An approximate line attractor in the hypothalamus encodes an aggressive state.

The hypothalamus regulates innate social behaviors, including mating and aggression. These behaviors can be evoked by optogenetic stimulation of specific neuronal subpopulations within MPOA and VMHvl, respectively. Here, we perform dynamical systems modeling of population neuronal activity in these nuclei during social behaviors. In VMHvl, unsupervised analysis identified a dominant dimension of neural activity with a large time constant (>50 s), generating an approximate line attractor in neural state space. Progression of the neural trajectory along this attractor was correlated with an escalation of agonistic behavior, suggesting that it may encode a scalable state of aggressiveness. Consistent with this, individual differences in the magnitude of the integration dimension time constant were strongly correlated with differences in aggressiveness. In contrast, approximate line attractors were not observed in MPOA during mating; instead, neurons with fast dynamics were tuned to specific actions. Thus, different hypothalamic nuclei employ distinct neural population codes to represent similar social behaviors.

Cell-type-specific population dynamics of diverse reward computations.

Computational analysis of cellular activity has developed largely independently of modern transcriptomic cell typology, but integrating these approaches may be essential for full insight into cellular-level mechanisms underlying brain function and dysfunction. Applying this approach to the habenula (a structure with diverse, intermingled molecular, anatomical, and computational features), we identified encoding of reward-predictive cues and reward outcomes in distinct genetically defined neural populations, including TH+ cells and Tac1+ cells. Data from genetically targeted recordings were used to train an optimized nonlinear dynamical systems model and revealed activity dynamics consistent with a line attractor. High-density, cell-type-specific electrophysiological recordings and optogenetic perturbation provided supporting evidence for this model. Reverse-engineering predicted how Tac1+ cells might integrate reward history, which was complemented by in vivo experimentation. This integrated approach describes a process by which data-driven computational models of population activity can generate and frame actionable hypotheses for cell-type-specific investigation in biological systems.

Mapping transcriptomic vector fields of single cells.

Single-cell (sc)RNA-seq, together with RNA velocity and metabolic labeling, reveals cellular states and transitions at unprecedented resolution. Fully exploiting these data, however, requires kinetic models capable of unveiling governing regulatory functions. Here, we introduce an analytical framework dynamo (https://github.com/aristoteleo/dynamo-release), which infers absolute RNA velocity, reconstructs continuous vector fields that predict cell fates, employs differential geometry to extract underlying regulations, and ultimately predicts optimal reprogramming paths and perturbation outcomes. We highlight dynamo's power to overcome fundamental limitations of conventional splicing-based RNA velocity analyses to enable accurate velocity estimations on a metabolically labeled human hematopoiesis scRNA-seq dataset. Furthermore, differential geometry analyses reveal mechanisms driving early megakaryocyte appearance and elucidate asymmetrical regulation within the PU.1-GATA1 circuit. Leveraging the least-action-path method, dynamo accurately predicts drivers of numerous hematopoietic transitions. Finally, in silico perturbations predict cell-fate diversions induced by gene perturbations. Dynamo, thus, represents an important step in advancing quantitative and predictive theories of cell-state transitions.

Tuning of plasma cell lifespan by competition explains the longevity and heterogeneity of antibody persistence.

Plasma cells that emerge after infection or vaccination exhibit heterogeneous lifespans; most survive for days to months, whereas others persist for decades, providing antigen-specific long-term protection. We developed a mathematical framework to explore the dynamics of plasma cell removal and its regulation by survival factors. Analyses of antibody persistence following hepatitis A and B and HPV vaccination revealed specific patterns of longevity and heterogeneity within and between responses, implying that this process is fine-tuned near a critical "flat" state between two dynamic regimes. This critical state reflects the tuning of rates of the underlying regulatory network and is highly sensitive to variation in parameters, which amplifies lifespan differences between cells. We propose that fine-tuning is the generic outcome of competition over shared survival signals, with a competition-based mechanism providing a unifying explanation for a wide range of experimental observations, including the dynamics of plasma cell accumulation and the effects of survival factor deletion. Our theory is testable, and we provide specific predictions.

Computation Through Neural Population Dynamics.

