Figure 1

Schematic of the second component $D_2^{\left(1\right)}\left(x_1,x_2\right)$ of a hypothetical drift vector. Black squares and error bars indicate estimates for $D_2^{\left(1\right)}\left(x_1^k,x_2^l\right)$ and their statistical errors for fixed values of $x_1^k$ and $x_2^l$. The solid horizontal line indicates the error-weighted average ${\mu }_l^w$ of the values of the coefficient over all bins in $x_1$ direction for a fixed value of $x_2^l$. The dashed vertical lines are the distances of these values from ${\mu }_l^w$.Reuse & Permissions

Figure 2

Dependence of $F^{\left(1\right)}$, $F^{\left(2\right)}$, and $M^{\left(2\right)}$ on the coupling strength. (a) Coupling in the deterministic part, (b) coupling in the stochastic part, (c) mixing of the noise processes, and (d) bidirectional coupling in the deterministic part with fixed ${ϵ}_2^{\left(1\right)}=0.7$. Symbols and error bars denote means and standard deviations derived from 100 realizations of the respective processes. Time series consisted of $N=50000$ data points. Parameters in Eq. (11): $\alpha =\beta =0.1$ for (a)–(c) and $\alpha =3$, $\beta =1$ for (d); $g_{11}=g_{22}=0.1$, $g_{12}=0.01$.Reuse & Permissions

Figure 3

Dependence on the coupling strength of $F^{\left(1\right)}$ in the case of a unidirectional coupling in the deterministic part (a), of $F^{\left(2\right)}$ in the case of a unidirectional coupling in the diagonal element of the stochastic part (b), and of $M^{\left(2\right)}$ in the case of a mixing of the noise processes (c) for different sizes $N$ of the analyzed time series. Symbols and error bars denote means and standard deviations derived from 100 realizations of the respective processes. $B=10$, $L=100$, and parameters in Eq. (11): $\alpha =\beta =0.1$, $g_{11}=g_{22}=0.1$, and $g_{12}=0.01$.Reuse & Permissions

Figure 4

Dependence of $F^{\left(1\right)}$ and $F^{\left(2\right)}$ (left) and $M^{\left(2\right)}$ (right) on the coupling strength $ϵ$ for two coupled Rössler dynamics. Symbols and error bars denote means and standard deviations derived from 100 realizations of the respective processes [cf. Eq. (12)]. Time series consisted of $N=10000$ data points. Note the break in the ordinate in the plot of $M^{\left(2\right)}$.Reuse & Permissions

Figure 5

(Color online) Exemplary results obtained from analyzing EEG data from a patient suffering from a left-sided focal epilepsy. (a) Section of the temporal evolution of the average interdependence between a given electrode contact and the remaining ones as quantified by ${\overline{F}}_c^{\left(1\right)}$ (top), ${\overline{F}}_c^{\left(2\right)}$ (middle), and ${\overline{M}}_c^{\left(2\right)}$ (bottom). The section started at 12:30 a.m. and covered 24 h. For ${\overline{F}}_c^{\left(1\right)}$ and ${\overline{F}}_c^{\left(2\right)}$ a positive value indicates that contact $c$ has—on average—a weaker influence on the remaining contacts than vice versa. (b) Electrode implantation scheme for left (L) and right (R) intrahippocampal depth electrodes. Each electrode is equipped with ten cylindrical contacts (diameter 2.5 mm, intercontact distance 4 mm). (c) Temporal averages ${⟨F_{c{c}^{\prime }}^{\left(1\right)}⟩}_t$ (left), ${⟨F_{c{c}^{\prime }}^{\left(2\right)}⟩}_t$ (middle), and ${⟨M_{c{c}^{\prime }}^{\left(2\right)}⟩}_t$ (right) calculated from the entire EEG recording (duration 7 days). Data from the seizure-free interval entered the calculations only. A color-coded matrix entry with row and column indices $c$ and $c^{\prime }$ represents the value of the respective measure calculated for an electrode contact combination $\left(c{c}^{\prime }\right)$, where $c,c^{\prime }\in \left\{R01,\dots ,R10,L01,\dots ,L10\right\}$. A positive value of ${⟨F_{c{c}^{\prime }}^{\left(1\right)}⟩}_t$ or ${⟨F_{c{c}^{\prime }}^{\left(2\right)}⟩}_t$ indicates a stronger influence of contact $c^{\prime }$ on contact $c$ than vice versa. Entries on the main diagonal are colored black.Reuse & Permissions

Figure 6

Predominant direction of interdependences between the hemisphere containing the epileptic focus (ipsilateral hemisphere, $I$) and the opposite (contralateral, $C$) hemisphere as quantified by the temporal and spatial averages ${⟨F_{IC}^{\left(1\right)}⟩}_t$ and ${⟨F_{IC}^{\left(2\right)}⟩}_t$. For all patients, the positive values indicate a stronger influence of the contralateral on the ipsilateral hemisphere than vice versa.Reuse & Permissions

Figure 7

Dependence of $F^{\left(1\right)}$, $F^{\left(2\right)}$, and $M^{\left(2\right)}$ on the coupling strength. Left: coupling of $x_2$ into $D_{11}^{\left(2\right)}$ and right: coupling of $x_1$ into $D_{22}^{\left(2\right)}$. Symbols and error bars denote means and standard deviations derived from 100 realizations of the respective processes. Time series consisted of $N=50000$ data points. Parameters in Eq. (11): $\alpha =\beta =0.1$, $g_{11}=g_{22}=0.1$, and $g_{12}=0.01$.Reuse & Permissions