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Strategies aimed at reducing the negative effects of long-term uncertainty and risk are common in biology, game theory, and finance, even if they entail a cost in terms of mean benefit. Here, we focus on the single mutant's invasion of a finite resident population subject to fluctuating environmental conditions. Thus, the game-theoretical model we analyze integrates environmental and demographic randomness, i.e., the two leading sources of stochasticity and uncertainty. We use simulations and mathematical analysis to study if and when strategies that either increase or reduce payoff variance across environmental states can enhance the mutant fixation probability. Variance aversion implies that the mutant pays insurance in terms of mean payoff to avoid worst-case scenarios. Variance-prone or gambling strategies, on the other hand, entail specialization, allowing the mutant to capitalize on transient favorable conditions, leading to a series of ``boom-and-bust'' cycles. Our analyses elucidate how the rate of change of environmental conditions and the shape of the probability distribution of possible states affect the possible most convenient strategies. We discuss how our results relate to the bet-hedging theory, which aims to reduce fitness variance rather than payoff variance. We also describe the analogies and differences between these similar yet distinct approaches.

Scale invariance profoundly influences the dynamics and structure of complex systems, spanning from critical phenomena to network architecture. Here, we propose a precise definition of scale-invariant networks by leveraging the concept of a constant entropy loss rate across scales in a renormalization-group coarse-graining setting. This framework enables us to differentiate between scale-free and scale-invariant networks, revealing distinct characteristics within each class. Furthermore, we offer a comprehensive inventory of genuinely scale-invariant networks, both natural and artificially constructed, demonstrating, e.g., that the human connectome exhibits notable features of scale invariance. Our findings open new avenues for exploring the scale-invariant structural properties crucial in biological and socio-technological systems., Comment: 6 pages, 3 figures and Supplemental Material

Recent analyses combining advanced theoretical techniques and high-quality data from thousands of simultaneously recorded neurons provide strong support for the hypothesis that neural dynamics operate near the edge of instability across regions in the brain. However, these analyses, as well as related studies, often fail to capture the intricate temporal structure of brain activity as they primarily rely on time-integrated measurements across neurons. In this study, we present a novel framework designed to explore signatures of criticality across diverse frequency bands and construct a much more comprehensive description of brain activity. Additionally, we introduce a method for projecting brain activity onto a basis of spatio-temporal patterns, facilitating time-dependent dimensionality reduction. Applying this framework to a magnetoencephalography dataset, we observe significant differences in both criticality signatures and spatio-temporal activity patterns between healthy subjects and individuals with Parkinson's disease.

Diverse equilibrium systems with heterogeneous interactions lie at the edge of stability. Such marginally stable states are dynamically selected as the most abundant ones or as those with the largest basins of attraction. On the other hand, systems with non-reciprocal (or asymmetric) interactions are inherently out of equilibrium, and exhibit a rich variety of steady states, including fixed points, limit cycles and chaotic trajectories. How are steady states dynamically selected away from equilibrium? We address this question in a simple neural network model, with a tunable level of non-reciprocity. Our study reveals different types of ordered phases and it shows how non-equilibrium steady states are selected in each phase. In the spin-glass region, the system exhibits marginally stable behaviour for reciprocal (or symmetric) interactions and it smoothly transitions to chaotic dynamics, as the non-reciprocity (or asymmetry) in the couplings increases. Such region, on the other hand, shrinks and eventually disappears when couplings become anti-symmetric. Our results are relevant to advance the knowledge of disordered systems beyond the paradigm of reciprocal couplings, and to develop an interface between statistical physics of equilibrium spin-glasses and dynamical systems theory.