Significant experimental, computational, and theoretical work has identified rich structure within the coordinated activity of interconnected neural populations. An emerging challenge now is to uncover the nature of the associated computations, how they are implemented, and what role they play in driving behavior. We term this computation through neural population dynamics. If successful, this framework will reveal general motifs of neural population activity and quantitatively describe how neural population dynamics implement computations necessary for driving goal-directed behavior. Here, we start with a mathematical primer on dynamical systems theory and analytical tools necessary to apply this perspective to experimental data. Next, we highlight some recent discoveries resulting from successful application of dynamical systems. We focus on studies spanning motor control, timing, decision-making, and working memory. Finally, we briefly discuss promising recent lines of investigation and future directions for the computation through neural population dynamics framework.

A geometrical model of cell fate specification in the mouse blastocyst.

The lineage decision that generates the epiblast and primitive endoderm from the inner cell mass (ICM) is a paradigm for cell fate specification. Recent mathematics has formalized Waddington's landscape metaphor and proven that lineage decisions in detailed gene network models must conform to a small list of low-dimensional stereotypic changes called bifurcations. The most plausible bifurcation for the ICM is the so-called heteroclinic flip that we define and elaborate here. Our re-analysis of recent data suggests that there is sufficient cell movement in the ICM so the FGF signal, which drives the lineage decision, can be treated as spatially uniform. We thus extend the bifurcation model for a single cell to the entire ICM by means of a self-consistently defined time-dependent FGF signal. This model is consistent with available data and we propose additional dynamic experiments to test it further. This demonstrates that simplified, quantitative and intuitively transparent descriptions are possible when attention is shifted from specific genes to lineages. The flip bifurcation is a very plausible model for any situation where the embryo needs control over the relative proportions of two fates by a morphogen feedback.

A mathematical framework for measuring and tuning tempo in developmental gene regulatory networks.

Embryo development is a dynamic process governed by the regulation of timing and sequences of gene expression, which control the proper growth of the organism. Although many genetic programmes coordinating these sequences are common across species, the timescales of gene expression can vary significantly among different organisms. Currently, substantial experimental efforts are focused on identifying molecular mechanisms that control these temporal aspects. In contrast, the capacity of established mathematical models to incorporate tempo control while maintaining the same dynamical landscape remains less understood. Here, we address this gap by developing a mathematical framework that links the functionality of developmental programmes to the corresponding gene expression orbits (or landscapes). This unlocks the ability to find tempo differences as perturbations in the dynamical system that preserve its orbits. We demonstrate that this framework allows for the prediction of molecular mechanisms governing tempo, through both numerical and analytical methods. Our exploration includes two case studies: a generic network featuring coupled production and degradation, with a particular application to neural progenitor differentiation; and the repressilator. In the latter, we illustrate how altering the dimerisation rates of transcription factors can decouple the tempo from the shape of the resulting orbits. We conclude by highlighting how the identification of orthogonal molecular mechanisms for tempo control can inform the design of circuits with specific orbits and tempos.

Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures.

A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space [Hopf, Commun. Appl. Maths 1, 303 (1948)]. The chaotic dynamics are shaped by the unstable simple invariant solutions populating the inertial manifold. The hope has been to turn this picture into a predictive framework where the statistics of the flow follow from a weighted sum of the statistics of each simple invariant solution. Two outstanding obstacles have prevented this goal from being achieved: 1) paucity of known solutions and 2) the lack of a rational theory for predicting the required weights. Here, we describe a method to substantially solve these problems, and thereby provide compelling evidence that the probability density functions (PDFs) of a fully developed turbulent flow can be reconstructed with a set of unstable periodic orbits. Our method for finding solutions uses automatic differentiation, with high-quality guesses constructed by minimizing a trajectory-dependent loss function. We use this approach to find hundreds of solutions in turbulent, two-dimensional Kolmogorov flow. Robust statistical predictions are then computed by learning weights after converting a turbulent trajectory into a Markov chain for which the states are individual solutions, and the nearest solution to a given snapshot is determined using a deep convolutional autoencoder. In this study, the PDFs of a spatiotemporally chaotic system have been successfully reproduced with a set of simple invariant states, and we provide a fascinating connection between self-sustaining dynamical processes and the more well-known statistical properties of turbulence.

Robust inference of causality in high-dimensional dynamical processes from the Information Imbalance of distance ranks.

