Dynamical evolution of spatiotemporal patterns in mammalian middle cortex

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Bistable spiral wave dynamics in electrically excitable media

We show that a positive feedback loop between sodium current inactivation and wave-front ramp-up speed causes a saddle-node bifurcation to result in bistable planar and spiral waves in electrically excitable media, in which both slow and fast waves are triggered by different stimulation protocols. Moreover, the two types of spiral wave conduction may interact to give rise to more complex spiral wave dynamics. The transitions between different spiral wave behaviors via saddle-node bifurcation can be a candidate mechanism for transitions widely seen in cardiac arrhythmias and neural diseases.

Leading basic modes of spontaneous activity drive individual functional connectivity organization in the resting human brain

Spontaneous activity of the human brain provides a window to explore intrinsic principles of functional organization. However, most studies have focused on interregional functional connectivity. The principles underlying rich repertoires of instantaneous activity remain largely unknown. We apply a recently proposed eigen-microstate analysis to three resting-state functional MRI datasets to identify basic modes that represent fundamental activity patterns that coexist over time. We identify five leading basic modes that dominate activity fluctuations. Each mode exhibits a distinct functional system-dependent coactivation pattern and corresponds to specific cognitive profiles. In particular, the spatial pattern of the first leading basis mode shows the separation of activity between the default-mode and primary and attention regions. Based on theoretical modelling, we further reconstruct individual functional connectivity as the weighted superposition of coactivation patterns corresponding to these leading basic modes. Moreover, these leading basic modes capture sleep deprivation-induced changes in brain activity and interregional connectivity, primarily involving the default-mode and task-positive regions. Our findings reveal a dominant set of basic modes of spontaneous activity that reflect multiplexed interregional coordination and drive conventional functional connectivity, furthering the understanding of the functional significance of spontaneous brain activity.

Kramers Rate Theory of Pacemaker Dynamics in Noisy Excitable Media

Rhythmic activities, which are usually driven by pacemakers, are common in biological systems. In noisy excitable media, pacemakers are self-organized firing clusters, but the underlying dynamics remains to be elucidated. Here we develop a Kramers rate theory of coupled cells to describe the firing properties of pacemakers and their dependence on coupling strength and system size and dimension. The theory captures accurately the simulation results of tissue models with stochastic Hodgkin-Huxley equations except when transitions from pacemakers to spiral waves occur under weak coupling.

Neuronal population dynamics during motor plan cancellation in nonhuman primates

To understand the cortical neuronal dynamics behind movement generation and control, most studies have focused on tasks where actions were planned and then executed using different instances of visuomotor transformations. However, to fully understand the dynamics related to movement control, one must also study how movements are actively inhibited. Inhibition, indeed, represents the first level of control both when different alternatives are available and only one solution could be adopted and when it is necessary to maintain the current position. We recorded neuronal activity from a multielectrode array in the dorsal premotor cortex (PMd) of monkeys performing a countermanding reaching task that requires, in a subset of trials, them to cancel a planned movement before its onset. In the analysis of the neuronal state space of PMd, we found a subspace in which activities conveying temporal information were confined during active inhibition and position holding. Movement execution required activities to escape from this subspace toward an orthogonal subspace and, furthermore, surpass a threshold associated with the maturation of the motor plan. These results revealed further details in the neuronal dynamics underlying movement control, extending the hypothesis that neuronal computation confined in an "output-null" subspace does not produce movements.

Slow waves form expanding, memory-rich mesostates steered by local excitability in fading anesthesia

In the arousal process, the brain restores its integrative activity from the synchronized state of slow wave activity (SWA). The mechanisms underpinning this state transition remain, however, to be elucidated. Here we simultaneously probed neuronal assemblies throughout the whole cortex with micro-electrocorticographic recordings in mice. We investigated the progressive shaping of propagating SWA at different levels of isoflurane. We found a form of memory of the wavefront shapes at deep anesthesia, tightly alternating posterior-anterior-posterior patterns. At low isoflurane, metastable patterns propagated in more directions, reflecting an increased complexity. The wandering across these mesostates progressively increased its randomness, as predicted by simulations of a network of spiking neurons, and confirmed in our experimental data. The complexity increase is explained by the elevated excitability of local assemblies with no modifications of the network connectivity. These results shed new light on the functional reorganization of the cortical network as anesthesia fades out.

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Complexity of isoflurane-induced slow waves reliably determines anesthesia level

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In deep anesthesia, the propagation strictly alternates between front-back-front patterns

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In light anesthesia, there is a continuum of directions and faster propagation

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Local excitability underpins the cortical reorganization in fading anesthesia

Energy-Based Hierarchical Clustering of Cortical Slow Waves in Multi-Electrode Recordings

The recent development of novel multi-electrode recording technologies has revealed the existence of traveling patterns of cortical activity in many species and under different states of awareness. Among these, slow activation waves occurring under sleep and anesthesia have been widely investigated as they provide unique insights into network features such as excitability, connectivity, structure, and dynamics of the cerebral cortex. Such characterization is usually based on clustering methods which are constrained by a priori assumptions as to the number of clusters to be used or rely on wave-by-wave pattern reconstruction. Here, we introduce a new computational tool based on modal analysis of fluid flows which is robustly applied to multivariate electrophysiological data from cortical networks, namely the Energy-based Hierarchical Waves Clustering method (EHWC). EHWC is composed of three main steps: (1) detecting the occurrence of global waves; (2) reducing the data dimensionality via singular value decomposition; (3) clustering hierarchically the singled-out waves. The analysis does not require the single-channel contribution to the waves, which is a typical bottleneck in this kind of analysis due to the unavoidable intrinsic variability of locally recorded activity. For testing and validation, here we used in vivo extracellular recordings from mice cortex under three different levels of anesthesia. As a result, we found slow waves with an increasing number of propagation modes as the anesthesia level decreases, giving an estimate of the increasing complexity of network dynamics. This and other wave's features replicate and extend the findings from previous literature, paving the way to extend the same approach to non-invasive electrophysiological recordings like EEG and fMRI used clinically for the characterization of brain dynamics and clinical stratification in brain lesions.

