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Journal of Computational Neuroscience

, Volume 44, Issue 2, pp 203–217 | Cite as

A mathematical model of recurrent spreading depolarizations

  • Cameron Conte
  • Ray Lee
  • Monica Sarkar
  • David TermanEmail author
Article

Abstract

A detailed biophysical model for a neuron/astrocyte network is developed in order to explore mechanisms responsible for the initiation and propagation of recurrent cortical spreading depolarizations. The model incorporates biophysical processes not considered in the earlier models. This includes a model for the Na+-glutamate transporter, which allows for a detailed description of reverse glutamate uptake. In particular, we consider the specific roles of elevated extracellular glutamate and K+ in the initiation, propagation and recurrence of spreading depolarizations.

Keywords

Spreading depolarization Neuron/astrocyte network Mathematical model 

Notes

Acknowledgments

This work was partially supported by the US National Science Foundation DMS Award 1410935. The first three authors all contributed equally to this work.

Compliance with ethical standards

Conflict of interests

The authors declare that they have no conflict of interest.

References

  1. Allaman, I., Bélanger, M, Magistretti, P.J. (2011). Astrocyte–neuron metabolic relationships: for better and for worse. Trends in Neurosciences, 34, 76–87.CrossRefPubMedGoogle Scholar
  2. Barres, B.A. (2008). The Mystery and Magic of Glia: A Perspective on Their Roles in Health and Disease.  https://doi.org/10.1016/j.neuron.2008.10.013.
  3. Barreto, E., & Cressman, J.R. (2011). Ion concentration dynamics as a mechanism for neuronal bursting. Journal of Biological Physics, 37, 361–373.CrossRefPubMedPubMedCentralGoogle Scholar
  4. Bennett, M.R., Farnell, L., Gibson, W.G. (2008). A quantitative model of cortical spreading depression due to purinergic and gap-junction transmission in astrocyte networks. Biophysical Journal, 95, 5648–5660.CrossRefPubMedPubMedCentralGoogle Scholar
  5. Chang, J.C., Brennan, K.C., He, D., Huang, H., Miura, R.M., et al. (2013). A mathematical model of the metabolic and perfusion effects on cortical spreading depression. PLoS ONE, 8, e70469.CrossRefPubMedPubMedCentralGoogle Scholar
  6. Chatton, J.-Y., Marquet, P., Magistretti, P. J. (2000). A quantitative analysis of L-glutamate-regulated Na+ dynamics in mouse cortical astrocytes: implications for cellular bioenergetics. European. J. Neuroscience, 12, 3843–3853.Google Scholar
  7. Chen, Y., & Swanson, R. A. (2003). Astrocytes and brain injury. Journal of Cerebral Blood Flow & Metabolism, 23, 137–149.CrossRefGoogle Scholar
  8. Choi, D.W., & Rothman, S.M. (1990). The role of glutamate neurotoxicity in hypoxic-ischemic neuronal death. Annu. Rev. Neuroscience, 13, 171–182.CrossRefGoogle Scholar
  9. Dahlem, M.A., Graf, R., Strong, A.J., Dreier, J.P., Dahlem, Y.A., Sieber, M., Hanke, W., Podoll, K., Sholl, E. (2010). Two-dimensional wave patterns of spreading depolarization: Retracting, re-entrant, and stationary waves. Physica D, 239, 889–903.CrossRefGoogle Scholar
  10. Danbolt, N.C. (2001). Glutamate uptake. Progress in Neurobiology, 65, 1–105.CrossRefPubMedGoogle Scholar
  11. Danbolt, N.C., Furness, D.N., Zhou, Y. (2016). Neuronal vs glial glutamate uptake: resolving the conundrum. Neurochem Int, 98, 29–45.CrossRefPubMedGoogle Scholar
