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How to Achieve Fast Entrainment? The Timescale to
Synchronization
Adria
´n E. Granada*, Hanspeter Herzel
Institute for Theoretical Biology, Humboldt-Universita
¨t zu Berlin, Berlin, Germany
Abstract
Entrainment, where oscillators synchronize to an external signal, is ubiquitous in nature. The transient time leading to
entrainment plays a major role in many biological processes. Our goal is to unveil the specific dynamics that leads to fast
entrainment. By studying a generic model, we characterize the transient time to entrainment and show how it is governed
by two basic properties of an oscillator: the radial relaxation time and the phase velocity distribution around the limit cycle.
Those two basic properties are inherent in every oscillator. This concept can be applied to many biological systems to
predict the average transient time to entrainment or to infer properties of the underlying oscillator from the observed
transients. We found that both a sinusoidal oscillator with fast radial relaxation and a spike-like oscillator with slow radial
relaxation give rise to fast entrainment. As an example, we discuss the jet-lag experiments in the mammalian circadian
pacemaker.
Citation: Granada AE, Herzel H (2009) How to Achieve Fast Entrainment? The Timescale to Synchronization. PLoS ONE 4(9): e7057. doi:10.1371/
journal.pone.0007057
Editor: Raya Khanin, University of Glasgow, United Kingdom
Received April 28, 2009; Accepted August 6, 2009; Published September 23, 2009
Copyright: ß2009 Granada, Herzel. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the Deutsche Forschungsgemeinschaft (SFB 618) and the EU-network Biosimulation. http://www.biologie.hu-berlin.de/
forschung/SFB_618/ and http://biosim-network.eu/biosimulation_/ The funders had no role in study design, data collection and analysis, decision to publish, or
preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: a.granada@biologie.hu-berlin.de
Introduction
Biological rhythms are ubiquitous in nature and are found in
diverse systems, from spiking neurons to animal populations with
periods ranging from milliseconds to years. Our everyday life
exhibits many behavioral and physiological oscillations that
interact with the external fluctuating environment. Biological
pacemakers typically interact with other oscillators including for
example coupled rhythms of heart, respiration and movement [1],
vocal fold oscillations [2] and singing duets of birds [3]. These
interactions can lead to mutual synchronization as in the collective
blinking of fireflies [4] and entrainment in which oscillators
synchronize to a common signal. An example of this is the left and
right birdsong control nuclei HVc that show synchronization in
the absence of interhemispheric connections [5]. Another example
is the entrainment of plant-leafs movements to the light-dark and
cold-warm cycles [6]. Complex interactions between multiple
oscillators are observed in the mammalian suprachiasmatic
nucleus (SCN), where mutual synchronization and entrainment
are combined. These tiny nuclei situated in the anterior
hypothalamus are responsible for controlling endogenous circadi-
an rhythms. Many different body functions like sleep-wake cycles
and body temperature rhythms are regulated by centrally
generated neuronal and hormonal activities. The SCN consists
of two nuclei of about ten thousand densely packed neurons and
generates a stable robust period of about 24 h. The SCN has the
striking ability of fast reentrainment as observed in jet-lag type
experiments, where after an abrupt phase shift of 6 h, the SCN
can be almost completely reentrained within one cycle [7–9]. Also
from the induced loss of rhythmicity in SCN slices after
application of tetrodotoxin (TTX, a voltage gated sodium channel
blocker), the SCN cells resynchronize within one cycle [10]. When
TTX is applied, the oscillations are lost at a single cell level but
after washing TTX out, the cells start oscillating again in a
synchronized manner after 1 day. Such short transients times are
remarkable, bearing in mind the large number of coupled
oscillators involved and the diversity of their initial conditions
and periods [11,12]. How synchronization and entrainment
mechanisms work within the SCN neurons is one of the main
open problems in the field of circadian rhythms. Furthermore, in
jet-lag and shift work schedules, the reentrainment time is of major
relevance and has been associated to a number of diseases, ranging
from sleep disorders to cancer [13–15].
Several mathematical models of SCN cells have been proposed
with an increasing complexity (using 7 up to 73 differential
equations [16–18]), none of which describes the short reentrain-
ment times in detail. Our goal in this present paper is to unveil the
specific dynamics that can lead to ultrafast entrainment. We
present a generic model to characterize transient times leading to
entrainment. This model is governed by two basic properties of the
oscillator: (a) the radial relaxation timescale and (b) the phase
velocity distribution around the limit cycle. When an oscillator is
perturbed, the radial relaxation timescale determines the rate of
convergence back to the unperturbed amplitude and it can hence
be associated with robustness towards amplitude fluctuations. The
phase velocity distribution determines the waveform of the
oscillation.
