Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering

Abstract

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from random initial configurations of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength. A sharp transition is proved to exist that separates a weak coupling regime of unclustered populations from a strong coupling phase where clusters of extensive size are formed. Each phenomena occurs with full probability in the thermodynamics limit. Moreover, the maximum number of asymptotic clusters is known to diverge linearly in this limit. In contrast, we show that with positive probability, the number of asymptotic clusters remains bounded, provided that the coupling strength is sufficiently large.

Appendices

Appendix A: Mean estimates for configurations in $\mathcal{T }_N$

Throughout the proofs, we use the following estimate on the mean $\frac{1}{N}\sum \nolimits _{k=1}^{N}x_k$ for a subset of configurations $\{x_k\}_{k=1}^{N-1}\in \mathcal{T }_N$ that has arbitrarily large probability measure. The estimate is a straightforward consequence of the Central Limit Theorem. It can be stated as follows. Recall that the symbol $\mathbb{P }$ denotes the law of a random variable.

Lemma 7.1

For every $\delta \in (0,1)$, we have $\lim \limits _{N\rightarrow \infty }\mathbb{P }(|\frac{1}{N} \sum \nolimits _{k=1}^{N}x_k-\frac{1+\eta }{2}|<\delta )=1$.

Proof

Let $N\in \mathbb{N },N>1$ be fixed and for every configuration $x=\{x_k\}_{k=1}^{N-1}$, let $S_{N-1}(x)=\frac{1}{N-1}\sum \limits _{k=1}^{N-1}x_k$. The quantity $S_{N-1}$ is regarded as a random variable with law $\mathbb{P }$.

Consider now the random process in the hypercube $[\eta ,1]^{N-1}$ endowed with the uniform measure $(1-\eta )^{-(N-1)}\mathrm{{Leb}}_{N-1}$. For this process, the law of $S_{N-1}$ is simply $(1-\eta )^{-(N-1)}\mathrm{{Leb}}_{N-1}\circ S_{N-1}^{-1}$. A standard argument (presented at the end of this proof below) shows that we have $\mathbb{P }=(1-\eta )^{-(N-1)}\mathrm{{Leb}}_{N-1}\circ S_{N-1}^{-1}$.

For the process in the hypercube, the quantity $S_{N-1}$ appears to be the normalized sum of i.i.d. random variables $x_i$ with Lebesgue distribution in $[\eta ,1]$. The corresponding mean value is $\frac{1+\eta }{2}$ and the variance is finite. By the Central Limit Theorem, we conclude that for every $p\in (0,1)$ there exists $c_p>0$ and $N_p\in \mathbb{N }$ such that

$$\begin{aligned}&\mathbb{P }\left( \left| S_{N-1}-\frac{1+\eta }{2}\right| \leqslant c_p/\sqrt{N-1}\right) \nonumber \\&\quad \quad =(1-\eta )^{-(N-1)}\mathrm{{Leb}}_{N-1} \left( \left| S_{N-1}-\frac{1+\eta }{2}\right| \leqslant c_p/\sqrt{N-1}\right) >p,\quad \forall N>N_p. \end{aligned}$$

In particular, for every $\delta \in (0,1)$, we can ensure that $|S_{N-1}-\frac{1+\eta }{2}|<\delta /2$ holds with probability larger than $p$, provided that $N>\max \{N_p,(2c_p/\delta )^2+1\}$ (so that we also have $c_p/\sqrt{N-1}< \delta /2$). Furthermore, the normalization $x_N=1$ yields the following inequality

$$\begin{aligned} \left| \frac{1}{N}\sum \limits _{k=1}^{N}x_k-\frac{1+\eta }{2}\right| \leqslant \left| S_{N-1}(x)-\frac{1+\eta }{2}\right| +\frac{1}{N}(1-S_{N-1}(x)),\quad \forall x\in \mathcal{T }_N. \end{aligned}$$

By taking $N>\max \{N_p,(2c_p/\delta )^2+1,2/\delta \}$ (so that we also have $1/N<\delta /2$), we can be sure that $|\frac{1}{N}\sum \nolimits _{k=1}^{N}x_k-\frac{1+\eta }{2}|<\delta$ whenever $|S_{N-1}-\frac{1+\eta }{2}|<\delta /2$. The Lemma then immediately follows.

