Predicting the SARS-CoV-2 effective reproduction number using bulk contact data from mobile phones.

Over the last months, cases of SARS-CoV-2 surged repeatedly in many countries but could often be controlled with nonpharmaceutical interventions including social distancing. We analyzed deidentified Global Positioning System (GPS) tracking data from 1.15 to 1.4 million cell phones in Germany per day between March and November 2020 to identify encounters between individuals and statistically evaluate contact behavior. Using graph sampling theory, we estimated the contact index (CX), a metric for number and heterogeneity of contacts. We found that CX, and not the total number of contacts, is an accurate predictor for the effective reproduction number R derived from case numbers. A high correlation between CX and R recorded more than 2 wk later allows assessment of social behavior well before changes in case numbers become detectable. By construction, the CX quantifies the role of superspreading and permits assigning risks to specific contact behavior. We provide a critical CX value beyond which R is expected to rise above 1 and propose to use that value to leverage the social-distancing interventions for the coming months.

Circuit-Specific Dendritic Development in the Piriform Cortex.

Dendritic geometry is largely determined during postnatal development and has a substantial impact on neural function. In sensory processing, postnatal development of the dendritic tree is affected by two dominant circuit motifs, ascending sensory feedforward inputs and descending and local recurrent connections. In the three-layered anterior piriform cortex (aPCx), neurons in the sublayers 2a and 2b display vertical segregation of these two circuit motifs. Here, we combined electrophysiology, detailed morphometry, and Ca2+ imaging in acute mouse brain slices and modeling to study circuit-specific aspects of dendritic development. We observed that determination of branching complexity, dendritic length increases, and pruning occurred in distinct developmental phases. Layer 2a and layer 2b neurons displayed developmental phase-specific differences between their apical and basal dendritic trees related to differences in circuit incorporation. We further identified functional candidate mechanisms for circuit-specific differences in postnatal dendritic growth in sublayers 2a and 2b at the mesoscale and microscale levels. Already in the first postnatal week, functional connectivity of layer 2a and layer 2b neurons during early spontaneous network activity scales with differences in basal dendritic growth. During the early critical period of sensory plasticity in the piriform cortex, our data are consistent with a model that proposes a role for dendritic NMDA-spikes in selecting branches for survival during developmental pruning in apical dendrites. The different stages of the morphologic and functional developmental pattern differences between layer 2a and layer 2b neurons demonstrate the complex interplay between dendritic development and circuit specificity.

Epidemics with mutating infectivity on small-world networks.

Epidemics and evolution of many pathogens occur on similar timescales so that their dynamics are often entangled. Here, in a first step to study this problem theoretically, we analyze mutating pathogens spreading on simple SIR networks with grid-like connectivity. We have in mind the spatial aspect of epidemics, which often advance on transport links between hosts or groups of hosts such as cities or countries. We focus on the case of mutations that enhance an agent's infection rate. We uncover that the small-world property, i.e., the presence of long-range connections, makes the network very vulnerable, supporting frequent supercritical mutations and bringing the network from disease extinction to full blown epidemic. For very large numbers of long-range links, however, the effect reverses and we find a reduced chance for large outbreaks. We study two cases, one with discrete number of mutational steps and one with a continuous genetic variable, and we analyze various scaling regimes. For the continuous case we derive a Fokker-Planck-like equation for the probability density and solve it for small numbers of shortcuts using the WKB approximation. Our analysis supports the claims that a potentiating mutation in the transmissibility might occur during an epidemic wave and not necessarily before its initiation.

Gap junctions set the speed and nucleation rate of stage I retinal waves.

