Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 18, 2015

How does spreading depression spread? Physiology and modeling

Bas-Jan Zandt ORCID logo EMAIL logo , Bennie ten Haken , Michel J.A.M. van Putten and Markus A. Dahlem

Abstract

Spreading depression (SD) is a wave phenomenon in gray matter tissue. Locally, it is characterized by massive redistribution of ions across cell membranes. As a consequence, there is sustained membrane depolarization and tissue polarization that depress any normal electrical activity. Despite these dramatic events, SD remains difficult to observe in humans noninvasively, which, for long, has slowed advances in this field. The growing appreciation of its clinical importance in migraine and stroke is therefore consistent with an increasing need for computational methods that tackle the complexity of the problem at multiple levels. In this review, we focus on mathematical tools to investigate the question of spread and its two complementary aspects: What are the physiological mechanisms and what is the spatial extent of SD in the cortex? This review discusses two types of models used to study these two questions, namely, Hodgkin-Huxley type and generic activator-inhibitor models, and the recent advances in techniques to link them.


Corresponding author: Bas-Jan Zandt, Department of Biomedicine, University of Bergen, Jonas Lies vei 91, Postboks 7800, N-5020 Bergen, Norway, e-mail: .

Acknowledgments

The authors thank the Fields Institute for hosting the workshop on Cortical Spreading Depression and Related Neurological Phenomena, allowing us to incorporate new recent insights in this review.

References

Aitken, P.G., Jing, J., Young, J., and Somjen, G.G. (1991). Ion channel involvement in hypoxia-induced spreading depression in hippocampal slices. Brain Res. 541, 7–11.10.1016/0006-8993(91)91067-BSearch in Google Scholar

Almeida, A.C.G., Texeira, H.Z., Duarte, M.A., and Infantosi, A.F.C. (2004). Modeling extracellular space electrodiffusion during Leão’s spreading depression. IEEE Trans. Biomed. Eng. 51, 450–458.10.1109/TBME.2003.821010Search in Google Scholar

Anderson, C.M. and Nedergaard, M. (2003). Astrocyte-mediated control of cerebral microcirculation. Trends Neurosci. 26, 340–344; author reply 344–245.10.1016/S0166-2236(03)00141-3Search in Google Scholar

Ayata, C. (2010). Cortical spreading depression triggers migraine attack: pro. Headache 50, 725–730.10.1111/j.1526-4610.2010.01647.xSearch in Google Scholar PubMed

Ayata, C. (2013). Spreading depression and neurovascular coupling. Stroke 44, S87–S89.10.1161/STROKEAHA.112.680264Search in Google Scholar PubMed

Barreto, E. and Cressman, J.R. (2011). Ion concentration dynamics as a mechanism for neuronal bursting. J. Biol. Phys. 37, 361–373.10.1007/s10867-010-9212-6Search in Google Scholar PubMed PubMed Central

Bastug, T. and Kuyucak, S. (2005). Temperature dependence of the transport coefficients of ions from molecular dynamics simulations. Chem. Phys. Lett. 408, 84–88.10.1016/j.cplett.2005.04.012Search in Google Scholar

Bazhenov, M., Timofeev, I., Fröhlich, F., and Sejnowski, T.J. (2008). Cellular and network mechanisms of electrographic seizures. Drug Discov. Today Dis. Models 5, 45–57.10.1016/j.ddmod.2008.07.005Search in Google Scholar PubMed PubMed Central

Bureš, J., Burešova, O., and Křivaˮek, J. (1974). The Mechanism and Applications of Leão’s Spreading Depression (New York: Academia).Search in Google Scholar

Chapuisat, G., Dronne, M.-A., Grenier, E., Hommel, M., and Boissel, J.-P. (2010). In silico study of the influence of intensity and duration of blood flow reduction on cell death through necrosis or apoptosis during acute ischemic stroke. Acta Biotheor. 58, 171–190.10.1007/s10441-010-9100-2Search in Google Scholar PubMed

Charles, A.C. and Baca, S.M. (2013). Cortical spreading depression and migraine. Nat. Rev. Neurol. 9, 637–644.10.1038/nrneurol.2013.192Search in Google Scholar PubMed

