Figure 1.
Frequency entrainment effects in a periodically forced neural mass model of a cortical area.
A frequency-detuning curve refers to the ratio of stimulus to characteristic mean response frequency plotted against the ratio of stimulus to intrinsic frequency, for the normalized stimulus amplitude of 1.5. The entrainment ranges around the intrinsic frequency and its subharmonics are shown in red and in blue/green. Such an entrainment around the intrinsic frequency can be found in photic driving experiments (e.g., [81]).
Figure 2.
Entrainment effect found in the experimental data and model.
For the experiment, the mean over subjects is shown as white lines and the region between the 5% and 95% quantiles is covered by red areas. For the model (black dots), the amplitude configuration that best fits the largest Lyapunov exponents of the experiments is used (see Comparison section in Materials and Methods). The entrainment effect is shown for A the stimulus frequency range of the model and B the stimulus range used in the experiments (green area in A).
Figure 3.
Complex behavior occurring in the periodically forced neural mass model of a single area.
Orbits, time series, and power spectra (columns) are shown for three configurations (rows) displaying (top-down) periodic (normalized input amplitude; normalized input frequency: 3.6301; 9.33·10−2), quasi-periodic (1.5; 7.59·10−2) and chaotic behavior (3.6301; 7.05·10−2). The orbits are in the state space of normalized postsynaptic potentials of pyramidal cells caused by both interneurons (x30) as well as at both excitatory and inhibitory interneurons caused by pyramidal cells (x31 and x32). The red circle represents the stable limit cycle (i.e., harmonic oscillation) arising from Andronov-Hopf bifurcations performed by the unperturbed system. The time series and the power spectra are shown for the normalized postsynaptic potentials of pyramidal cells (which are related to M/EEG). Periodic behavior is characterized by a closed orbit (limit cycle) and a discrete power spectrum with peaks at commensurable frequencies. Quasi-periodic behavior is characterized by trajectories forming an invariant n-dimensional torus and discrete power spectra with peaks at incommensurable frequencies. Chaotic behavior is indicated by a strange attractor, that is, a bounded attracting set in which all trajectories are unstable and nearby trajectories diverge locally from each other exponentially, and a broadband power spectrum.
Figure 4.
Largest Lyapunov exponent in parameter space.
The map shows the largest Lyapunov exponent λ1 as a function of stimulus amplitude and frequency, indicating the sensitivity of the periodically forced neural mass model to initial conditions. Positive exponents (magenta to black) reflect diverging trajectories irrespective of how close they are, and thus chaos in the system. They also measure the entropy of an attracting set for all cases because the maximum number of positive Lyapunov exponents for any parameter configuration is one. Zero exponents (white) indicate neutral stability, and negative exponents (cyan to yellow) reflect frequency locking. Arnol'd tongue structures (i.e., resonance zones) are indicated by negative largest Lyapunov exponents due to the phase locking between system kinetics and the stimulus. The red arrow indicates the amplitude for which the experimental data fits best (see Figure 8 and Comparison section in Materials and Methods). Several parameter regions with scattered, presumably fractal patterns of chaotic regime are shown at a finer resolution of normalized stimulus amplitude and frequency.
Figure 5.
Occurrence of quasi-periodic behavior forming a two-torus surface in state space.
Two-dimensional tori are indicated by two zero Lyapunov exponents (shown in black dots) in the parameter space of stimulus amplitude and frequency. The red arrow indicates the amplitude for which the experimental data fits best (see Figure 8 and Comparison section in Materials and Methods). The parameter regions of chaotic patterns that were selected for recomputing at a finer resolution (see Figure 4 and Figure 6) appear here mostly as white areas.
Figure 6.
Kaplan-Yorke dimension of the periodically forced neural mass model in parameter space.
The Kaplan-Yorke dimension DKY given by Equations (8) and (9) never goes above 1.7, thus hyperchaos does not exist in the model. The red arrow indicates the amplitude for which the experimental data fits best (see Figure 8 and Comparison section in Materials and Methods). Several parameter regions with scattered, presumably fractal, patterns of chaotic regime were selected for recomputing at a finer resolution of normalized stimulus amplitude and frequency.
Figure 7.
Experimental design of the flicker stimulation study.
The LEDs were powered for half of each period. The raise and decay time for the LEDs was measured to be 100 µs.
Figure 8.
Comparison of model and data from a photic driving experiment.
The largest normalized Lyapunov exponents calculated from the model show very good agreement with those obtained from experimental time series. The largest Lyapunov exponents for the average over all subjects and for the nearest neighbors of the model with the stimulus amplitude ζ that fits best (ζ = 3.6301) are plotted against the ratio of stimulus to intrinsic alpha frequency in (A). The largest Lyapunov exponents for the average over all subjects are normalized to the same range as for the model. The green area covers the standard deviation of the largest normalized Lyapunov exponents over all subjects. The comparison based on the minimization of the average of the minimum relative error (i.e., distance/maximum distance) between the normalized largest Lyapunov exponents of the model (comprising the four nearest neighbors) and of the experiment over stimulus frequencies is calculated for all stimulus amplitudes of the model. The mean error ε of model and average over subjects is shown in (B), where the red area covers the 5% and 95% quantiles. For the significant amplitude range, the quantiles are drawn in green. Significant amplitudes are consistent over subjects as shown in Table 1. For more details, see Comparison section in Materials and Methods.
Figure 9.
Bifurcation diagram for normalized stimulus amplitude ζ = 3.6301.
The vertical axis is the normalized postsynaptic potential x32 on pyramidal cells caused by inhibitory interneurons, that is, the coordinate of the intersection points (black dots) of trajectories with the Poincaré hyperplane after neglecting initial transients. The horizontal axis is the normalized stimulus frequency. The regimes are color-coded and indicated by the horizontal line. Periodic regimes (red) exist, for instance, for frequencies ranging from 0 to 5.34·10−2. In this range, the system appears to undergo a period-adding bifurcation cascade by decreasing the normalized stimulus frequency. Chaotic (blue) and quasi-periodic regimes (green) occur, for example, for frequencies ranging from 5.3·10−2 to 6.23·10−2 (scattered dots) and between 17.21·10−2 and 18.72·10−2. The classification can be also taken from Table 2.
Table 1.
Single subject comparison of largest Lyapunov exponents.
Table 2.
Dynamic regimes occurring for normalized stimulus amplitude ζ = 3.6301.