Spontaneous waves in the developing retina are essential in the formation of the retinotopic mapping in the visual system. From experiments in rabbits, it is known that the earliest type of retinal waves (stage I) is nucleated spontaneously, propagates at a speed of 451±91 µm/sec and relies on gap junction coupling between ganglion cells. Because gap junctions (electrical synapses) have short integration times, it has been argued that they cannot set the low speed of stage I retinal waves. Here, we present a theoretical study of a two-dimensional neural network of the ganglion cell layer with gap junction coupling and intrinsic noise. We demonstrate that this model can explain observed nucleation rates as well as the comparatively slow propagation speed of the waves. From the interaction between two coupled neurons, we estimate the wave speed in the model network. Furthermore, using simulations of small networks of neurons (N≤260), we estimate the nucleation rate in the form of an Arrhenius escape rate. These results allow for informed simulations of a realistically sized network, yielding values of the gap junction coupling and the intrinsic noise level that are in a physiologically plausible range.

New Insights on Temporal Lobe Epilepsy Based on Plasticity-Related Network Changes and High-Order Statistics.

Epilepsy is a disorder of the brain characterized by the predisposition to generate recurrent unprovoked seizures, which involves reshaping of neuronal circuitries based on intense neuronal activity. In this review, we first detailed the regulation of plasticity-associated genes, such as ARC, GAP-43, PSD-95, synapsin, and synaptophysin. Indeed, reshaping of neuronal connectivity after the primary, acute epileptogenesis event increases the excitability of the temporal lobe. Herein, we also discussed the heterogeneity of neuronal populations regarding the number of synaptic connections, which in the theoretical field is commonly referred as degree. Employing integrate-and-fire neuronal model, we determined that in addition to increased synaptic strength, degree correlations might play essential and unsuspected roles in the control of network activity. Indeed, assortativity, which can be described as a condition where high-degree correlations are observed, increases the excitability of neural networks. In this review, we summarized recent topics in the field, and data were discussed according to newly developed or unusual tools, as provided by mathematical graph analysis and high-order statistics. With this, we were able to present new foundations for the pathological activity observed in temporal lobe epilepsy.

Determining the Roles of Inositol Trisphosphate Receptors in Neurodegeneration: Interdisciplinary Perspectives on a Complex Topic.

It is well known that calcium (Ca2+) is involved in the triggering of neuronal death. Ca2+ cytosolic levels are regulated by Ca2+ release from internal stores located in organelles, such as the endoplasmic reticulum. Indeed, Ca2+ transit from distinct cell compartments follows complex dynamics that are mediated by specific receptors, notably inositol trisphosphate receptors (IP3Rs). Ca2+ release by IP3Rs plays essential roles in several neurological disorders; however, details of these processes are poorly understood. Moreover, recent studies have shown that subcellular location, molecular identity, and density of IP3Rs profoundly affect Ca2+ transit in neurons. Therefore, regulation of IP3R gene products in specific cellular vicinities seems to be crucial in a wide range of cellular processes from neuroprotection to neurodegeneration. In this regard, microRNAs seem to govern not only IP3Rs translation levels but also subcellular accumulation. Combining new data from molecular cell biology with mathematical modelling, we were able to summarize the state of the art on this topic. In addition to presenting how Ca2+ dynamics mediated by IP3R activation follow a stochastic regimen, we integrated a theoretical approach in an easy-to-apply, cell biology-coherent fashion. Following the presented premises and in contrast to previously tested hypotheses, Ca2+ released by IP3Rs may play different roles in specific neurological diseases, including Alzheimer's disease and Parkinson's disease.

Accurate Langevin approaches to simulate Markovian channel dynamics.

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

Degree Correlations Optimize Neuronal Network Sensitivity to Sub-Threshold Stimuli.

Information processing in the brain crucially depends on the topology of the neuronal connections. We investigate how the topology influences the response of a population of leaky integrate-and-fire neurons to a stimulus. We devise a method to calculate firing rates from a self-consistent system of equations taking into account the degree distribution and degree correlations in the network. We show that assortative degree correlations strongly improve the sensitivity for weak stimuli and propose that such networks possess an advantage in signal processing. We moreover find that there exists an optimum in assortativity at an intermediate level leading to a maximum in input/output mutual information.