Churchwell, K.B., Wright, S.H., Emma, F., Rosenberg, P.A., and Strange, K. (1996). NMDA receptor activation inhibits neuronal volume regulation after swelling induced by veratridine-stimulated Na+ influx in rat cortical cultures. J. Neurosci. 16, 7447–7457.10.1523/JNEUROSCI.16-23-07447.1996Search in Google Scholar

Cressman, J.R., Ullah, G., Ziburkus, J., Schiff, S.J., and Barreto, E. (2009). The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics. J. Comput. Neurosci. 26, 159–170.10.1007/s10827-008-0132-4Search in Google Scholar PubMed PubMed Central

Cressman, J.R., Ullah, G., Ziburkus, J., Schiff, S.J., and Barreto, E. (2011). Erratum to: The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics. J. Comput. Neurosci. 30, 781–781.10.1007/s10827-011-0333-0Search in Google Scholar

Dahlem, M.A. and Hadjikhani, N. (2009). Migraine aura: retracting particle-like waves in weakly susceptible cortex. PLoS One 4, e5007.10.1371/journal.pone.0005007Search in Google Scholar PubMed PubMed Central

Dahlem, M.A. and Isele, T.M. (2013). Transient localized wave patterns and their application to migraine. J. Math. Neurosci. 3, 7.10.1186/2190-8567-3-7Search in Google Scholar PubMed PubMed Central

Dahlem, M.A. and Müller, S.C. (2004). Reaction-diffusion waves in neuronal tissue and the window of cortical excitability. Ann. Phys. 13, 442–449.10.1002/andp.200410087Search in Google Scholar

Dahlem, M.A., Schmidt, B., Bojak, I., Boie, S., Kneer, F., Hadjikhani, N., and Kurths, J. (2014). Hot spots and labyrinths: why neuromodulation devices for episodic migraine should be personalized. Tech. Rep., PeerJ PrePrints.10.7287/peerj.preprints.515v1Search in Google Scholar

DiFrancesco, D. and Noble, D. (1985). A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos. Trans. R. Soc. Lond. B Biol. Sci. 307, 353–398.Search in Google Scholar

Dirnagl, U. and Pulsinelli, W. (1990). Autoregulation of cerebral blood flow in experimental focal brain ischemia. J. Cereb. Blood Flow Metab. 10, 327–336.10.1038/jcbfm.1990.61Search in Google Scholar PubMed

Doedel, E.J. and Oldeman, B.E. (2009). Auto-07P: Continuation and Bifurcation Software for Ordinary Differential Equations (Montreal, Canada: Concordia University).Search in Google Scholar

Dokos, S., Celler, B., and Lovell, N. (1993). Modification of DiFrancesco-noble equations to simulate the effects of vagal stimulation on in vivo mammalian sinoatrial node electrical activity. Ann. Biomed. Eng. 21, 321–335.10.1007/BF02368625Search in Google Scholar PubMed

Drake, C.T. and Iadecola, C. (2007). The role of neuronal signaling in controlling cerebral blood flow. Brain Lang. 102, 141–152.10.1016/j.bandl.2006.08.002Search in Google Scholar PubMed

Dreier, J.P. (2011). The role of spreading depression, spreading depolarization and spreading ischemia in neurological disease. Nat. Med. 17, 439–447.10.1038/nm.2333Search in Google Scholar

Dreier, J.P., Major, S., Pannek, H.-W., Woitzik, J., Scheel, M., Wiesenthal, D., Martus, P., Winkler, M.K.L., Hartings, J.A., Fabricius, M., et al. (2012). Spreading convulsions, spreading depolarization and epileptogenesis in human cerebral cortex. Brain 135, 259–275.10.1093/brain/awr303Search in Google Scholar

Dronne, M.-A., Boissel, J.-P., and Grenier, E. (2006). A mathematical model of ion movements in grey matter during a stroke. J. Theor. Biol. 240, 599–615.10.1016/j.jtbi.2005.10.023Search in Google Scholar

Dronne, M.-A., Grenier, E., Chapuisat, G., Hommel, M., and Boissel, J.-P. (2008). A modelling approach to explore some hypotheses of the failure of neuroprotective trials in ischemic stroke patients. Prog. Biophys. Mol. Biol. 97, 60–78.10.1016/j.pbiomolbio.2007.10.001Search in Google Scholar

