S, Markram H et al (2013) A biophysically detailed model of neocortical local field potentials predicts the critical role of active membrane currents. Neuron 79:375–390PubMedPubMedCentralGoogle Scholar
  45. Rose B, Loewenstein W (1976) Permeability of a cell junction and the local cytoplasmic free ionised calcium concentration; a study with aequorin. J Membr Biol 28:87–119PubMedGoogle Scholar
  46. Saez J, Connor J, Spray D, Bennett M (1989) Hepatocyte gap junctions are permeable to the second messenger, inositol 1,4,5-trisphosphate, and to calcium ions. Proc Natl Acad Sci USA 86:2708–2712PubMedGoogle Scholar
  47. Sandstede B (2002) Stability of travelling waves. Handb Dyn Syst 2:983–1055Google Scholar
  48. Sandstede B (2007) Evans function and nonlinear stability of traveling waves in neuronal network models. Int J Bifurc Chaos 17:2693–2704Google Scholar
  49. Schevon CA, Weiss SA, McKhann G, Goodman RR, Yuste R, Emerson RG, Trevelyan AJ (2012) Evidence of an inhibitory restraint of seizure activity in humans. Nat Commun 3:1060PubMedPubMedCentralGoogle Scholar
  50. Schmitz D, Schuchmann S, Fisahn A, Draguhn A, Buhl E et al (2001) Axo-axonal coupling: a novel mechanism for ultrafast neuronal communication. Neuron 31(5):831–840PubMedGoogle Scholar
  51. Sherman A, Rinzel J (1992) Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proc Nat Acad Sci USA 89:2471–2474PubMedGoogle Scholar
  52. Smith EH, Liou JY, Davis TS, Merricks EM, Kellis SS, Weiss SA, Greger B, House PA, McKhann GM, Goofman RR, Emerson RG, Bateman LM, Trevelyan AJ, Schevon CA (2016) The ictal wavefront is the spatiotemporal source of discharges during spontaneous human seizures. Nat Commun 7:11098PubMedPubMedCentralGoogle Scholar
  53. Stakgold I, Holst M (2011) Green’s functions and boundary value problems, 3rd edn. Wiley, New JerseyGoogle Scholar
  54. Steyn-Ross M, Steyn-Ross D, Wilson M, Sleigh JW (2007) Gap junctions mediate large-scale turing structures in a mean-field cortex driven by subcortical noise. Phys Rev E 76:011916Google Scholar
  55. Steyn-Ross M, Steyn-Ross D, Sleigh JW (2012) Gap junctions modulate seizures in a mean field-model of general anesthesia for the cortex. Cogn Neurodyn 6:215–225PubMedPubMedCentralGoogle Scholar
  56. Weiss SA, Banks GP, McKhann GMJ, Goodman RR, Emerson RG, Trevelyan AJ, Schevon C (2013) Ictal high frequency oscillations distinguish two types of seizure territories in humans. Brain 136:3796–808PubMedPubMedCentralGoogle Scholar
  57. Wilson H, Cowan J (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24PubMedPubMedCentralGoogle Scholar
  58. Zhang M, Ladas T, Qiu C, Shivacharan R, Gonzalez-Reyes L, Durand D (2014) Propagation of epileptiform activity can be independent of synaptic transmission, gap junctions, or diffusion and is consistent with electrical field transmission. J Neurosci 34(4):1409–1419PubMedPubMedCentralGoogle Scholar
  59. Zhao X, Robinson P (2015) Generalized seizures in a neural field model with bursting dynamics. J Comput Neurosci 39:197–216PubMedGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Formación Básica Disciplinaria, Instituto Politécnico NacionalUnidad Profesional Interdisciplinaria de Ingeniería Campus HidalgoSan Agustín TlaxiacaMexico
  2. 2.Department of Chemistry and BiochemistryUniversity of Notre DameNotre DameUSA

Personalised recommendations