Chaos: An Interdisciplinary Journal of Nonlinear Science

We have studied the emergence of slow relaxation oscillations in next generation neural mass models with spike frequency adaptation. Relaxation oscillations connect low firing state (Down state) to high firing state (Up state) via the slow adaptation. In the examined cases, the orbit relaxes towards the Up State via a sequence of collective damped oscillations (peaks of activity), thus revealing population bursting dynamics. The slower is the adaptation time scale the higher is the complexity (number of peaks) displayed by the relaxation oscillations. In particular, a chaos-induced spike-adding mechanism regulates the increase in the number of peaks. In analogy to what found in the Hidmarsh-Rose neuron model, two different types of chaotic behaviors have been identified: Population Spiking and Population Bursting Chaos. The increase of the adaptation strength leads to shorter (longer) Up (Down) state durations somehow mimicking the effect of charbachol in in vitro experiments, where spontaneous slow waves are observed. Indeed, the scenario depicted in [1], where an increase of the concentration of carbachol induces a transition from anesthesia-like to sleep-like dynamics is consistent with our results based on the variation of the adaptation strength. HighlightsO_LISpike Frequency Adaptation (SFA) promotes the emergence of Slow Relaxation Oscillations C_LIO_LISpike-adding mechanisms, controlled by SFA, lead to Relaxation Oscillations of increasing complexity C_LIO_LITwo types of chaotic behaviours: Population Spiking and Population Bursting Chaos C_LIO_LISFA regulates Up and Down States durations and their correlation C_LI

Despite extensive study of cellular mechanisms underlying long-term potentiation, no single specific protein or gene has been identified which encodes an individual unit of information, or memory bit. Indeed, the brain engram remains a knowledge gap. The theory of exclusion led us to cancel one-by-one several unrealistic biological options, suggesting that the explanation resides somewhere else. Superposition of up to concentric 300 myelin layers, spiraled, and highly compacted wrapping a single axon and each wrap could host hundreds to thousands of niches, as memory cells, collectively consisting of a massive array of cells. The disjointed 3D spatial superposition allows storage of charges, nodes not facing from a layer to next. The thickness of a single myelin layer ranges from 7.0 to 20 nm. The dimension scale is approximately the exact dimensions of the charge trap, the tunnel and dielectric also equipping current AI microchips. Stored charges are positive ions, with similar effect whether charges are negative or positive charges creating an electromagnetic field. To write data, following an action potential, this voltage applies to the control gates of the myelin layers producing an ionic charge injection. This causes charges to gain energy and tunnel through the myelin layer across Ranvier nodes, via quantum tunneling, and deep into the concentric myelin multilayers. This is creating an insulated trapping of K+ ions isolated from the system. In a long white matter tract bundle, the near-perfect isolation of millions of axons within compressed myelin wrap-ion channel K+/Na+ systems provides quantum coherence and precision of asynchronous firing property. The injected ionic charges (K+) become physically stuck in traps within the myelin layers. The K+ ions may not move freely, completely trapped after AP ceases. Mirroring a single-bit, single-level-cell, a trapped ionic charge (ions K+) may represent a 1, while an empty cell (absence of K+) represents a 0. The trial-and-error process, with a Bayesian inference which may have also been the core evolution of the learning human brain. Based on selected mathematical equations, we analyzed the general scheme on how deep learning may be embedded in the brain

Efficient operation of neural networks has been linked to criticality in their underlying non-equilibrium excitation dynamics. However, obtaining experimental evidence of this conjecture remains challenging due to limited control and undersampling in biological systems. Here, we experimentally explore neural network criticality using an ultracold Rydberg gas as a highly controllable simulator. We highlight the similarity of the excitation spreading via Rydberg facilitation and the synaptic connection of spiking activity of neurons, giving rise to distinct absorbing and active phases. We systematically explore and resolve criticality criteria, including power-law scaling of excitation avalanches and the emergence of universal avalanche shape collapse. Crucially, we implement a controlled gain mechanism to compensate for atom loss, mimicking metabolic resource replenishment and stabilizing the system in a controlled non-equilibrium steady state. We find peak temporal correlations at the critical point and stochastic oscillations with dragon king avalanches in the active phase, consistent with predictions for systems orbiting criticality. Our work establishes facilitated Rydberg gases as a platform for investigating criticality, resource dynamics, and emergent oscillations in neural networks.

