We start on the level of single neurons. Clearly it is very problematic to get data in this case before epileptic seizures so that we resort to model neurons. The main question will be whether we can predict a spike in the voltage time trace of the model neuron before it occurs. The FitzHugh-Nagumo (FHN) model [17], [18], [46] is a simplification of the Hodgkin-Huxley equations [47] which model the action potential in a neuron. We point out that the methods we are going to present here are going to apply to a much wider class of excitable neuronal models than the FHN equation such as the original Hodgkin-Huxley model [48] or the Morris-Lecar system [49] since these models have similar bifurcation structure and multiple time scale properties [15].

There are several forms of the FHN-equation [50]. One possible version suggested by FitzHugh is the Van der Pol-type [51] model(1)where represents voltage, is the recovery variable and , , are parameters. We think of as an external signal or applied current [52] and assume that the time scale separation satisfies so that is the fast variable and the slow variable. The dynamics of (1) can be understood using a fast-slow decomposition [53]–[55]. Setting in (1) yields a differential equation on the slow time scale defined on the algebraic constraintWe call the critical manifold; see Figure 1. Differentiating implicitly with respect to we find so that the differential equation on can be written aswhich we refer to as slow flow. Observe that the slow flow is not well-defined at the two points . Applying a time re-scaling to the fast time to (1) gives(2)

Setting in 2 gives the fast flow where implies that is viewed as a parameter in this context. Observe that consists of equilibrium points for the fast flow and that the points are fold (or saddle-node) bifurcation points [56] in this context. The critical manifold naturally splits into three partswhere are attracting equilibria and are repelling equilibria for the fast flow. We view as the refractory state and as the excited state for the neuron. For trajectories are concatenations of the fast and slow flows. We will consider two different situations for the parameters . In the first situation we chose the parameters so that (1) has a single equilibrium point on where is the -nullcline of the FHN-equation; see Figure 1(a1)–(a2). For suppose that ; then the slow flow moves the system to , a jump via the fast subsystem to occurs, the slow flow on brings the system to and another jump returns it to . This is the classical relaxation oscillation [55], [57]. However, in neuroscience one often also considers the excitable regime [15] where the global equilibrium for the system is stable and lies on close to ; see Figure 1(b1)–(b2). In this case, a trajectory of (1) can generate, depending on , at most one excursion/spike to the excitable state before returning to . Repeated spiking in the excitable regime can be obtained using the more general stochastic FHN-equation(3)where is delta-correlated white noise and is a parameter representing the noise level. We can now ask whether individual neuron spiking activity already has precursors. This viewpoint should provide new insights how neurons are able to control synchronization and how control failure occurs. Recent results on predicting critical transitions [20] suggest that statistical precursors can be used to predict events similar to spiking in neurons from a time series without knowing their exact location. The detailed mathematical theory can be found in [21], [30].

Here we present the first application of this theory in the context of single neurons. We want to predict a spiking transition from a neighborhood of to and consider the variance as an early-warning sign

Observe that we can view also as a function of , and write , since the mapping between and is bijective when restricting to . In the relaxation oscillation regime (see Figure 1(a1)–(a2)) and if are sufficiently small it can be shown [30], [58] that(4)for some constant and where is small. Therefore an increase in fast voltage-variable variance can potentially be used to predict and to control spiking if no equilibrium exists near . Here we extend the results of [30] by investigating the excitable regime. Figure 2 shows an average of the variance computed over 100 sample paths using a sliding window technique [21]. Figure 2(a) shows the relaxation oscillation regime where we can confirm the theoretical prediction (4).

The excitable regime is much more interesting since the equilibrium point can lead to a variety of distinct regimes depending on the noise level. In Figure 2(b) the noise is at an intermediate level so that deterministic oscillations around the equilibrium are visible in the variance before an escape; hence the prediction (4) is not a good prediction of a spike but one should rely on the oscillatory mechanism before escapes. In Figure 2(c) the noise is larger which provides a regularizing effect for the variance via noise-induced escapes. This relates to the well-known mechanism of coherence resonance [19]. In Figure 2(d) the noise is very small so that sample paths need exponentially long times to escape and are metastable near . This causes a decrease in variance and will make predictions very difficult. The different scaling regimes for noise level and time scale separation are discussed in more detail in [30], [59]–[61].

Based on our results we can conclude predictability of a spiking event and hence also its external control by input currents depend crucially on noise level and statistical properties of the state of a neuron. In particular, in the excitable state already a small change in the noise level or system parameters can result in a substantial loss of control due to unpredictable spiking. This could cause undesirable synchronization and continuous spiking. Let us point out that this is just one possible explanation for a potential prediction/control failure during epileptic seizures but our results show that prediction at neuronal level can already be extremely complicated. We proceed to look at the next scale in our analysis and move from single neurons to clusters/regions of neurons.

On the level of regions, we can start to analyze data obtained before epileptic seizures. The eight time series we use are described in detail in the Materials section. A natural extension of our previous strategy is to compute the variance for each time series using a sliding window technique and to understand the scaling laws associated with the variance on the cluster level.

