Skip to main content

Advertisement

SpringerLink
Log in

Attractor Dynamics and Thermodynamic Analogies in the Cerebral Cortex: Synchronous Oscillation, the Background EEG, and the Regulation of Attention

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

An Erratum to this article was published on 10 February 2011

Abstract

Ongoing changes in attention and cognition depend upon cortical/subcortical interactions, which select sequences of different spatial patterns of activation in the cortex.

It is proposed that each pattern of cortical activation permits evolution of electrocortical wave activity toward statistically stationary states, analogous to thermodynamic equilibrium. In each steady-state, neurons fire with an intrinsic Poisson spike probability and also with a bursting pattern related to network oscillations. Excitatory cell dendrites act as a regenerative reservoir in which pulse generation is balanced against dissipations.

Equilibria exhibit contrasting limits. One limit, at high cortical activation, generates widespread zero-lag synchrony among excitatory cells, with partial suppression of noise. Excitatory and inhibitory cells approach zero-lag local correlation, with 1/4 cycle lag-correlation at greater distances of separation. The high-activation limit defines a correlated system of attractor basins, capable of co-ordinating synaptic modifications and intracortical signal generation. Suppression of noise would enhance convergence about attractor basins in the manner of simulated annealing, while, conversely, the persistence of some noise prevents network paralysis by phase locking. At the opposite limit—that of low activation—spikes and waves have low cross- and auto-correlation, but have wide-spectrum sensitivity to inputs. It is hypothesised that cortical regions, transiently at equilibrium near these extremes, engage in interaction with each other and with subcortical systems, to generate ongoing sequences of attention and cognition.

This account is compatible with classical and recently observed experimental phenomena. The principle features inferred from a simplified linear mathematical account are reproduced in a more physiologically realistic and non-linear numerical simulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alexander, G. E., Crutcher, M. D., & DeLong, M. R. (1990). Basal ganglia-thalamocortical circuits: parallel substrates for motor, oculomotor, ‘prefrontal’ and ‘limbic’ functions. In Progress in brain research. Amsterdam: Elsevier.

    Google Scholar 

  • Amari, S.-I. (1974). A method of statistical neurodynamics. Kybernetik, 14, 201–215.

    MathSciNet  Google Scholar 

  • Amari, S.-I., & Maginu, K. (1988). Statistical neurodynamics of associative memory. Neural Netw., 1, 63–73.

    Article  Google Scholar 

  • Amari, S.-I., Yoshida, K., & Kanatani, K.-I. (1977). A mathematical foundation for statistical neurodynamics. SIAM J. Appl. Math., 33, 95–126.

    Article  MathSciNet  MATH  Google Scholar 

  • Amit, D. J. (1989). Modelling brain function. Cambridge: Cambridge University Press.

    Google Scholar 

  • Arbib, M. A. (Ed.) (1995). The handbook of brain theory and neural networks. Cambridge: MIT Press.

    Google Scholar 

  • Barrie, J. M., Freeman, W. J., & Lenhart, M. D. (1996). Spatiotemporal analysis of prepyriform, visual, auditory, and somesthetic surface EEGs in trained rabbits. J. Neurophysiol., 76, 520–539.

    Google Scholar 

  • beim Graben, P., Barrett, A., & Atmanspacher, H. (2009). Stability criteria for contextual emergence of macrostates in neural networks. Netw. Comput. Neural Syst., 20, 178–196.

    Article  Google Scholar 

  • Braitenberg, V., & Schüz, A. (1991). Anatomy of the cortex: statistics and geometry. Berlin, New York: Springer.

    Google Scholar 

  • Bressler, S. L., Coppola, R., & Nakamura, R. (1993). Episodic multiregional cortical coherence at multiple frequencies during visual task performance. Nature, 366, 153–156.

    Article  Google Scholar 

  • Chapman, C. L., Bourke, P. D., & Wright, J. J. (2002). Spatial eigenmodes and synchronous oscillation: coincidence detection in simulated cerebral cortex. J. Math. Biol., 45, 57–78.