Spatiotemporal characteristics in systems of diffusively coupled excitable slow-fast FitzHugh-Rinzel dynamical neurons

In this paper, we study an excitable, biophysical system that supports wave propagation of nerve impulses. We consider a slow-fast, FitzHugh-Rinzel neuron model where only the membrane voltage interacts diffusively, giving rise to the formation of spatiotemporal patterns. We focus on local, nonlinear excitations and diverse neural responses in an excitable one- and two-dimensional configuration of diffusively coupled FitzHugh-Rinzel neurons. The study of the emerging spatiotemporal patterns is essential in understanding the working mechanism in different brain areas. We derive analytically the coefficients of the amplitude equations in the vicinity of Hopf bifurcations and characterize various patterns, including spirals exhibiting complex geometric substructures. Furthermore, we derive analytically the condition for the development of antispirals in the neighborhood of the bifurcation point. The emergence of broken target waves can be observed to form spiral-like profiles. The spatial dynamics of the excitable system exhibits two- and multi-arm spirals for small diffusive couplings. Our results reveal a multitude of neural excitabilities and possible conditions for the emergence of spiral-wave formation. Finally, we show that the coupled excitable systems with different firing characteristics participate in a collective behavior that may contribute significantly to irregular neural dynamics.

Delayed global feedback in the genesis and stability of spatiotemporal excitation patterns in paced biological excitable media

Biological excitable media, such as cardiac or neural cells and tissue, exhibit memory in which a change in the present excitation may affect the behaviors in the next excitation. For example, a change in calcium (Ca²⁺) concentration in a cell in the present excitation may affect the Ca²⁺ dynamics in the next excitation via bi-directional coupling between voltage and Ca²⁺, forming a delayed feedback loop. Since the Ca²⁺ dynamics inside the excitable cells are spatiotemporal while the membrane voltage is a global signal, the feedback loop is then a delayed global feedback (DGF) loop. In this study, we investigate the roles of DGF in the genesis and stability of spatiotemporal excitation patterns in periodically-paced excitable media using mathematical models with different levels of complexity: a model composed of coupled FitzHugh-Nagumo units, a 3-dimensional physiologically-detailed ventricular myocyte model, and a coupled map lattice model. We investigate the dynamics of excitation patterns that are temporal period-2 (P2) and spatially concordant or discordant, such as subcellular concordant or discordant Ca²⁺alternans in cardiac myocytes or spatially concordant or discordant Ca²⁺ and repolarization alternans in cardiac tissue. Our modeling approach allows both computer simulations and rigorous analytical treatments, which lead to the following results and conclusions. When DGF is absent, concordant and discordant P2 patterns occur depending on initial conditions with the discordant P2 patterns being spatially random. When the DGF is negative, only concordant P2 patterns exist. When the DGF is positive, both concordant and discordant P2 patterns can occur. The discordant P2 patterns are still spatially random, but they satisfy that the global signal exhibits a temporal period-1 behavior. The theoretical analyses of the coupled map lattice model reveal the underlying instabilities and bifurcations for the genesis, selection, and stability of spatiotemporal excitation patterns.

Understanding the mechanisms of pattern formation in biological systems is of great importance. Here we investigate the dynamical mechanisms by which delayed global feedback affects excitation pattern formation and stability in periodically-paced biological excitable media, such as cardiac or neural cells and tissue. We focus on the formation and stability of the temporal period-2 and spatially in-phase and out-of-phase excitation patterns. Using models of different levels of complexity, we show that when the delayed global feedback is negative, only the spatially in-phase patterns are stable. When the feedback is positive, both spatially in-phase and out-of-phase patterns are stable, and the out-of-phase patterns are spatially random but satisfy that the global signals are temporal period-1 solutions.

Gamma-band correlations in the primary visual cortex

This paper generalizes and extends previous work on using neural field theory to quantitatively analyze the two-dimensional (2D) spatiotemporal correlation properties of gamma-band (30-70 Hz) oscillations evoked by stimuli arriving at the primary visual cortex, and modulated by patchy connectivities that depend on orientation preference (OP). Correlation functions are derived analytically for general stimulus and measurement conditions. The theoretical results reproduce a range of published experimental results. These include (i) the existence of two-point oscillatory temporal cross correlations with zero time lag between neurons with similar OP; (ii) the influence of spatial separation of neurons on the strength of the correlations; and (iii) the effects of differing stimulus orientations. They go beyond prior work by incorporating experimentally observed patchy projection patterns to predict the 2D correlation structure including both OP and ocular dominance effects, thereby relaxing assumptions of translational invariance implicit in prior one-dimensional analysis.

NFTsim: Theory and Simulation of Multiscale Neural Field Dynamics

A user ready, portable, documented software package, NFTsim, is presented to facilitate numerical simulations of a wide range of brain systems using continuum neural field modeling. NFTsim enables users to simulate key aspects of brain activity at multiple scales. At the microscopic scale, it incorporates characteristics of local interactions between cells, neurotransmitter effects, synaptodendritic delays and feedbacks. At the mesoscopic scale, it incorporates information about medium to large scale axonal ranges of fibers, which are essential to model dissipative wave transmission and to produce synchronous oscillations and associated cross-correlation patterns as observed in local field potential recordings of active tissue. At the scale of the whole brain, NFTsim allows for the inclusion of long range pathways, such as thalamocortical projections, when generating macroscopic activity fields. The multiscale nature of the neural activity produced by NFTsim has the potential to enable the modeling of resulting quantities measurable via various neuroimaging techniques. In this work, we give a comprehensive description of the design and implementation of the software. Due to its modularity and flexibility, NFTsim enables the systematic study of an unlimited number of neural systems with multiple neural populations under a unified framework and allows for direct comparison with analytic and experimental predictions. The code is written in C++ and bundled with Matlab routines for a rapid quantitative analysis and visualization of the outputs. The output of NFTsim is stored in plain text file enabling users to select from a broad range of tools for offline analysis. This software enables a wide and convenient use of powerful physiologically-based neural field approaches to brain modeling. NFTsim is distributed under the Apache 2.0 license.