Farr, H. and David, T. (2011). Models of neurovascular coupling via potassium and EET signalling. J. Theor. Biol. 286, 13–23.10.1016/j.jtbi.2011.07.006Search in Google Scholar

Feuerstein, G.Z. and Chavez, J. (2009). Translational medicine for stroke drug discovery: the pharmaceutical industry perspective. Stroke 40, S121–S125.10.1161/STROKEAHA.108.535104Search in Google Scholar

Florence, G., Dahlem, M.A., Almeida, A.-C.G., Bassani, J.W.M., and Kurths, J. (2009). The role of extracellular potassium dynamics in the different stages of ictal bursting and spreading depression: a computational study. J. Theor. Biol. 258, 219–228.10.1016/j.jtbi.2009.01.032Search in Google Scholar

Glitsch, H.G. (2001). Electrophysiology of the sodium-potassium-ATPase in cardiac cells. Physiol. Rev. 81, 1791–1826.10.1152/physrev.2001.81.4.1791Search in Google Scholar

Grafstein, B. (1963). Neural Release of Potassium During Spreading Depression. Brain Function. Cortical Excitability and Steady Potentials. M.A.B. Brazier, ed. (Berkeley: University of California Press), pp. 87–124.Search in Google Scholar

Hablitz, J.J. and Heinemann, U. (1989). Alterations in the microenvironment during spreading depression associated with epileptiform activity in the immature neocortex. Brain Res. Dev. Brain Res. 46, 243–252.10.1016/0165-3806(89)90288-5Search in Google Scholar

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., et al. (2011). Spreading depolarizations have prolonged direct current shifts and are associated with poor outcome in brain trauma. Brain 134, 1529–1540.10.1093/brain/awr048Search in Google Scholar PubMed

Heinemann, U. and Lux, H.D. (1977). Ceiling of stimulus induced rises in extracellular potassium concentration in the cerebral cortex of cat. Brain Res. 120, 231–249.10.1016/0006-8993(77)90903-9Search in Google Scholar

Helfferich, F. and Plesset, M. (1958). Ion exchange kinetics. a nonlinear diffusion problem. J. Chem. Phys. 28, 418–424.10.1063/1.1744149Search in Google Scholar

Hille, B. (2001). Ionic Channels of Excitable Membranes (Sunderland, MA: Sinauer). ISBN 9780878933211.Search in Google Scholar

Hoffmann, E.K., Lambert, I.H., and Pedersen, S.F. (2009). Physiology of cell volume regulation in vertebrates. Physiol. Rev. 89, 193–277.10.1152/physrev.00037.2007Search in Google Scholar PubMed

Hofmeijer, J. and van Putten, M.J.A.M. (2012). Ischemic cerebral damage: an appraisal of synaptic failure. Stroke 43, 607–615.10.1161/STROKEAHA.111.632943Search in Google Scholar PubMed

Howarth, C., Gleeson, P., and Attwell, D. (2012). Updated energy budgets for neural computation in the neocortex and cerebellum. J. Cereb. Blood Flow Metab. 32, 1222–1232.10.1038/jcbfm.2012.35Search in Google Scholar PubMed PubMed Central

Hübel, N. and Dahlem, M.A. (2014). Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs. PLos Comput. Biol. 10, e1003941.10.1371/journal.pcbi.1003941Search in Google Scholar PubMed PubMed Central

Hübel, N., Schöll, E., and Dahlem, M.A. (2014). Bistable dynamics underlying excitability of ion homeostasis in neuron models. PLoS Comput. Biol. 10, e1003551.10.1371/journal.pcbi.1003551Search in Google Scholar PubMed PubMed Central

Iadecola, C. and Nedergaard, M. (2007). Glial regulation of the cerebral microvasculature. Nat. Neurosci. 10, 1369–1376.10.1038/nn2003Search in Google Scholar PubMed

Idris, I. and Biktashev, V.N. (2008). Analytical approach to initiation of propagating fronts. Phys. Rev. Lett. 101, 244101.10.1103/PhysRevLett.101.244101Search in Google Scholar PubMed