Studying the transient time as a function of these two properties
will give us a general understanding of how fast can entrainment
be reached. Those two basic properties are inherent in every
oscillator and, therefore, such a concept can be applied to many
biological systems to predict the transient time to entrainment.
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Even more interestingly, one can infer properties of the underlying
oscillator from the observed transient times as we will show later.
Helpful insights derived from transients of an oscillatory system
have already been applied in heart cells studies [19]. Although we
focus on biological applications, the presented theory can be
applied to many other oscillatory systems undergoing entrainment.
Results
Timescales of entrainment
When a system is entrained, it reaches a stable phase relation
with the external rhythm and thus their phase difference becomes
constant (see Materials and Methods). The transient time it takes
to reach this stable phase relation depends on the initial conditions
(ICs), the entrainment signal and the properties of the oscillator.
An example of these transients for a generic circadian oscillator is
shown in Figure 1. Each initial condition has an associated initial
phase (see gray dots in Figure 1B and C), different initial phases
can lead to big differences in the transient time to entrainment.
Figure 1A shows the time evolution of two initial conditions,
ICslow and ICfast, leading to a long transient (pink) and a short
transient (blue) respectively. This can also be observed in a phase
evolution plot where ICslow needs 8 days to achieve a stable phase
relation whereas ICfast only 1 day (see Figure 1B). The
dependence on initial conditions for a specific circadian oscillator
model has already been studied [20]. A self-sustained oscillator is
Figure 1. Basic mechanisms involved in the entrainment of an oscillator. (A) Time series for two initial conditions, ICslow and ICfast, leading
to a long transient (pink) and a short transient (blue), respectively. The green bars represent the entrainment signal. (B) Phase evolution for both
initial conditions. ICfast entrain after 1 day while IC slow needs 8 days (fig. 1) (C) Oscillator limit cycle representation with 24 marked initial conditions
(gray). (D) Schematic representation of the entrainment region as a function of the entrainment amplitude and period (often termed 1:1 ‘‘Arnold
tongue’’). Gray scale represents different transients to entrainment zones within the entrainment region. The green dots represent the section of the
entrainment region (entrainment range) for a certain entrainment amplitude and in (E) their associated entrainment times are plotted as a function of
the entrainment period. Computational details of A,B and E are given in Materials and Methods.
doi:10.1371/journal.pone.0007057.g001
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able to entrain just to certain combinations of entrainment
amplitudes and periods that define the so-called entrainment
region or 1:1 ‘‘Arnold tongue’’ (see Figure 1D). In other words,
each entrainment amplitude entrains the system within a certain
period range, from a minimum Tmin
ðÞto a maximum Tmax
ðÞ
entrainment period, known as range of entrainment. Typically, at
the borders of this entrainment region the transients leading to
entrainment are much longer than those at the center (see
Figure 1E). In the following we focus on those inherent properties
of the oscillator that determine the transients.
Generic oscillator model
As will be seen, our results indicate that two characteristics of
the oscillator determine the transients: the radial relaxation time
and the phase velocity around the limit cycle. To better illustrate
the dependence of transient times on these two properties, we
introduce a simple model oscillator that can mimic various
oscillators. We use a generic circular oscillator of radius 1 and
period 1 (arbitrary units) so the results can be easily rescaled to
other systems. As a specific model, we introduce a variation of the
Poincare´ oscillator [21], given by
dr
dt ~lrn1{rðÞ,ð1Þ
dw
dt ~fwðÞ~ecos2pw
zoffset:ð2Þ
This oscillator can be smoothly switched from a sinusoidal
shape to a spike-like oscillator, while the radial relaxation can be
independently controlled (see Figure 2). Equation 1 describes the
radial evolution and has a stable orbit at r~1, with a radial
relaxation controlled by the parameters nand l. For n~0the
radial relaxation is exponential and for n§1the radial relaxation
is nonlinear. For l%1the radial relaxation time, tr~1
l, is long
and for l&1the radial relaxation time is short, sometimes referred
to as ‘‘sloppy’’ and ‘‘rigid’’ oscillators respectively (see Figure 2E
and F). Equation 2 describes the phase evolution or, in other
words, the velocity around the limit cycle, where econtrols the
velocity difference between the fastest (w~0) and slowest (w~0:5)
points. The ‘‘offset’’ is a small positive constant and guarantees
that the velocity is never zero (dw
dt=0). For e~0(i.e., dw
dt ~off set),
there are no velocity variations along the limit cycle, and the
oscillator is sinusoidal (see Figure 2A and B). For e&1, we have
large velocity differences along the limit cycle, leading to a spike-
like behavior (see Figure 2C and D).
