ANIMAL BEHAVIOUR, 2007, 74, 189e197
doi:10.1016/j.anbehav.2006.12.007
A behavioural shutdown can make sleeping safer: a strategic
perspective on the function of sleep
STEVEN L. LIMA * & NI EL S C. RA TTENBORG †
*Department of Ecology and Organismal Biology, Indiana State University, Terre Haute
yMax Planck Institute for Ornithology, Seewiesen
(Received 14 September 2006; initial acceptance 23 October 2006;
final acceptance 28 December 2006; published online 9 July 2007; MS. number: A10559)
Sleep appears to effect some sort of neural maintenance, but a complete theory of the function of sleep
must address why such maintenance requires a behavioural shutdown (or unconsciousness) that leaves
an animal vulnerable to predators. We present a simple, strategic model to determine the degree of sleep
that minimizes the risk of predation. We assume that the brain is composed of neural units that can, in
theory, ‘sleep’ independently of each other, and that a given neural unit must go offline for maintenance/
sleep. We also assume that the probability of detecting an attack depends on the fraction of neural units
that are awake. We found that having all neural units offline simultaneously (i.e. shutdown sleep) is often
the safest way to perform neural maintenance, even though partial sleep makes predators more detectable.
This counterintuitive result reflects the assumptions that, in a state of partial sleep, (1) neural maintenance
takes longer to complete and (2) predator detection is less effective than suggested by the proportion of
neural units online. Partial sleep is a possible outcome when the risk of attack increases as more neural
units are taken offline. Minimal sleep (with only one or a few units offline) is a possible outcome when
the attack rate while awake is substantially higher than when asleep. Partial sleep of a sort is known to
occur in some animals, but there is no apparent evidence for the idea of minimal sleep.
Ó 2007 The Association for the Study of Animal Behaviour. Published by Elsevier Ltd. All rights reserved.
Keywords: antipredator behaviour; optimality theory; predation; sleep; vigilance
Sleep (broadly defined) is a widespread behavioural phenomenon in the animal world (Campbell & Tobler 1984;
Tobler 2000; Rattenborg & Amlaner 2002). Many animals
will spend much of their lives in this behavioural state
(Campbell & Tobler 1984; Amlaner & Ball 1994; Zepelin
2000). Humans, for example, spend about one-third of
their lives asleep. Despite the ubiquitous nature of sleep,
however, the function or functions of sleep remain unclear (Rechtschaffen 1998; Rattenborg & Amlaner 2002;
Siegel 2005). Some researchers justifiably view the function of sleep as one of the most important unanswered
questions in biology (e.g. Krueger & Obál 2002).
The function of sleep is unknown not for lack of interest
in the topic. In fact, many functions have been proposed
Correspondence: S. L. Lima, Department of Ecology and Organismal
Biology, Indiana State University, Terre Haute, IN 47809, U.S.A.
(email: [email protected]). N. C. Rattenborg is at the Max
Planck Institute for Ornithology, Seewiesen, Postfach 1564, D-82305
Starnberg, Germany.
0003e 3472/07/$30.00/0
for sleep (Rechtschaffen 1998; Siegel 2005). Although varied and eclectic, most explanations for the function of
sleep are implicitly or explicitly based on one overriding
idea: sleep puts an animal in a particularly vulnerable
state; hence, there must be a good reason to sleep. Following Horne (1988) and Rechtschaffen (1998), these proposed functions might be categorized as related to the
body or brain. Hypotheses related to the effect that sleep
has on the body address factors such as energy conservation (Berger & Phillips 1995; Zepelin 2000), immune system function (Majde & Krueger 2005) and safety
(Meddis 1975; Webb 1975). Those related to the brain
deal one way or another with the possible maintenance
or restorative effects of sleep. Such ‘brain’ hypotheses
address metabolic activities (Benington & Heller 1995;
Basheer et al. 2004; Gip et al. 2004), or the maintenance
of synaptic function (Kavanau 1996; Krueger & Obál
2002; Cirelli 2005; Ganguly-Fitzgerald et al. 2006; Tononi
& Cirelli 2006), including neurogenesis (Guzman-Marin
et al. 2005) and memory consolidation and enhancement
(Stickgold & Walker 2005; Steriade 2006; Tononi & Cirelli
189
Ó 2007 The Association for the Study of Animal Behaviour. Published by Elsevier Ltd. All rights reserved.
