In vitro characterization of the human segmentation clock.

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 clock1,2. Although this oscillator has been well-characterized in model organisms1,2, 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 humans3. Here we show that human PSM cells derived in vitro-as well as those of the mouse4-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 signalling5. 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' models1. Our work identifying the human segmentation clock represents an important milestone in understanding human developmental biology.

Pattern Formation in Mesic Savannas.

We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study necessary conditions for pattern-forming bifurcations. Homogeneous solutions dominate the dynamics of the standard forest-savanna model, regardless of the length scales of the various spatial processes considered. However, several different pattern-forming scenarios are possible upon including spatial resource limitation, such as competition for water, soil nutrients, or herbivory effects. Using numerical simulations and continuation, we study the nature of the resulting patterns as a function of system parameters and length scales, uncovering subcritical pattern-forming bifurcations and observing significant regions of multistability for realistic parameter regimes. Finally, we discuss our results in the context of extant savanna-forest modeling efforts and highlight ongoing challenges in building a unifying mathematical model for savannas across different rainfall levels.

Reliability and robustness of oscillations in some slow-fast chaotic systems.

A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator (CPG), this paper proposes a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales. We show that systems displaying relaxation cycles while going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales and are, thus, consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may also generate erratic activity at slow timescales. We analyze these phenomena in detail in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics. We validate this numerically in three other models: a simple model of the crustacean CPG neural network, a discrete cubic map, and a continuous flow.

The use of AI in legal systems: determining independent contractor vs. employee status.

The use of artificial intelligence (AI) to aid legal decision making has become prominent. This paper investigates the use of AI in a critical issue in employment law, the determination of a worker's status-employee vs. independent contractor-in two common law countries (the U.S. and Canada). This legal question has been a contentious labor issue insofar as independent contractors are not eligible for the same benefits as employees. It has become an important societal issue due to the ubiquity of the gig economy and the recent disruptions in employment arrangements. To address this problem, we collected, annotated, and structured the data for all Canadian and Californian court cases related to this legal question between 2002 and 2021, resulting in 538 Canadian cases and 217 U.S. cases. In contrast to legal literature focusing on complex and correlated characteristics of the employment relationship, our statistical analyses of the data show very strong correlations between the worker's status and a small subset of quantifiable characteristics of the employment relationship. In fact, despite the variety of situations in the case law, we show that simple, off-the-shelf AI models classify the cases with an out-of-sample accuracy of more than 90%. Interestingly, the analysis of misclassified cases reveals consistent misclassification patterns by most algorithms. Legal analyses of these cases led us to identify how equity is ensured by judges in ambiguous situations. Finally, our findings have practical implications for access to legal advice and justice. We deployed our AI model via the open-access platform, https://MyOpenCourt.org/, to help users answer employment legal questions. This platform has already assisted many Canadian users, and we hope it will help democratize access to legal advice to large crowds.

Symmetry breaking in the embryonic skin triggers directional and sequential plumage patterning.

The development of an organism involves the formation of patterns from initially homogeneous surfaces in a reproducible manner. Simulations of various theoretical models recapitulate final states of natural patterns, yet drawing testable hypotheses from those often remains difficult. Consequently, little is known about pattern-forming events. Here, we surveyed plumage patterns and their emergence in Galliformes, ratites, passerines, and penguins, together representing the three major taxa of the avian phylogeny, and built a unified model that not only reproduces final patterns but also intrinsically generates shared and varying directionality, sequence, and duration of patterning. We used in vivo and ex vivo experiments to test its parameter-based predictions. We showed that directional and sequential pattern progression depends on a species-specific prepattern: an initial break in surface symmetry launches a travelling front of sharply defined, oriented domains with self-organising capacity. This front propagates through the timely transfer of increased cell density mediated by cell proliferation, which controls overall patterning duration. These results show that universal mechanisms combining prepatterning and self-organisation govern the timely emergence of the plumage pattern in birds.

Concurrent Thalamostriatal and Corticostriatal Spike-Timing-Dependent Plasticity and Heterosynaptic Interactions Shape Striatal Plasticity Map.