We develop a theoretical proposal linking vacuum stability and brain dynamics through superconductivity-inspired coherence, symmetry reduction, and the thermodynamic stabilization of low-entropy regimes. We take an unbroken SU(3) structure as a candidate stable residue of the low-temperature vacuum. At the neural level, we formulate a coarse-grained analog in which a two-fluid model with dissipative and coherence-supporting components describes brain dynamics. Specifically, the coherence-supporting component is proposed as a possible basis for the efficient binding and integration required to sustain a stable, unified conscious state. The proposal offers a common geometric language for relating physics and neuroscience with falsifiable signatures in coherence and state-dependent transitions. The main technical contribution is a computational algebraic model of conscious-state dynamics, where neural data are mapped to reconstructed state trajectories. Effective generators are inferred from those trajectories, and the two-fluid split is tested as a Cartan-root decomposition of su(3), with a rank-two commuting sector for coherence-preserving balance and six root directions for state transitions. This structure can be tested on neural data and contrasted with alternative dynamical models.

With 302 neurons and a rigorously characterized connectome, the nematode Caenorhabditis elegans represents a powerful model organism to study the fundamental roles of neuronal circuits in behavior. However, despite the breadth of research, many questions remain unanswered regarding how these organisms are able to successfully navigate their environment. Here, we present a biologically grounded dynamical circuit model for the investigation of sensory-guided behavior during C. elegans chemotaxis. Our mathematical model consists of the chemosensory neuron AWA, interneurons RIM and RIA, motor neurons, including SMDs and RMDs, and body wall muscles that provide proprioceptive feedback through stretch receptors. After optimization with an evolutionary algorithm, the model locomotes effectively toward a chemical attractant, realistically capturing nematode chemotactic behavior. Chemotaxis is ensured by sharp turns, which resemble the omega turns of living nematodes, as a key emergent property of the model. The sharp turning behavior is triggered by decreases in the concentration of the attractant. These result in reduced AWA activity, which in turn triggers disinhibition of RIM and subsequent changes in RIA oscillations. The ensuing coordinated changes in downstream motor neurons activity patterns produce sharp turns, which correct the nematodes path, so that the model worm heads toward the attractant, and remains at its proximity, after it reaches the gradient peak. The proposed framework, along with its emergent dynamics, provides new insights into the minimum requirements for C. elegans circuitry to display major features of its chemotactic behavior, including omega turns. In parallel, it generates experimentally testable hypotheses with respect to the participating neuronal elements.

Neural synchronization is fundamental to brain function and, when it becomes excessive, underlies pathological conditions such as epilepsy. Among brain regions, the temporal lobes, and the hippocampus in particular, exhibit the highest epileptogenic potential, with mesial temporal lobe epilepsy representing the most prevalent form of the condition in humans. Within the hippocampus, extracellular potassium dynamics are central to non-synaptic epileptiform activity, and astrocytic potassium buffering mechanisms have emerged as key regulators of network excitability. Yet the specific contributions of astrocytic gap-junction coupling and potassium spatial buffering to neuronal synchronization across different spatial scales remain poorly understood. To address this gap, we developed a microcircuit biophysical model consisting of two astrocyte-neuron modules, each comprising one astrocyte coupled to five neurons. Astrocyte-neuron interactions are mediated exclusively through shared extracellular potassium dynamics. Using a reduced astrocyte model that captures both local membrane and syncytial potassium buffering, we systematically investigated how astrocytic potassium handling shapes neuronal activity patterns and inter-module synchronization. Our results demonstrate that astrocytes prevent the emergence of pathological states -- such as sustained ictal activity and depolarization block, by stabilizing extracellular potassium levels. Furthermore, we show that astrocytic gap-junction coupling strength critically regulates phase synchronization between neuronal modules: stronger coupling promotes inter-module synchrony under physiological conditions, whereas impaired astrocytic function drives networks toward pathological hypersynchronization when extracellular potassium is elevated. These findings support the hypothesis that astrocytic networks impose modularity on hippocampal neuronal assemblies, and suggest that astrocytic connexins may represent a relevant therapeutic target in epilepsy and other disorders characterized by aberrant neural synchronization. Author summary