190
ANIMAL BEHAVIOUR, 74, 2
2006). With ongoing applications of functional genomics
to the study of sleep, we will undoubtedly see additional
maintenance-related hypotheses (see Tafti & Franken
2002; Mackiewicz & Pack 2003; Cirelli 2005). All of the
proposed functions have their adherents and detractors,
but there appears to be a consensus among sleep researchers that the primordial function of sleep is related
to neural maintenance (see Rattenborg & Amlaner 2002;
Siegel 2003; Cirelli 2005).
Two problems combine to obscure the function of sleep.
The first is the simple fact that an animal does nothing
(outwardly) while it sleeps. Thus, unlike other behaviours
that are obviously directed towards a clear goal (e.g. food
acquisition, mating, thermoregulation, etc.), a sleeping
animal does not readily indicate its ‘intent’ in engaging in
sleep. This situation has naturally led to the development of
techniques to probe the physiological and neural dynamics
of the sleeping brain for insights into the function of sleep.
These techniques have shown that the brain is hardly
‘turned-off’ during the unconsciousness of sleep, and that
many processes are taking place in the sleeping brain
(Steriade 2006). Many sleep researchers suspect that the
function of sleep will be realized only from studies at the
physiological or molecular/biochemical level (Rechtschaffen 1998; Benington 2000; Krueger & Obál 2002).
The second problem in understanding the function of
sleep is that many of the proposed functions would appear to
be achievable in the waking state as well (see Moorcroft 1995;
Rechtschaffen 1998). In other words, it is not clear why an
animal needs to be in such a vulnerable and unresponsive
state to achieve the various functions posited for sleep.
Some hypotheses, however, do address the reason for a behavioural shutdown. Some neural maintenance hypotheses
(Kavanau 1996; Krueger & Obál 2002; Tononi & Cirelli 2006)
posit that such a state enhances synaptic maintenance. The
immobilization hypothesis (Meddis 1975) states that shutdown sleep is simply a safer way to pass unproductive time
than is being awake and more active/detectable to predators.
This hypothesis is unique in suggesting that sleep is the safer
state (see also Lima et al. 2005).
We take an approach to explain the function of sleep that
focuses more on the second problem than the first, namely
the reason for a complete behavioural shutdown during
sleep. In other words, rather than using the traditional
approach of taking shutdown sleep as a given and then
attempting to explain function, we will assume a (general)
function, neural maintenance, and attempt to explain the
behavioural shutdown. Our approach is motivated by
a simple question. Why not take only a small portion of
the brain offline to sleep while keeping the rest of the brain
awake to achieve some degree of safety while sleeping? In
attempting to answer this question, we develop the first
formal evolutionary/strategic model to address the function of sleep, or, more specifically, the question of why the
vulnerable behavioural shutdown exists.
BASIC MODEL
We develop a model that suggests that shutdown (unconscious) sleep may often be the safest way to achieve
neural maintenance. Sleep shutdowns are enigmatic precisely because sleep compromises predator detection
(Lendrem 1984; Dukas & Clark 1995; Rattenborg et al.
1999; Gauthier-Clerc et al. 2002; Mathews et al. 2006;
see also Anderson 1998; Caro 2005; Lima et al. 2005),
hence we focus our model on sleep and predator avoidance. Furthermore, we take the position held by at least
the plurality of sleep researchers and assume that sleep
functions to enhance neural maintenance (Krueger &
Obál 2002; Siegel 2003, 2005; Hobson 2005; Tononi &
Cirelli 2006; see also Lesku et al. 2006). We assume that
an animal’s evolutionary ‘goal’ is to maximize its Darwinian fitness, which in our simple model is the equivalent of
maximizing survival. Our model is thus a strategic one in
the tradition of (phenotypic) evolutionary modelling
(Stephens & Krebs 1986; Mitchell & Valone 1990; Houston
& McNamara 1999; Brown 2001) based on the strategic
advantage of a given degree of sleep. This model is thus
not a mechanistic model, such as the ‘two process model’
used to predict the dynamics of sleep states (Borbély &
Achermann 2000). Our model also greatly simplifies brain
structure to clearly present the basic strategic principles
underlying our ideas. The general conceptual results outlined below, however, are nevertheless applicable to many
hypotheses about the maintenance-related functions of
sleep.
