Automated, predictive, and interpretable inference of Caenorhabditis elegans escape dynamics

Distilling identifiable and interpretable dynamic models from biological data

Mechanistic dynamical models allow us to study the behavior of complex biological systems. They can provide an objective and quantitative understanding that would be difficult to achieve through other means. However, the systematic development of these models is a non-trivial exercise and an open problem in computational biology. Currently, many research efforts are focused on model discovery, i.e. automating the development of interpretable models from data. One of the main frameworks is sparse regression, where the sparse identification of nonlinear dynamics (SINDy) algorithm and its variants have enjoyed great success. SINDy-PI is an extension which allows the discovery of rational nonlinear terms, thus enabling the identification of kinetic functions common in biochemical networks, such as Michaelis-Menten. SINDy-PI also pays special attention to the recovery of parsimonious models (Occam's razor). Here we focus on biological models composed of sets of deterministic nonlinear ordinary differential equations. We present a methodology that, combined with SINDy-PI, allows the automatic discovery of structurally identifiable and observable models which are also mechanistically interpretable. The lack of structural identifiability and observability makes it impossible to uniquely infer parameter and state variables, which can compromise the usefulness of a model by distorting its mechanistic significance and hampering its ability to produce biological insights. We illustrate the performance of our method with six case studies. We find that, despite enforcing sparsity, SINDy-PI sometimes yields models that are unidentifiable. In these cases we show how our method transforms their equations in order to obtain a structurally identifiable and observable model which is also interpretable.

A dynamical model of C. elegans thermal preference reveals independent excitatory and inhibitory learning pathways

Caenorhabditis elegans is capable of learning and remembering behaviorally relevant cues such as smells, tastes, and temperature. This is an example of associative learning, a process in which behavior is modified by making associations between various stimuli. Since the mathematical theory of conditioning does not account for some of its salient aspects, such as spontaneous recovery of extinguished associations, accurate modeling of behavior of real animals during conditioning has turned out difficult. Here, we do this in the context of the dynamics of the thermal preference of C. elegans. We quantify C. elegans thermotaxis in response to various conditioning temperatures, starvation durations, and genetic perturbations using a high-resolution microfluidic droplet assay. We model these data comprehensively, within a biologically interpretable, multi-modal framework. We find that the strength of the thermal preference is composed of two independent, genetically separable contributions and requires a model with at least four dynamical variables. One pathway positively associates the experienced temperature independently of food and the other negatively associates with the temperature when food is absent. The multidimensional structure of the association strength provides an explanation for the apparent classical temperature-food association of C. elegans thermal preference and a number of longstanding questions in animal learning, including spontaneous recovery, asymmetric response to appetitive vs. aversive cues, latent inhibition, and generalization among similar cues.

Model selection of chaotic systems from data with hidden variables using sparse data assimilation

Many natural systems exhibit chaotic behavior, including the weather, hydrology, neuroscience, and population dynamics. Although many chaotic systems can be described by relatively simple dynamical equations, characterizing these systems can be challenging due to sensitivity to initial conditions and difficulties in differentiating chaotic behavior from noise. Ideally, one wishes to find a parsimonious set of equations that describe a dynamical system. However, model selection is more challenging when only a subset of the variables are experimentally accessible. Manifold learning methods using time-delay embeddings can successfully reconstruct the underlying structure of the system from data with hidden variables, but not the equations. Recent work in sparse-optimization based model selection has enabled model discovery given a library of possible terms, but regression-based methods require measurements of all state variables. We present a method combining variational annealing-a technique previously used for parameter estimation in chaotic systems with hidden variables-with sparse-optimization methods to perform model identification for chaotic systems with unmeasured variables. We applied the method to ground-truth time-series simulated from the classic Lorenz system and experimental data from an electrical circuit with Lorenz-system like behavior. In both cases, we successfully recover the expected equations with two measured and one hidden variable. Application to simulated data from the Colpitts oscillator demonstrates successful model selection of terms within nonlinear functions. We discuss the robustness of our method to varying noise.

A lexical approach for identifying behavioural action sequences

Animals display characteristic behavioural patterns when performing a task, such as the spiraling of a soaring bird or the surge-and-cast of a male moth searching for a female. Identifying such recurring sequences occurring rarely in noisy behavioural data is key to understanding the behavioural response to a distributed stimulus in unrestrained animals. Existing models seek to describe the dynamics of behaviour or segment individual locomotor episodes rather than to identify the rare and transient sequences of locomotor episodes that make up the behavioural response. To fill this gap, we develop a lexical, hierarchical model of behaviour. We designed an unsupervised algorithm called "BASS" to efficiently identify and segment recurring behavioural action sequences transiently occurring in long behavioural recordings. When applied to navigating larval zebrafish, BASS extracts a dictionary of remarkably long, non-Markovian sequences consisting of repeats and mixtures of slow forward and turn bouts. Applied to a novel chemotaxis assay, BASS uncovers chemotactic strategies deployed by zebrafish to avoid aversive cues consisting of sequences of fast large-angle turns and burst swims. In a simulated dataset of soaring gliders climbing thermals, BASS finds the spiraling patterns characteristic of soaring behaviour. In both cases, BASS succeeds in identifying rare action sequences in the behaviour deployed by freely moving animals. BASS can be easily incorporated into the pipelines of existing behavioural analyses across diverse species, and even more broadly used as a generic algorithm for pattern recognition in low-dimensional sequential data.