Balanced neural networks -- in which excitatory and inhibitory inputs compensate each other on average -- give rise to a dynamical phase dominated by fluctuations called asynchronous state, crucial for brain functioning. However, structural disorder -- which is inherent to random networks -- can hinder such an excitation-inhibition balance. Indeed, structural and synaptic heterogeneities can generate extended regions in phase space akin to critical points, called Griffiths phases, with dynamical features very different from those of asynchronous states. Here, we study a simple neural-network model with tunable levels of heterogeneity able to display these two types of dynamical regimes -- i.e., asynchronous states and Griffiths phases -- putting them together within a single phase diagram. Using this simple model, we are able to emphasize the crucial role played by synaptic plasticity and homeostasis to re-establish balance in intrinsically heterogeneous networks. Overall, we shed light onto how diverse dynamical regimes, each with different functional advantages, can emerge from a given network as a result of self-organizing homeostatic mechanisms.

Laboratory experiments with bacterial colonies, under well-controlled conditions often lead to evolutionary diversification, where at least two ecotypes emerge from an initially monomorphic population. Empirical evidence suggests that such ''evolutionary branching'' occurs stochastically, even under fixed and stable conditions. This stochastic nature is characterized by: (i) occurrence in a significant fraction, but not all, of experimental settings, (ii) emergence at widely varying times, and (iii) variable relative abundances of the resulting subpopulations across experiments. Theoretical approaches to understanding evolutionary branching under these conditions have been previously developed within the (deterministic) framework of ''adaptive dynamics''. Here, we advance the understanding of the stochastic nature of evolutionary outcomes by introducing the concept of ''stochastic trade-offs'' as opposed to ''hard'' ones. The key idea is that the stochasticity of mutations occurs in a high-dimensional trait space and this translates into variability that is constrained to a flexible tradeoff curve. By incorporating this additional source of stochasticity, we are able to account for the observed empirical variability and make predictions regarding the likelihood of evolutionary branching under different conditions. This approach effectively bridges the gap between theoretical predictions and experimental observations, providing insights into when and how evolutionary branching is more likely to occur in laboratory experiments.

Deciphering the underpinnings of the dynamical processes leading to information transmission, processing, and storing in the brain is a crucial challenge in neuroscience. An inspiring but speculative theoretical idea is that such dynamics should operate at the brink of a phase transition, i.e., at the edge between different collective phases, to entail a rich dynamical repertoire and optimize functional capabilities. In recent years, research guided by the advent of high-throughput data and new theoretical developments has contributed to making a quantitative validation of such a hypothesis. Here we review recent advances in this field, stressing our contributions. In particular, we use data from thousands of individually recorded neurons in the mouse brain and tools such as a phenomenological renormalization group analysis, theory of disordered systems, and random matrix theory. These combined approaches provide novel evidence of quasi-universal scaling and near-critical behavior emerging in different brain regions. Moreover, we design artificial neural networks under the reservoir-computing paradigm and show that their internal dynamical states become near critical when we tune the networks for optimal performance. These results not only open new perspectives for understanding the ultimate principles guiding brain function but also towards the development of brain-inspired, neuromorphic computation.

During the last decades macroecology has identified broad-scale patterns of abundances and diversity of microbial communities and put forward some potential explanations for them. However, these advances are not paralleled by a full understanding of the dynamical processes behind them. In particular, abundance fluctuations of different species are found to be correlated, both across time and across communities in metagenomic samples. Reproducing such correlations through appropriate population models remains an open challenge. The present paper tackles this problem and points to sparse species interactions as a necessary mechanism to account for them. Specifically, we discuss several possibilities to include interactions in population models and recognize Lotka-Volterra constants as a successful ansatz. For this, we design a Bayesian inference algorithm to extract sets of interaction constants able to reproduce empirical probability distributions of pairwise correlations for diverse biomes. Importantly, the inferred models still reproduce well-known single-species macroecological patterns concerning abundance fluctuations across both species and communities. Endorsed by the agreement with the empirically observed phenomenology, our analyses provide insights on the properties of the networks of microbial interactions, revealing that sparsity is a crucial feature.