We assume that the brain is composed of distinct (but
unspecified) neural units that are linked to other such
units to perform various functions (see Fingelkurts et al.
2005 for related discussion). We do not specify the nature
of the neural units themselves, but they could be organized at any level from entire hemispheres down to neurons. We assume further that these units can sleep
separately from other units. Sleep in a given unit would require being taken offline much as envisioned by Krueger
& Obál (1993, 2002). We acknowledge that some degree
of neuronal synchrony gives rise to the EEG waves that
characterize nonrapid eye movement (NREM) sleep in
mammals and birds (Massimini et al. 2004; Rattenborg
2006; Steriade 2006), but there is also evidence that
NREM sleep develops independently in separate neural
units. For instance, in ‘drowsy’ monkeys (Macaca fascicularis), neurons in one portion of the cortex may show sleeplike activity while others remain awake and able to control
goal-directed behaviour (Pigarev et al. 1997). Sleep in separate neural units may also occur in various human sleep
disorders in which the boundary between sleep and wakefulness is blurred (Mahowald & Schenck 2001, 2005). Furthermore, recent work suggests that the intensity of NREM
sleep may differ between cortical areas depending on recent regional brain activity (Huber et al. 2004; Vyazovskiy
et al. 2004; Rector et al. 2005).
Assume a brain composed of N such neural units. For
simplicity and for mathematical tractability (but without
any loss of generality), assume that the brain sleeps n units
at a time (1 n N ). The proportion of the brain asleep at
a given time is thus p ¼ n/N. Shutdown sleep is indicated
by n ¼ N or p ¼ 1, whereas minimal sleep is indicated by
n ¼ 1 (this state would probably not be identified as sleep
per se). Assume that a given neural unit must be offline for
t units of time for maintenance. A sleeping unit cannot
LIMA & RATTENBORG: FUNCTION OF SLEEP
The probability of death given an attack while in the
sleeping state, ps(djA), is assumed to be a function of the
number of neural units that are awake or ‘online’. For simplicity, we initially assume (conservatively) that the probability of detecting and surviving attack is given by the
proportion of neural units that are online: (N n)/N.
Thus, the probability of death given an attack, ps(djA), is
given by 1 (N n)/N. The overall probability of survival
is therefore
PðsurvivalÞ ¼ exp as ð1 ðN nÞ=NÞTs
aa pa ðdjAÞTa
ð2Þ
We assume that pa(djA) ¼ 0 and that ps(djA) converges towards 0 for small n; that is, fully alert animals are safe from
predation (we modify this assumption below). Given this,
the probability of survival reduces to
PðsurvivalÞ ¼ expð as ð1 ðN nÞ=NÞTs Þ
ð3Þ
The important trade-off here is apparent. The time spent
sleeping Ts ¼ Nt/n decreases to a minimum at n ¼ N (complete shutdown sleep), but that time is also spent maximally
vulnerable to predatory attack ( ps(djA) ¼ 1). Surprisingly,
however, it can easily be shown that in equation (2), the
probability of survival is actually independent of n. That
is, equation (2) simplifies to P(survival) ¼ exp(ast). Thus,
given the assumptions underlying equation (3), it matters
little whether sleep is concentrated all into one period or
spread out over a much longer time.
This result of ‘indifferent’ sleep reflects the implicit
assumption that the N neural units function independently of one another. In other words, if the brain is
only 50% online, then it suffers only a 50% reduction of
functional capacity. Of course, the essence of brain
function is the interconnectedness of neural units, which
means that a 50% reduction in online units will generally
lead to a greater than 50% loss of functional capacity. This
effect can be represented simply by expressing the probability of surviving attack while partially asleep as ((N n)/
N )c. The parameter c determines the degree of loss in functional capacity as units are taken offline for maintenance;
c > 1 indicates this effect, with greater values having
a larger suppressive effect on functional capacity (Fig. 1).
