A Model for "Splitting" of Running-Wheel Activity in Hamsters
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Oda
JOURNAL
and Friesen
OF BIOLOGICAL
/ MODEL OF
RHYTHMS
SPLITTING
/ February 2002
A Model for “Splitting” of
Running-Wheel Activity in Hamsters
Gisele A. Oda2 and W. Otto Friesen1
Department of Biology, NSF Center for Biological Timing,
University of Virginia, Charlottesville, VA 22903-2477, USA
Abstract Splitting of locomotor activity rhythm in hamsters occurs when the
animals are exposed for several weeks to constant light. The authors propose a
mathematical model that explains splitting in terms of a switch in the sign of cou-
pling of two oscillators, from positive to negative, due to long-term exposure to
constant light. The model assumes that the two oscillators are not identical and
that the negative coupling strengths achieved by each individual animal are vari-
able. With these assumptions, the model provides a unified picture of all differ-
ent splitting patterns presented by the hamsters, provides an explanation for
why the two activity components cross each other during many patterns, and
explains why the phase difference achieved by the split components is often near
180°.
Key words SCN, computer simulations, mathematical models, circadian rhythms, cou-
pled oscillators, splitting
Pittendrigh and Daan (1976) proposed that the cir- activity dissociated, and animals began to run with
cadian clock that controls activity rhythms in ham- two different circadian periods until the activity bands
sters comprises at least two separate but mutually established a new phase relationship. In this case, the
coupled oscillators. According to this model, an E activity bands assumed a phase difference of 180°.
(evening) oscillator controls the beginning, whereas Under similar conditions, ground squirrel activity bands
an M (morning) oscillator controls the end of the activ- split temporarily and then phase-locked as a single
ity band in nocturnal hamsters. The length of the activ- band. As pointed out by Pavlidis (1978), the structure
ity band (α) reflects, in this model, the phase difference of the clock can be best unveiled from experiments
between these two oscillators. involving stimuli that are different from those seen by
The two-oscillator model for hamster pacemakers the organism in its natural environment.
was based mainly on the phenomenon of “splitting” The splitting phenomenon provided the primary
of activity (Pittendrigh and Daan, 1976) that was first motivation for assuming that evening and morning
described by Swade and Pittendrigh in experiments oscillators underlie circadian pacemakers in rodents.
with arctic ground squirrels under constant light (LL) Changes in the phase relationship between E and M
(Pittendrigh 1960) and then shown in a more system- could functionally account for the regulation of photo-
atic way when several hamsters were exposed to pro- periodic phenomena (Pittendrigh and Daan, 1976).
longed LL (Pittendrigh, 1960, 1993; Pittendrigh and Recent research has shown, however, that splitting
Daan, 1976). Under this condition, two clear bands of may result from a two-oscillator structure, distributed
1. To whom all correspondence should be addressed: Department of Biology, University of Virginia, Charlottesville, VA
22903-2477; e-mail: wof@virginia.edu.
2. Present address: Instituto de Física, Departamento de Física Aplicada, Universidade de São Paulo, São Paulo, SP, Brazil
05389-970.
JOURNAL OF BIOLOGICAL RHYTHMS, Vol. 17 No. 1, February 2002 76-88
© 2002 Sage Publications
76Oda and Friesen / MODEL OF SPLITTING 77
in the right and left sides of the SCN, that does not cor- We used two-dimensional, nonlinear differential
respond to the evening and morning oscillators believed equations that provide both phase and amplitude infor-
to underlie the photoperiodic phenomena (de la Iglesia mation and enable simulations for a wide range of
et al., 2000, Daan et al., 2001). Due to the novelty of this coupling strengths. The one-dimensional representa-
proposition, we analyzed splitting without consider- tion chosen by previous models restricted their analy-
ing the physical identities of the two oscillators. We sis to weak coupling. One of our findings is that the vari-
may call them evening and morning wherever it is able splitting patterns in hamsters involve weak to strong
useful to name them, but that should be interpreted coupling strengths, showing that multidimensional
simply as a late reflection of the initial motivation for analyses are crucial for understanding the phenomenon.
studies based on the original Pittendrigh and Daan We first directed our computer study toward an
model. understanding of how the period, amplitude, and
There have been several previous modeling studies phase relationship between the two oscillators depends
aimed at providing a theoretical basis for splitting. on the structural symmetry of the coupled system.
Daan and Berde (1978) introduced the first explicit From these “symmetry” studies, we isolated the ele-
quantitative structure to the model, presenting the ments necessary to simulate the splitting phenome-
first computer simulations of the behavior of coupled non in hamsters and carried out a series of explicit
E and M oscillators. Kawato and Suzuki (1980) extended computer experiments that embodied our general find-
the model using ordinary differential equations and ings in a series of specific, representative simulations.
proposed that the antiphasic relationship between E This methodology was chosen so that a unified picture
and M results from the weakening of coupling or of the splitting phenomenon, including all different
detuning of E and M periods—either of which can patterns, could be presented. Our study provides a
lead to bistability, with phase relationships either in mechanism that consistently explains the questions
phase and antiphasic. They used one-dimensional phase presented above for nocturnal hamsters in terms of a
oscillators in their analysis, which only describe the general model with a minimal set of assumptions.
