Abstract
Chapter describes the construction of population-based rate models under the assumption that the spiking of individual neurons is unimportant. The issue of how stochasticity at the single-cell level manifests itself at the population level is discussed, introducing topics such as balanced networks, Poisson statistics, and asynchronous states. Stochastic methods are then used to analyze bistability in a stochastic population model. Finally, the transition from spatially structured neural networks to continuum neural fields is highlighted. The latter take the form of nonlocal integrodifferential equations, in which the integral kernel represents the distribution of synaptic connections.
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References
Abbott, L.F., Marder, E.: Modelling small networks. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modelling, pp. 361–410, 2nd edn. MIT Press, Cambridge (1998)
Abbott, L.F., van Vresswijk, C.: Asynchronous states in networks of pulse–coupled oscillators. Phys. Rev. E 48(2), 1483–1490 (1993)
Abbott, L.F., Fahri, E., Gutmann, S.: The path integral for dendritic trees. Biol. Cybern. 66, 49–60 (1991)
Abbott, L.F., Varela, J.A., Sen, K., Nelson, S.B.: Synaptic depression and cortical gain control. Science 275, 220–224 (1997)
Aertsen, A.M.H.J., Gerstein, G.L., Habib, M.K., Palm, G.: Dynamics of neuronal firing correlation: modulation of ‘effective connectivity’. J. Neurophysiol. 61, 900–917 (1989)
Alexander, G., Crutcher, M.: Functional architecture of basal ganglia circuits: neural substrates of parallel processing. Trends. Neurosci. 13, 266–271 (1990)
Allard, J., Moginler, A.: Traveling waves in actin dynamics and cell motility. Curr. Opin. Cell Biol. 25, 107–115 (2013)
Amari, S.: Dynamics of pattern formation in lateral inhibition type neural fields. Biol. Cybern. 27, 77–87 (1977)
Andersen, S.S., Bi, G.Q.: Axon formation: a molecular model for the generation of neuronal polarity. Bioessays 22, 172–179 (2000)
Andersen, P., Morris, R., Amaral, D., Bliss, T., O’Keefe, J.: The Hippocampus Book. Oxford University Press, Oxford (2008)
Angelucci, A., Levitt, J.B., Walton, E.J.S., Hupe, J.M., Bullier, J., Lund, J.S.: Circuits for local and global signal integration in primary visual cortex. J. Neurosci. 22, 8633–8646 (2002)
Antic, S.D., Zhou, W.L., Moore, A.R., Short, S.M., Ikonomu, K.D.: The decade of the dendritic NMDA spike. J. Neurosci. Res. 88, 2991–3001 (2010)
Arimura, N., Kaibuchi, K.: Neuronal polarity: from extracellular signals to intracellular mechanisms. Nat. Rev. Neurosci 8, 194–205 (2007)
Armero, J., Casademunt, J., Ramirez-Piscina, L., Sancho, J.M.: Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise. Phys. Rev. E 58, 5494–5500 (1998)
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)
Atay, F.M., Hutt, A.: Stability and bifurcations in neural fields with finite propagation speed and general connectivity. SIAM J. Appl. Math. 65, 644–646 (2005)
Austin, T.D.: The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann. Appl. Prob. 18, 1279–1325 (2006)
Avoli, M., de Curtis, G.B.M.: Do interictal spikes sustain seizures and epileptogenesis. Epilepsy Currents 6, 203–207 (2006)
Baer, S.M., Rinzel, J.: Propagation of dendritic spikes mediated by excitable spines: a continuum theory. J. Neurophysiol. 65, 874–890 (1991)
Bair, W., Koch, C., Newsome, W.T., Britten, K.H.: Power spectrum analysis of bursting cells in area MT in the behaving monkey. J. Neurosci. 14, 2870–2893 (1994)
Baladron, J., Fasoli, D., Faugeras, O., Touboul, J.: Mean field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons. J. Math. Neuro. 2, 10 (2012)
Ballesteros-Yanez, I., Benavides-Piccione, R., Elston, G.N., Yuste, R., DeFelipe, J.: Density and morphology of dendritic spines in mouse neocortex. Neurosci. 138, 403–409 (2006)
Barabasi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Baras, F., Mansour, M., Malek, M., Pearson, J.E.: Microscopic simulation of chemical bistability in homogeneous systems. J. Chem. Phys. 105, 8257–8261 (1996)
Bart, E., Bao, S., Holcman, D.: Modeling the spontaneous activity of the auditory cortex. J. Comput. Neurosci. 19, 357–378 (2005)
Barthelemy, M., Barrat, A., Pastor-Satorras, R., Vespignani2, A.: Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett. 92, 178701 (2004)
Bazhenov, M., Timofeev, I., Steriade, M., Sejnowski, T.J.: Model of thalamocortical slow-wave sleep oscillations and transitions to activated states. J. Neurosci. 22, 8691–8704 (2002)
Bedard, C., Destexhe, A.: Macroscopic models of local field potentials the apparent 1/f noise in brain activity. Biophys. J. 96, 2589–2603 (2009)
Beggs, J.M., Plenz, D.: Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167–11177 (2003)
Beggs, J.M., Plenz, D.: Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures. J. Neurosci. 24, 5216–5229 (2004)
Beggs, J.M., Timme, N.: Being critical of criticality in the brain. Front. Physiol. 3 (163), 1–14 (2012)
Ben-Ari, Y.: Developing networks play a similar melody. Trend. Neurosci. 24, 353–360 (2001)
Ben-Jacob, E., Brand, H., Dee, G., Kramer, L., Langer, J.S.: Pattern propagation in nonlinear dissipative systems. Physica D 14, 348–364 (1985)
Ben-Yishai, R., Bar-Or, R.L., Sompolinsky, H.: Theory of orientation tuning in visual cortex. Proc. Nat. Acad. Sci. 92, 3844–3848 (1995)
Benda, J., Herz, A.V.M.: A universal model for spike-frequency adaptation. Neural Comput. 15, 2523–2564 (2003)
Benguria, R., Depassier, M.: Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation. Comm. Math. Phys. 175, 221–227 (1996)
Bennett, M.R., Farnell, L., Gibson, W.G.: A quantitative model of purinergic junctional transmission of calcium waves in astrocyte networks. Biophys. J. 89, 2235–2250 (2005)
Bennett, M.R., Farnell, L., Gibson, W.G.: A quantitative model of cortical spreading depression due to puringenic and gap-junction transmission in astrocyte networks. Biophys. J. 95, 5648–5660 (2008)
Benucci, A., Frazor, R., Carandini, M.: Standing waves and traveling waves distinguish two circuits in visual cortex. Neuron 55, 103–17 (2007)
Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: II-biological invasions and pulsating travelling fronts. J. Math. Biol. 51, 75–113 (2005)
Bernard, C., Anderson, A., Becker, A., Poolos, N.P., Beck, H., Johnston, D.: Acquired dendritic channelopathy in temporal lobe epilepsy. Science 305, 532–535 (2004)
Berridge, M.J.: Calcium signalling and cell proliferation. Bioessays 17, 491–500 (1995)
Berridge, M.J.: Neuronal calcium signaling. Neuron 21, 13–26 (1998)
Berridge, M.J., Lipp, P., Bootman, M.D.: The versatility and universality of calcium signalling. Nat. Rev. Mol. Cell Biol. 1, 11–21 (2000)
Bertram, R., Smith, G.D., Sherman, A.: Modeling study of the effects of overlapping calcium microdomains on neurotransmitter release. Biophys. J. 76, 735–750 (1999)
Blake, R.: A primer on binocular rivalry, including current controversies. Brain Mind 2, 5–38 (2001)
Blake, R., Logothetis, N.: Visual competition. Nat. Rev. Neurosci. 3, 1–11 (2002)
Blake, R., Wilson, H.R.: Binocular vision. Vis. Res. 51, 754–770 (2011)
Blakenship, A.G., Feller, M.B.: Mechanisms underlying spontaneous patterned activity in developing neural circuits. Nat. Rev. Neurosci. 11, 18–29 (2010)
Blasdel, G.G.: Orientation selectivity, preference, and continuity in monkey striate cortex. J. Neurosci. 12, 3139–3161 (1992)
Blasdel, G.G., Salama, G.: Voltage-sensitive dyes reveal a modular organization in monkey striate cortex. Nature 321, 579–585 (1986)
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006)
Bonhoeffer, T., Grinvald, A.: Orientation columns in cat are organized in pinwheel like patterns. Nature 364, 166–146 (1991)
Boonstra, T., Daffertshofer, A., Breakspear, M., Beek, P.: Multivariate time-frequency analysis of electromagnetic brain activity during bimanual motor learning. Neuroimage 36, 370–377 (2007)
Borisyuk, R., Kirillov, A.B.: Bifurcation analysis of a neural network model. Biol. Cybern. 66, 319–325 (1992)
Bosking, W.H., Zhang, Y., Schofield, B., Fitzpatrick, D.: Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J. Neurosci. 17, 2112–2127 (1997)
Boustani, S.E., Destexhe, A.: A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Comput. 21, 46–100 (2009)
Brackley, C.A., Turner, M.S.: Random fluctuations of the firing rate function in a continuum neural field model. Phys. Rev. E 75, 041913 (2007)
Brascamp, J.W., van Ee, R., Noest, A.J., Jacobs, R.H., van den Berg, A.V.: The time course of binocular rivalry reveals a fundamental role of noise. J. Vision 6, 1244–1256 (2006)
Breakspear, M., Roberts, J.A., Terry, J.R., Rodrigues, S., Mahant, N., Robinson, P.A.: A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb. Cortex 16, 1296–1313 (2006)
Bressloff, P.C.: Resonantlike synchronization and bursting in a model of pulse-coupled neurons with active dendrites. J. Comput. Neurosci. 6, 237–249 (1999)
Bressloff, P.C.: Synaptically generated wave propagation in excitable neural media. Phys. Rev. Lett. 82, 2979–2982 (1999)
Bressloff, P.C.: Traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons. J. Math. Biol. 40, 169–183 (2000)
Bressloff, P.C.: Traveling fronts and wave propagation failure in an inhomogeneous neural network. Physica D 155, 83–100 (2001)
Bressloff, P.C.: Bloch waves, periodic feature maps and cortical pattern formation. Phys. Rev. Lett. 89, 088101 (2002)
Bressloff, P.C.: Spatially periodic modulation of cortical patterns by long-range horizontal connections. Physica D 185, 131–157 (2003)
Bressloff, P.C.: Pattern formation in visual cortex. In: Chow, C.C., Gutkin, B., Hansel, D., Meunier, C., Dalibard, J. (eds.) Les Houches 2003: Methods and Models in Neurophysics, pp. 477–574. Elsevier, Amsterdam (2005)
Bressloff, P.C.: Stochastic neural field theory and the system-size expansion. SIAM J. Appl. Math. 70, 1488–1521 (2009)
Bressloff, P.C.: Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics. Phys. Rev. E 85, 051903 (2010)
Bressloff, P.C.: From invasion to extinction in heterogeneous neural fields. J. Math. Neurosci. 2(6), 1–27 (2012)
Bressloff, P.C.: Spatiotemporal dynamics of continuum neural fields. J. Phys. A 45, 033001 (109 pp.) (2012)
Bressloff, P.C.: Propagation of CaMKII translocation waves in heterogeneous spiny dendrites. J. Math. Biol. 66, 1499–1525 (2013)
Bressloff, P.C., Coombes, S.: Dynamics of strongly coupled spiking neurons. Neural Comput. 12, 91–129 (2000)
Bressloff, P.C., Cowan, J.D.: Amplitude equation approach to contextual effects in visual cortex. Neural Comput. 14, 493–525 (2002)
Bressloff, P.C., Cowan, J.D.: The visual cortex as a crystal. Physica D 173, 226–258 (2002)
Bressloff, P.C., Folias, S.E.: Front bifurcations in an excitatory neural network. SIAM J. Appl. Math. 65, 131–151 (2005)
Bressloff, P.C., Dwyer, V.M., Kearney, M.J.: Sum-over-paths approach to diffusion on trees. J. Phys. A 29, 1881–1896 (1996)
Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J.: Scalar and pseudoscalar bifurcations: pattern formation on the visual cortex. Nonlinearity 14, 739–775 (2001)
Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J., Wiener, M.: Geometric Visual Hallucinations, Euclidean Symmetry and the Functional Architecture of Striate Cortex. Phil. Trans. Roy. Soc. Lond. B 356, 299–330 (2001)
Bressloff, P.C., Folias, S.E., Prat, A., Li, Y.X.: Oscillatory waves in inhomogeneous neural media. Phys. Rev. Lett. 91, 178101 (2003)
Bressloff, P.C., Newby, J.: Quasi-steady state analysis of motor-driven transport on a two-dimensional microtubular network. Phys. Rev. E 83, 061139 (2011)
Bressloff, P.C., Newby, J.M.: Metastability in a stochastic neural network modeled as a velocity jump Markov process. SIAM J. Appl. Dyn. Sys. 12, 1394–1435 (2013)
Bressloff, P.C., Webber, M.: Neural field model of binocular rivalry waves. J. Comput. Neurosci. 32, 233–252 (2012)
Bressloff, P.C., Webber, M.A.: Front propagation in stochastic neural fields. SIAM J. Appl. Dyn. Syst. 11, 708–740 (2012)
Brown, R.J., Norcia, A.M.: A method for investigating binocular rivalry in real-time with the steady state VEP. Vision Res. 37, 2401–2408 (1997)
Brunel, N.: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci 8, 183–208 (2000)
Brunel, N., Hakim, V.: Fast global oscillations in networks of integrate–and–fire neurons with low firing rates. Neural Comput. 11, 1621–1671 (1999)
Buckwar, E., Riedler, M.G.: An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol. 63, 1051–1093 (2011)
