600
The results reported in the previous sections show a remarkable coexistence of
scale-601
invariant power-law structure for the durations of θ-bursts and a Weibull functional
602
form with a characteristic time scale for δ-burst durations (Figs. 2,3). This evidence
603
draws a strong parallel with far-from-equilibrium phenomena that are characterized
604
by bursting dynamics and abrupt transitions between active and quiet states, such
605
as avalanches and earthquakes (Munoz, 2018; Corral, 2004; Boffetta et al., 1999;
606
Paczuski et al., 2005). For instance, the intensity of avalanches and earthquakes
(ac-607
tive states) is also described by power-law distributions, while time intervals between
608
consecutive avalanches/earthquakes (quiet states) follow a generalized Gamma
distri-609
bution with a characteristic time scale (exponential tail). In this context Gamma is a
610
universal scaling function that is independent of spatial scales and minimum
magni-611
tude thresholds, and is consistently observed for a broad range of conditions despite
612
the large variability associated with phenomena such as avalanches and earthquakes
613
(Corral, 2004; Ribeiro et al., 2010; de Arcangelis et al., 2016).
614
In sleep dynamics, wake and brief arousals during sleep can be considered as
615
active states that, in rodents, are characterized by bursts in θ rhythms. Hence, we
616
hypothesize that a hierarchical structure, invariant across time scales, may underlie
617
the occurrence of θ-bursts, in analogy with non-equilibrium critical phenomena, and
618
investigate the relationship between the duration of θ-bursts and their temporal
619
occurrence (Fig. 10). To this end, we consider the time sequence of θ-bursts, and
620
we study the statistical features of the quiet times Δt separating consecutive bursts,
621
taking into account the duration dθ of each θ-burst (Fig. 10a). Since θ-bursts vary
622
in duration, we impose a threshold D0 representing the time scale of analysis, and
623
we define the quiet time Δtias the period from the end of θi-burst to the beginning
624
θi+1-burst. Thus, the statistical characteristics of Δtidepend on the threshold value
625
D0. We then obtain the probability distribution P (Δt; D0) of quiet times Δti for
626
different values of D0 (insets in Fig. 6c,d). With increasing threshold (scale of
627
observation) D0, the probability of longer Δti increases, while the probability of
628
short Δti decreases, leading to different curves for the distributions P (Δt; D0).
629
Visual inspection of the complex profile formed by the time sequence of θ-bursts
630
and their respective durations shows an apparent similarity when comparing short
631
segments of the profile with the entire sequence above a given threshold D0 (Fig.
632
6b). Presence of statistical self-similarity, observed after effective coarse-graining of
633
the profile, indicates a hierarchical structure across time scales D0 that
character-634
izes θ-bursts occurrence times and durations, and the associated quiet times. To
635
demonstrate statistical self-similarity in the quiet times, we systematically analyze
636
the functional form of the probability distributions P (Δt; D0) for different thresholds
637
D0 by rescaling each distribution by the average quiet timeΔtD0. Remarkably, we
638
find that all distribution curves collapse onto a single function G (Fig. 6c,d), defined
639
by the following scaling relation
640
P (Δt) =Δt−1· G(Δt/Δt). (10)
The scaling relation in Eq. 10 represents a mathematical expression of the statistical
641
self-similarity in the profile formed by the quiet times and θ-burst durations shown
642
in Fig. 6b. We find that the scaling function G(Δt/Δt) is well described by the
643
generalized Gamma distribution G(Δt/Δt; b, ν, p) =
644
p/bν(Δt/Δt)ν−1e−(Δt/bΔt)p/Γ(ν/p) (Stacy et al., 1962), where in our analysis Δt/Δt
645
is a dimensionless quiet time. The Gamma functional form is homogeneous (Ivanov et al.,
646
1996), i.e. rescaling the variable leaves the functional form unchanged. Such scaling
647
function indicates a hierarchical structure in the quiet times between consecutive
θ-648
bursts, independent of the scale of observation D0. In the limit of D0= 0, the quiet
649
time distribution P (Δt; D0) coincides with the distribution of δ-burst durations Pδ
650
(Figs. 2b and 3b,d) — a Weibull functional form that belongs to the same class of
651
homogeneous functions as the generalized Gamma.
652
Our analysis shows that the scaling relation in Eq. 10 and the associated Gamma
653
functional form for the quiet times are robust: (i) we find them in the 24 h period
654
(Fig. 10) as well as separately during light and dark periods, and (ii) they do not
655
significantly change with lesion in the VLPO (compare Fig. 10c and Fig. 10d).
656
Further, the presence of such hierarchical structure in quiet times indicates specific
657
temporal order in the occurrence of θ-bursts. To explicitly verify this, we randomly
658
reshuffle the sequence of θ-burst durations, while preserving the δ-burst durations
659
corresponding to quiet times at D0= 0, and we perform the analysis on the reshuffled
660
sequence to obtain quiet time distributions Prand(Δt; D0) for different thresholds D0.
661
In this case, after rescaling the distributions Prand(Δt; D0) by the average quiet
662
time ΔtD0, their curves collapse onto an exponential distribution (dashed lines in
663
Fig. 10c,d) — a hallmark of temporal independence between consecutive events
664
(Daley and Vere-Jones, 1988). This clearly demonstrates that temporal correlations
665
are intimately related to the existence of non-exponential scaling functions (Eq. 10)
666
(Corral, 2004; Daley and Vere-Jones, 1988).
667
Notably, a similar temporal organization characterized by coexistence of
power-668
law and generalized Gamma distribution has been reported for active states and quiet
669
times in a range of non-equilibrium systems self-tuning at criticality (earthquakes,
670
avalanches) (Munoz, 2018; Pruessner, 2012; Corral, 2004). Thus, our findings of
671
power-law distribution for θ-burst durations (Figs. 2,3) combined with a generalized
672
Gamma distribution for the quiet times between consecutive θ-bursts at different
673
scales of observation D0(Fig. 10) are a strong evidence in support of our hypothesis
674
that bursting activity of fundamental brain rhythms and the associated sleep
micro-675
architecture exhibit critical non-equilibrium behavior.
676
3.5 Long-range power-law correlations in the durations of θ
677