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1 Joural of Theoretical Biology 26 (29) Cotets lists available at ScieceDirect Joural of Theoretical Biology joural homepage: A ew mathematical model for the homeostatic effects of sleep loss o eurobehavioral performace Peter McCauley a, Leoid V. Kalachev b, Amber D. Smith a, Gregory Beleky a, David F. Diges c, Has P.A. Va Doge a, a Sleep ad Performace Research Ceter, Washigto State Uiversity, PO Box 149, Spokae, WA 9921, USA b Departmet of Mathematical Scieces, Uiversity of Motaa, Mathematics Buildig, Missoula, MT 9812, USA c Divisio of Sleep ad Chroobiology, Departmet of Psychiatry, Uiversity of Pesylvaia School of Medicie, 113 Blockley Hall, 423 Guardia Drive, Philadelphia, PA 1914, USA article ifo Article history: Received 13 May 28 Received i revised form 19 August 28 Accepted 1 September 28 Available olie 2 October 28 Keywords: Fatigue ad performace models Two-process model of sleep regulatio Coupled o-homogeeous first-order ordiary differetial equatios Cogitive performace Adeosie receptors abstract The two-process model of sleep regulatio makes accurate predictios of sleep timig ad duratio for a variety of experimetal sleep deprivatio ad ap sleep scearios. Upo extedig its applicatio to wakig eurobehavioral performace, however, the model fails to predict the effects of chroic sleep restrictio. Here we show that the two-process model belogs to a broader class of models formulated i terms of coupled o-homogeeous first-order ordiary differetial equatios, which have a dyamic repertoire capturig wakig eurobehavioral fuctios across a wide rage of wake/sleep schedules. We examie a specific case of this ew model class, ad demostrate the existece of a bifurcatio: for daily amouts of wakefuless less tha a critical threshold, eurobehavioral performace is predicted to coverge to a asymptotically stable state of equilibrium; whereas for daily wakefuless exteded beyod the critical threshold, eurobehavioral performace is predicted to diverge from a ustable state of equilibrium. Compariso of model simulatios to laboratory observatios of lapses of attetio o a psychomotor vigilace test (PVT), i experimets o the effects of chroic sleep restrictio ad acute total sleep deprivatio, suggests that this bifurcatio is a essetial feature of performace impairmet due to sleep loss. We preset three ew predictios that may be experimetally verified to validate the model. These predictios, if cofirmed, challege covetioal otios about the effects of sleep ad sleep loss o eurobehavioral performace. The ew model class implicates a biological system aalogous to two coected compartmets cotaiig iteractig compouds with timevaryig cocetratios as beig a key mechaism for the regulatio of psychomotor vigilace as a fuctio of sleep loss. We suggest that the adeosiergic euromodulator/receptor system may provide the uderlyig eurobiology. Published by Elsevier Ltd. 1. Itroductio Sleep deprivatio causes a wide rage of eurobehavioral performace deficits (Diges ad Kribbs, 1991; Baks ad Diges, 27). Various mathematical fatigue ad performace models have bee developed to predict such performace impairmet (Mallis et al., 24). However, scietific progress i this area has bee limited by difficulties predictig performace uder chroic coditios of partial sleep loss (Va Doge et al., 23; Va Doge, 24). Correspodig author. Tel.: ; fax: addresses: pmccauley@wsu.edu (P. McCauley), kalachev@mso.umt.edu (L.V. Kalachev), ambersmith@wsu.edu (A.D. Smith), beleky@wsu.edu (G. Beleky), diges@mail.med.upe.edu (D.F. Diges), hvd@wsu.edu (H.P.A. Va Doge). Most of the available fatigue ad performace models are based o the semial two-process model of sleep regulatio (Borbély, 1982; Daa et al., 1984). This model posits that sleep ad wakefuless are govered by two primary biological mechaisms: a homeostatic process that builds pressure for sleep durig wakefuless ad dissipates this pressure durig sleep (Borbély ad Acherma, 1999), ad a circadia process that modulates sleep pressure as a fuctio of time of day (Edgar et al., 1993). The two-process model has bee successful i predictig various aspects of sleep ad of wakig eurobehavioral fuctios across a rage of sleep ad sleep deprivatio paradigms (Borbély ad Acherma, 1999; Acherma, 24). For istace, it was show that wakig eurobehavioral fuctios could i may istaces be predicted by the arithmetic differece betwee the homeostatic pressure for sleep ad the circadia pressure for wakefuless (Acherma ad Borbély, 1994) /$ - see frot matter Published by Elsevier Ltd. doi:1.116/j.jtbi

2 228 P. McCauley et al. / Joural of Theoretical Biology 26 (29) Extedig the two-process model from its origial focus o sleep (Borbély, 1982) to iclude predictios of wakig fuctios has bee a goal for some time (Borbély ad Acherma, 1999; Diges ad Acherma, 1999), but efforts to achieve this goal have ot bee uiversally successful. Several studies have show that chroic sleep restrictio leads to cumulative icreases, progressig over days for a week or more, i sleep propesity ad Performace Performace eurobehavioral impairmet (Carskado ad Demet, 1981; Diges et al., 1997; Beleky et al., 23; Va Doge et al., 23) see Fig. 1a. The two-process model does ot accurately capture these icreasig deficits, predictig istead a stabilizatio of wakig eurobehavioral fuctios across days after just a few days of chroic sleep loss (Va Doge et al., 23) see Fig. 1b. Other fatigue ad performace models have similarly failed to predict the cumulative effects of chroic sleep restrictio (Va Doge, 24). Va Doge et al. (23) proposed a differet model, shiftig the emphasis from sleep loss to cumulative wake extesio or excess wakefuless. This subtle coceptual differece provided a parsimoious explaatio for the effects o wakig fuctios of both acute total sleep deprivatio ad chroic partial sleep deprivatio (Va Doge et al., 23; Va Doge ad Diges, 23b). Nevertheless, the excess wakefuless