Desynchrony and synchronisation underpinning sleep-wake cycles
EEPJ manuscript No. (will be inserted by the editor)
Desynchrony and synchronisation underpinning sleep-wake cycles
Dmitry E. Postnov , Ksenia O. Merkulova , and Svetlana Postnova , , Saratov State University, 83 Astrakhanskaya street, Saratov 410012, Russia School of Physics, University of Sydney, NSW 2006, Australia The University of Sydney Nano Institute (Sydney Nano), The University of Sydney, NSW 2006, Australia Charles Perkins Centre, The University of Sydney, NSW 2006, AustraliaReceived: date / Revised version: date
Abstract
Objectives:
This paper studies mechanisms of synchronisation and loss of synchrony among the three keyoscillatory processes controlling sleep-wake cycles in the human brain: the 24 h circadian oscillator, thehomeostatic sleep drive, and the environmental light-dark cycle. Synchronisation of these three rhythmspromotes sleep and brain clearance and is critical for human health. Their desynchrony, on the other hand,is associated with impaired performance and disease development, including cancer, cardiovascular diseaseand mental disorders.
Methods:
A biophysical model of arousal dynamics simulating sleep-wake cycles and circadian rhythmsis used as the study system. It is based on established neurobiological mechanisms controlling sleep-waketransitions and incorporates the three oscillatory processes. Nonlinear dynamics methods and synchroni-sation theory are used to numerically investigate model dynamics under conditions that are not easilyachievable in experiments. The role of homeostatic brain clearance rate in synchronisation is investigatedand selective turning on and off of coupling strengths between the oscillators allows us to determine theirrole in oscillators’ dynamics.
Results:
We find that the default state of the model corresponds to the endogenous homeostatic periodthat is far from ∼
24 h rhythm of the circadian and light-dark cycles. Combined action of light and cir-cadian oscillator on the homeostatic rhythm is required to achieve the typical sleep-wake pattern that isobserved in young healthy people. Change of homeostatic clearance rate is found to induce two types ofdesynchronisation: (i) fast clearance rates τ H < . τ H >
69 h maintain synchronisation between the homeostratic and circadian oscillators, but theperiod of both is different from that of the light-dark cycle. Between these regimes, all three rhythmsare synchronised under the studied conditions. The model predicts that the system is highly sensitive toexternal inputs to the neuronal populations of the sleep-wake switch, which affect the endogenous periodof the homeostatic oscillator and can lead to complete loss of sleep.
Conclusions:
Model dynamics show that loss of synchronisation, which is traditionally ascribed to impair-ment of the circadian oscillator, can be caused by changes in the homeostatic clearance rate of the brainor external input to the neuronal populations of the sleep-wake switch. This has significant implicationsfor understanding individual variability in sleep-wake patterns and in mechanisms of sleep and circadiandisorders, indicating that both the homeostatic and circadian mechanisms can be responsible for the sameclinical or behavioural presentation of a disease.
Sleep is a crucial time for clearance of toxic neuro-metabolites from the brain [1]. This process is driven by the recentlydiscovered glymphatic system - a brain-wide perivascular passageway that transports waste products out of the brainand in to the cerebrospinal fluid [2]. Glymphatic system is more active during sleep than wake [1] and its activationhas been linked to the slow wave activity (SWA) observed in the electroencephalography (EEG) during non-rapideye movement (NREM) sleep [3]. In line with these findings, sleep deprivation and selective suppression of slow waveactivity during sleep led to accumulation of waste products in the brain [4,5]. Like other physiological functions, theprocess of brain clearance is under circadian control. This is realised both indirectly via circadian regulation of sleep a [email protected] a r X i v : . [ phy s i c s . b i o - ph ] J a n time and quality [6,7,8] and directly via circadian distribution of CSF in the brain supported by aquaporin-4 channelsdynamics in astrocytes [9]. However, the overall complex system responsible for the clearance and accumulation ofneuro-toxic waste products during sleep-wake cycles is yet to be fully understood [3,10,11,9].The timing of sleep is under circadian control with highest NREM SWA power, and potentially fastest clearance,achieved when sleep appears during circadian rest phase (night for humans) [8]. The daily sleep-wake pattern dependson stable phase relationship between sleep homeostasis, circadian oscillator, and the environmental inputs (e.g., thelight-dark cycle) [6]. Sleep homeostasis reflects the sleep need, which increases during wakefulness and declines duringsleep [12]. The power of SWA in NREM sleep EEG is the current ’gold standard’ marker of sleep homeostasis [13]. Theexact mechanisms of sleep homeostasis are unknown, but thought to be associated with accumulation and clearanceof the somnogens, or toxic waste products, and changes in synaptic connectivity and astroglial calcium signalling inthe brain [14,15,16,11].The central circadian oscillator in the suprachiasmatic nucleus (SCN) of the hypothalamus controls the 24 hperiodicity of the sleep-wake cycles. It promotes wakefulness by counteracting the homeostatic need for sleep duringdaytime and enables consolidated sleep episodes during night in humans [12,7,6]. The phase of the circadian oscillator,in turn, is affected by timing and intensity of environmental and behavioural factors, such as light-dark cycle, meals, andphysical activity [6,17,18]. Phase alignment and synchronisation of these three rhythms: sleep homeostasis, circadianoscillator, and external drives, are critical for optimal sleep and health.Circadian misalignment is observed when sleep-wake cycles, circadian oscillator and/or environmental factors areout of sync with each other. Such misalignment of rhythms became common in the modern society with jetlag andshiftwork being an integral part of life for many people [19]. In these cases, misalignment is caused by changes inenvironmental factors and behaviour, which lead to desycnhronisation between the circadian oscillator and the sleephomeostat [6,19]. A rare example of circadian misalignment is spontaneous internal desynchrony, where long-termabsence of environmental inputs leads to desynchronisation of sleep and circadian oscillator - such conditions do notappear in the real life except for some blind individuals. In this case, the circadian oscillator functions at ∼
