A computational study of stimulus driven epileptic seizure abatement

Abstract

Active brain stimulation to abate epileptic seizures has shown mixed success. In spike-wave (SW) seizures, where the seizure and background state were proposed to coexist, single-pulse stimulations have been suggested to be able to terminate the seizure prematurely. However, several factors can impact success in such a bistable setting. The factors contributing to this have not been fully investigated on a theoretical and mechanistic basis. Our aim is to elucidate mechanisms that influence the success of single-pulse stimulation in noise-induced SW seizures. In this work, we study a neural population model of SW seizures that allows the reconstruction of the basin of attraction of the background activity as a four dimensional geometric object. For the deterministic (noise-free) case, we show how the success of response to stimuli depends on the amplitude and phase of the SW cycle, in addition to the direction of the stimulus in state space. In the case of spontaneous noise-induced seizures, the basin becomes probabilistic introducing some degree of uncertainty to the stimulation outcome while maintaining qualitative features of the noise-free case. Additionally, due to the different time scales involved in SW generation, there is substantial variation between SW cycles, implying that there may not be a fixed set of optimal stimulation parameters for SW seizures. In contrast, the model suggests an adaptive approach to find optimal stimulation parameters patient-specifically, based on real-time estimation of the position in state space. We discuss how the modelling work can be exploited to rationally design a successful stimulation protocol for the abatement of SW seizures using real-time SW detection.

Figure 1. Successful and unsuccessful auditory stimulation.

Clinical EEG recordings of successful (upper panel) and unsuccessful (bottom panel) seizure abatement by an auditory stimulus (arrow). Figure modified from with permissions.

Figure 2. Comparison between clinical and simulated EEG.

The clinical (left) and simulated (right) EEG are compared in various properties, such as the long-term time series (a,b), and seizure waveforms (c,d).

Figure 3. Three dimensional slice through the four dimensional basin of attraction of the background state.

Black line indicates the SWD attractor. Stars show the slice point positions. Coloured dots are located in the basin of attraction of the background state in three dimensional state space. Colouring is only included to enable better three dimensional visibility of the geometry of the basin. Red, green and blue intensities encode the three principal axes of the 3D plot. Additive colouring is used to plot off-axis positions (e.g. red & green contribute to the yellow areas between the first and second axis). The SWD basin of attraction is not specifically shown, as it is simply the part in state space that is not the background basin. (a) and (c) are three dimensional slices through the four dimensional state space. The slice point in the fourth dimension corresponds a single value of on the SWD attractor at time (red star) for (a) and (green star) for (c). (b) and (d) slices with the slice point corresponding to a single value of on the SWD attractor at time (red star) for (b) and (green star) for (d). (e) Corresponding time series showing the slice points. 3D Matlab.fig files are available for (a) and (b) in S1 File & S2 File.

Figure 4. Return probability in a 2D slice through state space.

Two dimensions of the four dimensional state space is visualised in the deterministic (left panel) and noise-driven (right panel) models. and are fixed at the value of the background fixed point. Return probabilities (colour code) are scanned in the variables. The red dot marks the position of the background fixed point.

Figure 5. Single pulse stimulation in a deterministic SWD system.

(a) Colour coded time series of one cycle of SWD. The green colour indicates a return to the background fixed point if stimulated at the colour-coded position (using a fixed stimulus amplitude). Blue/red arrows indicate stimulation points in (c)/(d). (b) Colour coded map of stimulation amplitude and timing in the same SWD cycle. The same colour code as in (a) has been used. The particular amplitude used for (a) has been outlined in a grey box. (c) Basin of attraction of the background state (coloured dots) in the projection for the stimulation point on the SWD cycle is shown together with the SWD attractor (black line). Blue arrow indicates the successful stimulus at this point, as it points into the basin of attraction. (d) Same as (c). Red arrow indicated the unsuccessful stimulus at this point as it does not point into the basin. Notice the change in axes between (c) and (d), the figures are rotated to aid visualisation, however, the stimulus direction is the same. 3D Matlab.fig files are available for (c) and (d) in S3 File & S4 File.

Figure 6. Stimulation in the noise-driven system.

(a) Two examples of the same simulation timing and amplitude in the same seizure, but with varied outcomes. Different noise inputs were used after the stimulation for the two examples. (b) The effect of different noise inputs (y axis) is scanned depending on the ensuing seizure duration (colour code). As in (a), different noise inputs were used following the stimulation. The stimulus amplitude was constant (−0.0825) for the whole scan. (c) The top bar indicates the success rate (derived from the data in (b)) at each stimulation time point, which is then applied as a colour code to the actual SWD time series. (d) The success rate is also scanned for different stimulation amplitudes. In essence the strip in (c) is a row in (d), indicated by a grey bar.

Figure 7. Delay embedding reconstruction of the SWD attractor.

The simulated seizure (left) and the clinical seizure (right) are reconstructed. The same reconstruction parameters (delay time and filter frequency cut-off) have been used for both simulated EEGs (Sim.) and clinical EEGs (EEG). F(…) indicates low-pass filtering of the simulated EEG, as explained in the Methods section. Time delays used are indicated in seconds on the axis label. (a, b) Reconstructed attractor, in this case corresponding to the phase space view (c.f. Fig. 5 (a) rotated). (c, d) Reconstructed attractor corresponding to the phase space view (c.f. Fig. 5 (c)). 3D Matlab.fig files are available for all subfigures in S5 File, S6 File, S7 File, & S8 File.

Figure 8. Schematic of a suggested single pulse stimulation protocol.

In the learning phase (top) success rate of state space targets are stored based on arbitrary stimulations during seizures. Once the state space is charted, the application phase (bottom) can use the information of the success rate of state space targets to deliver high success rate stimuli to abate SWD seizures.

Figure 9. Connectivity scheme of the model.

Excitatory (inhibitory) connections indicated in green (red). is the cortical pyramidal neural population, is the cortical inhibitory neural population, is the thalamocortical neural population, and is the thalamic reticular nucleus neural population.