Functional advantages of Lévy walks emerging near a critical point

How Lévy Flights Triggered by the Presence of Defectors Affect Evolution of Cooperation in Spatial Games

Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a Lévy flight. A Lévy flight is a kind of random walk but it is composed of many small movements with a few big movements. The role of Lévy flights for cooperation has been studied by Antonioni and Tomassini, who showed that Lévy flights promoted cooperation combined with conditional movements triggered by neighboring defectors. However, the optimal condition for neighboring defectors and how the condition changes with the intensity of Lévy flights are still unclear. Here, we developed an agent-based model in a square lattice where agents perform Lévy flights depending on the fraction of neighboring defectors. We systematically studied the relationships among three factors for cooperation: sensitivity to defectors, the intensity of Lévy flights, and population density. Results of evolutionary simulations showed that moderate sensitivity most promoted cooperation. Then, we found that the shortest movements were best for cooperation when the sensitivity to defectors was high. In contrast, when the sensitivity was low, longer movements were best for cooperation. Thus, Lévy flights, the balance between short and long jumps, promoted cooperation in any sensitivity, which was confirmed by evolutionary simulations. Finally, as the population density became larger, higher sensitivity was more beneficial for cooperation to evolve. Our study highlights that Lévy flights are an optimal searching strategy not only for foraging but also for constructing cooperative relationships with others.

Stochastic and deterministic dynamics in networks with excitable nodes

Networks of excitable systems provide a flexible and tractable model for various phenomena in biology, social sciences, and physics. A large class of such models undergo a continuous phase transition as the excitability of the nodes is increased. However, models of excitability that result in this continuous phase transition are based implicitly on the assumption that the probability that a node gets excited, its transfer function, is linear for small inputs. In this paper, we consider the effect of cooperative excitations, and more generally the case of a nonlinear transfer function, on the collective dynamics of networks of excitable systems. We find that the introduction of any amount of nonlinearity changes qualitatively the dynamical properties of the system, inducing a discontinuous phase transition and hysteresis. We develop a mean-field theory that allows us to understand the features of the dynamics with a one-dimensional map. We also study theoretically and numerically finite-size effects by examining the fate of initial conditions where only one node is excited in large but finite networks. Our results show that nonlinear transfer functions result in a rich effective phase diagram for finite networks, and that one should be careful when interpreting predictions of models that assume noncooperative excitations.

Scale-free behavioral dynamics directly linked with scale-free cortical dynamics

Naturally occurring body movements and collective neural activity both exhibit complex dynamics, often with scale-free, fractal spatiotemporal structure. Scale-free dynamics of both brain and behavior are important because each is associated with functional benefits to the organism. Despite their similarities, scale-free brain activity and scale-free behavior have been studied separately, without a unified explanation. Here, we show that scale-free dynamics of mouse behavior and neurons in the visual cortex are strongly related. Surprisingly, the scale-free neural activity is limited to specific subsets of neurons, and these scale-free subsets exhibit stochastic winner-take-all competition with other neural subsets. This observation is inconsistent with prevailing theories of scale-free dynamics in neural systems, which stem from the criticality hypothesis. We develop a computational model which incorporates known cell-type-specific circuit structure, explaining our findings with a new type of critical dynamics. Our results establish neural underpinnings of scale-free behavior and clear behavioral relevance of scale-free neural activity.

A guide to area-restricted search: a foundational foraging behaviour

Area-restricted search is the capacity to change search effort adaptively in response to resource encounters or expectations, from directional exploration (global, extensive search) to focused exploitation (local, intensive search). This search pattern is used by numerous organisms, from worms and insects to humans, to find various targets, such as food, mates, nests, and other resources. Area-restricted search has been studied for at least 80 years by ecologists, and more recently in the neurological and psychological literature. In general, the conditions promoting this search pattern are: (1) clustered resources; (2) active search (e.g. not a sit-and-wait predator); (3) searcher memory for recent target encounters or expectations; and (4) searcher ignorance about the exact location of targets. Because area-restricted search adapts to resource encounters, the search can be performed at multiple spatial scales. Models and experiments have demonstrated that area-restricted search is superior to alternative search patterns that do not involve a memory of the exact location of the target, such as correlated random walks or Lévy walks/flights. Area-restricted search is triggered by sensory cues whereas concentrated search in the absence of sensory cues is associated with other forms of foraging. Some neural underpinnings of area-restricted search are probably shared across metazoans, suggesting a shared ancestry and a shared solution to a common ecological problem of finding clustered resources. Area-restricted search is also apparent in other domains, such as memory and visual search in humans, which may indicate an exaptation from spatial search to other forms of search. Here, we review these various aspects of area-restricted search, as well as how to identify it, and point to open questions.