system at very low coupling strengths. Their model is
strongly supported by the evidence of hysteresis in the
splitting phenomenon of diurnal Tupaia (Hoffmann MODEL EQUATIONS
1971) because of the mathematical association between
hysteresis and bistability. We used coupled Pittendrigh-Pavlidis equations to
The previous models do not cover several aspects simulate the two oscillators. In these equations, R and
of splitting in nocturnal hamsters. To fill this gap in S are the state variables, and a, b, c, and d are the system
our understanding, we developed a phenomenological parameters. The effect of light is introduced by the
study based on computer simulations to answer some term L. Finally, K is a small nonlinear term (K = 1/[1
specific questions about splitting in hamsters. + 100R2]) formulated by W. T. Kyner (C. Pittendrigh
and W. T. Kyner, personal communication, 1991). A
1. There is a remarkable interindividual variability in the complete description of the state variables and param-
splitting patterns presented by hamsters maintained eters is presented in Oda et al. (2000). Here we only
in LL. A schematic diagram of the different patterns is make a further distinction between parameters of E
presented in Figure 1. According to the results of Ear- and M oscillators because in this work, we studied
nest and Turek (1982), 56% of hamsters exposed to LL
coupled identical and nonidentical oscillators.
presented splitting activity, and among these, the split-
ting pattern developed gradually in 75% of the indi- Evening oscillator (E):
viduals, whereas 25% presented an abrupt onset of
splitting. Why are there so many variable patterns, dR E / dt = R E − c E SE − bE SE2 + ( d E − L) + K
and is there any relationship between them?
2. One intriguing proposition was that the E and M com-
ponents of activity cross over each other during all dSE / dt = R E − aE SE + C ME SM .
patterns of splitting and sometimes even cross when
they do not split (Fig. 1A). This proposition was based Morning oscillator (M):
on visual characteristics that are used to discriminate
between the two components of activity and on the
dR M / dt = R M − c M SM − bM SM
2
+ ( d M − L) + K
compression of α (Pittendrigh and Daan, 1976). Do
they really cross and why?
dSM / dt = R M − aM SM + C EM SE78 JOURNAL OF BIOLOGICAL RHYTHMS / February 2002 Figure 1. Diagrams of different splitting patterns observed under LL conditions in hamsters. (A) Reduction in and increase in τ without splitting of activity into two components. (B) Decrease in α prior to splitting of the activity band into two components that subsequently refuse; presumed E and M components cross; little change in τ. (C) Splitting of activity into two bands that assume a permanent 180° out-of-phase relationship; presumed E and M components cross; little change in τ. (D) Temporary splitting of activity into two bands; α decreases prior to splitting; τ decreases during splitting. (E) Abrupt splitting of activity, proceeded by a progressive decrease in α; τ decreases following splitting. A-D reproduced, with permission, from Pittendrigh and Daan (1976); E reproduced, with permission, from Earnest and Turek (1982). Simulations were performed with CircadianDynamix, AE and AM. The total amplitude of the coupled system a computer program that is an extension of AT is the sum of these two terms (Oda et al., 2000). NeuroDynamix, originally developed to explore the In our previous simulations, setting the parameter properties of neurons and small neuronal networks L to a positive value simulated a pulse of light in both E (Friesen and Friesen, 1994). In this model, the analog and M oscillators (Oda et al., 2000). The straightfor- of running wheel activity occurs every time the vari- ward way to simulate constant light would, therefore, able R in either the M or E oscillator is above some be to set L to a constant, nonzero value. However, the threshold value, which we set to two-thirds of the high light intensity used to generate the PRC annihi- maximum amplitude of this variable. The length of the lated the oscillation. Lowering the amplitude of light total activity (α) reflects the phase difference between E intensity to simulate the effect of light adaptation, as and M. The onset of activity was assigned as the phase proposed by Daan and Pittendrigh (1976), damped reference point for circadian time (CT 12), as in behav- the amplitude and changed the period of the oscilla- ioral studies. With this choice of phase reference, we tion but did not provide any splitting phenomena. got positively correlated changes in the E and M phase Dissociation of the two oscillators can be simulated if difference and α for almost all simulated configura- differential effects of light on each oscillator are con- tions. Positive and negative coupling were determined sidered; however, phase locking at 180° and the vari- by the positive and negative signs, respectively, that able-splitting patterns cannot be explained in a con- precede the positive, linear coupling coefficients CEM cise way with this approach. Therefore, inclusion of a and CME. deeper modification of the coupled circadian system A central concept in this work is the distinction was considered in the model for simulating the effect between intrinsic properties of the component oscilla- of long-term exposure in LL. In this sense, we radically tors and the emergent properties of the coupled sys- separated the short- and long-term effects of constant tem. We used the terms τE and τM to denote the intrin- light on the system by even setting L to zero in the sim- sic periods of uncoupled E and M oscillators. The term ulations of the long-term effect to show that the simu- τEM describes the period of the coupled system when a lated splitting phenomenon had undoubtedly been stable phase relationship is established. The ampli- generated by this deeper modification without any tudes of individual E and M oscillators are denoted by synergy between the L-on and change of sign effects.
Oda and Friesen / MODEL OF SPLITTING 79
STRUCTURAL SYMMETRY OF TWO amplitude) and the coupling strength in either direc-
COUPLED OSCILLATOR SYSTEMS tion (Fig. 2).