24 h period,while sleep appears at much shorter (12-20 h) or longer (28-68 h) intervals [20,21,22,23,24]. Circadian misalignmentis a known risk factor for disease development, including metabolic, cardiovascular and neurological diseases [17,25].During circadian misalignment sleep is disturbed and appears at sub-optimal circadian phases. This likely leads todisturbances in brain drainage and clearance, which can result in long-term accumulation of toxins and developmentof disease [4,5,26].Mathematical models were developed to capture the interactions between sleep and circadian rhythms (reviewedin [27,28]). These were successful in simulating normal sleep, effects of sleep deprivation on sleep homeostasis andrecovery, sleep patterns of different mammals, and alertness dynamics, among other phenomena [27,28]. The modelof arousal dynamics [21,29,30,31], in particular, focused on the interaction between the sleep-wake cycles, circadianoscillator and the external driving force. It was tested against both laboratory and real-world experimental data and,in addition to the above phenomena, reproduced circadian misalignment dynamics observed during shiftwork, jetlag,spontaneous internal desycnhrony and forced desynchrony protocols [21,30,29,32]. However, physical mechanisms ofde-synchronisation in this model were not yet fully characterised.The model of arousal dynamics is composed of two coupled oscillators of different types (sleep homeostatic one andcircadian oscillator) with a external, usually periodic, driving forces. Such systems have been extensively studied innonlinear dynamics, synchronisation theory [33,34], which explains mechanisms underlying synchronisation in differenttypes of oscillating systems and provides mathematical tools for investigation of new systems.In this study we use nonlinear dynamics tools to (i) investigate synchronisation mechanisms in the model of arousaldynamics, (ii) compare these mechanisms to classical models of synchronisation, and (iii) study the role of homeostaticclearance of the brain during sleep on synchronisation and sleep-wake patterns. The paper is structured as follows. Inthe Methods section, we describe model equations and simplify the model to convert it to a more conventional formwhich was extensively studied in synchronisation theory. This will allow us to investigate sensitivity of the model todifferent parameters and compare synchronisation mechanisms in the model with those already established for otheroscillators. In the Results section, we investigate model’s synchronisation properties under conditions of differentcoupling configurations, form of external driving force (light-dark cycle), and varied properties of the homeostaticoscillator. Finally, in the last section, we discuss implications of our findings for physiology and modelling of sleep-wake cycles and for pathophysiology of circadian and sleep disorders.
We focus on the basic version of the model of arousal dynamics [29] as it contains all components relevant to studysynchronisation but excludes improvements and updates that do not affect on synchronisation. Later model extensionsintroducing dynamics of alertness [35], melatonin [30], and effects of light spectrum [31] do not change the coreoscillating processes in the model and will follow the same synchronisation mechanisms as the original model. Toenable comparison to well-studied oscillator models we reduce the number of equations in the homeostatic oscillator ur. Phys. J. Plus and simplify coupling between the oscillators. By doing so we demonstrate that the homeostatic oscillator in the modelcan be represented as the Fitzhugh-Nagumo (FHN) model, which is a well-known model in synchronisation theoryand nonlinear physics in general [36].
The model of arousal dynamics [29] is based on a combination of the model of ascending arousal system in the brainregulating the sleep-wake switch [37] and the dynamic circadian oscillator driven by light [38,39]. Schematic of the keymodel components is shown in Fig. 1. The light-dark cycle, i.e. the driving force, acts on the photoreceptors, P, in theeye. These, in turn, send an input to the circadian oscillator in the suprachiasmatic nucleus, SCN, of the hypothalamus,which has its own endogenous period of ∼
24 h. The homeostatic oscillator is composed of the homeostatic drive, H , andthe two mutually inhibitory neuronal populations: the sleep-active ventrolateral preoptic nucleus of the hypothalamus,VLPO, and the wake-active monoaminergic nuclei, MA, of the hypothalamus and brainstem [40]. VLPO receives asleep-promoting excitatory input from H and a wake-promoting inhibitory input from the SCN. MA controls theaccumulation and decay of H and determines the states of wake and sleep, S , which gate the light input to the brainand modulate the dynamics of the SCN through non-photic coupling. Parameters, A m and A v represent inputs fromother neuronal populations to the MA and VLPO, respectively, including those from the orexinergic and cholinergicnuclei. In the original model, these are assumed to be constant. SCN H MAVLPO
P SA m A v driving forceT=24h Photoreceptors Circadian oscillator Homeostatic oscillator
Inputs fromotherneural nuclei
Figure 1.