We use this generic limit cycle model instead of the widely used
phase models because amplitude dynamics will be of fundamental
importance in characterizing transients leading to entrainment. The
phase velocity around the limit cycle determines the temporal shape of
the oscillation (waveform), as illustrated in Figure 2B and D. The radial
relaxation rate ltogether with the degree of nonlinearity controlled by
the parameter ndetermines the timescale of convergence of perturbed
solutions to the limit cycle (see Figure 2E and F). It can be associated
with robustness towards amplitude fluctuations.
This oscillator, a modified Poincare´ oscillator [21], belongs to the
class of radial isochron limit cycles (RILC) due to its radial
symmetry (see Materials and Methods). Many examples of useful
biological insights based on RILC’s can be found elsewhere
[1,21,22]. Here we use the Winfree definition of isochrons as lines
in phase space leading to the same asymptotic phase. Thus all initial
conditions located on the same isochron will reach the limit cycle
with the same phase [23]. The intersections of the isochrons and the
limit cycle trajectory are the temporal phase points (see the dots in
Figure 2A and C). In the case of RILCs, the isochron structure in
the whole phase space can be deduced from the distribution of
temporal phases. Thus a sinusoidal oscillator has equally distributed
phase points and isochrons (see Figure 2A). A spike-like oscillator,
on the other hand, makes a rapid excursion along the fast branch to
spend most of its time at the slow branch. This time scale separation
generates an asymmetric distribution of isochrons at the limit cycle
by compressing them around the slow branch (see Figure 2C). The
isochron distribution will be essential for the general understanding
of the transient time to entrainment. As mentioned, our model was
designed such that the phase velocity around the limit cycle and the
radial relaxation time can be independently controlled to explore
their influence on transients. For clarity, the oscillator will be
rescaled to a period of 1 day and entrained with pulse-like
perturbations of 1 h length. Square waveform oscillators, like the
van der Pol oscillator in the relaxation regime, are not captured by
this fwðÞ. In order to simulate square waveform oscillators a new
fwðÞis introduced (see Figure S1 in Supporting Information).
Median time to entrainment
The time to entrainment depends strongly on the period ratio of
external and internal rhythms (detuning) and on the initial phase
(ICs). The internal period, such as the free-running period in
circadian biology, and phase of the oscillators are typically
unknown or difficult to measure. Therefore, we minimize the
effects of detuning and initial conditions by studying ensembles of
different external periods and initial phases. This allow us to
associate a characteristic STeTwith specific properties of an
oscillator. The median time to entrainment STeTis the median
value of 12 different Tes. We use 12 different external periods
evenly distributed within the range of entrainment to calculate the
median as shown in Figure 1D and E. Additionally, for each
external period, Teis taken as the median time from 24 uniformly
distributed initial temporal phases (see Figure 1D). By taking both
medians, we reduce the dependence on initial condition and
entrainment period significantly (see Materials and Methods).
We start our results discussion with the exponential radial
relaxation case (n~0in Equation 1) and describe the nonlinear case
at the end of this section. For the case of exponential relaxation, we
calculate the time to entrainment STeTfor different oscillator types
using a broad range of values of phase velocity parameters eand of
radial relaxation rates l, such that STeTshows significant variations
(see Figure 3). The entrainment signal was generated with short and
medium-sized square periodic pulses. Specifically, we used 1 h pulse
length with an amplitude of 0.8. This leads to a ‘‘range of
entrainment’’ similar to that observed in rat locomotor activity
under light pulse entrainment 24+2hðÞ[24]. As mentioned above,
the ‘‘range of entrainment’’ refers to the range of periods that a self-
sustained oscillator is capable of entraining by a 1:1 frequency ratio.