Buckwar, E., Riedler, M.G.: Laws of large numbers and langevin approximations for stochastic neural field equations. J. Math. Neurosci. 3(1), 1–54 (2012)
Buice, M.A., Chow, C.C.: Effective stochastic behavior in dynamical systems with incomplete information. Phys. Rev. E 84, 051120 (2011)
Buice, M., Cowan, J.D.: Field-theoretic approach to fluctuation effects in neural networks. Phys. Rev. E 75, 051919 (2007)
Buice, M., Cowan, J.D., Chow, C.C.: Systematic fluctuation expansion for neural network activity equations. Neural Comp. 22, 377–426 (2010)
Bures, J., Buresova, O., Krivanek, J.: The Mechanisms and Applications of Leao’s Spreading Depression of Electrical Activity. Academia, Prague (1974)
Butts, D.A., Feller, M.B., Shatz, C.J., Rokshar, D.S.: Retinal waves are governed by collective network properties. J. Neurosci. 19, 3580–3593 (1999)
Butz, E.G., Cowan, J.D.: Transient potentials in dendritic systems of arbitrary geometry. Biophys. J. 14, 661–689 (1974)
Buzsaki, G.: Hippocampal sharp waves: their origin and significance. Brain Res. 398, 242–252 (1986)
Buzsaki, G.: Theta oscillations in the hippocampus. Neuron 33, 325–40 (2002)
Buzsaki, G.: Rhythms of the Brain. Oxford University Press, Oxford (2006)
Byrne, J.H., Roberts, J.L.: From Molecules to Networks: An Introduction to Cellular and Molecular Neuroscience. Elsevier, Amsterdam (2004)
Cai, D., Tao, L., Shelley, M., McLaughlin, D.W.: An effective kinetic representation of fluctuation–driven neuronal networks with application to simple and complex cells in visual cortex. Proc. Natl. Acad. Sci. USA 101, 7757–7562 (2004)
Campbell, K., Gotz., M.: Radial glia: multi-purpose cells for vertebrate brain development. Trend. Neurosci. 25, 235–238 (2002)
Cantrell, R.S., Cosner, C.: Spatial Ecology Via Reaction-Diffusion Equations. Wiley, Chichester (2003)
Carlsson, A.E.: Dendritic actin filament nucleation causes traveling waves and patches. Phys. Rev. Lett. 104, 228102 (2010)
Carpenter, G.: A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Diff. Eqns. 23, 335–367 (1977)
Casti, A.R.R., Ormutag, A., Sornborger, A., Kaplan, E., Knight, B., Victor, J., Sirovich, L.: A population study of integrate–and–fire–or burst neurons. Neural Comput. 14, 957–986 (2002)
Charles, A.C., Merrill, J.E., Dickson, E.R., Sanderson, M.J.: Intercellular signaling in glial cells: calcium waves and oscillations in response to mechanical stimulation and glutamate. Neuron 6, 983–992 (1991)
Charles, A.C., Naus, C.C., Zhu, D., Kidder, G.M., Dirksen, E.R., Sanderson, M.J.: Intercellular calcium signaling via gap junctions in glioma cells. J. Cell. Biol. 118, 195–201 (1992)
Chen, X.: Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eqns. 2, 125 (1997)
Chen, K.C., Nicholson, C.: Spatial buffering of potassium ions in brain extracellular space. Biophys. J. 78, 2776–2797 (2000)
Cheng, H., Lederer, W.J.: Calcium sparks. Physiol. Rev. 88, 1491–1545 (2008)
Cheng, H., Lederer, M.R., Lederer, W.J., Cannell, M.B.: Calcium sparks and waves in cardiac myocytes. Am. J. Physiol. 270, C148–C159 (1996)
Chervin, R.D., Pierce, P.A., Connors, B.W.: Periodicity and directionality in the propagation of epileptiform discharges across neocortex. J. Neurophysiol. 60, 1695–1713 (1988)
Chialvo, D.: Emergent complex neural dynamics. Nat. Phys. 6, 744–750 (2010)
Chong, S.C., Tadin, D., Blake, R.: Endogenous attention prolongs dominance durations in binocular rivalry. J. Vision 5, 1044–1012 (2005)
Chow, C.C., Buice, M.: Path integral methods for stochastic differential equations. arXiv nlin/105966v1 (2011)
Chow, C.C., White, J.A.: Spontaneous action potentials due to channel fluctuations. Biophys. J. 71, 3013–3021 (1996)
Chu, P.H., Milton, J.G., Cowan, J.D.: Connectivity and the dynamics of integrate-and-fire neural networks. Int. J. Bif. Chaos 4, 237–243 (1994)
Chuquet, J., Hollander, L., Nimchinsky., E.A.: High-resolution in vivo imaging of the neurovascular unit during spreading depression. J. Neurosci. 27, 4036–4044 (2007)
Churchland, M.M., et al.: Stimulus onset quenches neural variability: a widespread cortical phenomenon. Nat. Neurosci. 13, 369–378 (2010)
Citri, A., Malenka, R.C.: Synaptic plasticity: multiple forms, functions, and mechanisms. Neuropsychopharmacol. 33, 18–41 (2008)
Cohen, A.H., Ermentrout, G.B., Kiermel, T., Kopell, N., Sigvardt, K.A., Williams, T.L.: Modeling of intersegmental coordination in the lamprey central pattern generator for motion. Trend. Neurosci. 15, 434–438 (1992)
Compte, A., Sanchez-Vives, M.V., McCormick, D.A., Wang, X.J.: Cellular and network mechanisms of slow oscillatory activity ( < 1 Hz) and wave propagations in a cortical network model. J. Neurophysiol. 89, 2707–2725 (2003)
Collingridge, G.L., Peineau, S., Howland, J.G., Wang, Y.T.: Long-term depression in the CNS. Nat. Rev. Neurosci. 11, 459–473 (2010)
Compte, A., Reig, R., Descalzo, V.F., Harvey, M.A., Puccini, G.D., Sanchez-Vives, M.V.: Spontaneous high-frequency (10–80 Hz) oscillations during up states in the cerebral cortex in vitro. J. Neurosci. 28, 13828–13844 (2008)
Connors, B.W., Amitai, Y.: Epilespy: Models, mechanisms and concepts. In: Schwartkroin, P.A. (ed.) Generation of Epileptiform Discharge by Local Circuits of Neocortex, pp. 388–423. Cambridge University Press, Cambridge (1993)
Connors, B.W., Long, M.A.: Electrical synapses in the mammalian brain. Annu. Rev. Neurosci. 27, 393–418 (2004)
Coombes, S.: The effects of ion pumps on the speed of traveling waves in the fire-diffuse-fire model of Ca2+ release. Bull. Math. Biol. 63, 1–20 (2001)
Coombes, S.: Dynamics of synaptically coupled integrate-and-fire-or-burst neuron. Phys. Rev. E 67, 041910 (2003)
Coombes, S., Bressloff, P.C.: Solitary waves in a model of dendritic cable with active spines. SIAM J. Appl. Math. 61, 432–453 (2000)
Coombes, S., Bressloff, P.C.: Saltatory waves in the spike-diffuse-spike model of active dendritic spines. Phys. Rev. Lett. 91, 028102 (2003)
Coombes, S., Bressloff, P.C. (eds.): Bursting: The Genesis of Rhythm in the Nervous System. World Scientific Press, London (2005)
Coombes, S., Laing, C.R.: Pulsating fronts in periodically modulated neural field models. Phys. Rev. E 83, 011912 (2011)
Coombes, S., Lord, G.J.: Intrinsic modulation of pulse-coupled integrate-and-fire neurons. Phys. Rev. E 56(5), 5809–5818 (1997)
Coombes, S., Owen, M.R.: Evans functions for integral neural field equations with Heaviside firing rate function. SIAM J. Appl. Dyn. Syst. 4, 574–600 (2004)
Coombes, S., Owen, M.R.: Bumps, breathers, and waves in a neural network with spike frequency adaptation. Phys. Rev. Lett. 94, 148102 (2005)
Coombes, S., Schmidt, H.: Neural fields with sigmoidal firing rates: approximate solutions. Discret. Contin. Dyn. Syst. Ser. A 28, 1369–1379 (2010)
Coombes, S., Timofeeva, Y.: Sparks and waves in a stochastic fire-diffue-fire model of Ca2+ release. Phy. Rev. E 68, 021915 (2003)
Coombes, S., Owen, M.R., Smith, G.D.: Mode-locking in a periodically forced integrate-and-fire-or-burst neuron model. Phys. Rev. E 64, 041914 (2001)
Coombes, S., Lord, G.J., Owen, M.R.: Waves and bumps in neuronal networks with axo-dendritic synaptic interactions. SIAM J. Appl. Dyn. Sys. 5, 552–574 (2003)
Coombes, S., Hinch, R., Timofeeva, Y.: Receptors, sparks and waves in a fire-diffuse-fire framework for Ca2+ release. Prog. Biophys. Mol. Biol. 85, 197–216 (2004)
Coombes, S., Timofeeva, Y., Svensson, C.M., Lord, G.J., Josic, K., Cox, S.J., Colbert, C.M.: Branching dendrites with resonant membrane: a “sum-over-trips” approach. Biol. Cybern. 97, 137–149 (2007)
Coombes, S., Venkov, N.A., Bojak, I., Liley, D.T.J., Laing, C.R.: Modeling electrocortical activity through improved local approximations of integral neural field equations. Phys. Rev. E 76, 051901 (2007)
Cornell-Bell, A.H., Finkbeiner, S.M., Cooper, M.S., Smith, S.J.: Glutamate induces calcium waves in cultured astrocytes: long-range glial signaling. Science 247, 470–473 (1990)
Dahlem, M.A., Chronicle, E.P.: A computational perspective on migraine aura. Prog. Neurobiol. 74, 351–361 (2004)
Dahlem, M.A., Muller, S.C.: Migraine aura dynamics after reverse retinotopic mapping of weak excitation waves in the primary visual cortex. Biol. Cybern. 88, 419–424 (2003)
Dale, N.: Dynamic ATP signalling and neural development. J. Physiol. 586, 2429–2436 (2008)
Deco, G., Jirsa, V., Robinson, P.A., Breakspear, M., Friston, K.: The dynamic brain: from spiking neurons to neural masses and cortical fields. PLoS Comput. Biol. 4, e1000092 (2008)
Deco, G., Jirsa, V.K., McIntosh., A.R.: Emerging concepts for the dynamical organization of resting-state activity in the brain. Nat. Rev. Neurosci. 12, 43–56 (2011)
Delaney, K.R., Galperin, A., Fee, M.S., Flores, J.A., Gervais, R., Tank, D.W., Kleinfeld, D.: Waves and stimulus-modulated dynamics in an oscillating olfactory network. Proc. Natl. Acad. Sci. USA 91, 669–673 (1994)
Delorme, V., Machacek, M., DerMardirossian, C., Anderson, K.L., Wittmann, T., Hanein, D., Waterman-Storer, C., Danuser, G., Bokoch, G.M.: Cofilin activity downstream of Pak1 regulates cell protrusion efficiency by organizing lamellipodium and lamella actin networks. Dev. Cell. 13, 646–662 (2007)
Derkach, V., Barria, A., Soderling, T.R.: Ca2+/calmodulin-kinase II enhances channel conductance of α-amino-3-hydroxyl-5-methyl-4-isoxazoleproprionate type glutamate receptors. Proc. Nat. Acad. Sci. (USA) 96, 3269–3274 (1999)
Destexhe, A., Contreras., D.: Neuronal computations with stochastic network states. Science 314, 85–90 (2006)
Destexhe, A., Sejnowski, T.J.: Thalamocortical Assemblies: How Ion Channels, Single Neurons and Large-Scale Networks Organize Sleep Oscillations. Oxford University Press, Oxford (2001)
Destexhe, A., Mainen, Z.F., Sejnowski, T.J.: Synthesis of models for excitable membranes synaptic transmission and neuromodulation using a common kinetic formalism. J. Comput. Neurosci. 1, 195–231 (1994)
Difato, F., Tsushima, H., Pesce, M., Benfenati, F., Blau, A., Chieregatti, E.: The formation of actin waves during regeneration after axonal lesion is enhanced by BDNF. Sci. Rep. 1 (183), 1–18 (2011)
Dockery, J.D., Keener, J.P.: Diffusive effects on dispersion in excitable media. SIAM J. Appl. Dyn. 49, 539–566 (1989)
Doubrovinski, K., Kruse, K.: Cytoskeletal waves in the absence of molecular motors. Europhys. Lett. 83, 18003 (2008)
van Drongelen, W., Lee, H.C., Stevens, R.L., Hereld, M.: Propagation of seizure-like activity in a model of neocortex. J. Clin. Neurophysiol. 24, 182–188 (2007)
Dyhrfjeld-Johnsen, J., Santhakumar, V., Morgan, R.J., Huerta, R., Tsimring, L., Soltesz, I.: Topological determinants of epileptogenesis in large-scale structural and functional models of the dentate gyrus derived from experimental data. J. Neurophysiol. 97, 1566–1587 (2007)
Dykman, M.I., Mori, E., Ross, J., Hunt, P.M.: Large fluctuations and optimal paths in chemical kinetics. J. Chem. Phys. A 100, 5735–5750 (1994)
Earnshaw, B.A., Bressloff, P.C.: Diffusion-activation model of CaMKII translocation waves in dendrites. J. Comput. Neurosci. 28, 77–89 (2010)
Ebert, U., van Saarloos, W.: Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Physica D 146, 1–99 (2000)
Edwards, J.R., Gibson, W.G.: A model for Ca2+ waves in networks of glial cells incorporating both intracellular and extracellular communication pathways. J. Theor. Biol. 263, 45–58 (2010)
Elgart, V., Kamenev, A.: Rare event statistics in reaction–diffusion systems. Phys. Rev. E 70, 041106 (2004)
Elmer, C.E., Vleck, E.S.V.: Analysis and computation of traveling waves solutions of bistable differential-difference equations. Nonlinearity 12, 771–798 (1999)
Ermentrout, G.B.: The behaviour of rings of coupled oscillators. Journal of Mathematical Biology 23, 55–74 (1985)
Ermentrout, G.B.: The analysis of synaptically generated travelling waves. J. Comput. Neurosci. 5, 191–208 (1998)
Ermentrout, G.B.: Neural networks as spatio-temporal pattern-forming systems. Rep. Prog. Phy. 61, 353–430 (1998)
Ermentrout, G.B., Cowan, J.: A mathematical theory of visual hallucination patterns. Bio. Cybern. 34, 137–150 (1979)
Ermentrout, G.B., Kleinfeld, D.: Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. Neuron 29, 33–44 (2001)
Ermentrout, G.B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators. SIAM J. Appl. Math. 15, 215–237 (1984)
Ermentrout, G.B., McLeod, J.B.: Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edin. A 123, 461–478 (1993)
Ermentrout, G.B., Terman, D.: Mathematical Foundations of Neuroscience. Springer, Berlin (2010)