The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully-connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here, we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types of phase transitions, including continuous ones with scaling behavior belonging to known universality classes -- such as directed percolation and tricritical directed percolation -- as well as novel ones. In particular, we show that under some special circumstances, at a so-called Hopf tricritical directed percolation transition, rather unconventional behavior including an anomalous breakdown of scaling emerges. These results broaden our knowledge of the possible types of critical behavior in networks of excitatory and inhibitory units and are of relevance to understanding avalanche dynamics in actual neuronal recordings. From a more general perspective, these results help extend the theory of non-equilibrium phase transitions into quiescent or absorbing states., Comment: 19 pages, 11 figures

We study standard and higher-order birth-death processes on fully connected networks, within the perspective of large-deviation theory (also referred to as Wentzel-Kramers-Brillouin (WKB) method in some contexts). We obtain a general expression for the leading and next-to-leading terms of the stationary probability distribution of the fraction of "active" sites, as a function of parameters and network size $N$. We reproduce several results from the literature and, in particular, we derive all the moments of the stationary distribution for the $q$-susceptible-infected-susceptible ($q-SIS$) model, i.e., a high-order epidemic model requiring of $q$ active ("infected") sites to activate an additional one. We uncover a very rich scenario for the fluctuations of the fraction of active sites, with non-trivial finite-size-scaling properties. In particular, we show that the variance-to-mean ratio diverges at criticality for $[1 \leq q\leq 3]$, with a maximal variability at $q=2$, confirming that complex-contagion processes can exhibit peculiar scaling features including wild variability and that the leading-order in a large-deviation approach does not suffice to describe them: next-to-leading terms are essential to capture the intrinsic singularity at the origin of systems with absorbing states., Comment: 10 pages, 4 figures

We propose and develop a new algorithm for trading wind energy in electricity markets, within an online learning and optimization framework. In particular, we combine a component-wise adaptive variant of the gradient descent algorithm with recent advances in the feature-driven newsvendor model. This results in an online offering approach capable of leveraging data-rich environments, while adapting to the nonstationary characteristics of energy generation and electricity markets, also with a minimal computational burden. The performance of our approach is analyzed based on several numerical experiments, showing both better adaptability to nonstationary uncertain parameters and significant economic gains.

A unique feature of neuromorphic computing is that memory is an implicit part of processing through traces of past information in the system's collective dynamics. The extent of memory about past inputs is commonly quantified by the autocorrelation time of collective dynamics. Based on past experimental evidence, a potential explanation for the underlying autocorrelations are close-to-critical fluctuations. Here, we show for self-organized networks of excitatory and inhibitory leaky integrate-and-fire neurons that autocorrelations can originate from emergent bistability upon reducing external input strength. We identify the bistability as a fluctuation-induced stochastic switching between metastable active and quiescent states in the vicinity of a non-equilibrium phase transition. This bistability occurs for networks with fixed heterogeneous weights as a consequence of homeostatic self-organization during development. Specifically, in our experiments on neuromorphic hardware and in computer simulations, the emergent bistability gives rise to autocorrelation times exceeding 500 ms despite single-neuron timescales of only 20 ms. Our results provide the first verification of biologically compatible autocorrelation times in networks of leaky integrate-and-fire neurons, which here are not generated by close-to-critical fluctuations but by emergent bistability in homeostatically regulated networks. Our results thereby constitute a new, complementary mechanism for emergent autocorrelations in networks of spiking neurons, with implications for biological and artificial networks, and introduces the general paradigm of fluctuation-induced bistability for driven systems with absorbing states.