Kager, H., Wadman, W.J., and Somjen, G.G. (2000). Simulated seizures and spreading depression in a neuron model incorporating interstitial space and ion concentrations. J. Neurophysiol. 84, 495–512.10.1152/jn.2000.84.1.495Search in Google Scholar PubMed

Kager, H., Wadman, W.J., and Somjen, G.G. (2002). Conditions for the triggering of spreading depression studied with computer simulations. J. Neurophysiol. 88, 2700–2712.10.1152/jn.00237.2002Search in Google Scholar PubMed

Kager, H., Wadman, W.J., and Somjen, G.G. (2007). Seizure-like afterdischarges simulated in a model neuron. J. Comput. Neurosci. 22, 105–128.10.1007/s10827-006-0001-ySearch in Google Scholar PubMed

Kapral, R. and Showalter, K., eds. (1995). Chemical Waves and Patterns (Dordrecht: Kluwer).10.1007/978-94-011-1156-0Search in Google Scholar

Keener, J. and Sneyd, J. (2010). Mathematical Physiology: I: Cellular Physiology. Interdisciplinary Applied Mathematics (Berlin: Springer). ISBN 9780387758473.Search in Google Scholar

Kozloski, J. and Wagner, J. (2011). An ultrascalable solution to large-scale neural tissue simulation. Front. Neuroinf. 5, 15.10.3389/fninf.2011.00015Search in Google Scholar PubMed PubMed Central

Krischer, K. and Mikhailov, A.S. (1994). Bifurcation to traveling spots in reaction-diffusion systems. Phys. Rev. Lett. 73, 3165–3168.10.1103/PhysRevLett.73.3165Search in Google Scholar PubMed

Krishnan, G.P. and Bazhenov, M. (2011). Ionic dynamics mediate spontaneous termination of seizures and postictal depression state. J. Neurosci. 31, 8870–8882.10.1523/JNEUROSCI.6200-10.2011Search in Google Scholar PubMed PubMed Central

Lashley, K. (1941). Patterns of cerebral integration indicated by scotomas of migraine. Arch. Neurol. Psychiatry 46, 331–339.10.1001/archneurpsyc.1941.02280200137007Search in Google Scholar

Lauritzen, M., Dreier, J.P., Fabricius, M., Hartings, J.A., Graf, R., and Strong, A.J. (2011). Clinical relevance of cortical spreading depression in neurological disorders: migraine, malignant stroke, subarachnoid and intracranial hemorrhage, and traumatic brain injury. J. Cereb. Blood Flow Metab. 31, 17–35.10.1038/jcbfm.2010.191Search in Google Scholar PubMed PubMed Central

Leão, A.A.P. (1947). Further observations on the spreading depression of activity in the cerebral cortex. J. Neurophysiol. 10, 409–414.10.1152/jn.1947.10.6.409Search in Google Scholar PubMed

Leonardi, M. and Raggi, A. (2013). Burden of migraine: international perspectives. Neurol. Sci. 34, S117–S118.10.1007/s10072-013-1387-8Search in Google Scholar PubMed

Lindquist, B.E. and Shuttleworth, C.W. (2012). Adenosine receptor activation is responsible for prolonged depression of synaptic transmission after spreading depolarization in brain slices. Neuroscience 223, 365–376.10.1016/j.neuroscience.2012.07.053Search in Google Scholar

Longsworth, L.G. (1953). Diffusion measurements, at 25°, of aqueous solutions of amino acids, peptides and sugars. J. Am. Chem. Soc. 75, 5705–5709.10.1021/ja01118a065Search in Google Scholar

Makarova, J., Ibarz, J.M., Canals, S., and Herreras, O. (2007). A steady-state model of spreading depression predicts the importance of an unknown conductance in specific dendritic domains. Biophys. J. 92, 4216–4232.10.1529/biophysj.106.090332Search in Google Scholar

Mangia, S., Giove, F., Tkác, I., Logothetis, N.K., Henry, P.-G., Olman, C.A., Maraviglia, B., Di Salle, F., and Uğurbil, K. (2009). Metabolic and hemodynamic events after changes in neuronal activity: current hypotheses, theoretical predictions and in vivo NMR experimental findings. J. Cereb. Blood Flow Metab. 29, 441–463.10.1038/jcbfm.2008.134Search in Google Scholar