Classification of Spatiotemporal Neural Activity Patterns in Brain Imaging Data

Various patterns of neural activity are observed in dynamic cortical imaging data. Such patterns may reflect how neurons communicate using the underlying circuitry to perform appropriate functions; thus it is crucial to investigate the spatiotemporal characteristics of the observed neural activity patterns. In general, however, neural activities are highly nonlinear and complex, so it is a demanding job to analyze them quantitatively or to classify the patterns of observed activities in various types of imaging data. Here, we present our implementation of a novel method that successfully addresses the above issues for precise comparison and classification of neural activity patterns. Based on two-dimensional representations of the geometric structure and temporal evolution of activity patterns, our method successfully classified a number of computer-generated sample patterns created from combinations of various spatial and temporal patterns. In addition, we validated our method with voltage-sensitive dye imaging data of Alzheimer's disease (AD) model mice. Our analysis algorithm successfully distinguished the activity data of AD mice from that of wild type with significantly higher performance than previously suggested methods. Our result provides a pragmatic solution for precise analysis of spatiotemporal patterns of neural imaging data.

'Blue' voltage-sensitive dyes for studying spatiotemporal dynamics in the brain: visualizing cortical waves

Among many distinct contributions made by Amiram Grinvald's group, the "Blue dyes" is a special gift for visualizing cortical population neuronal activity. The excitation wavelength of blue dyes has minimal overlap with the absorption of hemoglobin, and hence has minimal pulsation artifacts. This advantage leads to high signal-to-noise ratio optical recordings of cortical activity, with sensitivity as good as that of local field potential recordings. High sensitivity imaging allows for recording of spontaneous and evoked activity in single trials without spatial or temporal averaging, and has benefitted many scientists in their research projects. Single trial recording is particularly important for studying the cortex, because spontaneous and ongoing activities interact with sensory evoked events, creating rich dynamics in the wave patterns. Signal averaging in space and time would diminish the dynamic components in the patterns. Here, we discuss how the blue dyes help to achieve high-sensitivity voltage-sensitive dye imaging of spontaneous and evoked cortical activities. Spontaneous cortical activity has a constantly changing spatial pattern and temporal frequency, making it impossible to average in space and time. Amiran Grinvald's invention of blue dyes made it possible to examine the spatiotemporal patterns of cortical dynamics, about 15 years before the first useful genetically coded voltage proteins became available.

A spatially resolved network spike in model neuronal cultures reveals nucleation centers, circular traveling waves and drifting spiral waves

We show that in model neuronal cultures, where the probability of interneuronal connection formation decreases exponentially with increasing distance between the neurons, there exists a small number of spatial nucleation centers of a network spike, from where the synchronous spiking activity starts propagating in the network typically in the form of circular traveling waves. The number of nucleation centers and their spatial locations are unique and unchanged for a given realization of neuronal network but are different for different networks. In contrast, if the probability of interneuronal connection formation is independent of the distance between neurons, then the nucleation centers do not arise and the synchronization of spiking activity during a network spike occurs spatially uniform throughout the network. Therefore one can conclude that spatial proximity of connections between neurons is important for the formation of nucleation centers. It is also shown that fluctuations of the spatial density of neurons at their random homogeneous distribution typical for the experiments in vitro do not determine the locations of the nucleation centers. The simulation results are qualitatively consistent with the experimental observations.

Impact of bounded noise on the formation and instability of spiral wave in a 2D Lattice of neurons

Spiral waves in the neocortex may provide a spatial framework to organize cortical oscillations, thus help signal communication. However, noise influences spiral wave. Many previous theoretical studies about noise mainly focus on unbounded Gaussian noise, which contradicts that a real physical quantity is always bounded. Furthermore, non-Gaussian noise is also important for dynamical behaviors of excitable media. Nevertheless, there are no results concerning the effect of bounded noise on spiral wave till now. Based on Hodgkin-Huxley neuron model subjected to bounded noise with the form of Asin[ωt + σW(t)], the influences of bounded noise on the formation and instability of spiral wave in a two-dimensional (2D) square lattice of neurons are investigated in detail by separately adjusting the intensity σ, amplitude A, and frequency f of bounded noise. It is found that the increased intensity σ can facilitate the formation of spiral wave while the increased amplitude A tends to destroy spiral wave. Furthermore, frequency of bounded noise has the effect of facilitation or inhibition on pattern synchronization. Interestingly, for the appropriate intensity, amplitude and frequency can separately induce resonance-like phenomenon.

Separating Putative Pathogens from Background Contamination with Principal Orthogonal Decomposition: Evidence for Leptospira in the Ugandan Neonatal Septisome

Neonatal sepsis (NS) is responsible for over 1 million yearly deaths worldwide. In the developing world, NS is often treated without an identified microbial pathogen. Amplicon sequencing of the bacterial 16S rRNA gene can be used to identify organisms that are difficult to detect by routine microbiological methods. However, contaminating bacteria are ubiquitous in both hospital settings and research reagents and must be accounted for to make effective use of these data. In this study, we sequenced the bacterial 16S rRNA gene obtained from blood and cerebrospinal fluid (CSF) of 80 neonates presenting with NS to the Mbarara Regional Hospital in Uganda. Assuming that patterns of background contamination would be independent of pathogenic microorganism DNA, we applied a novel quantitative approach using principal orthogonal decomposition to separate background contamination from potential pathogens in sequencing data. We designed our quantitative approach contrasting blood, CSF, and control specimens and employed a variety of statistical random matrix bootstrap hypotheses to estimate statistical significance. These analyses demonstrate that Leptospira appears present in some infants presenting within 48 h of birth, indicative of infection in utero, and up to 28 days of age, suggesting environmental exposure. This organism cannot be cultured in routine bacteriological settings and is enzootic in the cattle that often live in close proximity to the rural peoples of western Uganda. Our findings demonstrate that statistical approaches to remove background organisms common in 16S sequence data can reveal putative pathogens in small volume biological samples from newborns. This computational analysis thus reveals an important medical finding that has the potential to alter therapy and prevention efforts in a critically ill population.