We introduce an approach which allows detecting causal relationships between variables for which the time evolution is available. Causality is assessed by a variational scheme based on the Information Imbalance of distance ranks, a statistical test capable of inferring the relative information content of different distance measures. We test whether the predictability of a putative driven system Y can be improved by incorporating information from a potential driver system X, without explicitly modeling the underlying dynamics and without the need to compute probability densities of the dynamic variables. This framework makes causality detection possible even between high-dimensional systems where only few of the variables are known or measured. Benchmark tests on coupled chaotic dynamical systems demonstrate that our approach outperforms other model-free causality detection methods, successfully handling both unidirectional and bidirectional couplings. We also show that the method can be used to robustly detect causality in human electroencephalography data.

Spiking at the edge: Excitability at interfaces in reaction-diffusion systems.

Excitable media, ranging from bioelectric tissues and chemical oscillators to forest fires and competing populations, are nonlinear, spatially extended systems capable of spiking. Most investigations of excitable media consider situations where the amplifying and suppressing forces necessary for spiking coexist at every point in space. In this case, spikes arise due to local bistabilities, which require a fine-tuned ratio between local amplification and suppression strengths. But, in nature and engineered systems, these forces can be segregated in space, forming structures like interfaces and boundaries. Here, we show how boundaries can generate and protect spiking when the reacting components can spread out: Even arbitrarily weak diffusion can cause spiking at the edge between two non-excitable media. This edge spiking arises due to a global bistability, which can occur even if amplification and suppression strengths do not allow spiking when mixed. We analytically derive a spiking phase diagram that depends on two parameters: i) the ratio between the system size and the characteristic diffusive length-scale and ii) the ratio between the amplification and suppression strengths. Our analysis explains recent experimental observations of action potentials at the interface between two non-excitable bioelectric tissues. Beyond electrophysiology, we highlight how edge spiking emerges in predator-prey dynamics and in oscillating chemical reactions. Our findings provide a theoretical blueprint for a class of interfacial excitations in reaction-diffusion systems, with potential implications for spatially controlled chemical reactions, nonlinear waveguides and neuromorphic computation, as well as spiking instabilities, such as cardiac arrhythmias, that naturally occur in heterogeneous biological media.

Modeling and dissociation of intrinsic and input-driven neural population dynamics underlying behavior.

Neural dynamics can reflect intrinsic dynamics or dynamic inputs, such as sensory inputs or inputs from other brain regions. To avoid misinterpreting temporally structured inputs as intrinsic dynamics, dynamical models of neural activity should account for measured inputs. However, incorporating measured inputs remains elusive in joint dynamical modeling of neural-behavioral data, which is important for studying neural computations of behavior. We first show how training dynamical models of neural activity while considering behavior but not input or input but not behavior may lead to misinterpretations. We then develop an analytical learning method for linear dynamical models that simultaneously accounts for neural activity, behavior, and measured inputs. The method provides the capability to prioritize the learning of intrinsic behaviorally relevant neural dynamics and dissociate them from both other intrinsic dynamics and measured input dynamics. In data from a simulated brain with fixed intrinsic dynamics that performs different tasks, the method correctly finds the same intrinsic dynamics regardless of the task while other methods can be influenced by the task. In neural datasets from three subjects performing two different motor tasks with task instruction sensory inputs, the method reveals low-dimensional intrinsic neural dynamics that are missed by other methods and are more predictive of behavior and/or neural activity. The method also uniquely finds that the intrinsic behaviorally relevant neural dynamics are largely similar across the different subjects and tasks, whereas the overall neural dynamics are not. These input-driven dynamical models of neural-behavioral data can uncover intrinsic dynamics that may otherwise be missed.

Discovering optimal kinetic pathways for self-assembly using automatic differentiation.

Macromolecular complexes are often composed of diverse subunits. The self-assembly of these subunits is inherently nonequilibrium and must avoid kinetic traps to achieve high yield over feasible timescales. We show how the kinetics of self-assembly benefits from diversity in subunits because it generates an expansive parameter space that naturally improves the "expressivity" of self-assembly, much like a deeper neural network. By using automatic differentiation algorithms commonly used in deep learning, we searched the parameter spaces of mass-action kinetic models to identify classes of kinetic protocols that mimic biological solutions for productive self-assembly. Our results reveal how high-yield complexes that easily become kinetically trapped in incomplete intermediates can instead be steered by internal design of rate-constants or external and active control of subunits to efficiently assemble. Internal design of a hierarchy of subunit binding rates generates self-assembly that can robustly avoid kinetic traps for all concentrations and energetics, but it places strict constraints on selection of relative rates. External control via subunit titration is more versatile, avoiding kinetic traps for any system without requiring molecular engineering of binding rates, albeit less efficiently and robustly. We derive theoretical expressions for the timescales of kinetic traps, and we demonstrate our optimization method applies not just for design but inference, extracting intersubunit binding rates from observations of yield-vs.-time for a heterotetramer. Overall, we identify optimal kinetic protocols for self-assembly as a powerful mechanism to achieve efficient and high-yield assembly in synthetic systems whether robustness or ease of "designability" is preferred.