Miura, R.M., Huang, H., and Wylie, J.J. (2013). Mathematical approaches to modeling of cortical spreading depression. Chaos 23, 046103.10.1063/1.4821955Search in Google Scholar

Mori, Y. (2014). A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression. arXiv preprint arXiv:1410.8391.Search in Google Scholar

Müller, M. and Somjen, G.G. (2000). Na(+) and K(+) concentrations, extra- and intracellular voltages, and the effect of TTX in hypoxic rat hippocampal slices. J. Neurophysiol. 83, 735–745.10.1152/jn.2000.83.2.735Search in Google Scholar

Nakamura, H., Strong, A.J., Dohmen, C., Sakowitz, O.W., Vollmar, S., Sué, M., Kracht, L., Hashemi, P., Bhatia, R., Yoshimine, T., et al. (2010). Spreading depolarizations cycle around and enlarge focal ischaemic brain lesions. Brain 133, 1994–2006.10.1093/brain/awq117Search in Google Scholar

Nicholson, C. and Phillips, J.M. (1981). Ion diffusion modified by tortuosity and volume fraction in the extracellular microenvironment of the rat cerebellum. J. Physiol. 321, 225–257.10.1113/jphysiol.1981.sp013981Search in Google Scholar

Nicholson, C. and Syková, E. (1998). Extracellular space structure revealed by diffusion analysis. Trends Neurosci. 21, 207–215.10.1016/S0166-2236(98)01261-2Search in Google Scholar

Nielsen, N., Wetterslev, J., Cronberg, T., Erlinge, D., Gasche, Y., Hassager, C., Horn, J., Hovdenes, J., Kjaergaard, J., Kuiper, M., et al. (2013). Targeted temperature management at 33°C versus 36°C after cardiac arrest. N. Engl. J. Med. 369, 2197–2206. PMID: 24237006.10.1056/NEJMoa1310519Search in Google Scholar PubMed

Noble, D. and Rudy, Y. (2001). Models of cardiac ventricular action potentials: iterative interaction between experiment and simulation. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 359, 1127–1142.10.1098/rsta.2001.0820Search in Google Scholar

Øyehaug, L., Østby, I., Lloyd, C.M., Omholt, S.W., and Einevoll, G.T. (2012). Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms. J. Comput. Neurosci. 32, 147–165.10.1007/s10827-011-0345-9Search in Google Scholar

Pearce, W.J. (1995). Mechanisms of hypoxic cerebral vasodilatation. Pharmacol. Ther. 65, 75–91.10.1016/0163-7258(94)00058-BSearch in Google Scholar

Pietrobon, D. and Moskowitz, M.A. (2014). Chaos and commotion in the wake of cortical spreading depression and spreading depolarizations. Nat. Rev. Neurosci. 15, 379–393.10.1038/nrn3770Search in Google Scholar

Plonsey, R. and Barr, R.C. (2007). Bioelectricity, 3rd ed. (Springer).Search in Google Scholar

Postnov, D., Postnov, D., and Schimansky-Geier, L. (2012). Self-terminating wave patterns and self-organized pacemakers in a phenomenological model of spreading depression. Brain Res. 1434, 200–211.10.1016/j.brainres.2011.10.001Search in Google Scholar

Qian, N. and Sejnowski, T. (1989). An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons. Biol. Cybern. 62, 1–15.10.1007/BF00217656Search in Google Scholar

Reggia, J.A. and Montgomery, D. (1994). Modeling cortical spreading depression. Proc. Annu. Symp. Comput. Appl. Med. Care 873–877.Search in Google Scholar

Reggia, J.A. and Montgomery, D. (1996). A computational model of visual hallucinations in migraine. Comput. Biol. Med. 26, 133–141.10.1016/0010-4825(95)00051-8Search in Google Scholar

Reshodko, L.V. and Bureš, J. (1975). Computer simulation of reverberating spreading depression in a network of cell automata. Biol. Cybern. 18, 181–189.10.1007/BF00326688Search in Google Scholar PubMed

