Across the living world, from cell biology and morphogenesis to behavior and cognition, system dynamics is often directed towards functionally important goals one might call target states. Dynamics towards such target states is never in equilibrium, always nonstationary, and typically swiftly over, terminating once the goal is reached. Complicating data-driven inference of effective dynamical models, trajectories of living systems homing in on target states are not only brief but exhibit initial conditions that are typically intrinsically heterogeneous and experimentally uncontrollable. Here we present a reverse-time stochastic dynamics theory for ensembles of target state directed processes, and uncover a universal space-time core of such ensembles that is independent of initial conditions. Examining this core structure, we demonstrate that the landscape of conceivable target state directed dynamics decomposes into sectors that qualitatively differ in their accessibility to data-driven inference. Ensembles undergo a phase transition from an opaque phase, in which their space-time core is uninformative about the underlying dynamics, to a transparent phase, in which the dynamics is directly encoded in its low-order moment functions. We use the reverse-time theory of target state directed processes to construct core-based methods for data-driven dynamical systems inference. Applied to biophysical models of cytokinesis, a directed process for which different regimes of actomyosin turnover cause distinct types of terminal dynamics, our approach precisely identifies the underlying molecular mechanism. Reverse-time ensemble theory establishes a versatile toolset for understanding target state directed biological processes and reveals that the mere presence of target states in living systems induces a novel kind of phase transition in dynamical systems inference.

The dynamics of living systems often serves the purpose of reaching functionally important target states. We previously proposed a theory to analyze stochastic biological dynamics evolving towards target states in reverse time. However, a large class of systems in biology can only be adequately described using state-dependent noise, which had not been discussed. For example, in gene regulatory networks, biochemical signaling networks or neuronal circuits, count fluctuations are the dominant noise component. We characterize such dynamics as an ensemble of target state aligned (TSA) trajectories and characterize its temporal evolution in reverse-time by generalized Fokker-Planck and stochastic differential equations with multiplicative noise. We establish the classification of boundary conditions for target state modeling for a wide range of power law dynamics, and derive a universal low-noise approximation of the final phase of target state convergence. Our work expands the range of theoretically tractable systems in biology and enables novel experimental design strategies for systems that involve target states.

Mesoscopic bio-systems typically evolve towards functionally important target states, such as cell-cycle checkpoints or decision boundaries for the release of specific behaviors. For the data-driven inference of the underlying directional out-of-equilibrium dynamics, we here develop a theory of target state aligned (TSA) ensembles. Target state alignment allows to analyze directional dynamics in reverse time, starting from the final conditions of the forward process. Knowledge about the initial conditions of the forward process is not required for the analysis. Our theory reveals whether and when such a system can be represented by a single, effective stochastic equation of motion. We show how, in these effective dynamics, genuine biological forces can be separated from spurious forces, which invariably arise from target state alignment. We apply our inference scheme to the example of cytokinetic ring constriction, and derive the universal low-noise and short-term behavior of TSA ensembles. Our theory establishes a transparent mathematical foundation for the analysis and inference of directed biological dynamics by target state alignment.

Finite time convergence to functionally important target states is a key component of many biological processes. We previously found that the terminal approach phase of such dynamics exhibits universal types of stochastic dynamics that differ qualitatively between noise-dominated and force-dominated regimes of the approach dynamics. While for the noise-dominated regime the approach dynamics is uninformative about the underlying force law, in the force-dominated regime it enables the accurate inference of the underlying dynamics. Biological systems often exhibit substantial parameter heterogeneity, for instance through copy number fluctuations of key molecules or variability in modulating factors. Here, we extend our theory of target state aligned (TSA) stochastic dynamics to investigate the impact of parameter heterogeneity in the underlying stochastic dynamics. We examine the approach to target states for a wide range of dynamical laws and additive as well as multiplicative noise. We find that the distinct regimes of noise-dominated and force-dominated dynamics strongly differ in their sensitivity to parameter heterogeneity. In the noise-dominated regime, TSA ensembles are insensitive to parameter heterogeneity in the force law, but sensitive to sample to sample heterogeneity in the diffusion constant. For force-dominated dynamics, both parameter heterogeneity in the force law and diffusion constant change the behaviour of the non-stationary statistics and in particular the two-time-covariance functions. In this regime, TSA ensembles provide a sensitive readout of parameter heterogeneity. Under natural conditions, parameter heterogeneity in many biological systems cannot be experimentally controlled or eliminated. Our results provide a systematic theoretical foundation for the analysis of target state directed dynamics in a large class of systems with substantial heterogeneity.

Real-world agents, such as humans, animals and robots, observe each other during interactions and choose their own actions taking the partners’ ongoing behaviour into account. Yet, classical game theory assumes that players act either strictly sequentially or strictly simultaneously (without knowing the choices of each other). To account for action visibility and provide a more realistic model of interactions under time constraints, we introduce a new game-theoretic setting called transparent game, where each player has a certain probability to observe the choice of the partner before deciding on its own action. Using evolutionary simulations, we demonstrate that even a small probability of seeing the partner’s choice before one’s own decision substantially changes evolutionary successful strategies. Action visibility enhances cooperation in a Bach-or-Stravinsky game, but disrupts cooperation in a more competitive iterated Prisoner’s Dilemma. In both games, strategies based on the “Win–stay, lose–shift” and “Tit-for-tat” principles are predominant for moderate transparency, while for high transparency strategies of “Leader-Follower” type emerge. Our results have implications for studies of human and animal social behaviour, especially for the analysis of dyadic and group interactions.