  12. Diekman, C.O., Fall, C.P., Lechleiter, J.D., Terman, D. (2013). Modeling the neuroprotective role of enhanced astrocyte mitochondrial metabolism during stroke. Biophysical Journal, 104, 1752–1763.CrossRefPubMedPubMedCentralGoogle Scholar
  13. Dirnagl, U., Iadecola, C., Moskowitz, M.A. (1999). Pathobiology of ischaemic stroke: An integrated view.  https://doi.org/10.1016/S0166-2236(99)01401-0.
  14. Dreier, J.P., & et al. (2006). Delayed ischaemic neurological deficits after subarachnoid haemorrhage are associated with clusters of spreading depolarizations. Brain, 129, 3224–3227.CrossRefPubMedGoogle Scholar
  15. Dreier, J.P. (2011). The role of spreading depression, spreading depolarization and spreading ischemia in neurological disease. Nat Med, 17, 439–447.CrossRefPubMedGoogle Scholar
  16. Enger, R., Tang, W., Vindedal, G.F., Jensen, V., Johannes Helm, P., Sprengel, R., Looger, L.L., Nagelhus, E.A. (2015). Dynamics of Ionic Shifts in Cortical Spreading Depression. Cerebral Cortex, 25, 4469–4476.CrossRefPubMedPubMedCentralGoogle Scholar
  17. Ermentrout, G.B. (2002). Simulating, analyzing and animating dynamical system: a guide to XPPAUT for researchers and students. SIAM 14.Google Scholar
  18. Ermentrout, G.B., & Terman, D.H. (2010). Mathematical foundations of neuroscience, volume 35 of nterdisciplinary Applied Mathematics. New York: Springer. xvi+ 422 pp.CrossRefGoogle Scholar
  19. Fall, C.P., & Keizer, J. (2001). Mitochondrial modulation of intracellular Ca2+ signaling. Journal of Theoretical Biology, 210, 151–165.CrossRefPubMedGoogle Scholar
  20. Fröhlich, F., & Bazhenov, M. (2006). Coexistence of tonic firing and bursting in cortical neurons. Physical Review E, 74, 031922.CrossRefGoogle Scholar
  21. Gorelova, N.A., & Bures, J. (1983). Spiral waves of spreading depression in the isolated chicken retina. Developmental Neurobiology, 14, 353–363.CrossRefGoogle Scholar
  22. Grafstein, B. (1956). Locus of propagation of spreading cortical depression. J Neurophysiology, 19, 308–316.CrossRefGoogle Scholar
  23. Hartings, J.A., Watanabe, T., Bullock, M.R., Okonkwo, D.O., Fabricius, M., Woitzik, J., Dreier, J.P., Puccio, A., Shutter, L.A., Pahl, C., Strong, A.J. (2011). Spreading depolarizations have prolonged direct current shifts and are associated with poor outcome in brain trauma. Brain, 134, 1529–1540.CrossRefPubMedGoogle Scholar
  24. Haydon, P.G. (2001). GLIA: listening and talking to the synapse. Nature Reviews Neuroscience, 2, 185–193.CrossRefPubMedGoogle Scholar
  25. Hille, B. (2001). Ion channels and excitable membranes, 3rd edn. Sunderland: Sinauer Associates.Google Scholar
  26. Hübel, N., & Dahlem, M.A. (2014). Dynamics from Seconds to Hours in Hodgkin-Huxley Model with Time-Dependent Ion Concentrations and Buffer Reservoirs. PLoS Computational Biology, 10(12), e1003941.CrossRefPubMedPubMedCentralGoogle Scholar
  27. Hübel, N., Schöll, E., Dahlem, M.A. (2014). Bistable Dynamics Underlying Excitability of Ion Homeostasis in Neuron Models. PLoS Computational Biology, 10(5), e1003551.CrossRefPubMedPubMedCentralGoogle Scholar
  28. Hübel, N., & Ullah, G. (2016). Anions Govern Cell Volume: A Case Study of Relative Astrocytic and Neuronal Swelling in Spreading Depolarization. PLoS One, 11(3), e0147060.CrossRefPubMedPubMedCentralGoogle Scholar
  29. Huguet, G., Joglekar, A., Matamba Messi, L., Buckalew, R., Wong, S., Terman, D. (2016). Neuroprotective Role of Gap Junctions in a Neuron Astrocyte Network Model. Biophysical J, 111, 452–462.CrossRefGoogle Scholar