The structural symmetry of the coupled oscillator
Methodology for Introducing Asymmetry
system has three aspects:
Introduction of Asymmetry in the
1. symmetry of the component oscillators (period and
amplitude), Properties of the Component Oscillators
2. symmetry of the coupling signs (positive and nega-
tive), and We altered the period and amplitude of one compo-
3. symmetry of coupling strengths. nent oscillator by manipulating the four parameters of
Pittendrigh-Pavlidis equations, one at a time. This
In Figure 2, we present a picture of the possible con- procedure is a mathematical analog of the experiment
figurations of the system, showing different degrees of carried out by Page and Nalovic (1992). In that experi-
internal symmetry. Configurations are shown, from ment, one eye of Bulla was treated with chemicals that
top to bottom, in the order of internal symmetry; thus, changed its period, while the other eye was left intact
the two cases of complete symmetry (A1, A2) are pre- to see how that change affected the general properties
sented at the top, and maximum asymmetry is at the of the coupled eye system. In our case, we repeated the
bottom of the figure (D). change in period and amplitude four times, manipu-
The objective of our initial simulations was to pro- lating a different parameter each time. This procedure
vide a general picture of the dependence of the emer- is necessary to verify that the changes in the system are
gent properties of the two-oscillator system on its due exclusively to changes in period and amplitude of
structural symmetry. We verified that the increase of one oscillator and independent of specific parameters.
asymmetry affected the following emergent proper- First, it was necessary to see how each of the param-
ties of the system: eters of the equation affected the intrinsic properties of
one oscillator before coupling was implemented. In
Figure 3, we show how the properties of one oscillator
1. τEM;
(e.g., the E oscillator) are affected by the changes in
2. ∆ΦEM (phase difference between E and M: ΦM-ΦE);
these parameters. We consider as a starting point the
3. α;
4. AE, AM, and AT; and nominal parameter set (aE = 0.85, bE = 0.3, cE = 0.8, and
5. phase leadership. We assign the term leading oscillator dE = 0.5), which gives a τE value close to 24 h and an
to that oscillator whose activity begins less than 180° amplitude of about 4 arbitrary units. Each point in Fig-
before the activity of the “lagging oscillator.” ure 3 represents the new values assumed by τE and AE
when one parameter is changed, leaving all the others
unaltered. For example, the point indicated by “dE =
SIMULATIONS 1.0” represents τE and AE values of the oscillator when
the parameter set is (aE = 0.85, bE = 0.3, cE = 0.8, and dE =
We began by examining the two cases of full struc- 1.0).
tural symmetry, that is, with two identical oscillators According to Figure 3, decreasing parameter aE or
coupled to each other either by mutually positive increasing parameter cE causes decreases in both τE
(+/+) connections or mutually negative connections and AE. Increasing parameter bE also causes a decrease
(–/–) of identical strength. Then we examined the case in τE but with a smaller decrease in AE compared to
of two identical oscillators coupled by a positive con- those due to aE and cE. Increasing parameter dE, on the
nection from one oscillator and negative connection other hand, leads to a decrease in τE but an increase in
from the other (+/–), both with same strength, corre- AE. Alterations in the parameters of the Pittendrigh-
sponding to an asymmetric system with respect to Pavlidis equations provide, therefore, a way of affect-
coupling sign. In each of the (+/+), (–/–), and (+/–) ing the intrinsic period and amplitude in each oscilla-
cases, we gradually and systematically added asym- tor. Each parameter set, most with a unique period
metry in the system by changing the properties of one and amplitude relationship, represents a different sys-
component oscillator (by changing the value of a spe- tem and, when b is altered, a different level of
cific model parameter that altered both period and nonlinearity.80 JOURNAL OF BIOLOGICAL RHYTHMS / February 2002
Figure 2. Schematic diagram of structural symmetry for two coupled oscillators. Three scenarios are illustrated: mutual positive coupling
(A1-D1), mutual negative coupling (A2-D2), and combined negative-positive coupling (A3-D3). The states of the oscillators before cou-
pling are represented by solid-line circles and after coupling by dotted circles. The radii of the filled circles are positively related to unit
oscillator period and amplitude before coupling and the radii of dotted circles to amplitude after coupling. In each oscillator pair, the
phase-leading oscillator appears at left. The distance between oscillators reflects the phase difference. Starting from the fully symmetric
configurations (A1, A2), asymmetry is incorporated first into coupling strength (B1, B2, and B3), then in oscillator properties (C1, C2, and
C3), and finally in both coupling strengths and oscillator properties (D1, D2, and D3). Note that maximum asymmetry occurs in A3 to D3,
where the sign of coupling also is reversed.
Introduction of Asymmetry in Coupling (b) asymmetric oscillators coupled by symmetric cou-
pling strengths, and
We varied the sign of coupling as already described (c) asymmetric oscillators with asymmetric couplingstrengths.
and manipulated coupling strength by assigning dif-
ferent values in four steps to the coupling coefficients, Step c included two substeps—a stronger coupling
such that CEM = CME = 0.2, 0.1, 0.05, or 0.01. effect exerted first by the oscillator with shorter intrin-
The progression from symmetric to asymmetric sic periods and second by the one with longer intrinsic
systems was carried out in three steps, as schematized periods. Simulations for the various values of oscilla-
in Figure 2, for each of the three major coupling config- tor parameters a, b, c, and d were performed for each of
urations ((+/+), (–/–), and (+/–)): the four coupling strengths to provide a broad picture
of the influences of parameters and coupling strengths
(a) identical oscillators with asymmetric coupling strengths, on system output.Oda and Friesen / MODEL OF SPLITTING 81
Any departure of the phase difference from 0° or 180°
is a reflection of asymmetry in the system.