Note that c < 1 indicates the unlikely situation in which
the online units function better when other units are offline. With the expression ((N n)/N )c, the overall equation for the probability of avoiding predation becomes
PðsurvivalÞ ¼ expð as ð1 ððN nÞ=NÞc ÞTs Þ
ð4Þ
The assumption of interdependence between units (c > 1)
leads to complete shutdown sleep as the optimal solution;
that is, all units sleep at the same time, thus n* ¼ N or
p* ¼ n*/N ¼ 1. This result holds for all c > 1, and is shown
for a few cases in Fig. 2; the greater the value of c, the stronger the relative benefit of shutdown sleep. An informal
proof that P(survival) is always maximal at n ¼ N (or
p ¼ 1) for c > 1 can be made with reference to Fig. 2.
When c ¼ 1, P(survival) ¼ exp(ast), which is independent
of n. With c > 1, the absolute value of the exponent in
equation (4) will exceed ast, hence P(survival) < exp(ast)
for all n < N and c > 1. However, when n ¼ N, P(survival) ¼ exp(ast) once again, regardless of the value of c.
Since exp(ast) is the maximum value of P(survival), it
follows that P(survival) is always maximal at n ¼ N when
c > 1. To put this graphically, P(survival) curves will increase with n and intersect at n ¼ N as in Fig. 2. We note
that this basic result of shutdown sleep will hold for any
function describing the probability of surviving attack
such that the effect of greater interconnectivity of brain
units suppresses predator detection below the level expected on the basis of the proportion of units awake (as in
Fig. 1). This shutdown result will also hold even if sleep in
a small proportion of units has little suppressive effect on
1
Probability of surviving attack
perform its normal functions. Let Ttot be the total time
available for sleep or nonsleep activities. We make no distinction between night and day in our model so that the
animal can be usefully awake at any time if that would
be the favoured solution. The total time spent in any degree of sleep (n 1) is Ts ¼ Nt/n. Ts cannot exceed Ttot,
which implies the condition that n Nt/Ttot. The time
spent fully awake is simply Ta ¼ Ttot Ts.
Our model is loosely based on strategic models of
antipredator vigilance or predator detection (Lima 1987;
McNamara & Houston 1992; see also Beauchamp 2003;
Fernández-Juricic et al. 2004). We specifically seek an
equation for the probability of surviving the period Ttot.
We assume that predatoreprey encounter rate can be
described as a Poisson process in which an encounter
can occur in any small time interval with equal probability. Under that assumption, the probability of survival has
the form exp(aipi(djA)Ti) (McNamara & Houston 1992),
where ai is the attack rate while in state i ¼ s (sleep,
n 1) or i ¼ a (awake), pi(djA) is the probability of death
given an attack, and Ti is the total time spent in state i.
Since the animal must survive both time spent awake
and asleep, the overall probability of survival over Ttot is
exp(asps(djA)Ts)exp(aapa(djA)Ta), which rearranges to
ð1Þ
PðsurvivalÞ ¼ exp as ps ðdjAÞTs aa pa ðdjAÞTa
0.8
0.6
1
2
0.4
4
8
0.2
16
0
0
0.2
0.4
0.6
0.8
1
Proportion of units awake
Figure 1. Probability of surviving attacks as a function of the proportion of neural units that are awake ((N n)/N ) for values of c as
shown. Increasing values of c have the effect of suppressing the
probability of surviving an attack even when much of the brain is
awake.
191
ANIMAL BEHAVIOUR, 74, 2
0.10
0.8
1
0.08
2
0.6
αs
P (survival)
1
0.06
2
4
4
0.04
0.4
8
8
16
0.02
0.2
0
5
10
n
15
20
0
0.2
0.4
0.6
Proportion asleep
0.8
1
Figure 2. Overall probability of surviving the time period Ttot as expressed in equation (4). Numbers beside curves represent values of c.
For c > 1, survival increases with an increase in the number of units
offline and sleeping (n). For c ¼ 1, survival is constant and independent of n. Other parameter values: N ¼ 20, t ¼ 10, T tot ¼ 100,
as ¼ 0.02.
