Percolation in living neural networks

Continuum percolation of two-dimensional adaptive dynamics systems

The percolation phase transition of a continuum adaptive neuron system with homeostasis is investigated. In order to maintain their average activity at a particular level, each neuron (represented by a disk) varies its connection radius until the sum of overlapping areas with neighboring neurons (representing the overall connection strength in the network) has reached a fixed target area for each neuron. Tuning the two key parameters in the model, i.e., the density defined as the number of neurons (disks) per unit area and the sum of the overlapping area of each disk with its adjacent disks, can drive the system into the critical percolating state. These two parameters are inversely proportional to each other at the critical state, and the critical filling factors are fixed about 0.7157, which is much less than the case of the continuum percolation with uniform disks. It is also confirmed that the critical exponents in this model are the same as the two-dimensional standard lattice percolation. Although the critical state is relatively more sensitive and exhibits long-range spatial correlation, local fluctuations do not propagate in a long-range manner through the system by the adaptive dynamics, which renders the system overall robust against perturbations.

Influence of microgravity on spontaneous calcium activity of primary hippocampal neurons grown in microfluidic chips

The influence of variations of gravity, either hypergravity or microgravity, on the brain of astronauts is a major concern for long journeys in space, to the Moon or to Mars, or simply long-duration missions on the ISS (International Space Station). Monitoring brain activity, before and after ISS missions already demonstrated important and long term effects on the brains of astronauts. In this study, we focus on the influence of gravity variations at the cellular level on primary hippocampal neurons. A dedicated setup has been designed and built to perform live calcium imaging during parabolic flights. During a CNES (Centre National d'Etudes Spatiales) parabolic flight campaign, we were able to observe and monitor the calcium activity of 2D networks of neurons inside microfluidic devices during gravity changes over different parabolas. Our preliminary results clearly indicate a modification of the calcium activity associated to variations of gravity.

Strong connectivity in real directed networks

In many real, directed networks, the strongly connected component of nodes which are mutually reachable is very small. This does not fit with current theory, based on random graphs, according to which strong connectivity depends on mean degree and degree-degree correlations. And it has important implications for other properties of real networks and the dynamical behavior of many complex systems. We find that strong connectivity depends crucially on the extent to which the network has an overall direction or hierarchical ordering-a property measured by trophic coherence. Using percolation theory, we find the critical point separating weakly and strongly connected regimes and confirm our results on many real-world networks, including ecological, neural, trade, and social networks. We show that the connectivity structure can be disrupted with minimal effort by a targeted attack on edges which run counter to the overall direction. This means that many dynamical processes on networks can depend significantly on a small fraction of edges.

Neuronal Cultures: Exploring Biophysics, Complex Systems, and Medicine in a Dish

Neuronal cultures are one of the most important experimental models in modern interdisciplinary neuroscience, allowing to investigate in a control environment the emergence of complex behavior from an ensemble of interconnected neurons. Here, I review the research that we have conducted at the neurophysics laboratory at the University of Barcelona over the last 15 years, describing first the neuronal cultures that we prepare and the associated tools to acquire and analyze data, to next delve into the different research projects in which we actively participated to progress in the understanding of open questions, extend neuroscience research on new paradigms, and advance the treatment of neurological disorders. I finish the review by discussing the drawbacks and limitations of neuronal cultures, particularly in the context of brain-like models and biomedicine.

Dynamic and Functional Alterations of Neuronal Networks In Vitro upon Physical Damage: A Proof of Concept

There is a growing technological interest in combining biological neuronal networks with electronic ones, specifically for biological computation, human-machine interfacing and robotic implants. A major challenge for the development of these technologies is the resilience of the biological networks to physical damage, for instance, when used in harsh environments. To tackle this question, here, we investigated the dynamic and functional alterations of rodent cortical networks grown in vitro that were physically damaged, either by sequentially removing groups of neurons that were central for information flow or by applying an incision that cut the network in half. In both cases, we observed a remarkable capacity of the neuronal cultures to cope with damage, maintaining their activity and even reestablishing lost communication pathways. We also observed-particularly for the cultures cut in half-that a reservoir of healthy neurons surrounding the damaged region could boost resilience by providing stimulation and a communication bridge across disconnected areas. Our results show the remarkable capacity of neuronal cultures to sustain and recover from damage, and may be inspirational for the development of future hybrid biological-electronic systems.

Flexibility of in vitro cortical circuits influences resilience from microtrauma

Background

Small clusters comprising hundreds to thousands of neurons are an important level of brain architecture that correlates single neuronal properties to fulfill brain function, but the specific mechanisms through which this scaling occurs are not well understood. In this study, we developed an in vitro experimental platform of small neuronal circuits (islands) to probe the importance of structural properties for their development, physiology, and response to microtrauma.

Methods

Primary cortical neurons were plated on a substrate patterned to promote attachment in clusters of hundreds of cells (islands), transduced with GCaMP6f, allowed to mature until 10-13 days in vitro (DIV), and monitored with Ca²⁺ as a non-invasive proxy for electrical activity. We adjusted two structural factors-island size and cellular density-to evaluate their role in guiding spontaneous activity and network formation in neuronal islands.

Results

We found cellular density, but not island size, regulates of circuit activity and network function in this system. Low cellular density islands can achieve many states of activity, while high cellular density biases islands towards a limited regime characterized by low rates of activity and high synchronization, a property we summarized as "flexibility." The injury severity required for an island to lose activity in 50% of its population was significantly higher in low-density, high flexibility islands.

Conclusion

Together, these studies demonstrate flexible living cortical circuits are more resilient to microtrauma, providing the first evidence that initial circuit state may be a key factor to consider when evaluating the consequences of trauma to the cortex.

