## Abstract

Nonlinear interactions among coupled cellular oscillators are likely to underlie a variety of complex rhythmic behaviours. Here we consider the case of one such behaviour, a doubling of rhythm frequency caused by the spontaneous splitting of a population of synchronized oscillators into two subgroups each oscillating in anti-phase (*phase-splitting*). An example of biological phase-splitting is the frequency doubling of the circadian locomotor rhythm in hamsters housed in constant light, in which the pacemaker in the suprachiasmatic nucleus (SCN) is reconfigured with its left and right halves oscillating in anti-phase. We apply the theory of coupled phase oscillators to show that stable phase-splitting requires the presence of negative coupling terms, through delayed and/or inhibitory interactions. We also find that the inclusion of real biological constraints (that the SCN contains a finite number of non-identical noisy oscillators) implies the existence of an underlying non-uniform network architecture, in which the population of oscillators must interact through at least two types of connections. We propose that a key design principle for the frequency doubling of a population of biological oscillators is inhomogeneity of oscillator coupling.

## 1. Introduction

Rhythmic behaviours of higher organisms can emerge from the temporal coordination of cellular oscillators within tissues. This kind of design may lead to complex states that arise from nonlinear mechanisms (Kuramoto 1984; Strogatz 1994; Winfree 2001), including dysrhythmias (e.g. chaotic activities and abrupt changes of frequency) and anomalous spatial patterns (e.g. turbulence and spiral waves). The control of intercellular synchronization within tissues is likely to underlie a variety of normal functions as well as certain pathologies.

One biologically important behaviour is an abrupt doubling of rhythm frequency; this may be normal, as seen with transitions in locomotor cadence (Grillner *et al.* 1979), or abnormal, as in cardiac tachyarrhythmias (Ritzenberg *et al.* 1984). Theoretical analyses of coupled oscillators have suggested a possible mechanism for frequency doubling. A group of synchronized cellular oscillators could double its frequency by spontaneously splitting into two subgroups, each subgroup oscillating with a common frequency but now in anti-phase (Schuster & Wagner 1989; Okuda 1993; Daido 1996). We call this phenomenon *phase-splitting*, by analogy with terminology used in mathematical physics (Basler *et al.* 1998; Auffeves *et al.* 2003). Our aim now is to better understand biological phase-splitting, and here we apply mathematical reasoning to infer the essential design principles, guided by the real constraints of an actual living tissue.

Recently, an example of phase-splitting behaviour in a well-characterized neural tissue has been demonstrated (de la Iglesia *et al.* 2000). The suprachiasmatic nucleus (SCN), a bilaterally paired cell group in the anterior hypothalamus of the mammalian brain, is the site of an endogenous timekeeping mechanism that regulates 24 hour (circadian) rhythmicity and its entrainment to day and night (Takahashi *et al*. 2001). It is a tissue clock composed of multiple single-cell circadian oscillators coupled together to generate a circadian output signal. In golden hamsters, as in other rodents, the SCN regulates the night-time expression of locomotor (wheel-running) activity during the light–dark (LD) cycle and its persistent rhythmicity in constant darkness (DD; figure 1*a*). Hamsters housed in constant light (LL) also show rhythmic locomotion, with a longer period than in DD (figure 1*b*). After a few months in LL, hamsters can exhibit a phenomenon known as ‘splitting’, in which an animal's single daily bout of locomotor activity dissociates into two components that each free-run with different periods until they become stably coupled approximately 180° (12 hours) apart (figure 1*b*). DD rapidly restores the split rhythm to its normal unsplit state (figure 1*b*). Recent data show that the SCN is dramatically reorganized in the split condition; its left and right halves continue to oscillate with a circadian period but now in anti-phase rather than in-phase (de la Iglesia *et al.* 2000). It has been proposed that this reconfigured phasing of cellular oscillators within the SCN tissue leads to the frequency doubling of the locomotor rhythm.

## 2. An idealized model of a population of limit-cycle oscillators

If individual cellular oscillators are governed by limit-cycle processes, and if the attraction to the limit cycle is strong relative to the coupling between oscillators, then the global behaviour of the population can be specified solely by the phases of the oscillators without regard to their amplitude of oscillation. This concept was originally presented as a conjecture by Winfree (1967) and subsequently supported by others using mathematical arguments (Kuramoto 1984; Strogatz 2000). Such a system can be formally described by(2.1)where *θ*_{i} represents the phase of the *i*th oscillator; *ω*_{i} denotes its intrinsic frequency; *K*_{ij} is the strength of coupling between the *i*th and the *j*th oscillator; and *f* specifies how the coupling varies with phase. Kuramoto (1984) and others (reviewed by Strogatz 2000) described oscillator synchronization in such a system, which also applies to cells that communicate strongly via multiple pulse interactions throughout the limit cycle (Ermentrout & Kopell 1991). If *f* is an odd periodic function (e.g. a sine function) and the coupling strength is symmetrical (i.e. *K*_{ij}=*K*_{ji}), then the overall frequency of the synchronized system (*Ω*) equals the average of the individual oscillator frequencies (i.e. ). Of note, this averaging principle appears to hold for the SCN (Liu *et al.* 1997).