Depinning of elastic systems advancing on disordered media can usually be described by the quenched Edwards-Wilkinson equation (qEW). However, additional ingredients such as anharmonicity and forces that can not be derived from a potential energy may generate a different scaling behavior at depinning. The most experimentally relevant is the Kardar-Parisi-Zhang (KPZ) term, proportional to the square of the slope at each site, which drives the critical behavior into the so-called quenched KPZ (qKPZ) universality class. We study this universality class both numerically and analytically: by using exact mappings we show that at least for $d=1,2$ this class encompasses not only the qKPZ equation itself, but also anharmonic depinning and a well-known class of cellular automata introduced by Tang and Leschhorn. We develop scaling arguments for all critical exponents, including size and duration of avalanches. The scale is set by the confining potential strength $m^2$. This allows us to estimate numerically these exponents as well as the $m$-dependent effective force correlator $\Delta(w)$, and its correlation length $\rho:= \Delta(0)/|\Delta'(0^+)|$. Finally we present a new algorithm to numerically estimate the effective ($m$-dependent) elasticity $c$, and the effective KPZ non-linearity $\lambda$. This allows us to define a dimensionless universal KPZ amplitude ${\cal A}:=\rho \lambda /c$, which takes the value ${\cal A}=1.10(2)$ in all systems considered in $d=1$. This proves that qKPZ is the effective field theory for all these models. Our work paves the way for a deeper understanding of depinning in the qKPZ class, and in particular, for the construction of a field theory that we describe in a companion paper., Comment: companion paper to arXiv:2207.09037. 19 pages, 17 figures. v2: accepted version

Brain functions require both segregated processing of information in specialized circuits, as well as integration across circuits to perform high-level information processing. One possible way to implement these seemingly opposing demands is by flexibly switching between synchronous and less synchronous states. Understanding how complex synchronization patterns are controlled by the interaction of network architecture and external perturbations is thus a central challenge in neuroscience, but the mechanisms behind such interactions remain elusive. Here, we utilise precision neuroengineering to manipulate cultured neuronal networks and show that a modular architecture facilitates desynchronization upon asynchronous stimulation, making external noise a control parameter of synchrony. Using spiking neuron models, we then demonstrate that external noise can reduce the level of available synaptic resources, which make intermodular interactions more stochastic and thereby facilitates the breakdown of synchrony. Finally, the phenomenology of stochastic intermodular interactions is formulated into a mesoscopic model that incorporates a state-dependent gating mechanism for signal propagation. Taken together, our results demonstrate a network mechanism by which asynchronous inputs tune the inherent dynamical state in structured networks of excitable units., Comment: 23 pages, 5 figures

The branching process is the minimal model for propagation dynamics, avalanches and criticality, broadly used in neuroscience. A simple extension of it, adding inhibitory nodes, induces a much-richer phenomenology, including, an intermediate phase, between quiescence and saturation, that exhibits the key features of "asynchronous states" in cortical networks. Remarkably, in the inhibition-dominated case, it exhibits an extremely-rich phase diagram, that captures a wealth of non-trivial features of spontaneous brain activity, such as collective excitability, hysteresis, tilted avalanche shapes, and partial synchronization, allowing us to rationalize striking empirical findings within a common and parsimonious framework.

The brain is in a state of perpetual reverberant neural activity, even in the absence of specific tasks or stimuli. Shedding light on the origin and functional significance of such a dynamical state is essential to understanding how the brain transmits, processes, and stores information. An inspiring, albeit controversial, conjecture proposes that some statistical characteristics of empirically observed neuronal activity can be understood by assuming that brain networks operate in a dynamical regime near the edge of a phase transition. Moreover, the resulting critical behavior, with its concomitant scale invariance, is assumed to carry crucial functional advantages. Here, we present a data-driven analysis based on simultaneous high-throughput recordings of the activity of thousands of individual neurons in various regions of the mouse brain. To analyze these data, we synergistically combine cutting-edge methods for the study of brain activity (such as a phenomenological renormalization group approach and techniques that infer the general dynamical state of a neural population), while designing complementary tools. This strategy allows us to uncover strong signatures of scale invariance that is "quasi-universal" across brain regions and reveal that all these areas operate, to a greater or lesser extent, near the edge of instability. Furthermore, this framework allows us to distinguish between quasi-universal background activity and non-universal input-related activity. Taken together, this study provides strong evidence that brain networks actually operate in a critical regime which, among other functional advantages, provides them with a scale-invariant substrate of activity covariances that can sustain optimal input representations., Comment: 22 pages, 5 figures

