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Entrainment of noise-induced and limit cycle oscillators under weak noise

Theoretical models that describe oscillations in biological systems are often either a limit cycle oscillator, where the deterministic nonlinear dynamics gives sustained periodic oscillations, or a noise-induced oscillator, where a fixed point is linearly stable with complex eigenvalues and addition of noise gives oscillations around the fixed point with fluctuating amplitude. We investigate how each class of model behaves under the external periodic forcing, taking the well-studied van der Pol equation as an example. We find that, when the forcing is additive, the noise-induced oscillator can show only one-to-one entrainment to the external frequency, in contrast to the limit cycle oscillator which is known to entrain to any ratio. When the external forcing is multiplicative, on the other hand, the noise-induced oscillator can show entrainment to a few ratios other than one-to-one, while the limit cycle oscillator shows entrain to any ratio. The noise blurs the entrainment in general, but clear entrainment regions for limit cycles can be identified as long as the noise is not too strong.

Biological systems present us with a wide range of oscillators, which include cell cycles, circadian rhythms, calcium oscillations, pace maker cells, and protein responses, but it is often a challenging task to identify the minimal models behind these oscillations. The proposed models are typically categorized into two classes: (i) Limit cycle oscillator, where fixed points are linearly unstable and the oscillations are described by stable limit cycles sustained by nonlinearities of the system which are deterministic. Noise can be added on the top of the deterministic oscillations. (ii) Noise-induced oscillator, where the fixed point is linearly stable for the system without noise and the system relaxes to the fixed point with damped oscillations when temporally perturbed. Addition of noise to this type of system is known to show sustained oscillations with fluctuating amplitudes. We propose a way to distinguish the two, by using the phenomenon of entrainment to a periodic perturbation. Taking the van der Pol equation with noise as an example, we show that entrainments to all the rational ratios are seen only in the limit cycle oscillator. In the case of the noise-induced oscillator with additive external forcing, the oscillator can entrain only at one-to-one ratio, meaning that the entrainment to other than the one-to-one ratio is the sign of the dominance of the limit cycle mechanism. When the external forcing is multiplicative, we find that the noise-induced oscillator with weak nonlinearity can show some entrainment ratios other than one-to-one, but not all the ratios.

I. INTRODUCTION
Biological systems present us with a bewildering fauna of oscillators: cell cycles 1 , circadian rhythms 2-5 , calcium oscillations 6 , pace maker cells 7 , protein responses [8][9][10][11][12][13][14] , and so on. Sometimes, however, it is hard to see what are the minimal models behind these oscillations. Typically, the models are categorized into two classes: (i) Limit cycle oscillator: The fixed point is linearly unstable and the oscillations are described by stable limit cycles sustained by nonlinearity of the system in the deterministic case 11,12 . Noise (e.g. molecular noise due to limited number of copy numbers) can be added on the top of the deterministic oscillations. (ii) Noise-induced oscillator: The fixed point is linearly stable for the system without noise and the system relaxes to the fixed point with damped oscillations when temporally perturbed. Addition of noise to such a system is known to show sustained oscillations with fluctuating amplitude 13,14 . For some systems, both limit cycle oscillators (i) and noise-induced oscillators (ii) are proposed as a mechanism for the oscillation [11][12][13] .
Here, we propose a way to distinguish the two, by using the phenomenon of entrainment to a periodic perturbation.
It is well known that, when an periodic perturbation is added to a deterministic limit cycle, the system's oscillation frequency ω will be entrained to the external frequency Ω with various rational numbers of frequencies ω/Ω = P/Q for all positive integers P and Q in a finite window of the external frequency Ω, where the width of the window depends on the amplitude of the external forcing 15,16 . Entrainment, also called mode-locking, has been observed in variety of physical systems during the last decades, from onset of turbulence 17 , Josephson junctions 18,19 , one-dimensional conductors 20 , semiconductors 21,22 and crystals 23 .
It has been predicted, and verified experimentally, that the mode-locking structure possesses certain universal properties 15,16 . In biological systems, entrainment has been investigated theoretically for circadian rhythms 4,5 as well as in model systems for protein responses 25 .
Experimental observation of entrainment in biological systems is often rather difficult due to noisy signals, but it has been observed for circadian rhythms 2,3 and synthetic genetic oscillators 24 .
In this paper, we study the difference in the entrainment behavior for the limit cycle oscillators and noise-induced oscillators.
Our main question is the following: Can we distinguish the two cases by means of the entrainment behavior? We employ the famous van der Pol equation with noise as an example, because there we can easily study both cases by changing parameters. We show that entrainments to all the rational ratios are seen only in the limit cycle oscillator. In the case of the noise-induced oscillator with additive external forcing, the oscillator can entrain only at one-to-one ratio, meaning that the entrainment to other than the one-to-one ratio is the sign of the dominance of the limit cycle mechanism.
When the external forcing is multiplicative, we find that the noise-induced oscillator with weak nonlinearity can show some entrainment ratios other than one-to-one, but not all the ratios.
To confirm the generality of the entrainment behavior for the limit cycle system under weak noise, we also study a biological example, the TNF-driven oscillating NF-κB system, and confirm that P/Q entrainments can be seen.