We wish to modify equation (2.1) to make explicit the possibility that there exist delays in the coupling terms. For example, intercellular communication within the SCN is not globally instantaneous. In addition, a circadian output signal (e.g. locomotion) feeds back to alter the phase of the SCN itself (Mrosovsky 1996), and the presence of light can modify the strength and timing of this feedback signal (Schaap & Meijer 2001). To incorporate such delays and feedback, we write(2.2)where represents the internal delay within the population and is the delay in the feedback of the population output upon itself; is the coupling term that specifies the intrinsic interaction between the *i*th and the *j*th oscillator; and is the coupling term that specifies feedback effects upon the oscillators.

In the idealized case, we consider *N* oscillators all uniformly coupled to each other (), uniformly receptive to a feedback signal () and with sine functions governing coupling between oscillators () and feedback upon the oscillators (). In the split condition, we define *θ*_{m} and *θ*_{n} as the phases of the oscillators within the two split subgroups and oscillating with a common frequency *Ω* but with their phases displaced by an angle *α*, where 0≤*α*≤*π*. We assume that the oscillators are nearly identical such that the deviation of the phases of the individual oscillators from the mean phase during synchronization is negligible, and we define as the difference in the mean frequencies of the two subgroups. Then, and . Substituting in equation (2.2), we obtainFor , *α*≈*π* cannot be a solution because *α* must be in the range of (−*π*/2, *π*/2), as defined by inverse sine functions. Hence, for phase-splitting (*α*≈*π*), and the solution is(2.3)The same reasoning and derivation has been presented previously by Schuster & Wagner (1989) for two coupled oscillators without feedback.

Equation (2.3) allows for phase-splitting (*α*≈*π*) with the conditions Δ*ω*≈0 and . This can be satisfied in a number of ways. For example, if the internal delay is negligible, then the feedback delay must be between *T*/4 and 3*T*/4 (where *T*=2*π*/*Ω*) and must be greater than . On the other hand, if both delays are negligible, then must be negative. This analysis shows that the presence of negative coupling terms, through delayed and/or inhibitory interactions, is necessary for phase-splitting in the idealized model.

## 3. Biological variability precludes phase-splitting in the idealized model

Equation (2.3) is derived for a population of nearly identical, noise-free oscillators, which is biologically implausible. In the SCN, individual cellular oscillators express intrinsic periods over a wide range of 20–28 hours (Welsh *et al.* 1995; Liu *et al.* 1997; Herzog *et al.* 1998; Honma *et al.* 1998). Each of these cellular oscillators exhibits cycle-to-cycle variability in the period of approximately 2 hours (Herzog *et al.* 2004). We now show that stable phase-splitting is not possible for the idealized model of equation (2.2) in the face of these known sources of biological variability.

If two subgroups of oscillators are each considered as single oscillator units, the only stable solution would be *α*→*π* (Schuster & Wagner 1989), and this would require that . Using the Chebyshev inequality (Papoulis 1984), it can be shown that this is achieved for non-identical oscillators only if the number of oscillators *N*→∞. In fact, the SCN contains no more than 20 000 oscillators (van den Pol 1980). Furthermore, it can be proved (see appendix A) that noise at any non-zero level will destabilize the split state of the idealized model. Figure 2 illustrates how the idealized phase-splitting (figure 2*a*) of equation (2.3) becomes destabilized when individual oscillators are non-identical (figure 2*b*) or when they are subjected to noisy perturbations of phase (figure 2*c*).

In actual experiments with hamsters, however, an observer might consider an animal stably split if the two locomotor activity bouts were to persist with a relatively unchanged *α*≈*π* for a finite observation period (e.g. 30 days). We ask whether there is any possible solution of the idealized model with a drift sufficiently small such that, for practical purposes, *α* would appear to be stable. In appendix A, we use stability analysis to estimate the minimum drift of *α* in the noiseless system and numerical simulations to incorporate noise-induced cycle-to-cycle variability in the period (Herzog *et al.* 2004). We found that, without noise, the minimum drift of *α* is 10 hours per week and the addition of noise causes loss of coherence of each split subgroup within 4 hours (see the end of appendix A for simulation parameters and results). Such a change in *α* over time and loss of coherence in the anti-phase solution would not be misinterpreted as stable splitting in the laboratory.