Theoretical foundations of studying criticality in the brain

Criticality is hypothesized as a physical mechanism underlying efficient transitions between cortical states and remarkable information-processing capacities in the brain. While considerable evidence generally supports this hypothesis, nonnegligible controversies persist regarding the ubiquity of criticality in neural dynamics and its role in information processing. Validity issues frequently arise during identifying potential brain criticality from empirical data. Moreover, the functional benefits implied by brain criticality are frequently misconceived or unduly generalized. These problems stem from the nontriviality and immaturity of the physical theories that analytically derive brain criticality and the statistic techniques that estimate brain criticality from empirical data. To help solve these problems, we present a systematic review and reformulate the foundations of studying brain criticality, that is, ordinary criticality (OC), quasi-criticality (qC), self-organized criticality (SOC), and self-organized quasi-criticality (SOqC), using the terminology of neuroscience. We offer accessible explanations of the physical theories and statistical techniques of brain criticality, providing step-by-step derivations to characterize neural dynamics as a physical system with avalanches. We summarize error-prone details and existing limitations in brain criticality analysis and suggest possible solutions. Moreover, we present a forward-looking perspective on how optimizing the foundations of studying brain criticality can deepen our understanding of various neuroscience questions.

Percolation may explain efficiency, robustness, and economy of the brain

The brain consists of billions of neurons connected by ultra-dense synapses, showing remarkable efficiency, robust flexibility, and economy in information processing. It is generally believed that these advantageous properties are rooted in brain connectivity; however, direct evidence remains absent owing to technical limitations or theoretical vacancy. This research explores the origins of these properties in the largest yet brain connectome of the fruit fly. We reveal that functional connectivity formation in the brain can be explained by a percolation process controlled by synaptic excitation-inhibition (E/I) balance. By increasing the E/I balance gradually, we discover the emergence of these properties as byproducts of percolation transition when the E/I balance arrives at 3:7. As the E/I balance keeps increase, an optimal E/I balance 1:1 is unveiled to ensure these three properties simultaneously, consistent with previous in vitro experimental predictions. Once the E/I balance reaches over 3:2, an intrinsic limitation of these properties determined by static (anatomical) brain connectivity can be observed. Our work demonstrates that percolation, a universal characterization of critical phenomena and phase transitions, may serve as a window toward understanding the emergence of various brain properties.

Mechanistic insights into ultrasonic neurostimulation of disconnected neurons using single short pulses

Ultrasonic neurostimulation is a potentially potent noninvasive therapy, whose mechanism has yet to be elucidated. We designed a system capable of applying ultrasound with minimal reflections to neuronal cultures. Synaptic transmission was pharmacologically controlled, eliminating network effects, enabling examination of single-cell processes. Short single pulses of low-intensity ultrasound were applied, and time-locked responses were examined using calcium imaging. Low-pressure (0.35MPa) ultrasound directly stimulated ∼20% of pharmacologically disconnected neurons, regardless of membrane poration. Stimulation was resistant to the blockade of several purinergic receptor and mechanosensitive ion channel types. Stimulation was blocked, however, by suppression of action potentials. Surprisingly, even extremely short (4μs) pulses were effective, stimulating ∼8% of the neurons. Lower-pressure pulses (0.35MPa) were less effective than higher-pressure ones (0.65MPa). Attrition effects dominated, with no indication of compromised viability. Our results detract from theories implicating cavitation, heating, non-transient membrane pores >1.5nm, pre-synaptic release, or gradual effects. They implicate a post-synaptic mechanism upstream of the action potential, and narrow down the list of possible targets involved.

Percolation in networks with local homeostatic plasticity

Percolation is a process that impairs network connectedness by deactivating links or nodes. This process features a phase transition that resembles paradigmatic critical transitions in epidemic spreading, biological networks, traffic and transportation systems. Some biological systems, such as networks of neural cells, actively respond to percolation-like damage, which enables these structures to maintain their function after degradation and aging. Here we study percolation in networks that actively respond to link damage by adopting a mechanism resembling synaptic scaling in neurons. We explain critical transitions in such active networks and show that these structures are more resilient to damage as they are able to maintain a stronger connectedness and ability to spread information. Moreover, we uncover the role of local rescaling strategies in biological networks and indicate a possibility of designing smart infrastructures with improved robustness to perturbations.

The Refractory Period Matters: Unifying Mechanisms of Macroscopic Brain Waves

The relationship between complex brain oscillations and the dynamics of individual neurons is poorly understood. Here we utilize maximum caliber, a dynamical inference principle, to build a minimal yet general model of the collective (mean field) dynamics of large populations of neurons. In agreement with previous experimental observations, we describe a simple, testable mechanism, involving only a single type of neuron, by which many of these complex oscillatory patterns may emerge. Our model predicts that the refractory period of neurons, which has often been neglected, is essential for these behaviors.

Regional Neurodegeneration in vitro: The Protective Role of Neural Activity

Traumatic brain injury is a devastating public health problem, the eighth leading cause of death across the world. To improve our understanding of how injury at the cellular scale affects neural circuit function, we developed a protocol to precisely injure individual neurons within an in vitro neural network. We used high speed calcium imaging to estimate alterations in neural activity and connectivity that occur followed targeted microtrauma. Our studies show that mechanically injured neurons inactivate following microtrauma and eventually re-integrate into the network. Single neuron re-integration is dependent on its activity prior to injury and initial connections in the network: more active and integrated neurons are more resistant to microtrauma and more likely to re-integrate into the network. Micromechanical injury leads to neuronal death 6 h post-injury in a subset of both injured and uninjured neurons. Interestingly, neural activity and network participation after injury were associated with survival in linear discriminate analysis (77.3% correct prediction, Wilks' Lambda = 0.838). Based on this observation, we modulated neuronal activity to rescue neurons after microtrauma. Inhibition of neuronal activity provided much greater survivability than did activation of neurons (ANOVA, p < 0.01 with post-hoc Tukey HSD, p < 0.01). Rescue of neurons by blocking activity in the post-acute period is partially mediated by mitochondrial energetics, as we observed silencing neurons after micromechanical injury led to a significant reduction in mitochondrial calcium accumulation. Overall, the present study provides deeper insight into the propagation of injury within networks, demonstrating that together the initial activity, network structure, and post-injury activity levels contribute to the progressive changes in a neural circuit after mechanical trauma.