Schematic of the model of arousal dynamics adapted from [29]. The two oscillators and the driving force (thelight-dark cycle) are highlighted with the grey background. Inputs from other neuronal populations, A m and A v are highlightedwith the pink background. The notations are: P - photoreceptors, SCN - suprachiasmatic nucleus of the hypothalamus, H -homeostatic drive, MA - monoaminergic nuclei of the hypothalamus and brainstem, VLPO - ventrolateral preoptic nucleus ofthe hypothalamus, S - arousal state, which can be either sleep or wake. Arrows indicate excitatory connections and bar-headedlines - inhibitory. The homeostatic oscillator is composed of the MA, VLPO and H and is described by the following equations: τ v dV v dt = ν vm Q ( V m ) − V v + ν vH H + A v + ν vC C ( X, Y ) , (1) τ m dV m dt = ν mv Q ( V v ) − V m + A m , (2) τ H dHdt = ν Hm Q ( V m ) − H, (3) where V v and V m are the mean voltages of the VLPO and MA populations, τ are the time constants of respectivevariables, ν ij are the coupling strengths from model component j to i . The states of sleep and wake, S , are determinedfrom the dynamics of V m : wake is registered when V m is above the threshold value V th ( V m > V th ) and sleep when V m ≤ V th .The mean firing rate Q is given by a sigmoid function of the mean voltage of a respective neuronal population Q ( V ) = Q max e ( Θ − V ) /σ (cid:48) , (4)where Q max is the maximum mean firing rate, Θ is half-activation threshold, and σ (cid:48) π/ √ ν vC C ( X, Y ), which is described in Sec. 2.1.3.The default parameter values are: τ v = 50 s, τ m = 50 s, τ H = 59 × ν vm = − . ν mv = − . ν Hm = 4 .
57 s, ν vh = 1 . ν vc = − . A v = − . A m = 1 . Q max = 100 Hz, Θ = 10 mV, σ (cid:48) = 3 mV. The circadian oscillator follows Van der Pol equations and is represented by two circadian variables X,Y τ x dXdt = Y + γ (cid:18) X + 43 X − X (cid:19) + C Xn + C Xp , (5) τ y dYdt = − (cid:18) δτ c (cid:19) X + C Y p , (6)where τ x = τ y scale the oscillator period to 24 hours, γ is the stiffness, τ c is the endogenous circadian period, and δ scales the period for consistency with experimental data [38]. Input functions C Xp , C Y p and C Xn describe photic(light-dependent) and non-photic (sleep-wake state-dependent) influences on the circadian oscillator.The default parameter values are: τ c = 24 . × τ x = 24 . × / (2 π ) s, τ y = 24 . × / (2 π ) s, γ = 0 . δ = 24 . × / . → homeostatic) The influence of the circadian oscillator on the homeostatic one is introduced with the term ν vC C ( X, Y ) in Eq.(1)where ν vC is the circadian coupling strength, X and Y are the variables of the circadian oscillator, and c , c , c are theweighting parameters adjusting the shape of the circadian drive. The circadian drive C ( X, Y ) is given as a nonlinearfunction of the circadian variables C ( X, Y ) = 0 . X + (cid:18) c X + c Y + c X + 2 (cid:19) . (7)The default parameter values are ν vc = − . c = 0 . c = 0 . c = 1 . Non-photic coupling (homeostatic → circadian) The non-photic coupling, C Xn simulates modulation of the circadian oscillator depending on model’s arousal state.Effectively this provides coupling from the homeostatic to the circadian oscillator: C Xn = ν Xn (cid:18) − S (cid:19) (1 − tanh ( rX )) , (8) S = U ( V m − V th ) , (9)where ν Xn is the non-photic coupling strength, and r regulates the timing of the non-photic effects. The state function, S , takes a value of unit during wake (when V m > V th ) and zero during sleep ( V m ≤ V th ). Here, U ( x ) is the unitfunction, U ( x ) = 1 if x >
0, and U ( x ) = 0 otherwise, V th is the voltage sleep threshold.The default parameter values are: ν Xn = 0 . r = 10, V th = − ur. Phys. J. Plus Photic coupling (light-dark cycle → circadian) Input from the external driving force, the light-dark cycle, to the circadian oscillator is given by the photic couplingfunctions C Xp and C Y p to each of the circadian variables, respectively. These depend on the fraction of activatedphotoreceptors P in the eye and photopic illuminance input I : C Xp = ν Xp α I (1 − P )(1 − (cid:15)X )(1 − (cid:15)Y ) , (10) C Y p = α I (1 − P )(1 − (cid:15)X )(1 − (cid:15)Y )( ν Y Y Y − ν Y X X ) , (11) τ p Pdt = α I (1 − P ) − βP, (12) α I = α S II + I (cid:114) II . (13)Parameters ν ij control coupling strength from j to i , (cid:15) modulates sensitivity of the photic drive to the circadianvariables, and τ p is the time constant of receptor activation. The fraction of photoreceptors ready to be activated is(1 − P ), which are converted to active state with the rate α I and converted from active to ready at rate β . Function α I introduces effects of light I on activation of photoreceptors and is gated to zero during sleep ( S = 0). Parameters α , I , I adjust the effects of light on activity of photoreceptors.We simulate 12/12 light-dark cycle using the unit function: I = I ext U (cid:18) sin (cid:18) π
24 ( t − . (cid:19)(cid:19) , (14)where I ext is constant, and t − . α = 0 . / . − , I = 100 lx, I = 9500 lx, (cid:15) = 0 . ν Xp = 37 ∗
60 s, ν Y Y = 12 . ∗
60 s, ν Y X = 20 . ∗
60 s, τ p = 1 s, β = 0 . /
