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Modeling focal epileptic activity in the Wilson-cowan model with depolarization block.

Meijer HG, Eissa TL, Kiewiet B, Neuman JF, Schevon CA, Emerson RG, Goodman RR, McKhann GM, Marcuccilli CJ, Tryba AK, Cowan JD, van Gils SA, van Drongelen W - J Math Neurosci (2015)

Bottom Line: We examined the effect of such a saturation in the Wilson-Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function.The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity.The online version of this article (doi:10.1186/s13408-015-0019-4) contains supplementary material 1.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Mathematics, MIRA Institute for Biomedical Engineering and Technical Medicine, University of Twente, Postbus 217, Enschede, 7500AE The Netherlands.

ABSTRACT

Unlabelled: Measurements of neuronal signals during human seizure activity and evoked epileptic activity in experimental models suggest that, in these pathological states, the individual nerve cells experience an activity driven depolarization block, i.e. they saturate. We examined the effect of such a saturation in the Wilson-Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function. We discuss experimental recordings during a seizure that support this substitution. Next we perform a bifurcation analysis on the Wilson-Cowan model with a Gaussian activation function. The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity. Analysis of coupled local networks then shows that such high activity can stay localized or spread. Specifically, in a spatial continuum we show a wavefront with inhibition leading followed by excitatory activity. We relate our model simulations to observations of spreading activity during seizures.

Electronic supplementary material: The online version of this article (doi:10.1186/s13408-015-0019-4) contains supplementary material 1.

No MeSH data available.


Related in: MedlinePlus

One parameter bifurcation diagram for  (top) and  (bottom). Colors indicate solution types: symmetric (black) and asymmetric (blue) steady states and symmetric (green) and in-phase asymmetric (red) and anti-phase asymmetric (light-blue) oscillations. Bifurcation labels are SN for saddle-node, PF for pitchfork, and H for Hopf. For the asymmetric branches, the upper part corresponds to one population, say , and then the lower part corresponds to the other population . The extremal values of E for quasi-periodic oscillations are indicated by purple lines. Thick lines indicate stable solution branches, thin dashed lines correspond to unstable branches
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Fig7: One parameter bifurcation diagram for (top) and (bottom). Colors indicate solution types: symmetric (black) and asymmetric (blue) steady states and symmetric (green) and in-phase asymmetric (red) and anti-phase asymmetric (light-blue) oscillations. Bifurcation labels are SN for saddle-node, PF for pitchfork, and H for Hopf. For the asymmetric branches, the upper part corresponds to one population, say , and then the lower part corresponds to the other population . The extremal values of E for quasi-periodic oscillations are indicated by purple lines. Thick lines indicate stable solution branches, thin dashed lines correspond to unstable branches

Mentions: Here we discuss the dynamical behavior for two coupled populations. Above we have discussed the bifurcation diagram for a single excitatory-inhibitory pair. We fix from now on to ensure the additional steady states exists. We choose two representative values for B with different dynamics for a single pair. For , we have two stable equilibria, one with high and the other with low excitatory activity. For , the stable high activity equilibrium remains, but the other attractor is a stable oscillation. This corresponds to areas 8 and 16 in Fig. 6. For both values, we construct a one parameter diagram by varying α the coupling strength between excitatory populations; see Fig. 7. Here, for continuity, we also show what happens for negative α, although this is not relevant neurophysiologically. Also, we omit several bifurcations and unstable branches that would obscure the presentation. The complete diagrams can be found in the supplementary material. Fig. 7


Modeling focal epileptic activity in the Wilson-cowan model with depolarization block.

Meijer HG, Eissa TL, Kiewiet B, Neuman JF, Schevon CA, Emerson RG, Goodman RR, McKhann GM, Marcuccilli CJ, Tryba AK, Cowan JD, van Gils SA, van Drongelen W - J Math Neurosci (2015)

One parameter bifurcation diagram for  (top) and  (bottom). Colors indicate solution types: symmetric (black) and asymmetric (blue) steady states and symmetric (green) and in-phase asymmetric (red) and anti-phase asymmetric (light-blue) oscillations. Bifurcation labels are SN for saddle-node, PF for pitchfork, and H for Hopf. For the asymmetric branches, the upper part corresponds to one population, say , and then the lower part corresponds to the other population . The extremal values of E for quasi-periodic oscillations are indicated by purple lines. Thick lines indicate stable solution branches, thin dashed lines correspond to unstable branches
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4385301&req=5

Fig7: One parameter bifurcation diagram for (top) and (bottom). Colors indicate solution types: symmetric (black) and asymmetric (blue) steady states and symmetric (green) and in-phase asymmetric (red) and anti-phase asymmetric (light-blue) oscillations. Bifurcation labels are SN for saddle-node, PF for pitchfork, and H for Hopf. For the asymmetric branches, the upper part corresponds to one population, say , and then the lower part corresponds to the other population . The extremal values of E for quasi-periodic oscillations are indicated by purple lines. Thick lines indicate stable solution branches, thin dashed lines correspond to unstable branches
Mentions: Here we discuss the dynamical behavior for two coupled populations. Above we have discussed the bifurcation diagram for a single excitatory-inhibitory pair. We fix from now on to ensure the additional steady states exists. We choose two representative values for B with different dynamics for a single pair. For , we have two stable equilibria, one with high and the other with low excitatory activity. For , the stable high activity equilibrium remains, but the other attractor is a stable oscillation. This corresponds to areas 8 and 16 in Fig. 6. For both values, we construct a one parameter diagram by varying α the coupling strength between excitatory populations; see Fig. 7. Here, for continuity, we also show what happens for negative α, although this is not relevant neurophysiologically. Also, we omit several bifurcations and unstable branches that would obscure the presentation. The complete diagrams can be found in the supplementary material. Fig. 7

Bottom Line: We examined the effect of such a saturation in the Wilson-Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function.The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity.The online version of this article (doi:10.1186/s13408-015-0019-4) contains supplementary material 1.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Mathematics, MIRA Institute for Biomedical Engineering and Technical Medicine, University of Twente, Postbus 217, Enschede, 7500AE The Netherlands.

ABSTRACT

Unlabelled: Measurements of neuronal signals during human seizure activity and evoked epileptic activity in experimental models suggest that, in these pathological states, the individual nerve cells experience an activity driven depolarization block, i.e. they saturate. We examined the effect of such a saturation in the Wilson-Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function. We discuss experimental recordings during a seizure that support this substitution. Next we perform a bifurcation analysis on the Wilson-Cowan model with a Gaussian activation function. The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity. Analysis of coupled local networks then shows that such high activity can stay localized or spread. Specifically, in a spatial continuum we show a wavefront with inhibition leading followed by excitatory activity. We relate our model simulations to observations of spreading activity during seizures.

Electronic supplementary material: The online version of this article (doi:10.1186/s13408-015-0019-4) contains supplementary material 1.

No MeSH data available.


Related in: MedlinePlus