A. van der Pol equation
Consider the following two-dimensional equation with noise: Here, d, σ, and B are parameters, and Γ i (t) are uncorrelated, statistically independent Gaussian white noise, satisfying First let us consider the deterministic case, σ = 0. The model has a fixed point at (x 1 , x 2 ) = (0, 0), and the eigenvalues around this fixed point are indicating that the system experiences a Hopf bifurcation at d = 0. When d < 0, the fixed point relaxes to the fixed point with damped oscillation with the angular frequency ω (d) = |d 2 − 4|/2, while when d > 0 and B > 0 the model shows a stable limit cycle (van der Pol oscillator).
In the stochastic case with σ > 0, however, the system shows a sustained oscillation even in the linearly stable case, d < 0, because the noise keeps activating the oscillation with frequency ω . This is the case of the linear p53 model introduced in Ref. 13 . When d > 0, σ > 0 adds fluctuations on top of the stable oscillation around the limit cycle.

B. Setup
We investigate the entrainment behavior of the model, focusing on the following three classes of parameter sets.
2. For the noise-induced oscillator, we consider the two subcategories.
(a) The linear system with a stable fixed point, with d < 0 and B = 0, i.e., the equations are linear in x and the fixed point is stable.
(b) The nonlinear system with a stable fixed point, with d < 0 and B > 0, i.e., the fixed point is linearly stable but the equations possess a nonlinear term.
When noise-induced oscillators are studied, normally only linear terms are considered. However, in reality, there are often nonlinear terms, which can play a role when distance from the stable fixed point |x| is sufficiently large. This is the reason why we consider both linear and nonlinear noise-induced oscillators.
When needed, numerical integration of stochastic differential equations are performed by using Euler method. Figure 1 shows the typical behavior of the model in each categories. The parameters are chosen so that the period and amplitude are in similar range. Without noise, the limit cycle is the only case with stable oscillation (Fig. 1a), while linear and nonlinear systems with a stable fixed point exhibit damped oscillations relaxing to the fixed point (Fig. 1bc).
When noise is added, the oscillation is perturbed for limit cycle oscillator (Fig. 1d); here the noise level is chosen so that the base oscillation is still recognizable. For linear and nonlinear noise-induced oscillators (Fig. 1ef), we observe oscillations with the expected angular frequency (ω (−0.1) ≈ 1). In order to demonstrate the difference between the two, we apply the exact same sequence of noises in both cases. We observe a bigger difference when linear noise-induced oscillator have large (|x| ≈ 1) amplitude, because the nonlinear term becomes more important. Naturally this effect depends on the value of B (data not shown).
We study these oscillators under the following two kinds of external periodic perturbation.
a. Additive forcing. The first case is an additive forcing, in the form oḟ with b. Multiplicative forcing. The second case is an multiplicative forcing (also called parametric forcing), in the form ofẋ where In the next section, we first present the behavior of the model under the additive forcing, and then show the parallel results for the multiplicative forcing.

III. RESULTS
A. Additive forcing

Linear case
In the case of the additive periodic forcing to a linear deterministic system, we have in where L is a coefficient matrix of the linearized equation, and A(t) is periodic function in time with a period T , satisfying A(t + T ) = A(t).
By expressing x(t) = d j=1 C j (t)u j , with using eigenvectors u j of the matrix L given by Lu j = λ j u j , we can show that in the long-time limit we have Note that (λ j ) < 0 because the fixed point x = 0 is stable. F n is defined by the Fourier where v t j is the left eigenvector. Therefore the solution will always be a periodic function of t with the period T in the long time limit, and contains only the frequencies that the external forcing has. In other words, the system will be always in a 1/1 entrained state if the perturbation is pure sine or cosine wave.
When Gaussian white noise is added to eq. (12), we havė In this case, we can evaluate the auto-correlation of C j (t) for large enough t 0 Namely, the response contains oscillations with the frequencies from forcing 2π/T and from the complex part of the eigenvalue λ j , and the amplitude of the latter is proportional to σ.

