mathematical model of network dynamics governing mouse sleep

Mathematical Model of Network Dynamics Governing MouseSleep–Wake Behavior

Cecilia G. Diniz Behn,1 Emery N. Brown,3 Thomas E. Scammell,2 and Nancy J. Kopell11Department of Mathematics and Center for BioDynamics, Boston University; 2Department of Neurology, Beth Israel DeaconessMedical Center; and 3 Department of Anesthesia and Critical Care, Massachusetts General Hospital, Boston, Massachusetts

Submitted 7 November 2006; accepted in final form 2 April 2007

Diniz Behn CG, Brown EN, Scammell TE, Kopell NJ. Mathemat-ical model of network dynamics governing mouse sleep–wakebehavior. J Neurophysiol 97: 3828–3840, 2007. First published April4, 2007; doi:10.1152/jn.01184.2006. Recent work in experimentalneurophysiology has identified distinct neuronal populations in therodent brain stem and hypothalamus that selectively promote wakeand sleep. Mutual inhibition between these cell groups has suggestedthe conceptual model of a sleep–wake switch that controls transitionsbetween wake and sleep while minimizing time spent in intermediatestates. By combining wake- and sleep-active populations with popu-lations governing transitions between different stages of sleep, a“sleep–wake network” of neuronal populations may be defined. Tobetter understand the dynamics inherent in this network, we created amodel sleep–wake network composed of coupled relaxation oscilla-tion equations. Mathematical analysis of the deterministic modelprovides insight into the dynamics underlying state transitions andpredicts mechanisms for each transition type. With the addition ofnoise, the simulated sleep–wake behavior generated by the modelreproduces many qualitative and quantitative features of mouse sleep–wake behavior. In particular, the existence of simulated brief awak-enings is a unique feature of the model. In addition to capturing theexperimentally observed qualitative difference between brief andsustained wake bouts, the model suggests distinct network mecha-nisms for the two types of wakefulness. Because circadian and otherfactors alter the fine architecture of sleep–wake behavior, this modelprovides a novel framework to explore dynamical principles that mayunderlie normal and pathologic sleep–wake physiology.

I N T R O D U C T I O N

In the 1970s McCarley and Hobson (1975) introduced adynamic model in which alternations between rapid eye move-ment (REM) sleep and non-REM (NREM) sleep were de-scribed by a “predator–prey”-like system composed of inhibi-tory NREM-active populations and excitatory REM-activepopulations. Since this early model, much physiologic andgenetic work has improved our understanding of the neuro-physiology involved in the control of sleep–wake behavior. Inparticular, mutually inhibitory sleep- and wake-active neuronalpopulations in the brain stem and hypothalamus are implicatedin the production of sleep and wake (see review in Saper et al.2005). Saper and colleagues (2001) proposed a conceptualmodel in which this mutual inhibition acts like a flip-flopswitch and governs the dynamics of sleep–wake transitions;however, formal modeling of behavioral state control has notyet integrated this new information.

When addressing the control of wake and sleep at theneuronal level, properties of neuronal firing and the resulting

structure of sleep–wake behavior must be considered on a finescale. This structure can be characterized by the number,duration, and organization of bouts of each state in addition tothe total percentage of time spent in each state. In contrast tothe single consolidated nighttime sleep period typically expe-rienced by adult humans, adult mice exhibit polyphasic sleep–wake behavior (Fig. 1). In addition, many species, includingmice and humans, normally experience brief awakeningsfrom sleep as well as sustained bouts of wakefulness asso-ciated with feeding and locomotor behavior. In the past,brief awakenings were often discounted as “noise,” but theyare now recognized as an essential element of sleep archi-tecture (Dijk and Kronauer 1999; Halasz et al. 2004; Lo etal. 2002, 2004).

Statistical analyses of the distributions of sleep and wakebout durations in multiple species established that sleep boutdurations follow an exponential distribution and wake boutdurations obey a “power law with a tail” distribution (Blum-berg et al. 2005; Lo et al. 2002, 2004). The distinctionbetween wake bout durations for which the power law holdsand those bout durations in the “tail” region defines anemergent, species-dependent threshold separating brief andsustained wake bouts; this threshold is about 2 min in mice.Furthermore, this distinction suggests that different mecha-nisms underlie brief and sustained wake bouts (Halasz et al.2004; Lo et al. 2004).

To examine the dynamics inherent to this circuitry, wedeveloped a deterministic model based on interactions amongwake-, sleep-, and REM-active populations. By consideringsleep–wake behavior in the context of neuronal populationactivity, we incorporate tenets derived from previous modelingof neuronal populations and sleep–wake behavior (Borbely1982; McCarley and Hobson 1975; McCarley and Massaquoi1986; Tobler et al. 1992; Wilson and Cowan 1972) whiletargeting the role of network dynamics in behavioral statecontrol. This model provides a novel framework for exploringdynamical principles that underlie normal sleep–wake physi-ology as well as physiology that may be altered with aging orin pathologic states like narcolepsy.

M E T H O D S

Summary of relevant physiology

Although much of the physiology of sleep–wake control issimilar across mammalian species, the dynamics of interaction mayvary. Therefore we restrict our attention to the mouse sleep–wake

Address for reprint requests and other correspondence: C. Diniz Behn,Department of Neurology, Beth Israel Deaconess Medical Center, 77 AvenueLouis Pasteur, Boston, MA 02115 (E-mail: [email protected]).

The costs of publication of this article were defrayed in part by the paymentof page charges. The article must therefore be hereby marked “advertisement”in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

J Neurophysiol 97: 3828–3840, 2007.First published April 4, 2007; doi:10.1152/jn.01184.2006.

3828 0022-3077/07 $8.00 Copyright © 2007 The American Physiological Society www.jn.org

network. The neuronal populations included in our definition ofthis network are the locus coeruleus (LC), tuberomammillarynucleus (TMN), dorsal raphe (DR), extended ventrolateral preopticnucleus (eVLPO), ventrolateral preoptic cluster (VLPO), lat-erodorsal tegmental nucleus (LDT), and pedunculopontine tegmen-tal nucleus (PPT). Each of these neuronal populations, summarizedin Table 1, demonstrates state-dependent firing profiles; causalrelationships between the activity of a given population and aparticular behavioral state were previously established throughextensive experimentation with site-specific lesions and neuro-transmitter agonists and antagonists (Saper et al. 2005).

We classify each physiologic neuronal population based on thestate (wake, sleep, or REM sleep) in which the population dem-onstrates a high level of activity and we associate the physiologicpopulation with an abstract wake-, sleep-, and REM sleep-promot-ing population in the model. We use the designations sleep- andREM-active (rather than NREM- and REM-active) because VLPOneurons exhibit high levels of activity during both NREM andREM sleep (Szymusiak et al. 1998). We do not include neuronal(sub-) populations that are equally active in wake and sleep (eitherNREM or REM sleep). For example, the LDT includes a subpopu-lation of neurons that is active mainly during REM sleep andanother subpopulation of neurons that is active during both wakeand REM sleep (Datta and Siwek 2002); in the theoretical contextof the reduced network, we consider the LDT to be REM-activeonly and we allow the purely wake-active populations to subsumethe role of wake-active LDT neurons.

