Abstract
The nonlinear wave phenomenon of cortical spreading depression (SD), which occurs in many brain structures, has mathematical similarities to neuronal spiking but on very different space and time scales. Its properties and previous modeling are briefly reviewed. A model consisting of a 6-component reaction–diffusion system in two space dimensions is described. With 3-parameter Poisson process sources of potassium ions representing extrusions due to the random firings of neurons, the model takes the form of a multi-component set of nonlinear stochastic partial differential equations. Assuming that in a restricted small area the sources have greater strength than background, the probability of an SD wave is found as a function of the patch size. Also investigated is the probability of elicitation of SD through the occurrence of a patch with compromised metabolic activity, as may occur by virtue of an infarct after stroke. The analysis proceeds in terms of the effect of relative decreases in the strength of ATP-dependent sodium–potassium exchange pump.
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References
Basarsky, T.A., Feighan, D., MacVicar, B.A.: Glutamate release through volume-activated channels during spreading depression. J. Neurosci. 19, 6439–6445 (1999)
Boissel, J.P., Ribba, B., Grenier, E., Chapuisat, G., Dronne, M.A.: Modelling methodology in physiopathology. Prog. Biophys. Molec. Biol. 97, 28–39 (2008)
Bonthius, D.J., Stringer, J.L., Lothman, E.W., Steward, O.: Spreading depression and reverberatory seizures induce the upregulation of mRNA for glial fibrillary acidic protein. Brain Res. 645, 215–224 (1994)
Broberg, M., Pope, K.J., Nilsson, M., Wallace, A., Wilson, J., Willoughby, J.O.: Preseizure increased gamma electroencephalographic activity has no effect on extracellular potassium or calcium. J. Neurosci. Res. 85, 906–918 (2007)
Chapuisat, G.: Discussion of a simple model of spreading depressions. ESAIM Proc. 18, 87–98 (2007)
Chapuisat, G., Dronneb, M.A., Grenier, E., Hommel, M., Gilquin, H., Boissel, J.P.: A global phenomenological model of ischemic stroke with stress on spreading depressions. Prog. Biophys. Molec. Biol. 97, 4–27 (2008)
Cucchiara, B., Detre, J.: Migraine and circle of Willis anomalies. Med. Hypoth. 70, 860–865 (2008)
Cui, Y., Katoaka, Y., Inui, T., Mochizuki, T., Onoe, H., Matsumura, K., Urade, Y., Yamada, H., Watanabe, Y.: Up-regulated neuronal COX-2 expression after cortical spreading depression is involved in non-REM sleep induction in rats. J. Neurosci. Res. 86, 929–936 (2008)
Dahlem, M.A., Schneider, F.M., Schöll, E.: Efficient control of transient wave forms to prevent spreading depolarizations. J. Theor. Biol. 251, 202–209 (2008)
Dahlem, M.A., Schneider, F.M., Schöll, E.: Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke. Chaos 18, 026,110 (2008)
Fabricius, M., Fuhr, S., Willumsen, L., Dreier, J.P., Bhatia, R., Boutelle, M.G., Hartings, J.A., Bullock, R., Strong, A.J., Lauritzen, M.: Association of seizures with cortical spreading depression and perinfarct depolarisations in the acutely injured human brain. Clin. Neurophysiol. 119, 1973–1984 (2008)
Fossier, P., Tauc, L., Baux, G.: Calcium transients and neurotransmitter release at an identified synapse. Trends Neurosci. 22, 161–166 (1999)
Garay, R.P., Garrahan, P.J.: The interaction of sodium and potassium with the sodium pump in red cells. J. Physiol. 231, 297–325 (1973)
Gardner-Medwin, A.R.: Possible roles of vertebrate neuroglia in potassium dynamics, spreading depression and migraine. J. Exp. Biol. 95, 111–127 (1981)
Gass, A., Ay, H., Szabo, K., Koroshetz, W.J.: Diffusion-weighted MRI for the small stuff: the details of acute cerebral ischaemia. Lancet Neurol. 3, 39–45 (2004)
Gorgi, A.: Spreading depression: a review of the clinical relevance. Brain Res. Rev. 38, 33–60 (2001)
Hadjikhani, N., Sanchez Del Rio, M., Wu, O., Schwartz, D., Bakker, D., Fischl, B., Kwong, K.K., Cutrer, F.M., Rosen, B.R., Tootell, R.B., Sorensen, A.G., Moskowitz, M.A.: Mechanisms of migraine aura revealed by functional MRI in human visual cortex. Proc. Natl. Acad. Sci. 98, 4687–4692 (2001)
Heidelberger, R., Heinemann, C., Neher, E., Matthews, G.: Calcium dependence of the rate of exocytosis in a synaptic terminal. Nature 371, 513–515 (1994)
Herreras, O.: Electrical prodromals of spreading depression void grafsteins potassium hypothesis. J. Neurophysiol. 94, 3656 (2005)
Hossman, K.A.: Cerebral ischemia: models, methods and outcomes. Neuropharmacology 55, 257–270 (2007)
Kager, H., Wadman, W.J., Somjen, G.G.: Simulated seizures and spreading depression in a neuron model incorporating interstitial space and ion concentrations. J. Neurophysiol. 84, 495,512 (2000)