It remains to show the equality of laws $\mathbb{P }=(1-\eta )^{-(N-1)}\mathrm{{Leb}}_{N-1}\circ S_{N-1}^{-1}$. First, notice that we have

$$\begin{aligned} \mathrm{{Leb}}_{N-1}\circ S_{N-1}^{-1}&=\mathrm{{Leb}}_{N-1}\circ (S_{N-1}|_{C_{N-1}})^{-1}\quad \text{ where }\ C_{N-1}\nonumber \\&=\left\{ x\in [0,1]^{N-1}: i\ne j\Rightarrow x_i\ne x_j\right\} . \end{aligned}$$

Indeed, any subset of $[\eta ,1]^{N-1}\setminus C_{N-1}$ has vanishing $\mathrm{{Leb}}_{N-1}$ measure. Moreover, we have $S_{N-1}\circ \sigma =S_{N-1}$ for every permutation of coordinates $\sigma$. Consequently, the following decomposition holds for every $\omega \in [\eta ,1]$

$$\begin{aligned} (S_{N-1}|_{C_{N-1}})^{-1}(\omega )=\bigcup _{\sigma \in \Pi _{N-1}}\sigma \circ (S_{N-1}|_{\mathcal{T }_{N}})^{-1}(\omega ) \end{aligned}$$

where $\Pi _{N-1}$ is the set of all permutations. By construction, the sets $\sigma \circ (S_{N-1}|_{\mathcal{T }_{N}})^{-1}(\omega )$ are pairwise disjoints. In addition, they all have the same $\mathrm{{Leb}}_{N-1}$ measure because permuting coordinates does not affect the volume. Since there are $(N-1)!$ permutations, it results that for every $\omega \in [\eta ,1]$, we have

$$\begin{aligned} \mathrm{{Leb}}_{N-1}\circ S_{N-1}^{-1}(\omega )\!=\!(N\!-\!1)!\mathrm{{Leb}}_{N-1}\circ (S_{N-1}|_{\mathcal{T }_{N}})^{-1}(\omega )\!=\!\frac{(N-1)!}{\alpha _N}\mathbb{P }(S_{N-1}=\omega ), \end{aligned}$$

where the last equality follows from the definition of the uniform distribution in Sect. 2. By integrating over $[\eta ,1]$, normalization then implies $\frac{(N-1)!}{\alpha _N(1-\eta )^{N-1}}=1$, viz. $(1-\eta )^{-(N-1)}\mathrm{{Leb}}_{N-1}\circ S_{N-1}^{-1}=\mathbb{P }$ as desired.$\square$

Appendix B: Compactness of the set of increasing functions

Throughout the proofs, we also often need to approximate the piecewise affine interpolation $\mathrm{{x}}_\mathrm{lin}$ of a configuration $x\in \mathcal{T }_N$ by a continuous and strictly increasing function chosen in a finite collection. Such approximation relies on the following statement. Let $\Vert \cdot \Vert _\infty$ denote the uniform norm of a function defined on $[0,1]$.

Proposition 8.1

For every $\delta >0$, there exists a finite collection $\{\mathrm{{x}}_{(i,\delta )}\}_{i=1}^{i_\delta }$ of continuous and strictly increasing functions such that, for every piecewise affine continuous increasing function $\mathrm{{x}}$, there exists $i\in \{1,\ldots ,i_\delta \}$ such that $\Vert \mathrm{{x}}-\mathrm{{x}}_{(i,\delta )}\Vert _\infty <\delta$.

This statement is a consequence of a similar property in the weaker $L^1$-norm, which we denote by $\Vert \cdot \Vert _1$.