Ryanodine Receptor Activation Induces Long-Term Plasticity of Spine Calcium Dynamics.

A key feature of signalling in dendritic spines is the synapse-specific transduction of short electrical signals into biochemical responses. Ca2+ is a major upstream effector in this transduction cascade, serving both as a depolarising electrical charge carrier at the membrane and an intracellular second messenger. Upon action potential firing, the majority of spines are subject to global back-propagating action potential (bAP) Ca2+ transients. These transients translate neuronal suprathreshold activation into intracellular biochemical events. Using a combination of electrophysiology, two-photon Ca2+ imaging, and modelling, we demonstrate that bAPs are electrochemically coupled to Ca2+ release from intracellular stores via ryanodine receptors (RyRs). We describe a new function mediated by spine RyRs: the activity-dependent long-term enhancement of the bAP-Ca2+ transient. Spines regulate bAP Ca2+ influx independent of each other, as bAP-Ca2+ transient enhancement is compartmentalized and independent of the dendritic Ca2+ transient. Furthermore, this functional state change depends exclusively on bAPs travelling antidromically into dendrites and spines. Induction, but not expression, of bAP-Ca2+ transient enhancement is a spine-specific function of the RyR. We demonstrate that RyRs can form specific Ca2+ signalling nanodomains within single spines. Functionally, RyR mediated Ca2+ release in these nanodomains induces a new form of Ca2+ transient plasticity that constitutes a spine specific storage mechanism of neuronal suprathreshold activity patterns.

Modulation of elementary calcium release mediates a transition from puffs to waves in an IP3R cluster model.

The oscillating concentration of intracellular calcium is one of the most important examples for collective dynamics in cell biology. Localized releases of calcium through clusters of inositol 1,4,5-trisphosphate receptor channels constitute elementary signals called calcium puffs. Coupling by diffusing calcium leads to global releases and waves, but the exact mechanism of inter-cluster coupling and triggering of waves is unknown. To elucidate the relation of puffs and waves, we here model a cluster of IP3R channels using a gating scheme with variable non-equilibrium IP3 binding. Hybrid stochastic and deterministic simulations show that puffs are not stereotyped events of constant duration but are sensitive to stimulation strength and residual calcium. For increasing IP3 concentration, the release events become modulated at a timescale of minutes, with repetitive wave-like releases interspersed with several puffs. This modulation is consistent with experimental observations we present, including refractoriness and increase of puff frequency during the inter-wave interval. Our results suggest that waves are established by a random but time-modulated appearance of sustained release events, which have a high potential to trigger and synchronize activity throughout the cell.

Functional regulation of neuronal nitric oxide synthase expression and activity in the rat retina.

In the nervous system within physiological conditions, nitric oxide (NO) production depends on the activity of nitric oxide synthases (NOSs), and particularly on the expression of the neuronal isoform (nNOS). In the sensory systems, the role of NO is poorly understood. In this study, we identified nNOS-positive cells in the inner nuclear layer (INL) of the rat retina, with distinct characteristics such as somata size, immunolabeling level and location. Employing mathematical cluster analysis, we determined that nNOS amacrine cells are formed by two distinct populations. We next investigated the molecular identity of these cells, which did not show colocalization with calbindin (CB), choline acetyltransferase (ChAT), parvalbumin (PV) or protein kinase C (PKC), and only partial colocalization with calretinin (CR), revealing the accumulation of nNOS in specific amacrine cell populations. To access the functional, circuitry-related roles of these cells, we performed experiments after adaptation to different ambient light conditions. After 24h of dark-adaptation, we detected a subtle, yet statistically significant decrease in nNOS transcript levels, which returned to steady-state levels after 24h of normal light-dark cycle, revealing that nNOS expression is governed by ambient light conditions. Employing electron paramagnetic resonance (EPR), we demonstrated that dark-adaptation decreases NO production in the retina. Furthermore, nNOS accumulation changed in the dark-adapted retinas, with a general reduction in the inner plexiform layer. Finally, computational analysis based on clustering techniques revealed that dark-adaptation differently affected both types of nNOS-positive amacrine cells. Taken together, our data disclosed functional regulation of nNOS expression and activity, disclosing new circuitry-related roles of nNOS-positive cells. More importantly, this study indicated unsuspected roles for NO in the sensory systems, particularly related to adaptation to ambient demands.