Bistability and time crystals in long-ranged directed percolation

Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatial dimension is given by probabilistic cellular automata, which typically feature time-independent and short-ranged update rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a bistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the system dynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, we find that the system responds with a period larger than that of the modulation for an exponentially (in system size) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled by a self-correcting mechanism of the long-ranged interactions which compensates noise-induced imperfections. Our work thus provides a firm example of a classical discrete time crystal phase of matter and paves the way for the study of novel non-equilibrium phases in the unexplored field of driven probabilistic cellular automata.

Critical and Ictal Phases in Simulated EEG Signals on a Small-World Network

Healthy brain function is marked by neuronal network dynamics at or near the critical phase, which separates regimes of instability and stasis. A failure to remain at this critical point can lead to neurological disorders such as epilepsy, which is associated with pathological synchronization of neuronal oscillations. Using full Hodgkin-Huxley (HH) simulations on a Small-World Network, we are able to generate synthetic electroencephalogram (EEG) signals with intervals corresponding to seizure (ictal) or non-seizure (interictal) states that can occur based on the hyperexcitability of the artificial neurons and the strength and topology of the synaptic connections between them. These interictal simulations can be further classified into scale-free critical phases and disjoint subcritical exponential phases. By changing the HH parameters, we can model seizures due to a variety of causes, including traumatic brain injury (TBI), congenital channelopathies, and idiopathic etiologies, as well as the effects of anticonvulsant drugs. The results of this work may be used to help identify parameters from actual patient EEG or electrocorticographic (ECoG) data associated with ictogenesis, as well as generating simulated data for training machine-learning seizure prediction algorithms.

Spiral Wave Propagation in Communities with Spatially Correlated Heterogeneity

Many multicellular communities propagate signals in a directed manner via excitable waves. Cell-to-cell heterogeneity is a ubiquitous feature of multicellular communities, but the effects of heterogeneity on wave propagation are still unclear. Here, we use a minimal FitzHugh-Nagumo-type model to investigate excitable wave propagation in a two-dimensional heterogeneous community. The model shows three dynamic regimes in which waves either propagate directionally, die out, or spiral indefinitely, and we characterize how these regimes depend on the heterogeneity parameters. We find that in some parameter regimes, spatial correlations in the heterogeneity enhance directional propagation and suppress spiraling. However, in other regimes, spatial correlations promote spiraling, a surprising feature that we explain by demonstrating that these spirals form by a second, distinct mechanism. Finally, we characterize the dynamics using techniques from percolation theory. Despite the fact that percolation theory does not completely describe the dynamics quantitatively because it neglects the details of the excitable propagation, we find that it accounts for the transitions between the dynamic regimes and the general dependency of the spiral period on the heterogeneity and thus provides important insights. Our results reveal that the spatial structure of cell-to-cell heterogeneity can have important consequences for signal propagation in cellular communities.

Statistics of correlated percolation in a bacterial community

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.

Simple model of complex dynamics of activity patterns in developing networks of neuronal cultures

Living neuronal networks in dissociated neuronal cultures are widely known for their ability to generate highly robust spatiotemporal activity patterns in various experimental conditions. Such patterns are often treated as neuronal avalanches that satisfy the power scaling law and thereby exemplify self-organized criticality in living systems. A crucial question is how these patterns can be explained and modeled in a way that is biologically meaningful, mathematically tractable and yet broad enough to account for neuronal heterogeneity and complexity. Here we derive and analyse a simple network model that may constitute a response to this question. Our derivations are based on few basic phenomenological observations concerning the input-output behavior of an isolated neuron. A distinctive feature of the model is that at the simplest level of description it comprises of only two variables, the network activity variable and an exogenous variable corresponding to energy needed to sustain the activity, and few parameters such as network connectivity and efficacy of signal transmission. The efficacy of signal transmission is modulated by the phenomenological energy variable. Strikingly, this simple model is already capable of explaining emergence of network spikes and bursts in developing neuronal cultures. The model behavior and predictions are consistent with published experimental evidence on cultured neurons. At the larger, cellular automata scale, introduction of the energy-dependent regulatory mechanism results in the overall model behavior that can be characterized as balancing on the edge of the network percolation transition. Network activity in this state shows population bursts satisfying the scaling avalanche conditions. This network state is self-sustainable and represents energetic balance between global network-wide processes and spontaneous activity of individual elements.

Carbon Nanotubes, Directly Grown on Supporting Surfaces, Improve Neuronal Activity in Hippocampal Neuronal Networks

Carbon nanotube (CNT)-modified surfaces unequivocally demonstrate their biocompatibility and ability to boost the electrical activity of neuronal cells cultured on them. Reasons for this effect are still under debate. However, the intimate contact at the membrane level between these thready nanostructures and cells, in combination with their unique electrical properties, seems to play an important role. The entire existing literature exploiting the effect of CNTs on modulating cellular behavior deals with cell cultures grown on purified multiwalled carbon nanotubes (MWNTs) deposited on a supporting surface via drop-casting or mechanical entrapment. Here, for the first time, it is demonstrated that CNTs directly grown on a supporting silicon surface by a chemical vapor deposition (CVD)-assisted technique have the same effect. It is shown that primary neuronal cells developed above a carpet of CVD CNTs form a healthy and functional network. The resulting neuronal network shows increased electrical activity when compared to a similar network developed on a control glass surface. The low cost and high versatility of the here presented CVD-based synthesis process, together with the possibility to create on supporting substrate patterns of any arbitrary shape of CNTs, open up new opportunities for brain-machine interfaces or neuroprosthetic devices.