model is ot useful for computatioal predictios of eurobehavioral impairmet, because it does ot explicitly state how recovery from the effects of prior sleep loss would be achieved. Alterative solutios were itroduced by Hursh et al. (24) ad by Johso et al. (24), who each icluded a additioal regulatory process modulatig their versios of the homeostatic process, i order for their models to accout for the cumulative effects of chroic sleep restrictio. Based o the approach proposed by Johso et al. (24), Aviash et al. (2) the exteded the origial two-process model. Their objective was to capture the effects of chroic sleep restrictio o wakig eurobehavioral performace (Va Doge et al., 23) while retaiig the successes of the origial two-process model i predictig other aspects of wakig fuctios ad sleep (Acherma, 24). As the preset paper builds upo this work, the mathematical basis is briefly reiterated here. The homeostatic process of the origial two-process model is typically represeted as a pair of differece equatios Performace Performace Fig. 1. Neurobehavioral performace observatios ad predictios by differet models. A total of 48 healthy youg adults were subjected to oe of four laboratory sleep deprivatio protocols (Va Doge et al., 23). Each protocol bega with several baselie days ivolvig 16 h scheduled wake time (SWT)/8 h time i bed (TIB); the last of these baselie days is labeled here as day. Subsequetly, 13 subjects were kept awake (24 h SWT/ h TIB) for three additioal days, for a total of 88 h awake (left paels), after which they received varied amouts of recovery sleep (ot show). The other subjects uderwet various doses of sleep restrictio for 14 cosecutive days, followed by two recovery days with 16 h SWT/8 h TIB (right paels). The sleep restrictio schedule ivolved 2 h SWT/4 h TIB per day for 13 subjects (circles; red); 18 h SWT/6 h TIB per day for aother 13 subjects (boxes; yellow); ad 16 h SWT/8 h TIB per day for the remaiig ie subjects (diamods; gree). Awakeig was scheduled at 7:3 each day. Neurobehavioral performace was tested every 2 h durig scheduled wakefuless usig the PVT, for which the umber of lapses (reactio times greater tha ms) was recorded. (a) Observed eurobehavioral performace for each test bout (dots represet group averages). The first two test bouts of each wakig period are omitted i order to avoid cofouds from sleep iertia. Gray bars idicate scheduled sleep periods. (b) Correspodig performace predictios accordig to the origial two-process model (Borbély ad Acherma, 1999), liearly scaled to the data. Data poits represet performace predictios at wake oset. Thi curves represet predictios withi days, but the focus here is o chages across days (dashed lies). Note the rapid stabilizatio across days predicted to occur i the chroic sleep restrictio coditios (right pael), which does ot match the observatios show i (a). (c) Correspodig predictios accordig to the exteded two-process model (Aviash et al., 2), liearly scaled to the data. Note the uder-predictio of performace impairmet i the total sleep deprivatio coditio (left pael) ad the over-predictio of the impairmet build-up across days i the 2 h SWT/4 h TIB coditio (right pael), relative to the actual data show i (a). (d) Correspodig predictios accordig to the ew model itroduced i this paper as defied by Eqs. (21) ad (26). Note the improved fit to the experimetal observatios across days for total sleep deprivatio (left pael), which is explored i more detail i Fig. 3, as well as for the 2 h SWT/4 h TIB coditio (right pael). Performace impairmet i the 18 h SWT/6 h TIB ad 16 h SWT/8 h TIB coditios (right pael) is uder-predicted. However, the group-average impairmet levels observed for these coditios are iflated due to a few outliers (Va Doge et al., 23).

3 P. McCauley et al. / Joural of Theoretical Biology 26 (29) (Borbély ad Acherma, 1999): S t 1 e ð Dt=trÞ ð1 S t Dt Þ durig wake, (1a) S t e ð Dt=t dþ S t Dt durig sleep. (1b) Here S is the homeostatic sleep pressure as a fuctio of time t; Dt is the time step; ad t r 4 ad t d 4 are time costats for the rise ad decay of the homeostatic process durig wakefuless ad sleep, respectively. The reaso the two-process model predicts excessively rapid stabilizatio of performace across days of sleep restrictio is related to the asymptotic properties of Eqs. (1). Specifically, the wake equatio teds to a steady state represeted by a upper asymptote U 1, while the sleep equatio teds to a steady state represeted by a lower asymptote V. This asymptotic behavior ca be demostrated by rewritig Eqs. (1): S t U t ðs t Dt U t Dt Þe ð Dt=trÞ durig wake, (2a) S t V t ðs t Dt V t Dt Þe ð Dt=t dþ durig sleep. (2b) The extesio of the two-process model by Aviash et al. (2) ivolved modulatig the homeostatic process through maipulatio of the asymptotes U ad V i Eqs. (2), as follows: U t U t Dt þ m r Dt durig wake, (3a) U t U t Dt þð1 U t Dt Þð1 e ð Dt=m d Þ Þ durig sleep, (3b) V t U t 1. (3c) Here m r 4 represets the slope of a liear rise of the asymptotes durig wakefuless, ad m d 4 represets the time costat of a expoetial decay of the asymptotes durig sleep. The model proposed by Aviash et al. (2) performed at capturig the cumulative deficits i eurobehavioral performace across days as iduced by chroic sleep restrictio, but at the cost of reduced accuracy i describig the magitude of the effects across days of acute total sleep deprivatio see Fig. 1c. However, it ca be show that the model of Eqs. (3) belogs to a broader class of homeostatic models based o the same priciples, which may offer further improvemets i predictig performace impairmet across days of sleep loss. I this paper, we itroduce this ew model class which to our kowledge has ot bee previously proposed i the published literature ad ivestigate its geeral dyamic properties. We do this first by examiig, across wake/sleep cycles, the predicted levels of performace at the oset of wakefuless ad