60 s − , I ext is chosen depending on a simulated protocol. To enable synchronisation analysis of the model we first investigate if the homeostatic oscillator can be simplified to alower-dimensional form. The time constants for the MA and VLPO are much smaller than for the homeostatic drive: τ m , τ v (cid:28) τ H . This means that the homeostatic oscillator is a fast-slow oscillator with the period driven by the slowtime constant and the slow movement on the limit cycle. Since the equation for V v includes coupling terms while theone for V m does not, we set τ m → V m are instantaneous, so τ m dV m /dt = 0 .Equation (2) can then be re-written as V m = ν mv Q ( V v ) + A m . (15)Substituting V m to the equations for V v and H we get τ v dV v dt = ν vm Q ( ν mv Q ( V v ) + A m ) − V v + ν vH H + A v + ν vC C, (16) τ H dHdt = ν Hm Q ( ν mv Q ( V v ) + A m ) − H. (17)Oscillator described in Eqs (16),(17) is a 2D oscillator, which is a simplified version of the 3D oscillator in Eqs(1)-(3).To characterise the phase plane of the 2D homeostatic oscillator independent of the circadian influence we set ν vC = 0. In this case, we can find the V v - and H -nullclines, which are V v − nullcline H = 1 ν vH { V v − ν vm Q ( ν mv Q ( V v ) + A m ) − A v } , (18) H − nullcline H = ν Hm Q ( ν mv Q ( V v ) + A m ) . (19)Figure 2(a) shows the nullclines on the H vs. V v plane and highlights the range responsible for the self-sustainedoscillations, i.e., the limit cycle in Fig. 2(b). In the range of self-sustained oscillations, the H -nullcline can be approx-imated with a line and the shape of the V v -nullcline can be described with a cubic parabola as shown with a dashedline in Fig. 2(b) - these are similar to nullclines of the FitzHugh-Nagumo (FHN) model [36,41].
0 20 40 60 80 100 0 20 40 60 80 100 v a r i ab l e s , a . u . v a r i ab l e s , a . u . Vv,mV
H H V v -nullcline V v -nullclineX Y XC(X,Y) K(Y) 10C Xn (X,S)t,h t,h1 1-1-1 (c) (d)-15 0 15 0 4-4-802010 11141213(a) (b)cubicH-nullcline H-nullclineVv ,mV Figure 2.
Dynamical features of the homeostatic oscillator and coupling between the circadian and homeostatic oscillators. (a)The H - and V v -nullclines of the homeostatic oscillator and (b) is the zoomed-in area critical for the self-sustained oscillationsthat is highlighted with a dashed rectangle in (a). Dashed arrows show the fast segments of the limit cycle and the direction ofvariables change. Slow segments are highlighted in magenta. The polynomial fitting for the V v -nullcline is denoted as ’cubic’, V cubic = 0 . z / − z ) + 12 .
3, where z = 0 . V v + 1. (c) Time-dependence of the variables relevant for the circadian input ν vC C ( X, Y ) acting on the homeostatic oscillator. The circadian drive is shown in magenta, and its approximation with K ( Y )is shown in blue. (d) The non-photic coupling C Xn ( X, S ) acting on the circadian oscillator. Factor of 10 is used for bettervisualisation alongside X variable. The mathematical form of the homeostatic oscillator is also identical to that of the FHN model. By making thefollowing substitutions v = V v , h = − H, γ h = 1 /τ H , and approximating the H -nullcline in Fig. 2(b) with a line having a slope a and offset bν Hm Q ( ν mv Q ( V v ) + A m ) /τ H = av + b, (20)we show that the 2D homeostatic oscillator in Eqs (16),(17) becomes ε v dvdt = F ( v ) − h ; dhdt = av + γ h h + b, (21)where ε v = τ v /ν vH , and F ( v ) = ( ν vm Q ( ν mv Q ( v ) + A m ) − v ) / ( ν vH ). The system of equations in (21) exactly matchesFHN model, which is considered one of the classical models in synchronisation theory and its oscillatory behaviour iswell characterised [34]. For example, the nullcine positions in Fig. 2(b) correspond to self-sustained oscillations arounda steady state, which is located at the crossing of the two nullclines. The oscillator’s period is determined by thelength of the slow segments on the V v -nullcline (same as v -nullcline) (highlighted in magenta). The time spent by the ur. Phys. J. Plus model in the fast segments, shown with dashed arrows have only a minor contribution to the period of oscillationsbut exact position of the fast segments determines the lengths of the slow ones. Changes in model parameters canlead to disappearence of self-sustained oscillations, which happens when the crossing of the two nullclines moves toeither one of the parabola extrema. This representation of the homeostatic oscillator is useful for understanding itssynchronisation properties, which we use in the analysis throughout this study. Circadian oscillator acts on the homeostatic one via the term ν vC C ( X, Y ) in Eqs (1) and (16), where circadian drive C ( X, Y ) is a nonlinear function of the circadian variables, Eq. (7).This means that the coupling is nonlinear andthat it may lead to appearance of new oscillation frequencies dependent on the degree of the polynomial. In thiswork, however, we focus on 1:1 synchronisation between the homeostatic and circadian oscillators, so the harmonicsintroduced by the nonlinear coupling are less relevant. This allows us to approximate the coupling term, C ( X, Y ) witha simpler linear function K ( Y ) dependent only on one circadian variable. C ( X, Y ) ≈ K ( Y ) K ( Y ) = c (1 − Y ) , (22)where