  30. 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, 495–512.CrossRefPubMedGoogle Scholar
  31. Kager, H., Wadman, W.J., Somjen, G.G. (2002). Conditions for the triggering of spreading depression studied with computer simulations. Journal of Neurophysiology, 88, 2700–2712.CrossRefPubMedGoogle Scholar
  32. Kager, H., Wadman, W.J., Somjen, G.G. (2007). Seizure-like afterdischarges simulated in a model neuron. Journal of Computational Neuroscience, 22, 105–128.CrossRefPubMedGoogle Scholar
  33. Lauritzen, M., Dreier, J.P., Fabricius, M., Hartings, J.A., Graf, R., et al. (2011). Clinical relevance of cortical spreading depression in neurological disorders: migraine, malignant stroke, subarachnoid and intracranial hemorrhage, and traumatic brain injury. Journal of Cerebral Blood Flow and Metabolism, 31, 17–35.CrossRefPubMedGoogle Scholar
  34. Leão, A.A.P. (1944). Spreading depression of activity in the cerebral cortex. Journal of Physiology, 7, 359–390.Google Scholar
  35. Levy, L.M., Warr, O., Attwell, D. (1998). Stoichiometry of the Glial Glutamate Transporter GLT-1 Expressed Inducibly in a Chinese Hamster Ovary Cell Line Selected for Low Endogenous Na-Dependent Glutamate Uptake. J Neuroscience, 18, 9620–9628.Google Scholar
  36. Li, Y.X., & Rinzel, J. (1994). Equations for InsP3 receptor-mediated [Ca2+]i oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. Journal of Theoretical Biology, 166, 461–473.CrossRefPubMedGoogle Scholar
  37. Ma, B., Bucklew, R., Du, Y., Kiyoshi, C.M., Alford, C.C., Wang, W., McTigue, Enyeart, J.J., Terman, D., Zhou, M. (2016). Gap junction coupling confers isopotentiality on astrocyte syncytium. Glia, 63, 214–226.CrossRefGoogle Scholar
  38. Moskowitz, M.A., Lo, E.H., Iadecola, C. (2011). The Science of Stroke: Mechanisms in Search of Treatments. Neuron Review, 67, 181–198.CrossRefGoogle Scholar
  39. Nakamura, H., Strong, A.J., Dohmen, C., Sakowitz, O.W., Vollar, S., Sue, M., Kracht, L., Hashemi, P., Bhatia, R., Yoshimine, T., Dreier, J.P., Dunn, A., Graf, R. (2010). Spreading depolarizations cycle around and enlarge focal ischaemic brain lesions. Brain, 133, 1994–2006.CrossRefPubMedPubMedCentralGoogle Scholar
  40. Nedergaard, M. (1994). Direct signaling from astrocytes to neurons in cultures of mammalian brain cells. Science, 263, 1768–71.CrossRefPubMedGoogle Scholar
  41. Nedergaard, M. (1996). Spreading depression as a contributor to ischemic brain damage. Advances in Neurology, 71, 75—83.Google Scholar
  42. Nedergaard, M., & Dirnagl, U. (2005). Role of glial cells in cerebral ischemia.  https://doi.org/10.1002/glia.20205.
  43. O’Connell, R., & Mori, Y. (2016). Effects of Glia in a Triphasic Continuum Model of Cortical Spreading Depression. Bull Math Biology, 78, 1943–1967.CrossRefGoogle Scholar
  44. Pietrobon, D., & Moskowitz, M. (2014). Chaos and commotion in the wake of cortical spreading depression and spreading depolarizations. Nature Reviews Neuroscience, 15, 379–93.CrossRefPubMedGoogle Scholar
  45. Risher, W.C., Ard, D., Yuan, J., Kirov, S.A. (2010). Recurrent spontaneous spreading depolarizations facilitate acute dendritic injury in the ischemic penumbra. The Journal of neuroscience : the official journal of the Society for Neuroscience, 30, 9859–9868.CrossRefGoogle Scholar
  46. Rossi, D.F., Oshima, T., Attwell, D. (2000). Glutamate release in severe brain ischaemia is mainly by reversed uptake. Nature, 403, 316–321.CrossRefPubMedGoogle Scholar