Layer-by-layer unsupervised clustering of statistically relevant fluctuations in noisy time-series data of complex dynamical systems.

Complex systems are typically characterized by intricate internal dynamics that are often hard to elucidate. Ideally, this requires methods that allow to detect and classify in an unsupervised way the microscopic dynamical events occurring in the system. However, decoupling statistically relevant fluctuations from the internal noise remains most often nontrivial. Here, we describe "Onion Clustering": a simple, iterative unsupervised clustering method that efficiently detects and classifies statistically relevant fluctuations in noisy time-series data. We demonstrate its efficiency by analyzing simulation and experimental trajectories of various systems with complex internal dynamics, ranging from the atomic- to the microscopic-scale, in- and out-of-equilibrium. The method is based on an iterative detect-classify-archive approach. In a similar way as peeling the external (evident) layer of an onion reveals the internal hidden ones, the method performs a first detection/classification of the most populated dynamical environment in the system and of its characteristic noise. The signal of such dynamical cluster is then removed from the time-series data and the remaining part, cleared-out from its noise, is analyzed again. At every iteration, the detection of hidden dynamical subdomains is facilitated by an increasing (and adaptive) relevance-to-noise ratio. The process iterates until no new dynamical domains can be uncovered, revealing, as an output, the number of clusters that can be effectively distinguished/classified in a statistically robust way as a function of the time-resolution of the analysis. Onion Clustering is general and benefits from clear-cut physical interpretability. We expect that it will help analyzing a variety of complex dynamical systems and time-series data.

Utilizing developmental dynamics for evolutionary prediction and control.

Understanding, predicting, and controlling the phenotypic consequences of genetic and environmental change is essential to many areas of fundamental and applied biology. In evolutionary biology, the generative process of development is a major source of organismal evolvability that constrains or facilitates adaptive change by shaping the distribution of phenotypic variation that selection can act upon. While the complex interactions between genetic and environmental factors during development may appear to make it impossible to infer the consequences of perturbations, the persistent observation that many perturbations result in similar phenotypes indicates that there is a logic to what variation is generated. Here, we show that a general representation of development as a dynamical system can reveal this logic. We build a framework that allows predicting the phenotypic effects of perturbations, and conditions for when the effects of perturbations of different origins are concordant. We find that this concordance is explained by two generic features of development, namely the dynamical dependence of the phenotype on itself and the fact that all perturbations must affect the developmental process to have an effect on the phenotype. We apply our theoretical framework to classical models of development and show that it can be used to predict the evolutionary response to selection using information of plasticity and to accelerate evolution in a desired direction. The framework we introduce provides a way to quantitatively interchange perturbations, opening an avenue of perturbation design to control the generation of variation.

Meaning from movement and stillness: Signatures of coordination dynamics reveal infant agency.

How do human beings make sense of their relation to the world and realize their ability to effect change? Applying modern concepts and methods of coordination dynamics, we demonstrate that patterns of movement and coordination in 3 to 4-mo-olds may be used to identify states and behavioral phenotypes of emergent agency. By means of a complete coordinative analysis of baby and mobile motion and their interaction, we show that the emergence of agency can take the form of a punctuated self-organizing process, with meaning found both in movement and stillness.

An impossibility theorem in game dynamics.

