Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction

Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh–Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.     

Reaction-diffusion waves of actin filament polymerization/depolymerization in Dictyostelium pseudopodium extension and cell locomotion

Cell surface movements and the intracellular spatial patterns and dynamics of actin filament (F-actin) were investigated in living and formalin-fixed cells of Dictyostelium discoideum by confocal microscopy. Excitation waves of F-actin assembly developed and propagated several micrometers at up to 26 microm/min in cells which had been intracellularly loaded with fluorescently labeled actin monomer. Wave propagation and extinction corresponded with the initiation and attenuation of pseudopodium extension and cell advance, respectively. The identification of chemical waves was supported by the ring, sphere, spiral and scroll wave patterns, which were observed in the extensions of fixed cells stained with phalloidin-rhodamine, and by the similar, asymmetrical [F-actin] distribution in wavefronts in living and fixed cells. These F-actin patterns and dynamics in Dictyostelium provide evidence for a new supramolecular state of actin, which propagates as a self-organized, reaction-diffusion wave of reversible F-actin assembly and affects pseudopodium extension. Actin's properties of oscillation and self-organization might also fundamentally determine the nature of the eukaryotic cell's reactions of adaptation, timing and signal response.     

Predicting the Distribution of Spiral Waves from Cell Properties in a Developmental-Path Model of Dictyostelium Pattern Formation

The slime mold Dictyostelium discoideum is one of the model systems of biological pattern formation. One of the most successful answers to the challenge of establishing a spiral wave pattern in a colony of homogeneously distributed D. discoideum cells has been the suggestion of a developmental path the cells follow (Lauzeral and coworkers). This is a well-defined change in properties each cell undergoes on a longer time scale than the typical dynamics of the cell. Here we show that this concept leads to an inhomogeneous and systematic spatial distribution of spiral waves, which can be predicted from the distribution of cells on the developmental path. We propose specific experiments for checking whether such systematics are also found in data and thus, indirectly, provide evidence of a developmental path.     

A field-theoretic approach to understanding scale-free neocortical dynamics

A mesoscopic field-theoretic approach is compared with neural network and brain imaging approaches to understanding brain dynamics. Analysis of high spatiotemporal resolution rabbit electroencephalogram (EEG) reveals neural fields in the form of spatial patterns in amplitude (AM) and phase (PM) modulation of gamma and beta carrier waves that serve to classify EEGs from trials with differing conditioned stimuli (CS+/−). Paleocortex exemplified by olfactory EEG has one AM–PM pattern at a time that forms by an input-dependent phase transition. Neocortex shows multiple overlapping AM–PM patterns before and during presentation of CSs. Modeling suggests that neocortex is stabilized in a scale-free state of self-organized criticality, enabling cooperative domains to form virtually instantaneously by phase transitions ranging in size from a few hypercolumns to an entire hemisphere. Self-organized local domains precede formation of global domains that supervene and contribute global modulations to local domains. This mechanism is proposed to explain Gestalt formation in perception.     

Spatiotemporal chemical dynamics in living cells: from information trafficking to cell physiology

Biological thought in the 20th century was dominated by the study of structures at increasingly minute levels. For biology to advance beyond structural reductionism and contribute its full measure to clinical care, living biological structures must be understood in the context of their collective chemical processes at the relevant chemical time-scales. Using high-speed fluorescence microscopy, we have studied intra- and inter-cellular signaling using shutter speeds ( approximately 100 ns) that remove the effects of wave motion and diffusion from optical images. By collecting a series of such images, stop-action movies of signal trafficking in living cells are created; these have revealed a new level of spatiotemporal chemical organization within cells. Numerous types of chemical waves have been found in living cells expressing a great variety of physical properties. In this article I will review some of these basic findings, discuss these events in the context of information trafficking, and illustrate the potential implications of this work in medicine.     