c ≈ . C ( X, Y ) and K ( Y ). Function K ( Y ) demonstrates maximum wake-promotingcircadian signal at the same time as C ( X, Y ) (maxima of both functions). There is discrepancy in the timing of theminimum wake-promoting signal, but this should not significantly affect frequency synchronisation of the oscillators.Frequency/period synchronisation is expected to have similar properties when K ( Y ) or C ( X, Y ) but the shape ofthe signal will affect phase synchronisation. This is why C ( X, Y ) is needed to reproduce nuances of the multitude ofexperimentally observed sleep phenomena as shown in the model of arousal dynamics.The approximation allows us to separate the oscillatory and constant components in the coupling. The former isresponsible for synchronisation of rhythms and the latter modulates the constant input to the VLPO and can thus beadded to A v which affects the period of the homeostatic oscillator. Thus change of ν vC has two effects: it affects thestrength of the circadian action on the homeostatic oscillator and it modulates the homeostatic period, which bothneed to be taken into consideration when analysing model dynamics.The non-photic coupling from the homeostatic oscillator to the circadian is shown in Fig. 2(d) and defined inEqs (8),(9). This coupling is weak compared to the magnitude of the circadian variables (note the factor of 10 forvisualisation in Fig. 2(d)) and is weaker than the coupling from the circadian oscillator to the homeostatic. Importantly,the non-photic term is negative during sleep (coinciding with increasing part of the X variable), which means thatthe coupling is likely to slow down the circadian oscillator.The photic coupling term describes the effects of the driving light-dark force on the circadian oscillator and isdefined by the functions C Xp and C Y p which, in turn, depend on several parameters and variables as described in Eqs(11),(10). Due to its complexity, the effects of this coupling on synchronisation need to be assessed numerically.
To collect information about the homeostatic and circadian periods, the simulation was run for 150 days of modeltime, and the last 100 days were used to calculate T S and T C . AS the starting points T C and T S calculation, conditions Y ( t ) = 0 and V v = V th were used, respectively. Maps of periods on the plane of two parameters were calculated bymeans of parallel computing, thus, each combination of parameters from the 400 ×
100 matrix was checked. ∼
24 h circadian rhythm
To investigate synchronisation properties of the homeostatic oscillator under influence of the circadian signal wecalculate Arnold tongues diagram at varied strength of the circadian coupling, ν vC , but zero non-photic and photiccoupling acting on the circadian oscillator ( I ext = 0 lx, ν Xn = 0). This approach is widely used in synchronisationtheory [33,34] and allows us to visualise synchronous states depending on the endogenous period of the homeostaticoscillator, controlled by τ H , and the strength of oscillatory coupling acting on it, ν vC . The endogenous period of thehomeostatic oscillator is controlled by its slowest variable, H , whose rate of change is determined by the time constant τ H . Indeed, if ν vC = 0, the period of the homeostatic oscillator can be approximated as T S = 1 . . τ H . τ H ,h ν vC ,h T S = T C T S = T C /2 Original modelSimpli fi ed coupling2D model + simpli fi ed coupling 2505101520 T S ,h
Figure 3.
Synchronisation map for the homeostatic oscillator in presence of one-directional coupling from the circadian oscil-lator. Both photic and non-photic inputs acting on the circadian oscillator are set to zero ( I ext = 0 lx, ν Xn = 0). All otherparameters are at their default values. Colourbar corresponds to the period of the homeostatic oscillator (sleep period, T S ) andyellow filled circle shows the default parameter values, τ H = 59 h and ν vC = − . Figure 3 shows the resulting synchronisation map for the homeostatic oscillator demonstrating T S at τ h = 0 . . . ν vC = 0 . . . − T C is constant across the map and is equal to 24.13h. This is because both photic and non-photic inputs are set to zero, so the circadian oscillator is not affected by thehomeostatic oscillator or by light. Note that the period is different from the default value observed under constantdarkness T C = 24 . τ C because the circadian oscillator was calibrated in presence of the non-photic input,while here it is set to zero.Behaviour of the three model versions is compared: (i) original model of arousal dynamics, (ii) original model withsimplified coupling, Eq. (22), and (iii) simplified model with 2D homeostatic oscillator, Eqs (16),(17) and simplifiedcoupling. For all model versions, the main synchronisation region, where T S = T C , occupies the largest part of thediagram. For both the original model and the model with simplified coupling, the synchronisation tongue starts at τ H = 88 .
13 h. For the model with 2D homeostatic oscillator, the tongue is shifted to τ H = 92 h but its shaperemains the same as seen by comparing the resonance lines (e.g., solid black and dashed magenta vs. dotted black).Interestingly, at ν vC ∈ [ − , − .
