Journal of Computational Neuroscience

, Volume 32, Issue 1, pp 147–165 | Cite as

Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms

  • Leiv ØyehaugEmail author
  • Ivar Østby
  • Catherine M. Lloyd
  • Stig W. Omholt
  • Gaute T. Einevoll


Exposed to a sufficiently high extracellular potassium concentration ([K + ]o), the neuron can fire spontaneous discharges or even become inactivated due to membrane depolarisation (‘depolarisation block’). Since these phenomena likely are related to the maintenance and propagation of seizure discharges, it is of considerable importance to understand the conditions under which excess [K + ]o causes them. To address the putative effect of glial buffering on neuronal activity under elevated [K + ]o conditions, we combined a recently developed dynamical model of glial membrane ion and water transport with a Hodgkin–Huxley type neuron model. In this interconnected glia-neuron model we investigated the effects of natural heterogeneity or pathological changes in glial membrane transporter density by considering a large set of models with different, yet empirically plausible, sets of model parameters. We observed both the high [K + ]o-induced duration of spontaneous neuronal firing and the prevalence of depolarisation block to increase when reducing the magnitudes of the glial transport mechanisms. Further, in some parameter regions an oscillatory bursting spiking pattern due to the dynamical coupling of neurons and glia was observed. Bifurcation analyses of the neuron model and of a simplified version of the neuron-glia model revealed further insights about the underlying mechanism behind these phenomena. The above insights emphasise the importance of combining neuron models with detailed astroglial models when addressing phenomena suspected to be influenced by the astroglia-neuron interaction. To facilitate the use of our neuron-glia model, a CellML version of it is made publicly available.


Potassium dynamics Positive feedback Spontaneous discharges Depolarisation block Glia 



We are grateful for the assistance of Pulasthi Mithraratne in the creation of the diagram in Fig. 1. We are also indebted to John Wyller for helpful discussions, and to Maxim Bazhenov, Giri Krishnan as well as two anonymous reviewers whose comments and suggestions helped improve the quality of the paper. The research has been partially supported by the Research Council of Norway through grants no 178143 and 178892.