    Article  MathSciNet  MATH  Google Scholar 

  • Eckhorn, R., Bauer, R., Jordon, W., Brosch, M., Kruse, W., Monk, M., & Reitboeck, H. J. (1988). Coherent oscillations: a mechanism of feature linking in the in the visual cortex? Biol. Cybern., 60, 121–130.

    Article  Google Scholar 

  • Freeman, W. J. (1975). Mass action in the nervous system. San Diego: Academic Press.

    Google Scholar 

  • Freeman, W. J. (1991). Predictions on neocortical dynamics derived from studies on paleocortex. In Induced rhythms of the brain. Boston: Birkhäuser.

    Google Scholar 

  • Freeman, W. J. (2003). The wave packet: an action potential for the 21st Century. J. Integr. Neurosci., 2, 3–30.

    Article  Google Scholar 

  • Freeman, W. J. (2004a). Origin, structure and role of background EEG activity. Part 1. Analytic amplitude. Clin. Neurophysiol., 115, 2077–2088.

    Article  Google Scholar 

  • Freeman, W. J. (2004b). Origin, structure and role of background EEG activity. Part 2. Analytic phase. Clin. Neurophysiol., 115, 2089–2107.

    Article  Google Scholar 

  • Freeman, W. J. (2005). Origin, structure and role of background EEG activity. Part 3. Neural frame classification. Clin. Neurophysiol., 116, 1118–1129.

    Article  Google Scholar 

  • Freeman, W. J. (2006). Origin, structure and role of background EEG activity. Part 4. Neural frame simulation. Clin. Neurophysiol., 117, 572–589.

    Article  Google Scholar 

  • Freeman, W. J. (2009). Freeman’s mass action. www.scholarpedia.org/article/Freeman%27s_mass_action_#.

  • Freeman, W. J., & Barrie, J. M. (2000). Analysis of spatial patterns of phase in neocortical gamma EEGs in rabbit. J. Neurophysiol., 84, 1266–1278.

    Google Scholar 

  • Fukushima, Y., Tsukada, M., Tsuda, I., Yamaguti, Y., & Kuroda, S. (2007). Spatial clustering property and self-similarity in membrane potentials of hippocampal CA1 neurons for spatio-temporal input sequence. Cogn. Neurodyn., 1, 305–316.

    Article  Google Scholar 

  • Gray, C. M., & Singer, W. (1989). Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc. Nat. Acad. Sci., 86, 1698–1702.

    Article  Google Scholar 

  • Gray, C. M., Konig, P., Engel, A. K., & Singer, W. (1989). Oscillatory responses in cat visual cortex exhibit intercolumnar synchronisation which reflects global stimulus properties. Nature, 388, 334–337.

    Article  Google Scholar 

  • Gray, C. M., Engel, A. K., Konig, P., & Singer, W. (1992). Synchronization of oscillatory neuronal responses in cat striate cortex: temporal properties. Vis. Neurosci., 8, 337–347.

    Article  Google Scholar 

  • Grossberg, S. (1980). How does a brain build a cognitive code? Psychol. Rev., 87, 1–51.

    Article  Google Scholar 

  • Haken, H. (1976). Principles of brain functioning. Berlin: Springer.

    Google Scholar 

  • Hasenstaub, A., Shu, Y., Haider, B., Kraushaar, U., Duque, A., & McCormick, D. A. (2005). Inhibitory postsynaptic potentials carry synchronized frequency information in active cortical networks. Neuron, 47, 423–435.

    Article  Google Scholar 

  • Hebb, D. O. (1949). The organization of behavior. New York: Wiley.

    Google Scholar 

  • Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci., 79, 2554–2558.

    Article  MathSciNet  Google Scholar 

  • Jirsa, V. K., & Haken, H. (1996). Field theory of electromagnetic brain activity. Phys. Rev. Lett., 77, 960–963.

