Pre-control characterization of hippocampal epileptic models

Abstract

This paper presents a dynamical characterization of epileptic seizures in animal models. Inter-hippocampal recordings of two animal models of seizures, kindling and pilocarpine, were analyzed by nonlinear analytic tools. The aim is to assess and differentiate pathophysiological states and behavioral phases of a status epilepticus. The achieved results indicates that stage V of Racine classification could be identified as the transition of dynamical indicators exhibit a monotonic decline up to this stage and an increase after that. Furthermore, concentration of data points on a small region of state space, achieved by our analysis, promises that a local nonlinear control may cause neuromodulation. This feasibility gets more strengthen by achievements of this paper on successful tracking of drifts of unstable periodic orbits at seizure onset. Nonlinear control algorithms could afterwards be designed to find suitable instances for inserting perturbations and steer the dynamics of system toward a desired dynamical operating mode.

Appendices

Appendix 1

Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and the largest one estimate the amount of chaos in a system. To estimate the maximal Lyapunov exponent, a robust method derived by [39] was employed. In this method, for each point $x_{i} \,$ in the phase space, neighbors within a small distance є, $x_{j} \,$, were found. Then, the trajectories of $x_{i} \,$ and all $x_{j} \,$ were allowed $\tau$ time steps to evolve, and the norm distances from $x_{i + \tau } \,$ to $x_{j + \tau } \,$ were calculated and averaged. The logarithm of the cumulative distance sum was then computed for each value of $\tau$. Subsequently, the resulting sum was averaged over all i. The mathematical representation of this description is

$$S(\tau ) = 1/N\sum\limits_{i = 1}^{N} {\ln \left[ {1/U_{i} \sum\limits_{{j \in U_{i} }} {\left\| {x_{i + \tau } - x_{j + \tau } } \right\|} } \right]}$$

(2)

where $U_{i}$ is the neighborhood of point $x_{i} \,$.

This computation was performed over some values of $\tau$, and the graph of S($\tau$) vs. $\tau$was plotted. The slope of the least squares fit line through the linear region of the graph was the estimate of the maximal Lyapunov exponent, λ.

Appendix 2

Lyapunov exponents offer a quantitative measure of the attractor’s stretching and folding mechanism. In chaos, there is a local exponential divergence of nearby orbits as well as an overall boundedness of trajectories. This measure elegantly evaluates the sensitive dependence on initial conditions by estimating the exponential divergence of nearby orbits. Such a divergence causes a lack of knowledge of the system’s behavior after a number of time steps, and conceptually the rate at which this uncertainty grows is expressed through the Lyapunov exponent. The algorithms used for estimating the LE commonly evaluate the average rate of the divergence occurring at different points of the trajectory. Indeed, the existence of at least one positive Lyapunov exponent in the system dynamics defines the directions of the local instabilities meaning the chaotic behavior.

The second invariant, correlation dimension, ignores the dynamics and concentrates on the properties of the attractor in the phase space. The correlation dimension, D2, is a measure of the geometric structure of the attractor and reflects the complexity of a dynamical process. It basically yields a lower boundary of the number of degrees of freedom required to describe the neuronal system and can be estimated using the scaling structure of the attractor. This measure is obtained by quantifying the spatial correlation between pairs of randomly chosen points on the attractor. Computing the correlation integral, which measures the probability of pairs of phase–space vectors being separate from each other at a distance less than or equal to a pre-defined value is the most common technique. The exponent of the correlation integral is known as the correlation dimension.

The last invariant employed here to capture the nonlinearity of the dynamics is Kolmogorov–Sinai entropy (KSE). The entropy is a measure of the (average) rate at which a system generates (or loses) information over time; therefore, its reciprocal value is proportional to the predictability time of the trajectory motion. The KSE describes the level of uncertainty about the future state of the system and therefore relates to predictability. Estimation of KSE¹⁷ involves covering the phase space with partitions which preserve the value of the entropy. A computational technique for calculating the entropy is tracing the trajectory evolution and registering into which cell of the partition the state of system falls at every point of time steps. To put it simply, the KSE is the probability rate of finding the system’s state in each cell.

Appendix 3

A statistical assessment of the differentiation between predefined behavioral phases was performed. The usefulness of this analysis is twofold: characterization of the differences between the five separable phases within a seizure and comparison of the reviewed nonlinear measures. Provided that the measures are generally consistent in characterizing the dynamics, the best invariant can be proposed for future works.

ANOVA was employed for inter-phase comparisons. This statistical test uses variation between the groups to decide whether the means for various groups are different, taking into account how many subjects there are in the groups. If the observed differences are large, then it is considered to be statistically significant. The statistical results for all the methods in this study are presented as mean ± var with p values in Table 1. Graphical representations are also given in the form of box plots. The following results were obtained: (1) excellent p values were achieved for all the methods, (2) the means of all nonlinear measures progressively declined from Phase 1 to Phase 3 but increased in the remaining phases, and (3) entropy turned out to be the best measure in identifying Phase 3 as the most distinctive phase in seizure evolution. Figure 11 depicts the results of this comparison by graphing the normalized mean values for entropy, Lyapunov exponent, and correlation dimension in one plot. It seems that the three measures are almost the same in characterizing the dynamical content of our recordings. However, this observation is not surprising as each method just takes a different view of the dynamics under investigation. An additional point observed in Fig. 11 is that the variation between measures in all recordings is the least at Phase 3. Another statistical evaluation was performed to further differentiate between the five phases of a status epilepticus. We tried to measure how much these stages are similar to nonseizure reference segments selected from those parts of the recording running before seizures (reference dataset). This statistic essentially captures the extent to which a given phase matches the segments in the reference dataset. For this assessment, 10 equally-sized data segments¹⁸ were randomly selected from non-seizure segments for each phase of each recording. Then, each nonlinear measure was calculated for all the reference segments. This was followed by averaging each measure over all segments.

Table 1 Averaged results of each nonlinear method for characterizing the defied epileptic phases in mean ± varwith the p value of the ANOVA analysis

Fig. 11

Mean normalized values of Lyapunov exponent, entropy and correlation dimension for each phase

Thus, we could compare entropy, Lyapunov exponent, and dimension with their nonseizure counterparts at each phase. This comparison was drawn through a similarity metric, i.e., the correlation test. Figure 12 summarizes the results of estimating correlation between the calculated measures for each phase and corresponding nonseizures segments. As can be seen, the dissimilarity to reference nonseizure segments is most obvious at Phase3, the interval at which Stage V of Racine is going on.

Fig. 12

The correlation of each phase with its non-seizure compartment averaged over 10 random segments