Ermentrout, G.B., Jalics, J.Z., Rubin, J.E.: Stimulus-driven traveling solutions in continuum neuronal models with a general smooth firing rate function. SIAM J. Appl. Math. 70, 3039–3064 (2010)
Escudero, C., Kamanev, A.: Switching rates of multistep reactions. Phys. Rev. E 79, 041149 (2009)
Escudero, C., Rodriguez, J.A.: Persistence of instanton connections in chemical reactions with time–dependent rates. Phys. Rev. E 77, 011130 (2008)
Evans, J.: Nerve axon equations IV: The stable and unstable impulse. Ind. Univ. Math. J. 24, 1169–1190 (1975)
Evans, L.C., Sougandis, P.E.: A PDE approach to geometric optics for certain semilinear parabolic equations. Ind. Univ. Math. J. 38, 141–172 (1989)
Faisal, A.A., Selen, L.P.J., Wolpert, D.M.: Noise in the nervous system. Nat. Rev. Neurosci. 9, 292 (2008)
Falcke, M.: On the role of stochastic channel behavior in intracellular Ca2+ dynamics. Biophys. J. 84, 42–56 (2003)
Falcke, M.: Reading the patterns in living cells - the physics of Ca2+ signaling. Adv. Phys. 53, 255–440 (2004)
Falcke, M., Tsimiring, L., Levine, H.: Stochastic spreading of intracellular Ca2+ release. Phys. Rev. E 62, 2636–2643 (2000)
Fath, G.: Propagation failure of traveling waves in a discrete bistable medium. Physica D 116, 176–190 (1998)
Faugeras, O., Touboul, J., Cessac, B.: A constructive mean–field analysis of multi–population neural networks with random synaptic weights and stochastic inputs. Frontiers in Comp. Neurosci. 3, 1–28 (2009)
Faugeras, O., Veltz, R., Grimbert, F.: Persistent neural states: Stationary localized activity patterns in nonlinear continuous n-population, q-dimensional neural networks. Neural Comput. 21, 147–187 (2009)
Fellin, T., Pascual, O., Gobbo, S., Pozzan, T., Haydon, P.G., Carmignoto, G.: Neuronal synchrony mediated by astrocytic glutamate through activation of extrasynaptic NMDA receptors. Neuron 43, 729–743 (2004)
Fellin, T., Pozzan, T., Carmignoto., G.: Purinergic receptors mediate two distinct glutamate release pathways in hippocampal astrocytes. J. Biol. Chem. 281, 4274–4284 (2006)
Ferster, D., Miller, K.: Neural mechanisms of orientation selectivity in the visual cortex. Annual review of neuroscience 23(1), 441–471 (2000)
Fianco, T.A., McCarthy, K.D.: Intracellular astrocyte calcium waves in situ increase the frequency of spontaneous AMPA receptor currents in CA1 pyramidal cells. J. Neurosci. 24, 722–732 (2004)
Firth, S.I., Wang, C.T., Feller, M.B.: Retinal waves: mechanisms and function in visual system development. Cell Calcium 37, 425–432 (2005)
Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369 (1937)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1960)
Fivaz, M., Bandara, S., Inoue, T., Meyer, T.: Robust neuronal symmetry-breaking by Ras-triggered local positive feedback. Curr. Biol. 18, 44–50 (2008)
Flynn, K.C., Pak, C.W., Shaw, A.E., Bradke, F., Bamburg, J.R.: Growth cone-like waves transport actin and promote axonogenesis and neurite branching. Dev. Neurobiol. 69, 761–779 (2009)
Fogelson, A., Zucker, R.S.: Presynaptic calcium diffusion from various arrays of single channels. implications for transmitter release and synaptic facilitation. Biophys. J. 48, 1003–1017 (1985)
Folias, S.E., Bressloff, P.C.: Breathing pulses in an excitatory neural network. SIAM J. Appl. Dyn. Syst. 3, 378–407 (2004)
Folias, S.E., Bressloff, P.C.: Breathers in two–dimensional neural media. Phys. Rev. Lett. 95, 208107 (2005)
Folias, S.E., Bressloff, P.C.: Stimulus–locked traveling pulses and breathers in an excitatory neural network. SIAM J. Appl. Math. 65, 2067–2092 (2005)
Fox, R., Rasche, F.: Binocular rivalry and reciprocal inhibition. Attention Percept. Psychophys. 5(4), 215–217 (1969)
Fox, R.F., Lu, Y.N.: Emergent collective behavior in large numbers of globally coupled independent stochastic ion channels. Phys. Rev. E 49, 3421–3431 (1994)
Franci, A., Drion, G., Sepulchre, R.: An organizing center in a planar model of neuronal excitability. SIAM J. Appl. Dyn. Sys. 11, 1698–1722 (2013)
Franks, K.M., Sejnowski, T.J.: Complexity of calcium signaling in synaptic spines. Bioessays 24, 1130–1144 (2002)
Freidlin, M.I.: Limit theorems for large deviations and reaction-diffusion equations. Ann. Prob. 13, 639–675 (1985)
Freidlin, M.I.: Geometric optics approach to reaction–diffusion equations. SIAM J. Appl. Math. 46, 222–232 (1986)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, New York (1984)
Friedman, A., Craciun, G.: A model of intracellular transport of particles in an axon. J. Math. Biol. 51, 217–246 (2005)
Friedman, A., Hu, B.: Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems. Ind. Univ. Math. J. 56, 2133–2158 (2007)
Fusi, S., Drew, P.J., Abbott, L.F.: Cascade models of synaptically stored memories. Neuron 45, 599–611 (2005)
Garcia-Ojalvo, J., Sancho, J.M.: External fluctuations in a pattern-forming instability. Phys. Rev. E 53, 5680–5689 (1996)
Gardiner, C.W.: Handbook of Stochastic Methods, 4th edn. Springer, Berlin (2009)
Gardner-Medwin, A.R.: Analysis of potassium dynamics in mammalian brain tissue. J. Physiol. (Lond) 335, 393–462 (1983)
Gartner, J., Freidlin, M.I.: On the propagation of concentration waves in periodic and random media. Soviet Math. Dokl. 20, 1282–1286 (1979)
Gerstner, W., van Hemmen, J.L.: Coherence and incoherence in a globally coupled ensemble of pulse–emitting units. Phys. Rev. Lett. 71(3), 312–315 (1993)
Gerstner, W., Kistler, W.: Spiking Neuron Models. Cambridge University Press, Cambridge (2002)
Gerstner, W., van Hemmen, J.L., Cowan, J.D.: What matters in neuronal locking? Neural Comput. 8, 1689–1712 (1996)
Gilbert, C.D.: Horizontal integration and cortical dynamics. Neuron 9, 1–13 (1992)
Gilbert, C.D., Das, A., Ito, M., Kapadia, M., Westheimer, G.: Spatial integration and cortical dynamics. Proc. Nat. Acad. Sci. USA 93, 615–622 (1996)
Gilbert, C.D., Wiesel, T.N.: Clustered intrinsic connections in cat visual cortex. J. Neurosci. 3, 1116–1133 (1983)
Gilboa, G., Chen, R., Brenner, N.: History–dependent multiple-time-scale dynamics in a single-neuron model. J. Neurosci. 25, 6479–6489 (2005)
Ginzburg, I., Sompolinsky, H.: Theory of correlations in stochastic neural networks. Phys. Rev. E 50, 3171–3191 (1994)
Glass, L., Mackey, M.C.: From Clocks to Chaos. Princeton University Press, Princeton (1988)
Godfrey, K.B., Eglen, S.J.: Theoretical models of spontaneous activity generation and propagation in the developing retina. Mol. Biosyst. 5, 1527–1535 (2009)
Godfrey, K.B., Swindale, N.V.: Retinal wave behavior through activity-dependent refractory periods. PLoS Comp. Biol. 3 (11), 2408–2420 (2007)
Goel, P., Sneyd, J., Friedman, A.: Homogenization of the cell cytoplasm: the calcium bidomain equations. SIAM Nultiscal. Model Simul. 5, 1045–1062 (2006)
Goldberg, J.L.: How does an axon grow? Genes Dev. 17, 941–958 (2003)
Golding, N.L., Staff, N.P., Spruston, N.: Dendritic spikes as a mechanism for cooperative long-term potentiation. Nature 418, 326–331 (2002)
Goldwyn, J.H., Shea-Brown, E.: The what and where of adding channel noise to the Hodgkin-Huxley equations. PLoS Comp. Biol. 7 (11), e1002247 (2011)
Golomb, D., Amitai, Y.: Propagating neuronal discharges in neocortical slices: Computational and experimental study. J. Neurophysiol. 78, 1199–1211 (1997)
Golomb, D., Ermentrout, G.: Bistability in pulse propagation in networks of excitatory and inhibitory populations. Phys. Rev. Lett. 86, 4179–4182 (2001)
Golomb, D., Ermentrout, G.B.: Continuous and lurching traveling waves in neuronal networks with spatially-decaying connectivity and delay. Proc. Natl. Acad Sci. USA 96, 13480–13485 (1999)
Golomb, D., Rinzel, J.: Clustering in globally coupled inhibitory neurons. Physica D 72, 259–282 (1994)
Golomb, D., Wang, X.J., Rinzel, J.: Propagation of spindle waves in a thalamic slice model. J. Neurophysiol. 75, 750–769 (1996)
Gomez, T.M., Spitzer, N.C.: Regulation of growth cone behavior by calcium: New dynamics to earlier perspectives. J. Neurobiol. 44, 174–183 (1999)
Gomez, T.M., Zheng, J.Q.: The molecular basis for calcium-dependent axon pathfinding. Nat. Rev. Neurosci. 7, 115–125 (2006)
Gonzalez-Billault, C., Munoz-Llancao, P., Henriquez, D.R., Wojnacki, J., Conde, C., Caceres, A.: The role of small GTPases in neuronal morphogenesis and polarity. Cytoskeleton 69, 464–485 (2012)
Gordon, G.R., Baimoukhametova, D.V., Hewitt, S.A., Kosala, W.R.A., Rajapaksha, J.S., Fisher, T.E., Bains, J.S.: Noropinephrine triggers release of glial ATP to increase postsynaptic efficacy. Nat. Neurosci. 8, 1078–1086 (2005)
Gorji, A.: Spreading depression: A review of the clinical relevance. Brain Res. Revs. 38, 33–60 (2001)
Gourley, S.A.: Travelling front solutions of a nonlocal fisher equation. J. Math Biol. 41, 272–284 (2000)
Grannan, E.R., Kleinfeld, D., Sompolinsky, H.: Stimulus dependent synchronization of neuronal assemblies. Neural Comput. 5, 550–569 (1993)
Gray, W.G., Lee, P.C.Y.: On the theorems for local volume averaging of multiphase systems. Int. J. Multiphase Flow 3, 333–340 (1977)
Greenstein, J.L., Hinch, R., Winslow, R.L.: Mechanisms of excitation-contraction coupling in an integrative model of the cardiac ventricular myocyte. Biophys. J. 90, 77–91 (2006)
Grindrod, P.: Patterns and Waves: The Theory and Application of Reaction-Diffusion Equations. Clarendon Press, Oxford (1991)
Grinvald, A., Lieke, E.E., Frostig, R.D., Hildesheim, R.: Cortical point-spread function and long-range lateral interactions revealed by real-time optical imaging of macaque monkey primary visual cortex. J. Neurosci. 14, 2545–2568 (1994)
Groff, J.R., DeRemigio, H., Smith, G.D.: Stochastic methods in neuroscience, chap. 2. In: Markov Chain Models of Ion Channels and Calcium Release Sites, pp. 29–64. Oxford University Press, Oxford (2009)
Gu, X., Olson, E.C., Spitzer, N.: Spontaneous neuronal calcium spikes and waves during early differentiation.. J. Neurosci. 14, 6325–6335 (1994)
Gu, X., Spitzer, N.C.: Breaking the code: regulation of neuronal differentiation by spontaneous calcium transients. Dev. Neurosci. 19, 33–41 (1997)
Guan, C.B., Xu, H.T., Jin, M., Yuan, X.B., Poo, M.M.: Long-range Ca2+ signaling from growth cone to soma mediates reversal of neuronal migration induced by Slit-2. Cell 129, 385–395 (2007)
Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)
Guthrie, P.B., Knappenberger, J., Segal, M., Bennett, M.V.,, Charles, A.C., Kater, S.B.: ATP released from astrocytes mediates glial calcium waves. J. Neurosci. 19, 520–528 (1999)
Haas, B., Schipke, C.G., Peters, O., Sohl, G., Willecke, K., Kettenmann, H.: Activity-dependent ATP-waves in the mouse neocortex are independent from astrocytic calcium waves. Cereb. Cortex 16, 237–246 (2006)
Hagberg, A., Meron, E.: Pattern formation in non–gradient reaction–diffusion systems: the effects of front bifurcations. Nonlinearity 7, 805–835 (1994)
Hagberg, A., Meron, E., Rubinstein, I., Zaltzman., B.: Controlling domain patterns far from equilibrium. Phys. Rev. Lett. 76, 427–430 (1996)
Halassa, M.M., Fellin, T., Haydon, P.G.: The tripartite synapse: roles for gliotransmission in health and disease. Trend. Mol. Med. 13, 54–63 (2007)
Hall, A.: Rho GTPases and the actin cytoskeleton. Science 279, 509–514 (1998)
Han, F., Caporale, N., Dan, Y.: Reverberation of recent visual experience in spontaneous cortical waves. Neuron 60, 321–327 (2008)
Hanggi, P., Grabert, H., Talkner, P., Thomas, H.: Bistable systems: master equation versus Fokker–Planck modeling. Z. Physik B 28, 135 (1984)
Hanggi, P., Talkner, P., Borkovec, M.: Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–341 (1990)
Hansel, D., Mato, G., Meunier, C.: Synchrony in excitatory neural networks. Neural Comput. 7, 2307–2337 (1995)
Hansen, M., Boltano, S., Dirksen, E.R., Sanderson, M.J.: Intercellular calcium signaling induced by extracellular adenosine 5-triphosphate and mechanical stimulation in airway epithelial cells. J. Cell. Sci. 106, 995–1004 (1993)
Hanson, M.G., Landmesser, L.T.: Chracaterization of the circuits that generate spontaneous episodes of activity in the early embryonic mouse spinal cord. J. Neurosci. 23, 587–600 (2003)
Hanson, P.I., Meyer, T., Stryer, L., Schulman, H.: Dual role of calmodulin in autophosphorylation of multifunctional cam kinase may underlie decoding of calcium signal. Neuron 12, 943–956 (1994)
Hardy, J.: A hundred years of Alzheimer’s disease research. Neuron 52, 3–13 (2006)
Harper, J.D., Lansbury, P.T. Jr.: Models of amyloid seeding in Alzheimer’s disease and scrapie: mechanistic truths and physiological consequences of the time-dependent solubility of amyloid proteins. Annu. Rev. Biochem. 66, 385–407 (1997)