Figure 3. Increase in attack rate while sleeping (as) as a function of
the proportion of neural units offline and sleeping (n/N ) as expressed in equation (5). Numbers besides curves indicate values of
b. Increasing values of b delay the increase in as until progressively
more of the brain is asleep. Other parameter values: N ¼ 20,
as,min ¼ 0.02, as,max ¼ 0.10.
predator detection, so long as an effect similar to that seen
in Fig. 1 holds for most degrees of sleep.
This maximal form of sleep may seem rather drastic
since staying awake to some extent (while sleeping) is
safer than the complete shutdown. However, in the bigger
temporal picture, it is safer overall to consolidate sleep
into one single block and maximize the time spent awake
to deal with predators. To put it differently, shutdown
sleep is the safest way to deal with the fact that neural
units must be taken offline for maintenance in an
interconnected and interdependent brain.
the increase in as with n should be gradual for low n and
increase more rapidly as n approaches N. A simple function for as with much flexibility in shape is
EXTENSIONS OF THE BASIC MODEL
Maximal sleep ( p* ¼ 1) is a robust outcome of the model
developed so far. Here, we explore some variations of the
basic model in which less-than-maximal sleep may be favoured. The two model variants below are based on opposing assumptions about the antipredatory consequences of
differing degrees of sleep.
Situations Favouring Partial Sleep
We show here that a significant increase in attack rate as
n increases will often favour some form of ‘partial’ sleep
(1/N < p* < 1). Animals that sleep in exposed situations
might be obviously vulnerable when in deeper states of
sleep, and thus experience higher attack rates as n increases. For instance, some large herbivores apparently
must lay down when in deep sleep, or may pass from partial eye closure in shallow sleep to full eye closure in deep
sleep (Ruckebusch 1972). Predators could easily take
advantage of such information and attack accordingly
(FitzGibbon 1989; Krause & Godin 1996).
Equation (4) still applies in this model variant. The
main difference with the foregoing model is that as is no
longer fixed but increases with n (vas/vn > 0). We impose
the restriction that v2as/vn2 > 0, reflecting the idea that
as ¼ ðas;max as;min Þðn=NÞb þ as;min
ð5Þ
where as,min is the minimum attack rate (for very small n),
as,max is the maximum attack rate experienced when
n ¼ N (maximal sleep), and n/N is the proportion of the
brain that is offline at a given time. The parameter b determines the convexity of the relationship (see Fig. 3); as b increases, the major increase in as is delayed until larger
values of n/N.
Partial sleep is a prominent outcome in this model
variant (Fig. 4). An increase in b generally favours an increase in the average proportion of the brain that is offline
1
Optimal proportion asleep
192
0.8
0.6
8
0.4
6
4
0.2
0
0
2
5
10
b
15
20
Figure 4. Optimal proportion of neural units asleep as a function of
the parameter b in equation (5). This parameter dictates the concavity of the relationship between the attack rate while sleeping (as) and
the proportion of neural units sleeping as per Fig. 3. Numbers beside
curves indicate values of c. Other parameter values: N ¼ 20, t ¼ 10,
Ttot ¼ 100, as,min ¼ 0.02, as,max ¼ 0.10.
LIMA & RATTENBORG: FUNCTION OF SLEEP
Situations Favouring Minimal Sleep
The reintroduction of aa into the analysis also leads to
considerable minimal sleep, and another route through
which p* is a function of c. We have so far assumed that
pa(djA) ¼ 0 and thus the value of aa did not enter into
the picture. Of course, for most animals pa(djA) > 0.
With this assumption the full model becomes
PðsurvivalÞ ¼ exp as ð1 ððN nÞ=nÞc ÞTs
aa pa ðdjAÞTa
ð6Þ
The n maximizing equation (6) takes on only two values:
minimal sleep (n* ¼ 1 or p* ¼ 1/N ) or maximal sleep
(n* ¼ N ). As one might expect, minimal sleep is favoured
when aa > as, whereas maximal sleep is favoured when
aa and as are more comparable in magnitude or when
as > aa. An increase in c also has the effect of favouring
maximal sleep over minimal sleep, since the former is favoured when there is little chance of detecting attack while
asleep. Below, we focus on the case of aa > as to explore
situations with much minimal sleep.