Positive Coupling (+/+)
Fully symmetric case (A1, Fig. 2). When the oscillators
are identical and coupling strength is the same in both
directions, an in-phase relationship (∆ΦEM = 0°)
occurs for all parameters tested. Because the phase
difference between E and M is always 0°, independent
of coupling strength, changes in α reflect changes only
in the shape of the state variable oscillations. For this
case, as coupling strength is increased, τEM, α, and AT
decrease.
For very strong coupling, the amplitude of both
Figure 3. Dependence of oscillator outputs τE and AE on parame- oscillators becomes very small, and the system damps
ters. The graphs depict the relationship between τE and AE of the and eventually oscillations cease. The period of the
evening oscillator derived from the Pittendrigh-Pavlidis equa- symmetrically coupled system is always shorter than
tions. Each parameter set (aE, bE, cE, dE) defines specific τE and AE
the intrinsic periods of the components.
values for the oscillator. The reference set (aE = 0.85, bE = 0.3, cE =
0.8, dE = 0.5) generates a system with τE ≅ 24 h and AE ≅ 4 units, indi-
cated by the asterisk. The points on the graph correspond to the Asymmetric cases (B1, C1, D1, Fig. 2). Introduction of
values attained by τE and AE for each parameter modification as
any asymmetry in the otherwise fully symmetric sys-
one parameter at a time was changed from the reference value set
to values shown. In our simulations of two coupled oscillators, tem, either by changing properties of the individual
one oscillator always incorporated the reference parameter set oscillators or altering one of the coupling terms, leads
indicated by the asterisk. Asymmetry in oscillator properties to a phase difference between E and M, departing
(period and amplitude) was achieved by setting parameters of the
from the 0° characteristic of the symmetric system.
second oscillator to the values shown. To make the scope of the
modeling manageable, we only explored the system with τ < 24 h.
The phase difference, nevertheless, is always small.
The largest phase differences are attained at very weak
coupling strengths, but activity always remains unsplit.
The period of the coupled system can be longer or
shorter than the intrinsic periods of the components,
depending on the nature of the asymmetry.
RESULTS In determining which oscillator will phase-lead,
two different situations must be considered. If coupling
strengths are identical, then the oscillator with the
General Behavior smaller intrinsic amplitude phase-leads (C1, Fig.
2). On the other hand, if oscillators are identical, then
Here we present our general simulations of sym- the oscillator with larger coupling input phase-leads.
metrical and asymmetrical coupled oscillators, pro- The amplitude of this oscillator decreases (B1, Fig. 2).
ceeding from positive coupling through negative cou- The greatest stability during asymmetrical coupling
pling and ending with negative-positive coupling occurs when the oscillator with lower amplitude exerts
interactions. the smaller coupling effect. Stability here means that
When two identical oscillators are coupled with the E and M oscillators achieve stable phase relationships
equal strength and sign, the oscillators attain either over the greatest range of coupling strengths. This lower
one of the only two symmetrical phase relationships, amplitude oscillator then phase-leads (D1, Fig. 2).
0° or 180°; there is no “phase-leading” oscillator in this No general statement can be made about the direc-
case. Our simulations showed that 0° and 180° phases tion of changes in coupling strength, τEM, α, and AT for
occur with fully symmetric (+/+) and (–/–) cases only. the asymmetric cases. Each specific set of parameters
Once symmetry is broken, however, there is a clear determines how these quantities depend on coupling
way to define which one of the oscillators phase-leads. strength.82 JOURNAL OF BIOLOGICAL RHYTHMS / February 2002
Negative Coupling (–/–) weak in asymmetric systems, phase locking does not
occur.
Fully symmetric case (A2, Fig. 2). Under negative The greatest stability during asymmetrical coupling
coupling and symmetrical conditions, oscillators E occurs when the oscillator with longer τ exerts the
and M attain a 180° antiphasic relationship when cou- stronger coupling effect. This oscillator, which in our
pling is strong. However, for weak coupling, there simulations also had the greatest amplitude, was phase-
exists an unstable in-phasic relationship, and for very leading (D2, Fig. 2). Depending on specific parame-
weak coupling bistability occurs, with both stable ters, such a system could exhibit split or fused activity
antiphasic and in-phasic relationships. Therefore, the bands. If this system exhibited fused activity, a split
specific phase relationships in this system are strongly pattern could be established by decreasing the asym-
dependent on the symmetry of component oscillators metry between constituent oscillators and the cou-
and the absolute values of the coupling strengths. pling strength or by increasing the strength of the neg-
As coupling strength is increased, τEM decreases ative coupling. No general statement can be made
and AT increases. The period of the symmetrically cou- about the direction of changes in coupling strength,
pled system is always longer than the intrinsic periods τEM, α, and AT for the asymmetric cases.
of components.