at the oset of sleep. These predictios are otioal the predictios at wake oset do ot accout for trasiet performace impairmet due to sleep iertia (e.g., Diges et al., 1981; Diges, 199), ad the predictios at sleep oset have o real meaig because the perso is asleep. However, they completely describe the model behavior across wake/sleep cycles, ad as such serve as useful achor poits to examie the patter of eurobehavioral performace chages over cosecutive days. Next, for a specific case of the ew model class, we compare model predictios to actual laboratory observatios of lapses of attetio o a psychomotor vigilace test (PVT; Diges ad Powell, 198; Dorria et al., 2; Lim ad Diges, 28), across cosecutive days of acute total sleep deprivatio or chroic partial sleep restrictio. Fially, based o the modelig results for PVT lapses of attetio, we ifer a possible eurobiological mechaism uderlyig the dyamic effects of sleep ad sleep loss o eurobehavioral performace. 2. A ew class of models formulated i terms of coupled ohomogeeous first-order ordiary differetial equatios 2.1. Defiig the ew model class Begiig with the origial two-process model (Acherma ad Borbély, 1994), we ca write model equatios for eurobehavioral performace as p ðtþ w ðtþ cðtþ for t 2½t ; t þ W Š ði:e:; durig wakeþ, q ðtþ s ðtþ cðtþ for t 2½t þ W ; t þ T Š ði:e:; durig sleepþ. (4a) (4b) The variables w ad s deote the homeostatic pressure durig wakefuless ad sleep, respectively, i the th wake/sleep cycle (i.e., day;,1, y). The fuctio c(t) is the origial circadia process (see Borbély ad Acherma, 1999). Further, t deotes the time of the begiig of the th wake/sleep cycle, T is the total duratio of the th cycle (such that t +1 t +T ), ad W is the duratio of wakefuless i the th cycle. We require that ow pt, where W T correspods to total sleep deprivatio. Fially, p ad q are the predictios for performace durig wakefuless ad sleep, respectively, i the th wake/sleep cycle. The predictios durig sleep are otioal; they are icluded strictly for cotiuity betwee cosecutive wake/sleep cycles. Here p ad q are coupled as follows: p ðt þ W Þq ðt þ W Þ, q ðt þ T Þp þ1 ðt þ1 Þ. (a) (b) The homeostatic process of Eqs. (1) may be writte i the form of a system of first-order ordiary differetial equatios (ODEs): dw dt 1 t r ðw 1Þ for t 2½t ; t þ W Š, (6a) ds dt 1 s t for t 2½t þ W ; t þ T Š. (6b) d Note that w ad s are still fuctios of time t, but to reduce clutter i later differetial equatios this is o loger idicated explicitly. From Eqs. (6) it follows that Eqs. (4) ca also be writte as a system of first-order ODEs: dp dt 1 t r p þ bðtþ for t 2½t ; t þ W Š, (7a) dq dt 1 q t þ gðtþ for t 2½t þ W ; t þ T Š, (7b) d where p ad q are agai coupled as per Eqs. (). The ohomogeeities b(t) ad g(t) represet the circadia process, ad may be geeralized to iclude other o-homeostatic iflueces o performace. The system of Eqs. (7) is a exact represetatio of the origial two-process model (Borbély ad Acherma, 1999). I the same maer, the exteded two-process model of Aviash et al. (2) ca be writte as a system of coupled o-homogeeous firstorder ODEs: _p _u þ 1=t r 1=t r p u " b # 1ðtÞ b 2 ðtþ for t 2½t ; t þ W Š, (8a)

4 23 P. McCauley et al. / Joural of Theoretical Biology 26 (29) _q 1=t d ð1=t d 1=m d Þ q g 1 ðtþ þ _v 1=m d v g 2 ðtþ for t 2½t þ W ; t þ T Š. (8b) Here u ad v are the levels of the upper ad lower asymptotes, respectively, i the th wake/sleep cycle. The o-homogeeities b 1 (t) ad g 1 (t) are bouded, oscillatory fuctios represetig the circadia process ad other o-homeostatic iflueces o performace. Likewise, b 2 (t) ad g 2 (t) are bouded, oscillatory fuctios represetig ay circadia or other o-homeostatic effects there might be o the levels of the upper ad lower asymptotes. Note that i this otatio, b 1 (t) ad b 2 (t) have absorbed the parameter m r (i.e., the slope of the liear rise of the upper asymptote durig wakefuless). Aalogous to Eqs. (), Eqs. (8) are coupled as follows: p ðt þ W Þ q ðt þ W Þ, (9a) u ðt þ W Þ v ðt þ W Þþd q ðt þ T Þ p þ1 ðt þ1 Þ, (9b) v ðt þ T Þ u þ1 ðt þ1 Þ d where d4 is the distace betwee the two asymptotes. For the exteded two-process model, d 1(Aviash et al., 2). Whe we write Eqs. (8) i geeralized form, it becomes clear that there is a asymmetry i the exteded two-process model of Aviash et al. (2): _p p " b # 1ðtÞ b 2 ðtþ a 11 a 12 þ for t 2½t _u u ; t þ W Š, (1a) " _q s # " 11 s 12 q g 1 þ ðtþ # for t 2½t _v s 22 v g 2 ðtþ þ W ; t þ T Š. (1b) Namely, Eq. (1b) for sleep has oe more parameter tha Eq. (1a) for wakefuless. Addig the correspodig coefficiet a 22 i Eq. (1a) geerates a useful ew model with a bifurcatio, which we will examie i detail i the ext sectio. Eqs. (1a) ad (1b) also each have room for aother parameter i their 2 2 coefficiet matrices (i.e., a 21 ad s 21, respectively). As such, we may defie our ew class of models, formulated i terms of coupled o-homogeeous first-order ODEs, by the followig geeralized equatios (which icorporate the origial ad exteded two-process models): _p p a 11 a 12 a 21 a 22 b 1 ðtþ b 2 ðtþ þ for t 2½t _u u ; t þ W Š, (11a) " _q s # " 11 s 12 q g 1 þ ðtþ # for t 2½t _v s 21 s 22 v g 2 ðtþ þ W ; t þ T Š. (11b) The couplig of these equatios is give by Eqs. (9). Of the ohomogeeities b 1 (t), b 2 (t), g 1 (t) ad g 2 (t) we oly require that they are bouded, oscillatory fuctios. They co-determie the profiles of performace chages withi wake/sleep cycles, i part through the circadia process, but this is beyod the focus of the preset paper. Of primary iterest are the a ad s coefficiet matrices, as they determie the dyamic behavior of the system across wake/ sleep cycles Dyamic properties of the ew model class For costat values of the a ad s coefficiets, the geeral solutio of the ODE system of Eqs. (11) is of the form (Derrick ad Grossma, 1997): p ðtþ c u ðtþ ðtþc 1 ðt p ðt Þ Þ þ u ðt Þ q ðtþ v ðtþ j ðtþj 1 ðt þ W Þ Z t þ tþw Z t t q ðt þ W Þ v ðt þ W Þ ds, g j