Fig 1.
Connectivity of the thalamocortical model.
Excitatory synapses are depicted by filled circles, inhibitory synapses by bars. Independent background noise entering the different populations is denoted by ϕn, and
, respectively. Stimulation is applied as an elevation in the mean of the background noise
of the thalamic relay population.
Fig 2.
Two-dimensional bifurcation analysis.
Here, we illustrate the bifurcation diagram of the isolated thalamus with respect to the two key parameters and
. The interaction between the currents incorporated into the thalamic module results in the emergence of two torus bifurcations via a blue sky catastrophe. They lead to spindle oscillations in the orange shaded regions. The left spindle regime (SI) is encased by a Hopf and a torus bifurcation, whereas the right spindle regime (SII) is constrained by two global bifurcations that are indicated by the dashed gray lines. The vertical line marks the emergence of the torus bifurcation, whereas the horizontal gray line marks the cusp bifurcation where the two saddle-nodes that accompany the left torus bifurcation vanish. The torus bifurcation on the right marks the transition from spindle oscillations to delta oscillations. The labeled points mark the parameter settings utilized in Fig 3, which are given in Table 1.
Table 1.
Parameter settings for the isolated thalamus.
Fig 3.
Dynamic modes of the isolated thalamic module.
Here, we illustrate the different dynamic modes the isolated thalamic module exhibits. The left panels depict the thalamic relay membrane voltage, whereas the right panels illustrate that of the thalamic reticular population. The parameter values are depicted in Fig 2 and given in Table 1. SI and SII: The isolated thalamus generates rhythmic spindle oscillations via a balanced interplay between IT and Ih. The length and the average time between spindles is governed by . CI and CII: Outside of the spindle regime fast oscillations generated by the T-type calcium currents dominate and Ih is unable to sufficiently depolarize the thalamic relay population to cease them. DI and DII: For strong hyperpolarization through ILK the thalamic module switches into low frequency delta oscillations.
Table 2.
TC parameter settings.
Fig 4.
Example time series of sleep stage N2.
Shown are membrane voltages of the cortical pyramidal (top) and the thalamic relay population (bottom). The spindle oscillations induced within the thalamic module project into the cortical module. While the spindle oscillations are generally induced by fluctuations in background noise, there is also a grouping between cortical KCs and thalamic spindles (see 7s-9s and 19s-21s). The grouping stems from the lack of depolarizing input during a cortical KC. Parameters are as in Table 2.
Fig 5.
Example time series of sleep stage N3.
Shown are membrane voltages of the cortical pyramidal (top) and the thalamic relay population (bottom). During N3 the model shows ongoing slow oscillatory activity. In contrast to sleep stage N2, SOs cannot be identified as isolated events. Furthermore, there are no isolated spindle oscillations and spindle activity is time-locked to SOs. Parameters are given in Table 2.
Fig 6.
Event triggered average potentials.
Averaged EEG signal (top) and fast spindle band power (bottom) time-locked to the negative peaks (t = 0 s) of all detected events from electrode Cz (black, left axis) and model output (red, right axis). (A) Detected KCs from data scored as sleep stage N2 (Experiment: 227,45 ± 19,22, Model: 238 events). (B) SO average from data scored as sleep stage N3 (Experiment: 983,64 ± 106,1, Model: 654 events). Each simulation was run for 3600 s with parameters set according to Table 2.
Fig 7.
The upper panel depicts in black the mean (± SEM) evoked potentials of human EEG data from electrode Cz during closed loop stimulation, time locked to the first stimulus (11 subjects, 245.6 ± 38.1 stimuli). In red the reproduction of the stimulation protocol with the model is shown (mean ± SD, 88 stimuli). The dashed line marks the stimulus onset. The lower panel shows the corresponding fast spindle power. Parameters used for model simulation are given in Table 2.
Fig 8.
Stimulation disturbs refractoriness.
The upper two panels depict the membrane voltages of the pyramidal and thalamic relay populations, respectively. In the third panel the conductivity of the Ih current is shown. (A) Example time series of an unperturbed train of SOs during sleep stage N3. The first two SOs lead to an activation of Ih, that slowly declines back to baseline levels. As Ih activation is still well above baseline, the third SO is unable to trigger a spindle response. During the fourth SO Ih activation is sufficiently low so that a spindle occurs. (B) Shown is an example of closed loop stimulation during sleep stage N3, with the dashed lines indicating stimulus onset. In contrast to the endogenous case, the depolarization of the thalamic relay population induced by the stimulation leads to a rapid decrease in Ih activation, so that the following SO triggers a spindle. Parameters as in Table 2.