Positive-Negative Coupling (+/–)
Asymmetric cases (B2, C2, D2, Fig. 2). Introduction of
asymmetry in component oscillator properties or in Asymmetric cases (A3, B3, C3, D3, Fig. 2). The oscilla-
coupling strengths leads to a departure of the phase tor that exerts negative coupling phase-leads, inde-
difference between E and M from the characteristic pendent of its values for period or amplitude. Thus,
180°. This departure is proportional to the extent of the when component oscillators and positive and nega-
asymmetry. Therefore, if the oscillators’ properties are tive coupling strengths are identical, the oscillator that
only slightly different, the system will still have two exerts the negative coupling attains a reduced ampli-
separate activity bands with a phase difference close tude and phase-leads (A3, Fig. 2).
to 180°. On the other hand, very asymmetric systems If the two oscillators are identical but coupling
can exhibit a single block of activity simply because strengths differ, maximum phase differences can be
the phase difference is not large enough to completely far from 180°, even when the negative coupling is
separate E and M activity bands. much stronger than the positive (B3, Fig. 2). The same
Strength of coupling is the other factor that influ- occurs if both positive and negative coupling strengths
ences the phase relationship (B2, Fig. 2). The stronger are identical but oscillators are different (C3, Fig. 2).
the coupling, the closer the phase difference is to 0° On the other hand, if the system is fully asymmet-
(+/+) or 180° (–/–). For weakly coupled systems, even ric, large phase differences close to 180° can arise only
a small asymmetry leads to a one-block activity pat- when the oscillator that exerts negative coupling has a
tern (fused E and M activity bands). A much greater smaller period and amplitude and the positive cou-
degree of asymmetry is needed in strongly coupled pling is extremely weak (D3a, Fig. 2).
systems for this to occur. Conversely, for two asym-
metric oscillators, the stronger the negative connec- General Conclusions about the
tion, the closer the phase difference is to 180°. The Phase Difference between E and M
period of the coupled system can be longer or shorter
than the intrinsic periods of the components, depend- Two factors define the closeness of the E-M phase
ing on the nature of the asymmetry. difference to 0° and 180° (Fig. 4):
In determining which oscillator will phase-lead,
two different situations must be considered. If (a) Symmetry of the system. The more symmetric the sys-
coupling strengths are identical, then the oscillator tem, with respect to oscillator properties and coupling
strengths, the closer the phase difference is to 0° (for
with larger τ phase-leads (C2, Fig. 2). On the other
positive coupling) or 180° (for negative coupling).
hand, if oscillators are identical, then the oscillator (b) Strength of coupling. The greater the coupling strength,
with larger coupling input phase-leads. The ampli- the closer the phase is to 0° (for positive coupling) or
tude of this oscillator decreases. When coupling is too 180° (for negative coupling).Oda and Friesen / MODEL OF SPLITTING 83
Guiding Elements for the Construction
of the Model for Splitting
Two general findings from our simulations, those
concerning phase difference and phase leadership,
form the building blocks of our model.
First, the configurations that provide oscillators
with phase differences close to 180° (i.e., with split
activity bands) are primarily those corresponding to
mutual negative coupling (A2, B2, C2, and D2 in Fig. 2).
The stronger the negative coupling and the more sym-
metric the oscillators, the closer the phase difference is
to 180°. Conversely, it is possible to have a nonsplit
activity pattern with mutually negative interactions if
the component oscillators are very asymmetric or if
coupling is very weak. The only other configuration
that generates phase differences close to 180° is the one
shown as D3a in Figure 2, which is the totally asym-
metric (+/–) case, where the oscillator with shorter τ and
lower amplitude exerts the stronger negative coupling
effect. For our further analysis of splitting, we confine
ourselves to the most general and simpler mutual neg-
ative case.
Second, when two oscillators with different intrin-
sic periods and amplitudes are coupled, they attain a
common period and establish a fixed phase relation- Figure 4. Schematic diagram of the phase relationships attained
by two coupled oscillators. (A) When coupling is (+/+), the abso-
ship. If coupling is positive, the component oscillator lute value of ∆ΦEM is small. The more nearly symmetric the
with lower intrinsic amplitude phase-leads, whereas oscillators or the stronger the coupling, the closer ∆ΦEM is to 0°.
if coupling is negative, the oscillator with greater intrin- (B) When coupling is (–/–), the absolute value of ∆ΦEM is large for
sic τ phase-leads, as shown in Figure 5. The period of strong coupling. The more nearly symmetric the oscillators or
the stronger the coupling, the closer the phase value is to 180°.
the coupled system, τEM, is different from the intrinsic
(C) When the (–/–) coupling is weak, bistability can occur and two
periods of the components. ∆ΦEM are possible: one close to 0° and the other to 180°. The more
We conclude from these extensive simulations that nearly symmetric the oscillators, the closer this pair of solutions
a combined influence of the intrinsic period and ampli- are to 0° and 180°. A perturbation of the system can shift the con-
figuration from one solution to the other. The absolute value of
tude of the constituent oscillators determines the main
∆ΦEM is represented by |∆ΦEM|.
emergent properties of the coupled system. Therefore,
the phase oscillator representation of coupled oscilla-
tors is inadequate for a complete analysis of the sys-
tem properties. When two different oscillator configu-
rations were examined in our simulations, we always An Explicit Model for Splitting of
chose the configuration in which the oscillator with Running-Wheel Activity in Hamsters
the shorter period also had a smaller amplitude. In
this way, we avoided dealing with nonsmooth, too We now turn from our exploration of the general
steep oscillators, which have very limited parameter properties of coupled oscillators to the specific prob-
ranges for stable coupling. In this sense, if coupling is lem presented by splitting of the activity bands in
positive, the oscillator with smaller intrinsic τ and hamsters into two distinct components. Under long-
lower intrinsic amplitude phase-leads, whereas if cou- term exposure to LL, hamsters exhibit a variety of
pling is negative, the oscillator with larger intrinsic τ behaviors (Fig. 1): (1) abrupt or slow onset of splitting,
and higher intrinsic amplitude phase-leads. (2) temporary splitting followed by rejoining of activ-84 JOURNAL OF BIOLOGICAL RHYTHMS / February 2002
phase difference and a change in phase leadership
between the two oscillators. First, the oscillators attain
new phase differences. The greater the coupling strength,
the closer the new phase difference is to 180°. Also, the
stronger the coupling, the more rapidly is this phase
relationship attained (briefer transients). Second, the
two oscillators exchange phase leadership, with M
leading after the change in coupling sign (Fig. 6C-E).