Mechanisms of intrinsic beating variability in cardiac cell cultures and model pacemaker networks

Heart rate variability (HRV) exhibits fluctuations characterized by a power law behavior of its power spectrum. The interpretation of this nonlinear HRV behavior, resulting from interactions between extracardiac regulatory mechanisms, could be clinically useful. However, the involvement of intrinsic variations of pacemaker rate in HRV has scarcely been investigated. We examined beating variability in spontaneously active incubating cultures of neonatal rat ventricular myocytes using microelectrode arrays. In networks of mathematical model pacemaker cells, we evaluated the variability induced by the stochastic gating of transmembrane currents and of calcium release channels and by the dynamic turnover of ion channels. In the cultures, spontaneous activity originated from a mobile focus. Both the beat-to-beat movement of the focus and beat rate variability exhibited a power law behavior. In the model networks, stochastic fluctuations in transmembrane currents and stochastic gating of calcium release channels did not reproduce the spatiotemporal patterns observed in vitro. In contrast, long-term correlations produced by the turnover of ion channels induced variability patterns with a power law behavior similar to those observed experimentally. Therefore, phenomena leading to long-term correlated variations in pacemaker cellular function may, in conjunction with extracardiac regulatory mechanisms, contribute to the nonlinear characteristics of HRV.     

Multiscale modeling of tumor growth induced by circadian rhythm disruption in epithelial tissue

We propose a multiscale chemo-mechanical model of cancer tumor development in epithelial tissue. The model is based on the transformation of normal cells into a cancerous state triggered by a local failure of spatial synchronization of the circadian rhythm. The model includes mechanical interactions and a chemical signal exchange between neighboring cells, as well as a division of cells and intercalation that allows for modification of the respective parameters following transformation into the cancerous state. The numerical simulations reproduce different dephasing patterns—spiral waves and quasistationary clustering, with the latter being conducive to cancer formation. Modification of mechanical properties reproduces a distinct behavior of invasive and localized carcinoma.     

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.     

Reconstruction of cellular variability from spatiotemporal patterns of Dictyostelium discoideum

Variability in cell properties can be an important driving mechanism behind spatiotemporal patterns in biological systems, as the degree of cell-to-cell differences determines the capacity of cells to locally synchronize and, consequently, form patterns on a larger spatial scale. In principle, certain features of spatial patterns emerging with time may be regulated by variability or, more specifically, by certain constellations of cell-to-cell differences. Similarly, measuring variability in a system (i.e. the spatial distribution of cell-cell differences) may help predict properties of later-stage patterns.     

Radial and spiral stream formation in Proteus mirabilis colonies

The enteric bacterium Proteus mirabilis, which is a pathogen that forms biofilms in vivo, can swarm over hard surfaces and form a variety of spatial patterns in colonies. Colony formation involves two distinct cell types: swarmer cells that dominate near the surface and the leading edge, and swimmer cells that prefer a less viscous medium, but the mechanisms underlying pattern formation are not understood. New experimental investigations reported here show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to previously-reported concentric rings. These new observations suggest that swimmers are motile and that indirect interactions between them are essential in the pattern formation. To explain these observations we develop a hybrid model comprising cell-based and continuum components that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams results from a swimming bias of the cells near the surface of the substrate. The spatial patterns generated from the model are in qualitative agreement with the experimental observations.     

Mechanobiology in cortical waves and oscillations

Cortical actin waves have emerged as a widely prevalent phenomena and brought pattern formation to many fields of cell biology. Cortical excitabilities, reminiscent of the electric excitability in neurons, are likely fundamental property of the cell cortex. Although they have been mostly considered to be biochemical in nature, accumulating evidence support the role of mechanics in the pattern formation process. Both pattern formation and mechanobiology approach biological phenomena at the collective level, either by looking at the mesoscale dynamical behavior of molecular networks or by using collective physical properties to characterize biological systems. As such they are very different from the traditional reductionist, bottom-up view of biology, which brings new challenges and potential opportunities. In this essay, we aim to provide our perspectives on what the proposed mechanochemical feedbacks are and open questions regarding their role in cortical excitable and oscillatory dynamics.     