LC, DR, and TMN are associated with the modeled wake-promot-ing population. The LC and DR are monoaminergic neuronal nucleilocated in the brain stem that have long been associated with promot-ing vigilance and maintaining muscle tone (Aston-Jones and Bloom1981; Aston-Jones et al. 1986; Hobson et al. 1975; McGinty andHarper 1976; Wu et al. 2003). The TMN is a wake-promotinghypothalamic nucleus that represents the sole source of histamine inthe brain (Sherin et al. 1998; Takahashi et al. 2006). Although orexinneurons are wake-active (Lee et al. 2005; Mileykovskiy et al. 2004)other aspects of their firing profiles and properties differ from the LC,DR, and TMN (Eggermann et al. 2003; Li et al. 2002). Therefore wemodel the effects of orexin signaling during wakefulness by state-dependently increasing the strength of inhibition from the wake- to thesleep-active populations.

The GABAergic/galaninergic VLPO and eVLPO (Chou et al. 2002;Lu et al. 2000; Sherin et al. 1998) are associated with the modeledsleep-promoting population. Although the distinction between theVLPO cluster and eVLPO is not unanimously recognized, Saper, Lu,and colleagues previously reported anatomic and functional differ-ences between the two: anatomic differences are based on cell densityand efferent projections, whereas functional differences suggest thatactivity (as measured by the expression of fos) in the VLPO core ishighly correlated with NREM sleep and activity in the eVLPO ishighly correlated with REM sleep (Lu et al. 2000, 2002).

Cholinergic signaling from neurons in LDT/PPT has been di-rectly linked to REM sleep in multiple experiments (Datta andSiwek 1997; Siegel 2005; Steriade et al. 1990; Thakkar et al.1998). Subpopulations in the LDT/PPT that are mainly activeduring REM sleep are associated with the modeled REM-promot-ing population. Although we do not include a separate populationof GABAergic REM-active neurons in our model, we include amechanism for increasing the strength of (GABAergic) inhibitionto the wake-active populations immediately preceding and duringeach REM bout. We attribute this increased inhibition to activationof the eVLPO, but other GABAergic populations (Lu et al. 2006;Maloney et al. 1999) could be responsible.

Because we are focusing on the dynamics of network interac-tions, the changes in state-dependent activity within these popula-tions are important. Experimental data suggest that transitions inneuronal firing rates are slower than the changes in the electroen-cephalogram (EEG). For example, in LC neurons, the mean firingrate decreases over the 100 s preceding NREM sleep; the transitionfrom low firing rates during NREM sleep to essentially silentbehavior during REM sleep occurs gradually; and firing rates jumpquickly at the onset of (or a few seconds before) wakefulness(Aston-Jones and Bloom 1981; Hobson et al. 1975). Transitionperiods have been considered in other populations in the sleep–wake network [DR (Lydic et al. 1983), TMN (Takahashi et al.2006), VLPO (Szymusiak et al. 1998), LDT/PPT (Steriade et al.1990)], but complete time course data have not been reported.Because each of the abstract wake-, sleep-, and REM sleep-promoting populations in the current model combines the featuresof several physiologic neuronal populations, our model does notreproduce firing rate or time course data. Instead, our modeldescribes heuristic activity levels ranging between 0 and 1 and

A B

FIG. 1. Experimentally observed alternations be-tween wake (W), sleep (S), and rapid eye movement(REM) sleep (R) in a behaving mouse. If brief awaken-ings are omitted (A), the polyphasic sleep–wake structureis evident. Including brief awakenings (B) illustrates thequalitative differences between brief and sustained awak-enings.

TABLE 1. Neuronal populations considered in our model

Active State Neuronal Population Neurotransmitters

Wake Dorsal raphe (DR) SerotoninLocus coeruleus (LC) NorepinephrineTuberomammillary nucleus (TMN) Histamine

Sleep Ventrolateral preoptic cluster (VLPO) GABA/GalaninExtended ventrolateral preoptic nucleus (eVLPO) GABA/Galanin

REM sleep only Laterodorsal tegmental nucleus (LDT) AcetylcholinePedunculopontine tegmental nucleus (PPT) Acetylcholine

3829MODELING MOUSE SLEEP–WAKE BEHAVIOR

J Neurophysiol • VOL 97 • JUNE 2007 • www.jn.org

transitions between levels of high and low activity are fast, similarto the changes observed in the EEG.

Model formulation

From the physiology reviewed earlier, we extracted a reducednetwork of three coupled wake-, sleep-, and REM-active neuronalpopulations (denoted W, S, and R, respectively) as shown in Fig. 2.Individual neurons in these populations demonstrate spontaneousfiring (Morairty et al. 2004; Sakai and Crochet 2000; Steriade et al.1990; Taddese and Bean 2002; Williams et al. 1984); because thepopulations tend to exhibit cohesive activity, this suggests spontane-ous activity of the population as a whole. In addition, cyclic attributesof brief awakenings and REM bouts were previously reported (Endoet al. 1998; Halasz et al. 2004). Based on these oscillatory andexcitable properties, a heuristic level of activity of each population inthe reduced network is modeled by relaxation oscillator equations(Morris and Lecar 1981; see APPENDIX for details).

Each population has associated vj and uj variables: vj describes theactivity of population j and uj is a recovery variable. Couplingbetween populations is based on experimentally determined anatomicconnections. The variables tS and tW are associated with coupling andare subsequently described in more detail. Homeostatic sleep drivesare modeled by scaling variables that modulate coupling effects overtime. The variable h is associated with homeostatic NREM drive; thevariables eV and r � rf � rs are associated with homeostatic REMdrive.

For population j � W, S, and R, the variables vj and uj are governedby relaxation oscillator equations

dvj

dt� f�vj,uj �

duj

dt� �g�vj, uj �

where 0 � � �� 1; the full model equations are presented in theAPPENDIX. Population-dependent parameters (Table A1 in the APPEN-DIX) determine slightly different intrinsic properties for W, S, and R.The “smallness” of � quantifies the order of magnitude separatingrates of change in vj and uj and introduces two timescales: t and �t.This permits the model to describe activity on multiple timescales: aspreviously described, population activity levels change quickly be-tween states and slowly within states.