This exchange in phase leadership may be attained
either with or without crossing of E and M bands,
depending on the phase at which coupling signs are
reversed. They always choose the shortest path to
Figure 5. Phase leadership exchanges when sign of coupling is attain the new phase difference, which may or may not
switched. (A) Time course of state variable R of free evening and be the crossing one. Finally, due to the new configura-
morning, showing their different intrinsic properties. E oscillator
has shorter period and lower amplitude. (B) In (+/+) coupled EM
tions, τEM of the coupled systems is different in all
system, E and M achieve a common new period and different cases.
amplitudes, and E phase-leads due to its shorter intrinsic period The final phase difference of the two oscillators is
and lower amplitude. (C) In (–/–) coupled EM system, M phase- unique and independent of where the switch occurs in
leads due to its longer period and high amplitude.
the moderate to strong coupled systems (Fig. 6A-C),
although the transient paths of the components do
depend on the phase of ∆. On the other hand, the two
ity bands, and (3) a decrease in α and an increase in τEM patterns corresponding to the same final, weak cou-
without splitting. Based on the two general principles pling shown in Figure 6D-E deserve special consider-
described above, our simulation of splitting of run- ation. Here bistability occurs because, when coupling
ning-wheel activity in hamsters relies on three is very weak, there are two stable phase relationships
assumptions: (Fig. 4C). The configuration that is actually realized
depends only on the system phase at the time of the
1. The E oscillator has a shorter intrinsic period and sign switch (note the two phases for ∆ in Fig. 6D-E). An
lower intrinsic amplitude than the M oscillator. appropriately timed perturbation by a simulated pulse
2. The effect of long-term exposure of hamsters to LL is of light can shift this system from one phase relation-
to switch the signs of the coupling from positive to
ship to the other. Note that the two final phase rela-
negative.
3. There is individual variability in the strength of the tions observed in Figure 6D-E are preceded by tran-
final negative coupling. sients of differing durations and that the period of the
coupled system depends on the final phase relation-
An additional, nonessential assumption that we adopt ships. A theoretical approach to this bistability phe-
for simplicity is that the coupling strengths are sym- nomenon is presented in Kawato and Suzuki (1980).
metrical. Overall, our approach was to assign a config- The five cases illustrated in Figure 6 correspond to
uration of the type C1 to the system before and of the variable patterns shown by individual hamsters
type C2 to the system after long-term exposure to LL maintained in LL. With a minimal set of assumptions,
(Figs. 2, 5). our model thus consistently provides a unified view of
In Figure 6, we show the simulated free-running all these patterns that initially seemed disconnected. If
activity of five individual hamsters incorporating the the negative coupling is very strong (Fig. 6A-B), abrupt
same initial symmetric positive coupling. Under this splitting takes place, and it is difficult to identify the E
positive coupling, the E oscillator phase-leads because and M components of activity. For moderate coupling
AE < AM and τE < τM. At the phases marked by ∆, the strengths, splitting occurs gradually, enabling track-
transition point between DD and LL, we abruptly ing of E and M components as they exchange phase
switched the coupling coefficient signs to negative leadership prior to splitting (Fig. 6C). If coupling is
values in all individuals. However, different final cou- weak, the final phase difference between E and M can
pling strengths are assigned to individuals (Fig. 6A-E). either be small (Fig. 6E) or large (Fig. 6D). In the former
In all cases, two general behavioral changes result case, the activity does not split, but phase leadership
from the switch in coupling—namely, a change in exchanges and α shortens. In the latter case, the activ-Oda and Friesen / MODEL OF SPLITTING 85 Figure 6. Splitting patterns that result from a switch in coupling sign. Five activity rhythms are simulated, all with the same initial condi- tion (DD), followed by a switch in sign and four final coupling strengths (LL). The E oscillator was configured to have shorter τ and lower amplitude than M. The DD condition is simulated by symmetric (+/+) coupling (C = CEM = CME = +0.2), which generates fused (E + M) activ- ity, with E phase-leading M, and τEM = 22.8 h. At the phases denoted by ∆, the sign of coupling was switched to symmetric negative (–/–) in all cases, but with different strengths. (A) CEM = CME = –0.2. Activity splits abruptly, E and M attain ∆ΦEM = 167°, τEM = 20.3 h. (B) CEM = CME = –0.1. Activity splits quickly, without E and M crossing, ∆ΦEM = 150°, τEM = 21.7 h. (C) CEM = CME = –0.08. Activity splits gradually, E and M cross before splitting, ∆ΦEM = 140°, τEM = 22.2 h. (D) CEM = CME = –0.05. Activity splits and then later rejoins, with E and M exchanging phase leadership, ∆ΦEM = 98°, τEM = 23.2 h. (E) CEM = CME = –0.05. Activity does not split, but phase leadership is exchanged, α shortens, and there is a change in τEM, ∆ΦEM = 4°, τEM = 24.2 h. Cases (D) and (E) correspond to two possible solutions for the same CEM = CME = –0.05 but are gen- erated by different phases of ∆, the former at the end and the latter at the beginning of activity. Throughout this series of simulations, parameters were as follows. For E: (aE = 0.85, bE = 0.3, cE = 1.0, dE = 0.5). For M: (aM = 0.85, bM = 0.3, cM = 0.8, dM = 0.5). The E and M components of activity are represented by thick and thin lines, respectively. When they overlap, only the E component is visible. ity may or may not split, depending on the degree of and final configurations are different, although they symmetry of the oscillators. For nearly symmetric E both show fused activity. and M, the final phase difference is large and compo- The bistability of two phase patterns—one closer to nent activities split. For highly asymmetric E and M, 0° and the other to 180°—that we observed also can the components split temporarily and then rejoin, explain