Sequestration of gene products through diverse mechanisms forms a fundamental layer of regulation in intracellular biochemical processes, including post-translational modification, promiscuous binding to genomic decoy sites, and partitioning into membraneless compartments formed through phase separation. Here, we develop a unified stochastic framework to quantify how such sequestration-type processes, when coupled to noisy gene expression, modulate cell-to-cell variation in protein levels. In this model, protein molecules reversibly switch between active (free) and inactive (sequestered) states, whose switching rates are arbitrary functions of the molecular counts. Using exact analytical calculations and the linear noise approximation, we derive expressions for the Fano factor of the active-protein level and identify fluctuation attenuation regimes in terms of the logarithmic sensitivities of the switching rates to protein abundances. We show that inactive-protein-dependent switching, of which genomic decoy binding is a natural example, can preserve Poisson-level fluctuations in the active-protein level under appropriate conditions. Enzymatic inactivation, a type of post-translational modification, emerges as a special case of active-protein-dependent sequestration, where greater responsiveness of the inactivation propensity attenuates active-protein fluctuations. In both decoy binding and enzymatic inactivation, protecting the inactive protein from decay lowers active-protein fluctuations. Finally, a noise-buffering regime associated with intracellular phase separation is recovered when the inactive-to-active switching rate depends inversely on the inactive-protein level. Together, the examples of genomic decoy binding, enzymatic inactivation, and intracellular phase separation suggest that noise buffering observed across diverse intracellular processes is rooted in a broader class of reversible sequestration mechanisms that attenuate protein-level fluctuations.

The basic reproduction number R0 confounds pathogen biology with adaptive human contact behavior. Earlier epidemiological--economic theory predicted a forward-looking behavioral contact response but could not test it in the absence of appropriate behavioral data. Using directly measured mobility as an observable proxy for contact, we (i) estimate the behavioral response function directly from data; (ii) show that the biology/behavior decomposition and hence the behavioral correction to R0 is not identified from an epidemic trajectory, the apparent constant-contact R0 being one endpoint of an observational-equivalence class that fits the factual curve identically yet diverges under counterfactual; and (iii) characterize that divergence ("what R0 deletes") as state-dependent, unimodal in counterfactual severity and vanishing when behavior saturates. We then show that, across US jurisdictions, the correction is empirically bounded because risk-responsiveness and behavioral non-saturation are confounded (r=-0.57, n=51): where behavior could compensate, it was already maximal, and where it was not maximal it did not respond. What R0 deletes is thus real and structurally characterizable yet empirically modest here, for reasons the framework itself supplies.

Understanding how network connectivity shapes neural representations is central to systems neuroscience. While dimensionality reduction methods uncover low-dimensional manifold structure in population recordings, a rigorous framework connecting manifold geometry to network mechanisms and information encoding remains lacking. We develop a differential geometric approach for analyzing neural manifolds in rate-based recurrent networks receiving tuned feedforward inputs. We derive expressions for the pullback metric of neural manifolds, showing how input tuning curves, feedforward and recurrent synaptic connectivity shape manifold geometry. Critically, we establish that the Fisher information matrix at steady states also has the structure of a pullback metric, directly linking intrinsic manifold geometry to stimulus discriminability and information encoding. For noise with slow temporal correlations propagated through the network, we show that recurrent effects on information geometry cancel: Fisher information depends only on the feedforward connectivity. Thus, feedforward connectivity critically determines representational geometry. As an example, we demonstrate that the representation of space by a module of hexagonal grid cells is approximately isometric for random distribution of grid phases. Moreover, a linear feedforward transformation can map spatially random input tuning curves into a population of hexagonal grid cells, forming a toroidal manifold. Thus, feedforward connectivity alone can generate structured spatial representations without requiring carefully tuned recurrent connectivity or continuous attractor dynamics. Recurrent connectivity, however, is shown to improve stimulus encoding under fast noise, thereby implementing a selective noise reduction.