From the class of relaxation oscillator equations, the Morris–Lecarequations were selected for their geometric properties rather than theirbiophysical interpretation; the equations have been rescaled to de-scribe heuristic activity levels ranging between 0 and 1. In particular,the shape of the nullclines associated with these equations is importantto our analysis. Typically, nullclines separate regions of the v–u-phaseplane in which u and v are increasing or decreasing and give insightinto the qualitative behavior of solutions of the system of equations.Taking all other variables to be fixed, nullclines Nv and Nu describethe values of v and u for which the equations dv/dt and du/dt are zero

Nv � ��v, u��dv/dt � 0�

Nu � ��v, u��du/dt � 0�

In the associated v–u-phase plane, Nv has a standard cubic shape,whereas Nu is sigmoidal.

Nullclines of this form may assume three possible distinct relativeconfigurations (Fig. 3). It is known from standard stability theory(Guckenheimer and Holmes 1983) that under these conditions, thetrajectory associated with network behavior will remain close to thepoint p at which Nv and Nu intersect if p is on the left or right branchof Nv; if p is on the middle branch of Nv, the trajectory will oscillatearound p. Thus the location of p specifies default behavior of theassociated population and varies with behavioral state:

● p on left branch of Nv: trajectory near p and (v, u) coordinates ofp correspond to low values of v; therefore this nullcline config-uration corresponds to low firing activity throughout the popu-lation.

● p on middle branch of Nv: trajectory oscillates around p with vtaking on high and low values; therefore this nullcline configu-ration corresponds to an oscillation between high and low firingactivity throughout the population.

● p on right branch of Nv: trajectory near p and (v, u) coordinatesof p correspond to high values of v; therefore this nullclineconfiguration corresponds to high firing activity throughout thepopulation.We assume that each population is tonically active in the absence of

coupling (p on right branch); coupling effects give rise to the othernullcline configurations and this geometry underlies the fundamentaldynamics of the network.

Connectivity of the model network

As previously mentioned, connectivity of the reduced network inFig. 2 is based on anatomical connectivity of the neuronal populationsof the sleep–wake network. For example, the experimentally identi-fied GABAergic projection from VLPO to TMN (Sherin et al. 1998)is represented as an inhibitory connection from S to W in the reducednetwork. Population-specific coupling effects, denoted by cj, areintroduced into the model equations through the dvj/dt equations asfollows

dvi

dt� f�vj, uj� � cj

where

cs � � gw�tw��1 � h� � gox�tw� � ns

cw � �� gs�ts� � gaLC��vw � aLC� � gevev � nw during NREM sleep

� gs�ts� � nw otherwise

cR � �� gs�ts� � gLC rev � nR during sleep� gw�tw� � nR during wake

Here h, r, and eV are scaling variables to be described below; nS, nW,and nR are noise terms; and gaLC

and aLC are constant parameters. The

FIG. 2. Model sleep–wake network includes wake-, sleep-, and REM-promoting populations. Note that the REM-promoting population is composedof subpopulations of laterodorsal tegmental nucleus/pedunculopontine tegmen-tal nucleus (LDT/PPT) that are active mainly during REM sleep. Connectivityof the network is based on known neural pathways: circles and arrows denoteinhibitory and excitatory connections, respectively. Mutual inhibition betweenthe wake- and sleep-promoting populations forms a sleep–wake switch (Saperet al. 2001). Reciprocal connections between the wake- and REM-promotingpopulations control non-REM (NREM)–REM alternation (McCarley and Hob-son 1975; McCarley and Massaquoi 1986). There are 2 projections from thesleep- to the REM-promoting population (Lu et al. 2002): one representsinhibition from ventrolateral preoptic cluster (VLPO) core to LDT/PPT; theother represents the net excitatory effect [mediated indirectly through locuscoeruleus (LC) and dorsal raphe (DR) or inhibitory interneurons in LDT] ofextended ventrolateral preoptic nucleus (eVLPO) activity on LDT/PPT neu-rons.

3830 DINIZ BEHN, BROWN, SCAMMELL, AND KOPELL


term gaLC(�W aLC) reflects the self-inhibition present in the

wake-promoting populations (McCarley and Hobson 1975) and con-tributes to their gradual shift from low activity during NREM sleep tosilence during REM sleep.

The saturating functions gW(tW) and gS(tS) describe the activation ofinhibition as a function of time elapsed since the onset of the wake(tW) or sleep (tS) bout, respectively. These functions describe themutual inhibition between W and S as well as the inhibition of R byboth W and S. The form of these functions enforces the state depen-dency of coupling effects: activation of the function is initiated by ahigh level of activity in the presynaptic population because neuronalactivity drives neurotransmitter release. The time course of activationof these saturating functions captures the hypothesized time course ofpopulation recruitment; this effect is absent from the relaxationoscillation equations themselves. The saturating function gOX(tW)describes the activation of orexinergic effects as a function of timeelapsed since the onset of the wake bout. The exact form of theseequations is given in the APPENDIX.

Coupling terms determine the state-dependent nullcline configura-tions of each population. During NREM sleep, the initial inhibitionfrom S to W causes the nullclines associated with W to assume anoscillatory (rather than inactive) configuration. This is consistent withthe spontaneous (intrinsic) activity exhibited by these populations(Taddese and Bean 2002; Williams et al. 1984). All other couplingterms impose an inactive configuration on the nullclines associatedwith the postsynaptic population. Our choice of parameters is basedon achieving this state-dependent geometry.

The behavioral state of the network is classified by the heuristicactivity level of W, S, and R. If vW or vR exceeds the “activitythreshold” delimiting the onset of population activity, then the net-work is assumed to be in wake or REM sleep, respectively. Thenetwork enters NREM sleep when vS exceeds the activity thresholdand vW and vR do not. By imposing additional conditions for NREMsleep, we eliminate ambiguity in scoring simulated behavioral states.For the simulation results we report, 0.5 is taken to be the “activitythreshold”; however, because onset and offset of activity in thesepopulations are fast, there is minimal dependence on the choice ofthreshold separating active and inactive states.

Homeostatic sleep drives and scaling variables

Initiation and consolidation of human sleep has been modeled asthe interaction of circadian and homeostatic drives known as processC and process S, respectively (Borbely 1982). Circadian drive influ-ences behavior according to time of day, whereas homeostatic sleepdrives help maintain fixed daily amounts of NREM and REM sleep bypromoting sleep in proportion to preceding waking behavior. Thetwo-process model has been adapted to other mammals (Tobler et al.1992), including those with polyphasic sleep–wake behavior, byidentifying species-specific time courses for homeostatic sleep driveand modifying effects of circadian drive.

Because sleep–wake behavior persists in the absence of a functionalcircadian pacemaker, we make the simplifying assumption that circa-dian drives constitute a modulation, rather than an intrinsic dynamicelement, of behavioral state control. Therefore in the present study, werestrict our focus to network dynamics driven by homeostatic effects.

