Coloured Noise from Stochastic Inflows in Reaction-Diffusion Systems

Abstract

In this paper, we present a framework for investigating coloured noise in reaction-diffusion systems. We start by considering a deterministic reaction-diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady-state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models, we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases, we show that the power spectrum of the coloured noise process and the power spectrum of the reaction-diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction-diffusion system.

Keywords: Chemical reaction networks; Coloured noise; Injectivity criterion; Power spectra; Reaction–diffusion.

Fig. 1

The two mechanisms contributing to stochastic inflows. The boxes on the left are usually treated as black boxes, resulting in some constant inflow modelled by the parameter $k_{\text{in}}$. We differentiate between experimental fluctuations, symbolised by a correlation $\tau$ in (a) and auxiliary networks described by $f_{\text{aux}}(\mathit{y})$ in case (b) (Color figure online)

Fig. 2

An overview of the deterministic and white noise behaviour of the system with parameters as in (42). The peak in 2c at zero spatial mode, $m=0$, indicates temporal oscillations, which result from deterministic limit cycle oscillations. Due to the choice of reaction kinetics, the temporal oscillations of species $x_1$ and $x_2$ are in phase (Color figure online)

Fig. 3

The patterns and power spectra for the species $x_1$ (left) and $x_2$ (right) with noise generated by an Ornstein–Uhlenbeck process and $\tau =100$. Spatial patterns become visible in the Ornstein–Uhlenbeck case which can be attributed to its dampening effect on temporal oscillations. This can be observed as a visible excitation of the $m=1$ mode in the power spectrum of $x_1$ with the dashed line representing the analytical prediction. The well-known phenomenon of polarity switching (Schumacher et al. 2013) is observed in the pattern of $x_1$. The spectra were averaged over 50 repetitions with $T=1000$ and subsystem size $\Omega =100$ (Color figure online)

Fig. 4

The patterns and power spectra for the species $x_1$ (left) and $x_2$ (right) with noise generated by a red noise process. As predicted, the spectra are dominated by the behaviour the $\omega =0$, which amounts to the stabilisation of a particular spatial mode. The species concentrations, however, go negative, indicating that a full nonlinear model needs to be used. The peak in the power spectrum for $x_2$ is a numerical artefact. The spectra were averaged over 50 repetitions with simulation time $T=200$. The subsystem size was $\Omega =5000$ (Color figure online)

Fig. 5

The patterns and power spectra for $x_1$ (left) and $x_2$ (right) with noise generated by a violet noise process. The discrepancy in the power spectra originates from the truncation of the Gaussian process. The negative species concentrations indicate that a nonlinear theory needs to be used to fully model a violet noise system. The spectra were averaged over 50 repetitions with simulation time $T=200$. The subsystem size was $\Omega =5000$ (Color figure online)

Fig. 6

A schematic of an auxiliary network and its input to the main system. This is a specific example of scenario (b) of Fig. 1 (Color figure online)

Fig. 7

The power spectra for the species $x_1$ (left) and $x_2$ (right) with noise generated by a stochastic predator–prey network (McKane and Newman 2005). The power spectra inherited a second peak from the subsystem which activates oscillatory modes on top of the deterministic oscillation frequency. The spectra were averaged over 50 repetitions and simulation time $T=200$ with m indexing the spatial mode. The subsystem size was $\Omega =100$ (Color figure online)

Fig. 8

The patterns generated for the species $x_1$ (left) and $x_2$ (right) with Ornstein–Uhlenbeck noise and $\tau =100$ for $x_1$ and $x_2$ with white noise. The patterns are phenomenologically similar to the Ornstein–Uhlenbeck patterns, however, with reduced amplitude. The power spectrum of $x_2$ is a mixture of white noise and Ornstein–Uhlenbeck noise. The power spectra were averaged over 50 simulations with simulation time $T=1000$ and subsystem size $\Omega =5000$ (Color figure online)