Protein-coated nanoparticles exhibit Lévy flights on a suspended lipid bilayer

Lateral diffusion of nano-objects on lipid membranes is a crucial process in cell biology. Recent studies indicate that nanoparticle lateral diffusion is affected by the presence of membrane proteins and deviates from Brownian motion. Gold nanoparticles (Au NPs) stabilized by short thiol ligands were dispersed near a free-standing bilayer formed in a 3D microfluidic chip. Using dark-field microscopy, the position of single NPs at the bilayer surface was tracked over time. Numerical analysis of the NP trajectories shows that NP diffusion on the bilayer surface corresponds to Brownian motion. The addition of bovine serum albumin (BSA) protein to the solution led to the formation of a protein corona on the NP surface. We found that protein-coated NPs show anomalous superdiffusion and that the distribution of their relative displacement obeys Lévy flight statistics. This superdiffusive motion is attributed to a drastic reduction in adhesive energies between the NPs and the bilayer in the presence of the protein corona. This hypothesis was confirmed by numerical simulations mimicking the random walk of a single particle near a weakly adhesive surface. These results may be generalized to other classes of nano-objects that experience adsorption-desorption behaviour with a weakly adhesive surface.

Living on fire: Deactivating fire coral polyps for larval settlement and symbiosis in the fire coral-associated barnacle Wanella milleporae (Thoracicalcarea: Wanellinae)

Symbiosis is increasingly recognized as being an important component in marine systems, and many such relationships are initiated when free-swimming larvae of one partner settle and become sedentary on a host partner. Therefore, several crucial questions emerge such as the larva's mechanism of locating a host, selection of substratum and finally settlement on the surface of its future partner. Here, we investigated these mechanisms by studying how larvae of the fire coral-associated barnacle Wanella milleporae move, settle and establish symbiosis with their host, Millepora tenera. Cyprids of W. milleporae possess a pair of specialized antennules with bell-shaped attachment discs that enable them to explore and settle superficially on the hostile surface of the fire coral. Intriguingly, the stinging polyps of the fire coral remain in their respective pores when the cyprids explore the fire coral surface. Even when cyprids come into contact with the nematocysts on the extended stinging polyps during the exploratory phase, no immobilization effects against the cyprids were observed. The exploratory phase of Wanella cyprids can be divided into a sequence of wide searching (large step length and high walking speed), close searching (small step length and low speed) and inspection behavior, eventually resulting in permanent settlement and metamorphosis. After settlement, xenogeneic interactions occur between the fire coral and the newly metamorphosed juvenile barnacle. This involved tissue necrosis and regeneration in the fire coral host, leading to a callus ring structure around the juvenile barnacle, enhancing survival rate after settlement. The complex exploratory and settlement patterns and interactions documented here represent a breakthrough in coral reef symbiosis studies to show how invertebrates start symbiosis with fire corals.

Mechanisms of Variability Underlying Odor-Guided Locomotion

Changes in locomotion mediated by odors (odor-guided locomotion) are an important mechanism by which animals discover resources important to their survival. Odor-guided locomotion, like most other behaviors, is highly variable. Variability in behavior can arise at many nodes along the circuit that performs sensorimotor transformation. We review these sources of variability in the context of the Drosophila olfactory system. While these sources of variability are important, using a model for locomotion, we show that another important contributor to behavioral variability is the stochastic nature of decision-making during locomotion as well as the persistence of these decisions: Flies choose the speed and curvature stochastically from a distribution and locomote with the same speed and curvature for extended periods. This stochasticity in locomotion will result in variability in behavior even if there is no noise in sensorimotor transformation. Overall, the noise in sensorimotor transformation is amplified by mechanisms of locomotion making odor-guided locomotion in flies highly variable.

Infinite density and relaxation for Lévy walks in an external potential: Hermite polynomial approach

Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(τ)=v_{0}/sqrt[τ]. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.

Intrinsic bursts facilitate learning of Lévy flight movements in recurrent neural network models

Isolated spikes and bursts of spikes are thought to provide the two major modes of information coding by neurons. Bursts are known to be crucial for fundamental processes between neuron pairs, such as neuronal communications and synaptic plasticity. Neuronal bursting also has implications in neurodegenerative diseases and mental disorders. Despite these findings on the roles of bursts, whether and how bursts have an advantage over isolated spikes in the network-level computation remains elusive. Here, we demonstrate in a computational model that not isolated spikes, but intrinsic bursts can greatly facilitate learning of Lévy flight random walk trajectories by synchronizing burst onsets across a neural population. Lévy flight is a hallmark of optimal search strategies and appears in cognitive behaviors such as saccadic eye movements and memory retrieval. Our results suggest that bursting is crucial for sequence learning by recurrent neural networks when sequences comprise long-tailed distributed discrete jumps.