Pattern Selection in Network of Coupled Multi-Scroll Attractors

Multi-scroll chaotic attractor makes the oscillator become more complex in dynamic behaviors. The collective behaviors of coupled oscillators with multi-scroll attractors are investigated in the regular network in two-dimensional array, which the local kinetics is described by an improved Chua circuit. A feasible scheme of negative feedback with diversity is imposed on the network to stabilize the spatial patterns. Firstly, the Chua circuit is improved by replacing the nonlinear term with Sine function to generate infinite aquariums so that multi-scroll chaotic attractors could be generated under appropriate parameters, which could be detected by calculating the Lyapunov exponent in the parameter region. Furthermore, negative feedback with different gains (D1, D2) is imposed on the local square center area A2 and outer area A1 of the network, it is found that spiral wave, target wave could be developed in the network under appropriate feedback gain with diversity and size of controlled area. Particularly, homogeneous state could be reached after synchronization by selecting appropriate feedback gain and controlled size in the network. Finally, the distribution for statistical factors of synchronization is calculated in the two-parameter space to understand the transition of pattern region. It is found that developed spiral waves, target waves often are associated with smaller factor of synchronization. These results show that emergence of sustained spiral wave and continuous target wave could be effective for further suppression of spatiotemporal chaos in network by generating stable pacemaker completely.

A biologically constrained, mathematical model of cortical wave propagation preceding seizure termination

Epilepsy--the condition of recurrent, unprovoked seizures--manifests in brain voltage activity with characteristic spatiotemporal patterns. These patterns include stereotyped semi-rhythmic activity produced by aggregate neuronal populations, and organized spatiotemporal phenomena, including waves. To assess these spatiotemporal patterns, we develop a mathematical model consistent with the observed neuronal population activity and determine analytically the parameter configurations that support traveling wave solutions. We then utilize high-density local field potential data recorded in vivo from human cortex preceding seizure termination from three patients to constrain the model parameters, and propose basic mechanisms that contribute to the observed traveling waves. We conclude that a relatively simple and abstract mathematical model consisting of localized interactions between excitatory cells with slow adaptation captures the quantitative features of wave propagation observed in the human local field potential preceding seizure termination.

Autapse-induced spiral wave in network of neurons under noise

Autapse plays an important role in regulating the electric activity of neuron by feedbacking time-delayed current on the membrane of neuron. Autapses are considered in a local area of regular network of neurons to investigate the development of spatiotemporal pattern, and emergence of spiral wave is observed while it fails to grow up and occupy the network completely. It is found that spiral wave can be induced to occupy more area in the network under optimized noise on the network with periodical or no-flux boundary condition being used. The developed spiral wave with self-sustained property can regulate the collective behaviors of neurons as a pacemaker. To detect the collective behaviors, a statistical factor of synchronization is calculated to investigate the emergence of ordered state in the network. The network keeps ordered state when self-sustained spiral wave is formed under noise and autapse in local area of network, and it independent of the selection of periodical or no-flux boundary condition. The developed stable spiral wave could be helpful for memory due to the distinct self-sustained property.

A combined method to estimate parameters of the thalamocortical model from a heavily noise-corrupted time series of action potential

A combined method composing of the unscented Kalman filter (UKF) and the synchronization-based method is proposed for estimating electrophysiological variables and parameters of a thalamocortical (TC) neuron model, which is commonly used for studying Parkinson's disease for its relay role of connecting the basal ganglia and the cortex. In this work, we take into account the condition when only the time series of action potential with heavy noise are available. Numerical results demonstrate that not only this method can estimate model parameters from the extracted time series of action potential successfully but also the effect of its estimation is much better than the only use of the UKF or synchronization-based method, with a higher accuracy and a better robustness against noise, especially under the severe noise conditions. Considering the rather important role of TC neuron in the normal and pathological brain functions, the exploration of the method to estimate the critical parameters could have important implications for the study of its nonlinear dynamics and further treatment of Parkinson's disease.

Critical avalanches and subsampling in map-based neural networks coupled with noisy synapses

Many different kinds of noise are experimentally observed in the brain. Among them, we study a model of noisy chemical synapse and obtain critical avalanches for the spatiotemporal activity of the neural network. Neurons and synapses are modeled by dynamical maps. We discuss the relevant neuronal and synaptic properties to achieve the critical state. We verify that networks of functionally excitable neurons with fast synapses present power-law avalanches, due to rebound spiking dynamics. We also discuss the measuring of neuronal avalanches by subsampling our data, shedding light on the experimental search for self-organized criticality in neural networks.

Selection of multiarmed spiral waves in a regular network of neurons

Formation and selection of multiarmed spiral wave due to spontaneous symmetry breaking are investigated in a regular network of Hodgkin-Huxley neuron by changing the excitability and imposing spatial forcing currents on the neurons in the network. The arm number of the multiarmed spiral wave is dependent on the distribution of spatial forcing currents and excitability diversity in the network, and the selection criterion for supporting multiarmed spiral waves is discussed. A broken spiral segment is measured by a short polygonal line connected by three adjacent points (controlled nodes), and a double-spiral wave can be developed from the spiral segment. Multiarmed spiral wave is formed when a group of double-spiral waves rotate in the same direction in the network. In the numerical studies, a group of controlled nodes are selected and spatial forcing currents are imposed on these nodes, and our results show that l-arm stable spiral wave (l = 2, 3, 4,...8) can be induced to occupy the network completely. It is also confirmed that low excitability is critical to induce multiarmed spiral waves while high excitability is important to propagate the multiarmed spiral wave outside so that distinct multiarmed spiral wave can occupy the network completely. Our results confirm that symmetry breaking of target wave in the media accounts for emergence of multiarmed spiral wave, which can be developed from a group of spiral waves with single arm under appropriate condition, thus the potential formation mechanism of multiarmed spiral wave in the media is explained.

