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Stimulus-induced Epileptic Spike-Wave Discharges in Thalamocortical Model with Disinhibition

View Article: PubMed Central - PubMed

ABSTRACT

Epileptic absence seizure characterized by the typical 2–4 Hz spike-wave discharges (SWD) are known to arise due to the physiologically abnormal interactions within the thalamocortical network. By introducing a second inhibitory neuronal population in the cortical system, here we propose a modified thalamocortical field model to mathematically describe the occurrences and transitions of SWD under the mutual functions between cortex and thalamus, as well as the disinhibitory modulations of SWD mediated by the two different inhibitory interneuronal populations. We first show that stimulation can induce the recurrent seizures of SWD in the modified model. Also, we demonstrate the existence of various types of firing states including the SWD. Moreover, we can identify the bistable parametric regions where the SWD can be both induced and terminated by stimulation perturbations applied in the background resting state. Interestingly, in the absence of stimulation disinhibitory functions between the two different interneuronal populations can also both initiate and abate the SWD, which suggests that the mechanism of disinhibition is comparable to the effect of stimulation in initiating and terminating the epileptic SWD. Hopefully, the obtained results can provide theoretical evidences in exploring dynamical mechanism of epileptic seizures.

No MeSH data available.


Related in: MedlinePlus

Bifurcation diagram.(a) Minima and maxima of time series for different values of k4. The blue and red lines represent the limit cycle and fixed point, respectively. The region indicated by yellow dashed rectangle represents the bistable parameter region between non-seizure and SWD states. The region indicated by white dashed rectangle represents the bistable region between non-seizure and clonic states. Pink and purple double arrows correspond to the SWD and clonic oscillations. Green double arrow indicates the kindling stimulation performed on the background resting state (stable focus) and the red arrow indicates the anti-kindling stimulation for the SWD discharges. (b) For the much small values of k4, there is one ustable focus and one stable limit cycle, all simulations converge to the simple tonic oscillations (stable limit cycle). The tonic oscillation disappears and evolves into the low saturated firings following a supercritical Hopf bifurcation (HB1). A bistable region exists with the stable focus and the stale limit cycle following a fold of cycles bifurcation (LPC1). The monostable SWD discharges and clonic oscillations exist following the subcritical Hopf bifurcation (HB2). Beyond the reappearance of the stable focus due to the subcritical Hopf bifurcation (HB3), another bistable region occurs. For much large values of k4 the bistable region disappears due to the second fold limit cycle bifurcation (LPC2).
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f7: Bifurcation diagram.(a) Minima and maxima of time series for different values of k4. The blue and red lines represent the limit cycle and fixed point, respectively. The region indicated by yellow dashed rectangle represents the bistable parameter region between non-seizure and SWD states. The region indicated by white dashed rectangle represents the bistable region between non-seizure and clonic states. Pink and purple double arrows correspond to the SWD and clonic oscillations. Green double arrow indicates the kindling stimulation performed on the background resting state (stable focus) and the red arrow indicates the anti-kindling stimulation for the SWD discharges. (b) For the much small values of k4, there is one ustable focus and one stable limit cycle, all simulations converge to the simple tonic oscillations (stable limit cycle). The tonic oscillation disappears and evolves into the low saturated firings following a supercritical Hopf bifurcation (HB1). A bistable region exists with the stable focus and the stale limit cycle following a fold of cycles bifurcation (LPC1). The monostable SWD discharges and clonic oscillations exist following the subcritical Hopf bifurcation (HB2). Beyond the reappearance of the stable focus due to the subcritical Hopf bifurcation (HB3), another bistable region occurs. For much large values of k4 the bistable region disappears due to the second fold limit cycle bifurcation (LPC2).