Stabilizing synaptic plasticity together with the neurons activity has remained a central challenge in theoretical neuroscience since the introduction of Hebbian learning principles. Classical Hebbian learning rules typically lead to unbounded synaptic growth, motivating the development of stabilization mechanisms such as normalization methods, BCM-type learning, synaptic scaling and others. While these approaches can prevent divergence, they can also exhibit different limitations e.g. resulting in too-sparse synaptic configurations or leading to poor scalability with increasing network size. A recently introduced meta-plasticity mechanism, termed annealed linear learning (ALL), dynamically reduces the learning rate as neuronal output increases, thereby preserving stable and interpretable fixed-point behavior of the output. However, the original formulation leads to an irreversible decay of the learning rate, preventing adaptation to changing environmental conditions. To address this, in the present study, we balance learning rate reduction at large outputs with recovery at small outputs and in addition introduce forgetting that gradually reduces synaptic weights. These extensions allow the system to discard outdated representations and adapt to novel input conditions. Analytical investigations demonstrate that the favorable output fixed-point properties of the original ALL framework are preserved under the extended rule. Furthermore, simulations with an artificial agent show that the proposed mechanism enables robust and fast re-learning and adaptation in changing environments.

SynGAP and PSD-95 undergo liquid-liquid phase separation in vitro, motivating models of the postsynaptic density (PSD) as a biomolecular condensate. However, how their association kinetics and intermolecular interactions shape activity-dependent partitioning remains unclear. We develop a multicomponent framework that couples equilibrium Flory-Huggins calculations in conserved total-composition space to four-species Cahn-Hilliard-reaction dynamics for free SynGAP, free PSD-95, the SynGAP-PSD-95 complex, and solvent. Complex formation and dissociation are represented by explicit forward and reverse activity-based mass-action rates. These calculations show that the equilibrium association constant and self- and cross-interaction energies jointly control the extent and composition of the two-phase region. For an illustrative LTP-like parameter switch, decreasing the equilibrium association constant and altering selected interaction energies redistributes SynGAP from the PSD-95-rich phase to the PSD-95-poor phase without eliminating the PSD-95-rich phase. A separate equilibrium comparison represents haploinsufficiency as a 50% reduction in total SynGAP. For the illustrative parameter sets, the reduced-SynGAP composition remains within the two-phase region but has different tie-line endpoints and phase compositions than the control composition.

The global potential of a chemical reaction network has many applications and is closely related to the stochastic detailed balance. However, many fundamental questions concerning stochastic detailed balance remain unresolved, such as whether it depends on the system volume and how to construct new systems that satisfy it. In this paper, we show that stochastic detailed balance may depend on the system volume. We therefore introduce four types of stochastic detailed balance according to their dependence on volume and rate constants, and systematically investigate the relationships among them. Our results distinguish detailed balance arising from particular choices of volume and parameters from that enforced by network structure, and identify conditions under which detailed balance at one volume extends to all volumes. We further obtain a class of networks satisfying stochastic detailed balance for every volume and every positive choice of rate constants, and construct new systems whose global potentials exhibit double-well structures.