Primer on Mathematical Modeling in C. elegans

Recently, applications of mathematical and computational models to biological processes have helped investigators to systematically interpret data, test hypotheses built on experimental data, generate new hypotheses, and guide the design of new experiments, protocols, and synthetic biological systems. Availability of diverse quantitative data is a prerequisite for successful mathematical modeling. The ability to acquire high-quality quantitative data for a broad range of biological processes and perform precise perturbation makes C. elegans an ideal model system for such studies. In this primer, we examine the general procedure of modeling biological systems and demonstrate this process using the heat-shock response in C. elegans as a case study. Our goal is to facilitate the initial discussion between worm biologists and their potential collaborators from quantitative disciplines.

Tracking changes in behavioural dynamics using prediction error

Automated analysis of video can now generate extensive time series of pose and motion in freely-moving organisms. This requires new quantitative tools to characterise behavioural dynamics. For the model roundworm Caenorhabditis elegans, body pose can be accurately quantified from video as coordinates in a single low-dimensional space. We focus on this well-established case as an illustrative example and propose a method to reveal subtle variations in behaviour at high time resolution. Our data-driven method, based on empirical dynamic modeling, quantifies behavioural change as prediction error with respect to a time-delay-embedded ‘attractor’ of behavioural dynamics. Because this attractor is constructed from a user-specified reference data set, the approach can be tailored to specific behaviours of interest at the individual or group level. We validate the approach by detecting small changes in the movement dynamics of C. elegans at the initiation and completion of delta turns. We then examine an escape response initiated by an aversive stimulus and find that the method can track return to baseline behaviour in individual worms and reveal variations in the escape response between worms. We suggest that this general approach—defining dynamic behaviours using reference attractors and quantifying dynamic changes using prediction error—may be of broad interest and relevance to behavioural researchers working with video-derived time series.

Inferring the structures of signaling motifs from paired dynamic traces of single cells

Individual cells show variability in their signaling dynamics that often correlates with phenotypic responses, indicating that cell-to-cell variability is not merely noise but can have functional consequences. Based on this observation, we reasoned that cell-to-cell variability under the same treatment condition could be explained in part by a single signaling motif that maps different upstream signals into a corresponding set of downstream responses. If this assumption holds, then repeated measurements of upstream and downstream signaling dynamics in a population of cells could provide information about the underlying signaling motif for a given pathway, even when no prior knowledge of that motif exists. To test these two hypotheses, we developed a computer algorithm called MISC (Motif Inference from Single Cells) that infers the underlying signaling motif from paired time-series measurements from individual cells. When applied to measurements of transcription factor and reporter gene expression in the yeast stress response, MISC predicted signaling motifs that were consistent with previous mechanistic models of transcription. The ability to detect the underlying mechanism became less certain when a cell's upstream signal was randomly paired with another cell's downstream response, demonstrating how averaging time-series measurements across a population obscures information about the underlying signaling mechanism. In some cases, motif predictions improved as more cells were added to the analysis. These results provide evidence that mechanistic information about cellular signaling networks can be systematically extracted from the dynamical patterns of single cells.

Making Sense of Computational Psychiatry

In psychiatry we often speak of constructing "models." Here we try to make sense of what such a claim might mean, starting with the most fundamental question: "What is (and isn't) a model?" We then discuss, in a concrete measurable sense, what it means for a model to be useful. In so doing, we first identify the added value that a computational model can provide in the context of accuracy and power. We then present limitations of standard statistical methods and provide suggestions for how we can expand the explanatory power of our analyses by reconceptualizing statistical models as dynamical systems. Finally, we address the problem of model building-suggesting ways in which computational psychiatry can escape the potential for cognitive biases imposed by classical hypothesis-driven research, exploiting deep systems-level information contained within neuroimaging data to advance our understanding of psychiatric neuroscience.

Targeted thermal stimulation and high-content phenotyping reveal that the C. elegans escape response integrates current behavioral state and past experience

The ability to avoid harmful or potentially harmful stimuli can help an organism escape predators and injury, and certain avoidance mechanisms are conserved across the animal kingdom. However, how the need to avoid an imminent threat is balanced with current behavior and modified by past experience is not well understood. In this work we focused on rapidly increasing temperature, a signal that triggers an escape response in a variety of animals, including the nematode Caenorhabditis elegans. We have developed a noxious thermal response assay using an infrared laser that can be automatically controlled and targeted in order to investigate how C. elegans responds to noxious heat over long timescales and to repeated stimuli in various behavioral and sensory contexts. High-content phenotyping of behavior in individual animals revealed that the C. elegans escape response is multidimensional, with some features that extend for several minutes, and can be modulated by (i) stimulus amplitude; (ii) other sensory inputs, such as food context; (iii) long and short-term thermal experience; and (iv) the animal's current behavioral state.