348
K. Lehnertz
techniques, parametric approaches provide an alternative way to estimate (single and
joint) properties of the power spectra [95]. Multivariate and/or vector autoregressive
models form the basis of techniques such as Granger causality [112, 144, 158] or
partial directed coherence [13, 135] that can index the direction of an interaction.
Bivariate nonlinear analysis techniques can be subdivided into two main categories
depending on the underlying concept for interaction: information-theory-based [52]
and synchronization-based techniques [24, 119, 151]. As with univariate nonlinear
methods, findings achieved with bivariate techniques need to be interpreted with
great care. Although extensions and new development of surrogate techniques can
help to avoid misinterpretations about the strength of an interaction [6, 80], causal
relationships are notoriously difficult to identify and we are still lacking reliable
surrogate techniques for directionality indices. Nevertheless, particularly estimates
for the strength of interaction (either using phase-based [110] or state-space-based
approaches [11, 36, 121]) were repeatedly shown to reliably identify a preseizure
state preceding focal-onset seizures (see, e.g., Refs. [1, 16, 38, 42, 69, 73–75, 83–
85, 105, 107, 108, 120, 137, 147, 168, 170]). Interestingly and probably contrary
to expectations, decreased levels of the strength of an interactions frequently characterized a preseizure state. Moreover, most of these studies identified interactions
between brain regions far off the epileptic focus and even from the opposite brain
hemisphere to indicate a preseizure state.
Multivariate analysis techniques (together with graph-theoretical concepts) provide a means to quantify structure and dynamical evolution of complex networks.
Network theory has contributed significantly to advance understanding of complex
systems, with wide applications in diverse fields, ranging from physics to biology
and medicine [3, 10, 23, 25, 30, 113, 124, 152]. In epileptology, the concept of
an epileptic network [15, 21, 82, 88, 125, 126, 148] received strong impetus from
network-theoretical concepts and multivariate analysis techniques. An epileptic network comprises anatomically, and more importantly, functionally connected cortical
and subcortical brain structures and regions. Seizures (and other related pathophysiologic dynamics) may emerge from, may spread via, and may be terminated by
network constituents that generate and sustain normal, physiological brain dynamics during the seizure-free interval [148]. So called functional brain networks are
supposed to reflect the interaction dynamics between brain regions [30] and such
a representation requires identification of vertices and edges. Vertices are usually
associated with sensors that are placed to sufficiently capture the dynamics of given
brain regions. Interactions between the vertices’ dynamics are considered as network
edges, and properties of interactions can be characterized with the aforementioned
bivariate analysis techniques. Graph theory then provides a large number of concepts
and measures to characterize macroscopic (such as clustering, average shortest path
length, or assortativity), mesoscopic (such as communities, motifs, or core-periphery
structure), and microscopic network properties (such as centrality of single vertices or
edges). Studies investigating macroscopic characteristics of time-evolving epileptic
networks provided first clues for certain network reconfigurations to promote the formation of a preseizure state [48, 70, 78, 138, 140]. Likewise, this approach allowed
to formulate a network mechanism by which most focal epileptic seizures stop spon-
K. Lehnertz
techniques, parametric approaches provide an alternative way to estimate (single and
joint) properties of the power spectra [95]. Multivariate and/or vector autoregressive
models form the basis of techniques such as Granger causality [112, 144, 158] or
partial directed coherence [13, 135] that can index the direction of an interaction.
Bivariate nonlinear analysis techniques can be subdivided into two main categories
depending on the underlying concept for interaction: information-theory-based [52]
and synchronization-based techniques [24, 119, 151]. As with univariate nonlinear
methods, findings achieved with bivariate techniques need to be interpreted with
great care. Although extensions and new development of surrogate techniques can
help to avoid misinterpretations about the strength of an interaction [6, 80], causal
relationships are notoriously difficult to identify and we are still lacking reliable
surrogate techniques for directionality indices. Nevertheless, particularly estimates
for the strength of interaction (either using phase-based [110] or state-space-based
approaches [11, 36, 121]) were repeatedly shown to reliably identify a preseizure
state preceding focal-onset seizures (see, e.g., Refs. [1, 16, 38, 42, 69, 73–75, 83–
85, 105, 107, 108, 120, 137, 147, 168, 170]). Interestingly and probably contrary
to expectations, decreased levels of the strength of an interactions frequently characterized a preseizure state. Moreover, most of these studies identified interactions
between brain regions far off the epileptic focus and even from the opposite brain
hemisphere to indicate a preseizure state.
Multivariate analysis techniques (together with graph-theoretical concepts) provide a means to quantify structure and dynamical evolution of complex networks.
Network theory has contributed significantly to advance understanding of complex
systems, with wide applications in diverse fields, ranging from physics to biology
and medicine [3, 10, 23, 25, 30, 113, 124, 152]. In epileptology, the concept of
an epileptic network [15, 21, 82, 88, 125, 126, 148] received strong impetus from
network-theoretical concepts and multivariate analysis techniques. An epileptic network comprises anatomically, and more importantly, functionally connected cortical
and subcortical brain structures and regions. Seizures (and other related pathophysiologic dynamics) may emerge from, may spread via, and may be terminated by
network constituents that generate and sustain normal, physiological brain dynamics during the seizure-free interval [148]. So called functional brain networks are
supposed to reflect the interaction dynamics between brain regions [30] and such
a representation requires identification of vertices and edges. Vertices are usually
associated with sensors that are placed to sufficiently capture the dynamics of given
brain regions. Interactions between the vertices’ dynamics are considered as network
edges, and properties of interactions can be characterized with the aforementioned
bivariate analysis techniques. Graph theory then provides a large number of concepts
and measures to characterize macroscopic (such as clustering, average shortest path
length, or assortativity), mesoscopic (such as communities, motifs, or core-periphery
structure), and microscopic network properties (such as centrality of single vertices or
edges). Studies investigating macroscopic characteristics of time-evolving epileptic
networks provided first clues for certain network reconfigurations to promote the formation of a preseizure state [48, 70, 78, 138, 140]. Likewise, this approach allowed
to formulate a network mechanism by which most focal epileptic seizures stop spon-