According to Figure 3, the longest time to entrainment is found
when the limit cycle has a sinusoidal temporal pattern and if the radial
relaxation time is long (box 1 in Figure 3). The radial relaxation time,
tr~1
l, is in this case much longer than all other involved time scales:
external periods *24 hðÞ, endogenous period (24 h) and pulse
duration (1 h). Such long radial relaxation times allow the entrainment
pulses to considerably perturb the trajectory of the limit cycle, leading
to an expanded entrained orbit (a representative scheme of the
mechanism is shown in Figure 4). Between the pulses, however, the
system has not enough time to relax back to the original unperturbed
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Figure 2. Limit cycle representations in phase space for a sinusoidal and spike-like oscillator together with their time series. (A)
Sinusoidal oscillator: limit cycle with 12 marked phase points (dots) and isochrons (rays). The intersection of each isochron with the limit cycle
determines the phase. (B) Temporal evolution of xvariable with parameters e~0,offset~0:02. (C) Spike-like oscillator, where most isochrons are
concentrated in a small region of the limit cycle. (D) Temporal evolution of the xvariable with parameters e~16,offset~0:02. Representations of both
oscillators (sinusoidal or spike-like) with short radial relaxation time (E) and long radial relaxation time (F) are also shown. Computational details are
given in Materials and Methods.
doi:10.1371/journal.pone.0007057.g002
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orbit. A sinusoidal oscillator (e~0) implies equally distributed isochrons
along the limit cycle and thus all isochrons diverge symmetrically from
thelimitcyclecenter.Whilethelimitcycleexpands,theisochrons
spread apart, so phase changes induced by the same pulse size
decrease. The phase change induced by a single pulse can be deduced
from the difference between the starting isochron, where the
perturbation starts, and the final isochron, where the trajectory is
located after a given perturbation (see pink arrows in Figure 4). The
combination of limit cycle expansion and equally distributed isochrons
reduces the effect of each pulse. Very few isochrons are crossed leading
to smaller phase changes per pulse. Consequently, it a rather long time
to reach the final stable phase. For illustrative purposes we use vertical
pulses in Figure 4, but similar features are observed with other types of
entrainment signals.
Surprisingly, the median time to entrainment STeTcan be
reduced up to 12-fold in our parameter range by changing
independently eor l. Keeping l%1but increasing esmoothly,
changes the sinusoidal waveform oscillator into a spike waveform
oscillator (see box 4 in Figure 3). The spike-like oscillator is also
known as relaxation oscillator due to its fast and slow branches.
Figure 3. Median times to entrainment SSSTeTTT as a function of the phase velocity distribution around the limit cycle eeeee and the radial
relaxation constant llllllll.Gray scale encodes the time to entrainment where black represents long STeTand white represents short STeT. Both axes
are plotted on logarithmic scales and n~0in Equation 1. Computational details are given in the Materials and Methods.
doi:10.1371/journal.pone.0007057.g003
Figure 4. Representative sinusoidal and spike-like limit cycles with long radial relaxation time. Isochrons are represented as thin rays
and perturbation pulses as pink arrows. (A) Unperturbed sinusoidal limit cycle trajectory (dashed small circle) and the expanded entrained limit cycle
(solid large circle) for the sinusoidal oscillator. Initially the pulse generates a phase change up to 4 h, but later the pulse phase shift is reduced to less
than 0.5 h. (B) Unperturbed spike-like limit cycle trajectory (dashed small circle) and the expanded entrained limit cycle (solid big circle) for the spike-
like oscillator. Initially the pulse generates phase advances up to 14 h and, after some pulses, the phase shifts are still 8 h. The original limit cycle
(r0~1) expands here about 7 times. The lower panels show the characteristic time series pattern of a sinusoidal and spike-like oscillator.
doi:10.1371/journal.pone.0007057.g004
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The oscillator spends most of its time on the slow branch, so most
stimuli are received on this branch. Spike-like oscillators have
lower isochron divergence angles from the origin (see Figure 4B).
This small isochron divergence allows considerable phase shifts of
pulses despite the expansion of the limit cycle. This isochron
clustering and, consequently, their low divergence angles allow the
system to reach the final stable phase much faster.
AsshowninFigure3,anincreaseintherelaxationratelleads to a
drastic reduction in the median transient time STeTas well. In this
case, the radial relaxation time is much shorter than the period keeping
the trajectory to the unperturbed limit cycle with keeping the trajectory
to the unperturbed limit cycle with r0~1.Thuspulsesinduce
considerable phase shifts for every given pulse and phase shifts are not
reduced due to limit cycle expansion (compare box 2 in Figure 3).