## 4. Stable biological phase-splitting requires non-uniform oscillator coupling

We now consider two constraints of the idealized model: the coupling functions ( and ) and the coupling parameters ( and ). Thus far, we have assigned the simplest odd periodic function (a sine function) for the coupling functions in equation (2.2). Hansel *et al.* (1993) have shown that stable anti-phase solutions can exist for a population of identical phase oscillators if the coupling function includes higher order harmonic terms. For such a solution to be relevant for circadian phase-splitting, the system must evolve from an initially synchronized state to the stable split condition. In appendix B we apply perturbation theory to determine whether such a transition is possible, and we find that the synchronized system of the idealized model (equation (2.2)) cannot evolve to the split state for any continuous or . This analysis does not exclude the possibility that phase-splitting might be achievable with discontinuous coupling functions, or that phase-splitting might be achievable via desynchronization to the incoherent state followed by instantaneous reconstitution of the anti-phase solution (Hansel *et al.* 1993). We know of no biological observations that would support the existence of such discontinuous behaviours in the SCN.

The idealized model also includes the constraint that the oscillator-to-oscillator coupling parameters and are uniform within the population. In fact, the SCN is markedly heterogeneous (Silver & Schwartz 2005), with multiple types of synaptic and non-synaptic connections. To analyse explicitly the impact of non-uniform network connectivity, we consider four coupled non-identical oscillators (figure 3*a*) whose behaviour is described by , where *ψ*_{i} is the phase displacement of the *i*th oscillator from the mean phase (*Ωt*) of the four-oscillator ensemble. The simplest non-uniform configuration of these oscillators is two subgroups defined by two different types of connections. Let and represent the coupling parameters within each subgroup, and and the coupling parameters across the two subgroups. We assign oscillators 1 and 3 to the first subgroup and oscillators 2 and 4 to the second subgroup, and define *α* as the mean phase difference across subgroups and *β* as the mean phase difference within each subgroup (figure 3*b*). If and if , it follows that and . Substituting for each of the four oscillators in equation (2.2), adding the solutions for each subgroup and subtracting between the subgroups, we obtain the following equations in place of equation (2.3):(4.1)and(4.2)For *α*≈*π*, the numerator of equation (4.1) must be approximately 0 while the denominator must be less than 0 (see §2). This is achieved by(4.3)and since *β* must be less than *α*, we obtain(4.4)Now consider a population *N* beyond four oscillators and with noisy perturbations *η*(*t*). We write(4.5)where and represent the mean fields corresponding to oscillators within and across the subgroups, respectively. We previously showed that, for uniform coupling , noise at any non-zero level will destabilize the split state of *N* identical oscillators. For non-uniform coupling of the simplest type (equation (4.5)) and using , we derive the stability criteria as(4.6)(4.7)For *α*=*π*,(4.8)Therefore, the system is stably split under the conditions of inequalities (4.3) and (4.4).

For *N* non-identical oscillators coupled by sine functions, the stability of phase-splitting of equation (4.5) can be estimated by numerical simulation with various intensities of noise and various coupling strengths and delays (see appendix C). We find that the existence of non-uniform network connectivity enables the system to remain stably split over a wide range of parameters, tested for up to 20 000 oscillators. The latency to onset of stable splitting (figure 4*a*) is prolonged by increasing the number of oscillators (figure 4*b*) or reducing the degree of heterogeneity, i.e. increasing towards (appendix C). Notably, over the range of *N*, the presence of noise reduces the latency to splitting (figure 4*b*). The angle *α* will be *π* if the mean frequency of the two subgroups is identical. A range of stable split angles around *π* is possible if the mean frequency of the two subgroups is different. For example, in a 20-oscillator network in which the mean frequency of the subgroups differs by 2 hours, , , , and hours, we find that the stable split angle is 167°.

Our numerical simulations show that the transition from synchronized to split states can exhibit different forms. An example similar to the most common transition seen in hamsters from synchronized to the split state (figure 1) is shown in figure 4*a*, in which there is a progressive separation of the phase difference between the two subgroups. The equation parameters influence the rate at which the two subgroups separate. Other transitions—arrhythmic or tri-branching—exist for certain parameter combinations and such transitions are rarely seen in hamster phase-splitting (see appendix D).