The striatum integrates inputs from the cortex and thalamus, which display concomitant or sequential activity. The striatum assists in forming memory, with acquisition of the behavioral repertoire being associated with corticostriatal (CS) plasticity. The literature has mainly focused on that CS plasticity, and little remains known about thalamostriatal (TS) plasticity rules or CS and TS plasticity interactions. We undertook here the study of these plasticity rules. We found bidirectional Hebbian and anti-Hebbian spike-timing-dependent plasticity (STDP) at the thalamic and cortical inputs, respectively, which were driving concurrent changes at the striatal synapses. Moreover, TS- and CS-STDP induced heterosynaptic plasticity. We developed a calcium-based mathematical model of the coupled TS and CS plasticity, and simulations predict complex changes in the CS and TS plasticity maps depending on the precise cortex-thalamus-striatum engram. These predictions were experimentally validated using triplet-based STDP stimulations, which revealed the significant remodeling of the CS-STDP map upon TS activity, which is notably the induction of the LTD areas in the CS-STDP for specific timing regimes. TS-STDP exerts a greater influence on CS plasticity than CS-STDP on TS plasticity. These findings highlight the major impact of precise timing in cortical and thalamic activity for the memory engram of striatal synapses.

On the complex dynamics of savanna landscapes.

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.

Progressive alignment of inhibitory and excitatory delay may drive a rapid developmental switch in cortical network dynamics.

Nervous system maturation occurs on multiple levels-synaptic, circuit, and network-at divergent timescales. For example, many synaptic properties mature gradually, whereas emergent network dynamics can change abruptly. Here we combine experimental and theoretical approaches to investigate a sudden transition in spontaneous and sensory evoked thalamocortical activity necessary for the development of vision. Inspired by in vivo measurements of timescales and amplitudes of synaptic currents, we extend the Wilson and Cowan model to take into account the relative onset timing and amplitudes of inhibitory and excitatory neural population responses. We study this system as these parameters are varied within amplitudes and timescales consistent with developmental observations to identify the bifurcations of the dynamics that might explain the network behaviors in vivo. Our findings indicate that the inhibitory timing is a critical determinant of thalamocortical activity maturation; a gradual decay of the ratio of inhibitory to excitatory onset time drives the system through a bifurcation that leads to a sudden switch of the network spontaneous activity from high-amplitude oscillations to a nonoscillatory active state. This switch also drives a change from a threshold bursting to linear response to transient stimuli, also consistent with in vivo observation. Thus we show that inhibitory timing is likely critical to the development of network dynamics and may underlie rapid changes in activity without similarly rapid changes in the underlying synaptic and cellular parameters.NEW & NOTEWORTHY Relying on a generalization of the Wilson-Cowan model, which allows a solid analytic foundation for the understanding of the link between maturation of inhibition and network dynamics, we propose a potential explanation for the role of developing excitatory/inhibitory synaptic delays in mediating a sudden switch in thalamocortical visual activity preceding vision onset.

Interplay of multiple pathways and activity-dependent rules in STDP.

Hebbian plasticity describes a basic mechanism for synaptic plasticity whereby synaptic weights evolve depending on the relative timing of paired activity of the pre- and postsynaptic neurons. Spike-timing-dependent plasticity (STDP) constitutes a central experimental and theoretical synaptic Hebbian learning rule. Various mechanisms, mostly calcium-based, account for the induction and maintenance of STDP. Classically STDP is assumed to gradually emerge in a monotonic way as the number of pairings increases. However, non-monotonic STDP accounting for fast associative learning led us to challenge this monotonicity hypothesis and explore how the existence of multiple plasticity pathways affects the dynamical establishment of plasticity. To account for distinct forms of STDP emerging from increasing numbers of pairings and the variety of signaling pathways involved, we developed a general class of simple mathematical models of plasticity based on calcium transients and accommodating various calcium-based plasticity mechanisms. These mechanisms can either compete or cooperate for the establishment of long-term potentiation (LTP) and depression (LTD), that emerge depending on past calcium activity. Our model reproduces accurately the striatal STDP that involves endocannabinoid and NMDAR signaling pathways. Moreover, we predict how stimulus frequency alters plasticity, and how triplet rules are affected by the number of pairings. We further investigate the general model with an arbitrary number of pathways and show that depending on those pathways and their properties, a variety of plasticities may emerge upon variation of the number and/or the frequency of pairings, even when the outcome after large numbers of pairings is identical. These findings, built upon a biologically realistic example and generalized to other applications, argue that in order to fully describe synaptic plasticity it is not sufficient to record STDP curves at fixed pairing numbers and frequencies. In fact, considering the whole spectrum of activity-dependent parameters could have a great impact on the description of plasticity, and a better understanding of the engram.

Local homeoprotein diffusion can stabilize boundaries generated by graded positional cues.