  1. Bazhenov, M., Timofeev, I., Steriade, M., & Sejnowski, T. J. (2004). Potassium model for slow (2–3 hz) in vivo neocortical paroxysmal oscillations. Journal of Neurophysiology, 92(2), 1116–1132. doi: 10.1152/jn.00529.2003.PubMedCrossRefGoogle Scholar
  2. Binder, D. K., & Steinhauser, C. (2006). Functional changes in astroglial cells in epilepsy. Glia, 54(5), 358–368. doi: 10.1002/glia.20394.PubMedCrossRefGoogle Scholar
  3. Boussouf, A., Lambert, R. C., & Gaillard, S. (1997). Voltage-dependent Na + -HCO\(_3^-\) cotransporter and Na + /H +  exchanger are involved in intracellular pH regulation of cultured mature rat cerebellar oligodendrocytes. Glia, 19(1), 74–84.PubMedCrossRefGoogle Scholar
  4. Chen, K. C., & Nicholson, C. (2000). Spatial buffering of potassium ions in brain extracellular space. Biophysical Journal, 78(6), 2776–2797. doi: 10.1016/S0006-3495(00)76822-6.PubMedCrossRefGoogle Scholar
  5. Cressman, J. R., Ullah, G., Ziburkus, J., Schiff, S. J., & Barreto, E. (2009). The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics. Journal of Computational Neuroscience, 26(2), 159–170. doi: 10.1007/s10827-008-0132-4.PubMedCrossRefGoogle Scholar
  6. Dronne, M., Boissel, J., & Grenier, E. (2006). A mathematical model of ion movements in grey matter during a stroke. Journal of Theoretical Biology, 240(4), 599–615. doi: 10.1016/j.jtbi.2005.10.023.PubMedCrossRefGoogle Scholar
  7. Dronne, M., Grenier, E., Dumont, T., Hommel, M., & Boissel, J. (2007). Role of astrocytes in grey matter during stroke: A modelling approach. Brain Research, 1138, 231–242. doi: 10.1016/j.brainres.2006.12.062.PubMedCrossRefGoogle Scholar
  8. Ermentrout, B., & Terman, D. (2010). Foundations of mathematical neuroscience. Berlin: Springer.CrossRefGoogle Scholar
  9. Fertziger, A. P., & Ranck, J. B. (1970). Potassium accumulation in interstitial space during epileptiform seizures. Experimental Neurology, 26(3), 571–585.PubMedCrossRefGoogle Scholar
  10. Florence, G., Dahlem, M. A., Almeida, A.-C. G., Bassani, J. W. M., & Kurths, J. (2009). The role of extracellular potassium dynamics in the different stages of ictal bursting and spreading depression: A computational study. Journal of Theoretical Biology, 258(2), 219–228. doi: 10.1016/j.jtbi.2009.01.032.PubMedCrossRefGoogle Scholar
  11. Frankenhäuser, B., & Hodgkin, A. L. (1956). The after-effects of impulses in the giant nerve fibres of loligo. The Journal of Physiology, 131(2), 341–376.Google Scholar
  12. Fröhlich, F., & Bazhenov, M. (2006). Coexistence of tonic firing and bursting in cortical neurons. Physical Review E, 74(3), 031922. doi: 10.1103/PhysRevE.74.031922.CrossRefGoogle Scholar
  13. Fröhlich, F., Bazhenov, M., Iragui-Madoz, V., & Sejnowski, T. J. (2008). Potassium Dynamics in the epileptic cortex: New insights on an old topic. The Neuroscientist, 14(5), 422–433. doi: 10.1177/1073858408317955.PubMedCrossRefGoogle Scholar
  14. Fröhlich, F., Bazhenov, M., Timofeev, I., Steriade, M., & Sejnowski, T. J. (2006). Slow state transitions of sustained neural oscillations by activity-dependent modulation of intrinsic excitability. The Journal of Neuroscience, 26(23), 6153–6162. doi: 10.1523/JNEUROSCI.5509-05.2006.PubMedCrossRefGoogle Scholar
  15. Grisar, T., Guillaume, D., & Delgado-Escueta, A. V. (1992). Contribution of Na + ,K + -ATPase to focal epilepsy: A brief review. Epilepsy Research, 12(2), 141–149.PubMedCrossRefGoogle Scholar
  16. Hahn, P. J., & Durand, D. M. (2001). Bistability dynamics in simulations of neural activity in high-extracellular-potassium conditions. Journal of Computational Neuroscience, 11(1), 5–18.PubMedCrossRefGoogle Scholar
  17. Hunter, P., Robbins, P., & Noble, D. (2002). The IUPS human physiome project. Pflügers Archiv: European Journal of Physiology, 445(1), 1–9. doi: 10.1007/s00424-002-0890-1.PubMedCrossRefGoogle Scholar
  18. Jensen, M. S., & Yaari, Y. (1997). Role of intrinsic burst firing, potassium accumulation, and electrical coupling in the elevated potassium model of hippocampal epilepsy. Journal of Neurophysiology, 77(3), 1224–1233.PubMedGoogle Scholar
  19. Johnston, D., & Wu, S. M-S. (2001). Foundations of cellular neurophysiology. Cambridge: MIT Press.Google Scholar
  20. Kager, H., Wadman, W. J., & Somjen, G. G. (2000). Simulated seizures and spreading depression in a neuron model incorporating interstitial space and ion concentrations. Journal of Neurophysiology, 84(1), 495–512.PubMedGoogle Scholar
  21. Kager, H., Wadman, W. J., & Somjen, G. G. (2002). Conditions for the triggering of spreading depression studied with computer simulations. Journal of Neurophysiology, 88(5), 2700–2712. doi: 10.1152/jn.00237.2002.PubMedCrossRefGoogle Scholar
  22. Kager, H., Wadman, W. J., & Somjen, G. G. (2007). Seizure-like afterdischarges simulated in a model neuron. Journal of Computational Neuroscience, 22(2), 105–128. doi: 10.1007/s10827-006-0001-y.PubMedCrossRefGoogle Scholar
  23. Keynes, R. D. (1951). The ionic movements during nervous activity. The Journal of Physiology, 114(1–2), 119–150.PubMedGoogle Scholar