A Transparent game is a game-theoretic setting that takes action visibility into account. In each round, depending on the relative timing of their actions, players have a certain probability to see their partner’s choice before making their own decision. This probability is determined by the level of transparency. At the two extremes, a game with zero transparency is equivalent to the classical simultaneous game, and a game with maximal transparency corresponds to a sequential game. Despite the prevalence of intermediate transparency in many everyday interactions such scenarios have not been sufficiently studied. Here we consider a transparent iterated Prisoner’s dilemma (iPD) and use evolutionary simulations to investigate how and why the success of various strategies changes with the level of transparency. We demonstrate that non-zero transparency greatly reduces the set of successful memory-one strategies compared to the simultaneous iPD. For low and moderate transparency the classical “Win - Stay, Lose - Shift” (WSLS) strategy is the only evolutionary successful strategy. For high transparency all strategies are evolutionary unstable in the sense that they can be easily counteracted, and, finally, for maximal transparency a novel “Leader-Follower” strategy outperforms WSLS. Our results provide a partial explanation for the fact that the strategies proposed for the simultaneous iPD are rarely observed in nature, where high levels of transparency are common.

A central problem in biomedical imaging is the automated segmentation of images for further quantitative analysis. Recently, fully convolutional neural networks, such as the U-Net, were applied successfully in a variety of segmentation tasks. A downside of this approach is the requirement for a large amount of well-prepared training samples, consisting of image - ground truth mask pairs. Since training data must be created by hand for each experiment, this task can be very costly and time-consuming. Here, we present a segmentation method based on cycle consistent generative adversarial networks, which can be trained even in absence of prepared image - mask pairs. We show that it successfully performs image segmentation tasks on samples with substantial defects and even generalizes well to different tissue types.

Stochastic processes that are randomly reset to an initial condition serve as a showcase to investigate non-equilibrium steady states. However, all existing results have been restricted to the special case of memoryless resetting protocols. Here, we obtain the general solution for the distribution of processes in which waiting times between reset events are drawn from an arbitrary distribution. This allows for the investigation of a broader class of much more realistic processes. As an example, our results are applied to the analysis of the efficiency of constrained random search processes.

Temporally disordered systems frequently exhibit complex intermittent dynamics accompanied by aging and weak ergodicity breaking. Here, we analyze the response behavior of temporally disordered dynamical processes by employing a functional representation. We find that the linear response to an external perturbation is history-dependent and explicitly calculate the response function for a sub-diffusive tracer particle confined in a harmonic potential. Our results demonstrate that the linear response function inherits the aging and weakly ergodic properties of the process, which has crucial implications for the interpretation of experimental measurements.

Spatial heterogeneity of a host population of mobile agents has been shown to be a crucial determinant of many aspects of disease dynamics, ranging from the proliferation of diseases to their persistence and to vaccination strategies. In addition, the importance of regional and structural differences grows in our modern world. Little is known, though, about the consequences when traits of a disease vary regionally. In this paper, we study the effect of a spatially varying per capita infection rate on the behaviour of livestock diseases. We show that the prevalence of an infectious livestock disease in a community of animals can paradoxically decrease owing to transport connections to other communities in which the risk of infection is higher. We study the consequences for the design of livestock transportation restriction measures and establish exact criteria to discriminate those connections that increase the level of infection in the community from those that decrease it.

Employing the path integral formulation of a broad class of anomalous diffusion processes, we derive the exact relations for the path probability densities of these processes. In particular, we obtain a closed analytical solution for the path probability distribution of a Continuous Time Random Walk (CTRW) process. This solution is given in terms of its waiting time distribution and short time propagator of the corresponding random walk as a solution of a Dyson equation. Applying our analytical solution we derive generalized Feynman?Kac formulae.

We investigate the geometric properties of two-dimensional continuous time random walks that are used extensively to model stochastic processes exhibiting anomalous diffusion in a variety of different fields. Using the concept of subordination, we determine exact analytical expressions for the average perimeter and area of the convex hulls for this class of non-Markovian processes. As the convex hull is a simple measure to estimate the home range of animals, our results give analytical estimates for the home range of foraging animals that perform sub-diffusive search strategies such as some Mediterranean seabirds and animals that ambush their prey. We also apply our results to Levy flights where possible.

Muller's ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process remains an open challenge. In contrast to previous works, we here study a Moran model of the ratchet with overlapping generations. Employing an approximation which describes the fittest individuals as one class and the rest as a second class, we obtain closed analytical expressions of the ratchet rate in the rare clicking regime. As a click in this regime is caused by a rare large fluctuation from a metastable state, we do not resort to a diffusion approximation but apply an approximation scheme which is especially well suited to describe extinction events from metastable states. This method also allows for a derivation of expressions for the quasi-stationary distribution of the fittest class. Additionally, we confirm numerically that the formulation with overlapping generations leads to the same results as the diffusion approximation and the corresponding Wright-Fisher model with non-overlapping generations.