The Nash equilibrium-a combination of choices by the players of a game from which no self-interested player would deviate-is the predominant solution concept in game theory. Even though every game has a Nash equilibrium, it is not known whether there are deterministic behaviors of the players who play a game repeatedly that are guaranteed to converge to a Nash equilibrium of the game from all starting points. If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this question becomes a problem in the theory of dynamical systems. We apply this theory, and in particular Conley index theory, to prove a general impossibility result: There exist games, for which all game dynamics fail to converge to Nash equilibria from all starting points. The games which help prove this impossibility result are degenerate, but we conjecture that the same result holds, under computational complexity assumptions, for nondegenerate games. We also prove a stronger impossibility result for the solution concept of approximate Nash equilibria: For a set of games of positive measure, no game dynamics can converge to the set of approximate Nash equilibria for a sufficiently small yet substantial approximation bound. Our results establish that, although the notions of Nash equilibrium and its computation-inspired approximations are universally applicable in all games, they are fundamentally incomplete as predictors of long-term player behavior.

A theoretical perspective on Waddington's genetic assimilation experiments.

Genetic assimilation is the process by which a phenotype that is initially induced by an environmental stimulus becomes stably inherited in the absence of the stimulus after a few generations of selection. While the concept has attracted much debate after being introduced by C. H. Waddington 70 y ago, there have been few experiments to quantitatively characterize the phenomenon. Here, we revisit and organize the results of Waddington's original experiments and follow-up studies that attempted to replicate his results. We then present a theoretical model to illustrate the process of genetic assimilation and highlight several aspects that we think require further quantitative studies, including the gradual increase of penetrance, the statistics of delay in assimilation, and the frequency of unviability during selection. Our model captures Waddington's picture of developmental paths in a canalized landscape using a stochastic dynamical system with alternative trajectories that can be controlled by either external signals or internal variables. It also reconciles two descriptions of the phenomenon-Waddington's, expressed in terms of an individual organism's developmental paths, and that of Bateman in terms of the population distribution crossing a hypothetical threshold. Our results provide theoretical insight into the concepts of canalization, phenotypic plasticity, and genetic assimilation.

The effects of Hh morphogen source movement on signaling dynamics.

Morphogens of the Hh family trigger gene expression changes in receiving cells in a concentration-dependent manner to regulate their identity, proliferation, death or metabolism, depending on the tissue or organ. This variety of responses relies on a conserved signaling pathway. Its logic includes a negative-feedback loop involving the Hh receptor Ptc. Here, using experiments and computational models we study and compare the different spatial signaling profiles downstream of Hh in several developing Drosophila organs. We show that the spatial distributions of Ptc and the activator transcription factor CiA in wing, antenna and ocellus show similar features, but are markedly different from that in the compound eye. We propose that these two profile types represent two time points along the signaling dynamics, and that the interplay between the spatial displacement of the Hh source in the compound eye and the negative-feedback loop maintains the receiving cells effectively in an earlier stage of signaling. These results show how the interaction between spatial and temporal dynamics of signaling and differentiation processes may contribute to the informational versatility of the conserved Hh signaling pathway.

Dynamic models augmented by hierarchical data: an application of estimating HIV epidemics at sub-national level.

Dynamic models have been successfully used in producing estimates of HIV epidemics at the national level due to their epidemiological nature and their ability to estimate prevalence, incidence, and mortality rates simultaneously. Recently, HIV interventions and policies have required more information at sub-national levels to support local planning, decision-making and resource allocation. Unfortunately, many areas lack sufficient data for deriving stable and reliable results, and this is a critical technical barrier to more stratified estimates. One solution is to borrow information from other areas within the same country. However, directly assuming hierarchical structures within the HIV dynamic models is complicated and computationally time-consuming. In this article, we propose a simple and innovative way to incorporate hierarchical information into the dynamical systems by using auxiliary data. The proposed method efficiently uses information from multiple areas within each country without increasing the computational burden. As a result, the new model improves predictive ability and uncertainty assessment.

Noninvasive inference methods for interaction and noise intensities of coupled oscillators using only spike time data.

Measurements of interaction intensity are generally achieved by observing responses to perturbations. In biological and chemical systems, external stimuli tend to deteriorate their inherent nature, and thus, it is necessary to develop noninvasive inference methods. In this paper, we propose theoretical methods to infer coupling strength and noise intensity simultaneously in two well-synchronized noisy oscillators through observations of spontaneously fluctuating events such as neural spikes. A phase oscillator model is applied to derive formulae relating each of the parameters to spike time statistics. Using these formulae, each parameter is inferred from a specific set of statistics. We verify these methods using the FitzHugh-Nagumo model as well as the phase model. Our methods do not require external perturbations and thus can be applied to various experimental systems.