Frequency and relative prevalence of calcium blips and puffs in a model of small IP3R clusters.

In this work, we model the local calcium release from clusters with a few inositol 1,4,5-trisphosphate receptor (IP3R) channels, focusing on the stochastic process in which an open channel either triggers other channels to open (as a puff) or fails to cause any channel to open (as a blip). We show that there are linear relations for the interevent interval (including blips and puffs) and the first event latency against the inverse cluster size. However, nonlinearity is found for the interpuff interval and the first puff latency against the inverse cluster size. Furthermore, the simulations indicate that the blip fraction among all release events and the blip frequency are increasing with larger basal [Ca(2+)], with blips in turn giving a growing contribution to basal [Ca(2+)]. This result suggests that blips are not just lapses to trigger puffs, but they may also possess a biological function to contribute to the initiation of calcium waves by a preceding increase of basal [Ca(2+)] in cells that have small IP3R clusters.

Diffusive spatio-temporal noise in a first-passage time model for intracellular calcium release.

The intracellular release of calcium from the endoplasmic reticulum is controlled by ion channels. The resulting calcium signals exhibit a rich spatio-temporal signature, which originates at least partly from microscopic fluctuations. While stochasticity in the gating transition of ion channels has been incorporated into many models, the distribution of calcium is usually described by deterministic reaction-diffusion equations. Here we test the validity of the latter modeling approach by using two different models to calculate the frequency of localized calcium signals (calcium puffs) from clustered IP3 receptor channels. The complexity of the full calcium system is here limited to the basic opening mechanism of the ion channels and, in the mathematical reduction simplifies to the calculation of a first passage time. Two models are then studied: (i) a hybrid model, where channel gating is treated stochastically, while calcium concentration is deterministic and (ii) a fully stochastic model with noisy channel gating and Brownian calcium ion motion. The second model utilises the recently developed two-regime method [M. B. Flegg, S. J. Chapman, and R. Erban, "The two-regime method for optimizing stochastic reaction-diffusion simulations," J. R. Soc., Interface 9, 859-868 (2012)] in order to simulate a large domain with precision required only near the Ca(2+) absorbing channels. The expected time for a first channel opening that results in a calcium puff event is calculated. It is found that for a large diffusion constant, predictions of the interpuff time are significantly overestimated using the model (i) with a deterministic non-spatial calcium variable. It is thus demonstrated that the presence of diffusive noise in local concentrations of intracellular Ca(2+) ions can substantially influence the occurrence of calcium signals. The presented approach and results may also be relevant for other cell-physiological first-passage time problems with small ligand concentration and high cooperativity.

Channel-based Langevin approach for the stochastic Hodgkin-Huxley neuron.

Stochasticity in ion channel gating is the major source of intrinsic neuronal noise, which can induce many important effects in neuronal dynamics. Several numerical implementations of the Langevin approach have been proposed to approximate the Markovian dynamics of the Hodgkin-Huxley neuronal model. In this work an improved channel-based Langevin approach is proposed by introducing a truncation procedure to limit the state fractions in the range of [0, 1]. The truncated fractions are put back into the state fractions in the next time step for channel noise calculation. Our simulations show that the bounded Langevin approaches combined with the restored process give better approximations to the statistics of action potentials with the Markovian method. As a result, in our approach the channel state fractions are disturbed by two terms of noise: an uncorrelated Gaussian noise and a time-correlated noise obtained from the truncated fractions. We suggest that the restoration of truncated fractions is a critical process for a bounded Langevin method.