the results presented in Ellis et al. (1982), who exchanging phase leadership only. This temporary found that a pulse of darkness could shift an individ- splitting followed by refusion was caused by the ini- ual hamster system under LL from a state of one activ- tial switch in sign alone. Previous models could only ity band to a split activity pattern. Either one of the reproduce this behavior by assuming an initial weak- configurations was possible for this individual that ening of coupling and then a sudden strengthening of had attained weak negative coupling; the choice this coupling, leaving the reason for this change to the depended on initial conditions. Thus, a single brief domain of biological unexplained complexities. perturbation could shift the system from one configu- These considerations explain complex activity pat- ration to the other. terns, such as those observed by Swade and Pittendrigh Our interpretation of splitting in the study by in arctic ground squirrels in LL (Pittendrigh, 1960). In Earnest and Turek (1982) is that 44% of their hamsters these experiments, two bands of activity dissociated, had weak negative coupling and did not split. Of the and the phase difference of the two bouts increased remaining individuals, 75% had moderate negative until reestablishing the original fused activity. We argue coupling (gradual onset of splitting), and only 25% that the final fusion of activity bouts is either due to the had strong negative coupling (abrupt onset of splitting). asymmetry of component oscillators or to weak cou- Finally, if coupling is even weaker then that shown pling. In either case, phase leadership is exchanged in Figure 6E, it is insufficient to phase-lock oscillators after dissociation of the two bands, so that the initial with different periods, which then run with relative
86 JOURNAL OF BIOLOGICAL RHYTHMS / February 2002
coordination (not shown). This pattern does not occur phase relationships are shown to be exclusively due to
with wild-type hamsters but does occur with tau- coupling sign switches, and variable transient times
mutant hamsters (Menaker et al., 1994). are shown to be due exclusively to variable final cou-
pling strengths. A slowly evolving switch in coupling
sign would not allow this distinction, mixing the causes
DISCUSSION of transients. All patterns can accommodate either an
abrupt or slowly evolving switch, except the abrupt
onset of splitting (Fig. 1E). This suggests that maybe
Limitations of the Model there is another source of variability in this phenome-
non—namely, the rate of switch in coupling sign. This
Our study of coupled oscillators is based on com-
rate might provide a source for the distinction between
puter simulations, which provide a phenomenological
short- and long-term effects of constant light on the
view of the behavior of the system as parameters are
system.
changed. More general and precise analytical methods
are, however, limited to the study of linear systems or
one-dimensional phase oscillators, which are good Main Aspects and Relationship
approximations only for weakly coupled oscillators. to Previous Models
Whereas analytical studies provide mathematically
explicit generalizations of oscillator properties, simu- It has long been assumed that splitting results from
lation-based studies can at most indicate that any spe- a phase dissociation of the two oscillators, either due
cific property is widely valid. To overcome this limita- to a weakening of coupling or detuning of the two
tion, we performed systematic and very extensive oscillators due to differential effects of constant light
explorations of parameter space for the Pittendrigh- on their periods (Pittendrigh and Daan, 1976; Daan
Pavlidis equations, manipulating the system with a and Berde, 1978). According to the latter interpreta-
wide set of values for the four parameters. Although tion, constant light caused the period of E to increase
our studies do not span the entire parameter space, the more than that of M, resulting in a temporary loss of
results presented here are generally valid for the broad coupling. The two oscillators free-run with M phase-
set of coupled oscillator systems represented within leading, due to its shorter intrinsic period, until lock-
this space. Furthermore, simulations using van der ing to 180°. An explanation for the antiphasic relation-
Pol oscillators in a narrower range of parameter space ship attained by the oscillators was based on the
provided the same general properties for phase differ- bistability of in-phasic and antiphasic relations in cou-
ences and phase leaderships between coupled oscilla- pled oscillators. This explanation was obtained by
tors observed using Pittendrigh-Pavlidis equations, analytical studies restricted to weakly coupled oscilla-
indicating the generality of our main findings (data tors, using phase-only oscillators (Kawato and Suzuki,
not shown). 1980). Diurnal Tupaia present splitting as the light
The degree of symmetry is only measurable in intensity is decreased below a threshold value. When
parts. We set three measures of this symmetry— this intensity is increased, the threshold value above
namely, the oscillator properties, the sign of coupling, which activity fuses again is not the same as the one in
and its strength. We can only say that one system is which splitting started. This experimental fact strongly
more or less symmetric than the other with respect to supports the previous model because of the mathe-
each of these measures separately. Furthermore, we matical association between hysteresis and bistability.
can only measure the degree of symmetry of compo- Nocturnal hamsters present splitting as a result of
nent oscillators with respect to each parameter—a, b, c, long-term effect of constant light; there is still no evi-
and d—separately. dence of the influence of light intensity on this phe-
This model does not account for the effect of light nomenon or on hysteresis. Furthermore, hamsters
intensity on the onset of splitting. This intensity has present variable splitting patterns that the previous
important implications for the phenomena, as shown model does not accommodate, especially the abrupt
experimentally for the diurnal animal Tupaia (Hoffmann, onset of splitting shown by Earnest and Turek (1982).