Numerical results
We now investigate numerically the entrainment behaviors for all three categories. First we demonstrate the behavior without noise, and then show how the noise modify this behavior.
a. Without noise. Figure 2 illustrates typical entrainment behaviors for additive forcing when noise is absent. With a limit cycle oscillator ( Fig. 2 a, d), the system's angular frequency can entrain to the external angular Ω with various ratios, while in the linear system, one-to-one entrainment occurs ( Fig. 2 b, e). The nonlinear system shows very similar behavior to the linear system, where we see only one-to-one entrainment ( Fig. 2 c, f).
In order to define the system's angular frequency in a simple way, we adopt the polar coordinate (r, θ) using as proposed in Ref. 26 . We define θ(t) so that (θ(t) − θ(0))/2π gives the winding number, i.e., how many times the orbit went around the fixed point by time t. The system's angular frequency is numerically calculated from for long enough T (typically 1000 times external forcing period). With this definition, Fig.   2(a) shows the entrainment of the ratio ω/Ω = 2/1, while Fig. 2(d) gives ω/Ω = 1/2.
b. With noise. The addition of noise blurs the entrainment behavior, as depicted in Fig. 3. For the limit cycle oscillator ( Fig. 3a and d), we can see that the noise makes the orbit irregular, which can make the phase to slip. In the linear noise-induced oscillator for small external angular frequency, we can clearly see that the noise induces the oscillation with angular frequency close to ω on top of one-to-one entrainment behavior (Fig. 3b), as expected from the auto-correlation eq. (13). When Ω is larger than ω , the external angular frequency is more visible, because the noise σ is small compared to the amplitude A for this case, though both frequencies should be present. The nonlinear noise-induced oscillator behaves again very similar to the linear case in entrainment behavior ( Fig. 3c and f). The visible difference is a suppression of large amplitude by the nonlinear term.
c. "Devil's staircase" and "Arnold's tongues". For deterministic limit cycles, the plot of ω/Ω vs Ω for a fixed amplitude of external forcing shows a infinitely complex structure with fractal nature, known as Devil's staircase 15,16,26 . For the present system of limit cycle oscillator without noise, this is also observed as shown in Fig. 4(a) (solid line). As noise increases, the phase slips occasionally, therefore narrow entrainment regions become harder to recognize (Fig. 4a, dashed and dotted line). For the systems with a stable fixed point, there is only one-to-one entrainment for the no noise case (Fig. 4b, solid line), while noise induced oscillation around the entrained solution will add some phase slips giving a change in the angular frequency when the entrainment is not so strong, resulting in an escape from the one-to-one ratio as shown in Fig. 4(b).
When entrainment regions for various values of ω/Ω are plotted in the A-Ω plain, it gives an "Arnold's tongue" structure for the deterministic limit cycles: The entrainment regions widen as the external forcing amplitude A grows, resulting in tongue-like shapes of the entrainment region -when A is large enough the tongues start to overlap 15,16 . This can be seen in the limit cycle oscillator without noise in Fig. 5 (a). When noise is added, the phase of the oscillator sometimes slips, resulting in narrower tongues (Fig. 5 b). For the noiseinduced oscillators (i.e. with a stable fixed point), there exists only 1/1 entrainment without noise, and with noise 1/1 entrainment is the only case that gives the tongue-like structure, both for the linear and nonlinear cases (Fig. 5 cd). We see other ratios of entrainment "regions", because for a given A with changing Ω, ω/Ω changes continuously outside of the entrainment region (e.g. Fig.4 b).
B. Multiplicative forcing

Linear case without noise
We next consider the multiplicative forcinġ where the matrix M(t) satisfies with T = 2π/Ω. It is known from Floquet theory 27 that the solution matrix of this equation is expressed as where and a general solution is the linear combinations of column vectors consisting of Q(t). The eigenvalues of the matrix Λ, called Floquet exponents, determine the stability of the solution: The solution will converge to the fixed point when the real parts of the Floquet exponents are all negative, and diverges if some Floquet exponent have positive real parts. Therefore, no entrainment behavior will be observed for a linear noise-induced oscillator without noise under multiplicative forcing.
In Fig.6, we show numerically calculated the maximum real part of the Floquet exponents λ R for (17) with (9) with d = −0.1 and B = 0, as a function of amplitude of forcing M and external frequency Ω. When λ R < 0 (dark blue region), |x| will exponentially decays to zero, otherwise |x| will diverge except for the marginal case λ R = 0.

