Scaling effects and spatio-temporal multilevel dynamics in epileptic seizures

Abstract

Epileptic seizures are one of the most well-known dysfunctions of the nervous system. During a seizure, a highly synchronized behavior of neural activity is observed that can cause symptoms ranging from mild sensual malfunctions to the complete loss of body control. In this paper, we aim to contribute towards a better understanding of the dynamical systems phenomena that cause seizures. Based on data analysis and modelling, seizure dynamics can be identified to possess multiple spatial scales and on each spatial scale also multiple time scales. At each scale, we reach several novel insights. On the smallest spatial scale we consider single model neurons and investigate early-warning signs of spiking. This introduces the theory of critical transitions to excitable systems. For clusters of neurons (or neuronal regions) we use patient data and find oscillatory behavior and new scaling laws near the seizure onset. These scalings lead to substantiate the conjecture obtained from mean-field models that a Hopf bifurcation could be involved near seizure onset. On the largest spatial scale we introduce a measure based on phase-locking intervals and wavelets into seizure modelling. It is used to resolve synchronization between different regions in the brain and identifies time-shifted scaling laws at different wavelet scales. We also compare our wavelet-based multiscale approach with maximum linear cross-correlation and mean-phase coherence measures.

Figure 1. Simulation of (3) with and using an Euler-Maruyama numerical SDE solver [?]; red curves are deterministic trajectories with and blue curves are sample paths with .

Systems have always been started at . The critical manifold is shown in grey and the -nullcline as a dashed black curve. (a) , the equilibrium for the full system lies on . (b) , the equilibrium lies on near the fold point . The deterministic trajectory has only one spike while noise-induced escapes produce repeated spiking for the stochastic system.

Figure 2. Average of the variance (black curves) over 100 sample paths starting for at up to a final time .

The green curves are fits of using (4) with fitting parameters and . Fixed parameter values are . (a) Relaxation oscillation regime with . (b) Excitable regime with ; sample paths can exhibit oscillations around the stable focus equilibrium which are visible in the variance. (c) Excitable regime with where larger noise regularizes the variance similar to (a). (d) Excitable regime where smaller noise does not allow fast escapes from and yields decreasing variance.

Figure 3. The eight plots show the average channel activity (top, blue) and the average of the inverse variance (bottom, black) for the eight time series ; the horizontal axis is the time axis where the labels correspond to the sample point number.

The sliding window length corresponds to the length of the initial gap in (5000 points). The green dots mark some local maxima of which correspond to local minima of . The fitted red curves are linear and demonstrate that the variance increases near the epileptic seizure. The black dashed vertical lines are inserted for orientation purposes, separating the two regions of low and high variance.

Figure 4. Time series of the fast variable (top, blue) and the associated inverse of the variance (bottom, black) for a Hopf critical transition model (6) with parameter values 7; cf. also Figure 3.

Figure 5. Maximum linear cross-correlation for eight pre-ictal time series .

Vertical lines indicate the approximate onset of the seizure attack.

Figure 6. Decrease of synchronization measures during a pre-ictal interval.

Left column: time series of maximum linear cross correlation during a pre-ictal (top) and an inter-ictal (middle) interval. Right column: time series of for three scales during a pre-ictal (top) and an inter-ictal (middle) period are depicted. Vertical dashed lines indicate the onset of the seizure attack. Averages over the first 150000 sample points of each time series indicate a distinct decrease of each synchronization measure during the pre-ictal interval (bottom row). Error bars show standard deviations.

Figure 7. Phase-locking measure for the eight time series .

Colors correspond to different scales. The vertical dashed lines indicate the approximate onset of the epileptic seizure attack.

Figure 8. Comparison between and mean phase coherence for patient 1.

Both measures based on phase-synchronization show a similar behavior with an increase around seizure onset time. Colored vertical lines indicate the beginning of the increase in synchronization near seizure onset (black dashed vertical line). The increase appears to be linear (grey dotted lines) and starts at different times for different scales.