ðtþj 1 ðsþ 1 ðsþ g 2 ðsþ c ðtþc 1 ðsþ b 1 ðsþ ds, b 2 ðsþ (12a) (12b) where c (t) ad j (t) are the respective fudametal solutios of the homogeeous parts of Eqs. (11). These fudametal solutios deped o the eigevalues l i ad the eigevectors [k i1 k i2 ] of the a ad s coefficiet matrices. The eigevalues l 1 ad l 2 ad the correspodig eigevectors of the a coefficiet matrix are foud by solvig: det! a 11 l i a 12, a 21 a 22 l i " a 11 l i a 12 # ki1 a 21 a 22 l i k i2 (13a). (13b) The process is aalogous for the eigevalues l 3 ad l 4 ad the correspodig eigevectors of the s coefficiet matrix. The fudametal solutio c (t) depeds o the real ad distict eigevalues l 1 ad l 2 foud by solvig Eqs. (13); ad the fudametal solutio j (t) depeds o the likewise derived real ad distict eigevalues l 3 ad l 4, as follows (Derrick ad Grossma, 1997): " c ðtþ k # 11e l1t k 21 e l2t k 12 e l 1t k 22 e l, (14a) 2t " j ðtþ k # 31e l 3t k 41 e l 4t k 32 e l 3t k 42 e l. (14b) 4t Note that while Eqs. (14) are sesitive to shiftig of the origi of the time variable t, the fuctios c (t) ad j (t) ed up beig used oly i products with their respective iverses, ad these socalled pricipal matrix solutios are ivariat to time traslatio. Havig foud the geeral solutio of the ODE system of Eqs. (11), differece equatios ca be derived for the predicted level of performace at the oset of each wake period ad at the oset of each sleep period. Although these predictios for wake oset do ot accout for trasiet effects of sleep iertia (Diges, 199), ad the predictios for sleep oset are otioal (sice the perso is asleep), they completely describe the model behavior across wake/sleep cycles. They therefore serve as useful achor poits to examie the patter of eurobehavioral performace chages across days. Usig Eqs. (9) ad (12), the differece equatios for performace at wake oset p (t ), ad for performace at sleep oset q (t +W ), ca be show to be give by p þ1 ðt þ1 Þ u þ1 ðt þ1 Þ j ðt þ T Þj 1 p ðt Þ u ðt Þ þ F, ðt þ W Þc ðt þ W Þc 1 ðt Þ (1a)

5 P. McCauley et al. / Joural of Theoretical Biology 26 (29) q þ1 ðt þ1 þ W þ1 Þ v þ1 ðt þ1 þ W þ1 Þ F c þ1 ðt þ1 þ W þ1 Þc 1 þ1 ðt þ1þj ðt þ T Þj 1 ðt þ W Þ q ðt þ W Þ þ G, v ðt þ W Þ! 1 j ðt þ T Þj 1 1 ðt þ W Þ d Z tþt g þ j ðt þ T Þj 1 ðsþ 1 ðsþ ds þ j g 2 ðsþ ðt þ T Þ tþw j 1 ðt þ W Þ Z tþw t b c ðt þ W Þc 1 ðsþ 1 ðsþ b 2 ðsþ ds, (1b) (1c)! 1 G c þ1 ðt þ1 þ W þ1 Þc 1 þ1 1 ðt þ1þ d Z tþ1 þw þ1 b þ c þ1 ðt þ1 þ W þ1 Þc 1 þ1 ðsþ 1 ðsþ ds t þ1 b 2 ðsþ þ c þ1 ðt þ1 þ W þ1 Þc 1 þ1 ðt þ1þ Z tþt g j ðt þ T Þj 1 ðsþ 1 ðsþ ds. (1d) g tþw 2 ðsþ Chages i eurobehavioral performace across wake/sleep cycles deped etirely o this system of differece equatios for performace at the osets of wakefuless ad sleep. Of particular iterest is whether the patter of chages i eurobehavioral performace across wake/sleep cycles ca display a steady state or equilibrium that is, whether the performace profile withi days ca be foud to repeat itself across days or across clusters of days whe a particular wake/sleep schedule is maitaied. This coditio of fixed wake duratio W ad fixed wake/sleep cycle duratio T is described by p þm ðt þm Þp ðt Þ, (16a) q þm ðt þm þ WÞ q ðt þ WÞ, (16b) where m 1, 2, y is the umber of wake/sleep cycles after which the performace patter repeats itself. If the oscillatio period t of the o-homogeeities b 1 (t), b 2 (t), g 1 (t) ad g 2 (t) equals the wake/ sleep cycle duratio T, as is the case uder coditios of circadia etraimet, the the steady state performace patter would be expected to repeat itself every day (i.e., m 1). If tat, the a beat pheomeo could occur i which the performace patter repeats itself every m days. Forced desychroy protocols (e.g., Dijk ad Czeisler, 1994) are based o this latter idea. Ideed, for fixed wake duratio W ad fixed wake/sleep cycle duratio T, ad assumig that the o-homogeeities oscillate with period t T, Eqs. (14), (1c) ad (1d) may become repetitive across wake/sleep cycles (where m 1). The equilibrium state for this case, which we deote as [p(t ) u(t )] ad [q(t +W) v(t +W)] for wakefuless ad for sleep, respectively, ca be derived by solvig Eqs. (1), which results i pðt Þ 1 1 FC F, (17a) uðt Þ 1 qðt þ WÞ 1 1 CF G, (17b) vðt þ WÞ 1 C cðt þ WÞc 1 ðt Þ, F jðt þ TÞj 1 ðt þ WÞ. (17c) (17d) Because of the matrix iversios embedded i Eqs. (17a) ad (17b), a state of equilibrium ca oly occur if 1 det FC a, (18a) 1 ad 1 det CF a. (18b) 1 We will examie this coditio for a specific case of the models defied by Eqs. (11), later i this paper. Provided a state of equilibrium is show to exist, the questio arises whether it is stable, that is, whether the model predictios would coverge to this state for a repetitive wake/sleep schedule. We ca say that the model is asymptotically stable (for m 1) or asymptotically periodic (for m41), if (Kelly ad Peterso, 21): " p þm ðt þm Þ lim pðt # Þ, (19a)!1 u þm ðt þm Þ uðt Þ lim!1 " q þm ðt þm þ WÞ qðt # þ WÞ, (19b) v þm ðt þm þ WÞ vðt þ WÞ eve if the startig values [p (t ) u (t )] ad [q (t ) v (t )] are ot already at equilibrium. Because Eqs. (1) are liear i [p (t ) u (t )] ad [q (t +W) v (t +W)], it ca be show (Kelly ad Peterso, 21) that states of equilibrium are asymptotically stable or periodic if all eigevalues L i of the system of Eqs. (1), whether real or complex, are iside the uit circle (i.e., L i o1). For fixed wake duratio W ad fixed wake/sleep cycle duratio T, ad assumig agai that the ohomogeeities oscillate with period t T (so that m 1), these eigevalues are foud by solvig the characteristic equatios: 1 det FC L i 1 1 det CF L i 1,. (2a) (2b) The eigevalues derived from Eq. (2a) are idetical to those derived from Eq. (2b); ad the equilibrium states [p(t ) u(t )] ad [q(t +W) v(t +W)] are either both asymptotically stable, or both ustable. Whe both are asymptotically stable, it meas that the predicted levels of performace at wake oset, at sleep oset, ad by extesio at all poits i betwee, coverge across days to a steady state repeated from day to day. Whe both are ustable, it implies that the predicted levels of performace at wake oset, at sleep oset, ad i betwee, diverge across days toward ifiity. Note that over days it grows progressively more difficult to maitai a wake/sleep schedule that iduces divergig icreases i performace deficits. I practice, sleep teds to break through ito scheduled wakefuless (e.g., Dora et al., 21) before performace becomes catastrophically impaired. 3. A model with a bifurcatio We ow cosider a particular case of the model of Eqs. (11): " _p a # " 11 a 12 p þ b # 1ðtÞ for t 2½t _u a 22 u b 2 ðtþ ; t þ W Š, (21a) _q _v " s # 11 s 12 q s 22 v þ " g 1 ðtþ # g 2 ðtþ for t 2½t þ W ; t þ T Š, (21b)