Differential effects of cholinergic and noradrenergic neuromodulation on spontaneous cortical network dynamics

Cholinergic and noradrenergic neuromodulation play a key role in determining overall behavioral state by shaping the underlying cortical network dynamics. The effects of these systems on synaptic and intrinsic cellular targets are quite diverse and a comprehensive understanding of how these neuromodulators regulate (spontaneous) cortical network activity has remained elusive. Here, we used multielectrode electrophysiology in vitro to investigate the effect of these neuromodulators on spontaneous network dynamics in acute slices of mouse visual cortex. We found that application of Carbachol (CCh) and Norepinephrine (NE) both enhanced the spontaneous network dynamics by increasing (1) the activity levels, (2) the temporal complexity of the network activity, and (3) the spatial complexity by decorrelating the network activity over a wide range of neuromodulator concentrations (1 μM, 10 μM, 50 μM, and 100 μM). Interestingly, we found that cholinergic neuromodulation was limited to the presence of CCh in the bath whereas the effects of NE, in particular for higher concentrations, induced plasticity that caused outlasting effects most prominently in the deep cortical layers. Together, these results provide a comprehensive network-level understanding of the similarities and differences of cholinergic and noradrenergic modulation of spontaneous network dynamics.

The formation mechanism of defects, spiral wave in the network of neurons

A regular network of neurons is constructed by using the Morris-Lecar (ML) neuron with the ion channels being considered, and the potential mechnism of the formation of a spiral wave is investigated in detail. Several spiral waves are initiated by blocking the target wave with artificial defects and/or partial blocking (poisoning) in ion channels. Furthermore, possible conditions for spiral wave formation and the effect of partial channel blocking are discussed completely. Our results are summarized as follows. 1) The emergence of a target wave depends on the transmembrane currents with diversity, which mapped from the external forcing current and this kind of diversity is associated with spatial heterogeneity in the media. 2) Distinct spiral wave could be induced to occupy the network when the target wave is broken by partially blocking the ion channels of a fraction of neurons (local poisoned area), and these generated spiral waves are similar with the spiral waves induced by artificial defects. It is confirmed that partial channel blocking of some neurons in the network could play a similar role in breaking a target wave as do artificial defects; 3) Channel noise and additive Gaussian white noise are also considered, and it is confirmed that spiral waves are also induced in the network in the presence of noise. According to the results mentioned above, we conclude that appropriate poisoning in ion channels of neurons in the network acts as ‘defects’ on the evolution of the spatiotemporal pattern, and accounts for the emergence of a spiral wave in the network of neurons. These results could be helpful to understand the potential cause of the formation and development of spiral waves in the cortex of a neuronal system.

TRANSITION OF ORDERED WAVES IN NEURONAL NETWORK INDUCED BY DIFFUSIVE POISONING OF ION CHANNELS

Normal physiological activities are often affected by some drugs, and some ion channels are blocked due to the katogene of drugs. This paper investigates the propagation of ordered waves in neuronal networks induced by diffusive poisoning, where the process is measured by increasing the number of neurons in the poisoned area of the networks. A coefficient of poisoning K is defined to measure the time units from one poisoned site to the adjacent site, a smaller K means that more neurons are poisoned in a certain period (a higher poisoning speed). A statistical factor of synchronization R in the two-dimensional array is defined to detect the transition of spiral waves induced by ion channel blocking. It is confirmed that the evolution of the spiral waves depends on the coefficient of poisoning K and number of poisoned neurons. Furthermore, breakup of the spirals is observed when weak channel noise is considered. Finally, the formation of the spiral wave induced by blocking the target wave with line defects is briefly discussed.

The role of inhibition in oscillatory wave dynamics in the cortex

Cortical oscillations arise during behavioral and mental tasks, and all temporal oscillations have particular spatial patterns. Studying the mechanisms that generate and modulate the spatiotemporal characteristics of oscillations is important for understanding neural information processing and the signs and symptoms of dynamical diseases of the brain. Nevertheless, it remains unclear how GABAergic inhibition modulates these oscillation dynamics. Using voltage-sensitive dye imaging, pharmacological methods, and tangentially cut occipital neocortical brain slices (including layers 3-5) of Sprague-Dawley rat, we found that GABAa disinhibition with bicuculline can progressively simplify oscillation dynamics in the presence of carbachol in a concentration-dependent manner. Additionally, GABAb disinhibition can further simplify oscillation dynamics after GABAa receptors are blocked. Both GABAa and GABAb disinhibition increase the synchronization of the neural network. Theta frequency (5-15-Hz) oscillations are reliably generated by using a combination of GABAa and GABAb antagonists alone. These theta oscillations have basic spatiotemporal patterns similar to those generated by carbachol/bicuculline. These results are illustrative of how GABAergic inhibition increases the complexity of patterns of activity and contributes to the regulation of the cortex.

Neural field theory of plasticity in the cerebral cortex

A generalized timing-dependent plasticity rule is incorporated into a recent neural field theory to explore synaptic plasticity in the cerebral cortex, with both excitatory and inhibitory populations included. Analysis in the time and frequency domains reveals that cortical network behavior gives rise to a saddle-node bifurcation and resonant frequencies, including a gamma-band resonance. These system resonances constrain cortical synaptic dynamics and divide it into four classes, which depend on the type of synaptic plasticity window. Depending on the dynamical class, synaptic strengths can either have a stable fixed point, or can diverge in the absence of a separate saturation mechanism. Parameter exploration shows that time-asymmetric plasticity windows, which are signatures of spike-timing dependent plasticity, enable the richest variety of synaptic dynamics to occur. In particular, we predict a zone in parameter space which may allow brains to attain the marginal stability phenomena observed experimentally, although additional regulatory mechanisms may be required to maintain these parameters.

Negative curvature boundaries as wave emitting sites for the control of biological excitable media

Understanding the interaction of electric fields with the complex anatomy of biological excitable media is key to optimizing control strategies for spatiotemporal dynamics in those systems. On the basis of a bidomain description, we provide a unified theory for the electric-field-induced depolarization of the substrate near curved boundaries of generalized shapes, resulting in the localized recruitment of control sites. Our findings are confirmed in experiments on cardiomyocyte cell cultures and supported by two-dimensional numerical simulations on a cross section of a rabbit ventricle.

Rainfall drives hydrocephalus in East Africa

OBJECT

Hydrocephalus is one of the most common brain disorders in children throughout the world. The majority of infant hydrocephalus cases in East Africa appear to be postinfectious, related to preceding neonatal infections, and are thus preventable if the microbial origins and routes of infection can be characterized. In prior microbiological work, the authors noted evidence of seasonality in postinfectious hydrocephalus (PIH) cases.