Numerical results
a. Without noise. Figure 7 shows the entrainment behaviors for multiplicative forcing.
For the limit cycle oscillator (Fig. 7 ac), there is no qualitative difference from the additive noise case, i.e., the system shows entrainment with various frequency ratio ω/Ω = P/Q. For the linear system with a stable fixed point without noise, on the other hand, the system can either decay to the fixed point (Fig. 7 b) or diverge (Fig. 7 d), which can be predicted from the Flouquet exponents (Fig.6). When nonlinear term is added, it does not prevent the decay (Fig. 7 c), but the divergent behavior is suppressed and system shows the entrainment behavior (Fig. 7 f). The frequency ratio ω/Ω is not necessarily 1/1; the example in Fig.   7(f) gives ω/Ω = 3/2.
b. With noise. When noise is added, the behavior changes drastically in the noiseinduced oscillators, as shown in. Fig. 8. The noise can induce the oscillation with the angular frequency close to ω for the case where the no-noise system would decay to the fixed point (Fig. 8 b, c). On the other hand, in the linear noise-induced oscillator, adding noise does not prevent divergence (Fig. 8 e). For the parameters where no-noise system would entrain, the noise blurs the entrainment due to occasional phase slip for both limit cycle oscillator (Fig. 8 ac) and nonlinear noise-induced oscillator (Fig. 8 f).
c. "Devil's staircase" and "Arnold's tongue". We also study the "Devil's staircase" for the multiplicative forcing. For the limit cycle oscillator without noise, we again see proper devil's staircase, where noise will blur the entrainment behaviors (Fig.9 a). For the noise-induced oscillators, only the nonlinear case is studied because the linear case may diverge depending on the parameter values. Without noise, we see discrete finite regions of entrainment ( Fig.9 b squares), while noise induces the oscillations in the decaying region resulting in a continuous line ( Fig.9 b dashed line).
The Arnold's tongue structure for the limit cycle is similar to those in the additive forcing case, as seen in Fig.10(a) and (b). The Arnold's tongues for all the entrainment ratios are observed without noise, and noise makes the regions smaller. For the nonlinear system with a stable fixed point without noise, there are entrainment regions for a few rational ratios, but the ones that appear are problem specific -for instance, in the present case, the ω/Ω = 1/3 is not observed at all in Fig.10(c). With noise (Fig.10d), the entrainment regions shrinks, but at the same time the system can occasionally pass the given ratio of ω/Ω, resulting in narrow line of "fake" entrainment.

IV. BIOLOGICAL EXAMPLE: ENTRAINMENT OF TNF-DRIVEN NF-κB SYSTEM
In this section, we study a biological example, TNF-driven NF-κB system, to confirm the generality of the entrainment behavior for the limit cycle system under weak noise.
The system has been studied for the deterministic case in ref. 25 . NF-κB is a transcription factor, and it has been verified experimentally that NF-κB level in the nucleus shows sharp oscillations after treatment with tumor-necrosis factor (TNF) 33,34 . The interaction network involves a negative feedback loop between the NF-κB and an inhibitor, IkBα, which is the main mechanism for the oscillations. TNF modulates the state of the IkBα and hence affects the oscillation. TNF can be added externally to the cell, therefore it can serve as a possible probe to study the entrainment, i.e., we can use TNF level as the external forcing term. In ref. 25 , the system was modeled by 5 dimensional coupled nonlinear ODEs, The variable x 1 denotes the nuclear NF-κB level, and [T N F ] denotes the TNF level, which we shall change to a periodically external forcing onto the system. The biological meaning of the variables and parameter values are summarized in Table I. Note that [T N F ] appears twice in eq. (25) in the terms multiplied with x, therefore this is an example of multiplicative forcing. We study this system with adding a Gaussian white noise in each term, i.e., This has been studied in the no-noise case by Jensen and Krishna 25 , and it was found that the entrainments of various ratios can occur, when the frequency of the NF-κB level is determined based on the frequency of the peaks. Figure 11(c) shows an example of 1/2 entrainment, for M T N F = 0.05 and Ω = 0.0297, in the deterministic case. With weak enough noise, the entrainment is maintained (Fig. 11e), but larger noise induces phase slips ( Fig. 11f), as has been seen in the Van der Pol system.
In Fig. 12, several Devil's staircases are shown with and without noise. In Ref. 25 the Arnold tongues have been calculated, and it has been demonstrated that general P/Q entrainments occur. A characteristic observation to this system is that the tongues overlap easier for larger external frequency; e.g., M T N F ≈ 0.04 for 1/3 and 1/2 tongues to overlap, while the 2/1 and 5/2 tongues do not overlap even at M T N F = 0.1. When the Devil's staircase are calculated for overlapping region, non-smooth or irregular jumps between the steps can be seen, and is thus dependent on initial conditions in general. This is visible in our data in Fig.12, for large M T N F and larger Ω. When weak noise is added, it enables the system to jump to other overlapping tongues, which results in irregular behavior around the entrainment regions. As the noise becomes larger, the entrainment is again smoothed away by phase slips.