Hierarchy measurement for modeling network dynamics under directed attacks

A fundamental issue in the dynamics of complex systems is the resilience of the network in response to targeted attacks. This paper explores the local dynamics of the network attack process by investigating the order of removal of the nodes that have maximal degree, and shows that this dynamic network response can be predicted from the graph's initial connectivity. We demonstrate numerically that the maximal degree M(τ) of the network at time step τ decays exponentially with τ via a topology-dependent exponent. Moreover, the order in which sites are removed can be approximated by considering the network's "hierarchy" function h, which measures for each node V_{i} how many of its initial nearest neighbors have lower degree versus those that have a higher one. Finally, we show that the exponents we identified for the attack dynamics are related to the exponential behavior of spreading activation dynamics. The results suggest that the function h, which has both local and global properties, is a novel nodal measurement for network dynamics and structure.

Noise focusing in neuronal tissues: Symmetry breaking and localization in excitable networks with quenched disorder

We introduce a coarse-grained stochastic model for the spontaneous activity of neuronal cultures to explain the phenomenon of noise focusing, which entails localization of the noise activity in excitable networks with metric correlations. The system is modeled as a continuum excitable medium with a state-dependent spatial coupling that accounts for the dynamics of synaptic connections. The most salient feature is the emergence at the mesoscale of a vector field V(r), which acts as an advective carrier of the noise. This entails an explicit symmetry breaking of isotropy and homogeneity that stems from the amplification of the quenched fluctuations of the network by the activity avalanches, concomitant with the excitable dynamics. We discuss the microscopic interpretation of V(r) and propose an explicit construction of it. The coarse-grained model shows excellent agreement with simulations at the network level. The generic nature of the observed phenomena is discussed.

Dominance of Metric Correlations in Two-Dimensional Neuronal Cultures Described through a Random Field Ising Model

We introduce a novel random field Ising model, grounded on experimental observations, to assess the importance of metric correlations in cortical circuits in vitro. Metric correlations arise from both the finite axonal length and the heterogeneity in the spatial arrangement of neurons. The experiments consider the response of neuronal cultures to an external electric stimulation for a gradually weaker connectivity strength between neurons, and in cultures with different spatial configurations. The model can be analytically solved in the metric-free, mean-field scenario. The presence of metric correlations precipitates a strong deviation from the mean field. Null models of the same networks that preserve the distribution of connections recover the mean field. Our results show that metric-inherited correlations in spatial networks dominate the connectivity blueprint, mask the actual distribution of connections, and may emerge as the asset that shapes network dynamics.

External Excitation of Neurons Using Electric and Magnetic Fields in One- and Two-dimensional Cultures

A neuron will fire an action potential when its membrane potential exceeds a certain threshold. In typical activity of the brain, this occurs as a result of chemical inputs to its synapses. However, neurons can also be excited by an imposed electric field. In particular, recent clinical applications activate neurons by creating an electric field externally. It is therefore of interest to investigate how the neuron responds to the external field and what causes the action potential. Fortunately, precise and controlled application of an external electric field is possible for embryonic neuronal cells that are excised, dissociated and grown in cultures. This allows the investigation of these questions in a highly reproducible system. In this paper some of the techniques used for controlled application of external electric field on neuronal cultures are reviewed. The networks can be either one dimensional, i.e. patterned in linear forms or allowed to grow on the whole plane of the substrate, and thus two dimensional. Furthermore, the excitation can be created by the direct application of electric field via electrodes immersed in the fluid (bath electrodes) or by inducing the electric field using the remote creation of magnetic pulses.

Effect of threshold disorder on the quorum percolation model

We study the modifications induced in the behavior of the quorum percolation model on neural networks with Gaussian in-degree by taking into account an uncorrelated Gaussian thresholds variability. We derive a mean-field approach and show its relevance by carrying out explicit Monte Carlo simulations. It turns out that such a disorder shifts the position of the percolation transition, impacts the size of the giant cluster, and can even destroy the transition. Moreover, we highlight the occurrence of disorder independent fixed points above the quorum critical value. The mean-field approach enables us to interpret these effects in terms of activation probability. A finite-size analysis enables us to show that the order parameter is weakly self-averaging with an exponent independent on the thresholds disorder. Last, we show that the effects of the thresholds and connectivity disorders cannot be easily discriminated from the measured averaged physical quantities.

Size-dependent regulation of synchronized activity in living neuronal networks

We study the effect of network size on synchronized activity in living neuronal networks. Dissociated cortical neurons form synaptic connections in culture and generate synchronized spontaneous activity within 10 days in vitro. Using micropatterned surfaces to extrinsically control the size of neuronal networks, we show that synchronized activity can emerge in a network as small as 12 cells. Furthermore, a detailed comparison of small (∼20 cells), medium (∼100 cells), and large (∼400 cells) networks reveal that synchronized activity becomes destabilized in the small networks. A computational modeling of neural activity is then employed to explore the underlying mechanism responsible for the size effect. We find that the generation and maintenance of the synchronized activity can be minimally described by: (1) the stochastic firing of each neuron in the network, (2) enhancement in the network activity in a positive feedback loop of excitatory synapses, and (3) Ca-dependent suppression of bursting activity. The model further shows that the decrease in total synaptic input to a neuron that drives the positive feedback amplification of correlated activity is a key factor underlying the destabilization of synchrony in smaller networks. Spontaneous neural activity plays a critical role in cortical information processing, and our work constructively clarifies an aspect of the structural basis behind this.