Harris, T.E.: The theory of branching processes. Dover, New York (1989)
Harris-White, M.E., Zanotti, S.A., Frautschy, S.A., Charles, A.C.: Spiral intercellular calcium waves in hippocampal slice cultures. J. Neurophysiol. 79, 1045–1052 (1998)
Hassinger, T.D., Guthrie, P.B., Atkinson, P.B., Bennett, M.V., Kater, S.B.: An extracellular signaling component in propagation of astrocytic calcium waves. Proc. Natl. Acad. Sci. USA 93, 13268–13273 (1996)
Hastings, S.P.: The existence of progressive wave solutions to the Hodgkin-Huxley equations. Arc. Rat. Mech. Anal. 60, 229–257 (1975)
Haydon, P.G.: Glia: listening and talking to the synapse. Nat. Rev. Neurosci. 2, 185–193 (2000)
Heitmann, S., Gong, P., Breakspear, M.: A computational role for bistability and traveling waves in motor cortex. Front. Comp. Neurosci. 6 (67), 1–15 (2012)
von Helmholtz, H.: Treatise on physiological optics. Dover, New York (1866)
Hennig, M.H., Adams, C., Willshaw, D., Sernagor, E.: Early-stage waves in the retinal network emerge close to a critical state transition between local and global functional connectivity. J. Neurosci. 29, 1077–1086 (2009)
Hill, S., Tononoi, G.: Modeling sleep and wakefulness in the thalamocortical system. J. Neurophysiol. 93, 1671–1698 (2005)
Hill, A.A.V., Masino, M.A., Calabrese, R.L.: Intersegmental coordination of rhythmic motor patterns. J. Neurophysiol. 90, 531–538 (2003)
Hillen, T., Othmer, H.: The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61, 751–775 (2000)
Hinch, E.J.: Perturbation Methods. Cambridge University Press, Cambridge (1991)
Hinch, R.: A mathematical analysis of the generation and termination of calcium sparks. Biophys. J. 86, 1293–1307 (2004)
Hinch, R., Chapman, S.J.: Exponentially slow transitions on a Markov chain: the frequency of calcium sparks. Eur. J. Appl. Math. 16, 427–446 (2005)
Hirsch, J.D., Gilbert, C.D.: Synaptic physiology of horizontal connections in the cat’s visual cortex. J. Physiol. Lond. 160, 106–154 (1991)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Hofer, T., Politi, A., Heinrich, R.: Intercellular Ca2+ wave propagation through gap-junctional Ca2+ diffusion: a theoretical study. Biophys. J. 80, 75–87 (2001)
Hofer, T., Venance, L., Giaume, C.: Control and plasticity of intercellular calcium waves in astrocytes: a modeling approach. J. Neurosci. 22, 4850–4859 (2002)
Holcman, D., Tsodyks, M.: The emergence of up and down states in cortical networks. PLoS Comp. Biol. 2(3), 0174–0181 (2006)
Holcman, D., Schuss, Z., Korkotian, E.: Calcium dynamics in dendritic spines and spine motility. Biophys. J. 87, 81–91 (2004)
Holmes, B.B., Diamond, M.I.: Cellular mechanisms of protein aggregate propagation. Curr. Opin. Neuro. 25, 721–726 (2012)
Holmes, E.E., Lewis, M.A., Banks, J.E., Veit, R.R.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, 17–29 (1994)
Holmes, W.R., Carlsson, A.E., Edelstein-Keshet, L.: Regimes of wave type patterning driven by refractory actin feedback: transition from static polarization to dynamic wave behaviour. Phys. Biol. 9, 046005 (2012)
Hoogland, T.M., Kuhn, B., Gobel, W., Huang, W., Nakai, J., Helmchen, F., Flint, J., Wang, S.S.: Radially expanding transglial calcium waves in the intact cerebellum. Proc. Natl. Acad. Sci. USA 106, 3496–3501 (2009)
Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two–state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)
Horton, A.C., Ehlers, M.D.: Neuronal polarity and trafficking. Neuron 40, 277–295 (2003)
Houweling, A.R., Bazhenov, M., Timofeev, I., Steriade, M., Sejnowski, T.J.: Homeostatic synaptic plasticity can explain post-traumatic epileptogenesis in chronically isolated neocortex. Cereb. Cortex 15, 834–845 (2005)
Hsu, D., Tang, A., Hsu, M., Beggs, J.M.: Simple spontaneously active Hebbian learning model: homeostatsis of activity and connectivity, and consequences for learining and epileptogenesis. Phys. Rev. E 76, 041909 (2007)
Huang, X., Troy, W.C., Yang, Q., Ma, H., Laing, C.R., Schiff, S.J., Wu, J.: Spiral waves in disinhibited mammalian neocortex. J. Neurosci. 24, 9897–9902 (2004)
Hubel, D.H., Wiesel, T.N.: Sequence regularity and geometry of orientation columns in the monkey striate cortex. J. Comp. Neurol. 158, 267–294 (1974)
Hubel, D.H., Wiesel, T.N.: Uniformity of monkey striate cortex: A parallel relationship between field size, scatter, and magnification factor. J. Comp. Neurol. 158, 295–306 (1974)
Hudson, A., Schulman, H.: Neuronal Ca2+/calmodulin-dependent protein kinase II: the role of structure and autoregulation in cellular function. Annu. Rev. Biochem. 71, 473–510 (2002)
Hutt, A., Bestehorn, M., Wennekers, T.: Pattern formation in intracortical neuronal fields. Network 14, 351–368 (2003)
Hutt, A., Longtin, A., Schimansky-Geier, L.: Additive noise-induces Turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation. Physica D 237, 755–773 (2008)
Iacobas, D.A., Suadicani, S.O., Spray, D.C., Scemes, E.: A stochastic two-dimensional model of intercellular Ca2+ wave spread in glia. Biophys. J. 90, 24–41 (2006)
Idan, S., Rinzel, J., Gordon, M.S. (eds.): The Theoretical Foundations of Dendritic Function: Selected Papers of Wilfrid Rall with Commentaries. MIT Press, Cambridge (1995)
Invernizzi, G., Papaleo, E., Sabate, R., Ventura, S.: Protein aggregation: Mechanisms and functional consequences. Int. J. Biochem. Cell Biol. 44, 1541–1554 (2012)
Izhikevich, E.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2006)
Jacobs, K.: Stochastic Processes for Physicists. Cambridge University Press, Cambridge (2010)
Jaffe, D.B., Brown, T.H.: Metabotropic glutamate receptor activation induces calcium waves within hippocampal dendrites. J. Neurophysiol. 72, 471–474 (1994)
Jahr, C., Stevens, C.: A quantitative description of NMDA receptor-channel kinetic behavior. J. Neurosci. 10, 1830–1837 (1990)
Jilkine, A., Maree, A.F.M., Edelstein-Keshet, L.: Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk. Bull. Math. Biol. 68, 1169–1211 (2007)
Jin, D.Z., Dragoi, V., Sur, M., Seung, H.S.: Tilt aftereffect and adaptation-induced changes in orientation tuning and visual cortex. J. Neurosci. 94, 4038–4050 (2005)
Jirsa, V.K., Haken, H.: A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics. Physica D 99, 503–526 (1997)
Jirsa, V.K., Kelso, J.A.S.: Spatiotemporal pattern formation in neural systems with heterogeneous topologies. Phys. Rev. E 62, 8462–8465 (2000)
van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)
Kang, M., Othmer, H.G.: Spatiotemporal characteristics of calcium dynamics in astrocytes. Chaos 19, 037116 (2009)
Kang, N., Xu, J., Xu, Q., Nedergaard, M., Kang, J.S.: Astrocytic glutamate release-induced transient depolarization and epileptiform discharges in hippocampal CA1 pyramidal neuron. J. Neurophysiol. 94, 4121–4130 (2005)
Kang, M., Heeger, D.J., Blake, R.: Periodic perturbations producing phase-locked fluctuations in visual perception. J. Vision 9(2):8, 1–12 (2009)
Kang, M., Lee, S.H., Kim, J., Heeger, D.J., Blake, R.: Modulation of spatiotemporal dynamics of binocular rivalry by collinear facilitation and pattern-dependent adaptation. J. Vision 10(11):3, 1–15 (2010)
Kapitula, T., Kutz, N., Sandstede, B.: The Evans function for nonlocal equations. Ind. Univ. Math. J. 53, 1095–1126 (2004)
Katz, L.C., Shatz, C.J.: Synaptic activity and the construction of cortical circuits. Science 274, 1133–1138 (1996)
Keener, J.P.: Waves in excitable media. SIAM J. Appl. Math. 39, 528–548 (1981)
Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math 31, 269–276 (1987)
Keener, J.P.: Homogenization and propagation in the bistable equation. Physica D 136, 1–17 (2000)
Keener, J.P.: Propagation of waves in an excitable medium with discrete release sites. SIAM J. Appl. Math. 61, 317–314 (2000)
Keener, J.P.: Stochastic calcium oscillations. Math. Med. Biol. 23, 1–25 (2006)
Keener, J.P., Newby, J.M.: Perturbation analysis of spontaneous action potential initiation by stochastic ion channels. Phy. Rev. E 84, 011918 (2011)
Keener, J., Sneyd, J.: Mathematical Physiology I: Cellular Physiology, 2nd edn. Springer, New York (2009)
Keener, J.P., Hoppensteadt, F.C., Rinzel, J.: Integrate–and–fire models of nerve membrane response to oscillatory input. SIAM J. Appl. Math. 41(3), 503–517 (1981)
Keizer, J., Smith, G.D.: Spark-to-wave transition: saltatory transmission of calcium waves in cardiac myocytes. Biophys. Chem. 72, 87–100 (1998)
Keizer, J., Smith, G.D., Ponce-Dawson, S., Pearson, J.E.: Saltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J. 75, 595–600 (1998)
Khazipov, R., Luhmann, H.J.: Early patterns of electrical activity in the developing cerebral cortex of humans and rodents. Trend. Neurosci. 29, 414–419 (2006)
Khovanov, I.A., Polovinkin, A.V., Luchinsky, D.G., McClintock, P.V.E.: Noise-induced escape in an excitable system. Phys. Rev. E 87, 032116 (2013)
Kilpatrick, Z.P., Bressloff, P.C.: Binocular rivalry in a competitive neural network with synaptic depression. SIAM J. Appl. Dyn. Syst. 9, 1303–1347 (2010)
Kilpatrick, Z.P., Bressloff, P.C.: Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network. Physica D 239, 547–560 (2010)
Kilpatrick, Z.P., Bressloff, P.C.: Spatially structured oscillations in a two-dimensional neuronal network with synaptic depression. J. Comp. Neurosci. 28, 193–209 (2010)
Kilpatrick, Z.P., Bressloff, P.C.: Stability of bumps in piecewise smooth neural fields with nonlinear adaptation. Physica D 239, 1048–1060 (2010)
Kilpatrick, Z.P., Folias, S.E., Bressloff, P.C.: Traveling pulses and wave propagation failure in inhomogeneous neural media. SIAM J. Appl. Dyn. Syst. 7, 161–185 (2008)
Kim, H.G., Connors, B.W.: Apical dendrites of the neocortex: correlation between sodium and calcium-dependent spiking and pyramidal cell morphology. J. Neurosci. 13, 5301–5311 (1993)
Kim, U., Bal, T., McCormick, D.A.: Spindle waves are propagating synchronized oscillations in the ferret LGN in vitro. J. Neurophysiol. 74, 1301–1323 (1995)
Kinezaki, N., Kawasaki, K., Takasu, F., Shigesada, N.: Modeling biological invasions into periodically fragmented environments. Theor. Popul. Biol. 64, 291–302 (2003)
Kishimoto, K., Amari, S.: Existence and stability of local excitations in homogeneous neural fields. J. Math. Biol. 7, 303–318 (1979)
Kistler, W.M., Seitz, R., van Hemmen, J.L.: Modeling collective excitations in cortical tissue. Physica D 114, 273–295 (1998)
Kleinfeld, D., Delaney, K.R., Fee, M.S., Flores, J.A., Tank, D.W., Gelperin, A.: Dynamics of propagating waves in the olfactory network of a terrestrial mollusc: an electrical and optical study. J. Neurophysiol. 72, 1402–1419 (1994)
Knessl, C., Mangel, M., Matkowsky, B.J., Schuss, Z., Tier, C.: An asymptotic solution of the Kramers-Moyal equation and first passage time problems for Markov jump processes. Phys. Rev. E 29, 3359–3369 (1984)
Knessl, C., Matkowsky, B.J., Schuss, Z., Tier, C.: An asymptotic theory of large deviations for Markov jump processes. SIAM J. Appl. Math. 46, 1006–1028 (1985)
Koch, C.: Cable theory in neurons with active, linearized membranes. Biol. Cybern. 50, 15–33 (1984)
Koch, C.: Biophysics of Computation. Oxford University Press, New York (1999)
Koch, C., Zador, A.M.: The function of dendritic dendritic spines. devices subserving biochemical rather than electrical compartmentalization. J. Neurosci. 13, 413–422 (1993)
Koch, C., Poggio, T., Torre, V.: Nonlinear interactions in a dendritic tree: localization, timing and role in information processing. Proc. Nat. Acad. Sci. (USA) 80, 2799–2802 (1983)
Kolmogorff, A., Petrovsky, I., Piscounoff, N.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscow Univ. Bull. Math. 1, 1–25 (1937)
Konur, S., Rabinowitz, D., Fenstermaker, V.L., Yuste, R.: Systematic regulation of spine sizes and densities in pyramidal neurons. J. Neurobiol. 56, 95–112 (2003)
Kopell, N., Ermentrout, G.B.: Symmetry and phase-locking in chains of weakly coupled oscillators. Comm. Pure Appl. Math. 39, 623–660 (1986)
Kramer, M.A., Cash, S.S.: Epilepsy as a disorder of cortical network organization. Neuroscientist 18, 360–372 (2012)
von Krosigk, M., Bal, T., McCormick, D.: Cellular mechanisms of a synchronized oscillation in the thalamus. Science 261, 361–364 (1993)
Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, New-York (1984)
Kurtz, T.G.: Limit theorems for a sequence of jump Markov processes approximating ordinary differential equations. J. Appl. Prob. 8, 344–356 (1971)