A central question in neuroscience is how the brains structural connectivity gives rise to its emergent, correlated dynamics. These large-scale dynamical correlations underlie functional networks that support cognitive functions. Here, we identify coupling correlation--the similarity between the input connectivity profiles of brain regions--as a key structural determinant of macroscopic neural dynamical correlation. Using dynamical mean-field theory (DMFT) and numerical simulations of random neural network models, we demonstrate that coupling correlation quantitatively governs dynamical correlation. The functional form of this structure-function mapping is dictated by the eigenvalue spectrum of the coupling correlation matrix: networks with bulk eigenspectra exhibit an exact linear relationship, whereas biologically plausible long-tailed spectra yield an approximately linear mapping except when the magnitude of coupling correlation approaches unity. Particularly, a long-tailed spectrum is necessary to reproduce the appropriate magnitude and size-invariance of coupling correlations observed in empirical data, thereby sustaining non-vanishing dynamical correlations that may support brain function in large systems. The theoretical prediction of approximate linearity is consistently validated using empirical datasets that include both structural coupling and neural dynamics in humans, mice, and Drosophila. Together, these results provide a mechanistic and quantitative framework linking macroscopic brain network structure to emergent neural dynamics--an essential step toward a theory of structure-function relationship in the brain. Significance StatementHow the brains wiring gives rise to its coordinated activity is a fundamental unsolved problem in neuroscience. Prior work has identified correlations between structural and functional connectivity, but these relationships lacked a mechanistic, first-principles explanation. Here, we derive an analytical framework using Dynamical Mean-Field Theory and random neural network models to show that a single structural statistic--coupling correlation, the similarity between the input connectivity profiles of brain regions--linearly and causally determines the magnitude of correlated neural dynamics. We further show that a long-tailed eigenvalue spectrum in biological structural connectivity is necessary to sustain the strong, size-invariant functional correlations observed across species. Validated in humans, mice, and Drosophila using multiple imaging and connectome modalities, this principle may provide a quantitative bridge between structural connectomics and emergent brain dynamics, with implications extending to a broad class of complex networked systems.

Primary sensory cortices often organize neurons with similar stimulus preference into spatially functional maps. Recent work in mouse primary visual cortex (V1) has established that neuronal tuning to the orientation of visual grating stimuli is organized into micro-clusters, where physically close neuron pairs (~ 20 {micro}m) share highly similar orientation preferences, but the organization is unstructured beyond this narrow range. This fine-scale organization is seemingly at odds with the underlying intracortical circuitry in mouse V1 whose spatial extent is an order of magnitude broader (100 ~ 200 {micro}m). In this study, we explore an activity-dependent synaptic plasticity model of spatially structured thalamo-cortical connectivity. We develop theory under asymptotic conditions specific for mouse V1, and derive concrete circuit conditions under which micro-clusters naturally develop. In particular, the recurrent interaction among V1 neurons requires an additional component over a micro-spatial scale, while the spatial profiles of balanced excitation and inhibition support an effective micro-scale interaction. Together, our results provide a developmental mechanism and analytical framework linking thalamo-cortical development, recurrent circuit structure, and the emergence of functional organization in primary visual cortex.

In a population model for an infectious disease, we consider the early stochastic dynamics of an emergent 'mutant' strain, appearing and spreading during an epidemic of another 'wildtype' strain. The mutant may not reach establishment in the host population. The time at which the mutant first appears determines its probability of establishment. We calculate this establishment probability with two methods. The first method assumes a classical branching process, with a constant transmission rate. The second method reflects the changing size of the pool of susceptible hosts, due to the dynamics of the wildtype. We find that susceptible depletion can substantially impact the establishment probability. We explore the consequences of this stochastic establishment on the "escape pressure" acting on a pathogen to produce immune escape variants. We find that the overall escape pressure rate depends strongly on the appearance time of the mutant, especially if the establishment probability is itself shaped by the continued spread of the wildtype. In most scenarios, the escape pressure rate (and thus, the risk of new escape variants) peaks slightly earlier than the prevalence of the wildtype strain. Integrating the escape pressure over time, we obtain the cumulative escape pressure generated by the wildtype epidemic. The relationship between the escape pressure and the vaccination coverage depends on the cross-immunity, due to susceptible depletion. For example, with intermediate cross-immunity, the risk of immune escape may be lowest at intermediate vaccination coverages. Thus, these results raise important considerations for vaccination strategies in response to novel outbreaks.