Lemma 8.2

For every $\delta >0$, there exists a finite collection $\{\mathrm{{x}}_{(i,\delta )}\}_{i=1}^{i_\delta }$ of continuous strictly increasing functions such that, for every piecewise affine continuous increasing function $\mathrm{{x}}$, there exists $i\in \{1,\ldots ,i_\delta \}$ such that $\Vert \mathrm{{x}}-\mathrm{{x}}_{(i,\delta )}\Vert _1<\delta$.

Proof of Lemma

By Helly Selection Theorem (Kolmogorov and Fomin 1999), the set of (right continuous) increasing functions from $[0,1]$ into itself is compact for the $L^1$-topology. Hence, for every $\delta >0$, there exists a finite collection $\{\tilde{\mathrm{{x}}}_{(i,\delta )}\}_{i=1}^{i_\delta }$ of (right continuous) increasing functions such that, for every piecewise affine continuous increasing function $\mathrm{{x}}$, there exists $i\in \{1,\ldots ,i_\delta \}$ such that $\Vert \mathrm{{x}}-\tilde{\mathrm{{x}}}_{(i,\delta )}\Vert _1<\delta /2$.

Let $h$ be a strictly increasing continuous function from $[-1,1]$ onto $[0,1]$. Then for each extended function $\tilde{\mathrm{{x}}}_{(i,\delta )}$ on $[-1,1]$ (where $\tilde{\mathrm{{x}}}_{(i,\delta )}(\omega )=0$ for $\omega <0$), consider the function $\mathrm{{x}}_{(i,\delta )}$ defined by the normalized convolution

$$\begin{aligned} \mathrm{{x}}_{(i,\delta )}(\omega )=\frac{(\tilde{\mathrm{{x}}}_{(i,\delta )}*h)(\omega )}{(\tilde{\mathrm{{x}}}_{(i,\delta )}*h)(1)},\quad \forall \omega \in [0,1] \end{aligned}$$

where $(u*h)(\omega )=\int _{\omega -1}^\omega u(\omega -\theta )~dh(\theta )$ (Lebesgue–Stieltjes integral). Each function $\mathrm{{x}}_{(i,\delta )}$ is continuous and strictly increasing from $[0,1]$ onto itself. Moreover, by taking $h$ sufficiently close to the Heaviside function $H$, one can ensure that $\Vert \mathrm{{x}}_{(i,\delta )}-\tilde{\mathrm{{x}}}_{(i,\delta )}\Vert _1<\delta /2$ for every $i\in \{1,\ldots ,i_\delta \}$ and the Lemma follows.

Indeed, if the sequence $\{h_n\}_{n\in \mathbb{N }}$ pointwise converges to $H$ on $[-1,1]$, Helly Convergence Theorem (Kolmogorov and Fomin 1999) implies that the sequence $\{(u*h_n)(\omega )\}_{n\in \mathbb{N }}$ converges to $(u*H)(\omega )=u(\omega )$ for every $\omega \in [0,1]$. Lebesgue dominated convergence then yields

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\int _0^1\frac{(u*h_n)(\omega )}{(u*h_n)(1)}~d\omega =\int _0^1 u \end{aligned}$$

from which the desired $L^1$-bound on the difference $\mathrm{{x}}_{(i,\delta )}-\tilde{\mathrm{{x}}}_{(i,\delta )}$ easily follows. $\square$

Proof of Proposition 8.1

According to the Lemma, it suffices to show that if $\{\mathrm{{x}}_n\}_{n\in \mathbb{N }}$ is a sequence of (strictly) increasing functions such that $\lim \limits _{n\rightarrow \infty }\Vert \mathrm{{x}}-\mathrm{{x}}_n\Vert _1=0$ where $\mathrm{{x}}$ is continuous, then $\lim \limits _{n\rightarrow \infty }\Vert \mathrm{{x}}-\mathrm{{x}}_n\Vert _\infty =0$. The proof is similar to that of Lemma B.3 in Coutinho and Fernandez (2004).