Kurtz, T.G.: Limit theorems and diffusion approximations for density dependent Markov chains. Math. Prog. Stud. 5, 67 (1976)
Lago-Fernandez, L.F., Huerta, R., Corbacho, F., Siguenza, J.A.: Fast response and temporal coherent oscillations in small-world networks. Phys. Rev. Lett. 84, 2758–2761 (2000)
Laing, C.R.: Spiral waves in nonlocal equations. SIAM J. Appl. Dyn. 4, 588–606 (2005)
Laing, C.R., Chow, C.C.: A spiking neuron model for binocular rivalry. J. Comput. Neurosci. 12, 39–53 (2002)
Laing, C.R., Troy, W.C.: PDE methods for nonlocal models. SIAM J. Appl. Dyn. Syst. 2, 487–516 (2003)
Lam, Y.W., Cohen, L.B., Wachowiak, M., Zochowski, M.R.: Odors elicit three different oscillations in the turtle olfactory bulb. J. Neurosci. 20, 749–762 (2000)
Lance, J.W.: Current concepts of migraine pathogenesis. Neurology 43, 11–15 (1993)
Largo, C., Cuevas, P., Somjen, G., Rio, R.M., Herreras, O.: The effect of depressing glial function in rat brain in situ on ion homeostasis, synaptic transmission, and neuronal survival. J. Neurosci. 16, 1219–1229 (1996)
Largo, C., Ibarz, J.M., Herreras, O.: Effects of the gliotoxin fluorocitrate on spreading depression and glial membrane potential in rat brain in situ. J. Neurophysiol. 78, 295–307 (1997)
Larkum, M.E., Zhu, J.J., Sakmann, B.: A new cellular mechanism for coupling inputs arriving at different cortical layers. Nature 398, 338–341 (1999)
Larkum, M.E., Nevian, T., Sandler, M., Polsky, A., Schiller, J.: Synaptic integration in tuft dendrites of layer 5 pyramidal neurons: a new unifying principle. Science 325, 756–760 (2009)
Lashley, K.: Patterns of cerebral integration indicated by scotomas of migraine. Arch. Neurol. Pyschiatry 46, 331–339 (1941)
Lauritsen, K.B., Zapperi, S., Stanley, H.E.: Self-organized branching processes: avalanche models with dissipation. Phys. Rev. E 54, 2483–2488 (1996)
Leao, A.A.P.: Spreading depression of activity in the cerebral cortex. J. Neurophysiol. 10, 409–414 (1944)
Leao, A.A.P., Morrisson, R.S.: Propagation of spreading cortical depression. J. Neurophysiol. 8, 33–45 (1945)
Lechleiter, J., Girard, S., Peralta, E., Clapham, D.: Spiral calcium wave propagation and annihilation in xenopus laevis oocytes. Science 252, 123–126 (1991)
Leclerc, C., Neant, I., Moreau, M.: The calcium: an early signal that initiates the formation of the nervous system during early embryogenesis. Front. Mol. Neurosci. 5 (64), 1–12 (2012)
Lee, S.H., Blake, R., Heeger, D.J.: Traveling waves of activity in primary visual cortex during binocular rivalry. Nat. Neurosci. 8, 22–23 (2005)
Lee, S.J., Escobedo-Lozoya, Y., Szatmari, E.M., Yasuda, R.: Activation of CaMKII in single dendritic spines during long-term potentiation. Nature 458, 299–304 (2009)
Lee, S.J., Lim, H.S., Masliah, E., Lee, H.J.: Protein aggregate spreading in neurodegerative diseases: problems and perspectives. Neuro. Res. 70, 339–348 (2011)
Lehky, S.R.: Binocular rivalry is not chaotic. Proc. Roy. Soc. Lond. Biol Sci. 259, 71–76 (1995)
Lehky, S.R.: No binocular rivalry in the LGN of alert macaque monkeys. Vis. Res. 36, 1225–1234 (1996)
Lehner, F.K.: On the validity of Fick’s law for transient diffusion through a porous medium. Chem. Eng. Sci. 34, 821–825 (1977)
Lemieux, L., Daunizeau, J., Walker, M.C.: Concepts of connectivity and human epileptic activity. Front. Syst. Neurosci. 5 (12), 1–13 (2011)
Leopold, D.A., Logothetis, N.K.: Activity changes in early visual cortex reflect monkeys’ percepts during binocular rivalry. Nature 379, 549–553 (1996)
LeVay, S., Nelson, S.B.: Columnar organization of the visual cortex. In: Leventhal, A.G. (ed.) The Neural Basis of Visual Function, pp. 266–315. CRC Press, Boca Raton (1991)
Levelt, W.J.M.: On Binocular Rivalry. Institute for Perception RVO–TNO, Soesterberg, The Netherlands (1965)
Leybaert, L., Paemeleire, K., Strahonja, A., Sanderson, M.J.: Inositol-trisphosphate-dependent intercellular calcium signaling in and between astrocytes and endothelial cells. Glia 24, 398–407 (1998)
Leybaert, L., Sanderson, M.J.: Intercellular Ca2+ waves: mechanisms and function. Physiol. Rev. 92, 1359–1392 (2012)
Li, Y., Rinzel, J.: Equations for InsP3 receptor-mediated calcium oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J. Theor. Biol. 166, 461–473 (1994)
Liley, D.J.T., Cadusch, P.J., Dafilis, M.P.: A spatially continuous mean field theory of electrocortical activity. Network 13, 67–113 (2002)
Linder, B., Garcia-Ojalvo, J., Neiman, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)
Lisman, J.E.: Relating hippocampal circuitry to function: recall of memory sequences by reciprocal dentate-CA3 interactions. Neuron 22, 233–242 (1999)
Lisman, J.: Long-term potentiation: outstanding questions and attempted synthesis. Phil. Trans. R. Soc. Lond. B 358, 829–842 (2003)
Lisman, J.E., Schulman, H., Cline, H.: The molecular basis of CaMKII function in synaptic and behavioral memory. Nat. Rev. Neurosci. 3, 175–190 (2002)
Litwin-Kumar, A., Doiron, B.: Slow dynamics and high variability in balanced cortical networks with clustered connections. Nat. Neurosci. 15, 1498–1505 (2012)
Liu, Q.S., Xu, Q., Kang, G.A.J., Nedergaard, M.: Astrocyte-mediated activation of neuronal kainate receptors. Proc. Natl. Acad Sci. USA 101, 3172–3177 (2004)
Llano, I., González, J., Caputo, C., Lai, F.A., Blayney, L.M., Tan, Y.P., Marty, A.: Presynaptic calcium stores underlie large-amplitude miniature IPSCs and spontaneous calcium transients. Nat. Neurosci. 3, 1256–1265 (2000)
Logothetis, N.K., Leopold, D.A., Sheinberg, D.L.: What is rivalling during binocular rivalry? Nature 380, 621–624 (1996)
London, M., Roth, A., Beeren, L., Hausser, M., Latham, P.E.: Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature 466, 123–127 (2010)
Loxley, P., Robinson, P.: Soliton model of competitive neural dynamics during binocular rivalry. Phys. Rev. Lett. 102(25), 258701 (2009)
Lubenov, E.V., Siapas, A.G.: Hippocampal theta oscillations are travelling waves. Nature 459, 534–539 (2009)
Luo, L.: Actin cytoskeleton regulation in neuronalmorphogenesis and structural plasticity. Ann. Rev. Cell Dev. Biol. 18, 601–635 (2002)
Ly, C., Tranchina, D.: Critical analysis of a dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Comput. 19, 2032–2092 (2007)
Madison, D.V., Nicoll, R.A.: Control of the repetitive discharge of rat CA1 pyramidal neurones in vitro. J. Physiol. 354, 319–331 (1984)
Magee, J.C., Johnston, D.: Characterization of single voltage-gated Na+ and Ca2+ channels in apical dendrites of rat CA1 pyramidal neurons. J. Physiol. 487, 67–90 (1995)
Maier, R.S., Stein, D.L.: Limiting exit location distribution in the stochastic exit problem. SIAM J. Appl. Math 57, 752–790 (1997)
Maier, N., Nimmrich, V., Draguhn, A.: Cellular and network mechanisms underlying spontaneous sharp wave–ripple complexes in mouse hippocampal slices. J. Physiol. 550, 873–887 (2003)
Major, G., Polsky, A., Denk, W., Schiller, J., Tank, D.W.: Spatiotemporally graded NMDA spike/plateau potentials in basal dendrites of neocortical pyramidal neurons. J. Neurophysiol. 99, 2584–2601 (2008)
Malach, R., Harel, Y.A.M., Grinvald, A.: Relationship between intrinsic connections and functional architecture revealed by optical imaging and in vivo targeted biocytin injections in primate striate cortex. Proc. Natl. Acad. Sci. USA 90, 0469–10473 (1993)
Malenka, R.C., Bear, M.F.: LTP and LTD: an embarrassment of riches. Neuron 44, 5–21 (2004)
Manita, S., Ross, W.N.: Synaptic activation and membrane potential changes modulate the frequency of spontaneous elementary Ca2+ releases events in the dendrites of pyramidal neurons. J. Neurosci. 29, 7833–7845 (2009)
Marder, E., Calabrese, R.L.: Principles of rhythmic motor pattern generation. Physiol. Rev. 76, 687–717 (1996)
Markram, H., Tsodyks, M.: Redistribution of synaptic efficacy between neocortical pyramidal neurons. Nature 382, 807–810 (1996)
Martins-Ferreira, H., Ribeiro, I.J.: Biphasic effects of gap junctional uncoupling agents on the propagation of retinal spreading depression. Braz. J. Med. Biol. Res. 28, 991–994 (1995)
Martins-Ferreira, H., Nedergaard, M., Nicholson, C.: Perspectives on spreading depression. Brain Res. Rev. 32, 215–234 (2000)
Masel, J., Jansen, V.A.A., Nowak, M.A.: Quantifying the kinetic parameters of prion replication. Biophys. Chem. 77, 139–15 (1999)
Masuda, N., Aihara, K.: Global and local synchrony of coupled neurons in small-world networks. Biol. Cybern. 90, 302–309 (2004)
Mataga, N., Mizuguchi, Y., Hensch, T.K.: Experience-dependent pruning of dendritic spines in visual cortex by tissue plasminogen activator. Neuron 44, 1031–1041 (2004)
Matthaus, F.: Diffusion versus network models as descriptions for the spread of prion diseasess in the brain. J. Theor. Biol. 240, 104–113 (2006)
Mattia, M., Guidice, P.D.: Population dynamics of interacting spiking neurons. Phys. Rev. E 66, 051917 (2002)
Matus, A.: Actin plasticity in dendritic spines. Science 290, 754–758 (2000)
Mayer, M.L., Westbrook, G.L., Guthrie, P.B.: Voltage-dependent block by Mg2+ of NMDA responses in spinal cord neurone. Nature 309, 261–263 (1984)
McCormick, D.A., Contreras, D.: On the cellular and network bases of epileptic seizures. Annu. Rev. Physiol. 63, 815–846 (2001)
McLaughlin, T., Torborg, C.L., Feller, M.B., O’Leary, D.D.M.: Retinotopic map refinement requires spontaneous retinal waves during a brief critical period of development. Neuron 40, 1147–1160 (2003)
Meinhardt, H.: Orientation of chemotactic cells and growth cones: models and mechanisms. J. Cell. Sci. 112, 2867–2874 (1999)
Meisel, C., Storch, A., Hallmeyer-Elgner, S., Bullmore, E., Gross, T.: Failure of adaptive self-organized criticality during epileptic seizure attacks. PLoS Comp. Biol. 8, e1002312 (2012)
Melamed, O., Barak, O., Silberberg, G., Markram, H., Tsodyks, M.: Slow oscillations in neural networks with facilitating synapses. J. Comp. Neurosci. 25, 308–316 (2008)
Menchon, S.A., Gartner, A., Roman, P., Dotti, C.G.: Neuronal (bi)polarity as a self-organized process enhanced by growing membrane. PLoS One 6(9), e24190 (2011)
Mendez, V., Fort, J., Rotstein, H.G., Fedotov, S.: Speed of reaction-diffusion fronts in spatially heterogeneous media. Phys. Rev. E 68, 041105 (2003)
Mendez, V., Fedotov, S., Horsthemke, W.: Reaction-Transport Systems. Springer, Berlin (2010)
Meng, X., Huguet, G., Rinzel, J.: Type III excitability, slope sensitivity and coincidence detection. Disc. Cont. Dyn. Systs, 32, 2729–2757 (2012)
Meunier, C., d’Incamps, B.L.: Extending cable theory to heterogeneous dendrites. Neural Comput. 20, 1732–1775 (2008)
Meyer, C., van Vreeswijk, C.: Temporal correlations in stochastic networks of spiking neurons. Neural Comput. 14, 369–404 (2002)
Mikhailov, A.S., Davydov, V.A., Zykov, V.S.: Complex dynamics of spiral waves and motion of curves. Physica D 70, 1–39 (1994)
Miller, J.P., Rall, W., Rinzel, J.: Synaptic amplification by active membrane in dendritic spines. Brain Res. 325, 325–330 (1985)
Miller, L.D., Golarai, J.J.P., Connor, J.: Ca2+ release from intracellular stores induced by afferent stimulation of CA3 pyramidal neurons in hippocampal slices. J. Neurophysiol. 76, 554–562 (1996)
Millhauser, G.L., Salpeter, E.E., Oswald, R.E.: Diffusion model of ion–channel gating and the origin of power–law distributions from single–channel recordings. Proc. Natl. Acad. Sci. USA 85, 1503–1507 (1988)
Milton, J., Jung, P.: Epilepsy as a Dynamic Disease. Springer, Berlin (2003)
Mirollo, R.E., Strogatz, S.H.: Synchronisation of pulse–coupled biological oscillators. SIAM J. Appl. Math. 50(6), 1645–1662 (1990)
Mitchison, T., Kirschner., M.: Cytoskeletal dynamics and nerve growth. Neuron 1, 761–772 (1988)
Miura, R., Huang, H., Wylie, J.: Cortical spreading depression: An enigma. Eur. Phys. J. Special Topics 147, 287–302 (2007)
Mogilner, A., Oster, G.: Cell motility driven by actin polymerization. Biophys. J. 71, 3030–3045 (1996)
Moldakarimov, S., Rollenhagen, J.E., Olson, C.R., Chow, C.C.: Competitive dynamics in cortical responses to visual stimuli. J. Neurophysiol. 94, 3388–3396 (2005)
Moreno-Bote, R., Rinzel, J., Rubin, N.: Noise-induced alternations in an attractor network model of perceptual bistability. J. Neurophysiol. 98, 1125–1139 (2007)
Morgan, R.J., Soltesz, I.: Nonrandom connectivity of the epileptic dentate gyrus predicts a major role for neuronal hubs in seizures. Proc. Natl. Acad. Sci. USA 105, 6179–6184 (2008)
Mori, Y., Jilkine, A., Edelstein-Keshet, L.: Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys. J. 94, 3684–3697 (2008)
