Search

In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested in free-running (i.e., non-computer-controlled) systems. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of the theory of dynamical systems with symmeries in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of four microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming homogeneity across the oscillators, theory predicts four categories of stable spatiotemporal phase-locked periodic states and four categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed that three of the four phase-locked states were displaced from their idealized positions and, in the ensemble of measurements, appeared as clusters of different shapes and sizes, and that one of the predicted states was absent. We also observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity in intrinsic frequency. We further elucidate how different patterns of heterogeneity impact each attractor differently through a bifurcation analysis. We show that examining bifurcations along invariant manifolds provides a general framework for developing intuition about how chemical-specific dynamics interact with topology in the presence of heterogeneity that can be applied to other oscillators in other topologies.

Development of cortical regions with precise, sharp, and regular boundaries is essential for physiological function. However, little is known of the mechanisms ensuring these features. Here, we show that determination of the boundary between neocortex and medial entorhinal cortex (MEC), two abutting cortical regions generated from the same progenitor lineage, relies on COUP-TFI (chicken ovalbumin upstream promoter-transcription factor I), a patterning transcription factor with graded expression in cortical progenitors. In contrast with the classical paradigm, we found that increased COUP-TFI expression expands MEC, creating protrusions and disconnected ectopic tissue. We further developed a mathematical model that predicts that neuronal specification and differential cell affinity contribute to the emergence of an instability region and boundary sharpness. Correspondingly, we demonstrated that high expression of COUP-TFI induces MEC cell fate and protocadherin 19 expression. Thus, we conclude that a sharp boundary requires a subtle interplay between patterning transcription factors and differential cell affinity., (Copyright © 2021 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).)

The segmental organization of the vertebral column is established early in embryogenesis, when pairs of somites are rhythmically produced by the presomitic mesoderm (PSM). The tempo of somite formation is controlled by a molecular oscillator known as the segmentation clock 1,2 . Although this oscillator has been well-characterized in model organisms 1,2 , whether a similar oscillator exists in humans remains unknown. Genetic analyses of patients with severe spine segmentation defects have implicated several human orthologues of cyclic genes that are associated with the mouse segmentation clock, suggesting that this oscillator might be conserved in humans 3 . Here we show that human PSM cells derived in vitro-as well as those of the mouse 4 -recapitulate the oscillations of the segmentation clock. Human PSM cells oscillate with a period two times longer than that of mouse cells (5 h versus 2.5 h), but are similarly regulated by FGF, WNT, Notch and YAP signalling 5 . Single-cell RNA sequencing reveals that mouse and human PSM cells in vitro follow a developmental trajectory similar to that of mouse PSM in vivo. Furthermore, we demonstrate that FGF signalling controls the phase and period of oscillations, expanding the role of this pathway beyond its classical interpretation in 'clock and wavefront' models 1 . Our work identifying the human segmentation clock represents an important milestone in understanding human developmental biology.

Simple mathematical models can exhibit rich and complex behaviors. Prototypical examples of these drawn from biology and other disciplines have provided insights that extend well beyond the situations that inspired them. Here, we explore a set of simple, yet realistic, models for savanna-forest vegetation dynamics based on minimal ecological assumptions. These models are aimed at understanding how vegetation interacts with both climate (a primary global determinant of vegetation structure) and feedbacks with chronic disturbances from fire. The model includes three plant functional types-grasses, savanna trees, and forest trees. Grass and (when they allow grass to persist in their subcanopy) savanna trees promote the spread of fires, which in turn, demographically limit trees. The model exhibits a spectacular range of behaviors. In addition to bistability, analysis reveals ( i ) that diverse cyclic behaviors (including limit and homo- and heteroclinic cycles) occur for broad ranges of parameter space, ( ii ) that large shifts in landscape structure can result from endogenous dynamics and not just from external drivers or from noise, and ( iii ) that introducing noise into this system induces resonant and inverse resonant phenomena, some of which have never been previously observed in ecological models. Ecologically, these results raise questions about how to evaluate complicated dynamics with data. Mathematically, they lead to classes of behaviors that are likely to occur in other models with similar structure., Competing Interests: The authors declare no conflict of interest.

Loss of neurons in the neocortex is generally thought to result in a final reduction of cerebral volume. Yet, little is known on how the developing cerebral cortex copes with death of early-born neurons. Here, we tackled this issue by taking advantage of a transgenic mouse model in which, from early embryonic stages to mid-corticogenesis, abundant apoptosis is induced in the postmitotic compartment. Unexpectedly, the thickness of the mutant cortical plate at E18.5 was normal, due to an overproduction of upper layer neurons at E14.5. We developed and simulated a mathematical model to investigate theoretically the recovering capacity of the system and found that a minor increase in the probability of proliferative divisions of intermediate progenitors (IPs) is a powerful compensation lever. We confirmed experimentally that mutant mice showed an enhanced number of abventricular progenitors including basal radial glia-like cells and IPs. The latter displayed increased proliferation rate, sustained Pax6 expression and shorter cell cycle duration. Altogether, these results demonstrate the remarkable plasticity of neocortical progenitors to adapt to major embryonic insults via the modulation of abventricular divisions thereby ensuring the production of an appropriate number of neurons., (© The Author 2016. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.)

The timing of cortical neurogenesis has a major effect on the size and organization of the mature cortex. The deletion of the LIM-homeodomain transcription factor Lhx2 in cortical progenitors by Nestin-cre leads to a dramatically smaller cortex. Here we report that Lhx2 regulates the cortex size by maintaining the cortical progenitor proliferation and delaying the initiation of neurogenesis. The loss of Lhx2 in cortical progenitors results in precocious radial glia differentiation and a temporal shift of cortical neurogenesis. We further investigated the underlying mechanisms at play and demonstrated that in the absence of Lhx2, the Wnt/β-catenin pathway failed to maintain progenitor proliferation. We developed and applied a mathematical model that reveals how precocious neurogenesis affected cortical surface and thickness. Thus, we concluded that Lhx2 is required for β-catenin function in maintaining cortical progenitor proliferation and controls the timing of cortical neurogenesis.

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean-field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit (1995): their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.