Imaging and characterization of transitions in biofilm morphology via anomalous diffusion following environmental perturbation

Microorganisms form macroscopic structures for the purpose of environmental adaptation. Sudden environmental perturbations induce dynamics that cause bacterial biofilm morphology to transit to another equilibrium state, thought to be related to anomalous diffusion processes. Here, detecting the super-diffusion characteristics would offer a long-sought goal for a rapid detection method of biofilm phenotypes based on their dynamics, such as growth or dispersal. In this paper, phase-sensitive Doppler optical coherence tomography (OCT) and dynamic light scattering (DLS) are combined to demonstrate wide field-of-view and label-free internal dynamic imaging of biofilms. The probability density functions (PDFs) of phase displacement of the backscattered light and the dynamic characteristics of the PDFs are estimated by a simplified mixed Cauchy and Gaussian model. This model can quantify the super-diffusion state and estimate the dynamic characteristics and macroscopic responses in biofilms that may further describe dispersion and growth in biofilm models.

Synchronized attachment and the Darwinian evolution of coronaviruses CoV-1 and CoV-2

CoV2019 has evolved to be much more dangerous than CoV2003. Experiments suggest that structural rearrangements dramatically enhance CoV2019 activity. We identify a new first stage of infection that precedes structural rearrangements by using biomolecular evolutionary theory to identify sequence differences enhancing viral attachment rates. We find a small cluster of mutations which show that CoV-2 has a new feature that promotes much stronger viral attachment and enhances contagiousness. The extremely dangerous dynamics of human coronavirus infection is a dramatic example of evolutionary approach of self-organized networks to criticality. It may favor a very successful vaccine. The identified mutations can be used to test the present theory experimentally.

Weierstrassian Lévy walks are a by-product of crawling

Weierstrassian Lévy walks are one of the simplest random walks which do not satisfy the central limit theorem and have come to epitomize scale invariance even though they were initially regarded as being a mathematical abstraction. Here, I show how these Lévy walks can be generated intrinsically as a by-product of crawling, a common but ancient form of locomotion. This may explain why Weierstrassian Lévy walks provide accurate representations of the movement patterns of a diverse group of molluscs-certain mussels, mud snails and limpets. I show that such movements are not specific to molluscs as they are also evident in Drosophila larvae. The findings add to the growing realization that there are many idiosyncratic, seemingly accidental pathways to Lévy walking. And that the occurrence of Lévy walks need not be attributed to the execution of an advantageous searching strategy.

Lévy walk dynamics explain gamma burst patterns in primate cerebral cortex

Lévy walks describe patterns of intermittent motion with variable step sizes. In complex biological systems, Lévy walks (non-Brownian, superdiffusive random walks) are associated with behaviors such as search patterns of animals foraging for food. Here we show that Lévy walks also describe patterns of oscillatory activity in primate cerebral cortex. We used a combination of empirical observation and modeling to investigate high-frequency (gamma band) local field potential activity in visual motion-processing cortical area MT of marmoset monkeys. We found that gamma activity is organized as localized burst patterns that propagate across the cortical surface with Lévy walk dynamics. Lévy walks are fundamentally different from either global synchronization, or regular propagating waves, because they include large steps that enable activity patterns to move rapidly over cortical modules. The presence of Lévy walk dynamics therefore represents a previously undiscovered mode of brain activity, and implies a novel way for the cortex to compute. We apply a biophysically realistic circuit model to explain that the Lévy walk dynamics arise from critical-state transitions between asynchronous and localized propagating wave states, and that these dynamics yield optimal spatial sampling of the cortical sheet. We hypothesise that Lévy walk dynamics could help the cortex to efficiently process variable inputs, and to find links in patterns of activity among sparsely spiking populations of neurons.

Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments

The frequency of occurrence of step length in the migratory behaviour of various organisms, including humans, is characterized by the power law distribution. This pattern of behaviour is known as the Lévy walk, and the reason for this phenomenon has been investigated extensively. Especially in humans, one possibility might be that this pattern reflects the change in self-confidence in one’s chosen behaviour. We used simulations to demonstrate that active assumptions cause changes in the confidence level in one’s choice under a situation of lack of information. More specifically, we presented an algorithm that introduced the effects of learning and forgetting into Bayesian inference, and simulated an imitation game in which two decision-making agents incorporating the algorithm estimated each other’s internal models. For forgetting without learning, each agents’ confidence levels in their own estimation remained low owing to a lack of information about the counterpart, and the agents changed their hypotheses about the opponent frequently, and the frequency distribution of the duration of the hypotheses followed an exponential distribution for a wide range of forgetting rates. Conversely, when learning was introduced, high confidence levels occasionally occurred even at high forgetting rates, and exponential distributions universally turned into power law distribution.

Lévy noise-driven escape from arctangent potential wells

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Lévy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Lévy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.