Boundary formation in the developing neuroepithelium decides on the position and size of compartments in the adult nervous system. In this study, we start from the French Flag model proposed by Lewis Wolpert, in which boundaries are formed through the combination of morphogen diffusion and of thresholds in cell responses. In contemporary terms, a response is characterized by the expression of cell-autonomous transcription factors, very often of the homeoprotein family. Theoretical studies suggest that this sole mechanism results in the formation of boundaries of imprecise shapes and positions. Alan Turing, on the other hand, proposed a model whereby two morphogens that exhibit self-activation and reciprocal inhibition, and are uniformly distributed and diffuse at different rates lead to the formation of territories of unpredictable shapes and positions but with sharp boundaries (the 'leopard spots'). Here, we have combined the two models and compared the stability of boundaries when the hypothesis of local homeoprotein intercellular diffusion is, or is not, introduced in the equations. We find that the addition of homeoprotein local diffusion leads to a dramatic stabilization of the positioning of the boundary, even when other parameters are significantly modified. This novel Turing/Wolpert combined model has thus important theoretical consequences for our understanding of the role of the intercellular diffusion of homeoproteins in the developmental robustness of and the changes that take place in the course of evolution.

Enhanced Abventricular Proliferation Compensates Cell Death in the Embryonic Cerebral Cortex.

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.

Lhx2 regulates the timing of ß-catenin-dependent cortical neurogenesis.

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/ß-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 ß-catenin function in maintaining cortical progenitor proliferation and controls the timing of cortical neurogenesis.

The hemodynamic signal as a first-order low-pass temporal filter: Evidence and implications for neuroimaging studies.

Neuronal activation triggers local changes in blood flow and hemoglobin oxygenation. These hemodynamic signals can be recorded through functional magnetic resonance imaging or intrinsic optical imaging, and allows inferring neural activity in response to stimuli. These techniques are widely used to uncover functional brain architectures. However, their accuracy suffers from distortions inherent to hemodynamic responses and noise. The analysis of these signals currently relies on models of impulse hemodynamic responses to brief stimuli. Here, in order to infer precise functional architectures, we focused on integrated signals associated to the dynamic response of functional maps. To this end, we recorded orientation and direction maps in cat primary visual cortex and compared two protocols: the conventional episodic stimulation technique and a continuous, periodic stimulation paradigm. Conventional methods show that the dynamics of activation and deactivation of the functional maps follows a linear first-order differential equation representing a low-pass filter. Comparison with the periodic stimulation methods confirmed this observation: the phase shifts and magnitude attenuations extracted at various frequencies were consistent with a low-pass filter with a 5s time constant. This dynamics presumably reflects the variations in deoxyhemoglobin mediated by arterial dilations. This dynamics open new avenues in the analysis of neuroimaging data that differs from common methods based on the hemodynamic response function. In particular, we demonstrate that inverting this first-order low-pass filter minimized the distortions of the signal and enabled a much faster and accurate reconstruction of functional maps.

On the Dynamical Interplay of Positive and Negative Affects.

Emotional disorders and psychological flourishing are the result of complex interactions between positive and negative affects that depend on external events and the subject's internal representations. Based on psychological data, we mathematically model the dynamical balance between positive and negative affects as a function of the response to external positive and negative events. This modeling allows the investigation of the relative impact of two leading forms of therapy on affect balance. The model uses a delay differential equation to analytically study the bifurcation diagram of the system. We compare the results of the model to psychological data on a single, recurrently depressed patient who was administered the two types of therapies considered (coping focused versus affect focused). The model leads to the prediction that stabilization at a normal state may rely on evaluating one's emotional state through a historical ongoing emotional state rather than in a narrow present window. The simple mathematical model proposed here offers a theoretical framework for investigating the temporal process of change and parameters of resilience to relapse.

Pinwheel-dipole configuration in cat early visual cortex.

In the early visual cortex, information is processed within functional maps whose layouts are thought to underlie visual perception. However, the precise organization of these functional maps as well as their interrelationships remain unsettled. Here, we show that spatial frequency representation in cat early visual cortex exhibits singularities around which the map organizes like an electric dipole potential. These singularities are precisely co-located with singularities of the orientation map: the pinwheel centers. To show this, we used high resolution intrinsic optical imaging in cat areas 17 and 18. First, we show that a majority of pinwheel centers exhibit in their neighborhood both semi-global maximum and minimum in the spatial frequency map (i.e. extreme values of the spatial frequency in a hypercolumn). This contradicts pioneering studies suggesting that pinwheel centers are placed at the locus of a single spatial frequency extremum. Based on an analogy with electromagnetism, we proposed a mathematical model for a dipolar structure, accurately fitting optical imaging data. We conclude that a majority of orientation pinwheel centers form spatial frequency dipoles in cat early visual cortex. Given the functional specificities of neurons at singularities in the visual cortex, it is argued that the dipolar organization of spatial frequency around pinwheel centers could be fundamental for visual processing.