The initial response of populations to the SARS-CoV-2 virus reduced the incidence of COVID-19 cases. However, this success was shorted lived once most populations relaxed most restrictions, resulting in an increase in infections. This feedback contributed to additional pandemic waves. The temporal unfolding of behavioural changes in populations present a challenge to mathematical models for disease dynamics. Coupled behaviour-disease models with varying levels of complexity accounting for several factors have been used to capture behavioural dynamics and SARS-CoV-2 transmission, with varying results. To study the impact of model complexity on the predictive power of models, here we formulate five coupled behaviour-disease models with varying structure and number of parameters. We fit the models to SARS-CoV-2 infection incidence and stringency of control interventions from five European countries in the first wave, and study how well these fitted models predict the second wave. We show that models with more parameters do not necessarily have a greater ability to explain and predict key features of a pandemic wave. Hence, our results show that a relatively simple coupled behaviour-disease model with important parameters can do an adequate job of providing information about the pandemic wave. Additionally, our findings show that complex models can be country-specific, working better for some countries and poorly for others. We conclude that modellers should not always opt for the most complicated possible models, if the data do not support their use.

Cancer remains a global health challenge requiring sophisticated understanding of tumor-immune dynamics for effective treatment design. Mathematical oncology has emerged as a rapidly evolving interdisciplinary field that uses mathematical models to enhance our understanding of cancer dynamics, including tumor growth, metastasis, and treatment response. This paper presents a comprehensive multiscale framework integrating patient-specific data, machine learning, and optimal control for personalized immunotherapy design. We develop a hybrid model that combines deterministic dynamics with stochastic elements and time delays, capturing the inherent variability and temporal lags in biological processes. The model incorporates biologically realistic Holling Type-II functional responses and is validated against longitudinal clinical data from 100+ cancer patients and patient-derived organoid experiments. Using deep neural networks with Bayesian regularization, we learn patient-specific parameter distributions from clinical biomarkers and predict treatment responses with high accuracy. Our optimal control framework, incorporating clinical constraints and toxicity limits, generates personalized treatment protocols that stabilize otherwise unstable dynamics. The framework establishes a new paradigm for precision immuno-oncology, bridging mathematical theory, computational methods, and clinical practice. Author summaryCancer remains one of the leading causes of death worldwide, and the immune system plays a crucial role in controlling tumor growth. However, the complex interactions between tumor cells and immune cells make it difficult to predict how individual patients will respond to immunotherapy. In this work, we develop a mathematical framework that integrates patient-specific data, machine learning, and optimal control to design personalized immunotherapy strategies. Our model captures the realistic dynamics of tumor-immune interactions by incorporating biologically relevant features such as time delays (representing immune response lags) and stochastic effects (representing biological variability). Using deep learning, we estimate patient-specific parameters from clinical biomarkers, enabling personalized predictions of treatment outcomes. We validate our framework against data from over 100 cancer patients and patient-derived organoid experiments, demonstrating excellent agreement. Our optimal control approach generates personalized treatment protocols that stabilize otherwise unstable tumor dynamics, achieving 78% tumor reduction compared to 52% for standard-of-care protocols. These findings suggest that therapies targeting immunological thresholds may be as important as those directly killing tumor cells, providing a new perspective for immunotherapy design. This framework bridges mathematical theory, computational methods, and clinical practice, offering a pathway toward truly personalized cancer treatment.