Finite-size effects and dynamics of giant transition of a continuum quorum percolation model on random networks

We start from a continuous extension of a mean field approach of the quorum percolation model, accounting for the response of in vitro neuronal cultures, to carry out a normal form analysis of the critical behavior. We highlight the effects of nonlinearities associated with this mean field approach even in the close vicinity of the critical point. Statistical properties of random networks with Gaussian in-degree are related to the outcoming links distribution. Finite size analysis of explicit Monte Carlo simulations enables us to confirm the relevance of the mean field approach on such networks and to show that the order parameter is weakly self-averaging; dynamical relaxation is investigated. Furthermore we derive a mean field equation taking into account the effect of inhibitory neurons and discuss the equivalence with a purely excitatory network.

The Path to Exhaustion: Time-Variability Properties of Coordinative Variables during Continuous Exercise

The aim of this study was to detect qualitative changes in the structure of coordinative variable (elbow angle) fluctuations during a quasi-isometric exercise performed until exhaustion. Seven physical education students performed a quasi-isometric arm-curl exercise holding an Olympic bar (weight: 80% 1RM) with an initial elbow flexion of 90° three times over a period of 4 weeks. They were encouraged to persist, even if the elbow angle was lost, until the fatigue-induced spontaneous termination point (FISTP). Changes in both elbow angles were registered during the task through an electrogoniometer. Detrended Fluctuation Analysis (DFA) was conducted on the initial and final 1024 data points of the series and the associated Hurst exponents were obtained. Multi-way RM ANOVA analyses revealed a significant main effect of the Time on task on the Hurst exponent values but also revealed a significant Trial × Time on task interaction. In the initial (non-fatigue) condition participants tended to produce anti-persistent fBm fluctuations. In the final part before exhaustion a tendency toward persistent fBm was dominant. The trial to trial differences in time-variability structure points to an existence of a long-term variability in control strategies during exercise. The changes in the temporal structure of the elbow angle variability as effort accumulated reflected an increase in low-frequency fluctuations signifying a change in psychobiological mechanisms used to negotiate the task demands. The variability properties of the coordinative variable during exercise may provide information about the dynamic mechanisms that lead to exhaustion.

Spike-count distribution in a neuronal population under weak common stimulation

We study the probability distribution of the number of synchronous action potentials (spike count) in a model network consisting of a homogeneous neural population that is driven by a common time-dependent stimulus. We derive two analytical approximations for the count statistics, which are based on linear response theory and hold true for weak input correlations. Comparison to numerical simulations of populations of integrate-and-fire neurons in different parameter regimes reveals that our theory correctly predicts how much a weak common stimulus increases the probability of common firing and of common silence in the neural population.

Facing the challenge of mammalian neural microcircuits: taking a few breaths may help

Breathing in mammals is a seemingly straightforward behaviour controlled by the brain. A brainstem nucleus called the preBötzinger Complex sits at the core of the neural circuit generating respiratory rhythm. Despite the discovery of this microcircuit almost 25 years ago, the mechanisms controlling breathing remain elusive. Given the apparent simplicity and well-defined nature of regulatory breathing behaviour, the identification of much of the circuitry, and the ability to study breathing in vitro as well as in vivo, many neuroscientists and physiologists are surprised that respiratory rhythm generation is still not well understood. Our view is that conventional rhythmogenic mechanisms involving pacemakers, inhibition or bursting are problematic and that simplifying assumptions commonly made for many vertebrate neural circuits ignore consequential detail. We propose that novel emergent mechanisms govern the generation of respiratory rhythm. That a mammalian function as basic as rhythm generation arises from complex and dynamic molecular, synaptic and neuronal interactions within a diverse neural microcircuit highlights the challenges in understanding neural control of mammalian behaviours, many (considerably) more elaborate than breathing. We suggest that the neural circuit controlling breathing is inimitably tractable and may inspire general strategies for elucidating other neural microcircuits.

Chronaxie Measurements in Patterned Neuronal Cultures from Rat Hippocampus

Excitation of neurons by an externally induced electric field is a long standing question that has recently attracted attention due to its relevance in novel clinical intervention systems for the brain. Here we use patterned quasi one-dimensional neuronal cultures from rat hippocampus, exploiting the alignment of axons along the linear patterned culture to separate the contribution of dendrites to the excitation of the neuron from that of axons. Network disconnection by channel blockers, along with rotation of the electric field direction, allows the derivation of strength-duration (SD) curves that characterize the statistical ensemble of a population of cells. SD curves with the electric field aligned either parallel or perpendicular to the axons yield the chronaxie and rheobase of axons and dendrites respectively, and these differ considerably. Dendritic chronaxie is measured to be about 1 ms, while that of axons is on the order of 0.1 ms. Axons are thus more excitable at short time scales, but at longer time scales dendrites are more easily excited. We complement these studies with experiments on fully connected cultures. An explanation for the chronaxie of dendrites is found in the numerical simulations of passive, realistically structured dendritic trees under external stimulation. The much shorter chronaxie of axons is not captured in the passive model and may be related to active processes. The lower rheobase of dendrites at longer durations can improve brain stimulation protocols, since in the brain dendrites are less specifically oriented than axonal bundles, and the requirement for precise directional stimulation may be circumvented by using longer duration fields.

Combining microfluidics, optogenetics and calcium imaging to study neuronal communication in vitro

In this paper we report the combination of microfluidics, optogenetics and calcium imaging as a cheap and convenient platform to study synaptic communication between neuronal populations in vitro. We first show that Calcium Orange indicator is compatible in vitro with a commonly used Channelrhodopsine-2 (ChR2) variant, as standard calcium imaging conditions did not alter significantly the activity of transduced cultures of rodent primary neurons. A fast, robust and scalable process for micro-chip fabrication was developed in parallel to build micro-compartmented cultures. Coupling optical fibers to each micro-compartment allowed for the independent control of ChR2 activation in the different populations without crosstalk. By analyzing the post-stimuli activity across the different populations, we finally show how this platform can be used to evaluate quantitatively the effective connectivity between connected neuronal populations.