Mori, Y., Jilkine, A., Edelstein-Keshet, L.: Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization. SIAM J. Appl. Math. 71, 1401–1427 (2011)
Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. J. Biophys. 35, 193–213 (1981)
Mulleners, W.M., Chronicle, E.P., Palmer, J.E., Koehler, P.J., Vredeveld, J.W.: Visual cortex excitability in migraine with and without aura. Headache 41, 565–572 (2001)
Muller, L., Destexhe, A.: Propagating waves in thalamus, cortex and the thalamocortical system: experiments and models. J. Physiol. (Paris) 106, 222–238 (2012)
Mullins, O.J., Hackett, J.T., Buchanan, J.T., Friesen, W.O.: Neuronal control of swimming behavior: Comparison of vertebrate and invertebrate model systems. Prog. Neurobiol. 93, 244–269 (2011)
Murray, J.D.: Mathematical Biology, vol. I, II. Springer, Berlin (2002)
Naeh, T., Klosek, M.M., Matkowsky, B.J., Schuss, Z.: A direct approach to the exit problem. SIAM J. Appl. Math. 50, 595–627 (1990)
Nagumo, J., Armoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. Inst. Radio Eng. 50, 2061–2070 (1964)
Nakamura, T., Barbara, J.G., Nakamura, K., Ross, W.N.: Synergistic release of Ca2+ from IP3 sensitive stores evoked by synaptic activation of mGluRs paired with backpropagating action potentials. Neuron 24, 727–737 (1999)
Nedergaard, M., Cooper, A.J., Goldman, S.A.: Gap junctions are required for the propagation of spreading depression. J. Neurobiol. 28, 433–444 (1995)
Nedergaard, M., Ransom, B., Goldman, S.A.: New roles for astrocytes: redefining the functional architecture of the brain. Trend. Neurosci. 26, 523–530 (2003)
Netoff, T.I., Schiff, S.J.: Decreased neuronal synchronization during experimental seizures. J. Neurosci. 22, 7297–7307 (2002)
Netoff, T.I., Clewely, R., Arno, S., Keck, T., White, J.A.: Epilepsy in small-world networks. J. Neurosci. 24, 8075–8083 (2004)
Neukirchen, D., Bradke, F.: Neuronal polarization and the cytoskeleton. Sem. Cell. Dev. Biol. 22, 825–833 (2011)
Newby, J.M., Bressloff, P.C.: Local synaptic signalling enhances the stochastic transport of motor-driven cargo in neurons. Phys. Biol. 7, 036004 (2010)
Newby, J.M., Bressloff, P.C.: Quasi-steady state reduction of molecular-based models of directed intermittent search. Bull. Math. Biol. 72, 1840–1866 (2010)
Newby, J., Chapman, J.: Metastable behavior in Markov processes with internal states: breakdown of model reduction techniques. J. Math. Biol. published online, DOI: 10.1007/s00285-013-0723-1 (2013)
Newby, J.M., Bressloff, P.C., Keeener, J.P.: The effect of potassium channels on spontaneous action potential initiation by stochastic ion channels. Phys. Rev. Lett. 111, 128–101 (2013)
Newman, E.A.: Propagation of intercellular calcium waves in retinal astrocytes and Muller cells. J. Neurosci. 21, 2215–2223 (2001)
Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002)
Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)
Newman, E.A., Zahs, K.R.: Calcium waves in retinal glial cells. Science 275, 844–847 (1997)
Nicholson, C.: Diffusion and related transport mechanisms in brain tissue. Rep. Prog. Phys. 64, 815–884 (2001)
Nimchinsky, E.A., Sabatini, B.L., Svoboda, K.: Structure and function of dendritic spines. Ann. Rev. Physiol. 64, 313–353 (2002)
Nita, D.A., Cisse’, Y., Timofeev, I., Steriade, M.: Increased propensity to seizures after chronic cortical deafferentation in vivo. J. Neurophysiol. 95, 902–913 (2006)
Noble, J.V.: Geographic and temporal development of plagues. Nature 250, 726–729 (1974)
Novikov, E.A.: Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290 (1965)
Nowak, L., Bregestovski, P., Ascher, P., Herbet, A., Prochiantz, A.: Magnesium gates glutamate-activated channels in mouse central neurones. Nature 307, 462–465 (1984)
Nunez, P.I.: Neocortical dynamics and human EEG rhythms. Oxford University Press, New York (1995)
Nykamp, D., Tranchina, D.: A population density method that facilitates large–scale modeling of neural networks: analysis and application to orientation tuning. J. Comp. Neurosci. 8, 19–50 (2000)
Obermayer, K., Blasdel, G.G.: Geometry of orientation and ocular dominance columns in monkey striate cortex. J. Neurosci. 13, 4114–4129 (1993)
O’Donnell, M., Chance, R.K., Bashaw, G.J.: Axon growth and guidance: Receptor regulation and signal transduction. Ann. Rev. Neurosci. 32, 383–412 (2009)
O’Donovan, M.J.: The origin of spontaneous activity in developing networks of the vertebrate nervous system. Curr. Opin. Neuro. 9, 94–104 (1999)
O’Donovan, M.J., Chub, N., Wenner, P.: Mechanisms of spontaneous activity in the developing retina. J. Neurobiol. 37, 131–145 (1998)
Ohira, T., Cowan, J.D.: Stochastic neurodynamics and the system size expansion. In: Ellacott, S., Anderson, I.J. (eds.) Proceedings of the First International Conference on Mathematics of Neural Networks, pp. 290–294. Academic, New York (1997)
O’Keefe, J., Dostrovsky, J.: The hippocampus as a spatial map. preliminary evidence from unit activity in the freely moving rat. Brain Res. 34, 171–174 (1971)
O’Keefe, J., Recce, M.L.: Phase relationship between hippocampal place units and the EEG theta rhythm. Hippocampus 3, 317–330 (1993)
Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, New York (2007)
Omurtag, A., Knight, B.W., Sirovich, L.: On the simulation of large populations of neurons. J. Comput. Neurosci. 8, 51–63 (2000)
Onsum, M., Rao, C.V.: A mathematical model for neutrophil gradient sensing and polarization. PLoS Comput. Biol. 3, e36 (2007)
Ooyen, A.V., Graham, B.P., Ramakers, G.J.A.: Competition for tubulin between growing neurites during development. Neurocomputing 38, 73–78 (2001)
Oray, S., Majewska, A., Sur, M.: Dendritic spine dynamics are regulated by monocular deprivation and extracellular matrix degredation. Neuron 44, 1021–1030 (2004)
Othmer, H., Dunbar, S., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)
Otsuji, M., Ishihara, S., Co, C., Kaibuchi, K., Mochizuki, A., Kuroda, S.: A mass conserved reaction-diffusion system captures properties of cell polarity. PLoS Comput. Biol. 3, e108 (2007)
Pak, C.W., Flynn, K.C., Bamburg, J.R.: Actin-binding proteins take the reins in growth cones. Nat. Rev. Neurosci. 9, 136–147 (2008)
Pakdaman, K., Thieullen, M., Wainrib, G.: Fluid limit theorems for stochastic hybrid systems with application to neuron models. J. Appl. Prob. 24, 1 (2010)
Panatier, A., Theodosis, D.T., Mothet, J.P., Touquet, B., Pollegioni, L., Poulain, D.A., Oliet, S.H.: Glia-derived D-serine controls NMDA receptor activity and synaptic memory. Cell 125, 775–784 (2006)
Panja, D.: Effects of fluctuations on propagating fronts. Phys. Rep. 393, 87–174 (2004)
Papanicolaou, G.C.: Asymptotic analysis of transport processes. Bull. Amer. Math. Soc. 81, 330–392 (1975)
Parga, N., Abbott, L.F.: Network model of spontaneous activity exhibiting synchronous transitions between up and down states. Front. Neurosci. 1, 57–66 (2007)
Parker, I., Ivorra, I.: Localized all-or-none calcium liberation by inositol triphosphate. Science 250, 977–979 (1990)
Parker, I., Yao, Y.: Regenerative release of calcium from functionally discrete subcellular stores by inositol triphosphate. Proc. Roy. Soc. Lond. B 246, 269–274 (1991)
Parpura, V., Basarsky, T.A., Liu, F., Jeftinija, K., Jeftinija, S., Haydon, P.G.: Glutamate-mediated astrocyte-neuron signalling. Nature 369, 744–747 (1994)
Parri, H.R., Gould, T.M., Crunelli, V.: Spontaneous astrocytic Ca2+ oscillations in situ drive NMDAR-mediated neuronal excitation. Nat. Neurosci. 4, 803–81 (2001)
Pascual, O., Casper, K.B., Kubera, C., Zhang, J., Revilla-Sanchez, R., Sul, J.Y., Takano, H., Moss, S.J., McCarthy, K., Haydon, P.G.: Astrocytic purinergic signaling coordinates synaptic networks. Science 310, 113–116 (2005)
de Pasquale, F., Gorecki, J., Poielawski., J.: On the stochastic correlations in a randomly perturbed chemical front. J. Phys. A 25, 433 (1992)
Paullet, J.E., Ermentrout, G.B.: Stable rotating waves in two-dimensional discrete active media. SIAM J. Appl. Math. 54, 1720–1744 (1994)
Pavliotis, G.A., Stuart, A.M.: Multiscale Methods: Averaging and Homogenization. Springer, New York (2008)
Pearson, J.E., Ponce-Dawson, S.: Crisis on skid row. Physica A 257, 141–148 (1998)
Pearson, R., Catsicas, M., Becker, D., Mobbs, P.: Purinergic and muscarinic modulation of the cell cycle and calcium signaling in the chick retinal ventricular zone. J. Neurosci. 22, 7569–7579 (2002)
Pearson, R.A., Dale, N., Llaudet, E., Mobbs, P.: ATP released via gap junction hemichannels from the pigment epithelium regulates neural retinal progenitor proliferation. Neuron 46, 731–744 (2005)
Pereda, E., Quiroga, R., Bhattacharya, J.: Nonlinear multivariate analysis of neurophysiological signals. Prog. Neurobiol. 77, 1–37 (2005)
Peters, O., Schipke, C.G., Hashimoto, Y., Kettenmann, H.: Different mechanisms promote astrocyte Ca2+ waves and spreading depression in the mouse neocortex. J. Neurosci. 23, 9888–9896 (2003)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)
Pinder, G.F., Gray, W.G.: Essentials of Multiphase Flow and Transport in Porous Media. Wiley, New York (2008)
Pinto, D., Ermentrout, G.B.: Spatially structured activity in synaptically coupled neuronal networks I: traveling fronts and pulses. SIAM J. Appl. Math 62, 206–225 (2001)
Pinto, D., Ermentrout, G.B.: Spatially structured activity in synaptically coupled neuronal networks II: lateral inhibition and standing pulses. SIAM J. Appl. Math. 62, 226–243 (2001)
Pinto, D., Jackson, R.K., Wayne, C.E.: Existence and stability of traveling pulses in a continuous neuronal network. SIAM J. Appl. Dyn. Syst. 4, 954–984 (2005)
Pinto, D., Patrick, S.L., Huang, W.C., Connors, B.W.: Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms. J. Neurosci. 25, 8131–8140 (2005)
Plenz, D., Thiagarajan, T.C.: The organizing principles of neuronal avalanches: cell assemblies in the cortex? Trend. Neurosci. 30, 101–110 (2007)
Polleux, F., Snider, W.: Initiating and growing an axon. Cold Spring Harb Perspect Biol 2, a001925 (2010)
Polonsky, A., Blake, R., Braun, J., Heeger, D.J.: Neuronal activity in human primary visual cortex correlates with perception during binocular rivalry. Nat. Neurosci. 3, 1153–1159 (2000)
Ponce, S.J., Keizer, J., Pearson, J.E.: Fire-diffuse-fire model of dynamics of intracellular calcium waves. Proc. Nat. Acad. Sci. USA 96, 6060–6063 (1999)
Prechtl, J.C., Cohen, L.B., Pesaran, B., Mitra, P.P., Kleinfeld, D.: Visual stimuli induce waves of electrical activity in turtle cortex. Proc. Natl. Acad. Sci. USA 94, 7621–7626 (1997)
Prusiner, S.B.: Prions. Proc. Nat. Acad. Sci. (USA) 95, 13363–13383 (1998)
Rakic, P.: Elusive radial glial cells: historical and evolutionary perspective. Glia 43, 19–32 (2003)
Rall, W., Snir, H.A.: Cable theory for dendritic neurons. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modelling, 2nd edn. MIT Press, Cambridge (1998)
Ramirez, O.A., Couve, A.: The endoplasmic reticulum and protein trafficking in dendrites and axons. Trend. Cell. Biol. 21, 219–227 (2011)
Rangan, A.V., Kovacic, G., Cai, D.: Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train. Phys. Rev. E 77, 041915 (2008)
Reed, M.C., Venakides, S., Blum, J.J.: Approximate traveling waves in linear reaction-hyperbolic equations. SIAM J. Appl. Math. 50(1), 167–180 (1990)
Renart, A., de la Rocha, J., Bartho, P., Hollender, L., Parga, N., Reyes, A., Harris, K.D.: The asynchronous state in cortical circuits. Science 327, 587–590 (2010)
Rhodes, P.: The properties and implications of NMDA spikes in neocortical pyramidal cells. J. Neurosci. 26, 6704–6715 (2006)
Richardson, K.A., Schiff, S.J., Gluckman, B.J.: Control of traveling waves in the mammalian cortex. Phys. Rev. Lett. 94, 028103 (2005)
Richardson, M.P.: Large scale brain models of epilepsy: dynamics meets connectomics. Epilepsy (2012)
Rinzel, J., Maginu, K.: Kinematic analysis of wave pattern formation in excitable media. In: Pacault, A., Vidal, C. (eds.) Non-Equilibrium Dynamics in Chemical Systems. Springer, Berlin (1984)
Rinzel, J., Terman, D.: Propagation phenomena in a bistable reaction–diffusion system. SIAM J. Appl. Math. 42, 1111–1137 (1982)
Rinzel, J., Terman, D., Wang, X., Ermentrout, B.: Propagating activity patterns in large-scale inhibitory neuronal networks. Science 279, 1351–1355 (1998)
Robinson, P.A.: Patchy propagator, brain dynamics, and the generation of spatially structured gamma oscillations. Phys. Rev. E 73, 041904 (2006)