1971). Our computer simulation of two-dimensional oscilla-
Finally, we assumed, for the sake of simplicity, that tors spans a larger range of coupling strengths, broad-
the switch in coupling sign is instantaneous, rather ening the picture of coupled oscillator behaviors. Fur-
than a slowly evolving process (Fig. 6). Changes in thermore, we tried to avoid a common danger ofOda and Friesen / MODEL OF SPLITTING 87 simulation-based models—namely, that of constructing (Daan et al., 2001). In light of the conjecture that split- reproductions of experiments based only on ting is a phenomenon whose anatomical or mechanis- nonsystematic trial-and-error parameter adjustments. tic origins do not coincide with those of E and M oscil- Instead, we first generated a general picture of the lators, our proposition that different mechanisms may behavior of coupled oscillators, differing in structural underlie the splitting phenomenon in hamsters and symmetry, by a systematic modification of parame- Tupaia is not unreasonable. ters. We then searched in that background for an One implication of our study is that the two split- ordered picture that could accommodate all splitting ting oscillators in hamsters must necessarily have dif- patterns. As a result, we verified that indeed we need ferent intrinsic periods and amplitudes. Their asym- to include a wide range of coupling strengths, from metry was not necessary to explain the differential weak to strong, to understand the splitting phenome- shifts of the beginning and end of activity due to light non in nocturnal hamsters with all its variants. pulses in hamsters (Oda et al., 2000), but it is an inte- In our model, all splitting patterns were generated gral element in our model to account for all the vari- by one novel mechanism—a switch in coupling sign. able splitting patterns. Another implication of this This mechanism was introduced as the main assump- model is that, in hamsters, the oscillators do not disso- tion in our simulations based on the general picture of ciate due to loss of entrainment. Instead, they assume coupled oscillators. The successful simulation of all new phase relationships as determined by negative splitting patterns, which the previous models could coupling, which alters their amplitudes. If phase- not accommodate, strongly indicates that a switch in shifting experiments were performed for split compo- coupling sign might be the main long-term effect of nents under LL, the results would not provide the constant light on the pacemaker of nocturnal ham- intrinsic PRCs of the two oscillators because the sters. On the other hand, our model does not explain phase-response curve depends on the amplitude of the hysteresis presented by diurnal Tupaia, which sup- the oscillator, and that amplitude is modified by the ported the previous models. Perhaps different mecha- negative coupling that exists between split components. nisms underlie splitting in hamsters and Tupaia, both To summarize, we propose that splitting in ham- of which lead to antiphasic activity patterns. sters is caused by a change in coupling sign between The existence and location of dual circadian oscilla- pacemaker component oscillators: a switch from posi- tors have been demonstrated in invertebrates (Page, tive values in LD to negative values following pro- 1985; Koehler and Fleissner, 1978; Roberts and Block, longed exposure to LL. If the two component oscilla- 1985). It is known that isolated neurons within the tors are indeed located in left and right SCN, their SCN of rodents already exhibit circadian clock func- interactions by excitatory and inhibitory synapses (Wil- tions (Welsh et al., 1995; Liu et al., 1997; Herzog et al., son and Cowan, 1972) could be tested. In addition, the 1998), suggesting that a population comprising thou- strength of interactions could be tested to determine sands of oscillators underlies the mammalian circa- where there is a correlation between synaptic poten- dian clock. Several studies have tried to identify the E tial amplitude and patterns of splitting. and M oscillators among subpopulations of these SCN neurons. In some studies, the right and left SCN were examined (Pickard and Turek, 1982; Davis and Gorski, ACKNOWLEDGMENTS 1984; Zlomanczuk et al., 1991). The circadian rhythm in SCN electrophysiological activity, recorded as a sin- We thank Professor Michael Menaker for reviewing gle daytime peak in hamster hypothalamic slices, shows a preliminary draft of the manuscript and Drs. Shin two distinct peaks when slices are cut in the horizontal Yamazaki and Manuel Miranda-Anaya for helpful plane (Jagota et al., 2000). Finally, a more recent study discussions. G.A.O. was supported by a postdoctoral showed that, under split conditions, the left and right fellowship from Fundação de Amparo à Pesquisa do SCN oscillate in antiphase, demonstrating that these Estado de São Paulo (97/13910-0), and W.O.F. was might comprise the two oscillators that the splitting supported by the National Science Foundation (IBN phenomenon unveils (de la Iglesia et al., 2000). These 97- 23320). We especially acknowledge support from two oscillators, however, might not correspond to the the NSF Center for Biological Timing. ultimate evening and morning oscillators that are now Methodological details are available upon mail conjectured to be located at a deeper, molecular level request.
88 JOURNAL OF BIOLOGICAL RHYTHMS / February 2002
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