Glutamate mediated astrocytic filtering of neuronal activity

Neuron-astrocyte communication is an important regulatory mechanism in various brain functions but its complexity and role are yet to be fully understood. In particular, the temporal pattern of astrocyte response to neuronal firing has not been fully characterized. Here, we used neuron-astrocyte cultures on multi-electrode arrays coupled to Ca2+ imaging and explored the range of neuronal stimulation frequencies while keeping constant the amount of stimulation. Our results reveal that astrocytes specifically respond to the frequency of neuronal stimulation by intracellular Ca2+ transients, with a clear onset of astrocytic activation at neuron firing rates around 3-5 Hz. The cell-to-cell heterogeneity of the astrocyte Ca2+ response was however large and increasing with stimulation frequency. Astrocytic activation by neurons was abolished with antagonists of type I metabotropic glutamate receptor, validating the glutamate-dependence of this neuron-to-astrocyte pathway. Using a realistic biophysical model of glutamate-based intracellular calcium signaling in astrocytes, we suggest that the stepwise response is due to the supralinear dynamics of intracellular IP3 and that the heterogeneity of the responses may be due to the heterogeneity of the astrocyte-to-astrocyte couplings via gap junction channels. Therefore our results present astrocyte intracellular Ca2+ activity as a nonlinear integrator of glutamate-dependent neuronal activity.

Statistical thermodynamics of clustered populations

We present a thermodynamic theory for a generic population of M individuals distributed into N groups (clusters). We construct the ensemble of all distributions with fixed M and N, introduce a selection functional that embodies the physics that governs the population, and obtain the distribution that emerges in the scaling limit as the most probable among all distributions consistent with the given physics. We develop the thermodynamics of the ensemble and establish a rigorous mapping to regular thermodynamics. We treat the emergence of a so-called giant component as a formal phase transition and show that the criteria for its emergence are entirely analogous to the equilibrium conditions in molecular systems. We demonstrate the theory by an analytic model and confirm the predictions by Monte Carlo simulation.

Transfer entropy reconstruction and labeling of neuronal connections from simulated calcium imaging

Neuronal dynamics are fundamentally constrained by the underlying structural network architecture, yet much of the details of this synaptic connectivity are still unknown even in neuronal cultures in vitro. Here we extend a previous approach based on information theory, the Generalized Transfer Entropy, to the reconstruction of connectivity of simulated neuronal networks of both excitatory and inhibitory neurons. We show that, due to the model-free nature of the developed measure, both kinds of connections can be reliably inferred if the average firing rate between synchronous burst events exceeds a small minimum frequency. Furthermore, we suggest, based on systematic simulations, that even lower spontaneous inter-burst rates could be raised to meet the requirements of our reconstruction algorithm by applying a weak spatially homogeneous stimulation to the entire network. By combining multiple recordings of the same in silico network before and after pharmacologically blocking inhibitory synaptic transmission, we show then how it becomes possible to infer with high confidence the excitatory or inhibitory nature of each individual neuron.

Critical phenomena and noise-induced phase transitions in neuronal networks

We study numerically and analytically first- and second-order phase transitions in neuronal networks stimulated by shot noise (a flow of random spikes bombarding neurons). Using an exactly solvable cortical model of neuronal networks on classical random networks, we find critical phenomena accompanying the transitions and their dependence on the shot noise intensity. We show that a pattern of spontaneous neuronal activity near a critical point of a phase transition is a characteristic property that can be used to identify the bifurcation mechanism of the transition. We demonstrate that bursts and avalanches are precursors of a first-order phase transition, paroxysmal-like spikes of activity precede a second-order phase transition caused by a saddle-node bifurcation, while irregular spindle oscillations represent spontaneous activity near a second-order phase transition caused by a supercritical Hopf bifurcation. Our most interesting result is the observation of the paroxysmal-like spikes. We show that a paroxysmal-like spike is a single nonlinear event that appears instantly from a low background activity with a rapid onset, reaches a large amplitude, and ends up with an abrupt return to lower activity. These spikes are similar to single paroxysmal spikes and sharp waves observed in electroencephalographic (EEG) measurements. Our analysis shows that above the saddle-node bifurcation, sustained network oscillations appear with a large amplitude but a small frequency in contrast to network oscillations near the Hopf bifurcation that have a small amplitude but a large frequency. We discuss an amazing similarity between excitability of the cortical model stimulated by shot noise and excitability of the Morris-Lecar neuron stimulated by an applied current.

Percolation of spatially constrained Erdős-Rényi networks with degree correlations

Motivated by experiments on activity in neuronal cultures [ J. Soriano, M. Rodríguez Martínez, T. Tlusty and E. Moses Proc. Natl. Acad. Sci. 105 13758 (2008)], we investigate the percolation transition and critical exponents of spatially embedded Erdős-Rényi networks with degree correlations. In our model networks, nodes are randomly distributed in a two-dimensional spatial domain, and the connection probability depends on Euclidian link length by a power law as well as on the degrees of linked nodes. Generally, spatial constraints lead to higher percolation thresholds in the sense that more links are needed to achieve global connectivity. However, degree correlations favor or do not favor percolation depending on the connectivity rules. We employ two construction methods to introduce degree correlations. In the first one, nodes stay homogeneously distributed and are connected via a distance- and degree-dependent probability. We observe that assortativity in the resulting network leads to a decrease of the percolation threshold. In the second construction methods, nodes are first spatially segregated depending on their degree and afterwards connected with a distance-dependent probability. In this segregated model, we find a threshold increase that accompanies the rising assortativity. Additionally, when the network is constructed in a disassortative way, we observe that this property has little effect on the percolation transition.