5] mV, the model with simplified coupling produces higher number of synchronisationregions than the other two model versions as seen by the multiple tongues outlined by the dashed magenta lines in thetop left region of the map. Importantly, however, the key synchronisation behaviour of the original model is conservedin simplified versions, especially for weak coupling, ν vC ∈ [ − . ,
0] mV.Endogenous homeostatic period at the default value of τ H = 59 h and ν vC = 0 mV, is found to be T S = 16 . ≈ . T C . This is a significant difference in the oscillators periods, and, normally, synchronisation of twooscillators with such different periods requires either strong coupling strength or external driving force. However,amplitude of the coupling term ν vC C ( X, Y ) is only 5% of that for the variable V v on which it acts. It is, thus, notsurprising that the default state of the model (yellow filled circle in Fig. 3) is outside the main synchronisation rangefor all model versions in absence of the light-dark driving force and non-photic coupling. This means that the circadianand homeostatic oscillators in the original model at the default parameter values have tendency to be asynchronousin absence of external driving forces. ur. Phys. J. Plus In this section we show that synchronisation of all three rhythms, the homeostatic, circadian and the light-dark cycleis only observed in a small range of τ H values, while synchronisation of two of these three rhythms is more likely.The non-photic coupling introduces disturbances in the periodicity of the circadian oscillator by dragging it awayfrom the endogenous value of ≈ . τ H . This is seen in Fig. 4(a) where T C and T S are shown for the case of default non-photic coupling while the circadian coupling and light input are set tozero ( ν vC = 0 mV, I ext = 0 lx). In the resonant areas where T C /T S = 1 , , T C /T S = 3), the circadianperiod follows the period of the homeostatic oscillator. This is characteristic behaviour for the phase/frequency lockingmechanism of synchronisation [34]. The non-photic coupling does not lead to large synchronisation areas as observedfor the circadian coupling, but it is clear that it supports synchronisation of the homeostatic and circadian oscillators. c T C =T S TsTc20 40 60 8023.624.4 (a)(b) (c) τ h=88 h τ h=65 h τ h=59 h τ h=40 h τ h=30 h τ h ,h τ h ,h T c , s , h T c , s , h Clock time,h
030 0 24 D a y nu m be r , a . u . C =2T S T C =4T S T C =3T S Figure 4.
Effects of non-photic coupling and light on synchronisation of the homeostatic and circadian oscillators. All resultsare for the original model. (a) Effect of τ H on circadian period, T C (red line), and homeostatic period, T S (dashed black line),in presence on non-photic coupling. Circadian and photopic coupling are set to zero, ν vc = 0 mV, I ext = 0 lx.(b) Dependence of T C and T S on τ H in presence of all couplings and external light-dark cycle I ext = 80 lx. (d) Raster plots for selected examplesof τ H (values shown in panel titles) demonstrating 30-days dynamics of sleep times (blue lines) and circadian marker (onset ofmelatonin synthesis, red triangles) against clock time. Yellow indicates wakefulness. In all panels, simulations were run for 150days. The periods presented in (a) and (b) are averaged over the last 100 days of the simulations, and the rasters in (c) areshown for the last 30 days. Figure 4(b) shows similar calculations for T C and T S to those in Fig. 4(a) but, this time, in the full model of arousaldynamics and in presence of all coupling terms and the external light-dark cycle, I ext = 80 lx, with a period of 24 h.In this case, the circadian oscillator is synchronised with the light dark cycle ( T C = 24 h) but not the homeostatic oscillator ( T S <<
24 h) at most values of τ H < . τ H >
69 h, where the circadian and homeostatic oscillator are synchronised ( T C = T S ), but their periods are differentfrom the 24-hour rhythm of the light-dark cycle and increase with the increase of τ H . Looking back at Fig. 3 whereonly circadian coupling is present, this range of τ H at ν vC = − . T S = T C for τ H ∈ [69 , τ H . At τ H = 30 h, T C /T S ≈ τ H = 40 h there is a resonance with T C /T S ≈ τ H = 59 h, regular sleep-wake cyclesare observed with one 8-hour sleep episode per day starting at ≈ τ H to 65 h results in the shift of the sleepepisodes to later time while the timing of melatonin onset shifts only slightly resulting in a larger time gap betweenmelatonin onset and sleep start. However, all the rhythms remain synchronised. At τ H = 88 h, the sleep wake cyclesare synchronised with the circadian oscillator (phase difference between sleep and melatonin onset is constant) butboth are different from the 24 h period of the light-dark cycle. Previous sections focused on the role of τ H and coupling terms in synchronisation. However, position of nullclines inFig. 2 and oscillators properties also depend on other parameters. In particular, parameters A m and A v representexternal neuronal inputs to the homeostatic oscillator from other brain nuclei and are likely to be varied dependingon an individual and their physical or mental state. In this section we study how changes in A m and A v affect thedynamics. First, we do it for the blocked coupling from the circadian oscillator, at ν vC = 0, in order to see revealthe intrinsic features of the homeostatic oscillator. Second, we do it in the full model with circadian and non-photiccouplings but in absence of driving force ( I ext = 0). It allows one to assess the contribution from reciprocal couplingbetween two oscillators.Figure 5(a)-right shows response of T S to changes in A m and A v in absence of circadian input to the homeostaticoscillator. As seen from the direction of the contour lines in the map, the change of A m has stronger effect on T S thanthe change of A v . However, at the default parameter values (point B) the homeostatic oscillator is sensitive to bothparameters. Decrease of A v starting at point B leads to slowing of the oscillations until they disappear completely atpoint C. Increase of A v leads to similar behaviour and disappearance of oscillations at