The spike-like oscillator with short radial relaxation time is
optimal as far as time to entrainment is concerned, because
isochrons are concentrated in the slow branch without suffering
from limit cycle expansion (see box 3 in Figure 3). In this case,
perturbations induce large phase jumps, leading quickly to a stable
phase from almost any initial condition.
In the present example, we used entrainment pulse amplitudes
of 0.8, but qualitatively similar results are also observed with
smaller amplitudes and also with sinusoidal perturbations (see
Figure S2 in Supporting Information). As shown in Figure S2, the
entrainment signal amplitude and waveform do not play a major
role in determining the transient to entrainment.
Oscillators with highly nonlinear radial relaxation exhibit a much
shorter median time to entrainment STeTas shown in Figure S2C
and D in Supporting Information. This property is captured by our
model using n~1,2,3 ... in Equation 1. The normal form of limit
cycles arising via supercritical Hopf bifurcations corresponds to
n~2. Due to this strong nonlinearity, perturbations to the limit
cycle trajectory relax rapidly back to the unperturbed limit cycle
reducing considerably the limit cycle expansion effect (see Figure
S2D). In the following, we relate our theory to a specific biological
rhythm to gain insight into the properties of the system.
Fast entrainment in the mammalian circadian pacemaker
Physiological and behavioral processes in most organisms are
synchronized with a 24 h day-night rhythm. Mammals have a
central pacemaker located in the hypothalamic suprachiasmatic
nucleus (SCN) that orchestrates circadian rhythms for the whole
body. The SCN consists of two nuclei of about ten thousand
densely packed neurons and generates a stable robust period of
circa 24 h. This stable neuronal and hormonal rhythm regulates
many different body functions. Cells within the SCN have an
endogenous molecular clock based on a network of interlocking
feedback loops of genes and proteins. The intercellular coupling
between individual neurons generates not only a robust 24 h
collective self-sustained rhythm under constant conditions (com-
plete darkness) but also confers robustness against mutations [25].
The suprachiasmatic nucleus has a heterogeneous complex
architecture. There is spatial heterogeneity, and individual
neurons differ in their neuropeptide expression, light responsive-
ness, phase, and free running period [11,12,26]. For example,
individual periods of dispersed cells span over 20 to 30.9 h with an
average period of 24:1+1:4 h mean+SDðÞ. In organotypic slice
cultures, periods range from 22.4 to 26.7 h with an average of
24:2+0:7h [12]. Surprisingly, despite this complexity, the SCN
exhibits fast reentrainment. In jet-lag type experiments the SCN
can be almost completely reentrained within one day after an
abrupt phase shift of 6 h [7–9]. Advanced microscopic techniques
allow single cell bioluminescence measurements of clock proteins
at intervals as short as 20 min. These measurements display almost
sinusoidal oscillations [10,25]. However, bioluminescence mea-
surements provide only smoothed time series of specific reporter
constructs and thus it is not entirely clear how sinusoidal the
underlying core oscillator is. From our generic model, we predict
that the observed fast reentrainment can be achieved in the
following ways: (i) Sinusoidal waveform oscillator with relative
short radial relaxation times (box 2 in Figure 3); (ii) spike-like
oscillations with long relaxation time (box 4 in Figure 3); or (iii) a
spike-like waveform and short radial relaxation time (box 3 in
Figure 3). If SCN cells are self-sustained sinusoidal oscillators, we
predict that the SCN cell oscillators have a short radial relaxation
time. The radial relaxation time can be experimentally determined
via a nonlinear fit to a time series in which an amplitude relaxation
can be observed as in Figure 1A. The nonlinear fit can be done
with the ansatz tðÞ~e{lt:sin 2pt
Tzw0
, where tis time, Tis the
oscillation period, w0the initial phase difference and lis the radial
relaxation from which the radial relaxation time tr~1
lcan be
directly obtained. In the vicinity of the limit cycle, the radial
relaxation rate can be directly connected to the Floquet exponents.
Large Floquet exponents (short radial relaxation times) have
already been predicted on the basis of robustness studies using
different clock models [27] and by optimizing a specific feedback
model [28]. Our generic approach is based on one single
characteristic, namely, the transient time to entrainment, and
thus our prediction is independent of specific model assumptions.
Most SCN cell models assume self-sustained oscillation, but
experimental data [29,30] and theoretical predictions [31,32]
suggest an alternative scenario, where most SCN cells might
behave as damped oscillators. Detailed characterization of the
transient time to entrainment with a mixture of sustained and
damped oscillators is beyond the scope of the present work.