  47. Rossi, D., Brady, J.D., Mohr, C. (2007). Astrocyte metabolism and signaling during brain ischemia. Nature Neuroscience, 10, 1377–1386.CrossRefPubMedGoogle Scholar
  48. Santos, E., Scholl, M., Sanchez-Porras, R., Dahlem, M.A., Silos, H., Unterberg, A., Dickhaus, H., Sakowitz, O.W. (2014). Radial, spiral and reverberating waves of spreading depolarization occur in the gyrencephalic brain. NeuroImage, 99, 244–255.CrossRefPubMedGoogle Scholar
  49. Shapiro, B.E. (2001). Osmotic forces and gap junctions in spreading depression: A computational model. Journal of Computational Neuroscience, 10, 99–120.CrossRefPubMedGoogle Scholar
  50. Shibata, M., & Bures, J. (1974). Optimum topographical conditions for reverberating cortical spreading depression in rats. Developmental Neurobiology, 5, 107–118.CrossRefGoogle Scholar
  51. Somjen, G.G. (2001). Mechanisms of spreading depression and hypoxic spreading depression-like depolarization. Physiological Reviews, 81, 1065–1096.CrossRefPubMedGoogle Scholar
  52. Somjen, G.G., Kager, H., Wadman, W.J. (2008). Computer simulations of neuron-glia interactions mediated by ion flux. Journal of Computational Neuroscience, 25, 349–365.CrossRefPubMedGoogle Scholar
  53. Sonnewald, U., Westergaard, N., Schousboe, A. (1997). Glutamate transport and metabolism in astrocytes. Glia, 21, 56.CrossRefPubMedGoogle Scholar
  54. Strong, A.J., & Dardis, R. (2005). Depolarisation phenomena in traumatic and ischaemic brain injury. Advances and Technical Standards in Neurosurgery, 30, 3–49.CrossRefPubMedGoogle Scholar
  55. Szatkowski, M., Barbour, B., Attwell, D. (1990). Nonvesicular release of glutamate from glial cells by reversed electrogenic glutamate uptake. Nature, 348, 443–446.CrossRefPubMedGoogle Scholar
  56. Tuckwell, H.C., & Miura, R.M. (1978). A mathematical model for spreading cortical depression. Biophysical Journal, 23, 257–276.CrossRefPubMedPubMedCentralGoogle Scholar
  57. Ullah, G., Wei, Y., Dahlem, D.A., Wechselberger, M., Schiff, S.J. (2015). The role of cell volume in the dynamics of seizure, spreading depression, and anoxic depolarization. PLOS Computational Biology, 11(8), 2015.CrossRefGoogle Scholar
  58. Van Harreveld, A. (1959). Compounds in brain extracts causing spreading depression of cerebral cortical activity and contraction of crustacean muscle. J Neurochem, 3, 300–315.CrossRefGoogle Scholar
  59. Wang, F., Smith, N., Xu, Q., Fujita, T., Baba, A., et al. (2012). Astrocytes modulate neural network activity by Ca2+-dependent uptake of extracellular K+. Science Signaling, 5, ra26–ra26.PubMedPubMedCentralGoogle Scholar
  60. Wei, Y., Ullah, G., Schiff, S.J. (2014). Unification of neuronal spikes, seizures, and spreading depression. The Journal of Neuroscience, 34, 11733–11743.CrossRefPubMedPubMedCentralGoogle Scholar
  61. Yao, W., Huang, H., Miura, R. (2011). A continuum neuronal model for the instigation and propagation of cortical spreading depression. Bulletin of Mathematical Biology, 73, 2773– 2790.CrossRefPubMedGoogle Scholar
  62. Zandt, B.J., ten Haken, B., van Dijk, J.G., van Putten, M.J.A.M. (2011). Neural dynamics during anoxia and the ”wave of death”. PLoS ONE, 6, e22127.CrossRefPubMedPubMedCentralGoogle Scholar
  63. Zhao, Y., & Rempe, D.A. (2010). Targeting Astrocytes for Stroke Therapy. Neurotherapeutics, 7, 439–451.CrossRefPubMedPubMedCentralGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsOhio State UniversityColumbusUSA

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