Abstract
In this chapter we examine the dynamical behavior of a spatially homogeneous two-dimensional model of the cortex that incorporates membrane potential, synaptic flux rates and long- and short-range synaptic input, in two spatial dimensions, using parameter sets broadly realistic of humans and rats. When synaptic dynamics are included, the steady states may not be stable. The bifurcation structure for the spatially symmetric case is explored, identifying the positions of saddle–node and sub- and supercritical Hopf instabilities. We go beyond consideration of small-amplitude perturbations to look at nonlinear dynamics. Spatially-symmetric (breathing mode) limit cycles are described, as well as the response to spatially-localized impulses. When close to Hopf and saddle–node bifurcations, such impulses can cause traveling waves with similarities to the slow oscillation of slow-wave sleep. Spiral waves can also be induced. We compare model dynamics with the known behavior of the cortex during natural and anesthetic-induced sleep, commenting on the physiological significance of the limit cycles and impulse responses.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Ch. 9 of the present volume.
- 2.
See Ch. 12 of the present volume.
References
Colrain, I.M.: The K-complex: A 7-decade history. Sleep 28, 255–273 (2006)
Freeman, W.J.: Predictions on neocortical dynamics derived from studies in paleocortex. In: E. Basar, T.H. Bullock (eds.), Induced Rhythms of the Brain, chap. 9, pp. 183–199, Birkhaeuser, Boston (1992)
Golomb, D., Amitai, Y.: Propagating neuronal discharges in neocortical slices: Computational and experimental study. J. Neurophys. 78, 1199–1211 (1997)
Huang, X., Troy, W.C., Yang, Q., Ma, H., Laing, C.R., Schiff, S.J., Wu, J.Y.: Spiral waves in disinhibited mammalian neocortex. J. Neurosci. 24, 9897–9902 (2004), doi:10.1523/jneurosci.2705-04.2004
Hutt, A., Bestehorn, M., Wennekers, T.: Pattern formation in intracortical neuronal fields. Network 14, 351–368 (2003)
Jirsa, V.K., Haken, H.: A field theory of electromagnetic brain activity. Phys. Rev. Lett. 77, 960–963 (1996), doi:10.1103/PhysRevLett.77.960
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastc Differential Equations. Springer, Berlin (1992)
Kramer, M.A., Kirsch, H.E., Szeri, A.J.: Pathalogical pattern formation and epileptic seizures. J. Roy. Soc. Interface 2, 113 (2005), doi:10.1098/rsif.2004.0028
Liley, D.T.J., Cadusch, P.J., Wright, J.J.: A continuum theory of electro-cortical activity. Neurocomp. 26–27, 795–800 (1999), doi:10.1016/S0925-2312(98)00149-0
Massimini, M., Huber, R., Ferrarelli, F., Hill, S., Tononi, G.: The sleep slow oscillation as a traveling wave. J. Neurosci. 24, 6862–6870 (2004), doi:10.1523/jneurosci.1318-04.2004
Numminen, J., Makela, J.P., Hari, R.: Distributions and sources of magnetoencephalographic K-complexes. Electroencephal. Clin. Neurophysl. 99, 544–555 (1996), doi:10.1016/S0013-4694(96)95712-0
Nunez, P.L.: The brain wave function: A model for the EEG. Math. Biosci. 21, 279–297 (1974), doi:10.1016/0025-5564(74)90020-0
Osan, R., Ermentrout, B.: Two dimensional synaptically generated travelling waves in a theta-neuron neural network. Neurocomp. 38–40, 789–795 (2001)
Rennie, C.J., Wright, J.J., Robinson, P.A.: Mechanisms for cortical electrical activity and emergence of gamma rhythm. J. Theor. Biol. 205, 17–35 (2000), doi:10.1006/jtbi.2000.2040
Robinson, P.A., Rennie, C.J., Rowe, D.L., ’Connor, S.C., Wright, J.J., Gordon, E., Whitehouse, R.W.: Neurophysical modeling of brain dynamics. Neuropsychopharmacol. 28, S74–S79 (2003), doi:10.1038/sj.npp.1300143
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), doi:10.1103/PhysRevE.56.826
Steriade, M., Timofeev, I., Grenier, F.: Natural waking and sleep states: A view from inside neocortical neurons. J. Neurophysiol. 85, 1969–1985 (2001)
Steyn-Ross, D.A., Steyn-Ross, M.L., Sleigh, J.W., Wilson, M.T., Gillies, I.P., Wright, J.J.: The sleep cycle modelled as a cortical phase transition. J. Biol. Phys. 31, 547–569 (2005), doi:10.1007/s10867-005-1285-2
Steyn-Ross, M.L., Steyn-Ross, D.A., Sleigh, J.W.: Modelling general anaesthesia as a first-order phase transition in the cortex. Progress in Biophysics and Molecular Biology 85, 369–385 (2004), doi:10.1016/j.pbiomolbio.2004.02.001
Wilson, M.T., Sleigh, J.W., Steyn-Ross, D.A., Steyn-Ross, M.L.: General anesthetic-induced seizures can be explained by a mean-field model of cortical dynamics. Anesthesiology 104, 588–593 (2006), doi:10.1097/00000542-200603000-00026
Wilson, M.T., Steyn-Ross, D.A., Sleigh, J.W., Steyn-Ross, M.L., Wilcocks, L.C., Gillies, I.P.: The K-complex and slow oscillation in terms of a mean-field cortical model. J. Comp. Neurosci. 21, 243–257 (2006), doi:10.1007/s10827-006-7948-6
Wilson, M.T., Steyn-Ross, M.L., Steyn-Ross, D.A., Sleigh, J.W.: Predictions and simulations of cortical dynamics during natural sleep using a continuum approach. Phys. Rev. E 72, 051910 1–14 (2005), doi:10.1103/PhysRevE.72.051910
Wright, J.J., Liley, D.T.J.: Dynamics of the brain at global and microscopic scales: Neural networks and the EEG. Behav. Brain Sci. 19, 285–316 (1996)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Wilson, M., Steyn-Ross, M., Steyn-Ross, D., Sleigh, J., Gillies, I., Hailstone, D. (2010). What can a mean-field model tell us about the dynamics of the cortex?. In: Steyn-Ross, D., Steyn-Ross, M. (eds) Modeling Phase Transitions in the Brain. Springer Series in Computational Neuroscience, vol 4. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0796-7_10
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0796-7_10
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0795-0
Online ISBN: 978-1-4419-0796-7
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)