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Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.

Chemical reaction networks are a popular formalism for modeling biological processes which supports both a deterministic and a stochastic interpretation based on ordinary differential equations and continuous-time Markov chains, respectively. In most cases, these models do not enjoy analytical solution, thus typically requiring expensive computational methods based on numerical solvers or stochastic simulations. Exact model reduction techniques can be used as an aid to lower the analysis cost by providing reduced networks that preserve the dynamics of interest to the modeler. We hereby consider a family of techniques for both deterministic and stochastic networks which are based on equivalence relations over the species in the network, leading to a coarse graining which provides the exact aggregate time-course evolution for each equivalence class. We present a large-scale empirical assessment on the BioModels repository by measuring their compression capability over 667 models. Through a number of selected case studies, we also show their ability in yielding physically interpretable reductions that can reveal dynamical patterns of the bio-molecular processes under consideration.

Synchronisation is an emergent phenomenon observable in nature. Natural synchronising systems have inspired the development of protocols for achieving coordination in a diverse range of distributed dynamic systems. Spontaneously synchronising systems can be mathematically modelled as coupled oscillators. In this paper we present a novel approach using model checking to reason about achieving synchrony for different models of synchronisation. We describe a general, formal population model where oscillators interact at discrete moments in time, and whose cycles are sequences of discrete states. Using the probabilistic model checker Prism, we investigate the influence of various parameters of the model on the likelihood of, and time required for, achieving synchronisation.

We consider the problem of mining signal temporal logical requirements from a dataset of regular (good) and anomalous (bad) trajectories of a dynamical system. We assume the training set to be labeled by human experts and that we have access only to a limited amount of data, typically noisy. We provide a systematic approach to synthesize both the syntactical structure and the parameters of the temporal logic formula using a two-steps procedure: first, we leverage a novel evolutionary algorithm for learning the structure of the formula; second, we perform the parameter synthesis operating on the statistical emulation of the average robustness for a candidate formula w.r.t. its parameters. We compare our results with our previous work [{BufoBSBLB14] and with a recently proposed decision-tree [bombara_decision_2016] based method. We present experimental results on two case studies: an anomalous trajectory detection problem of a naval surveillance system and the characterization of an Ineffective Respiratory effort, showing the usefulness of our work., Comment: 16 pages, 3 figure

Many complex systems can be described by population models, in which a pool of agents interacts and produces complex collective behaviours. We consider the problem of verifying formal properties of the underlying mathematical representation of these models, which is a Continuous Time Markov Chain, often with a huge state space. To circumvent the state space explosion, we rely on stochastic approximation techniques, which replace the large model by a simpler one, guaranteed to be probabilistically consistent. We show how to efficiently and accurately verify properties of random individual agents, specified by Continuous Stochastic Logic extended with Timed Automata (CSL-TA), and how to lift these specifications to the collective level, approximating the number of agents satisfying them using second or higher order stochastic approximation techniques.