Selective sleep-deprivation protocols suggest the existence of sep-arate homeostatic sleep drives for NREM and REM sleep. Agents ofREM sleep homeostasis remain unknown, but several factors havebeen proposed to mediate NREM homeostatic drive (Porkka-Heis-kanen et al. 2000; Strecker et al. 2000). The best studied of these isadenosine. Adenosine concentrations rise during both spontaneouswaking and waking during sleep deprivation; during sleep theseconcentrations fall in several brain regions including the preoptic areaof the hypothalamus (Porkka-Heiskanen et al. 2000). High adenosinelevels reduce the frequency of inhibitory postsynaptic potentials(IPSPs) on VLPO neurons (Chamberlin et al. 2003; Morairty et al.2004), thus providing a mechanism for the production of sleep.Adenosine may also act through other mechanisms (Scammell et al.2001), particularly in basal forebrain (Porkka-Heiskanen et al. 1997;Strecker et al. 2000), and much remains to be learned about sleep-promoting factors in general.

Based on our current understanding of adenosine, we model ho-meostatic sleep drive with a variable h that rises during wakefulness,falls during sleep, and scales the strength of inhibition from W to S.Such a mechanism is also consistent with adenosine acting in the basalforebrain (Porkka-Heiskanen et al. 1997). The behavior of h iseffectively described by

dh

dt� �

�1 � h�

�hw

� nh during wakefulness

� h

�hJ

� nh during sleep

where �hWand �hS

are time constants controlling the growth and decayof h during wake and sleep, respectively; and nh is a noise term. Toobtain the appropriate reduction in inhibition from W to S, gW(tW) ismultiplied by (1 h) in cW; thus the strength of inhibition from W toS varies inversely with h.

In the absence of an identified agent of REM homeostasis, ourimplementation of homeostatic REM drive is based on phenomeno-logical observations pertaining to REM sleep (Benington 2002; Endoet al. 1997, 1998; Franken 2002; Le Bon et al. 2002; McCarley andHobson 1975). Experiments suggest that the occurrence of REM sleepis associated with both excitation (or disinhibition) of REM-activepopulations and a gating mechanism involving a reduction of activityin wake-active populations (Franken 2002; McCarley and Hobson1975). Therefore the initiation of REM sleep in our model depends ona REM-promoting force r and a scaling variable eV that increases thestrength of r when W is inactive. Based on a model proposed byFranken (2002), the variable r is composed of two processes, rf and rs,acting on different timescales: r � rf � rs. The fast process, denotedrf in our model, is involved in timing REM sleep within a sleep boutand is governed by

drf

dt� � �r� � rf �

�rf

during NREM sleep

0 otherwise

where r� and �rfare parameters describing the maximal value and time

constant of rf , respectively; rf is reset to zero by the occurrence of aREM bout.

A B CFIG. 3. Nullclines Nv and Nu associated with

Morris-Lecar-type equations have cubic and sig-moidal shapes, respectively. The relative config-uration of these nullclines is specified by thebranch of Nv on which the intersection point p islocated: left branch (A), middle branch (B), orright branch (C).



The slow process, denoted rs in our model, regulates the dailyamount of REM sleep and is governed by the equation

drs

dt� �

� 1

rsdwn

during REM sleep

1

rsup

otherwise

where parameters rsup and rsdwn control the rate of growth and decayof rs.

The gating mechanism involving reduced activity in wake-activepopulations is modeled with the variable eV: this variable scales bothinhibition to W and excitation to R (refer to definition of cW and cR).Our implementation of an eV-scaled excitation to R is based onobservations that eVLPO fos expression is correlated with REM sleep(Lu et al. 2002) and represents disinhibition of R through indirectpathways: eVLPO neurons inhibit monoaminergic populations repre-sented by W and other populations (e.g., GABAergic neurons inventrolateral periaqueductal gray matter) that inhibit LDT/PPT (Lu etal. 2002). The variable eV is governed by the equation

dev

dt� � �1 � ev�

�rv

when W is inactive

0 otherwise

where �eVis a parameter controlling the rate of growth of eV; the

occurrence of a REM bout resets eV to 0.

Numerical implementation

The resulting model consists of 12 differential equations and 40parameters. To reproduce the “noisiness” inherent in any biologicalsystem, we introduced stochastic terms into four of the differentialequations in our system: dh/dt and dvj/dt for j � W, S, and R. Thestochastic terms were normally distributed with mean 0 and variance0.25, 1, 1, and 20, respectively; at each time step, new values weregenerated for the stochastic terms and the differential equations weremodified accordingly.

All simulations were run on a Linux workstation using the ordinarydifferential equation (ODE) solver XPPAUT (developed by GB Er-mentrout and available at (ftp://ftp.math.pitt.edu/pub/bardware). ARunge–Kutta integration method with step size 0.01 (min) was usedfor all simulations. Simulation output was analyzed using code writtenin MATLAB (The MathWorks, Natick, MA).

R E S U L T S

In the absence of noise, our deterministic model networkgave rise to a stereotypical sequence of behavior: a sustainedwake bout and an extended sleep bout composed of two sleepcycles (alternating between NREM and REM sleep) and fourbrief awakenings (Fig. 4). The geometry of the system permit-ted analysis of this solution to identify the dynamic mecha-nisms underlying state transitions and it suggested separatemechanisms for brief and sustained wake bouts. This approachalso allowed us to formally evaluate model sensitivity toparameters. With the addition of noise, our parameter regimegenerated simulated behavior consistent with experimentallyobserved mouse sleep–wake behavior.

Model geometry and predictions

To better understand dynamic principles of the network andmechanisms of transitions between states, we analyzed themathematical structure of the network equations in the absenceof noise. Complete mathematical details are beyond the scope

of this paper, although the METHODS section provides an over-view of our approach. By exploiting both the separation oftimescales within the network equations and the state-depen-dent relevance of variables, we were able to reduce the dimen-sion of the system in a state- and population-dependent man-ner. Through this reduction, we identified a sequence of low-dimensional systems that approximated the behavior of the fullsystem and provided a structured framework for understandingmechanisms of state transitions and parameter dependency inour model.

Mechanisms for state transitions

Each low-dimensional system includes the population vari-ables (vj and uj) associated with one of the wake-, sleep-, orREM-active populations. Thus within these systems we couldexamine network dynamics in terms of nullclines associatedwith population variables. This analysis resulted in the follow-ing predicted mechanisms for each state transition.

The transition from wake to (NREM) sleep is initiated by thehomeostatic NREM sleep drive, described by the variable h.During wakefulness h increases and the strength of inhibitionfrom W to S varies inversely with h. When the strength ofinhibition is no longer sufficient to prevent the onset of activityin S, the “flip-flop” switch flips: activation of S initiatesinhibition from S to W and transitions the network fromsustained wake to sleep (Fig. 5). In the coupled oscillatorliterature, this mechanism is known as “intrinsic escape” (Skin-ner et al. 1994; Wang and Rinzel 1992).

