Coupling regularizes individual units in noisy populations

The regularity of a noisy system can modulate in various ways. It is well known that coupling in a population can lower the variability of the entire network; the collective activity is more regular. Here, we show that diffusive (reciprocal) coupling of two simple Ornstein-Uhlenbeck (O-U) processes can regularize the individual, even when it is coupled to a noisier process. In cellular networks, the regularity of individual cells is important when a select few play a significant role. The regularizing effect of coupling surprisingly applies also to general nonlinear noisy oscillators. However, unlike with the O-U process, coupling-induced regularity is robust to different kinds of coupling. With two coupled noisy oscillators, we derive an asymptotic formula assuming weak noise and coupling for the variance of the period (i.e., spike times) that accurately captures this effect. Moreover, we find that reciprocal coupling can regularize the individual period of higher dimensional oscillators such as the Morris-Lecar and Brusselator models, even when coupled to noisier oscillators. Coupling can have a counterintuitive and beneficial effect on noisy systems. These results have implications for the role of connectivity with noisy oscillators and the modulation of variability of individual oscillators.

Figure 1

Coupling effects on the regularity of an individual O-U process. (a) $\text{var}\left(X_1\right)$ is plotted for ${\sigma }_2^2=5$ fixed, and ${\sigma }_1^1=0,2,4,5$. Simulations (gray) match the theory [black, Eq. (2)]. $K^{\ast }=\frac{1+\sqrt{3}}{5}$ for ${\sigma }_1^2=2$. (b) Similar to (a) except ${\sigma }_2^2=50$ fixed and ${\sigma }_1^2=15,20,25$. As ${\sigma }_1$ decreases, $K^{\ast }$ decreases for a fixed value of ${\sigma }_2^2$.

Figure 2

Coupling reduces variability of phase models when ${\sigma }_1={\sigma }_2$. Monte Carlo simulations show for many coupling functions and (small) noise levels that coupling regularizes the individual oscillator. a) $H=-\text{sin}\left(2\pi \psi \right)$ antiphase coupling with ${\Delta }_{\alpha }=-\text{sin}\left(2\pi \theta \right)$. (b) $H=\text{sin}\left(2\pi \psi \right)$ synchronous coupling with ${\Delta }_{\beta }=1-\text{cos}\left(2\pi \theta \right)$. (c) $H=\text{sin}\left(-4\pi \psi \right)$ so $\psi =\frac{1}{4},\frac{3}{4}$ are stable states without noise and with ${\Delta }_{\alpha }$.

Figure 3

Decrease in variability with coupling. (a) The dominant term with small coupling $-{\int }_0^1\left[H^{\prime }\left(\psi \right)+3H\left(\psi \right)\right]p\left(\psi \right)d\psi$ is negative for synchronous [(a)-i] and antiphase coupling [(a)-ii] with the two canonical PRCs: ${\Delta }_{\alpha }=-\text{sin}\left(2\pi \theta \right)$ and ${\Delta }_{\beta }=1-\text{cos}\left(2\pi \theta \right)$, respectively. $\psi ={\theta }_2-{\theta }_1$ (mod 1), $K=0.02$, ${\sigma }_1={\sigma }_2=0.1$. (b) Monte Carlo simulations of the variance of $T$ for different coupling values $K$ qualitatively matches the asymptotic theory [Eq. (10)] for small parameters $\left({\sigma }_1={\sigma }_2\right)$. [(b)-i] ${\Delta }_{\alpha }$ with antiphase coupling $H=-\text{sin}\left(2\pi \psi \right)$ (in ${\theta }_1$). [(b)-ii] ${\Delta }_{\beta }$ with synchronous coupling $H=\text{sin}\left(2\pi \psi \right)$.

Figure 4

Comparison of theory and simulations of phase models for ${\sigma }_1\ne {\sigma }_2$ matches well. ${\sigma }_2$ is fixed at 0.1 and ${\sigma }_1$ is varied between 0 to 0.1. (a) For ${\Delta }_{\alpha }$ with antiphase coupling, $\text{var}\left(T\right)$ for ${\theta }_1$ is compared for different $K$ values. (b) Same as (a) except with ${\Delta }_{\beta }$ and synchronous coupling. The gray curve is both panels corresponds to when ${\sigma }_1={\sigma }_2$; it is the same blue curve in Fig. 3b. The behavior is qualitatively the same as the full oscillator systems in that coupling to a noisier oscillator can decrease the variability.