Revett, K., Ruppin, E., Goodall, S., and Reggia, J.A. (1998). Spreading depression in focal ischemia: a computational study. J. Cereb. Blood Flow Metab. 18, 998–1007.10.1097/00004647-199809000-00009Search in Google Scholar PubMed

Richards, W. (1971). The fortification illusions of migraines. Sci. Am. 224, 88–96.10.1038/scientificamerican0571-88Search in Google Scholar PubMed

Schenk, C., Or-Guil, M., Bode, M., and Purwins, H.-G. (1997). Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains. Phys. Rev. Lett. 78, 3781.10.1103/PhysRevLett.78.3781Search in Google Scholar

Schock, S.C., Munyao, N., Yakubchyk, Y., Sabourin, L.A., Hakim, A.M., Ventureyra, E.C.G., and Thompson, C.S. (2007). Cortical spreading depression releases ATP into the extracellular space and purinergic receptor activation contributes to the induction of ischemic tolerance. Brain Res. 1168, 129–138.10.1016/j.brainres.2007.06.070Search in Google Scholar PubMed

Shapiro, B.E. (2001). Osmotic forces and gap junctions in spreading depression: a computational model. J. Comput. Neurosci. 10, 99–120.10.1023/A:1008924227961Search in Google Scholar

Skou, J.C. (1974). The (Na++K+) activated enzyme system and its relationship to transport of sodium and potassium. Q. Rev. Biophys. 7, 401–434.10.1017/S0033583500001475Search in Google Scholar PubMed

Somjen, G.G. (2001). Mechanisms of spreading depression and hypoxic spreading depression-like depolarization. Physiol. Rev. 81, 1065–1096.10.1152/physrev.2001.81.3.1065Search in Google Scholar PubMed

Somjen, G.G. (2004). Ions in the Brain – Normal Function, Seizures, and Stroke (Oxford: Oxford University Press), chap. 18.Search in Google Scholar

Somjen, G.G., Kager, H., and Wadman, W.J. (2009). Calcium sensitive non-selective cation current promotes seizure-like discharges and spreading depression in a model neuron. J. Comput. Neurosci. 26, 139–147.10.1007/s10827-008-0103-9Search in Google Scholar PubMed

Sotero, R.C. and Trujillo-Barreto, N.J. (2008). Biophysical model for integrating neuronal activity, EEG, fMRI and metabolism. Neuroimage 39, 290–309.10.1016/j.neuroimage.2007.08.001Search in Google Scholar PubMed

Syková, E. and Nicholson, C. (2008). Diffusion in brain extracellular space. Physiol. Rev. 88, 1277–1340.10.1152/physrev.00027.2007Search in Google Scholar PubMed PubMed Central

Teixeira, H.Z., de Almeida, A.-C.G., Infantosi, A.F.C., Vasconcelos, M.A., and Duarte, M.A. (2004). Simulation of the effect of Na+ and Cl- on the velocity of a spreading depression wave using a simplified electrochemical model of synaptic terminals. J. Neural. Eng. 1, 117–126.10.1088/1741-2560/1/2/007Search in Google Scholar PubMed

Trayanova, N.A. (2011). Whole-heart modeling: applications to cardiac electrophysiology and electromechanics. Circ. Res. 108, 113–128.10.1161/CIRCRESAHA.110.223610Search in Google Scholar PubMed PubMed Central

Tuckwell, H.C. (1980). Predictions and properties of a model of potassium and calcium ion movements during spreading cortical depression. Int. J. Neurosci. 10, 145–164.10.3109/00207458009160493Search in Google Scholar

Tuckwell, H.C. (1981). Simplified reaction-diffusion equations for potassium and calcium ion concentrations during spreading cortical depression. Int. J. Neurosci. 12, 95–107.10.3109/00207458108985794Search in Google Scholar

Tuckwell, H.C. and Miura, R.M. (1978). A mathematical model for spreading cortical depression. Biophys. J. 23, 257–276.10.1016/S0006-3495(78)85447-2Search in Google Scholar

Ullah, G., Cressman, J.R., Jr., Barreto, E., and Schiff, S.J. (2009). The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states. II. Network and glial dynamics. J. Comput. Neurosci. 26, 171–183.10.1007/s10827-008-0130-6Search in Google Scholar

