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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

Dynamics of the asymmetric in-phase oscillation. Left: Time-series of the activity of excitatory and inhibitory populations of the asymmetric in-phase oscillation for  and . Middle: corresponding time-series of the input currents. Right: The dynamical range of input currents J along the excitatory (blue) and inhibitory (black) activation functions
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Fig8: Dynamics of the asymmetric in-phase oscillation. Left: Time-series of the activity of excitatory and inhibitory populations of the asymmetric in-phase oscillation for and . Middle: corresponding time-series of the input currents. Right: The dynamical range of input currents J along the excitatory (blue) and inhibitory (black) activation functions

Mentions: Starting from with , we first follow the symmetric low steady state (black line) around . Increasing α, it becomes unstable at a saddle-node at . Following the symmetric branch, we get to the high steady state. It is stable between the two pitchfork bifurcations at and at . From an unstable asymmetric steady state emerges, which becomes stable at a saddle-node bifurcation at . For this stable asymmetric equilibrium with high coupling strength, one excitatory population drives the other into depolarization block. The asymmetric steady state near is unstable, but becomes stable at a saddle-node at . Then decreasing α from this saddle-node, we encounter a supercritical Hopf bifurcation H at . Here we find a solution branch of stable asymmetric in-phase limit cycles which ends in a saddle-node homoclinic bifurcation. The periodic orbit has small amplitude fluctuations (maximal amplitude ≈0.015) with high excitatory activity in one population. The amplitude in the other population is much larger as large as 0.2; see also Fig. 8 for a time-series. For this branch we have also plotted the range of input currents along the activation functions. It shows for population 1 that the input current is quite high but of small amplitude. For population 2 the values are lower but the ranges are larger. Since the EEG does not capture the spikes and filters out the DC-component, in an experiment this would give the counter-intuitive result of high spiking activity accompanied with low amplitude EEG output, whereas, in contrast, its neighbor has low spiking activity but a markedly higher amplitude EEG output. Fig. 8


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)

Dynamics of the asymmetric in-phase oscillation. Left: Time-series of the activity of excitatory and inhibitory populations of the asymmetric in-phase oscillation for  and . Middle: corresponding time-series of the input currents. Right: The dynamical range of input currents J along the excitatory (blue) and inhibitory (black) activation functions
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig8: Dynamics of the asymmetric in-phase oscillation. Left: Time-series of the activity of excitatory and inhibitory populations of the asymmetric in-phase oscillation for and . Middle: corresponding time-series of the input currents. Right: The dynamical range of input currents J along the excitatory (blue) and inhibitory (black) activation functions
Mentions: Starting from with , we first follow the symmetric low steady state (black line) around . Increasing α, it becomes unstable at a saddle-node at . Following the symmetric branch, we get to the high steady state. It is stable between the two pitchfork bifurcations at and at . From an unstable asymmetric steady state emerges, which becomes stable at a saddle-node bifurcation at . For this stable asymmetric equilibrium with high coupling strength, one excitatory population drives the other into depolarization block. The asymmetric steady state near is unstable, but becomes stable at a saddle-node at . Then decreasing α from this saddle-node, we encounter a supercritical Hopf bifurcation H at . Here we find a solution branch of stable asymmetric in-phase limit cycles which ends in a saddle-node homoclinic bifurcation. The periodic orbit has small amplitude fluctuations (maximal amplitude ≈0.015) with high excitatory activity in one population. The amplitude in the other population is much larger as large as 0.2; see also Fig. 8 for a time-series. For this branch we have also plotted the range of input currents along the activation functions. It shows for population 1 that the input current is quite high but of small amplitude. For population 2 the values are lower but the ranges are larger. Since the EEG does not capture the spikes and filters out the DC-component, in an experiment this would give the counter-intuitive result of high spiking activity accompanied with low amplitude EEG output, whereas, in contrast, its neighbor has low spiking activity but a markedly higher amplitude EEG output. Fig. 8

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