Graph-facilitated resonant mode counting in stochastic interaction networks

Abstract

Oscillations in dynamical systems are widely reported in multiple branches of applied mathematics. Critically, even a non-oscillatory deterministic system can produce cyclic trajectories when it is in a low copy number, stochastic regime. Common methods of finding parameter ranges for stochastically driven resonances, such as direct calculation, are cumbersome for any but the smallest networks. In this paper, we provide a systematic framework to efficiently determine the number of resonant modes and parameter ranges for stochastic oscillations relying on real root counting algorithms and graph theoretic methods. We argue that stochastic resonance is a network property by showing that resonant modes only depend on the squared Jacobian matrix J², unlike deterministic oscillations which are determined by J By using graph theoretic tools, analysis of stochastic behaviour for larger interaction networks is simplified and stochastic dynamical systems with multiple resonant modes can be identified easily.

Keywords: chemical reaction networks; graph theoretic methods; quasi-cycles; resonant modes; sturm chains.

Figure 1.

The directed graph associated with J² of the n = 3 autocatalytic network. The edges have weights: , and .

Figure 2.

A trajectory of the n = 3 autocatalytic network with parameter values α = β = 0.1, r = 1 and Ω = 5000. The smooth, decaying curve is the numerical solution of the ODE system (2.9) and the oscillating trajectory is the stochastic trajectory. (Online version in colour.)

Figure 3.

The power spectrum of the stochastic n = 3 autocatalytic network with parameter values α = β = 0.1, r = 1 and Ω = 5000. The smooth line represents the analytic curve, calculated from equation (14) in [13], and the dotted blue line is the average power spectrum of 200 simulations. Following [39], we normalized the spectra such that they have unit area. (Online version in colour.)

Figure 4.

The phase diagram for the α = β slice of the parameter space of the n = 3 autocatalytic network. We identified two connected regions, one where stochastic oscillations are possible and one where the power spectrum is flat. (Online version in colour.)

Figure 5.

The phase diagram for the α = β slice of the parameter space of the n = 5 autocatalytic network. We identified three connected regions, one where there are two stochastic modes, one with only one mode and where no oscillations are possible. (Online version in colour.)

Figure 6.

A trajectory of the n = 5 autocatalytic network with parameter values α = β = 0.01, r = 4 and Ω = 10 000. The smooth decaying curve is the numerical solution of the ODE system (4.1) and the blue curve is the stochastic trajectory. (Online version in colour.)

Figure 7.

The power spectrum of the stochastic n = 5 autocatalytic network with parameter values α = β = 0.01, r = 4 and Ω = 10 000. The smooth green line represents the analytic curve, calculated from equation (14) in [13], and the dotted blue line is the average power spectrum of 500 simulations. Following [39], we normalized the spectra such that they have unit area. (Online version in colour.)