The existence of minimal sleep in maximizing equation
(6) reflects not only the condition that aa > as, but also the
assumption that as applies to a sleeping animal regardless
of the proportion of neural units that are offline. Under
these circumstances, minimal sleep represents a solution
in which maximizing time asleep maximizes safety. However, since minimal sleep is more likely to resemble waking than it is sleeping, it seems more reasonable to
assume that as increases towards aa as n (or p) decreases.
Such an assumption is at the heart of the immobilization
hypothesis for the function of sleep (Meddis 1975, 1977;
see also Lima et al. 2005). This hypothesis holds that sleep
has no maintenance function, but simply prevents an animal from responding to various stimuli and thus attracting the attention of predators. We have shown elsewhere
(Lima et al. 2005) that the immobilization hypothesis can
explain shutdown sleep under certain circumstances, and
it has some bearing on the present model.
We assume here that as will be minimal (at as,min) when the
animal is maximally asleep (n ¼ N ), and that as increases at
an accelerating rate as n approaches 0; thus vas/vn < 0 and
v2as/vn2 > 0. A useful function meeting these criteria is
as ¼ ðaa as;min Þð1 n=NÞd þ as;min
ð7Þ
which is a modification of equation (5). As before, the exponent (d ) determines the convexity of the relationship between as and n/N; as d increases, the accelerated increase in
as occurs at progressively smaller values of n/N (see Fig. 5).
Even with aa > as, maximal and partial sleep are likely
outcomes under the above assumptions, but minimal
sleep is still prominent (Fig. 6). Here, both as and aa are
nonzero, but the results depend only on the ratio of attack
rates aa/as,min (see also Lima & Bednekoff 1999). For all attack ratios, two general trends are apparent (Fig. 6). First,
an increase in d leads to a drop in p* (¼n*/N ). This result
again reflects the benefits of maximizing time in the safer
state of sleep, and the fact that an increase in d means that
the increase in as occurs at increasingly smaller values of p
(Fig. 5). Generally speaking, d must be fairly large to effect
minimal sleep. Second, an increase in c, or a decrease in
the ability to detect attack while partially asleep, leads to
a greater degree of sleep as seen earlier. The general effect
of an increase in the attack ratio aa/as,min is to decrease the
degree of sleep, again reflecting the safer state of sleep. Furthermore, at low attack ratios, the degree of sleep follows
a step function from maximal to minimal sleep (with increasing d, Fig. 6a). These step functions give way to
a more gradual decrease in the degree of sleep as the attack
ratio increases (Fig. 6aec).
GENERAL DISCUSSION
An understanding of the function of sleep will require
a two-pronged approach. First, we need to know what is
0.020
0.016
0.012
1
αs
( p* ¼ n*/N ). Recall that higher values of c represent a decreasing ability to detect and respond to attack while in
a state of partial sleep. Relatively safe sleep (low c values)
favour minimal sleep when b is low (i.e. when b is small,
an increase in n will lead to a marked increase in as). However, maximal sleep ( p* ¼ 1) is favoured for low b when
predators are relatively undetectable when asleep (higher
c values). Beyond these low values of b, p* values converge
to an increasing function of b. This latter result reflects the
value in shutdown sleep outlined in the basic model. As
b increases, as is effectively constant over a large range
of n/N (Fig. 3), which favours an increase in n. This effect
is eventually countered by a rapid increase in as. The increase in p* with b reflects the fact that this rapid increase
occurs at progressively larger values of b. Finally, we note
that the existence of minimal sleep (as in Fig. 4) reflects
a relatively large value of as,max. A smaller difference between as,max and as,min (equation 5) both favours an overall increase in p* and largely eliminates p* curves
originating at minimal sleep.
2
0.008
4
8
0.004
16
0
0
0.2
0.4
0.6
Proportion asleep
0.8
1
Figure 5. Decrease in attack rate while sleeping (as) as a function of
the proportion of neural units offline and sleeping (n/N ) as expressed in equation (7). Numbers beside curves indicate values of
d. An increase in d delays the increase in as until a progressively
smaller proportion of the brain is asleep. Other parameter values:
N ¼ 20, as,min ¼ 0.002, aa ¼ 0.02.