6 232 P. McCauley et al. / Joural of Theoretical Biology 26 (29) where a 11 o ad s 11 o, ad where a 11 aa 22 ad s 11 as 22. The couplig of these equatios is give by Eqs. (9). As before, we require that the o-homogeeities b 1 (t), b 2 (t), g 1 (t) ad g 2 (t) are bouded, oscillatory fuctios. Expedietly, we also assume that the o-homogeeities oscillate with period t T. Per Eqs. (13), the (real ad distict) eigevalues of the a ad s coefficiet matrices are: l 1 a 11 o; l 2 a 22 ; l 3 s 11 o; ad l 4 s 22. Through Eqs. (14), these eigevalues determie the existece of states of equilibrium as assessed usig Eqs. (18). Uder coditios of fixed wake duratio W ad fixed wake/sleep cycle duratio T, Eqs. (18) reduce to the followig sole iequality: ð1 e a 11W e s 11ðT WÞ Þð1 e a 22W e s 22ðT WÞ Þa. If both a parameters ad both s parameters are egative, this iequality is satisfied ad thus states of equilibrium exist for all owpt (both for performace at wake oset ad for performace at sleep oset). If either a 22 Xors 22 X, however, there may be a critical amout of daily wakefuless W c, with ow c pt, for which o equilibrium exists: s 22 W c T. (22) s 22 a 22 To assess the stability of the states of equilibrium whe they do exist, we solve Eqs. (2), which results i the followig eigevalues: L 1 e a 11W e s 11ðT WÞ, (23a) L 2 e a22w e s22ðt WÞ. (23b) If all a ad s parameters i Eqs. (23) are egative, the ol i o1 for both eigevalues, meaig that the equilibrium states (which the exist for all owpt) are always asymptotically stable. Sice the model system cosidered here is liear, this stability is global (i.e., the predictios coverge to equilibrium regardless of iitial coditios) (Verhulst, 2). If a 22 (the key parameter distiguishig the model give by Eqs. (21) from the exteded two-process model) is positive, there are three possibilities for L 2. I order of icreasig amout of wake extesio (i.e., greater sleep restrictio), these possibilities are: for WoW c, we fid that ol 2 o1, which implies globally asymptotically stable states of equilibrium; for W W c, o equilibrium exists (see above); for W 4 W c, we fid that L 2 4 1, which implies that the states of equilibrium are ustable. Performace Fig. 2. Illustratio of the model give by Eqs. (21) usig parameter values selected to illustrate its bifurcatig dyamic behavior. The figure shows model predictios at wake oset (data poits) for 16 days (,1, y,1) of fixed duratio T 24 h, assumig a costat period t 24 h for the o-homogeeities. The thi curves represet the predictios withi days usig the o-homogeeities give by Eqs. (26) but the profile of chages across days (dashed lies) as determied by the a ad s coefficiet matrices i Eqs. (21) is of primary iterest here. Each predictio curve correspods to a differet amout of daily wakefuless: W 16 h (diamods; gree), W 18 h (boxes; yellow), W 2 h (circles; red), W 22 h (dowward triagles; gray), ad W 24h (i.e., total sleep deprivatio) (upward triagles; black). Light gray areas idicate octural sleep periods. I this illustratio, the model parameter values are itetioally selected such that the bifurcatio threshold occurs at W c 2 h (i.e., 4 h daily sleep). For daily wake duratios below this bifurcatio threshold (gree ad yellow), the model coverges to a asymptotically stable equilibrium, meaig that performace impairmet ultimately levels off. For daily wake duratios beyod the bifurcatio threshold (gray ad black), the model diverges from a ustable equilibrium, meaig that performace impairmet teds to escalate. At exactly the bifurcatio value W W c (red), there is o equilibrium state, resultig i a asymptotically liear build-up of performace impairmet across days. f s 12ðe s 11ðT WcÞ e s 22ðT WcÞ Þe a 22Wc s 11 s 22 þ a 12ðe a 11Wc e a 22Wc Þe s 11ðT WcÞ a 11 a 22, (24c) Thus, for a 22 4, the model behavior is such that if the amout of wakefuless W i each wake/sleep cycle exceeds a critical threshold W c, the model flips from a state i which performace predictios coverge toward a asymptotically stable equilibrium, to a state i which performace predictios diverge away from a ustable equilibrium. This qualitative chage i dyamic behavior implies a bifurcatio, as illustrated i Fig. 2. It is istructive to study the model behavior whe daily wakefuless is kept costat at the bifurcatio value: W W c. Here, the geeralized iterative system of Eqs. (1) assumes the followig specific form: p þ1 ðt þ1 Þ u þ1 ðt þ1 Þ þ F, eðða 22s 11 a 11 s 22 Þ=ða 22 s 22 ÞÞT f p ðt Þ 1 u ðt Þ q þ1 ðt þ1 þ W c Þ eðða 22s 11 a 11s 22Þ=ða 22 s 22ÞÞT g v þ1 ðt þ1 þ W c Þ 1 q ðt þ W c Þ þ G, v ðt þ W c Þ (24a) (24b) g s 12ðe s 11ðT WcÞ e s 22ðT WcÞ Þe a 11Wc s 11 s 22 þ a 12ðe a 11Wc e a 22Wc Þe s 22ðT WcÞ a 11 a 22. (24d) The solutio of this system teds to a straight lie as -N. The chage across days for performace at wake oset ad sleep oset is defied, respectively, by slopes M p ad M q : ff 2 M p 1 e ðða 22s 11 a 11 s 22 Þ=ða 22 s, (2a) 22 ÞÞT gg 2 M q 1 e ðða 22s 11 a 11 s 22 Þ=ða 22 s, (2b) 22 ÞÞT where F 2 ad G 2 are the secod elemet of vectors F ad G i Eqs. (1). From Eqs. (2) it follows that the slopes of chage across days are ot ecessarily the same for performace at wake oset ad performace at sleep oset.