V. SUMMARY AND DISCUSSION
Our motivation behind this work was to ask: Can one by applying an external periodic forcing and studying entrainment determine whether an oscillating system is driven by a linear mechanism (noise induced oscillator) or a non-linear mechanism (limit cycle oscillator)? Our answer to this question is generally yes. Our obtained results on entrainment behavior of oscillators are summarized in Table II.
When the forcing is additive, there is clear difference between the limit cycle oscillators and the noise-induced oscillators. The former can entrain to any frequency ratio, while the latter shows only one-to-one entrainment. Therefore, if one see entrainment to P/Q = 1 ratio, under additive forcing, it is a sign of limit cycle oscillator.
When the forcing is multiplicative, the non-linear noise-induced oscillators can also show P/Q = 1 ratio entrainment, but not necessarily all of the rational ratios. If the system is noise-induced oscillator and the non-linear term is small, one might be able to capture the diverging tendency of the amplitude, because saturation happens when the amplitude is large enough to make the non-linear term relevant. In such a case, one might see big difference in amplitude for a fixed M with varying Ω.
We thus urge experiment to be performed on oscillating biological systems. It is well known that some proteins (p53, NF-kB, Wnt) can oscillate in cells under stress responses.
In the case of p53, both non-linear 11,12 and linear models have been proposed 13  Finally, we would like to briefly comment on "noise-induced" oscillations by mechanisms other than the linear model studied here. It has been long known that, when noise is added to excitable system with a stable fixed point, regular oscillatory behaviour can be observed at a certain level of noise (coherence resonance) 28,29 . Since the nonlinearity plays an important role in an oscillation, such a system shows mode-locking behaviour similar to the deterministic nonlinear oscillators 30 . More recently, in gene network models with negative feedback, it has been shown that the noise due to finiteness of the number of molecules can modify the condition for oscillatory behaviour 31 or enhance the oscillation 32 .
It would also be interesting to see the entrainment behaviour in such systems.     The horizontal axis is the external frequency Ω, and the vertical axis is the forcing amplitude M .
Entrainment is defined as ω/Ω is within 1% of the given value. For (c), the exponentially decaying case were excluded numerically by the following way: The equations are integrated with initial condition x(1) = 1 and x(2) = 0, and if the average amplitude for 390π/Ω < t < 400π/Ω is less than 90% of the average amplitude for 200π/Ω < t < 210π/Ω, then the solution is excluded.   Limit cycle entrainment to any P/Q entrainment to any P/Q with phase slips A entrainment to any P/Q entrainment to any P/Q with phase slips M Linear one-to-one entrainment * one-to-one entrainment * with phase slips A noise-induced decay or diverge noise-induced oscillation with ∼ ω M or diverge Nonlinear one-to-one entrainment * one-to-one entrainment * with phase slips A noise-induced small decay or noise-induced oscillation with ∼ ω or M some P/Q entrainment some P/Q entrainment with phase slips * All the frequencies contained in the forcing can be observed. regions are calculated from the frequency of the peaks in the the deterministic case. In the finite noise case, we define the nuclear NF-κB peak as follows: We first determined the maximum value N max and the minimum value N min of x 1 of the steady state in the deterministic simulation for the given parameters. We then calculate two thresholds, N H = (N max + N min )/2 and N L = (N max + 3N min )/4. Next we perform the corresponding simulation with finite σ. We define a switching event from the "low" state to "high" state when x 1 exceeds N H , while the reverse switching happens when x 1 becomes smaller than N L . The number of peaks are calculated from how often the "high" states are reached. This way we can filter out the wiggly motion due to the noise and thus define the overall peak.