## 5. Discussion

Here we have sought the essential design principles for a population of synchronized coupled oscillators to spontaneously exhibit stable phase-splitting. We found that the inclusion of biological reality—a finite number of non-identical noisy oscillators—implies the existence of an underlying non-uniform network architecture, in which the population of oscillators must interact through at least two types of connections. This connectivity requirement follows from our assumption of continuous coupling functions. Whether there exist discontinuous coupling functions or instantaneous transitions that would obviate the connectivity requirement for phase-splitting of biological oscillators is a question for further mathematical study. We have also assumed that coupling among SCN cellular oscillators is weak relative to the strength of attraction to each cell's limit-cycle oscillation. This assumption is indirectly supported by the observations of Liu *et al.* (1997), but further work is needed to test whether SCN cells can act as phase oscillators and to estimate the strength and timing of coupling using quantitative experimental and modelling approaches at the interface between intracellular oscillations of ‘clock’ genes (Forger & Peskin 2003; Leloup & Goldbeter 2003; Becker-Weimann *et al.* 2004; Ueda *et al.* 2005) and intercellular coupling of neural activities (Pavlidis 1971; Carpenter & Grossberg 1983; Díez-Noguera 1994; Antle *et al.* 2003; Kunz & Achermann 2003; Nakao *et al.* 2004; Gonze *et al.* 2005; Bush & Siegelmann 2006; Liu *et al.* 2007; Sim & Forger 2007; To *et al.* 2007).

Our analyses show how coupling strengths and delays must interrelate in order to functionally partition the population into two split subgroups. The theoretical requirements imposed by inequalities (4.3) and (4.4) could be satisfied by a number of parameter combinations. In the case of hamster phase-splitting, the split subgroups are the left and right halves of the paired SCN, but higher order split clusters may exist (Tavakoli-Nezhad & Schwartz 2005; Yan *et al.* 2005). We also note that the heterogeneity requirement for stable phase-splitting could be satisfied by feedback coupling acting differentially on the oscillator network, which (in the simplest case of only two types of feedback connections) would yield a canonical phase equation whose form is identical to (4.5).

So how might LL induce phase-splitting of the hamster SCN? One possibility would be that the transition from DD to LL leads to a change in sign of from positive to negative; if all the delays are small relative to the overall circadian period, then conditions for stable phase-splitting are satisfied. Indeed, Oda & Friesen (2002), expanding on earlier models (Daan & Berde 1978; Kawato & Suzuki 1980), have considered this idea and performed numerical simulations using coupled relaxation oscillators. An alternative possibility would be that LL does not alter intrinsic SCN coupling but instead induces phase-splitting by altering feedback coupling . This would require that and are both positive in LL (as they are in DD to achieve synchronization). Since (Liu *et al.* 1997), all delays in DD must be small relative to the overall circadian period (see appendix A) and must be greater than (inequalities (4.4) and (4.8)). Under these conditions, the feedback delay required for stable phase-splitting can be estimated, given that . If , must be between and ; if , must be between and .

It is important to realize that, although LL is defined as constant light, what the animal actually perceives might be an entirely different matter. Typically measured are rest and activity cycles, but not intermittent sleep bouts, during which time the eyes are closed and the animal assumes a tightly curled, shielded sleeping posture tucked in the bedding. Thus, we predict that the overall photic input to the SCN during the rest phase of an animal in LL must be less than that during the active phase when the animal is awake running in the wheel. Running in the light in LL would therefore generate an oscillating, phase-shifted photic input to the SCN when compared with the usual situation in an LD cycle, in which a nocturnally active animal like the hamster restricts its running bouts to the dark. Our analysis shows that the long latency to splitting onset reflects nonlinearities in the system (equation (4.5)) and does not require any change in parameters during the LL incubation period. However, latency is parameter dependent (figures 4 and 5; appendix C) and there is at least one example in the literature in which latency is dramatically reduced. Administration of a 6 hour dark pulse timed to precede the expected onset of locomotor activity in LL has been reported to induce immediate splitting (Duncan & Deveraux 2000). If reflects the amplitude of a sinusoidal photic input, then an increase in at such a dark-to-light transition could trigger more rapid phase-splitting.

The central nervous system contains many bilaterally symmetrical neural oscillators in addition to the SCN, including those that underlie respiratory, autonomic and locomotor activities, and frequency and phase control are critical to their function. We suggest that this natural midline symmetry provides an architectural partition that could promote frequency doubling by a phase-splitting mechanism.

## Acknowledgments

The hamster wheel running data used in this article are from experiments that were approved by the Institutional Animal Care and Use Committee of the University of Massachusetts Medical School.

We thank Leon Glass for early discussions on circadian splitting and Steven Strogatz, Bard Ermentrout and Daniel Forger for reviewing a previous version of the manuscript. This study was supported in part by the National Institutes of Health R01 grants NS046605 and HL71884. The contents of this article are solely the responsibility of the authors and do not necessarily represent the official views of the NIH.

## Footnotes

- Received October 1, 2007.
- Accepted November 17, 2007.

- © 2007 The Royal Society