The Goodwin oscillator
The Goodwin oscillator [33] is a minimal model that describes
the oscillatory negative feedback regulation of a protein which
inhibits its own transcription. It provides a basic description of the
central components in the circadian oscillators of Neurospora,
Drosophila, and mammals [31,34]. In this model, a clock gene
mRNA (x) produces a clock protein (y) which, in turn, activates a
transcriptional inhibitor (z). Here we study a version of the
Goodwin oscillator successfully used to model data from dermal
fibroblasts from skin biopsies of human subjects [35]. The aim of
that study was to investigate whether different types of behavior,
early (‘‘larks’’) or late chronotypes (‘‘owls’’), have different clock
properties in dermal fibroblasts. The model equations are:
dx
dt ~Vsn
snzzn{dxx,ð3Þ
dy
dt ~byx{dyy,ð4Þ
dz
dt ~bzy{dzz:ð5Þ
This model describes the time evolution of mRNA (x), of a
cytosolic clock protein (y) and a nuclear clock protein (z).
Concentrations of these are measured in arbitrary units (a.u.). This
model has mostly linear kinetics with production rate constants
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by~0:4and by~0:06 and degradation rate constants dx~0:12,
dy~0:24,anddz~0:12, all rate constants in h{1. The Hill function,
which expresses the transcription rate that is inhibited by the nuclear
clock protein (z), has a maximal rate V~1a.u./h, a half-saturation
point s~1a.u., and a Hill coefficient which we vary between
n~10:8and n~22:3. Our aim is to compare our generic predictions
discussed above with a biochemical oscillator model. All eight
parameters have an influence on the dynamics of the system. It has
been shown that the Hill coefficient has a strong influence on the
oscillator properties [35]. Therefore, we choose the Hill coefficient as
the parameter to calculate the associated time to entrainment STeT.
Indeed, simulations confirm that the Hill coefficient has a strong
effect on the time to entrainment. In Figure 5A, an increase from
n~10:8to n~22:3reduces the time to entrainment 5-fold. The
entrainment signal was applied to all three dynamical variables in
turn: to the cytosolic protein concentration, to mRNA concentration
and to nuclear protein concentration. We observe qualitatively
similar results in all three cases. Furthermore, in order to relate these
transient times to our generic models, we extract for each Hill
coefficient in the Goodwin model the velocity variations along the
limit cycle parameter eand the Floquet exponent associated with the
radial relaxation timescale parameter l. In this way, we can project
these values on our plots for three models of radial oscillators (see
Figure 5B, C and D). Interestingly, the Hill coefficient changes both
velocity variations along the limit cycle eand the radial relaxation
timescale l. Importantly, these two parameters govern the transients
also in this higher-dimensional biochemical model. This is a
demonstration that biochemical models are amaneable for studies
using the concept developed in this paper.
In addition to the results presented above we also checked if our
findings hold for square-waveform oscillators, alternative entrain-
ment signals and the more general scenario of mutually coupled
oscillators (see Figures S1, S2 and S3 in Supporting Information).
In all three cases we obtained qualitatively similar results in
agreement with our concept.
Discussion
To our knowledge, ours is the first study that characterizes the
transient time to entrainment in terms of the oscillator properties.
Figure 5. Median time to entrainment SSTeTT for the Goodwin model and the comparison with three radial oscillators models. (A)
Median time to entrainment for the Goodwin oscillator as a function of the Hill coefficient. STeTwas calculated for an entrainment signal applied
separately to each variable: the cytosolic protein concentration (pink), the mRNA concentration (dark violet) and the nuclear protein concentration
(brown). (B) STeTfor a linear oscillator and the values of land eextracted from the Goodwin oscillator (green). (C) STeTfor a Poincare
´oscillator and
the values of land eextracted from the Goodwin oscillator (green). (D) STeTfor a Hopf-like oscillator and the values of land eextracted from the
Goodwin oscillator (green). STeTwas calculated as in Figure 3 with relative pulse strength 0.4.s.
doi:10.1371/journal.pone.0007057.g005
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Entrainment can be regarded as a particular case of synchroni-
zation with unidirectional coupling between the oscillators.