Robinson, P.A., Rennie, C.J., Wright, J.J.: Propagation and stability of waves of electrical activity in the cerebral cortex. Phys. Rev. E 56, 826–840 (1997)
Robinson, P.A., Rennie, C.J., Wright, J.J., Bahramali, H., Gordon, E., Rowe, D.I.: Prediction of electroencephalographic spectra from neurophysiology. Phys. Rev. E 63, 021903 (2001)
Robinson, P.A., Rennie, C.J., Rowe, D.L.: Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Phys. Rev. E 65, 041924 (2002)
Rocco, A., Ebert, U., van Saarloos, W.: Subdiffusive fluctuations of “pulled” fronts with multiplicative noise. Phys. Rev. E. 65, R13–R16 (2000)
Rockland, K.S., Lund, J.: Intrinsic laminar lattice connections in primate visual cortex. J. Comp. Neurol. 216, 303–318 (1983)
Rose, J., Jin, S.X., Craig, A.M.: Heterosynaptic molecular dynamics: locally induced propagating synaptic accumulation of CaM Kinase II. Neuron 61, 351–358 (2009)
Ross, C.A., Poirier, M.A.: Protein aggregation and neurodegenerative disease. Nat. Med. 10(Suppl.), S10–S17 (2004)
Ross, W.N.: Understanding calcium waves and sparks in central neurons. Nat. Rev. Neurosci. 13, 157–168 (2012)
Roxin, A., Riecke, H., Solla, S.A.: Self-sustained activity in a small-world network of excitable neurons. Phys. Rev. Lett. 92, 198101 (2004)
Rubin, J.E.: A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discret. Contin. Dyn. Syst. 10, 925–940 (2004)
Rubin, J., Terman, D.: Geometric singular perturbation analysis of neuronal dynamics. In: Fiedler, B. (ed.) Handbook of Dynamical Systems: Towards Applications, vol. 2. Elsevier, Amsterdam (2002)
Rubin, J.E., Terman, D.: Explicit maps to predict activation order in multiphase rhythms of a coupled cell network. J. Math. Neurosci. 2, 4 (2012)
Rubin, J., Wechselberger, M.: Giant squid-hidden canard: the 3d geometry of the Hodgkin–Huxley model. Biol. Cybern. 97, 5–32 (2007)
Rubin, J., Wechselberger, W.: The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales. Chaos 18, 015105 (2008)
Rubinov, M., Sporns, O.: Complex network measures of brain connectivity: Uses and interpretations. Neuroimage 52, 1059–1069 (2010)
Ruthel, G., Banker, G.: Actin-dependent anterograde movement of growth-cone-like structures along growing hippocampal axons: A novel form of axonal transport? Cell Motil. Cytoskel. 40, 160–173 (1998)
Ruthel, G., Banker, G.: Role of moving growth cone-like wave structures in the outgrowth of cultured hippocampal axons and dendrites. J. Neurobiol. 39, 97–106 (1999)
van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)
Saez, J.C., Berthoud, V.M., Branes, M.C., Martinez, A.D., Beyer., E.C.: Plasma membrane channels formed by connexins: their regulation and functions. Physiol. Rev. 83, 1359–1400 (2003)
Sagues, F., Sancho, J.M., Garcia-Ojalvo, J.: Spatiotemporal order out of noise. Rev. Mod. Phys. 79, 829–882 (2007)
Saitoh, T., Schwartz, J.H.: Phosphorylation-dependent subcellular translocation of a Ca2+/calmodulin-dependent protein kinase produces an autonomous enzyme in aplysia neurons. J. Cell. Biol. 100, 835–842 (1985)
Samuels, D.C., Hentschel, H.G.E., Fine, A.: The origin of neuronal polarization: a model of axon formation. Phil. Trans. R. Soc. Lond. B 351, 1147–1156 (1996)
Sanchez-Vives, M.V., McCormick, D.A.: Cellular and network mechanisms of rhythmic recurrent activity in neocortex. Nat. Neurosci. 3, 1027–1034 (2000)
Sanderson, M.J., Charles, A.C., Dirksen, E.R.: Mechanical stimulation and intercellular communication increases intracellular Ca2+ in epithelial cells. Cell. Regul. 1, 585–596 (1990)
Sandstede, B.: Stability of travelling waves. In: Handbook of Dynamical Systems, vol. 2, pp. 983–1055. North-Holland, Amsterdam (2002)
Sandstede, B.: Evans functions and nonlinear stability of traveling waves in neuronal network models. Int. J. Bifur. Chaos Appl. Sci. Eng. 17, 2693–2704 (2007)
Sato, T.K., Nauhaus, I., Carandini, M.: Traveling waves in visual cortex. Neuron 75, 218–229 (2012)
Scemes, E., Giaume, C.: Astrocyte calcium waves: what they are and what they do. Glia 54, 716–725 (2006)
Schiller, J., Schiller, Y., Stuart, G., Sakmann, B.: Calcium action potentials restricted to distal apical dendrites of rat neocortical pyramidal neurons. J. Physiol. 505, 605–616 (1997)
Schiller, J., Major, G., Koester, H.J., Schiller, Y.: NMDA spikes in basal dendrites of cortical pyramidal neurons. Nature 404, 285–289 (2000)
Schimansky-Geier, L., Mikhailov, A.S., Ebeling., W.: Effects of fluctuations on plane front propagation in bistable nonequilibrium systems. Ann. Phys. 40, 277 (1983)
Schock, S.C., Munyao, N., Yakubchyk, Y., Sabourin, L.A., Hakim, A.M., Ventureyra, E.C.G., 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)
Schuss, Z.: Theory and applications of stochastic processes: an analytical approach. In: Applied Mathematical Sciences, vol. 170. Springer, New York (2010)
Schuster, H.G., Wagner, P.: A model for neuronal oscillations in visual cortex. Biol. Cybern. 64, 77–82 (1990)
Schutz, P., Bode, M., Purwins, H.G.: Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities. Physica D 82, 382–397 (1995)
Schwartz, E.: Spatial mapping in the primate sensory projection: analytic structure and relevance to projection. Biol. Cybern. 25, 181–194 (1977)
Seely, J., Chow, C.C.: Role of mutual inhibition in binocular rivalry. J. Neurophysiol. 106, 2136–2150 (2011)
Seifert, G.: Astrocyte dysfunction in neurological disorders; a molecular perspective. Nat. Rev. Neurosci. 7, 194–206 (2006)
Selkoe, D.J.: Folding proteins in fatal way. Nature 426, 900–904 (2003)
Selkoe, D.J.: Cell biology of protein misfolding: the examples of Alzheimer’s and Parkinson’s diseases. Nat. Cell Biol. 6, 1054–1061 (2004)
Shadlen, M.N., Newsome, W.T.: Noise, neural codes and cortical organization. Curr. Opin. Neurobiol. 4, 569–579 (1994)
Shapiro, B.E.: Osmotic forces and gap junctions in spreading depression: A computational model. J. Comp. Neurosci. 10, 99–120 (2001)
Shen, K., Meyer, T.: Dynamic control of CaMKII translocation and localization in hippocampal neurons by NMDA receptor simulation. Science 284, 162–166 (1999)
Shen, K., Tereul, M.N., Subramanian, K., Meyer, T.: CaMKIIβ functions as an F-actin targeting module that localizes CaMKIIα/β heterooligomers to dendritic spines. Neuron 21, 593–606 (1998)
Shepherd, G.M.: The dendritic spine: a multifunctional unit. J. Neurophysiol. 75, 2197–2210 (1996)
Shepherd, G.M., Brayton, R.K., Miller, J.P., Segev, I., Rinzel, J., Rall, W.: Signal enhancement in distal cortical dendrites by means of interactions between active dendritic spines. Proc. Natl. Acad. Sci. USA 82, 2192–2195 (1985)
Shew, W.L., Plenz, D.: The functional benefits of criticality in the cortex. Neuroscientist 19, 88–100 (2012)
Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30, 143–160 (1986)
Shigesada, N., Kawasaki, K.: Biological invasions: theory and practice. Oxford University Press, Oxford (1997)
Shimada, T., Toriyama, M., Uemura, K., Kamiguchi, H., Sugiura, T., Watanabe, N., Inagaki, N.: Shootin1 interacts with actine retrograde flow and L1-CAM to promote axon outgrowth. J. Cell. Biol. 181, 817–829 (2008)
Shpiro, A., Moreno-Bote, R., Rubin, N., Rinzel, J.: Dynamical characteristics common to neuronal competition models. J. Neurophysiol. 97, 37–54 (2007)
Shpiro, A., Curtu, R., Rinzel, J., Rubin, N.: Balance between noise and adaptation in competition models of perceptual bistability. J Comp. Neurosci. 27, 462–473 (2009)
Shuai, J.W., Jung, P.: Stochastic properties of Ca2+ release of Inositol 1,4,5-Triphosphate receptor clusters. Biophys. J. 83, 87–97 (2002)
Shusterman, V., Troy, W.C.: From baseline to epileptiform activity: a path to synchronized rhythmicity in large-scale neural networks. Phys. Rev. E 77, 061911 (2008)
Silva, J.S.D., Dotti, C.G.: Breaking the neuronal sphere: regulation of the actin cytoskeleton in neuritogenesis. Nat. Rev. Neurosci. 3, 694–704 (2002)
Silva, F.H.L.D., Blanes, W., Kalitzin, S.N., Parra, J., Suffczynski, P., Velis, D.N.: Dynamical diseases of brain systems: different routes to epileptic seizures. IEEE Trans. Biomed. Eng. 50, 540–548 (2003)
Sincich, L.C., Blasdel, G.G.: Oriented axon projections in primary visual cortex of the monkey. J. Neurosci. 21, 4416–4426 (2001)
Sinha, S.R.: Basic mechanisms of sleep and epilespy. J. Clin. Neurophysiol. 28, 103–110 (2011)
Sjostrom, P.J., Rancz, E.A., Roth, A., Hausser, M.: Dendritic excitability and synaptic plasticity. Physiol. Rev. 88, 769–840 (2008)
Skinner, F., Kopell, N., Marder, E.: Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. J. Comput. Neurosci. 1, 69–87 (1994)
Smaily, M.E., Hamel, F., Roques, L.: Homogenization and influence of fragmentation in a biological invasion model. Discret. Contin. Dyn. Syst. Ser. A 25, 321–342 (2009)
Smith, G.D.: Modeling the stochastic gating of ion channels. In: Fall, C., Marland, E.S., wagner, J.M., Tyson, J.J. (eds.) Computational Cell Biology, chap. 11. Springer, New York (2002)
Smith, G.D., Cox, C.L., Sherman, S.M., Rinzel, J.: Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-burst model. J. Neurophysiol. 83, 588–610 (2000)
Sneyd, J., Charles, A.C., Sanderson, M.J.: A model for the propagation of intercellular calcium waves. Amer. J. Physiol. cell physiol. 266, C293–C302 (1994)
Sneyd, J., Wetton, B.T., Charles, A.C., Sanderson, M.J.: Intercellular calcium waves mediated by diffusion of inositol trisphosphate: a two-dimensional model. Am. J. Physiol. Cell Physiol. 268, C1537–C1545 (1995)
Sneyd, J., Dale, P.D., Duffy, A.: Traveling waves in buffered systems: applications to calcium waves. SIAM J. Appl. Math. 58, 1178–1192 (1998)
Softky, W.R., Koch, C.: Cortical cell should spike regularly but do not. Neural Comput. 4, 643–646 (1992)
Soltesz, I., Staley, K.: Computational Neuroscience in Epilepsy. Academic, San Diego (2008)
Somers, D.C., Nelson, S., Sur, M.: An emergent model of orientation selectivity in cat visual cortical simple cells. J. Neurosci. 15, 5448–5465 (1995)
Somjen, G.G.: Mechanisms of spreading depression and hypoxic spreading depression-like depolarization. Physiol. Rev. 81, 1065–1096 (2001)
Soula, H., Chow, C.C.: Stochastic dynamics of a finite-size spiking neural network. Neural Comput. 19, 3262–3292 (2007)
Sporns, O.: Networks of the Brain. MIT Press, Cambridge (2010)
Staley, K.J., Dudek, F.E.: Interictal spikes and epileptogenesis. Epilepsy Currents 6, 199–202 (2006)
Stam, C.J., Reijeneveld, J.C.: Graph theoretical analysis of complex networks in the brain. Nonlin. Biomed. Phys. 1 (3), 1–19 (2007)
Steinberg, B., Wang, Y.Q., Huang, H., Miura, R.: Spatial buffering mechanism: Mathematical model and computer simulations. Math. Biosci. Eng. 2, 675–702 (2005)
Steriade, M., McCormick, D.A., Sejnowski, T.J.: Thalamocortical oscillations in the sleeping and aroused brain. Science 262, 679–685 (1993)
Steyn-Ross, M.L., Sleigh, D.A.S.R.J.W., Whiting, D.R.: Theoretical predictions for spatial covariance of the electroencephalographic signal during the anesthetic-induced phase transition: Increased correlation length and emergence of spatial self-organization. Phys. Rev. E 68, 021902 (2003)
Stocker, M., Krause, M., Pedarzani, P.: An apamin-sensitive Ca2+-activated K+ current in hippocampal pyramidal neurons. Proc. Natl. Acad. Sci. USA 96, 4662–4667 (1999)
Strack, S., Choi, S., Lovinger, D.M., Colbran, R.J.: Translocation of autophosphorylated calcium/calmodulin-dependent protein kinase II to the postsynaptic density. J. Biol. Chem. 272, 13467–13470 (1997)
Stuart, G.J., Sakmann, B.: Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature 367, 69–72 (1994)
Stuart, G., Spruston, N., Hausser, M.: Dendrites. Oxford University Press, New York (1999)
Stuart, G., Spruston, N., Hausser, M. (eds.): Dendrites. Oxford University Press, Oxford (2007)
Subramanian, K., Narang, A.: A mechanistic model for eukaryotic gradient sensing: spontaneous and induced phosphoinositide polarization. J. Theor. Biol. 231, 49–67 (2004)
Sun, W., McConnell, E., Pare, J.F., Xu, Q., Chen, M., Peng, W., Lovatt, D., Han, X., Y.Smith, Nedergaard, M.: Glutamate-dependent neuroglial calcium signaling differs between young and adult brain. Science 339, 197–200 (2013)
Sutherland, G.R., McNaughton, B.: Memory trace reactivation in hippocampal and neocortical neuronal ensembles. Curr. Opin. Neurobiol. 10, 180–186 (2000)
Swillens, S., Dupont, G., Combettes, L., Champeil, P.: From calcium blips to calcium puffs: theoretical analysis of the requirement for interchannel communication. Proc. Nat. Acad. Sci. (USA) 96, 13750–13755 (1999)