METHODS

The geographical address of 696 consecutive children with PIH who were treated over 6 years was fused with satellite rainfall data for the same time period. A comprehensive time series and spatiotemporal analysis of cases and rainfall was performed.

RESULTS

Four infection-onset peaks were found to straddle the twice-yearly rainy season peaks, demonstrating that the infections occurred at intermediate levels of rainfall.

CONCLUSIONS

The findings in this study reveal a previously unknown link between climate and a neurosurgical condition. Satellite-derived rainfall dynamics are an important factor in driving the infections that lead to PIH. Given prior microbial analysis, these findings point to the importance of environmental factors with respect to preventing the newborn infections that lead to PIH.

Neural field theory with variance dynamics

Previous neural field models have mostly been concerned with prediction of mean neural activity and with second order quantities such as its variance, but without feedback of second order quantities on the dynamics. Here the effects of feedback of the variance on the steady states and adiabatic dynamics of neural systems are calculated using linear neural field theory to estimate the neural voltage variance, then including this quantity in the total variance parameter of the nonlinear firing rate-voltage response function, and thus into determination of the fixed points and the variance itself. The general results further clarify the limits of validity of approaches with and without inclusion of variance dynamics. Specific applications show that stability against a saddle-node bifurcation is reduced in a purely cortical system, but can be either increased or decreased in the corticothalamic case, depending on the initial state. Estimates of critical variance scalings near saddle-node bifurcation are also found, including physiologically based normalizations and new scalings for mean firing rate and the position of the bifurcation.

Interactions between two propagating waves in rat visual cortex

Sensory-evoked propagating waves are frequently observed in sensory cortex. However, it is largely unknown how an evoked propagating wave affects the activity evoked by subsequent sensory inputs, or how two propagating waves interact when evoked by simultaneous sensory inputs. Using voltage-sensitive dye imaging, we investigated the interactions between two evoked waves in rat visual cortex, and the spatiotemporal patterns of depolarization in the neuronal population due to wave-to-wave interactions. We have found that visually-evoked propagating waves have a refractory period of about 300 ms, within which the response to a subsequent visual stimulus is suppressed. Simultaneous presentation of two visual stimuli at different locations can evoke two waves propagating toward each other, and these two waves fuse. Fusion significantly shortens the latency and half-width of the response, leading to changes in the spatial profile of evoked population activity. The visually-evoked propagating wave may also be suppressed by a preceding spontaneous wave. The refractory period following a propagating wave and the fusion between two waves may contribute to visual sensory processing by modifying the spatiotemporal profile of population neuronal activity evoked by sensory events.

COLLECTIVE BEHAVIORS OF SPIRAL WAVES IN THE NETWORKS OF HODGKIN-HUXLEY NEURONS IN PRESENCE OF CHANNEL NOISE

Collective behaviors of spiral waves in the networks of Hodgkin-Huxley neuron are investigated. A stable rotating spiral wave can be developed to occupy the quiescent areas in networks of neurons by selecting appropriate initial values for the variables in the networks of neurons. In our numerical studies, most neurons are quiescent and finite (few) numbers of neurons are selected with different values to form a spiral seed. In this way, neurons communicating are carried by propagating spiral wave to break through the quiescent domains (areas) in networks of neurons. The effect of membrane temperature on the formation of spiral wave is investigated by selecting different fixed membrane temperatures in the networks, and it is found that a spiral wave cannot be developed if the membrane temperature is close to a certain threshold. A quantitative factor of synchronization is defined to measure the statistical properties and collective behaviors of the spiral wave. And a distinct phase transition, which indicates the critical condition for spiral survival, is observed in the sudden changing point of the factors of synchronization curve vs. certain bifurcation parameter. Internal noise is introduced into ion channels (channel noise) with the Langevin method. It is found that a stable rotating spiral wave is developed and the spiral wave is robust to weak channel noise (the membrane patch is not small). The spiral wave can not grow up and the stable rotating spiral wave encounters instability in presence of strong channel noise. Coherence resonance-like behavior is observed in calculating the factors of synchronization in presence of channel noise.

Spiral wave dynamics in neocortex

Although spiral waves are ubiquitous features of nature and have been observed in many biological systems, their existence and potential function in mammalian cerebral cortex remain uncertain. Using voltage-sensitive dye imaging, we found that spiral waves occur frequently in the neocortex in vivo, both during pharmacologically induced oscillations and during sleep-like states. While their life span is limited, spiral waves can modify ongoing cortical activity by influencing oscillation frequencies and spatial coherence and by reducing amplitude in the area surrounding the spiral phase singularity. During sleep-like states, the rate of occurrence of spiral waves varies greatly depending on brain states. These results support the hypothesis that spiral waves, as an emergent activity pattern, can organize and modulate cortical population activity on the mesoscopic scale and may contribute to both normal cortical processing and to pathological patterns of activity such as those found in epilepsy.

Lattice Boltzmann study of pattern formation in reaction-diffusion systems

Pattern formation in reaction-diffusion systems is of great importance in surface micropatterning [Grzybowski et al., Soft Matter 1, 114 (2005)], self-organization of cellular micro-organisms [Schulz et al., Annu. Rev. Microbiol. 55, 105 (2001)], and in developmental biology [Barkai et al., FEBS Journal 276, 1196 (2009)]. In this work, we apply the lattice Boltzmann method to study pattern formation in reaction-diffusion systems. As a first methodological step, we consider the case of a single species undergoing transformation reaction and diffusion. In this case, we perform a third-order Chapman-Enskog multiscale expansion and study the dependence of the lattice Boltzmann truncation error on the diffusion coefficient and the reaction rate. These findings are in good agreement with numerical simulations. Furthermore, taking the Gray-Scott model as a prominent example, we provide evidence for the maturity of the lattice Boltzmann method in studying pattern formation in nonlinear reaction-diffusion systems. For this purpose, we perform linear stability analysis of the Gray-Scott model and determine the relevant parameter range for pattern formation. Lattice Boltzmann simulations allow us not only to test the validity of the linear stability phase diagram including Turing and Hopf instabilities, but also permit going beyond the linear stability regime, where large perturbations give rise to interesting dynamical behavior such as the so-called self-replicating spots. We also show that the length scale of the patterns may be tuned by rescaling all relevant diffusion coefficients in the system with the same factor while leaving all the reaction constants unchanged.