7 P. McCauley et al. / Joural of Theoretical Biology 26 (29) Model simulatios Compariso to data from sleep restrictio experimets To compare the model give by Eqs. (21) to actual performace observatios uder coditios of sleep loss, we fit it to groupaverage data of performace lapses o the PVT (Diges ad Powell, 198; Dorria et al., 2; Lim ad Diges, 28) froma study of healthy youg adults exposed to chroic sleep restrictio or total sleep deprivatio with W 16, 18, 2, or 24 h (Va Doge et al., 23). These data are show i Fig. 1a. For the o-homogeeities, we make use of the circadia process c(t) defied by Borbély ad Acherma (1999), applied to the performace predictios p ad q (but ot the asymptotes u ad v ), as follows: b 1 ðtþ kcðt yþþm for t 2½t b 2 ðtþ ; t þ W Š, (26a) Performace g 1 ðtþ g 2 ðtþ kcðt yþþm t 2½t þ W ; t þ T Š. (26b) Here k ad m are parameters scalig the circadia process, ad y is a phase parameter shiftig it i time. For the iitial coditios [p (t ) u (t )] we estimate the values correspodig to the equilibrium state at W 16 h, which characterizes the baselie coditio i the study. Further, t 7. h (i.e., 7:3), ad T ad t are fixed at 24 h. Usig least-squares regressio o all 44 data poits show i Fig. 1a, we fid the followig parameter estimates: 8 a 11 a 12 :13 :929 a 22 :743 s 11 s 12 2:17 :872 ; s 22 :397 >< >: d 19:8; k :86; m :472; y 12:7; p ðt Þ4:49; u ðt Þ29:9: ; (27) The resultig PVT performace predictios are show i Fig. 1d, ad the predictios for the total sleep deprivatio coditio (W 24 h) are explored i more detail i Fig. 3. With the parameter estimates of Eqs. (27), the model explais 72.4% of the variace i the group-average data of Fig. 1a. It fits substatially to the data tha the origial two-process model (Fig. 1b, explaied variace 22.6%) ad the exteded twoprocess model (Fig. 1c, explaied variace 38.4%). Evaluatio of Eq. (22) give the parameter estimates i Eqs. (27) idicates that there must be a bifurcatio at W c 2.2 h. That is, the model should flip from a state of covergece to a state of divergece whe daily wakefuless is icreased to more tha 2.2 h (i.e., whe daily sleep is reduced to less tha 3.8 h). This property ca be verified by comparig model predictios to the group-average observatios of PVT performace i aother study of chroic sleep restrictio, with W 1, 17, 19 or 21 h (Beleky et al., 23). These data are show i Fig. 4a. We use the model of Eqs. (21) agai, apply the o-homogeeities defied i Eqs. (26), set t 7. h (i.e., 7:) i accordace with the study desig (Beleky et al., 23), ad fix all model parameters at their previously estimated values give i Eqs. (27). Applyig liear scalig to accout for ay irrelevat differeces i absolute Fig. 3. Detailed examiatio of the performace predictios uder coditios of total sleep deprivatio. The ew model defied by Eqs. (21) ad (26) with the parameter estimates give by Eqs. (27) has a bifurcatio at W c 2.2 h, implyig that predictios for performace i the total sleep deprivatio coditio (i.e., W 24 h4w c ; see Fig. 1a, top left pael) should exhibit divergig (i.e., escalatig) performace impairmet across days. However, the actual predictios (see Fig. 1d, bottom left pael) would seem to suggest a covergig patter. This ca be explaied by simultaeously cosiderig the performace predictios p (black dashed curve), the level of the ustable equilibrium state p (dotted horizotal lie), ad the upper asymptote u (gray dashed curve). Sice a 22 4, the upper asymptote u icreases expoetially across days. Thus, withi wakig episodes, performace p is icreasigly draw upwards. O the other had, the equilibrium level p is located above the iitial performace value p (t ). Thus, divergece from this ustable equilibrium would etail a drive dowwards. Here, the et result is that performace impairmet is predicted to icrease across days, but i a deceleratig maer (cf. Va Doge et al., 23). If wakefuless were maitaied for additioal days, though, the performace predictios would cross the ustable equilibrium state ad the diverge from it upwards, exposig the typical escalatig behavior for W4W c i this model (see the illustratio i Fig. 2, black upward triagles). performace outcomes (e.g., due to variatios i populatio characteristics or performace testig coditios), we fid the scalig factor to be 1.17 suitably close to 1. The correspodig performace predictios are show i Fig. 4b. They explai 72.2% of the variace i the data, ad fit well to the observed performace chages across days. Note that the W 21 h coditio shows a diverget profile i both observatios ad predictios (Fig. 4a ad b), which is ot see i the Wp19 h coditios i this study. This qualitative differece idicates the presece of a bifurcatio. Ideed, o the basis of fittig the model to the data i Fig. 1a, we had predicted that a bifurcatio should occur at W c 2.2 h (see above). The goodess-of-fit of our model to the data i Fig. 4a is cosistet with this predictio, ad provides a first validatio of the model itroduced i this paper New predictios for chroic sleep restrictio ad recovery The value of a ew model is determied, i part, by ay falsifiable ew predictios it makes. Here we preset three specific predictios that ca be tested i laboratory experimets, ad that will have cosiderable theoretical ad/or practical impact if cofirmed.