The biological principles of swarm intelligence

The roots of swarm intelligence are deeply embedded in the biological study of self-organized behaviors in social insects. From the routing of traffic in telecommunication networks to the design of control algorithms for groups of autonomous robots, the collective behaviors of these animals have inspired many of the foundational works in this emerging research field. For the first issue of this journal dedicated to swarm intelligence, we review the main biological principles that underlie the organization of insects’ colonies. We begin with some reminders about the decentralized nature of such systems and we describe the underlying mechanisms of complex collective behaviors of social insects, from the concept of stigmergy to the theory of self-organization in biological systems. We emphasize in particular the role of interactions and the importance of bifurcations that appear in the collective output of the colony when some of the system’s parameters change. We then propose to categorize the collective behaviors displayed by insect colonies according to four functions that emerge at the level of the colony and that organize its global behavior. Finally, we address the role of modulations of individual behaviors by disturbances (either environmental or internal to the colony) in the overall flexibility of insect colonies. We conclude that future studies about self-organized biological behaviors should investigate such modulations to better understand how insect colonies adapt to uncertain worlds.     

Pattern formation of the coenobial algae Pediastrum biwae Negoro

It is suggested that flat colony patterns of the coenobial green algae Pediastrum biwae Negoro can be determined only by some properties of the zoospores without any other control system as a whole. These suggestions are made through observations of colony patterns and colony formation during asexual reproduction and digital electronic computer simulation. The zoospore can be regarded as a sphere which has two C-sites (presumptive sites for connection) and one H-site (presumptive site for horn formation) on its equator. This sphere swims rapidly at random in a transparent vesicle and undergoes a series of changes in its properties: (i) the regions along the equator might come to have a slight affinity for each other, which could cause the arrangement of the zoospores into a plane; (ii) the two C-sites might become connection sites and the zoospores, therefore, might form strings, leading to the characteristic pattern of the adult colony (e.g. a few concentric circular strings, a spiral string and so on); and finally, (iii) the H-site of the zoospore determines whether it grows to a horn or not, i.e. the zoospore grows into a horn cell or a triangular cell according to the absence or presence of other zoospores which prevent its growth.     

Microscopic spiral waves reveal positive feedback in subcellular calcium signaling.

The regenerative Ca(2+)-induced Ca2+ release mechanism is an important amplifier of signal transduction in diverse cells. In heart muscle cells, this mechanism contributes to the Ca2+ transient activating the mechanical contraction, but it is also believed to drive Ca2+ waves propagating within the cytosol. We investigated the subcellular Ca2+ distribution in heart muscle cells during spontaneous Ca2+ release using laser scanning confocal microscopy with a ratiometric fluorescent indicator technique. Besides planar Ca2+ waves with linear propagation, sequences of confocal optical sections also revealed spiral Ca2+ waves spinning around a subcellular core at approximately 1 Hz. Although the Ca2+ spirals were continuous processes they frequently exhibited an apparently oscillatory output function into the elongated cell body. These oscillatory waves emanating from the spiral at regular intervals were formally considered to be short outer segments of the spiral but could not be distinguished from planar Ca2+ waves propagating along the longitudinal cell axis. The complex spatiotemporal pattern of spiral Ca2+ waves implies the participation of an active process exhibiting a large degree of positive feedback, most likely the Ca(2+)-induced Ca2+ release mechanism.     

From Cell Differentiation to Cell Collectives: Bacillus subtilis Uses Division of Labor to Migrate

The organization of cells, emerging from cell–cell interactions, can give rise to collective properties. These properties are adaptive when together cells can face environmental challenges that they separately cannot. One particular challenge that is important for microorganisms is migration. In this study, we show how flagellum-independent migration is driven by the division of labor of two cell types that appear during Bacillus subtilis sliding motility. Cell collectives organize themselves into bundles (called “van Gogh bundles”) of tightly aligned cell chains that form filamentous loops at the colony edge. We show, by time-course microscopy, that these loops migrate by pushing themselves away from the colony. The formation of van Gogh bundles depends critically on the synergistic interaction of surfactin-producing and matrix-producing cells. We propose that surfactin-producing cells reduce the friction between cells and their substrate, thereby facilitating matrix-producing cells to form bundles. The folding properties of these bundles determine the rate of colony expansion. Our study illustrates how the simple organization of cells within a community can yield a strong ecological advantage. This is a key factor underlying the diverse origins of multicellularity.     