Many of the amazing functional capabilities of the brain are collective properties stemming from the interactions of large sets of individual neurons. In particular, the most salient collective phenomena in brain activity are oscillations, which require the synchronous activation of many neurons. Here, we analyse parsimonious dynamical models of neural synchronisation running on top of synthetic networks that capture essential aspects of the actual brain anatomical connectivity such as a hierarchical-modular and core-periphery structure. These models reveal the emergence of complex collective states with intermediate and flexible levels of synchronisation, halfway in the synchronous-asynchronous spectrum. These states are best described as broad Griffiths-like phases, i.e. an extension of standard critical points that emerge in structurally heterogeneous systems. We analyse different routes (bifurcations) to synchronisation and stress the relevance of 'hybrid-type transitions' to generate rich dynamical patterns. Overall, our results illustrate the complex interplay between structure and dynamics, underlining key aspects leading to rich collective states needed to sustain brain functionality., Comment: 13 pages, 2 figures. Accepted to be published in Philos. Trans. Royal Soc. A

Ecological and evolutionary dynamics have been historically regarded as unfolding at broadly separated timescales. However, these two types of processes are nowadays well documented to much more tightly than traditionally assumed, especially in communities of microorganisms. With this motivation in mind, here we scrutinize recent experimental results showing evidence of rapid evolution of tolerance by lag in bacterial populations that are periodically exposed to antibiotic stress in laboratory conditions. In particular, the distribution of single-cell lag times evolves its average value to approximately fit the antibiotic-exposure time. Moreover, the distribution develops right skewed heavy tails, revealing the presence of individuals with anomalously large lag times. Here, we develop a parsimonious individual-based model mimicking the actual demographic processes of the experimental setup. Individuals are characterized by a single phenotypic trait: their intrinsic lag time, which is transmitted with variation to the progeny. The model (in a version in which the amplitude of phenotypic variations grows with the parent\'s lag time) is able to reproduce quite well the key empirical observations. Furthermore, we develop a general mathematical framework allowing us to describe with good accuracy the properties of the stochastic model by means of a macroscopic equation, which generalizes the Crow Kimura equation in population genetics. From a broader perspective, this work represents a benchmark for the mathematical framework designed to tackle much more general eco-evolutionary problems, thus paving the road to further research avenues.

We consider a two-stage generation scheduling problem comprising a forward dispatch and a real-time re-dispatch. The former must be conducted facing an uncertain net demand that includes non-dispatchable electricity consumption and renewable power generation. The latter copes with the plausible deviations with respect to the forward schedule by making use of balancing power during the actual operation of the system. Standard industry practice deals with the uncertain net demand in the forward stage by replacing it with a good estimate of its conditional expectation (usually referred to as a point forecast), so as to minimize the need for balancing power in real time. However, it is well known that the cost structure of a power system is highly asymmetric and dependent on its operating point, with the result that minimizing the amount of power imbalances is not necessarily aligned with minimizing operating costs. In this paper, we propose a bilevel program to construct, from the available historical data, a prescription of the net demand that does account for the power system's cost asymmetry. Furthermore, to accommodate the strong dependence of this cost on the power system's operating point, we use clustering to tailor the proposed prescription to the foreseen net-demand regime. By way of an illustrative example and a more realistic case study based on the European power system, we show that our approach leads to substantial cost savings compared to the customary way of doing.

Shedding light onto how biological systems represent, process and store information in noisy environments is a key and challenging goal. A stimulating, though controversial, hypothesis poses that operating in dynamical regimes near the edge of a phase transition, i.e. at criticality or the "edge of chaos", can provide information-processing living systems with important operational advantages, creating, e.g., an optimal trade-off between robustness and flexibility. Here, we elaborate on a recent theoretical result, which establishes that the spectrum of covariance matrices of neural networks representing complex inputs in a robust way needs to decay as a power-law of the rank, with an exponent close to unity, a result that has been indeed experimentally verified in neurons of the mouse visual cortex. Aimed at understanding and mimicking these results, we construct an artificial neural network and train it to classify images. Remarkably, we find that the best performance in such a task is obtained when the network operates near the critical point, at which the eigenspectrum of the covariance matrix follows the very same statistics as actual neurons do. Thus, we conclude that operating near criticality can also have -- besides the usually alleged virtues -- the advantage of allowing for flexible, robust and efficient input representations.