point A. These dynamics areexplained by the nullclines in Fig. 5(a)-left. For both points A and C the H − and V v − nullclines cross at one of theextrema of the V v − nullcline. This situation is well-studied in FHN model and corresponds to a transition from self-sustained oscillations to excitable dynamics [36,41]. Mathematically, this corresponds to presence of the supercriticalAndronov-Hopf bifurcation with the so-called Canard explosion [42]. Thus oscillations disappear at A m values to theleft of points A and C, and more generally to the left of the yellow line in the map.The T S map changes significantly when circadian coupling is set to its default value, ν vC = − . I ext isstill zero). In this case, the map shows a step-like structure where T S lingers at fixed resonant values of T S = nT C ,where n = 1 , , ... , while A m and A v are changed until it transitions to the next resonance, Fig. 5(b). This set offrequency/phase-locked states makes the effect of A v , and especially, A m changes highly dependent on its specificchoice: from negligibly weak within the resonant areas to abrupt changes at their borders.Similar to Fig. 5(a), oscillations disappear at low A m and increase of A m leads to higher T S , while A v has minoreffect on T S . The location of the default values of A m and A v is shown in Fig. 5(c), which makes it clear that in thedefault state the model is more sensitive to small decrease of A v than changes of A m or increase of A v . With thedefault parameter values sitting on the border of the T S = T C resonance, decrease of A v would further desynchronisethe system. We have applied methods of nonlinear dynamics to study synchronisation in the model of arousal dynamics and showedthat the key model element, the 3D homeostatic oscillator can be simplified to 2D form without significant change inits synchronisation properties. By approximating one of the nullclines of the 2D oscillator with a line, we showed thatthe 2D homeostatic oscillator is equivalent to the well-known Fitzhugh-Nagumo model, which is a widely used model in ur. Phys. J. Plus (a) 0 10 20 30 40 6050 708090-15-2 0 5ABC A v +A m + A m -A v - -11-92420 0.81.8T c c c (c)-10 4 H , h v ,mV H , h H , h T S ,h T S , h T S , h A v , m V A v ,mV A m ,mV A v ,mVA m ,mVA m ,mV Figure 5.
Effects of A m and A v on the homeostatic period, T S . (a) Map of periods for the homeostatic oscillator, T S (right)and nullclines (left) for representative points A, B, and C in the map in absence of circadian coupling, ν vC = 0. Colourcodingin the map indicates different values of T S with the contours indicating lines of equal T S . Numbers indicate relevant T S valuesin hours. H -nullcline is shown in red and V v -nullcline in blue. (b) Map of T S in presence of default circadian and non-photiccoupling, but zero light, I ext = 0 lx.(c) Zoomed-in area of (b) showing location of the default values of A m = 1 . A v = − . nonlinear physics [36,41]. The circadian oscillator, on the other hand, is modelled with the Van der Pol oscillator [38],the only difference from the classical Van der Pol oscillator [43,44] is the higher degree of polynomial used to describevariable X - seven instead of three. Taking into consideration the coupling between the two oscillator, the full modelcan thus be described as a periodically forced and reciprocally coupled two 2D self-sustained oscillators, one showingsmooth oscillations, and another - fast-slow dynamics. Mathematically, such system is represented by trajectoriesmoving on a 3D torus [45,46], and the resonances between the homeostatic, circadian and light-dark cycles fit wellwith this paradigm [47]. The default state of the model of arousal dynamics represents sleep-wake and circadian dynamics for a typical healthyyoung individual (or a group average). The model has been validated against >
50 experimental datasets and success-fully reproduces variety of sleep phenomena [29,35,30]. From synchronisation point of view, the default state of themodel corresponds to the global resonance 1:1:1. Interestingly, however, this state is far from parameter values wherethe homeostatic oscillator has a period close to the circadian oscillator and the light-dark cycle. Instead the defaultstate of the model is located at the border of synchronisation range, where, in absence of the circadian input, theendogenous homeostatic period is T S ≈
16 h. In this uncoupled state multiple short ( << The fact that in absence of light but with intact circadian coupling, the model functions at the border of thesynchronous regime means that it is very close to the so-called spontaneous internal desynchrony (SID) - a phenomenonthat was experimentally observed in people living for extended periods of time in constant darkness. During SID thecircadian oscillator usually has period close to 24 h, whereas sleep appears with much shorter (12-20 h) or longer(28-68 h) period [22,23,49,20,21,24] - similar to the dynamics seen in our study. This ease of transition from normalsleep-wake cycles under the 24 h light-dark cycles to SID under darkness in humans indicates that the brain operatesclose to a bifurcation point and only presence of environmental driving force allows it to have stable and synchronisedperiodic activity. The model of arousal dynamics was not originally designed to reproduce SID, but its ability to doso with only a small nudge towards lower τ H and/or lower ν vC supports the current set of default parameters. If themodel default state was deep in the synchronous regime, it would have been difficult to achieve SID with physiologicallyjustified parameter changes.Our results predict two main types of desynchrony in the model. First is at τ H ∈ [0 ,
58] h, where the circadianoscillator with a typical, experimentally confirmed [50], endogenous period of 24.1-24.2 h is entrained to the 24 h lightdark cycle, but the homeostatic oscillator is asynchronous. In this case, melatonin synthesis onset appears at a fixedtime every day and at the correct time of day (physiological range between 19:00 to 01:00 [51]) but sleep times arenot phase locked with it. From the appearance of sleep-wake cycles and melatonin rhythms, especially at non-resonantvalues of τ H , this regime is easily confused with the one where circadian coupling is zero. The mechanism, however, isdifferent because the circadian coupling is intact but it is the homeostatic time constant that causes the desynchrony.Second, is at τ H >
