A Formal Approach for Tuning Stochastic Oscillators
Paolo Ballarini, Mahmoud Bentriou, Paul-Henry Cournède
TL;DR
The paper addresses calibrating noisy stochastic oscillators by identifying regions of the parameter space $\Theta$ that yield a target mean period $\bar{t_p}^{(obs)}$ with positive probability. It introduces a formal period-meter encoded as a Hybrid Automaton ${\cal A}^{\bar{t_p}^{(obs)}}_{per}$ and embeds it into an Automaton-ABC framework within HASL-SMC to measure distance dist$(\sigma_A,n,\bar{t_p}^{(obs)})$ combining the mean period and its variance: $\text{dist}(\sigma_A,n,\bar{t_p}^{(obs)}) = \min\left(\frac{|\bar{t_p}(n)-\bar{t_p}^{(obs)}|}{\bar{t_p}^{(obs)}}, \frac{\sqrt{s^2_{t_p}(n)}}{\bar{t_p}^{(obs)}}\right)$. The method is demonstrated on a three-way synthetic oscillator and the Repressilator, showing posterior concentration in parameter subspaces that achieve the target period and highlighting parameter sensitivities. The approach offers automated, generalizable tuning of stochastic oscillators with a clear separation between model dynamics and the period-meter, and it points to future work incorporating amplitude criteria via a peak-detector automaton.
Abstract
Periodic recurrence is a prominent behavioural of many biological phenomena, including cell cycle and circadian rhythms. Although deterministic models are commonly used to represent the dynamics of periodic phenomena, it is known that they are little appropriate in the case of systems in which stochastic noise induced by small population numbers is actually responsible for periodicity. Within the stochastic modelling settings automata-based model checking approaches have proven an effective means for the analysis of oscillatory dynamics, the main idea being that of coupling a period detector automaton with a continuous-time Markov chain model of an alleged oscillator. In this paper we address a complementary aspect, i.e. that of assessing the dependency of oscillation related measure (period and amplitude) against the parameters of a stochastic oscillator. To this aim we introduce a framework which, by combining an Approximate Bayesian Computation scheme with a hybrid automata capable of quantifying how distant an instance of a stochastic oscillator is from matching a desired (average) period, leads us to identify regions of the parameter space in which oscillation with given period are highly likely. The method is demonstrated through a couple of case studies, including a model of the popular Repressilator circuit.