In the two-process model of sleep-wake regulation, circadian-modulated thresholds time every sleep onset and awakening, yet they remain free parameters rather than quantities derived from neuronal dynamics. This limitation leaves the framework unable to predict how sleep patterns change when neuromodulatory drive is altered, as in aging and narcolepsy. Here we close that gap by deriving closed-form, circadian-modulated threshold expressions from the Phillips-Robinson model with explicit orexinergic excitation of the wake-promoting population. Within this single threshold geometry, aging and narcolepsy appear as opposite deformations along the orexinergic axis: age-related hyperexcitability of orexin neurons elevates the sleep-onset boundary and creates a fragility regime in which minor nocturnal disturbances trigger premature awakenings, whereas orexin loss depresses the same boundary toward the wake-onset threshold and produces the rapid state fragmentation of narcolepsy. Concurrently, reduced circadian amplitude compresses the inter-threshold corridor, advancing sleep onset and shortening sleep duration. These results convert the classical two-process thresholds from descriptive conveniences into mechanistic organizers of sleep-wake dynamics across healthy aging and orexin deficiency. Author summaryThe classical two-process model of sleep relies on a pair of switching thresholds that have been imposed by hand rather than derived from the neurons that actually stabilize sleep and wakefulness. Here we remove this limitation: starting from a biophysical mean-field model of the sleep- and wake-promoting populations, and adding the orexin system that stabilizes arousal, we derive the sleep-onset and awakening thresholds analytically from the bifurcation geometry of the underlying dynamical system. These closed-form thresholds depend explicitly on circadian phase, homeostatic state, and orexinergic tone, revealing that aging and narcolepsy are opposite deformations of a single threshold corridor along one orexinergic axis. In aging, orexin hyperexcitability raises the sleep-onset barrier past a sharp "arousal fragility boundary," beyond which a minor disturbance triggers irreversible awakening; in narcolepsy, orexin loss collapses the same barrier and fragments sleep while paradoxically preserving total sleep time. We further find, contrary to common assumption, that orexin sustains wakefulness chiefly by raising the barrier to falling asleep rather than by resisting awakening. This work turns phenomenological sleep thresholds into physics-derived organizers of behavior, providing a mechanistic bridge from neuronal circuitry to whole-organism sleep dynamics.

A novel mathematical framework to define the threshold of action potentials in excitable cells is presented. Unlike previously applied methods that rely on approximations or bifurcations, the approach focuses on the geometry of membrane potential trajectories. The changes in concavity during the upstroke of an action potential can be directly obtained from a time series of voltages. The concavity criterion is then extended to models based on autonomous dynamical systems where the changes in concavity can be obtained analytically from a curve of inflection points in phase space. The inflection point manifold defines a region required for excitability: all the orbits that cross it contain action potentials, and all the trajectories that contain action potentials are in it. This analytical principle can then be used to define excitability in a dynamical system, and also a measure of excitability that enables quantification and comparisons of excitability across dynamical system. The measure provides a way to compare the excitabilities of systems that model neurons with different electrophysiological phenotypes and consider different stimulus conditions. The traditionally vague physiological concept of electrical excitability is transformed into a rigorous analytical description by considering the time-dependent curvature of the membrane potential. The criterion is robust across smooth, single compartment models of electrical excitability and can be can be extended to single compartment models in higher dimensions, and multicompartment models as well.

The brain is the most captivating chef doeuvre of nature. Naturally then, the mind wonders about the process that births such a fascinating organ. Neurodevelopment is a complex yet robust phenomenon that conceals answers to our questions in its intricacies. In an attempt to shed some light on this matter, we study the developing brain connectome of the nematode, C. elegans across the post-embryonic phase. A tiny organism with only around 200 neurons comprising its brain and yet a diverse array of behaviors to display, it makes for a great model. Starting with most of its head neurons already present at hatching, the worm brain accumulates numerous more synaptic connections increasing the edge density. It maintains a weak connectivity throughout thereby, balancing global communication as well as hierarchy. At the mesoscopic level, we find that the core has a conserved backbone of persistent neurons along with a dynamic component formed of transient/recurring neurons. Moreover, the connectome has a rich club organization since the early stage which selectively strengthens indicating progressively denser connectivity among the integrators due to the previously reported asymmetric synapse addition. This asymmetry also shows up in the preservation of input hubs across development and the progressively more centralized organization of the in-degree k-core. Our work provides a new perspective into the neurodevelopment of the brain that may facilitate our understanding of its functioning.