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Pattern formation in a mean field model of electrocortical activity

BMC Neuroscience201314 (Suppl 1) :P149

https://doi.org/10.1186/1471-2202-14-S1-P149

  • Published:

Keywords

  • Periodic Orbit
  • Hopf Bifurcation
  • Dynamical System Approach
  • Damp Wave Equation
  • Snap Shot

Mean field models of cortical activity describe the electrical potentials and interactions of cortical neuron populations, coarse-grained over the scale of a few macro columns. Such models can be analysed as dynamical systems, in particular as time-dependent partial differential or integro-differential equations, which may have complicating aspects such as explicit time delays and stochastic terms. We analyse a model of intermediate complexity, formulated by Liley et al. [1]. This model describes an excitatory and an inhibitory population in a simple geometry, on which the effect of long-range connections is represented by a damped wave equations.

Although somewhat rudimentary from a physiological point of view, this model has been shown to predict several features of electrocortical dynamics rather well (see, e.g., [2, 3] and refs therein) and is challenging to analyse mathematically. It consists of fourteen coupled partial differential equations with strong nonlinearities.

Where previous analysis on this model, and similar mean-field models, has used drastic simplifications, such as reduction to zero or one spatial dimensions or a single population, we developed tools for parsing the dynamics of the full-fledged equations [4]. Using the open-source library PETSc [5], we have implemented fully implicit time-stepping for the field equations and the tangent linear model, as well as arclength continuation for equilibrium and time-periodic solutions. All computations are performed in parallel using domain decomposition.

In the current application of these tools, we focus on physiologically interesting γ-range activity [3]. This activity is triggered by a Hopf bifurcation under small variations of the local inhibitory to inhibitory connection density. We computed the saddle-type periodic orbit that regulates the transient dynamics of perturbations to the base equilibrium state. Two snap shots of this orbit, computed on a 12.8 by 12.8 cm domain, with 0.5 mm resolution, are shown in Figure 1. The period of this orbit corresponds to a 12 Hz oscillation, whereas the final, attracting, state has a strong 40 Hz peak in the power spectrum [3].
Figure 1
Figure 1

Partial bifurcation diagram near the primary instability, which is a subcritical Hopf bifurcation.

In ongoing work we are investigating bifurcations of this periodic orbit, which can turn completely stable or give rise to more complicated solutions, such as quasi-periodic or chaotic. This dynamical systems approach to the analysis of mean-field models should give us more insight in the complex model behaviour.
Figure 2
Figure 2

Two snap shots of the excitatory membrane potential along the saddle periodic orbit. The potential ranges from -67 mV (dark red) to -52 mV (dark blue).

Authors’ Affiliations

(1)
Faculty of Science, University of Ontario Institute of Technology, Oshawa, Ontario, L1H 7K4, Canada

References

  1. Liley DTJ, Cadusch PJ, Dafilis MP: A spatially continuous mean field theory of electrocortical activity. Network: comput Neural syst. 2002, 13: 67-113.View ArticleGoogle Scholar
  2. Coombes S: Large-scale neural dynamics: Simple and complex. NeuroImage. 2010, 52: 731-739. 10.1016/j.neuroimage.2010.01.045.View ArticlePubMedGoogle Scholar
  3. Bojak I, Liley DTJ: Self-organized 40 Hz synchronization in a physiological theory of EEG. Neurocomput. 2007, 70: 2085-2090. 10.1016/j.neucom.2006.10.087.View ArticleGoogle Scholar
  4. Green KR, van Veen L: Open-source tools for dynamical analysis of Liley's mean-field cortex model. arXiv:1210.4784Google Scholar
  5. Balay S, et al: PETSc users manual. Argonne National Laboratory. ANL-25/11R3.2, [http://www.mcs.anl.gov/petsc]

Copyright

© Veen and Green; licensee BioMed Central Ltd. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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