193
ANIMAL BEHAVIOUR, 74, 2
1
(a)
0.8
0.6
4
2
0.4
0.2
1
0
1
Optimal proportion asleep
194
(b)
0.8
0.6
4
2
0.4
0.2
1
0
1
(c)
0.8
0.6
4
2
0.4
0.2
0
0
1
5
10
15
20
d
Figure 6. Optimal proportion of neural units asleep as a function of
the parameter d in equation (7). This parameter dictates the concavity of the relationship between the attack rate while sleeping (as,min)
and the proportion of neural units sleeping as per Fig. 5. The attack
ratio (aa/as,min) varies across panels as follows: (a) 5, (b) 7.5, and
(c) 10. Values shown in the figures indicate values of c. Other parameter values and variables: N ¼ 20, t ¼ 10, Ttot ¼ 100, as,min ¼ 0.002,
aa ¼ 0.20, and pa(djA) ¼ 0.10.
going on within the brain during sleep. Second, we need
to know why those specific tasks would best be accomplished via the unconsciousness of a behavioural shutdown. To address this second issue, we have chosen to
take the approach of assuming that some sort of neurological maintenance is occurring in the brain during sleep
(Krueger & Obál 2002; Stickgold & Walker 2005; Steriade
2006; Tononi & Cirelli 2006). Specifically, we assumed
that a given neural unit of the brain must go offline for
maintenance. We then explored the factors that favour
shutdown sleep or work against that option. We focused
on the options that minimize the risk of predation, since
predation is undoubtedly a significant cost of sleep (in
an evolutionary sense). Furthermore, our simple models
removed most restrictions on sleep time in an effort to
explore fully the consequences of various forms of sleep
for predator evasion.
Our main insight is that maximal or shutdown sleep is
often the safest way to perform the assumed maintenance
required by neural units. This is somewhat counterintuitive because a shutdown is clearly the form of sleep that
leaves an animal most vulnerable to predatory attack.
Partial sleep, during which a subset of units goes offline,
might seem to be a better and safer option. However,
interconnected functionality between neural units means
that the brain functions disproportionately more poorly
as more units go offline for maintenance. Furthermore,
partial sleep can be a prolonged affair as time is needed to
cycle through maintenance for all neural units. A prolonged period of less-than-effective antipredator functionality may not be a favoured solution. Thus, the overall
safest way to sleep is often to have all neural units offline
and engaged in maintenance at the same time. Quick and
efficient maintenance can be seen in this light as a major
function of shutdown sleep (see also Moorcroft 1995).
This result is analogous to that from models of antipredator behaviour in which zero vigilance (maximal feeding)
can be favoured when vigilance scanning is not very effective at detecting attack (Lima 1987; Brown 1999).
Maximal sleep is not always the optimal way to sleep
under the risk of predation. Partial sleep (in which some
neural units are online while others are offline) is
a prominent aspect of sleep when we assumed that the
risk of attack increases as more units are offline. Even
under these assumptions, however, maximal sleep is
observed if the increase in attack rate is not too great, or
c (the interconnected-functionality parameter) is large.
Minimal sleep (with only one neural unit offline at a given
time) is a more prominent outcome when the attack rate
while awake is substantially higher than when asleep.
The key here is that maximizing time in the low-attackrate mode of sleep will overwhelm the above benefits of
maximal sleep. Maximal or partial sleep can be reinstated
by assuming that the attack rate while asleep rises to that
while awake as fewer neural units are offline (or as relative
wakefulness increases). As before, partial or maximal sleep
becomes a more likely outcome as c increases.
In most mammals, sleep is typically viewed as a largely
unconscious, whole brain affair (Rattenborg et al. 2000;
Krueger & Obál 2002; Massimini et al. 2005; but see below), which would most obviously correspond to our state
of maximal sleep. Partial sleep may occur to some extent
in primates, but it is largely a transitional state between
wakefulness and sleep (Pigarev et al. 1997), or associated
with pathological states (Mahowald & Schenck 2001,
2005). This apparent lack of partial sleep suggests that,
for most mammals, in the context of our model, c is effectively quite high and sleep is inherently more dangerous
than being awake. It seems likely that the effective c of
the brain is greatly underestimated by our simple
LIMA & RATTENBORG: FUNCTION OF SLEEP
modelling approach, in which we effectively assumed
much functional redundancy among units regarding predator detection. It does, however, seem unlikely that all
mammals would fall into the range of parameters favouring maximal sleep. In particular, sleep may well be
a lower-attack-rate option for many mammals that sleep
in secure burrows, a situation that would favour partial
or minimal sleep. Regarding the relative danger of sleep
versus wakefulness, few data exist with which to make
such an assessment (Caro 2005; Lima et al. 2005).