8 234 P. McCauley et al. / Joural of Theoretical Biology 26 (29) The first ew predictio focuses o the effectiveess of ap sleep as a meas to sustai performace across days. It has previously bee show that a sigle 2 h ap ca mitigate performace impairmet across exteded periods of wakefuless (Diges et al., 1988), but it is ot kow whether a 2 h ap take every day (i.e., W 22 h) ca maitai performace at reasoable levels across days. Our ew model defied by Eqs. (21) ad (26) with the parameter values give by Eqs. (27) predicts that sice daily wake duratio exceeds the bifurcatio threshold (W c 2.2 h), performace deficits should escalate, ad thus a daily 2 h ap would ot suffice to maitai reasoable levels of performace. As show i Fig., which compares the predicted effects of the 2 h ap schedule to those of total sleep deprivatio, the daily ap would mitigate performace impairmet substatially i the first few days (cf. Va Doge ad Diges, 23b), but i later days Performace this beeficial effect would icreasigly dimiish. After 8 days with a 2 h ap each day, the predicted level of performace impairmet approaches that of 3 days with total sleep deprivatio; ad if it were possible to cotiue the ap schedule much loger, the effectiveess of the 2 h daily ap would essetially be lost. Thus, our predictio is that a daily 2 h ap is ot effective as a meas to sustai performace across days. The secod ew predictio pits the ew model defied by Eqs. (21) agaist the oly other quatitative, sleep/wake physiology-based model of the effects of chroic sleep restrictio o eurobehavioral performace: the excess wakefuless model (Va Doge et al., 23). I that model, performace impairmet across days is posited to be proportioal to the cumulative amout of wakefuless exceedig a maximum period of stable wakefuless x (where xe16 h if prior sleep duratio exceeds 4 h). This is similar to the cocept of cumulative sleep debt (e.g., Demet, 26), but coceptually distict from the modelig framework itroduced i the preset paper. Our predictio ivolves the importat questio of how much sleep is eeded to recover from performace impairmet iduced by prior chroic sleep restrictio (e.g., Lamod et al., 27). The excess wakefuless model would predict that as log as wake duratio exceeds x, performace will cotiue to deteriorate. O the cotrary, the ew model defied by Eqs. (21) ad (26) with the parameter values give by Eqs. (27) would predict that whe wake duratio is less tha the bifurcatio threshold W c, performace levels should coverge to a state of equilibrium, ad thus some recovery could occur if wake duratio is shorter tha what was maitaied i the prior days of chroic sleep loss. As a specific example, cosider a sceario ivolvig days of wake extesio to 2 h per day (i.e., 4 h sleep daily), followed by a day with wake extesio to just 18 h (i.e., 6 h sleep). The opposig model predictios are illustrated i Fig. 6. The excess wakefuless model predicts that performace deteriorates progressively across the days with 2 h wakefuless, ad cotiues to deteriorate albeit at a slower rate followig the day with oly 18 h wakefuless. Our ew model also predicts progressive performace degradatio across the days with 2 h wakefuless. However, the state of equilibrium for 18 h awake occurs at a lower level tha the performace degradatio reached after days with 2 h wakefuless. Therefore, the ew model forecasts some degree of recovery after the 18 h wakefuless day. This predictio may seem couterituitive cosiderig that stayig awake for Performace Fig. 4. Experimetal observatios ad predictios by our ew model for eurobehavioral performace impairmet. A total of 66 healthy youg adults were subjected to oe of four laboratory sleep deprivatio protocols (Beleky et al., 23). Each protocol bega with several baselie days ivolvig 16 h scheduled wake time (SWT)/8 h time i bed (TIB); the last of these baselie days is labeled here as day. The subjects subsequetly uderwet various doses of sleep restrictio for seve cosecutive days, followed by three recovery days with 16 h SWT/8 h TIB. The sleep restrictio schedule ivolved 21 h SWT/3 h TIB per day for 13 subjects (circles; purple); 19 h SWT/ h TIB per day for 13 subjects (boxes; orage); 17 h SWT/7 h TIB per day for 14 subjects (diamods; brow); ad 1 h SWT/9 h TIB per day for 16 subjects (triagles; blue). Awakeig was scheduled at 7: each day. Neurobehavioral performace was tested daily at 9:, 12:, 1: ad 21: usig the PVT. I the 19 h SWT/ h TIB coditio a additioal test bout occurred at midight, ad i the 21 h SWT/3 h TIB coditio yet aother oe took place 2 h after midight. (a) Observed eurobehavioral performace (PVT lapses) for each test bout (dots represet group averages). The first test bout of each wakig period is omitted i order to avoid cofouds from sleep iertia. Gray bars idicate scheduled sleep periods. (b) Correspodig performace predictios accordig to the ew model defied by Eqs. (21) ad (26). Parameter estimates are fixed at the values of Eqs. (27), as previously estimated usig the data i Fig. 1a. Data poits represet predictios at wake oset; thi curves represet predictios withi days. The focus here is o chages across days (dashed lies). Note that the model predictios across the seve days of sleep restrictio accurately capture the qualitative chage from covergece (i.e., levelig off of performace impairmet) i the 1, 17 ad 19 h SWT coditios, to divergece (i.e., disproportioately rapid escalatio of performace impairmet) i the 21 h SWT/3 h TIB coditio.

9 P. McCauley et al. / Joural of Theoretical Biology 26 (29) Performace the iterveig 14 h wake/1 h sleep day should provide (ear-)complete recovery to baselie performace (Baks et al., 27a), effectively udoig the impairmet icurred by the prior sleep loss. Thus, the performace profile see durig the secod -day period of wake extesio might be expected to be similar to that see durig the first -day period of wake extesio. The dyamics of the ew model, however, imply that the sigle 14 h wake/1 h sleep day should be see as a itermittet perturbatio i a exteded series of days with wake extesio to 2 h per day. Thus, the model predicts that the 1 h recovery sleep cofers oly a short-lastig performace improvemet, after which performace further deteriorates as it cotiues to coverge to the asymptotically stable equilibrium associated with W 2 h. This predictio is illustrated i Fig. 7. Prelimiary evidece from the laboratory study examiig the sceario at had suggests that ideed there is substatial carry-over of performace impairmet from the first -day period with daily wake extesio to the secod (Baks et al., 27b), providig tetative support for the ew model.. Discussio.1. Model implicatios Fig.. Predictio of the effectiveess of a sigle 2 h octural period of ap sleep each day for maitaiig performace across days. The figure shows predicted performace at wake oset (boxes) ad withi each of the days (thi curve) across 8 days with a ap scheduled from 2:4 util 4:4 daily (gray bars). For compariso, predicted performace across 4 days of total sleep deprivatio is show as well (triagles represet performace at 4:4, which is the same time as scheduled wake oset i the ap coditio). Both coditios are iitiated after awakeig from 8 h baselie sleep at 7:3 (dashed vertical lies idicate 7:3 at 24 h icremets). The performace predictios, derived from the ew model give by Eqs. (21), (26) ad (27) ad expressed i terms of the umber of lapses o the PVT, idicate that a daily 2 h ap is ot effective at maitaiig reasoable levels of performace across multiple days. (For a discussio of the predictios for the total sleep deprivatio coditio, see Fig. 3, but ote that the