It has been recently discovered that the measles virus can wipe out the adaptive immune system, destroying B lymphocytes and reducing the diversity of non-specific B cells of the infected host. In particular, this implies that previously acquired immunization from vaccination or direct exposition to other pathogens could be erased in a phenomenon named "immune amnesia", whose effects can become particularly worrisome given the actual rise of anti-vaccination movements. Here we present the first attempt to incorporate immune amnesia into standard models of epidemic spreading. In particular, we analyze diverse variants of a model that describes the spreading of two concurrent pathogens causing measles and another generic disease: the SIR-IA model. Analytical and computational studies confirm that immune amnesia can indeed have important consequences for epidemic spreading, significantly altering the vaccination coverage required to reach herd-immunity for concurring infectious diseases. More specifically, we uncover the existence of novel propagating and endemic phases which are induced by immune amnesia, that appear both in fully-connected and more structured networks, such as random networks and power-law degree-distributed ones. In particular, the transitions from a quiescent state into these novel phases can become rather abrupt in some cases that we specifically analyze. Furthermore, we discuss the meaning and consequences of our results and their relation with, e.g., immunization strategies, together with the possibility that explosive types of transitions may emerge, making immune-amnesia effects particularly dramatic. This work opens the door to further developments and analyses of immune amnesia effects, contributing, more generally, to the theory of interacting epidemics on complex networks.

Brain networks can be defined and explored through their connectivity. Here, we analyzed the relationship between structural connectivity (SC) across 2,514 regions that cover the entire brain and brainstem, and their dynamic functional connectivity (DFC). To do so, we focused on a combination of two metrics: the first assesses the degree of SC-DFC similarity and the second is the intrinsic variability of the DFC networks over time. Overall, we found that cerebellar networks have a smaller DFC variability than other networks in the brain. Moreover, the internal structure of the cerebellum could be clearly divided in two distinct posterior and anterior parts, the latter also connected to the brainstem. The mechanism to maintain small variability of the DFC in the posterior part of the cerebellum is consistent with another of our findings, namely, that this structure exhibits the highest SC-DFC similarity relative to the other networks studied. By contrast, the anterior part of the cerebellum also exhibits small DFC variability but it has the lowest SC-DFC similarity, suggesting a different mechanism is at play. Because this structure connects to the brainstem, which regulates sleep cycles, cardiac and respiratory functioning, we suggest that such critical functionality drives the low variability in the DFC. Overall, the low variability detected in DFC expands our current knowledge of cerebellar networks, which are extremely rich and complex, participating in a wide range of cognitive functions, from movement control and coordination to executive function or emotional regulation. Moreover, the association between such low variability and structure suggests that differentiated computational principles can be applied in the cerebellum as opposed to other structures, such as the cerebral cortex., Comment: 19 pages, 6 figures

The human cortex is never at rest but in a state of sparse and noisy neural activity that can be detected at broadly diverse resolution scales. It has been conjectured that such a state is best described as a critical dynamical process -- whose nature is still not fully understood -- where scale-free avalanches of activity emerge at the edge of a phase transition. In particular, some works suggest that this is most likely a synchronization transition, separating synchronous from asynchronous phases. Here, by investigating a simplified model of coupled excitable oscillators describing the cortex dynamics at a mesoscopic level, we investigate the possible nature of such a synchronization phase transition. Within our modeling approach, we conclude that -- in order to reproduce all key empirical observations, such as scale-free avalanches and bistability, on which fundamental functional advantages rely -- the transition to collective oscillatory behavior needs to be of an unconventional hybrid type, with mixed features of type-I and type-II excitability, opening the possibility for a particularly rich dynamical repertoire., Comment: 7 pages, 4 figures, and Supplementary Information. Accepted to be published in PRR