    Article  Google Scholar 

  • Jirsa, V. K., & Stefanescu, R. A. (2010). Neural population codes capture biologically realistic large scale network dynamics. This Special Issue, BMB.

  • Kandel, E. R., Schwartz, J. H., & Jessell, T. M. (1991). Principles of neural science (3rd edn.). Englewood Cliffs: Prentice-Hall International (p. 276 and assoc.).

    Google Scholar 

  • Kay, J. W., & Phillips, W. A. (2010). Coherent infomax as a computational goal for neural systems. This Special Issue, BMB.

  • Kirkpatrick, S., Gellatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671–680.

    Article  MathSciNet  Google Scholar 

  • Liley, D. T. J., & Wright, J. J. (1994). Intracortical connectivity of pyramidal and stellate cells: estimates of synaptic densities and coupling symmetry. Netw. Comput. Neural Syst., 5, 175–189.

    Article  MATH  Google Scholar 

  • Mathworld (2009). http://mathworld.wolfram.com/DifferentialEntropy.html.

  • Miltner, W. H., Braun, C., Arnold, M., Witte, H., & Taube, E. (1999). Coherence of gamma-band EEG activity as a basis for associative learning. Nature, 397, 434–436.

    Article  Google Scholar 

  • Mizraji, E., & Lin, J. (2010). Logic in a dynamic brain. This Special Issue, BMB.

  • Neuenschwander, S., & Singer, W. (1996). Long range synchronisation of oscillatory light responses in the cat retina and lateral geniculate nucleus. Nature, 379, 728–733.

    Article  Google Scholar 

  • Nunez, P. L. (1981). Electric fields of the brain. London: Oxford University Press.

    Google Scholar 

  • Nunez, P. L. (1995). Neocortical dynamics and human EEG rhythms. London: Oxford University Press.

    Google Scholar 

  • Pitts, W., & McCulloch, W. S. (1943). A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys., 5, 115–133.

    Article  MathSciNet  MATH  Google Scholar 

  • Potthast, R. P., beim Graben, P., & Wright, J. J. (2010). Emergence of cortical maps through synaptic competition and cooperation dynamics. This Special Issue, BMB.

  • Rennie, C. J., Wright, J. J., & Robinson, P. A. (2000). Mechanisms of cortical electrical activity and the emergence of gamma rhythm. J. Theor. Biol., 205, 17–35.

    Article  Google Scholar 

  • Rennie, C. J., Robinson, P. A., & Wright, J. J. (2002). Unified neurophysical model of EEG spectra and evoked potentials. Biol. Cybern., 86, 457–471.

    Article  MATH  Google Scholar 

  • Robinson, P. A., Wright, J. J., & Rennie, C. J. (1998). Synchronous oscillations in the cerebral cortex. Phys. Rev. E, 57, 4578–4588.

    Article  Google Scholar 

  • Saddy, D., beim Graben, P., Drenhaus, H., & Frisch, S. (2004). Distinguishing process from content in language processing: a new answer to an old question. In Proceedings of the 8th experimental chaos conference, American Institute of Physics. ISBN 0-7354-0226-4.

  • Scheibel, M. E., & Scheibel, B. A. (1970). Elementary processes in selected thalamic and cortical subsystems—the structural substrates. In Schmitt, F. O. (Ed.), The neurosciences: 2nd study program (pp. 443–457). New York: Rockefeller University Press.

    Google Scholar 

  • Schillen, T. B., & Konig, P. (1994). Binding by temporal structure in multiple feature domains of an oscillatory neural network. Biol. Cybern., 70, 397–405.

    Article  Google Scholar 

  • Scholl, D. A. (1956). The organization of the cerebral cortex. New York: Wiley.

    Google Scholar 

  • Singer, W. (1994). Putative functions of temporal correlations in neocortical processing. In Large scale neuronal theories of the brain. Cambridge: MIT Press.

    Google Scholar 

  • Singer, W., & Gray, C. M. (1995). Visual feature integration and the temporal correlation hypothesis. Ann. Rev. Neurosci., 18, 555–586.