timig of wake oset is ot the same.) 18 h followig multiple baselie days with 16 h wakefuless (8 h sleep) actually leads to performace degradatio (Va Doge et al., 23). Yet, prelimiary evidece from a ogoig laboratory study (Baks et al., 2) suggests that some recovery does occur with 18 h wakefuless (6 h sleep) i this chroic sleep restrictio sceario, supportig the ew model over the excess wakefuless model. The third ew predictio cocers the recycle issue, which derives from the questio of whether or ot there is ay carryover of performace impairmet from past sleep restrictio whe begiig a ew period of sleep restrictio followig limited time for recovery. Let s cosider a laboratory study curretly uderway (Baks et al., 27b), which ivolves a period of days with wake extesio to 2 h per day (i.e., 4 h sleep daily), followed by a day with oly 14 h of wakefuless (i.e., 1 h time for recovery sleep), followed by aother period of days with wake extesio to 2 h per day. Iitial experimetal evidece would suggest that The regulatio of sleep, wakefuless ad performace ivolves a array of possible eurobiological mechaisms (e.g., Porkka- Heiskae et al., 1997; Krueger ad Obál, 23; Fuller et al., 26), ad is ot fully uderstood. Noetheless, at the behavioral level, the circadia compoet has bee captured by models with relatively few degrees of freedom (see Idic et al., 26). We believe the same may be possible for the sleep homeostatic compoet. Usig evidece from laboratory studies with multiple days of sleep loss (Figs. 1a, 4a), we showed that the homeostatic regulatio of eurobehavioral performace ca be described by meas of a system of coupled o-homogeeous first-order ODEs with oly a few additioal degrees of freedom relative to the homeostatic process postulated i the origial two-process model (Acherma ad Borbély, 1994; Borbély ad Acherma, 1999). Our ew model does iclude a additioal compoet, modulatig the homeostatic process across days ad weeks, as prompted by fidigs from chroic sleep restrictio experimets demostrated to be icogruet with the origial two-process model (Va Doge et al., 23; Va Doge, 24). Yet, the model structure itroduced i this paper is essetially still composed of a homeostatic process ad a circadia process. Coceptually, therefore, the ew model remais compatible with the priciples of sleep regulatio istatiated i the origial twoprocess model (Borbély, 1982). The dyamics of the ew model across days are pricipally govered by the a ad s coefficiet matrices i the homogeeous part of the differetial equatios (the homeostatic process), while the chages withi days are primarily govered by the o-homogeeities (the circadia process). These model compoets also iteract, i agreemet with laboratory observatios of a oliear iteractio betwee the homeostatic ad circadia processes (Dijk et al., 1992; Va Doge ad Diges, 23a). Two semial laboratory studies first highlighted the eed for fudametally ew model developmet beyod the two-process model i order to accout for the wakig eurobehavioral cosequeces of chroic sleep loss (Beleky et al., 23; Va Doge et al., 23). However, these two studies previously drew markedly differet coclusios about the dyamics of eurobehavioral impairmet across days of sleep restrictio. I their study with 7 days of systematic sleep restrictio, Beleky et al. (23) reported a plateau of cogitive impairmet whe sleep was

10 236 P. McCauley et al. / Joural of Theoretical Biology 26 (29) Performace Fig. 6. Opposig predictios from two models regardig recovery followig chroic sleep restrictio. The figure shows performace predictios at wake oset (boxes) for five days with 2 h wakefuless ad 4 h sleep per day, followed by oe day with 18 h wakefuless ad 6 h recovery sleep. Gray areas idicate octural sleep periods. (a) Predictios for performace chages across days accordig to the excess wakefuless model (Va Doge et al., 23). This model predicts that performace deteriorates progressively across the five days with 2 h wake/4 h sleep, ad cotiues to deteriorate at a slower rate followig the day with 18 h wake/6 h sleep. (b) Predictios for performace chages across days accordig to the model defied by Eqs. (21), (26) ad (27). This ew model also predicts that performace deteriorates progressively across the five days with 2 h wake/4 h sleep, but forecasts a modest relative performace improvemet followig the day with 18 h wake/6 h sleep. (Note that sleep iertia is ot accouted for i these predictios.) restricted to 7 or h per day, as well as icomplete recuperatio at the ed of the study after 3 days with 8 h time i bed for recovery sleep. They hypothesized that chroic sleep loss iduces loglastig adaptive chages i the brai s respose to sleep loss, leadig to stabilized reduced performace uder coditios of sleep restrictio at the cost of dimiished maximal performace capacity followig recovery sleep. I cotrast, i their study with 14 days of sleep restrictio, Va Doge et al. (23) oted that performace cotiued to degrade whe sleep was restricted to 6 or 4 h per day, with o evidece of adaptatio across the study period. I the preset paper, the two data sets (Figs. 1a ad 4a) are examied i a sigle aalytical framework. Usig PVT performace lapses as a well-validated outcome measure (Dorria et al., 2) for both studies, o covicig evidece of a impairmet plateau is foud i either data set. Yet, our modelig results idicate that stabilizatio of performace impairmet would occur evetually, beyod the duratio of the two experimets. Furthermore, the modelig outcomes suggest that several days with recovery sleep would be eeded to restore performace to baselie levels. Experimets curretly uderway (Weseste et al., 2; Baks et al., 27a, b) will shed further light o the time course of post-deprivatio recovery. Mathematical examiatio of the dyamics of the ew model defied by Eqs. (21) revealed a uaticipated emerget model property: a bifurcatio ivolvig a critical amout of wakefuless which, if exceeded, chages the model behavior from a state of covergece toward a asymptotically stable equilibrium, to a state of divergece away from a ustable equilibrium (as illustrated i Fig. 2). This feature, previously alluded to (Beleky et al., 23; Va Doge ad Diges, 23b) but as yet ot cosidered explicitly, tured out to capture a essetial aspect of the ature of performace impairmet due to sleep loss. Usig data from the chroic sleep restrictio ad total sleep deprivatio experimets documeted by Va Doge et al. (23) (Fig. 1a), we estimated the critical wakefuless threshold to occur at 2.2 h (i.e., at 3.8 h daily sleep). This estimate was supported by data from the chroic sleep restrictio study of Beleky et al. (23), who observed escalatig performace impairmet whe wakefuless was exteded to 21 h per day (3 h daily sleep coditio i Fig. 4a). The sigificace of the bifurcatio i the ew model implies that other two-process-based models of performace impairmet due to chroic sleep loss (Hursh et al., 24; Johso et al., 24; Aviash et al., 2), which do ot possess the bifurcatio property, must have a more limited rage of applicability tha the ew model. The excess wakefuless model (Va Doge et al., 23), which is based o the fudametally differet cojecture that performace impairmet across days is proportioal to the cumulative amout of wakefuless i excess of a ratio determied by the precedig sleep period, does ot a priori have this same limitatio of scope (Va Doge ad Diges, 23b). However, the excess wakefuless model ad the model itroduced i the preset paper make cotradictory predictios for performace impairmet after a period of chroic sleep restrictio followed by a limited amout of recovery sleep (Fig. 6).