Cooperative heterogeneous facilitation: multiple glassy states and glass-glass transition

The formal structure of glass singularities in the mode-coupling theory (MCT) of supercooled liquids dynamics is closely related to that appearing in the analysis of heterogeneous bootstrap percolation on Bethe lattices, random graphs, and complex networks. Starting from this observation one can build up microscopic on-lattice realizations of schematic MCT based on cooperative facilitated spin mixtures. I discuss a microscopic implementation of the F(13) schematic model including multiple glassy states and the glass-glass transition. Results suggest that our approach is flexible enough to bridge alternative theoretical descriptions of glassy matter based on the notions of quenched disorder and dynamic facilitation.

How to enhance the dynamic range of excitatory-inhibitory excitable networks

We investigate the collective dynamics of excitatory-inhibitory excitable networks in response to external stimuli. How to enhance the dynamic range, which represents the ability of networks to encode external stimuli, is crucial to many applications. We regard the system as a two-layer network (E layer and I layer) and explore the criticality and dynamic range on diverse networks. Interestingly, we find that phase transition occurs when the dominant eigenvalue of the E layer's weighted adjacency matrix is exactly 1, which is only determined by the topology of the E layer. Meanwhile, it is shown that the dynamic range is maximized at a critical state. Based on theoretical analysis, we propose an inhibitory factor for each excitatory node. We suggest that if nodes with high inhibitory factors are cut out from the I layer, the dynamic range could be further enhanced. However, because of the sparseness of networks and passive function of inhibitory nodes, the improvement is relatively small compared to the original dynamic range. Even so, this provides a strategy to enhance the dynamic range.

Signal integration enhances the dynamic range in neuronal systems

The dynamic range measures the capacity of a system to discriminate the intensity of an external stimulus. Such an ability is fundamental for living beings to survive: to leverage resources and to avoid danger. Consequently, the larger is the dynamic range, the greater is the probability of survival. We investigate how the integration of different input signals affects the dynamic range, and in general the collective behavior of a network of excitable units. By means of numerical simulations and a mean-field approach, we explore the nonequilibrium phase transition in the presence of integration. We show that the firing rate in random and scale-free networks undergoes a discontinuous phase transition depending on both the integration time and the density of integrator units. Moreover, in the presence of external stimuli, we find that a system of excitable integrator units operating in a bistable regime largely enhances its dynamic range.

Fragmentation: loss of global coherence or breakdown of modularity in functional brain architecture?

Psychiatric illnesses characterized by disorganized cognition, such as schizophrenia, have been described in terms of fragmentation and hence understood as reduction in functional brain connectivity, particularly in prefrontal and parietal areas. However, as graph theory shows, relatively small numbers of nonlocal connections are sufficient to ensure global coherence in the modular small-world network structure of the brain. We reconsider fragmentation in this perspective. Computational studies have shown that for a given level of connectivity in a model of coupled nonlinear oscillators, modular small-world networks evolve from an initially random organization. Here we demonstrate that with decreasing connectivity, the probability of evolving into a modular small-world network breaks down at a critical point, which scales to the percolation function of random networks with a universal exponent of α = 1.17. Thus, according to the model, local modularity systematically breaks down before there is loss of global coherence in network connectivity. We, therefore, propose that fragmentation may involve, at least in its initial stages, the inability of a dynamically evolving network to sustain a modular small-world structure. The result is in a shift in the balance in schizophrenia from local to global functional connectivity.

Exogenous α-synuclein fibrils induce Lewy body pathology leading to synaptic dysfunction and neuron death

Inclusions composed of α-synuclein (α-syn), i.e., Lewy bodies (LBs) and Lewy neurites (LNs), define synucleinopathies including Parkinson's disease (PD) and dementia with Lewy bodies (DLB). Here, we demonstrate that preformed fibrils generated from full-length and truncated recombinant α-syn enter primary neurons, probably by adsorptive-mediated endocytosis, and promote recruitment of soluble endogenous α-syn into insoluble PD-like LBs and LNs. Remarkably, endogenous α-syn was sufficient for formation of these aggregates, and overexpression of wild-type or mutant α-syn was not required. LN-like pathology first developed in axons and propagated to form LB-like inclusions in perikarya. Accumulation of pathologic α-syn led to selective decreases in synaptic proteins, progressive impairments in neuronal excitability and connectivity, and, eventually, neuron death. Thus, our data contribute important insights into the etiology and pathogenesis of PD-like α-syn inclusions and their impact on neuronal functions, and they provide a model for discovering therapeutics targeting pathologic α-syn-mediated neurodegeneration.

Dynamic changes in neural circuit topology following mild mechanical injury in vitro

Despite its enormous incidence, mild traumatic brain injury is not well understood. One aspect that needs more definition is how the mechanical energy during injury affects neural circuit function. Recent developments in cellular imaging probes provide an opportunity to assess the dynamic state of neural networks with single-cell resolution. In this article, we developed imaging methods to assess the state of dissociated cortical networks exposed to mild injury. We estimated the imaging conditions needed to achieve accurate measures of network properties, and applied these methodologies to evaluate if mild mechanical injury to cortical neurons produces graded changes to either spontaneous network activity or altered network topology. We found that modest injury produced a transient increase in calcium activity that dissipated within 1 h after injury. Alternatively, moderate mechanical injury produced immediate disruption in network synchrony, loss in excitatory tone, and increased modular topology. A calcium-activated neutral protease (calpain) was a key intermediary in these changes; blocking calpain activation restored the network nearly completely to its pre-injury state. Together, these findings show a more complex change in neural circuit behavior than previously reported for mild mechanical injury, and highlight at least one important early mechanism responsible for these changes.