  24. Lauf, P. K., & Adragna, N. C. (2000). K-Cl cotransport: Properties and molecular mechanism. Cellular Physiology and Biochemistry, 10(5–6), 341–354.PubMedCrossRefGoogle Scholar
  25. Lebovitz, R. M. (1996). Quantitative examination of dynamic interneuronal coupling via single-spike extracellular potassium ion transients. Journal of Theoretical Biology, 180(1), 11–25. doi: 10.1006/jtbi.1996.0074.PubMedCrossRefGoogle Scholar
  26. Lloyd, C., Halstead, M., & Nielsen, P. (2004). CeIIML: Its future, present and past. Progress in Biophysics & Molecular Biology, 85(2–3), 433–450. doi: 10.1016/j.pbiomolbio.2004.01.004.CrossRefGoogle Scholar
  27. Lux, H. D., Heinemann, U., & Dietzel, I. (1986). Ionic changes and alterations in the size of the extracellular space during epileptic activity. Advances in Neurology, 44, 619–639.PubMedGoogle Scholar
  28. Nadkarni, S., Jung, P., & Levine, H. (2008). Astrocytes optimize the synaptic transmission of information. PLoS Computational Biology, 4(5), e1000088. doi: 10.1371/journal.pcbi.1000088.CrossRefGoogle Scholar
  29. Newman, E. A. (1991). Sodium-bicarbonate cotransport in retinal Müller (glial) cells of the salamander. The Journal of Neuroscience, 11(12), 3972–3983.PubMedGoogle Scholar
  30. Orkand, R. K., Nicholls, J. G., & Kuffler, S. W. (1966). Effect of nerve impulses on the membrane potential of glial cells in the central nervous system of amphibia. Journal of Neurophysiology, 29(4), 788–806.PubMedGoogle Scholar
  31. Østby, I., Øyehaug, L., Einevoll, G. T., Nagelhus, E. A., Plahte, E., Zeuthen, T., et al. (2009a). Astrocytic mechanisms explaining neural-activity-induced shrinkage of extraneuronal space. PLoS Computational Biology, 5(1), e1000272. doi: 10.1371/journal.pcbi.1000272.CrossRefGoogle Scholar
  32. Østby, I., Øyehaug, L., Einevoll, G. T., Ottersen, O. P., & Omholt, S. W. (2009b). Modeling of astrocytic mechanisms explaining neural activity-induced shrinkage of extracellular space and clearance of excess extracellular potassium. Meeting abstract, 2nd INCF Congress of Neuroinformatics, Pilsen, Czech Republic, 6–8 September 2009.Google Scholar
  33. Pangrsic, T., Potokar, M., Haydon, P. G., Zorec, R., & Kreft, M. (2006). Astrocyte swelling leads to membrane unfolding, not membrane insertion. Journal of Neurochemistry, 99(2), 514–523. doi: 10.1111/j.1471-4159.2006.04042.x.PubMedCrossRefGoogle Scholar
  34. Park, E., & Durand, D. M. (2006). Role of potassium lateral diffusion in non-synaptic epilepsy: A computational study. Journal of Theoretical Biology, 238(3), 666–682. doi: 10.1016/j.jtbi.2005.06.015.PubMedCrossRefGoogle Scholar
  35. Postnov, D. E., Ryazanova, L. S., Brazhe, N. A., Brazhe, A. R., Maximov, G. V., Mosekilde, E., et al. (2008). Giant glial cell: New insight through mechanism-based modeling. Journal of Biological Physics, 34(3–4), 441–457. doi: 10.1007/s10867-008-9070-7.PubMedCrossRefGoogle Scholar
  36. Rose, C. R., Kovalchuk, Y., Eilers, J., & Konnerth, A. (1999). Two-photon Na +  imaging in spines and fine dendrites of central neurons. Pflügers Archiv: European Journal of Physiology, 439(1–2), 201–207.PubMedCrossRefGoogle Scholar
  37. Somjen, G. G. (2004). Ions in the brain: Normal function, seizures, and stroke. Oxford: Oxford University Press.Google Scholar
  38. Somjen, G. G., Kager, H., & Wadman, W. J. (2008). Computer simulations of neuron-glia interactions mediated by ion flux. Journal of Computational Neuroscience, 25(2), 349–365. doi: 10.1007/s10827-008-0083-9.PubMedCrossRefGoogle Scholar
  39. Ullah, G., & Schiff, S. J. (2010). Assimilating seizure dynamics. PLoS Computational Biology, 6(5), e1000776. doi: 10.1371/journal.pcbi.1000776.CrossRefGoogle Scholar
  40. Volman, V., Ben-Jacob, E., & Levine, H. (2007). The astrocyte as a gatekeeper of synaptic information transfer. Neural Computation, 19(2), 303–326. doi: 10.1162/neco.2007.19.2.303.PubMedCrossRefGoogle Scholar
  41. Ziburkus, J., Cressman, J. R., Barreto, E., & Schiff, S. J. (2006). Interneuron and pyramidal cell interplay during in vitro seizure-like events. Journal of Neurophysiology, 95(6), 3948–3954. doi: 10.1152/jn.01378.2005.PubMedCrossRefGoogle Scholar
  42. Zuckermann, E. C., & Glaser, G. H. (1970). Activation of experimental epileptogenic foci: Action of increased K +  in extracellular spaces of the brain. Archives of Neurology, 23(4), 358–364.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Leiv Øyehaug
    • 1
    Email author
  • Ivar Østby
    • 1
  • Catherine M. Lloyd
    • 2
  • Stig W. Omholt
    • 3
  • Gaute T. Einevoll
    • 1
  1. 1.Centre for Integrative Genetics (CIGENE), Department of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway
  2. 2.Auckland Bioengineering InstituteThe University of AucklandAucklandNew Zealand
  3. 3.Centre for Integrative Genetics (CIGENE), Department of Animal and Aquacultural SciencesNorwegian University of Life SciencesÅsNorway

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