Figure 1.
The model includes interactions between the sleep-active ventrolateral preoptic area of the hypothalamus (VLPO), the wake-active monoaminergic brainstem nuclei (MA), and the orexinergic neurons of the lateral hypothalamic area (Orx), as well as the circadian () and homeostatic (
) drives. Arrows indicate interactions between the populations, as well as the pathways of the circadian and homeostatic drives, and represent either excitatory (
) or inhibitory (
) interactions. A All modeled interactions are shown, including the mutual inhibition between VLPO and MA [27], inhibition of Orx by VLPO [8], and excitatory input from Orx to MA [72]. The circadian drive,
, which originates in the suprachiasmatic nucleus (SCN), is afferent to both VLPO (inhibition) and Orx (excitation) [4], while the homeostatic sleep drive,
, which increases during wake and decreases during sleep, disinhibits VLPO [63]. Example two day time traces for normal sleep-wake behavior are annotated below the
and
drives. B The model can be mathematically reduced to the core dynamics of mutual inhibition between the sleep-active VLPO and wake-active MA groups. In this representation, net drives,
and
, to VLPO and MA, respectively, control the arousal state dynamics. This reduced representation is used throughout this work to visualize and understand the model dynamics.
Table 1.
Nominal model parameter values.
Figure 2.
Model dynamics represented in terms of the net drives to the sleep-active VLPO, , and the wake-active MA,
.
A Three distinct regions of –
space are: (i) wake: at low
and high
a stable wake state exists, (ii) sleep: at high
and low
a stable sleep state exists, and (iii) bistable (shaded): at intermediate
and
wake and sleep states are simultaneously stable and transient noise can produce lasting changes of state. Simulated 5-h time series and
–
plots for fixed points in this space are shown in the remaining figures. Time series are plotted for average firing rates of the VLPO,
(red), and the MA,
(blue). In the
–
plots, we include
nullclines (solid lines),
nullclines (dashed lines), stable equilibriums (solid circles), saddle points (open circles), and the separatrix (dotted black line); see File S1 for definitions and numerical details. B
is high and
is low; a single stable wake state exists. C
is high and
is low; a single stable sleep state exists. D In the bistable region at high
and
, thresholds for transitions between wake and sleep are high and hence state transitions are extremely improbable: the system remains either awake or asleep depending on its initial state (on timescales relevant to the current dynamics). E In the bistable region nearer the sleep bifurcation boundary, transitions from wake to sleep are more probable than transitions from sleep to wake. F In the bistable region at low
and
, thresholds for transitions between sleep and wake are low and simulated time series are highly fragmented.
Figure 3.
Noise-free model simulations represented as trajectories in terms of net drives to the VLPO, , and MA,
, and as time series.
A The bistable region is shaded blue, and the wake and sleep regions are labeled. The trajectory for normal dynamics (i.e., including Orx) forms a loop and is plotted using black (wake) and gray (sleep). The trajectory for the model without Orx is a small oscillation at low and
, and is labeled ‘no Orx’. The trajectory for the original Phillips-Robinson model occurs at fixed
mV and is shown semi-transparent for comparison (note that the wake trajectory of the Phillips-Robinson model extends beyond the limits of the figure to
mV). When Orx is removed from the model, the system oscillates at low
and
, where thresholds for transitions between wake and sleep are low. However, with Orx in the model, the wake and sleep states are stabilized: Orx is active during wake, increasing
, and Orx is suppressed during sleep, decreasing
, thereby moving the system away from the bistable region where state transitions can occur and promoting consolidated wake and sleep episodes. Circadian input to Orx modulates waking arousal levels:
is lower in the early morning and increases to a maximum at the circadian maximum, then decreases through the afternoon and evening. Two-day time series for noise-free model dynamics (including Orx) are also plotted as: B Firing rates
(black),
(blue), and
(green, dashed), C Net drives to the VLPO,
[black, Eq. (7)], and the MA,
[gray, Eq. (8)], and D Drives
[black, Eq. (5)] and
[gray, Eq. (6)]. Approximate clock times for a typical sleep-wake schedule are given as a guide, and sleep periods are shaded.
Figure 4.
Removing Orx from the model produces fragmented sleep-wake time series characteristic of the narcoleptic phenotype.
Simulated 24-h time series are plotted for A Normal dynamics including Orx (i.e., mV s) for
(blue),
(green), and
(orange), and B Fragmented dynamics with Orx removed from the model (i.e.,
). Periods of sleep, with
(black), and wake, with
(white), are shown in the strip above the main plot. When
is reduced, the system moves from a regime in which Orx stabilizes extended wake and sleep bouts, to a regime characterized by low waking arousal levels and increased fragmentation due to a lowering of the threshold for state transitions.
Figure 5.
Model dynamics as a function of orexin levels, corresponding to the model parameter .
A Periods of sleep (black) and wake (white) are plotted as a function of across two-day model simulations. B The circadian drive,
, versus time. Various statistics taken from the model output are plotted as a function of
as the mean (solid)
standard deviation (dotted) measured across a 25 day model simulation (following a 3 day equilibration period), for C Total sleep duration per day, D Number of state transitions per day, E Duration of sleep bouts, F
during wake (blue) and
during sleep (black), and G Homeostatic sleep drive,
.
Figure 6.
Dependence of sleep-to-wake and wake-to-sleep transitions on the timescale for Orx dynamics, .
Time series for the firing rates of MA, (blue), VLPO,
(black), and Orx,
(green), are plotted for the sleep-to-wake (A–C) and wake-to-sleep (D–F) transitions for
s (A, B),
min (B, E), and
min (C, F), as a function of time relative to the change of state. The plots were produced by averaging 50 model runs relative to the time of the state transition; one standard deviation about the mean is shown dotted. The approximate steady state firing rate for
is annotated as a dashed purple line in A–C, and for
in D–F. The parameter
selectively tunes the duration of the sleep-to-wake transition but has minimal effect on the wake-to-sleep transition. This gradual wake transition can be linked to the clinical phenomenon of sleep inertia.