Recall that the onset of inhibition from S to W results in anoscillatory configuration of W nullclines. Thus in contrast tothe inactivity observed in S when it is inhibited by W, activityin W alternates between high and low levels in an intrinsicrelaxation oscillation. We consider the short intervals of highactivity in W to describe brief awakenings; therefore the modelsuggests that the occurrence of brief awakenings is linked tointrinsic excitability of the wake-active populations. This isconsistent with the theory that regular periods of arousal duringsleep act as a safety measure to offset the vulnerability ofdecreased sensory perception during sleep (Halasz et al. 2004).

A

B

C

D

E

FIG. 4. Solution of the deterministic system for variables vS (A), vW (B), andvR (C). D: all 3 variables are plotted on the same axes to compare their relativedynamics. E: hypnogram shows the conversion from model output to simulatedsleep–wake behavior.



As the sleep bout progresses, short-term homeostatic REMpressure increases the strength of inhibition to W and theexcitation to R. When inhibition to W is sufficiently strong, theW nullclines move to an inactive configuration and oscillationin W activity ceases. The complete cessation of activity in Wresults in further disinhibition of R; if homeostatic REM sleepdrive is sufficiently strong, a REM bout occurs.

In our model, a REM bout corresponds to a single excursion(of relaxation oscillation-type) in R; thus the form and durationof the REM bout result from intrinsic properties of REM-activepopulations. A combination of excitation and disinhibitionpermits the intrinsic escape of R; however, the activation of Rcompletely discharges short-term REM pressure and reduceslong-term REM pressure: the resulting drop in total REMpressure reverses both the excitation to R and the disinhibitionof W. Thus R returns to relative inactivity after its excursionand the REM bout activates W: depending on the state of thenetwork, this may generate a post-REM brief awakening or afull transition to sustained wake.

The transition from NREM sleep to wake involves a moresubtle mechanism than those associated with other state tran-

sitions. As previously described, brief episodes of activity in Wresult in pulses of inhibition from W to S. In contrast to theexpected behavior of a pure flip-flop switch, this activity doesnot necessarily transition the network from sleep to sustainedwake: instead, switching between states depends on thestrength of this pulse. If the pulse sufficiently depresses activityin S, the network will transition from sleep to wake; otherwise,the activity in W self-terminates with the falling phase of theoscillation and the sleep bout continues. These possibilities areillustrated in Fig. 6.

The strength of the pulse of inhibition from W to S duringa brief awakening is determined by the variable h describinghomeostatic NREM sleep drive. As h recovers over thecourse of the sleep bout, the strength of these pulses in-creases; thus the value of h during the brief awakeningdetermines whether the network will switch to sustainedwake or remain in sleep. Thus the transition from NREMsleep to sustained wake requires both sufficient recovery inh and the initiation of a brief awakening. This suggests asignificant role for intrinsic excitability in the reversibilityof sleep.

FIG. 5. The transition from wake to NREM sleep can beunderstood as intrinsic escape of S: as homeostatic NREMsleep drive h increases, the strength of inhibition from W toS decreases. As a result, the intersection point of the S-nullclines, pS, moves down the left branch of the nullclineNv

S; because ps is quasi-stable on the left branch of NvS, it is

tracked by the trajectory. When ps crosses the left curve ofknees KL, it loses quasi-stability. The trajectory ceases totrack ps and jumps to the right branch of Nv

S. The onset ofactivity in S associated with this trajectory jump initiates thetransition from wake to NREM sleep.

FIG. 6. During NREM sleep, the trajectory is near thequasi-stable point pS on the right branch of the nullcline Nv

S.At the onset of a brief awakening, inhibition from W to S isinitiated. As this inhibition pushes Nv

S down, the trajectorytracks the point pS until pS reaches the right curve of knees,KR, and loses quasi-stability. When pS loses quasi-stability,the trajectory begins to jump to the left. Because the briefawakening (self-) terminates, Nv

S returns to its initial posi-tion after a specified duration. The resulting behavior of thenetwork depends on the position of the trajectory in relationto Nv

S at the termination of the brief awakening: if thetrajectory is below Nv

S, then the direction of the vector fieldacting on the trajectory is switched, and the trajectoryreturns to pS on the right branch of Nv

S; if the trajectory isabove Nv

S, then the vector field near the trajectory is notaffected, and the trajectory continues to jump to the left.These cases correspond to the continuation and termination,respectively, of the sleep bout.



Model sensitivity to parameters

Our model parameters fall into three categories based on thevariables they modify: parameters associated with intrinsicproperties of W, S, or R; parameters involved in couplingeffects; and parameters associated with homeostatic sleepdrives. Parameters and their typical values are listed in TablesA1 and A2 of the APPENDIX.

Mathematical understanding of state transitions in the modelprovides insights into the role of each parameter in networkdynamics. Thus rather than attempting to explore model ro-bustness in a high-dimensional parameter space, we evaluatethe effects of parameter variation using our analysis of themodel dynamics.

As long as the geometry of the equations is preserved,parameter changes have relatively minor effects on networkbehavior. Changes to the first category of parameters alter theshape of the nullclines associated with each population;nullcline shape (both cubic and sigmoidal), and thus modelbehavior, is very robust to minor changes in these parameters.The second and third categories of parameters determine state-dependent nullcline configurations and intrastate changes tothese configurations. Minor variation of these parameters doesnot significantly alter network dynamics; however, if parame-ter variation results in a significant change in the number ofbrief awakenings observed in a stereotypical sequence ofbehavior, quantitative measures of sleep–wake behavior willbe affected.

For example, the number of brief awakenings in simulatedsleep–wake behavior varies inversely with the maximalstrength of inhibition from S to W (gS max); network behavior isless sensitive to changes in the maximal strength of inhibitionfrom W to S (gW max). If gS max is increased or decreased by �2or 3%, altered sleep–wake behavior will be reflected bychanges in a range of measures: distributions of wake andNREM bout durations (fewer brief awakenings results in con-solidation of NREM bouts), the number and mean durations ofwake and NREM bouts, and percentage of time spent in eachstate. If the increase in gS max is sufficient (�10% of baselinevalue) to eliminate all brief awakenings, then the network losesits transition mechanism from sleep to sustained wake andexhibits constant sleep.

Similarly, changing the rate of activation of inhibition fromW to S (�W) affects the number of brief awakenings in eachstereotypical sequence of behavior. If activation is instanta-neous or the rate is increased by �40% of its baseline value,brief awakenings (with concurrent activity in W and S) areeliminated and the network rapidly oscillates between wakeand NREM sleep. If the rate is decreased, additional brief

awakenings occur. Network behavior is not sensitive tochanges in the rate of activation from S to W (�S).