Therefore, similar features observed in our results might be
expected in other synchronization scenarios (see Supporting
Information S1). The time to synchronization for a network of
oscillators has been studied for several systems using analytical
approaches [36–40] and numerical simulations [41–43]. Most
synchronization studies focused on specific model oscillators at the
network level and derived scaling laws associated with the number
of oscillators. In [41] and [43], the synchronization rate of
different conductance based models (Hodgkin-Huxley type
models) was studied. Both studies showed that, a spike-like
oscillator reached a synchronized state much more rapidly than
a sinusoidal oscillator. Interestingly, it has been shown that
synchronization can be achieved in a few cycles by relaxation
oscillators [44] and by more sinusoidal ‘‘repressilators’’ [45].
Generally in models, the radial relaxation time and the phase
velocity cannot be controlled independently. Therefore, changing
the waveform pattern generally also changes the radial relaxation
time, compounding the contributions of both properties and
confusing the interpretation. Indeed, we observed this in the case
of the Goodwin oscillator while increasing the Hill coefficient (see
Figure 4C). Our goal was to reach a general understanding of the
transient to entrainment based on topological representations. We
use numerical simulations to exemplify our basic ideas. The model
independent results can be related to most previously conducted
studies.
Under the assumption that SCN cells are self-sustained
sinusoidal oscillators, we predict that single cell oscillators have a
short radial relaxation time. However, we cannot exclude that
some SCN cells are spike-like oscillators and exhibit short
transients this way. In fact, each SCN cell is a complex molecular
oscillator and certain variables might exhibit a sinusoidal shape
while others might have a spike-like shape. Perhaps, the pathway
governing transients might be associated with a spike-like
components. Time scale separations that support this view can
be inferred from the rapid reentrainment observed in the SCN.
Experiments with light pulses show that some core components of
the SCN are able to respond to light within 1 h [46,47].
In summary, we have shown how the time to entrainment is
governed by the interplay of the radial relaxation time and the
phase velocity distribution around the limit cycle. The time to
entrainment STeTmight be considered as an essential dynamical
feature of an oscillator. In many systems, this quantity can be more
easily extracted from experimental data than other related
dynamical features such as Floquet exponents or isochron
distributions. The median transient time to entrainment can be
used to infer properties of the underlying oscillator from the
observed transient times.
Materials and Methods
Model oscillator
The oscillator was designed to explore how the median time to
entrainment STeTdepends on a few generic parameters that are
applicable to a big class of oscillators. In Equation 2, fwðÞ
describes the phase evolution, where the parameter econtrols the
ratio between the slowest and fastest velocities around the limit
cycle. For e~0,fwðÞ~offset results in a sinusoidal oscillation
(dashed blue line in Figure 6), for e&1, a spike-like oscillation is
generated (black line in Figure 6) and for a new fwðÞwe obtain a
square-waveform oscillator (pink curve in Figure 6). The
parameter lcontrols the radial relaxation time independently of
the phase dynamics. In the vicinity of the limit cycle, lcan be
associated with the Floquet exponents, and ewith the isochron
structure [23] of the limit cycle. This model allows us to create a
spike-like oscillator with arbitrary Floquet exponents.
A modified Poincare´ oscillator is also known as radial isochron
limit cycle due to the radial structure of its isochrons. The phase
dynamics fwðÞis independent of the radial variable r. Isochrons
can be analytically calculated in some simple cases [21] or
otherwise extracted with numerical approaches [48], but these
approaches are not needed here since the isochrons can be
projected directly from the temporal phase points plotted in
Figure 3. Isochrons are a powerful tool to understand the phase
changes induced by perturbations [49].
The simulations were carried out using the equations 1 and 2
with n~0and in Cartesian coordinates:
dx
dt ~Tf
24 l:x1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2zy2
p{1
"#
{ye1
2zx
x2zy2
zoffset
()
,ð6Þ
Figure 6. Phase velocity dw
dt as a function of the phase for different fwwðÞðÞðÞðÞ.econtrols the velocity gap between the fastest and slowest points
and the parameter ‘‘offset’’ guarantees that the velocity is never zero. The black line corresponds to a spike-like oscillator, the pink line corresponds to
the square-waveform oscillator and the dashed blue line corresponds to a sinusoidal oscillator with a constant phase velocity around the limit cycle.
doi:10.1371/journal.pone.0007057.g006
How to Entrain Fast?