A model of feedback control for the charge-balanced suppression of epileptic seizures

Here we present several refinements to a model of feedback control for the suppression of epileptic seizures. We utilize a stochastic partial differential equation (SPDE) model of the human cortex. First, we verify the strong convergence of numerical solutions to this model, paying special attention to the sharp spatial changes that occur at electrode edges. This allows us to choose appropriate step sizes for our simulations; because the spatial step size must be small relative to the size of an electrode in order to resolve its electrical behavior, we are able to include a more detailed electrode profile in the simulation. Then, based on evidence that the mean soma potential is not the variable most closely related to the measurement of a cortical surface electrode, we develop a new model for this. The model is based on the currents flowing in the cortex and is used for a simulation of feedback control. The simulation utilizes a new control algorithm incorporating the total integral of the applied electrical potential. Not only does this succeed in suppressing the seizure-like oscillations, but it guarantees that the applied signal will be charge-balanced and therefore unlikely to cause cortical damage.

Neurophysiological changes with age probed by inverse modeling of EEG spectra

OBJECTIVE

To investigate age-associated changes in physiologically-based EEG spectral parameters in the healthy population.

METHODS

Eyes-closed EEG spectra of 1498 healthy subjects aged 6-86 years were fitted to a mean-field model of thalamocortical dynamics in a cross-sectional study. Parameters were synaptodendritic rates, cortical wave decay rates, connection strengths (gains), axonal delays for thalamocortical loops, and power normalizations. Age trends were approximated using smooth asymptotically linear functions with a single turning point. We also considered sex differences and relationships between model parameters and traditional quantitative EEG measures.

RESULTS

The cross-sectional data suggest that changes tend to be most rapid in childhood, generally leveling off at age 15-20 years. Most gains decrease in magnitude with age, as does power normalization. Axonal and dendritic delays decrease in childhood and then increase. Axonal delays and gains show small but significant sex differences.

CONCLUSIONS

Mean-field brain modeling allows interpretation of age-associated EEG trends in terms of physiological processes, including the growth and regression of white matter, influencing axonal delays, and the establishment and pruning of synaptic connections, influencing gains.

SIGNIFICANCE

This study demonstrates the feasibility of inverse modeling of EEG spectra as a noninvasive method for investigating large-scale corticothalamic dynamics, and provides a basis for future comparisons.

Data assimilation for heterogeneous networks: the consensus set

Data assimilation in dynamical networks is intrinsically challenging. A method is introduced for the tracking of heterogeneous networks of oscillators or excitable cells in a nonstationary environment, using a homogeneous model network to expedite the accurate reconstruction of parameters and unobserved variables. An implementation using ensemble Kalman filtering to track the states of the heterogeneous network is demonstrated on simulated data and applied to a mammalian brain network experiment. The approach has broad applicability for the prediction and control of biological and physical networks.

Propagating waves of activity in the neocortex: what they are, what they do

The development of voltage-sensitive dyes (VSD) and fast optical imaging techniques have brought us a new tool for examining spatiotemporal patterns of population neuronal activity in the neocortex. Propagating waves have been observed during almost every type of cortical processing examined by VSD imaging or electrode arrays. These waves provide subthreshold depolarization to individual neurons and increase their spiking probability. Therefore, the propagation of the waves sets up a spatiotemporal framework for increased excitability in neuronal populations, which can help to determine when and where the neurons are likely to fire. In this review, first discussed is propagating waves observed in various systems and possible mechanisms for generating and sustaining these waves. Then discussed are wave dynamics as an emergent behavior of the population activity that can, in turn, influence the activity of individual neurons. The functions of spontaneous and sensory-evoked waves remain to be explored. An important next step will be to examine the interaction between dynamics of propagating waves and functions in the cortex, and to verify if cortical processing can be modified when these waves are altered.

Mean-field modeling of the basal ganglia-thalamocortical system. I Firing rates in healthy and parkinsonian states

Parkinsonism leads to various electrophysiological changes in the basal ganglia-thalamocortical system (BGTCS), often including elevated discharge rates of the subthalamic nucleus (STN) and the output nuclei, and reduced activity of the globus pallidus external (GPe) segment. These rate changes have been explained qualitatively in terms of the direct/indirect pathway model, involving projections of distinct striatal populations to the output nuclei and GPe. Although these populations partly overlap, evidence suggests dopamine depletion differentially affects cortico-striato-pallidal connection strengths to the two pallidal segments. Dopamine loss may also decrease the striatal signal-to-noise ratio, reducing both corticostriatal coupling and striatal firing thresholds. Additionally, nigrostriatal degeneration may cause secondary changes including weakened lateral inhibition in the GPe, and mesocortical dopamine loss may decrease intracortical excitation and especially inhibition. Here a mean-field model of the BGTCS is presented with structure and parameter estimates closely based on physiology and anatomy. Changes in model rates due to the possible effects of dopamine loss listed above are compared with experiment. Our results suggest that a stronger indirect pathway, possibly combined with a weakened direct pathway, is compatible with empirical evidence. However, altered corticostriatal connection strengths are probably not solely responsible for substantially increased STN activity often found. A lower STN firing threshold, weaker intracortical inhibition, and stronger striato-GPe inhibition help explain the relatively large increase in STN rate. Reduced GPe-GPe inhibition and a lower GPe firing threshold can account for the comparatively small decrease in GPe rate frequently observed. Changes in cortex, GPe, and STN help normalize the cortical rate, also in accord with experiments. The model integrates the basal ganglia into a unified framework along with an existing thalamocortical model that already accounts for a wide range of electrophysiological phenomena. A companion paper discusses the dynamics and oscillations of this combined system.