Figure 3 shows the results of this computation. We plot the inverse of the variance for since this makes it easier to understand the scaling of near the seizure point at . Vertical lines are drawn for orientation purposes in Figure 3 separating a region of low variance from a high-variance regime, giving an indication where the seizure roughly occured; see also Materials. Furthermore, for we have marked several local maxima which have been found by subdividing each time series into 20 equal time intervals and checking whether the local maximum in is also a maximum for . All four plots have several important features in common:

The next problem to consider is what types of dynamical models can reproduce the behavior we have observed from the data analysis i.e. we look for a model for the variance in clusters/regions of neurons that displays the observed oscillatory behavior and scaling law. At first glance, the dynamics in Figures 3(a)–(h) could be interpreted as a summation of voltage traces from Figure 2(b) i.e. of neurons that are (almost) in synchrony where the coherent spiking originates from the noise-induced escape of a spiral sink. However, the real problem in understanding the dynamical mechanism of epileptic seizures is shown in Figure 4 where we also plot the inverse of the variance near a critical transition. The similarities to the data in Figure 3 are clear; all four observations (A)–(D) also apply in Figure 4. The data in Figure 4 have been generated using a simple model for a Hopf critical transition [21], [30]:(6)where are independent white noise processes that satisfy for . The model (6) was first analyzed in the context of delayed Hopf bifurcation [62], [63]. Observe that the deterministic part of the fast variables is the normal form of a (subcritical) Hopf bifurcation [64]. The slow variable can also be viewed as time since . For the simulation in Figure 3 we have chosen(7)with a deterministic initial condition . It is known that near a sub- or supercritical Hopf bifurcation a scaling law of the form (5) holds [30]. Obviously the scales differ between Figure 3 and Figure 4 but those can be re-scaled to match. Therefore we have found a dynamical model that could potentially explain the qualitative features of a single variance time series for a cluster of neurons.

It is very important to note that we have obtained the conjecture that a Hopf bifurcation is involved in the transition to a seizure without a detailed biophysical model. In fact, several mean-field models for various types of epileptic seizures do exhibit Hopf bifurcations [31]–[35] that form a boundary between a regular equilibrium (non-seizure) an oscillatory (seizure) regime. However, there are several mean-field models available [14] and also other bifurcation mechanisms have been identified to play a role near seizure onset [65].

Our methods also have another important implication regarding the distinction between a preictal and a proictal state [36]. From a dynamical perspective, it was suggested that one can differentiate between models that show a distinct preictal state with a parameter driving the system to a bifurcation or systems showing a proictal state where noise-induced escapes play a dominant role [6]. A subcritical Hopf bifurcation is a model that can interpolate between the two cases. Consider (6) in the following two cases:

Obviously there is a continuum of possibilities in between these two situations [60], [69] depending on the scaling of noise and time scale separation. In fact, the results shown in Figure 2 illustrate the variation in such a continuum situation for the saddle-node bifurcation. A study of intermediate regimes for all bifurcations, including the Hopf bifurcations on a mean-field level, could certainly be carried out similar to the strategy employed in [30]. Let us also point out that several models have been proposed to account for this problem in the context of epileptic seizures [70]. However, these models are usually based on introducing global dynamics as well as using global measures, such as interseizure intervals, for validation. Historically similar dynamical systems attempts have been made in other disciplines, for example for multi-mode oscillations [71] in chemistry. Later on, it turned out [53] that the local mechanisms and scaling laws are much more important as they often form the truly mathematically generic [72] building blocks of the dynamics. The Hopf bifurcation normal form (6) as well as the local dynamics near the fast subsystem saddle-node bifurcation in (1) are the most generic - i.e. codimension 1 [72], [73] - phenomena available. Therefore it is absolutely necessary to investigate the link between these phenomena and epileptic seizures first as demonstrated by our scaling law results.

In the preceding two sections we investigated neuronal dynamics at different spatial scales, from single model neurons to neurophysiological data from clusters of neurons, using the variance as a univariate measure. In the following section, we will focus on the dynamics from many clusters of neurons encompassing a larger spatial scale. In systems with spatial degrees of freedom an increase in the noise level can produce spatiotemporal order characterized by more regular activity patterns [74], [75].

In contrast to the previous, bivariate measures for the activity between different clusters will be used. Bivariate measures can take into account the correlation of two signals. Information about the correlation of neuronal activity between different anatomical regions can give insights into the state of the network as a whole. With regard to epilepsy, correlation based measures such as mean phase coherence (MPC) and maximum linear cross correlation (MLCC) have yielded promising results in identifying pre-ictal states [29], [44], [76], [77].

In this section, we will start by considering the maximum linear cross correlation for the ECoG data used in the preceding parts, reviewing and confirming some recent observations. We will then continue to extend the bivariate analysis to wavelet-based synchronization measures able to resolve pairwise correlations at different frequency bands. We will compare these results to those obtained using MLCC and MPC. Our focus is again on multiscale character of the system with the goal of identifying scaling relationships at each level of observation.