Self-organizing circuit assembly through spatiotemporally coordinated neuronal migration within geometric constraints

Background

Neurons are dynamically coupled with each other through neurite-mediated adhesion during development. Understanding the collective behavior of neurons in circuits is important for understanding neural development. While a number of genetic and activity-dependent factors regulating neuronal migration have been discovered on single cell level, systematic study of collective neuronal migration has been lacking. Various biological systems are shown to be self-organized, and it is not known if neural circuit assembly is self-organized. Besides, many of the molecular factors take effect through spatial patterns, and coupled biological systems exhibit emergent property in response to geometric constraints. How geometric constraints of the patterns regulate neuronal migration and circuit assembly of neurons within the patterns remains unexplored.

Methodology/Principal Findings

We established a two-dimensional model for studying collective neuronal migration of a circuit, with hippocampal neurons from embryonic rats on Matrigel-coated self-assembled monolayers (SAMs). When the neural circuit is subject to geometric constraints of a critical scale, we found that the collective behavior of neuronal migration is spatiotemporally coordinated. Neuronal somata that are evenly distributed upon adhesion tend to aggregate at the geometric center of the circuit, forming mono-clusters. Clustering formation is geometry-dependent, within a critical scale from 200 µm to approximately 500 µm. Finally, somata clustering is neuron-type specific, and glutamatergic and GABAergic neurons tend to aggregate homo-philically.

Conclusions/Significance

We demonstrate self-organization of neural circuits in response to geometric constraints through spatiotemporally coordinated neuronal migration, possibly via mechanical coupling. We found that such collective neuronal migration leads to somata clustering, and mono-cluster appears when the geometric constraints fall within a critical scale. The discovery of geometry-dependent collective neuronal migration and the formation of somata clustering in vitro shed light on neural development in vivo.     

Perspective: network-guided pattern formation of neural dynamics

The understanding of neural activity patterns is fundamentally linked to an understanding of how the brain''s network architecture shapes dynamical processes. Established approaches rely mostly on deviations of a given network from certain classes of random graphs. Hypotheses about the supposed role of prominent topological features (for instance, the roles of modularity, network motifs or hierarchical network organization) are derived from these deviations. An alternative strategy could be to study deviations of network architectures from regular graphs (rings and lattices) and consider the implications of such deviations for self-organized dynamic patterns on the network. Following this strategy, we draw on the theory of spatio-temporal pattern formation and propose a novel perspective for analysing dynamics on networks, by evaluating how the self-organized dynamics are confined by network architecture to a small set of permissible collective states. In particular, we discuss the role of prominent topological features of brain connectivity, such as hubs, modules and hierarchy, in shaping activity patterns. We illustrate the notion of network-guided pattern formation with numerical simulations and outline how it can facilitate the understanding of neural dynamics.     

Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave.     

Collective behavior in cancer cell populations

In recent years the argument has been made that malignant tumors represent complex dynamic and self‐organizing biosystems. Furthermore, there is increasing evidence that collective cell migration is common during invasion and metastasis of malignant tumors. Here, we argue that cancer systems may be capable of developing multicellular collective patterns that resemble evolved adaptive behavior known from other biological systems including collective sensing of environmental conditions and collective decision‐making. We present a concept as to how these properties could arise in tumors and why the emergence of such swarm‐like patterns would confer advantageous properties to the spatiotemporal expansion of tumors, and consequently, why understanding and ultimately targeting such collectivity should be of interest for basic and clinical cancer research alike.