Parsimony, Exhaustivity and Balanced Detection in Neocortex.

The layout of sensory brain areas is thought to subtend perception. The principles shaping these architectures and their role in information processing are still poorly understood. We investigate mathematically and computationally the representation of orientation and spatial frequency in cat primary visual cortex. We prove that two natural principles, local exhaustivity and parsimony of representation, would constrain the orientation and spatial frequency maps to display a very specific pinwheel-dipole singularity. This is particularly interesting since recent experimental evidences show a dipolar structures of the spatial frequency map co-localized with pinwheels in cat. These structures have important properties on information processing capabilities. In particular, we show using a computational model of visual information processing that this architecture allows a trade-off in the local detection of orientation and spatial frequency, but this property occurs for spatial frequency selectivity sharper than reported in the literature. We validated this sharpening on high-resolution optical imaging experimental data. These results shed new light on the principles at play in the emergence of functional architecture of cortical maps, as well as their potential role in processing information.

Anti-Hebbian plasticity drives sequence learning in striatum.

Spatio-temporal activity patterns have been observed in a variety of brain areas in spontaneous activity, prior to or during action, or in response to stimuli. Biological mechanisms endowing neurons with the ability to distinguish between different sequences remain largely unknown. Learning sequences of spikes raises multiple challenges, such as maintaining in memory spike history and discriminating partially overlapping sequences. Here, we show that anti-Hebbian spike-timing dependent plasticity (STDP), as observed at cortico-striatal synapses, can naturally lead to learning spike sequences. We design a spiking model of the striatal output neuron receiving spike patterns defined as sequential input from a fixed set of cortical neurons. We use a simple synaptic plasticity rule that combines anti-Hebbian STDP and non-associative potentiation for a subset of the presented patterns called rewarded patterns. We study the ability of striatal output neurons to discriminate rewarded from non-rewarded patterns by firing only after the presentation of a rewarded pattern. In particular, we show that two biological properties of striatal networks, spiking latency and collateral inhibition, contribute to an increase in accuracy, by allowing a better discrimination of partially overlapping sequences. These results suggest that anti-Hebbian STDP may serve as a biological substrate for learning sequences of spikes.

Topological and dynamical complexity of random neural networks.

Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown and, similar to spin glasses, shall be fundamentally related to the behavior of the system. In this Letter, we investigate the explosion of complexity arising near that phase transition. We show that the mean number of equilibria undergoes a sharp transition from one equilibrium to a very large number scaling exponentially with the dimension on the system. Near criticality, we compute the exponential rate of divergence, called topological complexity. Strikingly, we show that it behaves exactly as the maximal Lyapunov exponent, a classical measure of dynamical complexity. This relationship unravels a microscopic mechanism leading to chaos which we further demonstrate on a simpler disordered system, suggesting a deep and underexplored link between topological and dynamical complexity.

Heterogeneous connections induce oscillations in large-scale networks.

Realistic large-scale networks display a heterogeneous distribution of connectivity weights that might also randomly vary in time. We show that, depending on the level of heterogeneity in the connectivity coefficients, different qualitative macroscopic and microscopic regimes emerge. We evidence, in particular, generic transitions from stationary to perfectly periodic phase-locked regimes as the disorder parameter is increased, both in a simple model treated analytically and in a biologically relevant network made of excitable cells.

Exact event-driven implementation for recurrent networks of stochastic perfect integrate-and-fire neurons.

In vivo cortical recording reveals that indirectly driven neural assemblies can produce reliable and temporally precise spiking patterns in response to stereotyped stimulation. This suggests that despite being fundamentally noisy, the collective activity of neurons conveys information through temporal coding. Stochastic integrate-and-fire models delineate a natural theoretical framework to study the interplay of intrinsic neural noise and spike timing precision. However, there are inherent difficulties in simulating their networks' dynamics in silico with standard numerical discretization schemes. Indeed, the well-posedness of the evolution of such networks requires temporally ordering every neuronal interaction, whereas the order of interactions is highly sensitive to the random variability of spiking times. Here, we answer these issues for perfect stochastic integrate-and-fire neurons by designing an exact event-driven algorithm for the simulation of recurrent networks, with delayed Dirac-like interactions. In addition to being exact from the mathematical standpoint, our proposed method is highly efficient numerically. We envision that our algorithm is especially indicated for studying the emergence of polychronized motifs in networks evolving under spike-timing-dependent plasticity with intrinsic noise.