Mentions: In Fig. 7, the maxima and minima of the model output for different values of the parameter k4 are shown. Compared Fig. 7(a) to 7(b), we can see that for the much small values (k4 < ≈ 0.7, left side of figure) there is one unstable focus and one stable limit cycle, hence all the simulations converge to the simple tonic oscillations (stable limit cycle). For the less small values (0.7 < ≈ k4 < ≈ 0.99), the tonic oscillations of the system disappear and evolve into the low saturated firings, with only one stable focus. This follows a supercritical Hopf bifurcation at k4 ≈ 0.7 (HB1, Fig. 7(b)). However, for the less large values of k4 (0.99 < ≈ k4 < ≈ 1.14, the area of Fig. 7(a) indicated by the yellow dashed rectangle) a bistable region exists with the stable focus and the stale limit cycle. This arises following a fold of cycles bifurcation (LPC1, Fig. 7(b)) at k4 ≈ 0.99 with giving birth to one stable limit cycle and one unstable. Then the system transits from the low saturated firings to the bistable states between the non-seizure state and SWD oscillations. Consecutively, when k4 ≈ > 1.14, the stable focus loses its stability due to another subcritical Hopf bifurcation (HB2, Fig. 7(b)) at k4 ≈ 1.14, and the monostable SWD discharges and simple slow waves (clonic) oscillations exist in the region of 1.14 < ≈ k4 < ≈ 1.48. Beyond the reappearance of the stable focus due to the subcritical Hopf bifurcation (HB3, Fig. 7(b)), another bistable region indicated by the white dashed rectangle (Fig. 7(a)) between stable focus and slow wave (clonic) oscillation exists at 1.48 < ≈ k4 < ≈ 1.64. However, because of the absence of SWD, this bistable region will not be particularly elaborated. For much large values of k4 (1.64 < ≈ k4 < 2) the limit cycles disappears due to the second fold limit cycle bifurcation (LPC2, Fig. 7(b)) at k4 ≈ 1.64 and all the simulations converge to the steady high saturated firing state. In addition, in the region immediately preceding the bifurcation at k4 ≈ 1 complex excitable transients occur lasting several seconds.


Stimulus-induced Epileptic Spike-Wave Discharges in Thalamocortical Model with Disinhibition
Bifurcation diagram.(a) Minima and maxima of time series for different values of k4. The blue and red lines represent the limit cycle and fixed point, respectively. The region indicated by yellow dashed rectangle represents the bistable parameter region between non-seizure and SWD states. The region indicated by white dashed rectangle represents the bistable region between non-seizure and clonic states. Pink and purple double arrows correspond to the SWD and clonic oscillations. Green double arrow indicates the kindling stimulation performed on the background resting state (stable focus) and the red arrow indicates the anti-kindling stimulation for the SWD discharges. (b) For the much small values of k4, there is one ustable focus and one stable limit cycle, all simulations converge to the simple tonic oscillations (stable limit cycle). The tonic oscillation disappears and evolves into the low saturated firings following a supercritical Hopf bifurcation (HB1). A bistable region exists with the stable focus and the stale limit cycle following a fold of cycles bifurcation (LPC1). The monostable SWD discharges and clonic oscillations exist following the subcritical Hopf bifurcation (HB2). Beyond the reappearance of the stable focus due to the subcritical Hopf bifurcation (HB3), another bistable region occurs. For much large values of k4 the bistable region disappears due to the second fold limit cycle bifurcation (LPC2).
© Copyright Policy - open-access
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC5120301&req=5

