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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

Phase portraits for Gaussian FRF. Characteristic phase portraits for all 19 regions for a single local population. Numbers correspond to parameter values in areas as in Fig. 5. Red indicates equilibrium or limit cycle, stable manifolds are green, unstable manifolds are blue and orbits yellow
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Fig6: Phase portraits for Gaussian FRF. Characteristic phase portraits for all 19 regions for a single local population. Numbers correspond to parameter values in areas as in Fig. 5. Red indicates equilibrium or limit cycle, stable manifolds are green, unstable manifolds are blue and orbits yellow

Mentions: For a complete understanding of the bifurcation diagram for the Gaussian case, we have generated characteristic phase portraits for all 19 parameter regions; see Fig. 6. Starting in region 1, we find a single low stable equilibrium. Crossing a saddle-node bifurcation to areas 2 or 5, two equilibria with high excitatory activity appear. Whereas in area 2 depolarization block plays a role, in area 5 the coupling is too low for depolarization block to occur and the inhibitory population is active too. Next, crossing saddle-node bifurcations to area 3, there is a single stable node again, while in area 4 we have three equilibria, one saddle, one with stable low activity and one with high excitatory and high inhibitory activity, different from the one in area 2. On the saddle-node bifurcation curves we find, in total, four Bogdanov–Takens (BT) bifurcations. From each BT-point a Hopf curve emerges and each of these ends up in another BT-point. Along a Hopf bifurcation we find degeneracies where the Hopf bifurcation changes from super- to subcritical. Here a limit point of cycle (LPC) bifurcation curve emerges that ends in a point where the saddle along a homoclinic curve is a neutral saddle (NH). The homoclinic curves either end in saddle-node homoclinics (SNIC) or connect to another BT-point. The parameter region for which we find stable oscillations, is made up of areas 7, 10, 11, 14, 16, 19, and it is delineated by Hopf, homoclinic, LPC and SNIC bifurcation curves. All other transitions involve unstable invariant sets, and therefore we do not discuss them. Phase portraits in areas 1&3, 2&4&5&18, 12&13&17, 9&15, 10&16 and 11&14 are structurally equivalent, but are shown for completeness as the amount of inhibitory activity varies. Fig. 6


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)

Phase portraits for Gaussian FRF. Characteristic phase portraits for all 19 regions for a single local population. Numbers correspond to parameter values in areas as in Fig. 5. Red indicates equilibrium or limit cycle, stable manifolds are green, unstable manifolds are blue and orbits yellow
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig6: Phase portraits for Gaussian FRF. Characteristic phase portraits for all 19 regions for a single local population. Numbers correspond to parameter values in areas as in Fig. 5. Red indicates equilibrium or limit cycle, stable manifolds are green, unstable manifolds are blue and orbits yellow
Mentions: For a complete understanding of the bifurcation diagram for the Gaussian case, we have generated characteristic phase portraits for all 19 parameter regions; see Fig. 6. Starting in region 1, we find a single low stable equilibrium. Crossing a saddle-node bifurcation to areas 2 or 5, two equilibria with high excitatory activity appear. Whereas in area 2 depolarization block plays a role, in area 5 the coupling is too low for depolarization block to occur and the inhibitory population is active too. Next, crossing saddle-node bifurcations to area 3, there is a single stable node again, while in area 4 we have three equilibria, one saddle, one with stable low activity and one with high excitatory and high inhibitory activity, different from the one in area 2. On the saddle-node bifurcation curves we find, in total, four Bogdanov–Takens (BT) bifurcations. From each BT-point a Hopf curve emerges and each of these ends up in another BT-point. Along a Hopf bifurcation we find degeneracies where the Hopf bifurcation changes from super- to subcritical. Here a limit point of cycle (LPC) bifurcation curve emerges that ends in a point where the saddle along a homoclinic curve is a neutral saddle (NH). The homoclinic curves either end in saddle-node homoclinics (SNIC) or connect to another BT-point. The parameter region for which we find stable oscillations, is made up of areas 7, 10, 11, 14, 16, 19, and it is delineated by Hopf, homoclinic, LPC and SNIC bifurcation curves. All other transitions involve unstable invariant sets, and therefore we do not discuss them. Phase portraits in areas 1&3, 2&4&5&18, 12&13&17, 9&15, 10&16 and 11&14 are structurally equivalent, but are shown for completeness as the amount of inhibitory activity varies. Fig. 6

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