Koester, H.J., Sakmann, B.: Calcium dynamics associated with action potentials in single nerve terminals of pyramidal cells in layer 2/3 of the young rat neocortex. J. Physiol. 529, 625–646 (2000)
Kramer, M.A., Szeri, A.J., Sleigh, J.W., Kirsch, H.E.: Mechanisms of seizure propagation in a cortical model. J. Comput. Neurosci. 22, 63–80 (2007)
Larrosa, B., Pastor, J., López-Aguado, L., Herreras, O.: A role for glutamate and glia in the fast network oscillations preceding spreading depression. Neuroscience 141, 1057–1068 (2006)
Lauritzen, M.: Pathophysiology of the migraine aura: the spreading depression theory. Brain 117, 191–210 (1994)
Leão, A.A.P.: Spreading depression of activity in the cerebral cortex. Neurophysiology 7, 359–390 (1944)
Leão, A.A.P., Morison, R.S.: Propagation of spreading cortical depression. J. Neurophysiol. 8, 33–45 (1945)
Makarova, J., Ibarz, J.M., Canals, S., Herreras, O.: A steady-state model of spreading depression predicts the importance of an unknown conductance in specific dendritic domains. Biophys. J. 92, 4216–4232 (2007)
Makarova, J., Makarov, V.A., Herreras, O.: Generation of sustained field potentials by gradients of polarization within single neurons: a macroscopic model of spreading depression. J. Neurophysiol. 103, 2446–2457 (2010)
Mayevsky, A., Doron, A., Manor, T., Meilin, S., Zarchin, N., Ouaknine, G.E.: Cortical spreading depression recorded from the human brain. Brain Res. 740, 268–274 (1996)
Milner, P.M.: Note on a possible correspondence between the scotomas of migraine and spreading depression of leĂŁo. Electroencephalogr. Clin. Neurophysiol. 10, 705 (1958)
Obrenovitch, T.P., Zilkha, E.: Inhibition of cortical spreading depression by L-701,324, a novel antagonist at the glycine site of the N-methyl-D-aspartate receptor complex. Br. J. Pharmacol. 117, 931–937 (1996)
Rodgers, C.I., Armstrong, G.A.B., Robertson, R.M.: Coma in response to environmental stress in the locust: a model for cortical spreading depression. J. Insect. Physiol. 56, 980–990 (2010)
Rovira, A., Grivé, E., Rovira, A., Alvarez-Sabin, J.: Distribution territories and causative mechanisms of ischemic stroke. Eur. Radiol. 15, 416–426 (2005)
Schiller, J., Helmchen, F., Sakmann, B.: Spatial profile of dendritic calcium transients evoked by action potentials in rat neocortical pyramidal neurones. J. Physiol. 487, 583–600 (1995)
Schock, S.C., Munyao, N., Yakubchyk, Y., Sabourin, L.A., Hakim, A.M., Ventureyra, E.C., Thompson, C.S.: 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 (2007)
Shapiro, B.E.: Osmotic forces and gap junctions in spreading depression: a computational model. J. Comp. Neurosci. 10, 99–120 (2001)
Somjen, G.G.: Mechanisms of spreading depression and hypoxic spreading depression-like depolarization. Physiol. Rev. 81, 1065–1096 (2001)
Somjen, G.G., Kager, H., Wadman, W.J.: Calcium sensitive nonselective cation current promotes seizure-like discharges and spreading depression in a model neuron. J. Comp. Neurosci. 26, 139–147 (2009)
Teixeira, H.Z., Alvarenga, D.J., Almeida, A.C.G., Rodrigues, A.M., Duarte, M.A.: Parallelization of the electrodiffusion mechanism of the computational model of spreading depression. Computational Science and Engineering, Proceedings of 11th International Conference 2008, IEEE Conference publications. DOI: 10.1109/CSE.2008.12 pp. 261–266 (2008)
Torok, T.L.: Electrogenic Na + /Ca2 + -exchange of nerve and muscle cells. Prog. Neurobiol. 82, 287–347 (2007)
Tuckwell, H.C.: Predictions and properties of a model of potssium and calcium ion movements during spreading cortical depression. Int. J. Neurosci. 10, 145–165 (1980)
Tuckwell, H.C.: Mathematical modeling of spreading cortical depression: spiral and reverberating waves. Am. Inst. Phys. Conf. Proc. 1028, 46–64 (2008)
Tuckwell, H.C., Hermansen, C.L.: Ion and transmitter movements during spreading cortical depression. Int. J. Neurosci. 12, 109–135 (1981)
Tuckwell, H.C., Miura, R.M.: A mathematical model for spreading cortical depression. Biophys. J. 23, 257–276 (1978)
Van Harreveld, A.: Two mechanisms for spreading depression in the chicken retina. J. Neurobiol. 9, 419–431 (1978)
Yingst, D.R., Davis, J., Schiebinger, R.: Effects of extracellular calcium and potassium on the sodium pump of rat adrenal glomerulosa cells. Am. J. Physiol. Cell Physiol. 280, C119–C125 (2001)
Zoremba, N., Homola, A., Rossaint, R., Syková, E.: Brain metabolism and extracellular space diffusion parameters during and after transient global hypoxia in the rat cortex. Exp. Neurol. 203, 34–41 (2007)
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Tuckwell, H.C. (2013). Stochastic Modeling of Spreading Cortical Depression. In: Bachar, M., Batzel, J., Ditlevsen, S. (eds) Stochastic Biomathematical Models. Lecture Notes in Mathematics(), vol 2058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32157-3_8
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