69 h where the homeostatic and circadian oscillators are synchronised T S = T C but are differentfrom the 24 h of the light-dark cycle. In this case the onsets of melatonin synthesis and of sleep are phase locked incorrect relationship (melatonin preceding sleep) but the phase angle increases with increase of τ H and both melatoninand sleep shift to later and later time every day. The desynchronisation patterns discussed above may be linked to those observed in circadian rhythm sleep disorders[52]. These include diseases where (i) sleep appears several hours earlier than conventional or desired sleep time - this isknown as advanced phase sleep disorder, ASPD; (ii) sleep appears substantially later than the conventional sleep time -delayed sleep phase disorder, DSPD; and (iii) sleep does not follow circadian rhythms and instead of being consolidatedinto a single episode per day, there are several shorter sleep episodes that appear at random times throughout day andnight - this is knowm as irregular sleep-wake rhythm, ISWR [52]. In all these diseases, it is challenging (sometimesimpossible) to maintain socially-conventional schedules of work and commitments and results in further disturbancesof sleep and health [53]. All these diseases are generally ascribed to disturbances in the circadian system but theirexact mechanisms are unknown [52]. Our study shows that all these circadian sleep patterns can also be obtained bychanging τ H while the circadian system remains unchanged. We predict that ISWR dynamics would be observed atvery short, non-resonant τ H , e.g., τ H <
40 h, ASPD would be seen at moderatly short values of τ H just below thedefault state, and DSPD dynamics would be seen at τ H >
69 h. Similarly, it is commonly assumed that timing ofmelatonin marker is caused by changes in the circadian system. Our study shows, that change in τ H can cause advanceand delay of melatonin onset relative to clock time and to sleep onset, without any changes in the cicradian oscillatoritself. These are critical insights which may help us better understand mechanisms of these diseases and develop betterdiagnosis and treatment procedures. Future research should thus focus on both the circadian- and homeostatic-drivenpathways to these diseases.Finally, we showed that parameters A m and A v responsible for inputs from other neuronal populations to the MAand the VLPO also affect the period of the homeostatic oscillator and synchronisation. This means that these otherpopulations, e.g., orexin neurons, may likewise be involved in either support of synchronisation or desynchronisation ofthe homeostatic and circadian oscillators and the light-dark cycle. In practical sense it means that information on sleep-wake patterns and melatonin timing during circadian rhythms sleep disorders is insufficient for understanding whichsystem components led to desynchronisation. Interestingly, low values of A m lead to complete loss of self-sustainedoscillations - a situation that is not observed at variation of the other parameters studied here. In real life such completeloss of sleep is observed in fatal familial insomnia - a rare genetic neurodegenerative disorder characterised by completeinability to sleep and loss of some autonomic functions (e.g., temperature control), which has no cure and ultimatelyleads to death [54]. Our study predict that it may be caused by degeneration of neuronal populations acting on theMA, which should be studied further in the future. In this study we have considered only two light profiles: one is a constant darkness with I ext = 0 lx and the other isthe 12/12 light-dark cycle with I ext = 80 lx resulting in darkness at night between 20:00 and 08:00 and sinusoidally ur. Phys. J. Plus modulated light during daytime (08:00 to 20:00) peaking at 80 lx. Intensity of light and shape of light profile (timing)strongly affect the dynamics of the circadian oscillator [6], which has been extensively studied in circadian models,e.g. [31]. However, systematic studies of how light intensity and timing affect synchronisation of all three rhythmsare lacking and need to be performed in the future. This will aid in better understanding of dynamics and design ofoptimal light schedules for such common examples of circadian misalignment like shiftwork and jetlag.Majority of sleep models, including the model of arousal dynamics, are deterministic. To ensure a stable stateof the model at different parameter values we had to perform simulations of hundreds of days of sleep-wake cycles.However, in real life our sleep-wake cycles are affected by numerous random processes and neither sleep times normelatonin markers are identical from day to day. Mathematically, this means that stochastic model of sleep-wakecycles needs to be implemented accounting for different sources of randomness, both endogenous (e.g., fluctuationsin dynamics of neuronal populations [55]) and external (e.g., changes of light, stress, meals, exercise [56]). Additionof these endogenous dynamics, in particular those of neuronal populations will allow to bridge the gap between thecurrent models of sleep-wake cycles and brain clearance mechanisms during sleep, which has recently been shown tobe under control of some of the same populations as implemented in the model of arousal dynamics [57]. Acknowledgments
This research was supported by the Russian Ministry of Education and Science, project
Conflicts of Interest
DEP, KOM, and SP have no conflicting interests to declare. In interest of full disclosure: SP served as a Theme Leaderand previously as a Project Leader in the CRC for Alertness, Safety and Productivity which funded development ofthe model of arousal dynamics. She reports research grants from Qantas Airways Ltd and Alertness CRC, which arenot related to this paper.
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