Most of the parameters associated with homeostatic sleepdrives (suggestively denoted �i) are time constants describingthe evolution of variables that scale coupling strengths in thenetwork. Based on the mechanisms of transitions previouslydiscussed, we expect all transitions, with the exception of thetransition from the extended sleep bout to the sustained wakebout, to depend smoothly on these parameters. Simulationresults support this conclusion: numbers of all bout types andmean durations of sustained wake and NREM bouts vary withparameters associated with coupling and homeostatic sleepdrives.

The exception in smooth dependency arises because, in theabsence of noise, the transition from sleep to sustained wakedepends on activation of W in the form of a brief awakening.This brief awakening may be intrinsic or may occur followingREM sleep; however, transitions to sustained wake are mostoften caused by post-REM brief awakenings (in both simulatedand experimental data). Therefore the duration of extendedsleep bouts tends to change in (discrete) units of sleep cyclesrather than continuously with varying parameters.

Simulated mouse sleep–wake behavior

With the addition of noise, our model circuitry reproducesexperimentally observed sleep–wake behavior including pro-portions of wake, NREM sleep, and REM sleep; number andduration of bouts; and realistic sleep architecture as measuredby transition probabilities. Figure 7 summarizes 2 h of simu-lated mouse behavior and can be qualitatively compared withFig. 1. In Figs. 8 and 9, we compare the simulated sleep–wakebehavior described by our model with previously publishedexperimental data from eight male C57BL/6J mice (Mochizukiet al. 2004); a discussion of methods used to collect these datais described in the original paper. Simulated behavior wasgenerated with eight runs of the model under a single set ofparameters with different initial conditions and random noise.

Daily percentages of wake, NREM sleep, and REM sleep

Our model reproduces experimentally observed daily (24-h)percentages of time spent in wake, NREM sleep, and REMsleep (Fig. 8A). By considering percentages of time spent ineach behavioral state over 24 h, we mitigate circadian depen-dency to reveal underlying homeostatic equilibria. The modelproduces slightly less REM sleep than is observed experimen-tally; the source of this difference is a discrepancy in theduration, rather than the raw number, of REM bouts between

A B

FIG. 7. Simulated mouse sleep–wake behavior from themodel demonstrates both the polyphasic nature of mouse sleep–wake behavior (A) and the qualitative differences between briefand sustained wake bouts (B). Note the similarities to Fig. 1.



the model and the experimental data. The close agreementbetween our model results and experimental data for dailypercentages of wake, NREM sleep, and REM sleep suggeststhat the time constants of homeostatic drives in the model aretuned to biophysically reasonable values.

Number and duration of bouts

Because mouse sleep–wake behavior is polyphasic, dailyamounts of wake, NREM sleep, and REM sleep are composedof many bouts of varying lengths. Scoring behavior in 10-sepochs, Mochizuki and colleagues identified mean numbersand durations for bouts of each behavioral type. As describedin the INTRODUCTION, we will consider brief and sustained wakebouts separately.

The mean durations (Fig. 8B) and average numbers (Fig. 8C)of bouts of each behavioral type show good agreement betweenthe model and the experimental data, although the model hasfewer brief wake bouts than the experimental data. The dis-crepancy between the simulated and experimental mean dura-tions of sustained wake bouts probably occurs because ourmodel lacks circadian effects that promote very long wakebouts. Mean bout durations for the other behavioral states showclose agreement.

Because bout durations are nonnormally distributed (Lo etal. 2004), we compare the distributions of simulated andexperimentally observed bout lengths (Fig. 9). The modelcaptures the relative frequency of each type of wake bout. Wemeasure durations in minutes instead of seconds because veryextended wake bout durations occur during the dark period.The slightly lower number of brief (0- to 2-min) wake bouts inmodel data could be explained by the absence of brief awak-enings triggered by uncontrolled environmental factors (occa-sional noise, vibration, etc.).

There was good qualitative agreement between the modeland experimental data in the distribution of NREM boutdurations: bouts of NREM sleep were most likely to be �2 minin duration and nearly all bouts of NREM sleep fell in the rangefrom 0 to 8 min in duration. There was a lower incidence ofNREM bouts of 2- to 4-min duration and a higher incidence ofNREM bouts of 4- to 8-min duration in the model comparedwith the experimental data; additional noise in the model,particularly random external stimuli, may fracture some ofthese longer bouts into bouts of shorter duration.

Most REM bouts fall in the first two bins (0–40 and 40–80 s)in both the model output and the experimental data. The absencein model output of longer REM bouts observed in the experimen-tal data accounts for the discrepancy between the percentage ofREM sleep in the simulated and experimental data. This clusteringof REM bout durations occurs because REM bouts are modeled asa single excursion of the variable vR; thus the duration of theexcursion is controlled robustly by intrinsic properties of theequations governing vR and uR.

Organization of sleep–wake behavior

To quantify the organization of sleep–wake behavior, wecomputed probabilities of transitioning from one behavioralstate to another (given that a state transition occurred). Stereo-typed sleep–wake patterns that emerge are often conservedacross species: some transitions (e.g., wake3 REM sleep) arenever seen in normal behavior and can be considered “inap-propriate,” whereas others (e.g., NREM sleep 3 REM sleep)are common. Transition probabilities for simulated and exper-imental data (Fig. 8D) show similar structure; no inappropriatestate transitions are produced by the model.

In addition to specific transitions from one state to another,stereotypical sequences of behavior are observed experimen-

A B

C D

FIG. 8. Percentage of time spent in each be-havioral state (A), mean durations of bouts (B),and number of bouts (C) of each type are similarbetween experimental and simulated data (n � 8mice are compared with 8 runs of the model).Organization of simulated sleep–wake behavior,as measured by the probability of transitionsbetween each pair of states, is also similar to thatobserved experimentally (D).



tally and clinically. One such sequence is the progression fromNREM sleep to REM sleep to brief wakefulness and back toNREM sleep (Dijk and Kronauer 1999). This sequence isobserved repeatedly in our simulated behavior.

D I S C U S S I O N

We have shown that a network of three coupled relaxationoscillators with variable connectivity can be used to simulatequalitative and quantitative features of mouse sleep–wakebehavior including brief and sustained bouts of wake, NREMsleep, and REM sleep. Our model demonstrates fundamentalnetwork dynamics generated by the circuitry underlying be-havioral state control and we predict mechanisms for each statetransition based on formal analysis of our model equations.Because our model development was based on experimentallyestablished physiology, our model has many implications forunderstanding sleep–wake behavior.

Fine architecture of sleep–wake behavior

A major feature of our model is its ability to describe the finearchitecture of sleep–wake behavior and, in particular, theincidence of brief awakenings. Brief awakenings are an impor-tant indicator of sleep quality and abnormal numbers anddurations of brief awakenings have been associated with in-somnia (Anderson et al. 2005), sleep apnea (Loredo et al.1999), and compromised histamine H1 receptor signaling(Huang et al. 2006). Although some brief awakenings occur inresponse to short, random sensory stimuli from uncontrolledenvironmental factors, the high incidence of brief awakeningsthroughout sleep suggests that environmental factors are notexclusively responsible.

