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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

Bifurcation diagrams for Gaussian (top) and sigmoid (bottom) activation function. The background input B and the coupling parameter  are varied. Other parameters as in Sect. 2. Bifurcation curves are indicated with color: saddle-node (blue), Hopf (red), limit point of cycles (black), homoclinic to saddle (green). The red dashed line indicates a neutral saddle, which is not a bifurcation but here an LPC emerges from a homoclinic bifurcation. Note that our diagram for the sigmoid differs from [22] as we have modified the activation functions
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Fig5: Bifurcation diagrams for Gaussian (top) and sigmoid (bottom) activation function. The background input B and the coupling parameter are varied. Other parameters as in Sect. 2. Bifurcation curves are indicated with color: saddle-node (blue), Hopf (red), limit point of cycles (black), homoclinic to saddle (green). The red dashed line indicates a neutral saddle, which is not a bifurcation but here an LPC emerges from a homoclinic bifurcation. Note that our diagram for the sigmoid differs from [22] as we have modified the activation functions

Mentions: The additional steady state is a robust feature that coexists with the normal dynamical repertoire of the Wilson–Cowan model with a sigmoid. To show this, consider the bifurcation diagram in the -parameter plane as shown in Fig. 5. We have chosen to vary these parameters as this combination controls the level of activity of the populations and the strength of the feedback loop, and hence the dynamics, i.e. stable steady states and periodic oscillations. An earlier study [22] also presented a bifurcation analysis for the sigmoid case varying these parameters. Hence we can compare the two diagrams, where most bifurcation curves are similar. Our shift in thresholds , results in a larger region with stable oscillations than in [22] for both Gaussian and sigmoid. For the Gaussian we see that there is an additional saddle-node bifurcation curve, not present for the sigmoid, which corresponds to the additional steady state. It is characterized by high values of and to lower values of B, such that the excitatory population can drive the inhibitory population into depolarization block. Fig. 5


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)

Bifurcation diagrams for Gaussian (top) and sigmoid (bottom) activation function. The background input B and the coupling parameter  are varied. Other parameters as in Sect. 2. Bifurcation curves are indicated with color: saddle-node (blue), Hopf (red), limit point of cycles (black), homoclinic to saddle (green). The red dashed line indicates a neutral saddle, which is not a bifurcation but here an LPC emerges from a homoclinic bifurcation. Note that our diagram for the sigmoid differs from [22] as we have modified the activation functions
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig5: Bifurcation diagrams for Gaussian (top) and sigmoid (bottom) activation function. The background input B and the coupling parameter are varied. Other parameters as in Sect. 2. Bifurcation curves are indicated with color: saddle-node (blue), Hopf (red), limit point of cycles (black), homoclinic to saddle (green). The red dashed line indicates a neutral saddle, which is not a bifurcation but here an LPC emerges from a homoclinic bifurcation. Note that our diagram for the sigmoid differs from [22] as we have modified the activation functions
Mentions: The additional steady state is a robust feature that coexists with the normal dynamical repertoire of the Wilson–Cowan model with a sigmoid. To show this, consider the bifurcation diagram in the -parameter plane as shown in Fig. 5. We have chosen to vary these parameters as this combination controls the level of activity of the populations and the strength of the feedback loop, and hence the dynamics, i.e. stable steady states and periodic oscillations. An earlier study [22] also presented a bifurcation analysis for the sigmoid case varying these parameters. Hence we can compare the two diagrams, where most bifurcation curves are similar. Our shift in thresholds , results in a larger region with stable oscillations than in [22] for both Gaussian and sigmoid. For the Gaussian we see that there is an additional saddle-node bifurcation curve, not present for the sigmoid, which corresponds to the additional steady state. It is characterized by high values of and to lower values of B, such that the excitatory population can drive the inhibitory population into depolarization block. Fig. 5

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