U.S. National Institutes of Health (2014). The Intravascular Cooling in the Treatment of Stroke 2/3 Trial (ICTuS2/3). http://clinicaltrials.gov/show/NCT01123161. Accessed June 6, 2014.Search in Google Scholar

van der Worp, H.B., Macleod, M.R., Kollmar, R., and European Stroke Research Network for Hypothermia. (2010). Therapeutic hypothermia for acute ischemic stroke: ready to start large randomized trials? J. Cereb. Blood Flow Metab. 30, 1079–1093.10.1038/jcbfm.2010.44Search in Google Scholar

van Putten, M.J.A.M. (2009). Essentials of Neurophysiology – Basic Concepts and Clinical Applications for Scientists and Engineers (Springer).Search in Google Scholar

Vatov, L., Kizner, Z., Ruppin, E., Meilin, S., Manor, T., and Mayevsky, A. (2006). Modeling brain energy metabolism and function: a multiparametric monitoring approach. Bull. Math. Biol. 68, 275–291.10.1007/s11538-005-9008-1Search in Google Scholar

Vincent, M. and Hadjikhani, N. (2007). Migraine aura and related phenomena: beyond scotomata and scintillations. Cephalalgia 27, 1368–1377.10.1111/j.1468-2982.2007.01388.xSearch in Google Scholar

Wiener, N. and Rosenblueth, A. (1946). The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specially in cardiac muscle. Arch. Inst. Cardiol. Mex. 16, 205–243.Search in Google Scholar

Woesler, R., Schütz, P., Bode, M., Or-Guil, M., and Purwins, H.-G. (1996). Oscillations of fronts and front pairs in two-and three-component reaction-diffusion systems. Phys. Nonlinear Phenom. 91, 376–405.10.1016/0167-2789(95)00270-7Search in Google Scholar

Yao, W., Huang, H., and Miura, R.M. (2011). A continuum neuronal model for the instigation and propagation of cortical spreading depression. Bull. Math. Biol. 73, 2773–2790.10.1007/s11538-011-9647-3Search in Google Scholar PubMed

Zandt, B.-J. (2014). Neuronal Activity and Ion Homeostasis in the Hypoxic Brain. PhD. thesis (Enschede: University of Twente).Search in Google Scholar

Zandt, B.-J., ten Haken, B., van Dijk, J.G., and van Putten, M.J.A.M. (2011). Neural dynamics during anoxia and the “wave of death”. PLoS One 6, e22127.10.1371/journal.pone.0022127Search in Google Scholar PubMed PubMed Central

Zandt, B.-J., Stigen, T., Ten Haken, B., Netoff, T., and van Putten, M.J.A.M. (2013a). Single neuron dynamics during experimentally induced anoxic depolarization. J. Neurophysiol. 110, 1469–1475.10.1152/jn.00250.2013Search in Google Scholar PubMed

Zandt, B.-J., ten Haken, B., and van Putten, M.J.A.M. (2013b). Diffusing substances during spreading depolarization: analytical expressions for propagation speed, triggering, and concentration time courses. J. Neurosci. 33, 5915–5923.10.1523/JNEUROSCI.5115-12.2013Search in Google Scholar PubMed PubMed Central

Zel’dovich, Y.B. and Frank-Kamenetskii, D.A. (1938). Theory of thermal flame propagation. Zh. Fiz. Khim. 12, 100–105.Search in Google Scholar

Zonta, M., Angulo, M.C., Gobbo, S., Rosengarten, B., Hossmann, K.-A., Pozzan, T., and Carmignoto, G. (2003). Neuron-to-astrocyte signaling is central to the dynamic control of brain microcirculation. Nat. Neurosci. 6, 43–50.10.1038/nn980Search in Google Scholar PubMed

Received: 2014-10-6
Accepted: 2014-10-31
Published Online: 2015-2-18
Published in Print: 2015-4-1

©2015 by De Gruyter

Downloaded on 31.1.2023 from https://www.degruyter.com/document/doi/10.1515/revneuro-2014-0069/html
Scroll Up Arrow