Previous sleep modeling approaches did not address theoccurrence of brief awakenings in sleep–wake behavior. Inclassical two-process models (Borbely 1982; Tobler et al.1992), the durations of wake and sleep bouts were governed bythe time constants of homeostatic sleep drive and circadianphase; because the durations of brief awakenings cannot belinked logically to sleep homeostasis, they could not occur inthis framework. The Chou (2003) stochastic firing rate modelof the sleep–wake switch included brief awakenings but used amodeling approach that could not shed light on the dynamicmechanisms associated with the simulated behavior.

Tamakawa and colleagues (2006) recently proposed a modelof rat sleep–wake behavior based on the physiology of sleep–wake circuitry. Using formal neuron models of ten neuronalpopulations, including those described in our model, theyproduced simulated rat sleep–wake behavior that matchedexperimentally determined mean durations of wake and sleepbouts. Based on the observation that simulated activity in theten populations is essentially identical within state-dependentgroups corresponding to wake, NREM sleep, REM sleep, andboth wake and REM sleep, this is described as a “quartetmodel.” As previously mentioned, we found it unnecessary toinclude a population with high levels of activity during wakeand REM sleep. Both the quartet model and our model includetwo different homeostatic sleep drives: their SS1 variable issimilar to our h variable, although it acts directly on the VLPOpopulation; their SS2 variable increases during wake and REMsleep and modulates the population representing NREM sleep-active GABAergic neurons in the median preoptic area, thusfunctioning differently from our rs and rf variables. Althoughthe quartet model is consistent with general measures ofsleep–wake behavior, such as mean duration and percentage oftime in each state, additional work was needed to develop amodeling framework that could describe the fine architectureof sleep–wake behavior.

By specifying an oscillatory regime for the equations asso-ciated with wake-active neuronal populations, our model in-cludes a mechanism for wake bouts with durations that aregoverned by intrinsic properties of the population rather thanhomeostatic sleep drive. Thus simulated sleep–wake behaviorreflects the structure and organization of mouse sleep–wakebehavior; in particular, distributions of bout durations and thedistinction between brief and sustained wake bouts are repro-duced by the model. The absence of environmentally inducedbrief awakenings in our model may be responsible for theslightly lower number of brief awakenings in the simulatedbehavior compared with the experimental data.

A

B

C

FIG. 9. Distributions of wake (A), NREM sleep (B), and REM sleep (C)bout durations in simulated wake–sleep behavior capture the features of thedistributions of bout durations in experimental data.



Concurrent activity in wake- and sleep-active populations

By implementing the mutual inhibition of the flip-flopswitch in the context of coupled relaxation oscillators, ourapproach captures the key feature of a sleep–wake switch—two stable states with minimal time spent in intermediate states(Saper et al. 2001). A pure flip-flop switch mechanism issubject to inappropriate transitions in the presence of noise; inbehaving animals, such inappropriate transitions would trans-late to potentially dangerous switches between wake and sleep.Our model also includes time courses associated with inhibi-tion onset that maintain the robustness and stability necessaryin a physiological system.

In our model there are certain conditions under whichconcurrent activity in wake- and sleep-active populations canoccur. In particular, simulated brief awakenings occur when Wbecomes active for a short time during an extended sleep bout,but high homeostatic sleep drive prevents a transition intosustained wake. This theoretical mechanism for brief awaken-ings predicts that wake-promoting and VLPO neurons may beactive concurrently during brief awakenings in mice.

Unfortunately, existing experimental data do not address theissue of concurrent activity in wake- and sleep-active popula-tions. Unit recordings (Szymusiak et al. 1998) and fos immu-nostaining (Sherin et al. 1996) established that VLPO neuronsare active during sleep, although the long half-life of fos cannotresolve VLPO activity during brief awakenings and unit re-cordings have been analyzed only during stable wake. ThusVLPO neurons may remain active during brief awakenings andadditional experiments are needed to test this prediction.

A NREM–REM flip-flop switch?

Lu and colleagues (2006) recently proposed a conceptualmodel for NREM–REM alternation involving mutual inhibitionbetween GABAergic REM-on (sublaterodorsal nucleus) andREM-off (ventrolateral part of the periaqueductal gray matter andthe lateral pontine tegmentum) populations. In this framework,increased activity in eVLPO inhibits monoaminergic populations,thereby removing excitation from REM-off populations. Theresulting decrease of activity in REM-off populations results indisinhibition (and activation) of REM-on populations.

Our approach to NREM–REM alternation during a sleepcycle is based on the classical view of interacting (REM-off)monoaminergic populations (W) and REM-on cholinergic popu-lations (R) (McCarley and Hobson 1975). However, our imple-mentation is consistent with the theory that increased activity inthe eVLPO results in disinhibition of REM-on neurons becausewe model increased inhibition of W over the course of the sleepcycle as a result of increased eVLPO activity.

In contrast to the idea of the NREM–REM flip-flop switch,the disinhibition of R in our model is not mediated throughREM-off neurons. Although the intermediate REM-off popu-lation is not necessary for modeling normal sleep–wake be-havior, REM-off neurons may be involved in other regulatoryaspects of REM phenomena (Lu et al. 2006). Therefore wemay need to extend our model to include a REM-off interme-diary population in future work.

Technical considerations and future directions

By using relaxation oscillator equations to model each pop-ulation, we imposed an assumption of fast transitions between

high- and low-activity levels as demarcated by the knees of thecubic nullclines associated with v-variables. Although themodel predicts some gradual changes in activity, such as theslow increase in activity of VLPO neurons preceding a transi-tion to NREM sleep, it does not capture the timescales ofchanges on the level of single- or multiunit firing rates. Asadditional experiments establish time courses of activity forevery population in the sleep–wake network, future models ofthe network could incorporate these refinements. By describingthe time courses of transitions, such models would representincreasingly complex network dynamics and may offer animportant framework for exploring the relative dynamics andcausal implications of firing rate changes among these popu-lations.

The period of relaxation oscillations is relatively robust tonoise, so the durations of simulated brief awakenings and REMbouts tend to be normally distributed around the characteristicperiod of the oscillations associated with the W and R equa-tions, respectively. In experimental data, the durations of briefawakenings and REM bouts follow power law and exponentialdistributions, respectively. Therefore although our simple im-plementation of noise was sufficient to reproduce the majorstatistical features of mouse sleep–wake behavior, relaxationoscillator formalism limits our ability to fully capture all theproperties of the distributions of bout durations. Formal anal-ysis of the implementation of noise— including an assessmentof the type of noise used and the placement of stochastic termsin the model equations—might improve some of these statis-tical features of the simulated data.