Figure 5

Reciprocal coupling reduces the variability of spike times of Morris-Lecar model. (a) Morris-Lecar (ML) phase plane with $g_p=5\frac{\mu \text{A}}{{\text{cm}}^2}$ and ${\tilde{\sigma }}_1={\tilde{\sigma }}_2=5\frac{\mu \text{A}\text{ms}}{{\text{cm}}^2}$. (b) Probability density of ISI for various values of electrical coupling $g_p=0$ (black), 0.25 (gray), and 1 (dashed-black) $\text{mS}/{\text{cm}}^2$ with ${\tilde{\sigma }}_1={\tilde{\sigma }}_2=5\frac{\mu \text{A}\text{ms}}{{\text{cm}}^2}$. (c) The variance and $\text{CV}=\sqrt{\text{var}\left(T\right)}/⟨T⟩$ of the spike times vs. $g_p$ with various noise values $\left({\tilde{\sigma }}_1={\tilde{\sigma }}_2\right)$. The curves are not necessarily monotonic, but coupling always regularizes the ISI compared with the uncoupled system. (d) $\text{var}\left(T\right)$ of ${\text{ML}}_1$ vs $g_p$ with ${\tilde{\sigma }}_2=10\frac{\mu \text{A}\text{ms}}{{\text{cm}}^2}$ fixed, and ${\tilde{\sigma }}_1$ varying between 0 to $10\frac{\mu \text{A}\text{ms}}{{\text{cm}}^2}$. Notice for ${\tilde{\sigma }}_1<{\tilde{\sigma }}_2$, coupling can reduce the variability of the first ML oscillator even though it is coupled to a noisier oscillator $\left({\tilde{\sigma }}_1=6,8\frac{\mu \text{A}\text{ms}}{{\text{cm}}^2}\right)$.

Figure 6

Morris-Lecar cells coupled via inhibitory synapses. In this absence of noise, the two cells settle to an antiphase state. Synaptic coupling regularizes the spike times of the individual cells. The effect is not dramatic because the noise and coupling parameters have to be quite small for there to be a well-defined phase on the limit cycle. The noise levels are equal and dotted black lines are $\text{var}\left(T\right)$ for $\tilde{K}=0$.

Figure 7

Coupling reduces variability in a network of Morris-Lecar oscillators. Since ${\text{ML}}_1$ is coupled to four other MLs, each $g_p$ is scaled to $\frac{1}{4}{g}_p$ for comparison to Fig. 5. (a) $\text{var}\left(T\right)$ of a single ML decreases with nonzero $g_p$. The noise values of each oscillator are the same, with the legend used in Fig. 5c. (b) The noise of the first ML $\left(7\frac{\mu \text{A}\text{ms}}{{\text{cm}}^2}\right)$ is less than the other four MLs $\left(10\frac{\mu \text{A}\text{ms}}{{\text{cm}}^2}\right)$, yet coupling still decreases the variance of it’s spike times.

Figure 8

Brusselator model of oscillating chemical concentrations. (a) Phase plane of concentrations $\left(x_1,y_1\right)$ with $\tilde{K}=0.02$, $\tilde{\sigma }=0.05$ for both Brusselators. (b) Variance of period with different coupling strengths $\tilde{K}$ for ${\tilde{\sigma }}_1={\tilde{\sigma }}_2=\tilde{\sigma }$. Diffusive coupling regularizes the period of each cycle (dotted black line $\text{var}\left(T\right)$ for $\tilde{K}=0$). (c) Fixed ${\tilde{\sigma }}_2=0.06$, variance of period of first Brusselator with different coupling strengths, varying ${\tilde{\sigma }}_1$ between 0 and 0.06. Again, coupling can reduce the variability of the first Brusselator even though it is coupled to a noisier oscillator $\left({\tilde{\sigma }}_1=0.045,0.055\right)$. The gray curve ${\tilde{\sigma }}_1=0.06$ is the top curve in (b) (dashed-black).