Termination of Ca²+ release for clustered IP3R channels.

In many cell types, release of calcium ions is controlled by inositol 1,4,5-trisphosphate (IP3) receptor channels. Elevations in Ca²âº concentration after intracellular release through IP3 receptors (IP3R) can either propagate in the form of waves spreading through the entire cell or produce spatially localized puffs. The appearance of waves and puffs is thought to implicate random initial openings of one or a few channels and subsequent activation of neighboring channels because of an "autocatalytic" feedback. It is much less clear, however, what determines the further time course of release, particularly since the lifetime is very different for waves (several seconds) and puffs (around 100 ms). Here we study the lifetime of Ca²âº signals and their dependence on residual Ca²âº microdomains. Our general idea is that Ca²âº microdomains are dynamical and mediate the effect of other physiological processes. Specifically, we focus on the mechanism by which Ca²âº binding proteins (buffers) alter the lifetime of Ca²âº signals. We use stochastic simulations of channel gating coupled to a coarse-grained description for the Ca²âº concentration. To describe the Ca²âº concentration in a phenomenological way, we here introduce a differential equation, which reflects the buffer characteristics by a few effective parameters. This non-stationary model for microdomains gives deep insight into the dynamical differences between puffs and waves. It provides a novel explanation for the different lifetimes of puffs and waves and suggests that puffs are terminated by Ca²âº inhibition while IP3 unbinding is responsible for termination of waves. Thus our analysis hints at an additional role of IP3 and shows how cells can make use of the full complexity in IP3R gating behavior to achieve different signals.

Traveling echo waves in an array of excitable elements with time-delayed coupling.

We study time-delayed coupling in one-dimensional arrays of discrete excitable FitzHugh-Nagumo elements. For small time delays in an excitatory coupling, noise-triggered spikes undergo a slight prolongation. For larger delay times, the coupling to adjacent elements leads to a splitting of spikes and an "echo" effect that produces dampened series of spikes. For very large delay times, a transition to a stationary regime appears, as the spikes are sustained by mutual excitations, resulting in antiphase synchronization of all elements. We characterize several features exhibited by the system, such as speed and duration of pulses as well as bifurcations leading to the stationary regime.

Mechanogenetic coupling of Hydra symmetry breaking and driven Turing instability model.

The freshwater polyp Hydra can regenerate from tissue fragments or random cell aggregates. We show that the axis-defining step ("symmetry breaking") of regeneration requires mechanical inflation-collapse oscillations of the initial cell ball. We present experimental evidence that axis definition is retarded if these oscillations are slowed down mechanically. When biochemical signaling related to axis formation is perturbed, the oscillation phase is extended and axis formation is retarded as well. We suggest that mechanical oscillations play a triggering role in axis definition. We extend earlier reaction-diffusion models for Hydra regrowth by coupling morphogen transport to mechanical stress caused by the oscillations. The modified reaction-diffusion model reproduces well two important experimental observations: 1), the existence of an optimum size for regeneration, and 2), the dependence of the symmetry breaking time on the properties of the mechanical oscillations.

Phenomenological model of weakly damped Faraday waves and the associated mean flow.

A phenomenological model of parametric surface waves (Faraday waves) is introduced in the limit of small viscous dissipation that accounts for the coupling between surface motion and slowly varying streaming and large-scale flows (mean flow). The primary bifurcation of the model is to a set of standing waves (stripes, given the functional form of the model nonlinearities chosen here). Our results for the secondary instabilities of the primary wave show that the mean flow leads to a weak destabilization of the base state against Eckhaus and transverse amplitude modulation instabilities, and introduces a longitudinal oscillatory instability which is absent without the coupling. We compare our results with recent one-dimensional amplitude equations for this system systematically derived from the governing hydrodynamic equations.