The interaction between the circadian system and sleep–wake behavior has been studied extensively and recent exper-iments established some of the neural circuitry involved in thisinteraction. Our model does not yet capture circadian effects,such as hourly variations in the amounts of wake and sleep,extended wake bouts at the onset of the dark period, andnonuniform distribution of REM sleep (Mochizuki et al. 2004).Future studies will integrate circadian effects into this modeland should provide an excellent opportunity to explore how thecircadian system modulates sleep–wake dynamics.

Our model provides a novel framework for understandingdynamics observed in altered sleep–wake behavior, especiallywhen physiologic observations have analogs in the modelparameters. For example, disruption of sleep with aging maybe linked to a loss of VLPO neurons (Gaus et al. 2002). In ourmodel, this would correspond to a decrease in the parametergS max representing the maximal strength of inhibition from S toW, and we find that decreasing gS max increases the number ofbrief awakenings in simulations. As another example, it re-mains unclear how a loss of orexin signaling causes thebehavioral state instability associated with narcolepsy (Mochi-zuki et al. 2004). Because narcolepsy is essentially a disorderof sleep–wake dynamics, our model framework is well suitedfor investigating the functional role of orexin neurons. A fulltheoretical understanding of the dynamic effects of changes inthe model may suggest mechanisms of age- and orexin-relatedchanges in sleep–wake behavior.

In summary, we have shown that a model of three coupledrelaxation oscillators can generate the network dynamics asso-ciated with behavioral state control to realistically simulatemouse sleep–wake behavior. By directly addressing the dy-namics of the sleep–wake network, this modeling approach



may provide predictions and insights that improve our under-standing of normal and pathologic sleep–wake behavior and itsunderlying physiology.

A P P E N D I X

The activity of each population is modeled by modifiedMorris–Lecar equations. Morris–Lecar equations are a lower-dimensional form of conductance-based Hodgkin–Huxley-typesingle-cell models (Hodgkin and Huxley 1952). A single gat-ing variable on the potassium current yields a two-dimensionalsystem of equations in the standard notation

dvi

dt� Ii � gL�vi � EL� � gCa m��vi��vi � ECa� � gKui�vi � EK�

dui

dt� �u��vi� � ui ⁄ �ui

�vi�

where Ii, gL, EL, gCa, ECa, gK, EK, and are constant param-eters (Morris and Lecar 1981) and m�, u�, and �ui

are functionsdefined below. Because we are describing population activityrather than membrane voltage, we rescaled these equationswith the affine transformation A(v) � (v � 65)/130 � v. Thenv assumes values between 0 and 1, and

v� �1

130v�

By rescaling parameters, grouping terms appropriately, andadding coupling terms ci, we obtain the following expressions

dvi

dt� Ii � ��i � im��vi�vi � �i

2� i�m��vi� � ��ivi � 0.5�i � �i�ui � ci

dui

dt� i�u��vi� � ui ⁄ �u\i�vi�

where Ii, �i, i, i, �i, �i, and i are constant parameters(parameter values are listed in Table A1); the coupling terms ciare given in the text; and the functions u�� and �u� aredefined below. For convenience, we will suppress the “hat”notation, thereby letting vi and ui denote the rescaled variables.

As previously described, we chose Morris–Lecar relaxationoscillation equations for the geometry of the nullclines associ-

ated with vi and ui for i � W, S, R. Because these equations areusually associated with single-cell activity rather than popula-tion activity, the timescales associated with our modified re-laxation oscillator equations are much slower than the time-scales involved in the original Morris–Lecar equations.

The saturating functions describing the onset of inhibitionfrom S and W, respectively, are given by the following equa-tions

gS�tS� � gSmaxtanh ��StS�

gW�tW� � gWmaxtanh ��WtW�

with constant parameters gS max, gW max, �S, and �W. Thevariables ts and tw measure time elapsed since the onset of thecurrent wake or sleep bout and are governed by

dts

dt� �1 during sleep

0 during wake

dtw

dt� �1 during wake

0 during sleep

The saturating function describing the onset of orexinergiceffects is given by

gOX�tw� � OXs�tanh �OXONtW� � 2�tanh �OXONtW�5�

with constant parameters OXS and OXON. The important ele-ment of this functional form is a change of concavity thatallows gOX(tW) to remain very low for the first minute ofactivation.

The equations for m�(vi), u�(vi), and �u(vi) are given, respec-tively, by

m��vi� � �1 � exp� � ��vi � �m�⁄�m�1

u��vi� � ��u, uw��1 � exp� � ��vi � ��u,uw��⁄��u, uw��1

�u�vi� � �cosh ��vi � ��u, uw�� ⁄ �2��u, uw��1

with constant parameters �{u,uw}, �{m,u,uw}, and �{m,u,uw}.The state-dependent equations for h, rs, rf , and eV were

given in the main text. The parameter values used to generatesimulated behavioral data are listed in Tables A1 and A2.

TABLE A2. Parameters and typical parameter values associatedwith coupling between wake-, sleep-, and REM-active populations(W, S, and R) and homeostatic sleep drives

Coupling Homeostatic Sleep Drives

Role Parameter Value Role Parameter Value

Maximalstrength gsmax

2.8 NREM drive �hS15.0

gWmax2.69 �hW

100.0gLC 1.92 Fast REM �rf

15.0geV

0.538 r� 0.75gaLC

0.009 Pre-REM/REM �eV10.0

Direction ofeffect aLC 30.0 Slow REM rsup 800.0

Rate ofactivation �S 6.0 rsdwn 40.0

�W 3.0 Orexin OXS 0.033OXon 0.7

TABLE A1. Parameters associated with the wake-, sleep-, and REM-active neuronal populations (W, S, and R) and typical parametervalues as adapted and rescaled from Morris–Lecar formalism

S Population W Population R Population

Parameter Value Parameter Value Parameter Value

IS 3.4 IW 4.16 IR 3.6�S 4.0 �W 4.0 �R 4.0S 8.0 W 8.0 R 8.0S 7.385 W 13.539 R 7.385�S 8.0 �W 8.3 �R 8.0�S 5.169 �W 5.491 �R 5.169 S 1.2 W 0.85 R 0.045

Parameters associated with m�, u�, and �u

�m 63.8 � 130.0 �m 7.0�u 1.0 �uw 2.0 �u 4.0�u 55.0 �uw 57.8 �uw 7.0

Parameter values from Morris and Lecar (1981).



A C K N O W L E D G M E N T S

We thank Dr. Takatoshi Mochizuki for experimental data on mouse sleep–wake behavior.

Present address of C. Diniz Behn: Division of Sleep Medicine, HarvardMedical School, and the Department of Neurology, Beth Israel DeaconessMedical Center, Boston, MA.

G R A N T S

This work was supported by the Burroughs Wellcome Fund; the NationalPhysical Science Consortium; National Science Foundation Grant DMS-0211505; and National Institutes of Health Grants DAO-15644, MH-62589,and HL-60292.

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mathematical model of network dynamics governing mouse sleep

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