Figure 1.
Sleep-wake cycles generated by the two-process model.
With the parameters as in [10], Figure 3:
(a)
(b)
(c)
The times when sleep occurs (
decreasing) are shaded.
Figure 2.
The PR model and typical solutions.
(a) Diagrammatic description of the PR model showing the links between the VLPO, MA, the homeostatic and the circadian processes. (b), (c) and (d) show typical timeseries for the level of the homeostat, , and the firing rates of the MA and VLPO,
and
, respectively. The times where sleep occurs are shaded.
Figure 3.
The PR model and the PR switch model.
(a) The dashed (black) line shows the firing function given by equation (5); the thicker (red) line shows the portion that is used for the ‘normal’ PR cycle. (b) A magnified version of (a). The thin (blue) line shows the switch function (8). Panels (c),(d) and (e) show the behaviour of the homeostat, , and the firing rates
and
for the PR model (solid line) and the PR model with the hard switch (dashed line). The switch parameters are
the mean firing rate of the neural population during wake; all other parameters are listed in the Tables section.
Figure 4.
The two-process model compared to the two PR models.
(a) Comparison of the PR switch model with the two-process model. (b) Comparison of the PR model with the two-process model. Crosses show the two-process model; solid line the PR model and (blue) dashed line the PR switch model.
Figure 5.
The one-dimensional map for the two-process model.
(a) A single trajectory of the two-process model showing successive times of sleep onset. (b) Trajectories of the two-process model for different initial sleep onset times. Each different sleep onset time results in a different sequence, , but each sequence rapidly converges to the same sleep onset time, of
modulo 1 day. (c) A zoom of (b), showing only the trajectories for
and
(d) First return map for the two-process model. The black line shows the return map, in other words for any value of sleep onset time on day
,
it shows the onset time of sleep on day
,
. The grey diagonal line is the line along which
. One typical trajectory is plotted for
showing the rapid convergence to the periodic cycle where
modulo 1 day, the point at which the return map and the diagonal line intersect. The discontinuity is a result of the fact that neighbouring values of
exist that lead to very different values for
, as shown in (c). Parameter values for the two-process model are based on the PR model for the human sleep-wake cycle and can be found in (15).
Figure 6.
Varying the homeostatic constant.
Using the two-process model with parameters as indicated in (15). Figures (a)–(d) give sleep-wake cycles for different values of the homeostatic time constant (
illustrating that reducing
results in more daily sleep episodes. (e) Sleep regions (shaded) as a function of
. Note that the circadian maximum occurs at
Figure 7.
Sleep-wake cycles with a two day period.
Solutions of the two-process model showing periodicity on the period of two days. (a) (b)
All other parameters are as in Figure 6 and can be found in (15).
Figure 8.
Sleep timing in the two-process model.
The upper and lower thresholds are moved simultaneously via and
with (a)
(b)
and all other parameters as in (15). Note that mean VLPO drive equals
for consistency with [24]. Sleep regions are shaded.
Figure 9.
Normal and deprived sleep in the PR model.
(a) Sleep-wake cycle showing the MA firing rate as a function of the drive to the VLPO
. Over one cycle
oscillates between high and low values. When
is low,
is high and the model is in a wake state. When
is high,
is low and the model is in a sleep state. The transitions from wake to sleep and sleep to wake occur at
and
respectively. The size of the hysteresis loop depends on
, shrinking to nothing for
and for
(b) The path of
and
in the
plane.
and
do not exist for values of
that are either less than
or greater than
. Consequently for
or
increasing
will result in a smooth change from high
(wake) to low
(sleep) instead of the jump from one state to the other shown in (a). (c) A blow up of (b), with the ‘normal’ sleep-wake cycle superimposed. (d) The
plane showing the wake trajectory in a sleep deprivation experiment.
Figure 10.
Sleep deprivation and the wake effort.
(a) The two-process model, showing the typical trajectory of the homeostatic pressure during a sleep deprivation experiment. Using the wake effort concept of [25] suggests that the upper threshold moves simultaneously: the dashed line shows the position of the upper threshold after 4 days. (b) The difference between the homeostatic pressure and the value at the ‘normal’ threshold, . (c) The wake effort computed from the two-process model (10) (solid line), the PR model as in [25] using
(crosses), the PR model with
(dashed line). (d) The dependence of
, the upper asymptote, on time for the three different cases shown in (c). The downward spikes indicate that the model gets very close to falling asleep, hence
gets very close to 0.
Table 1.
Typical parameter values for the PR model and the equivalent parameters for the PR model with a hard switch.