PLoS ONE | www.plosone.org 8 September 2009 | Volume 4 | Issue 9 | e7057
dy
dt ~Tf
24 l:y1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2zy2
p{1
"#
zxe1
2zx
x2zy2
zoff set
()
zK:Ht{ton
ðÞ{Ht{tof f
ðÞ½
ð7Þ
Here Tf, the unscaled period, depends on the parameters l,e
and offset. As discussed below we rescaled the period to 24 h by
choosing an appropriate scaling factor Tf. The entrainment signal
HtðÞis the Heaviside step function, Kis the pulse strength, the
pulse start time is ton~tent :m, with tent the entrainment period,
m~1,2,3 ... and toff ~ton z1h is the pulse end time.
Time to entrainment
Our numerical experiments were designed to reduce depen-
dencies on initial conditions and entrainment frequency. In
Figure 3 we calculated the STeTfor a wide range of land e
values. Each point of the plot was calculated following the same
numerical protocol: 1) Choose a parameter combination (l,e)of
interest and rescale the system to a period T0~24 h. 2) Calculate
the range of entrainment and choose entrainment frequencies
equidistributed within this range. 3) Choose initial temporal phases
ICi. The 24 h temporal initial phases are located around the
unperturbed limit cycle (gray dots in Figure 1B and C), i.e. each
initial condition is given by ICi~xwi
ðÞ,ywi
ðÞ½, where wi~T0
24
i
with i~1...24. 4) Start the simulations with periodic 1 h vertical
pulses and calculate the instantaneous phase difference between
the oscillator and the train of pulses for a total duration of 500 days
(see Figure 1B). 5) The time to entrainment is considered to be
reached if the mean phase difference of eight consecutive cycles is
smaller than 5 minutes. Otherwise, no entrainment is detected. 6)
Repeat steps 3–5 for the 24 different temporal phase initial
conditions and then take their median value Te (see Figure 1C). 7)
Repeat steps 2–6 for 12 evenly distributed frequencies within the
total range of entrainment and then take their median value STeT
(see Figure 1E). 8) Choose another combination of land eand
restart the protocol.
Supporting Information
Supporting Information S1
Found at: doi:10.1371/journal.pone.0007057.s001 (0.06 MB
PDF)
Figure S1 Square waveform oscillator, its time series and the
median time to entrainment. (A) Square waveform oscillator: limit
cycle with 24 marked phase points (dots) and isochrons (rays). The
intersection of each isochron with the limit cycle determines the
phase and (B) the temporal evolution of x variable with parameters
e= 1, offset = 0.02, n = 0 and (C) the median time to entrainment
,T
e
.as a function of the phase velocity around the limit cycle, e,
and radial relaxation constant, l, for pulse entrainment. Gray
scales refer to the median time to entrainment, where black
represents long and white short ,T
e
..
Found at: doi:10.1371/journal.pone.0007057.s002 (0.72 MB TIF)
Figure S2 Median time to entrainment ,T
e
.for different
entrainment signals and oscillators, under soft-pulses entrainment
and under medium-sized-pulses for a nonlinear oscillator and for a
Hopf oscillator. (A) Entrainment under sinusoidal perturbations
with amplitude 0.05. (B) Entrainment under pulse perturbation
with amplitude 0.4. (C) Entrainment under 1 h pulse perturbation
with amplitude 0.8 for a nonlinear radial relaxation oscillator. (D)
Entrainment under 1 h pulse perturbation with amplitude 0.8 for
a Hopf oscillator. The median time to entrainment is plotted as a
function of the phase velocity around the limit cycle, e, and radial
relaxation constant, l. Gray scales refer to the median time to
entrainment, where black represents long and white short ,T
e
..
Both axes are plotted using logarithmic scales.}
Found at: doi:10.1371/journal.pone.0007057.s003 (1.23 MB TIF)
Figure S3 Time to synchronization for two coupled oscillators.
(A) Time to synchronization of two coupled ‘‘sloppy’ oscillators as
a function of their transition from sinusoidal to a spike-like
oscillator (B) Time to synchronization of two sinusoidal oscillators
as a function of their transition from ‘‘sloppy’ to ‘‘rigid’ oscillator.
See Supporting Information for model details.}
Found at: doi:10.1371/journal.pone.0007057.s004 (0.83 MB TIF)
Acknowledgments
We thank Pa˚l Westermark, Arkady Pikovsky and Achim Kramer for
stimulating discussions and Anmar Khadra and Manuela Benary for
carefully reading the manuscript.
Author Contributions
Conceived and designed the experiments: AEG HH. Performed the
experiments: AEG. Analyzed the data: AEG. Contributed reagents/
materials/analysis tools: HH. Wrote the paper: AEG.
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