By contradiction, assume there exist $\delta >0$ and a subsequence $\{\mathrm{{x}}_{n_i}\}_{i\in \mathbb{N }}$ (with $\lim \limits _{i\rightarrow \infty }n_i=\infty$) such that

$$\begin{aligned} \Vert \mathrm{{x}}-\mathrm{{x}}_{n_i}\Vert _\infty \geqslant \delta ,\quad \forall i\in \mathbb{N }. \end{aligned}$$

Accordingly, there exists $\omega _i\in [0,1]$ for every $i$ such that

$$\begin{aligned} {\text{ either }}\ \mathrm{{x}}(\omega _i)\geqslant \mathrm{{x}}_{n_i}(\omega _i)+\delta \quad \text{ or }\ \mathrm{{x}}(\omega _i)\leqslant \mathrm{{x}}_{n_i}(\omega _i)-\delta . \end{aligned}$$

By taking a subsequence if necessary, we can assume to have either $\mathrm{{x}}(\omega _i)\geqslant \mathrm{{x}}_{n_i}(\omega _i)+\delta$ for all $i\in \mathbb{N }$ or $\mathrm{{x}}(\omega _i)\leqslant \mathrm{{x}}_{n_i}(\omega _i)-\delta$ for all $i\in \mathbb{N }$.

Assume to be in the first case. The second case can be treated similarly. Since $\omega _i\in [0,1]$ for all $i$, there exists a convergent subsequence. W.l.o.g. assume that we have $\lim \limits _{i\rightarrow \infty }\omega _i=\omega _\infty$.

By compactness, the function $\mathrm{{x}}$ is uniformly continuous. Let then $\gamma >0$ be small enough so that we have

$$\begin{aligned} |\mathrm{{x}}(\omega )-\mathrm{{x}}(\omega +\gamma )|<\delta /2,\quad \forall \omega \in [0,1-\gamma ]. \end{aligned}$$

Let now $\tilde{\omega }\in (\omega _\infty -\delta /2,\omega _\infty )$ be such that $\lim \limits _{i\rightarrow \infty }\mathrm{{x}}_{n_i}(\tilde{\omega })=\mathrm{{x}}(\tilde{\omega })$. (The existence of $\tilde{\omega }$ is a consequence of $L^1$-convergence.) Convergence to $\omega _\infty$ and the choice of $\tilde{\omega }$ imply that we simultaneously have

$$\begin{aligned} |\omega _{i}-\omega _\infty |<\gamma /2\quad \text{ and }\quad \tilde{\omega }<\omega _i,\quad \text{ and } \text{ hence } |\tilde{\omega }-\omega _i|<\gamma , \end{aligned}$$

provided that $i$ is sufficiently large. The last inequality implies that $\mathrm{{x}}(\tilde{\omega })-\delta /2\geqslant \mathrm{{x}}(\omega _i)-\delta$ and thus $\mathrm{{x}}(\tilde{\omega })-\delta /2\geqslant \mathrm{{x}}_{n_i}(\omega _i)$ by the initial assumption. Monotonicity of the $\mathrm{{x}}_{n_i}$ and the middle inequality above then yield $\mathrm{{x}}(\tilde{\omega })-\delta /2\geqslant \mathrm{{x}}_{n_i}(\tilde{\omega })$. By taking the limit $i\rightarrow \infty$, we obtain from the convergence at $\tilde{\omega }$ that $-\delta /2\geqslant 0$, which is impossible.

Appendix C: Intensive number of clusters for trajectories starting on equidistant configurations

In this section, we examine the fate at strong coupling, of trajectories initiated from equidistant configurations (or initial conditions close to equidistant configurations) and prove that their asymptotic number of clusters must be intensive. This property is an immediate consequence of the following technical statement.

Lemma 9.1

Let $\epsilon >\frac{2}{1-\eta }$ and consider the trajectory started from $x_i=\eta +(1-\eta )\frac{i-1}{N-1}\ (i=1,\ldots ,N)$.