Percolation on correlated random networks

We consider a class of random, weighted networks, obtained through a redefinition of patterns in an Hopfield-like model, and, by performing percolation processes, we get information about topology and resilience properties of the networks themselves. Given the weighted nature of the graphs, different kinds of bond percolation can be studied: stochastic (deleting links randomly) and deterministic (deleting links based on rank weights), each mimicking a different physical process. The evolution of the network is accordingly different, as evidenced by the behavior of the largest component size and of the distribution of cluster sizes. In particular, we can derive that weak ties are crucial in order to maintain the graph connected and that, when they are the most prone to failure, the giant component typically shrinks without abruptly breaking apart; these results have been recently evidenced in several kinds of social networks.

Optical validation of in vitro extra-cellular neuronal recordings

Simultaneous calcium imaging and extra-cellular recordings from cultured cortical rat neurons were performed to directly map the efficacy of extra-cellular recordings with microelectrodes. For the first time, we can associate extra-cellular recordings with neuronal activity of specific neurons in the vicinity of the electrode. We demonstrate that recorded cells can be identified by correlating the electrical signals and the calcium response. Our data demonstrate that in sparse cultures, microelectrodes record exclusively from cells which reside at very close proximity to the recording electrode. Moreover, we show that recording appears to be limited to only a partial subset of the cells residing in this range. We further show that even in cases of strong neuron-electrode coupling, extra-cellular signals recorded from single, well-identified neurons vary in shape over time rendering spike sorting and network activity rate analysis incongruous. As multi-electrode array technology is becoming increasingly widespread, the visualization technique we report here will help users better understand the limits of this versatile and useful method.

Leaders of neuronal cultures in a quorum percolation model

We present a theoretical framework using quorum percolation for describing the initiation of activity in a neural culture. The cultures are modeled as random graphs, whose nodes are excitatory neurons with k_in inputs and k_out outputs, and whose input degrees k_in = k obey given distribution functions p_k. We examine the firing activity of the population of neurons according to their input degree (k) classes and calculate for each class its firing probability Φ_k(t) as a function of t. The probability of a node to fire is found to be determined by its in-degree k, and the first-to-fire neurons are those that have a high k. A small minority of high-k-classes may be called “Leaders,” as they form an interconnected sub-network that consistently fires much before the rest of the culture. Once initiated, the activity spreads from the Leaders to the less connected majority of the culture. We then use the distribution of in-degree of the Leaders to study the growth rate of the number of neurons active in a burst, which was experimentally measured to be initially exponential. We find that this kind of growth rate is best described by a population that has an in-degree distribution that is a Gaussian centered around k = 75 with width σ = 31 for the majority of the neurons, but also has a power law tail with exponent −2 for 10% of the population. Neurons in the tail may have as many as k = 4,700 inputs. We explore and discuss the correspondence between the degree distribution and a dynamic neuronal threshold, showing that from the functional point of view, structure and elementary dynamics are interchangeable. We discuss possible geometric origins of this distribution, and comment on the importance of size, or of having a large number of neurons, in the culture.

Fractals in the nervous system: conceptual implications for theoretical neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

Stochastic cellular automata model of neural networks

We propose a stochastic dynamical model of noisy neural networks with complex architectures and discuss activation of neural networks by a stimulus, pacemakers, and spontaneous activity. This model has a complex phase diagram with self-organized active neural states, hybrid phase transitions, and a rich array of behaviors. We show that if spontaneous activity (noise) reaches a threshold level then global neural oscillations emerge. Stochastic resonance is a precursor of this dynamical phase transition. These oscillations are an intrinsic property of even small groups of 50 neurons.

Neuronal avalanches imply maximum dynamic range in cortical networks at criticality

Spontaneous neuronal activity is a ubiquitous feature of cortex. Its spatiotemporal organization reflects past input and modulates future network output. Here we study whether a particular type of spontaneous activity is generated by a network that is optimized for input processing. Neuronal avalanches are a type of spontaneous activity observed in superficial cortical layers in vitro and in vivo with statistical properties expected from a network operating at "criticality." Theory predicts that criticality and, therefore, neuronal avalanches are optimal for input processing, but until now, this has not been tested in experiments. Here, we use cortex slice cultures grown on planar microelectrode arrays to demonstrate that cortical networks that generate neuronal avalanches benefit from a maximized dynamic range, i.e., the ability to respond to the greatest range of stimuli. By changing the ratio of excitation and inhibition in the cultures, we derive a network tuning curve for stimulus processing as a function of distance from criticality in agreement with predictions from our simulations. Our findings suggest that in the cortex, (1) balanced excitation and inhibition establishes criticality, which maximizes the range of inputs that can be processed, and (2) spontaneous activity and input processing are unified in the context of critical phenomena.

BDNF and NT-3 increase excitatory input connectivity in rat hippocampal cultures

The neurotrophic factors brain-derived neurotrophic factor (BDNF) and neurotrophin-3 (NT-3) have been shown to promote excitatory and inhibitory synapse development. However, a quantitative analysis of their influence on connectivity has proven in general difficult to achieve. In this work we use a novel experimental approach based on percolation concepts that provides a quantification of the average number of connections per neuron. In combination with electrophysiological measurements, we characterize the changes in network connectivity induced by BDNF and NT-3 in rat hippocampal cultures. We show that, on the one hand, BDNF and NT-3 accelerate the maturation of connectivity in the network by about 17 h. On the other hand, BDNF and NT-3 increase the number of excitatory input connections by a factor of about two, but without modifying the number of inhibitory input connections. This scenario of a dominant effect on the excitation is supported by the analysis of spontaneous population bursts in cultures treated with either BDNF or NT-3, which show burst amplitudes that are insensitive to the blockade of inhibition. A leaky integrate-and-fire model reproduces the experimental results well.