Closed-loop feedback control and bifurcation analysis of epileptiform activity via optogenetic stimulation in a mathematical model of human cortex

Optogenetics provides a method of neuron stimulation that has high spatial, temporal, and cell-type specificity. Here we present a model of optogenetic feedback control that targets the inhibitory population, which expresses light-sensitive channelrhodopsin-2 channels, in a mean-field model of undifferentiated cortex that is driven to seizures. The inhibitory population is illuminated with an intensity that is a function of electrode measurements obtained via the cortical model. We test the efficacy of this control method on seizurelike activity observed in two parameter spaces of the cortical model that most closely correspond to seizures observed in patients. We also compare the effect of closed-loop and open-loop control on seizurelike activity using a less-complicated ordinary differential equation model of the undifferentiated cortex in parameter space. Seizurelike activity is successfully suppressed in both parameter planes using optimal illumination intensities less likely to have adverse effects on cortical tissue.

Figure 1

Seizures in the ${\text{P}}_{ee}-{\mathrm{\Gamma }}_e$ space. ${\mathrm{P}}_{ee}$ is gradually increased then decreased in time with a maximum value of 548.0 as shown in the figure on the extreme right. ${\mathrm{\Gamma }}_e$ is held constant at 0.0008 with $\alpha =1.6$. All other parameters are similar to baseline parameters in Ref. [11]. The mean soma potential (${\tilde{h}}_e$) at a point, and traveling seizure waves depicted by a one-dimensional slice of the two-dimensional cortex are shown in the first two figures on the left.

Figure 2

Effect of closed-loop optogenetic control on seizures in the ${\text{P}}_{ee}-{\mathrm{\Gamma }}_e$ space using a temporal modulation of ${\mathrm{P}}_{ee}$ in time shown in Fig. 1. ${\mathrm{\Gamma }}_e$ is held constant at 0.0008 with $\alpha =1.6$. All other parameters are similar to baseline parameters in Ref. [11]. Figures on the extreme left and right show the variation with time of $h_e$ and intensity, respectively, at the same point. The red dot-dashed line in the figure on the extreme right indicates the open-loop intensity required to suppress a fully formed seizures at $P_{ee}=548.0$. Gains used: $K_P=0.4$ and $K_I=3.6$.

Figure 3

Seizures in the ${\text{P}}_{ee}-{\mathrm{\Gamma }}_e$ space. ${\mathrm{P}}_{ee}$ is gradually increased then decreased in time with a maximum value of 548.0 as shown by the figure in the panel on the right. ${\mathrm{\Gamma }}_e$ is held constant at 0.00066 with $\alpha =1.15$. All other parameters are similar to baseline parameters in Ref. [11]. The mean soma potential (${\tilde{h}}_e$) at a point and traveling seizure waves depicted by a one-dimensional slice of the two-dimensional cortex are shown in the first two figures on the left.

Figure 4

Effect of closed-loop optogenetic control on seizures in the ${\text{P}}_{ee}-{\mathrm{\Gamma }}_e$ space using a temporal modulation of ${\mathrm{P}}_{ee}$ in time shown in Fig. 1. ${\mathrm{\Gamma }}_e$ is held constant at 0.00066 with $\alpha =1.15$. All other parameters are similar to baseline parameters in Ref. [11]. Figures on the extreme left and right show the variation with time of $h_e$ and intensity, respectively, at the same point. The red dot-dashed line in the figure on the extreme right indicates the open-loop intensity required to suppress a fully formed seizures at $P_{ee}=548.0, {\mathrm{\Gamma }}_e=0.0008$, and $\alpha =1.6$. Gains used: $K_P=0.4$ and $K_I=3.6$, which are the same gains used to suppress the robust seizure presented in Sec. 3a.

Figure 5

Comparison between the full SPDE model and the simpler ODE model. Left: Response diagram for the hyperexcited cortex with subcortical input $P_{ee}=548.0$. Amplitude of noise from stochastic subcortical inputs was reduced by an order of magnitude to 0.1 to demonstrate the effect of spatial terms on the dynamics of the model. The red jagged lines indicate maximum and minimum values of $h_e$ over 2 s for different ${\mathrm{\Gamma }}_e$, which is the influence of the synaptic input on the mean soma potential of the excitatory population, using the SPDE model. Black lines indicate the bifurcation diagram for the ODE model. Dashed and solid lines indicate unstable and stable fixed points, respectively. Maximum and minimum values of $h_e$ during stable (dot-dashed) and unstable (dashed) limit cycles arising from a subcritical Hopf bifurcation (asterisk) are also shown. Right: Comparison of the SPDE and ODE models in parameter space, with gray regions indicating seizure causing areas. The stochastic inputs in the SPDE model enhance the seizure area (indistinct boundary marked in gray) in parameter space when compared to the ODE model (darker region with sharp boundaries). Here we use $\alpha =1.6$, which corresponds to the amplitude of noise used with the SPDE model elsewhere in the manuscript.

Figure 6

Bifurcation analysis using an unstimulated cortex (left panel) and a cortex stimulated with light of constant 10 mW/${\mathrm{mm}}^2$ intensity in the open-loop configuration (center panel). Asterisks indicate subcritical Hopf bifurcations that lead to unstable limit cycles. Time integration of the combined cortico optogenetic model with PI control to track the maximum and minimum values of the excitatory mean soma potential is shown on the right. The red triangle indicates the start of oscillatory activity despite closed-loop control being triggered. The dashed black lines in the rightmost panel indicate minimum and maximum values of ${\tilde{h}}_e$ during oscillatory activity arising despite the use of closed-loop control. Gains used $K_P=0.4$ and $K_I=3.6$.

Figure 7

Seizures in the $M_e-{\mathrm{\Gamma }}_i$ space using a variation of $M_e$ in time with a maximum value of $-12.5$ as shown in the panel on the right. ${\mathrm{\Gamma }}_i$ is held constant at 0.085. All other parameters are similar to baseline parameters in Ref. [11]. The mean soma potential (${\tilde{h}}_e$) at a point and traveling seizure waves depicted by a one-dimensional slice of the two-dimensional cortex are shown in the first two figures on the left.

Figure 8

Effect of closed-loop optogenetic control on seizures in the $M_e-{\mathrm{\Gamma }}_i$ space. $M_e$ is varied in time with a maximum value of $-12.5$ as shown in Fig. 7. ${\mathrm{\Gamma }}_i$ is held constant at 0.085 with $\alpha =1.6$. All other parameters are similar to baseline parameters in Ref. [11]. The left and right panels show the variation with time of ${\tilde{h}}_e$ and intensity, respectively at the same point. The red dot-dash line in the right panel indicates the open-loop intensity required to suppress a fully formed seizures at $g_e=-12.5$. Gains used: $K_P=1.0$ and $K_I=0.64$.

Figure 9

Seizure-prone areas in the $M_e-{\mathrm{\Gamma }}_i$ parameter space with no optogenetic stimulation. Here dark regions represent values of $M_e$ and ${\mathrm{\Gamma }}_i$ which produce oscillations in the mean soma potential $h_e$. The gray scale shading indicates the variation in $h_e$ during oscillatory behavior. These results were obtained using the dimensionless ODE system of equations.