• (i) For every $\ell \in \mathbb{N }$ and there exist $\rho _{\ell }\in (0,1)$ and $M_{\ell }\in \mathbb{N }$ such that for every $N>M_{\ell }$, the cluster size $K_\ell$ at $\ell$th firing satisfies $K_\ell \geqslant \lceil \rho _{\ell } N\rceil$, unless the accumulated reset size $K_1+\cdots +K_\ell =N$.

• (ii) We have $\rho _{\ell +1}>\rho _{\ell }$ for every $\ell$.

Naturally, property (ii) implies the existence of $L_\epsilon$ such that $\sum \limits _{\ell =1}^{L_\epsilon }\rho _\ell \geqslant 1$. Property (i) then forces $K_1+\cdots +K_{L_\epsilon }=N$ for every $N>M_{L_\epsilon }$. Thus, for every $N\in \mathbb{N }$, when starting from the equidistant configuration, the asymptotic number of clusters cannot exceed $\max \{L_\epsilon ,M_\epsilon \}$.

With a bit of additional effort, one can show that a similar upper bound applies to every trajectory started from configurations in some $\ell ^\infty$-neighborhood of the equidistant configuration. (However, this neighborhood has vanishing measure $\mu$ in the thermodynamics limit.) Therefore, our result indicates that for every $\epsilon >\frac{2}{1-\eta }$ (a threshold that is larger but close to $\frac{2}{1+\eta }$), for every population size, there is positive probability $\mu$ to obtain an intensive number of clusters in the long time limit.

Proof

We begin by showing the extensive bound on the size $K_1$ of the first firing cluster. Explicit calculations show that the quantity involved in the definition of $K_1$ in Sect. 3.2 is given by

$$\begin{aligned} \frac{1}{N}\sum _{k=j+1}^N(x_k-x_j)= \frac{1-\eta }{2}\left( 1-\frac{j}{N}\right) \left( 1- \frac{j-2}{N-1}\right) . \end{aligned}$$

Using $\frac{j-2}{N-1}<\frac{j}{N}$ yields $K_1\geqslant \max \{j\in \{1,\ldots ,N\} : (1-\frac{j}{N})^2\geqslant (1-\rho _1)^2\}$ where $\rho _1\in (0,1)$ is such that $(1-\rho _1)^2=\frac{2}{(1-\eta )\epsilon }$. This quantity $\rho _1$ exists for every $\epsilon >\frac{2}{1-\eta }$. It follows that $K_1\geqslant \lceil \rho _{1} N\rceil$ for all $N\in \mathbb{N }$ as desired.

For $\ell >1$, we proceed by induction. Assume that we have already proved that for $i=1,\ldots ,\ell$, we have $K_i\geqslant \lceil \rho _{i} N\rceil$ with $\rho _i\in (0,1)$ provided that $N$ is sufficiently large. Then, the reasoning at the beginning of the proof of Lemma 5.2 applies here; hence Eq. (9) is a lower bound for $K_{L+1}$. Using the expression of the equidistant configuration, it easily follows that $K_{\ell +1}\geqslant \lfloor \frac{\rho _\ell }{1-\eta }(N-1)\rfloor$ (provided that $K_1+\cdots K_\ell + \lfloor \frac{\rho _\ell }{1-\eta }(N-1)\rfloor \leqslant N$).

Let then $M_{\ell +1}$ be sufficiently large so that $\lfloor \frac{\rho _\ell }{1-\eta }(N-1)\rfloor \geqslant \lceil \frac{\rho _\ell }{1-1.1\eta }N\rceil$ for all $N>M_{\ell +1}$. Then, we clearly have $K_{L+1}\geqslant \lceil \rho _{\ell +1} N\rceil$ for all $N>M_{\ell +1}$ , where $\rho _{\ell +1}=\frac{\rho _\ell }{1-1.1\eta }>\rho _{\ell }$. The induction follows. $\square$