Mean-field modeling of the basal ganglia-thalamocortical system. II Dynamics of parkinsonian oscillations

Neuronal correlates of Parkinson's disease (PD) include a shift to lower frequencies in the electroencephalogram (EEG) and enhanced synchronized oscillations at 3-7 and 7-30 Hz in the basal ganglia, thalamus, and cortex. This study describes the dynamics of a recent physiologically based mean-field model of the basal ganglia-thalamocortical system, and shows how it accounts for many key electrophysiological correlates of PD. Its detailed functional connectivity comprises partially segregated direct and indirect pathways through two populations of striatal neurons, a hyperdirect pathway involving a corticosubthalamic projection, thalamostriatal feedback, and local inhibition in striatum and external pallidum (GPe). In a companion paper, realistic steady-state firing rates were obtained for the healthy state, and after dopamine loss modeled by weaker direct and stronger indirect pathways, reduced intrapallidal inhibition, lower firing thresholds of the GPe and subthalamic nucleus (STN), a stronger projection from striatum to GPe, and weaker cortical interactions. Here it is shown that oscillations around 5 and 20 Hz can arise with a strong indirect pathway, which also causes increased synchronization throughout the basal ganglia. Furthermore, increased theta power with progressive nigrostriatal degeneration is correlated with reduced alpha power and peak frequency, in agreement with empirical results. Unlike the hyperdirect pathway, the indirect pathway sustains oscillations with phase relationships that coincide with those found experimentally. Alterations in the responses of basal ganglia to transient stimuli accord with experimental observations. Reduced cortical gains due to both nigrostriatal and mesocortical dopamine loss lead to slower changes in cortical activity and may be related to bradykinesia. Finally, increased EEG power found in some studies may be partly explained by a lower effective GPe firing threshold, reduced GPe-GPe inhibition, and/or weaker intracortical connections in parkinsonian patients. Strict separation of the direct and indirect pathways is not necessary to obtain these results.

Neuronal synchrony: peculiarity and generality

Synchronization in neuronal systems is a new and intriguing application of dynamical systems theory. Why are neuronal systems different as a subject for synchronization? (1) Neurons in themselves are multidimensional nonlinear systems that are able to exhibit a wide variety of different activity patterns. Their "dynamical repertoire" includes regular or chaotic spiking, regular or chaotic bursting, multistability, and complex transient regimes. (2) Usually, neuronal oscillations are the result of the cooperative activity of many synaptically connected neurons (a neuronal circuit). Thus, it is necessary to consider synchronization between different neuronal circuits as well. (3) The synapses that implement the coupling between neurons are also dynamical elements and their intrinsic dynamics influences the process of synchronization or entrainment significantly. In this review we will focus on four new problems: (i) the synchronization in minimal neuronal networks with plastic synapses (synchronization with activity dependent coupling), (ii) synchronization of bursts that are generated by a group of nonsymmetrically coupled inhibitory neurons (heteroclinic synchronization), (iii) the coordination of activities of two coupled neuronal networks (partial synchronization of small composite structures), and (iv) coarse grained synchronization in larger systems (synchronization on a mesoscopic scale).

On how network architecture determines the dominant patterns of spontaneous neural activity

In the absence of sensory stimulation, neocortical circuits display complex patterns of neural activity. These patterns are thought to reflect relevant properties of the network, including anatomical features like its modularity. It is also assumed that the synaptic connections of the network constrain the repertoire of emergent, spontaneous patterns. Although the link between network architecture and network activity has been extensively investigated in the last few years from different perspectives, our understanding of the relationship between the network connectivity and the structure of its spontaneous activity is still incomplete. Using a general mathematical model of neural dynamics we have studied the link between spontaneous activity and the underlying network architecture. In particular, here we show mathematically how the synaptic connections between neurons determine the repertoire of spatial patterns displayed in the spontaneous activity. To test our theoretical result, we have also used the model to simulate spontaneous activity of a neural network, whose architecture is inspired by the patchy organization of horizontal connections between cortical columns in the neocortex of primates and other mammals. The dominant spatial patterns of the spontaneous activity, calculated as its principal components, coincide remarkably well with those patterns predicted from the network connectivity using our theory. The equivalence between the concept of dominant pattern and the concept of attractor of the network dynamics is also demonstrated. This in turn suggests new ways of investigating encoding and storage capabilities of neural networks.

The source of afterdischarge activity in neocortical tonic-clonic epilepsy

Tonic-clonic seizures represent a common pattern of epileptic discharges, yet the relationship between the various phases of the seizure remains obscure. Here we contrast propagation of the ictal wavefront with the propagation of individual discharges in the clonic phase of the event. In an in vitro model of tonic-clonic epilepsy, the afterdischarges (clonic phase) propagate with relative uniform speed and are independent of the speed of the ictal wavefront (tonic phase). For slowly propagating ictal wavefronts, the source of the afterdischarges, relative to a given recording electrode, switched as the wavefront passed by, indicating that afterdischarges are seeded from wavefront itself. In tissue that has experienced repeated ictal events, the wavefront generalizes rapidly, and the afterdischarges in this case show a different "flip-flop" pattern, with frequent switches in their direction of propagation. This same flip-flop pattern is also seen in subdural EEG recordings in patients suffering intractable focal seizures caused by cortical dysplasias. Thus, in both slowly and rapidly generalizing ictal events, there is not a single source of afterdischarge activity: rather, the source is continuously changing. Our data suggest a complex view of seizures in which the ictal event and its constituent discharges originate from distinct locations.

Kalman filter control of a model of spatiotemporal cortical dynamics

Recent advances in Kalman filtering to estimate system state and parameters in nonlinear systems have offered the potential to apply such approaches to spatiotemporal nonlinear systems. We here adapt the nonlinear method of unscented Kalman filtering to observe the state and estimate parameters in a computational spatiotemporal excitable system that serves as a model for cerebral cortex. We demonstrate the ability to track spiral wave dynamics, and to use an observer system to calculate control signals delivered through applied electrical fields. We demonstrate how this strategy can control the frequency of such a system, or quench the wave patterns, while minimizing the energy required for such results. These findings are readily testable in experimental applications, and have the potential to be applied to the treatment of human disease.