    Article  Google Scholar 

  • Steriade, M., Timofeev, I., & Grenier, F. (2001). Natural waking and sleep states: a view from inside cortical neurons. J. Neurophysiol., 85, 1969–1985.

    Google Scholar 

  • Steyn-Ross, D. A., & Steyn-Ross, M. L. (Eds.) (2009). Modeling phase transitions in the brain. Berlin: Springer. ISBN: 978-1-4419-0795-0 (in press).

    Google Scholar 

  • Steyn-Ross, M. L., Steyn-Ross, D. A., Sleigh, J. W., & Wilson, M. T. (2010). A mechanism for ultra-slow oscillations in the cortical default network. This Special Issue, BMB.

  • Stryker, M. P. (1989). Is grandmother an oscillation? Nature, 388, 297–298.

    Article  Google Scholar 

  • Traub, R. D., Whittington, M. A., Stanford, I. M., & Jefferys, J. G. R. (1996). A mechanism for generation of long-range synchronous fast oscillations in the cortex. Nature, 383, 621–624.

    Article  Google Scholar 

  • Tsuda, I., & Kuroda, S. (2004). A complex systems approach to an interpretation of dynamic brain activity II: does Cantor coding provide a dynamic model for the formation of episodic memory? In Erdi, P. et al. (Eds.), LCNS : Vol. 3146. Cortical dynamics. Berlin: Springer.

    Google Scholar 

  • Tsukada, M., & Fukushima, Y. (2010). A context sensitive mechanism in hippocampal CA1 networks. This Special Issue, BMB.

  • van Rotterdam, A., Lopes da Silva, F. H., van den Ende, J., Viergever, M. A., & Hermans, A. J. (1982). A model of the spatio-temporal characteristics of the alpha rhythm. Bull. Math. Biol., 44, 283–305.

    Article  MathSciNet  MATH  Google Scholar 

  • von der Malsburg, C. (1983). How are nervous structures organised? In Basar, E., Flohr, H., Haken, H., & Mandell, A. J. (Eds.), Synergetics of the brain. Berlin, Heidelberg, New York: Springer.

    Google Scholar 

  • Whittington, M. A., Faulkner, H. J., Doheny, H. C., & Traub, R. D. (2000). Neuronal fast oscillations as a target site for psychoactive drugs. Pharmacol. Ther., 86, 171–190.

    Article  Google Scholar 

  • Wilson, H. R., & Cowan, J. D. (1973). A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13, 55–80.

    Article  Google Scholar 

  • Wright, J. J. (2009a). Generation and control of cortical gamma: findings from simulation at two scales. Neural Netw., 22, 373–384.

    Article  Google Scholar 

  • Wright, J. J. (2009b). Cortical phase transitions: properties demonstrated in continuum simulations at mesocopic and macroscopic scales. New Math. Nat. Comput., 5, 159–183.

    Article  MATH  Google Scholar 

  • Wright, J. J., & Liley, D. T. J. (1996). Dynamics of the brain at global and microscopic scales. Neural networks and the EEG. Behav. Brain Sci., 19, 285–320.

    Article  Google Scholar 

  • Wright, J. J., Bourke, P. D., & Chapman, C. L. (2000). Synchronous oscillation in the cerebral cortex and object coherence: simulation of basic electrophysiological findings. Biol. Cybern., 83, 341–353.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Liggins Institute and Department of Psychological Medicine, University of Auckland, Auckland, New Zealand

    J. J. Wright

Authors
  1. J. J. Wright
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. J. Wright.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s11538-011-9639-3

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wright, J.J. Attractor Dynamics and Thermodynamic Analogies in the Cerebral Cortex: Synchronous Oscillation, the Background EEG, and the Regulation of Attention. Bull Math Biol 73, 436–457 (2011). https://doi.org/10.1007/s11538-010-9562-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-010-9562-z

Keywords

Search

Navigation