f7: Bifurcation diagram.(a) Minima and maxima of time series for different values of k4. The blue and red lines represent the limit cycle and fixed point, respectively. The region indicated by yellow dashed rectangle represents the bistable parameter region between non-seizure and SWD states. The region indicated by white dashed rectangle represents the bistable region between non-seizure and clonic states. Pink and purple double arrows correspond to the SWD and clonic oscillations. Green double arrow indicates the kindling stimulation performed on the background resting state (stable focus) and the red arrow indicates the anti-kindling stimulation for the SWD discharges. (b) For the much small values of k4, there is one ustable focus and one stable limit cycle, all simulations converge to the simple tonic oscillations (stable limit cycle). The tonic oscillation disappears and evolves into the low saturated firings following a supercritical Hopf bifurcation (HB1). A bistable region exists with the stable focus and the stale limit cycle following a fold of cycles bifurcation (LPC1). The monostable SWD discharges and clonic oscillations exist following the subcritical Hopf bifurcation (HB2). Beyond the reappearance of the stable focus due to the subcritical Hopf bifurcation (HB3), another bistable region occurs. For much large values of k4 the bistable region disappears due to the second fold limit cycle bifurcation (LPC2).
Mentions: In Fig. 7, the maxima and minima of the model output for different values of the parameter k4 are shown. Compared Fig. 7(a) to 7(b), we can see that for the much small values (k4 < ≈ 0.7, left side of figure) there is one unstable focus and one stable limit cycle, hence all the simulations converge to the simple tonic oscillations (stable limit cycle). For the less small values (0.7 < ≈ k4 < ≈ 0.99), the tonic oscillations of the system disappear and evolve into the low saturated firings, with only one stable focus. This follows a supercritical Hopf bifurcation at k4 ≈ 0.7 (HB1, Fig. 7(b)). However, for the less large values of k4 (0.99 < ≈ k4 < ≈ 1.14, the area of Fig. 7(a) indicated by the yellow dashed rectangle) a bistable region exists with the stable focus and the stale limit cycle. This arises following a fold of cycles bifurcation (LPC1, Fig. 7(b)) at k4 ≈ 0.99 with giving birth to one stable limit cycle and one unstable. Then the system transits from the low saturated firings to the bistable states between the non-seizure state and SWD oscillations. Consecutively, when k4 ≈ > 1.14, the stable focus loses its stability due to another subcritical Hopf bifurcation (HB2, Fig. 7(b)) at k4 ≈ 1.14, and the monostable SWD discharges and simple slow waves (clonic) oscillations exist in the region of 1.14 < ≈ k4 < ≈ 1.48. Beyond the reappearance of the stable focus due to the subcritical Hopf bifurcation (HB3, Fig. 7(b)), another bistable region indicated by the white dashed rectangle (Fig. 7(a)) between stable focus and slow wave (clonic) oscillation exists at 1.48 < ≈ k4 < ≈ 1.64. However, because of the absence of SWD, this bistable region will not be particularly elaborated. For much large values of k4 (1.64 < ≈ k4 < 2) the limit cycles disappears due to the second fold limit cycle bifurcation (LPC2, Fig. 7(b)) at k4 ≈ 1.64 and all the simulations converge to the steady high saturated firing state. In addition, in the region immediately preceding the bifurcation at k4 ≈ 1 complex excitable transients occur lasting several seconds.

View Article: PubMed Central - PubMed

ABSTRACT

Epileptic absence seizure characterized by the typical 2&ndash;4&thinsp;Hz spike-wave discharges (SWD) are known to arise due to the physiologically abnormal interactions within the thalamocortical network. By introducing a second inhibitory neuronal population in the cortical system, here we propose a modified thalamocortical field model to mathematically describe the occurrences and transitions of SWD under the mutual functions between cortex and thalamus, as well as the disinhibitory modulations of SWD mediated by the two different inhibitory interneuronal populations. We first show that stimulation can induce the recurrent seizures of SWD in the modified model. Also, we demonstrate the existence of various types of firing states including the SWD. Moreover, we can identify the bistable parametric regions where the SWD can be both induced and terminated by stimulation perturbations applied in the background resting state. Interestingly, in the absence of stimulation disinhibitory functions between the two different interneuronal populations can also both initiate and abate the SWD, which suggests that the mechanism of disinhibition is comparable to the effect of stimulation in initiating and terminating the epileptic SWD. Hopefully, the obtained results can provide theoretical evidences in exploring dynamical mechanism of epileptic seizures.

No MeSH data available.


Related in: MedlinePlus