Partial sleep of a sort does exist in birds. We have shown
(Rattenborg et al. 1999, 2000) that birds sleep unihemispherically to some extent, especially when the perceived
risk of predation is relatively high. Unihemispheric sleep
has also been observed in cetaceans and a few other
aquatic mammals (Lyamin et al. 2004; Siegel 2005), and
might exist in some reptiles (Rattenborg et al. 2000).
This unihemispheric form of partial sleep may ultimately
reflect the fact that the cerebral hemispheres in birds are
not as functionally integrated as they are in most mammals (Vallortigara & Rogers 2005). In the context of our
model, unihemispheric sleep may represent a low c condition between neural units (cerebral hemispheres), which
would favour partial sleep. In cetaceans, the comparatively small size of the corpus callosum (the main route
for interhemispheric communication in mammals) is
also consistent with this idea (Tarpley & Ridgway 1994;
Rattenborg et al. 2000).
An important question here is what constitutes observable partial sleep. Clearly, the sleeping brain is not
completely ‘off’. Threatening and other meaningful stimuli above a certain threshold can quickly arouse the brain
to the waking state (see Coenen 1998), so some neural entities must be awake in some sense to monitor the environment. We also know that the stimulus intensity
needed to terminate sleep is lower in certain states of
sleep: does this fact imply that these more ‘vigilant’ states
of sleep (Lima et al. 2005) qualify as some sort of partial
sleep? Perhaps some degree of partial sleep is the norm
in the sleeping brain.
Minimal sleep (with just a single or small number of
offline units) is an interesting theoretical possibility, but
there is no obvious empirical support for its existence
(such a form of sleep might not even be identified as sleep
per se). Minimal sleep in our model reflects the assumption that a given unit can undergo maintenance in a fixed
period of time (t) independent of other units. It is perhaps
more realistic to assume that maintenance becomes more
efficient as more neural units are taken offline for maintenance. Such an effect can be incorporated into the above
models by assuming that t is a decreasing function of the
number of units offline. Under this assumption, shutdown sleep is an almost uniform outcome, and may
hold even if c < 1 to some extent. Furthermore, if sleep
functions to maintain the neural connections between
the neural units themselves (e.g. Krueger & Obál 2002),
then minimal sleep is clearly not a likely outcome.
Our simple model is apparently the first formal evolutionary/strategic model to address the function of sleep.
This lack of theoretical attention reflects two simple facts.
First, beyond a few studies on antipredator vigilance in
sleeping animals (e.g. Lendrem 1984; Rattenborg et al.
1999; Gauthier-Clerc et al. 2002), behavioural ecologists
have paid little attention to sleep as a behavioural phenomenon (see also Elgar et al. 1988; Dukas & Clark
1995; Christe et al. 1996). For instance, models of daily activity patterns make no differentiation between rest and
sleep, both of which occur when nothing else can be
done (e.g. McNamara et al. 1994; but see Clark & Dukas
2000; Pravosudov & Lucas 2000). On the other side of
things, students of sleep are typically clinically oriented
or highly mechanistic in their approach to the subject,
and few have considered the strategic (evolutionary) consequences of sleep as a behaviour. Some sleep researchers
have even argued that the function of sleep cannot be understood without a detailed mechanistic understanding of
what goes on during sleep at the cellular and molecular
level (see Rechtschaffen 1998). Perhaps this sentiment
will prevail, but if so, then sleep would uniquely be the
only trait in the animal world for which strategic evolutionary considerations would be without insight. We believe that an integration of this traditionally mechanistic
approach with a more strategic approach will ultimately
be more successful in elucidating the function of sleep.
We offer this simple model as a first step in that direction.
Acknowledgments
We thank J. Lesku and C. Amlaner for helpful comments
and discussions. We also thank the Department of Ecology
and Organismal Biology at Indiana State University for
support. N.C.R. was supported by the Max Planck Society.
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