Swindale, N.V.: The development of topography in the visual–cortex: A review of models. Network 7, 161–274 (1996)
Syed, M.M., Lee, S., He, S., Zhou, Z.J.: Spontaneous waves in the ventricular zone of developing mammalian retina. J. Neurophysiol. 91, 1999–2009 (2004)
Syed, M.M., Lee, S., Zheng, J., Zhou, Z.J.: Stage-dependent dynamics and modulation of spontaneous waves in the developing rabbit retina. J. Physiol. 560, 533–549 (2004)
Sykova, E., Nicholson, C.: Diffusion in brain extracellular space. Physiol. Rev. 88, 1277–1340 (2008)
Tabak, J., Senn, W., O’Donovan, M.J., Rinzel, J.: Modeling of spontaneous activity in developing spinal cord using activity-dependent depression in an excitatory network. J. Neurosci. 20, 3041–3056 (2000)
Tauber, U.C.: Field-theory approaches to nonequilibrium dynamics. Lect. Notes Phys. 716, 295–348 (2007)
Taylor, A., Cottrell, G., Kristan Jr, W.: Analysis of oscillations in a reciprocally inhibitory network with synaptic depression. Neural comput. 14(3), 561–581 (2002)
Terman, D.H., Ermentrout, G.B., Yew, A.C.: Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math 61, 1578–1604 (2003)
Thul, R., Coombes, S., Smith, G.D.: Sensitisation waves in a bidomain fire-diffuse-fire model of intracellular Ca2+ dynamics. Physica D 238, 2142–2152 (2009)
Thul, R., Smith, G.D., Coombes, S.: A bidomain threshold model of propagating calcium waves. J. Math. Biol. 56, 435–463 (2008)
Tian, G.F.,, Hooman, A., Takano, T., Xu, Q., Peng, W., Lin, J., Oberheim, N., Lou, N., Zielke, R., Kang, J., Nedergaard, M.: An astrocytic basis of epilepsy. Nat. Med. 11, 973–981 (2005)
Timofeeva, Y.: Traveling waves in a model of quasi-active dendrites with active spines. Physica D 239, 494–503 (2010)
Timofeeva, Y., Lord, G.J., Coombes, S.: Spatio-temporal filtering properies of a dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework. J. Comput. Neurosci. 21, 293–306 (2006)
Tong, F., Engel, S.A.: Interocular rivalry revealed in the human cortical blind-spot representation. Nature 411, 195–199 (2001)
Tong, F., Meng, M., Blake, R.: Neural bases of binocular rivalry. Trend. Cog. Sci. 10, 502–511 (2006)
Toriyama, M., Shimada, T., Kim, K.B., Mitsuba, M., Nomura, E., Katsuta, K., Sakumura, Y., Roepstorff, P., Inagaki, N.: Shootin1: a protein involved in the organization of an asymmetric signal for neuronal polarization. J. Cell Biol. 175, 147–157 (2006)
Toriyama, M., Sakumura, Y., Shimada, T., Ishii, S., Inagaki, N.: A diffusion-based neurite length-sensing mechanism involved in neuronal symmetry breaking. Mol. Syst. Biol. 6, 1–16 (2010)
Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)
Touboul, J., Destexhe, A.: Can power-law scaling and neuronal avalanches arise from stochastic dynamics? PLoS One 5, 14 (2010)
Touboul, J., Hermann, G., Faugeras, O.: Noise–induced behaviors in neural mean field dynamics. SIAM J. Appl. Dyn. Sys. 11, 49–81 (2012)
Tritsch, N.X., Yi, E., Gale, J.E., Glowatzki, E., Bergles, D.E.: The origin of spontaneous activity in the developing auditory system. Nature 450, 50–55 (2007)
Troy, W.C., Shusterman, V.: Patterns and features of families of traveling waves in large-scale neuronal networks. SIAM J. Appl. Dyn. Syst. 6, 263–292 (2007)
Tsai, J.C., Sneyd, J.: Are buffers boring? uniqueness and asymptotical sttability of traveling wave fronts in the buffered bistable equation. J. Math. Biol. 54, 513–553 (2007)
Tsai, J.C., Sneyd, J.: Traveling waves in the buffered FitzHugh-Nagumo model. SIAM J. Appl. Math. 71, 1606–1636 (2011)
Tsodyks, M., Pawelzik, K., Markram, H.: Neural networks with dynamic synapses. Neural Comput. 10, 821–835 (1998)
Tuckwell, H.C., Miura, R.M.: A mathematical model for spreading cortical depression. Biophys. J. 23, 257–276 (1978)
Vecchia, D., Pietrobon, D.: Migrain: a disorder of brain excitatory-inhibitory balance? Trend. Neurosci. 35, 507–520 (2012)
Veen, M.V., Pelt, J.V.: Neuritic growth rate described by modelling microtubule dynamics. Bull. Math. Biol. 56, 249–273 (1994)
Vellela, M., Qian, H.: Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited. J. R. Soc. Interface 6, 925–940 (2009)
Veltz, R., Faugeras, O.: Local/global analysis of the stationary solutions of some neural field equations. SIAM J. Appl. Dyn. Syst. 9, 954–998 (2010)
Vicker, M.G.: Eukaryotic cell locomotion depends on the propagation of self-organized reaction-diffusion waves and oscillations of actin filament assembly. J. Expt. Cell Res. 275, 54–66 (2002)
Vladimirski, B.B., Tabak, J., ODonovan, M.J., Rinzel, J.: Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field. J. Comp. Neurosci. 25, 39–63 (2008)
Vogels, T.P., Abbott, L.F.: Signal propagation and logic gating in networks of integrate-and-fire neurons. J. Neurosci. 25, 10786–10795 (2005)
Volman, V., Sejnowski, T.J., Bazhenov, M.: Topological basis of epileptogenesis in a model of severe cortical trauma. J. Neurophysiol. 106, 1933–1942 (2011)
Volman, V., Bazhenov, M., Sejnowski, T.J.: Computational models of neuron-astrocyte interaction in epilepsy. Front. Comp. Neurosci. 6(58), 1–10 (2012)
Volpert, V., Petrovskii, S.: Reaction-diffusion waves in biology. Phys. Life Rev. 6, 267–310 (2009)
van Vreeswijk, C.: Partial synchronization in populations of pulse-coupled oscillators. Phys. Rev. E. 54(5), 5522–5537 (1996)
van Vreeswijk, C., Sompolinsky, H.: Chaotic balanced state in a model of cortical circuits.. Neural Comput. 10, 1321–1371 (1998)
van Vreeswijk, C., Ermentrout, G.B., Abbott, L.F.: When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci. 1, 313–321 (1994)
Vuyst, E.D., Wang, N., Decrock, E., Bock, M.D., Vinken, M., Moorhem, M.V., Lai, C., Culot, M., Rogiers, V., Cecchelli, R., Naus, C.C., Evans, W.H., Leybaert, L.: Ca2+ regulation of connexin 43 hemichannels in C6 glioma and glial cells. Cell Calcium 46, 176–187 (2009)
Wagner, J., Sneyd, J.: Effects of rapid buffers on Ca2+ diffusion and Ca2+ oscillations. Biophys. J. 67, 447–456 (1994)
Wainrib, G., Thieullen, M., Pakdaman, K.: Reduction of stochastic conductance-based neuron models with time-scales separation. J. Comput. Neurosci. 32, 327–46 (2012)
Walther, G.R., Maree, A.F., Edelstein-Keshet, L., Grieneisen, V.A.: Deterministic versus stochastic cell polarisation through wave-pinning. Bull. Math. Biol. 74, 2570–2599 (2012)
Wang, X.J.: Decision making in recurrent neuronal networks. Neuron 60, 215–234 (2008)
Wang, X.J., Rinzel, J.: Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput. 4, 84–97 (1992)
Wang, S.S., Denk, W., Hausser, M.: Coincidence detection in single dendritic spines mediated by calcium release. Nat. Neurosci. 3, 1266–1273 (2000)
Wang, Z., Haydon, P.G., Yeung, E.S.: Direct observation of calcium-independent intercellular ATP signaling in astrocytes.. Anal. Chem. 72, 2001–2007 (2000)
Ward, M.J., Lee, J.: On the asymptotic and numerical-analyses of exponentially ill-conditioned singularly perturbed boundary value problems. Stud. Appl. Math. 94, 271–326 (1995)
Watt, A.J.: Traveling waves in developing cerebellar cortex mediated by asymmetrical purkinje cell connectivity. Nat. Neurosci. 12, 463–473 (2009)
Watts, D.J., Strogatz, S.H.: Collective dynamics of “small-world” networks. Nature 393, 440–442 (1998)
Webber, M., Bressloff, P.C.: The effects of noise on binocular rivalry waves: a stochastic neural field model. J. Stat. Mech 3, P03001 (2013)
Weinberger, H.F.: On spreading speeds and traveling waves for growth and migration in a periodic habitat. J. Math. Biol. 45, 511–548 (2002)
Weiner, O.D., Marganski, W.A., Wu, L.F., Altschuler, S.J., Kirschner, M.W.: An actin-based wave generator organizes cell motility. PLoS Biol. 5, 2053–2063 (2007)
Weissman, T.A., Riquelme, P.A., Ivic, L., Flint, A.C., Kriegstein, A.R.: Calcium waves propagate through radial glial cells and modulate proliferation in the developing neocortex. Neuron 43, 647–661 (2004)
Werry, E.L., Liu, G.J., Bennett., M.R.: Glutamate-stimulated ATP release from spinal cord astrocytes is potentiated by substance P. J. Neurochem. 99, 924–936 (2006)
Wetherington, J., Serranoa, G., Dingledine, R.: Astrocytes in the epileptic brain. Neuron 58, 168–178 (2008)
White, J., Southgate, E., Thomson, J., Brenner, S.: The structure of the nervous system of the nematode Caenorhabditis elegans. Phil. Trans. R. Soc. Lond. B Biol. Sci. 314, 1–340 (1986)
White, J.A., Rubinstein, J.T., Kay, A.R.: Channel noise in neurons. Trends. Neurosci. 23, 131–137 (2000)
Wiley, D.A., Strogatz, S.H., Girvan, M.: The size of the sync basin. Chaos 16, 015103 (2006)
Williams, G.S.B., Huertas, M.A., Sobie, E.A., Jafri, M.S., Smith, G.D.: A probability density approach to modeling local control of calcium-induced calcium release in cardiac myocytes. Biophys. J. 92, 2311–2328 (2007)
Williams, G.S.B., Huertas, M.A., Sobie, E.A., Jafri, M.S., Smith, G.D.: Moment closure for local control models of calcium-induced calcium release in cardiac myocytes. Biophys. J. 95, 1689–1703 (2008)
Wilson, H.R.: Computational evidence for a rivalry hierarchy in vision. Proc. Natl. Acad Sci. USA 100, 14499–144503 (2003)
Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–23 (1972)
Wilson, H.R., Cowan, J.D.: A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80 (1973)
Wilson, M.A., McNaughton, B.L.: Reactivation of hippocampal ensemble memories during sleep. Science 265, 676–679 (1994)
Wilson, H.R., Blake, R., Lee, S.H.: Dynamics of traveling waves in visual perception. Nature 412, 907–910 (2001)
Winfree, A.: The Geometry of Biological Time. Springer, New York (1980)
Wissner-Gross, Z.D., Scott, M.A., Steinmeyer, J.D., Yanik, M.F.: Synchronous symmetry breaking in neurons with different neurite counts. PLoS One 8 (2), e54905 (2013)
Wolszon, L.R., Rehder, V., Kater, S.B., Macagno, E.R.: Calcium wave fronts that cross gap junctions may signal neuronal death during development. J. Neurosci. 14, 3437–3448 (1994)
Wong, R.O.L.: Retinal waves and visual development. Ann. Rev. Neurosci. 22, 29–47 (1999)
Wu, J.: Propagating waves of activity in the neocortex: what they are, what they do. Neuroscientist 14, 487–502 (2008)
Wu, J.Y., Guan, L., Tsau, Y.: Propagating activation during oscillations and evoked responses in neocortical slices. J. Neurosci. 19, 5005–5015 (1999)
Wu, J., Guan, L., Bai, L., Yang, Q.: Spatiotemporal properties of an evoked population activity in rat sensory cortical slices. J. Neurophysiol. 86, 2461–74 (2001)
Wylie, J.J., Miura, R.M.: Traveling waves in coupled reaction-diffusion models with degenerate sources. Phys. Rev. E 74, 021909 (2006)
Xin, J.: Front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000)
Xu, W., Huang, X., Takagaki, K., Wu, J.Y.: Compression and reflection of visually evoked cortical waves. Neuron 55, 119–129 (2007)
Yang, E., Schulman, H.: Structural examination of autoregulation of multifunctional calcium/calmodulin-dependent protein kinase II. J. Biol. Chem. 274, 26199–26208 (1999)
Yao, Y., Choi, J., Parker, I.: Quantal puff of intracellular Ca2+ evoked by inositol triphosphate in Xenopus oocytes. J. Physiol. 482, 533–553 (1995)
Yoshioka, T., Blasdel, G.G., Levitt, J.B., Lund, J.S.: Relation between patterns of intrinsic lateral connectivity, ocular dominance, and cytochrome oxidase–reactive regions in macaque monkey striate cortex. Cereb. Cortex 6, 297–310 (1996)
Young, G.W.D., Keizer, J.: A single pool IP3-receptor model for agonist stimulated Ca2+ oscillations. Proc. Nat. Acad. Sci. (USA) 89, 9895–9899 (1992)
Yuste, R., Bonhoeffer, T.: Morphological changes in dendritic spines associated with LTP. Ann. Rev. Neurosci. 24, 1071–1089 (2001)
Yuste, R., Denk, W.: Dendritic spines as a basic functional units of neuronal integration. Nature 375, 682–684 (1995)
Zahradnikova, A., Zahradnik, I.: A minimal gating model for the cardiac Ca2+ release channel. Biophys. J. 71, 2996–3012 (1996)
Zhang, L.: On the stability of traveling wave solutions in synaptically coupled neuronal networks. Diff. Int. Eq. 16, 513–536 (2003)
Zimmermann, H.: Purinergic signaling in neural development. Sem. Cell Dev. Biol. 22, 194–204 (2011)
Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena, 4th edn. Oxford University Press, Oxford (2002)
Zinner, B.: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Diff. Eqns. 96, 1–27 (1992)
Zucker, R.S.: Short term synaptic plasticity. Ann. Rev. Neurosci. 12, 13–31 (1989)
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Bressloff, P.C. (2014). Population Models and Neural Fields. In: Waves in Neural Media. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8866-8_6
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