Coupled map lattice simulation of epileptogenesis in hippocampal slices

Bioelectric activity of a nervous tissue and its synchronization with formating epileptiform bursts are simulated by a coupled map lattice. The functional units of the map located in the lattice sites represent neural masses which consist of current sources and sinks. The sources lead to depolarization of neurons, and sinks provide hyperpolarization. The map describes a single variable – the bioelectric potential. This potential is created by the interplay of all current sources and sinks in the neural masses. The neural masses are diffusively coupled with each other both by electrotonic influence and synaptic coupling. Both mechanisms mentioned are suggested to be essential for the formation of synchronous bursts. The transition from chaotic activity to bursts was studied. Received: 30 September 1997 / Accepted in revised from: 11 March 1998     

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.     

Spectral properties of an N-pole helical structure consisting of like-charged equal-size dust grains

The oscillations and stability of an N-pole helical structure that consists of like-charged equal-size dust grains and is confined in a plasma in an axisymmetric potential well are studied theoretically. Self-confining structures, as well as their linear collective modes corresponding to three coupled types of grain displacements (change in the radius of the structure, change in the distance between neighboring lattice planes of the structure, and angular displacements in the lattice planes), are found. On the whole, the coupled oscillations have the form of wormlike perturbations. Dispersion relations for the oscillation modes of helical structures composed of N interwoven helices are derived and solved numerically.     

BDNF boosts spike fidelity in chaotic neural oscillations

Oscillatory activity and its nonlinear dynamics are of fundamental importance for information processing in the central nervous system. Here we show that in aperiodic oscillations, brain-derived neurotrophic factor (BDNF), a member of the neurotrophin family, enhances the accuracy of action potentials in terms of spike reliability and temporal precision. Cultured hippocampal neurons displayed irregular oscillations of membrane potential in response to sinusoidal 20-Hz somatic current injection, yielding wobbly orbits in the phase space, i.e., a strange attractor. Brief application of BDNF suppressed this unpredictable dynamics and stabilized membrane potential fluctuations, leading to rhythmical firing. Even in complex oscillations induced by external stimuli of 40 Hz (gamma) on a 5-Hz (theta) carrier, BDNF-treated neurons generated more precisely timed spikes, i.e., phase-locked firing, coupled with theta-phase precession. These phenomena were sensitive to K252a, an inhibitor of tyrosine receptor kinases and appeared attributable to BDNF-evoked Na(+) current. The data are the first indication of pharmacological control of endogenous chaos. BDNF diminishes the ambiguity of spike time jitter and thereby might assure neural encoding, such as spike timing-dependent synaptic plasticity.     

Bradykinin and muscarine induce Ca(2+)-dependent oscillations of membrane potential in rat glioma cells indicating a rhythmic Ca2+ release from internal stores: thapsigargin and 2,5-di(tert-butyl)-1, 4-benzohydroquinone deplete InsP3-sensitive Ca2+ stores in glioma and in neuroblastoma-glioma hybrid cells.

Continuous superfusion of rat glioma cells with medium containing bradykinin (from 0.2 nM) induced a transient hyperpolarization followed by regular hyperpolarizing oscillations of the membrane potential. Similar repetitive hyperpolarizing oscillations were caused by extracellularly applied bradykinin or muscarine or by intracellularly injected GTP-gamma-S. The frequency of the oscillations was 1 per minute at bradykinin concentrations ranging from 0.2 nM to 2 microM, but the amplitude and duration increased with rising peptide concentration. The muscarine-induced oscillations were blocked by atropine. In the presence of extracellular Ca2+, the substances thapsigargin, 2,5-di(tert-butyl)-1,4-benzohydroquinone (tBuBHQ), and ionomycin reversibly suppressed the bradykinin-induced oscillations. Thapsigargin and tBuBHA, which are known to block the Ca2+ ATPase of endoplasmic reticulum, caused a transient rise in cytosolic Ca2+ activity, monitored with Fura-2, in suspensions of rat glioma cells or of mouse neuroblastoma-rat glioma hybrid cells. After a transient Ca2+ rise caused by thapsigargin, tBuBHQ, or ionomycin, the Ca2+ response to bradykinin which is known to be due to release of Ca2+ from internal stores was suppressed. This indicates that thapsigargin and tBuBHQ deplete internal Ca2+ stores as already seen previously for ionomycin. Thus, the inhibition of the membrane potential oscillations by thapsigargin, tBuBHQ, and ionomycin indicates that the oscillations are associated with activation of InsP3-sensitive Ca2+ stores. In some cells composite oscillation patterns which consisted of two independent oscillations with different amplitudes that overlapped additively were seen. We discuss that this pattern and the concentration dependency of the oscillations could be due to "quantal" Ca2+ release from stores with different inositol 1,4,5-triphosphate sensitivities. Subsidence of the oscillations after omission of extracellular Ca2+ seems to be due to a lack of replenishment of the intracellular stores with Ca2+, which comes from the extracellular compartment.     

Propagation of cytoplasmic Ca2+ oscillations in clusters of pancreatic beta-cells exposed to glucose

Digital image analysis was employed for resolving the temporal and spatial variations of the cytoplasmic Ca2+ concentration ([Ca2+]i) in pancreatic beta-cells loaded with the Ca(2+)-indicator Fura-2. Glucose-stimulated individual beta-cells exhibited large amplitude oscillations of [Ca2+]i with a mean frequency of 0.33 min-1. When Ca2+ diffusion was restricted by increasing the Ca2+ buffering capacity, the sugar-induced rise of [Ca2+]i preferentially affected the peripheral cytoplasm. When glucagon was present glucose also caused less prominent oscillations with about a 10-fold higher frequency superimposed on an elevated [Ca2+]i. In small clusters of 6-14 cells the average frequency of the large amplitude oscillations increased to 0.60 min-1. The clusters were found to contain micro-domains of electrically coupled cells with synchronized oscillations. After increasing the glucose concentration, adjacent domains became functionally coupled. The oscillations originated from different cells in the cluster. Also the fast glucagon-dependent oscillations were synchronized between cells and had different origins. The results indicate that coupling of beta-cells leads to an increased frequency of the large amplitude oscillations, and that the oscillatory characteristics are determined collectively among electrically coupled beta-cells rather than by particular pacemaker cells. In the light of these data it is necessary to reconsider the previous ideas that glucose-induced oscillations of membrane potential and [Ca2+]i require coupling between many beta-cells, and that the peak [Ca2+]i values reached during oscillations should increase with the size of the coupled cluster.     

Preventing extinction and outbreaks in chaotic populations

Interactions in ecological communities are inherently nonlinear and can lead to complex population dynamics including irregular fluctuations induced by chaos. Chaotic population dynamics can exhibit violent oscillations with extremely small or large population abundances that might cause extinction and recurrent outbreaks, respectively. We present a simple method that can guide management efforts to prevent crashes, peaks, or any other undesirable state. At the same time, the irregularity of the dynamics can be preserved when chaos is desirable for the population. The control scheme is easy to implement because it relies on time series information only. The method is illustrated by two examples: control of crashes in the Ricker map and control of outbreaks in a stage-structured model of the flour beetle Tribolium. It turns out to be effective even with few available data and in the presence of noise, as is typical for ecological settings.     

In situ measurements of external pH and optical density oscillations in Dictyostelium discoideum aggregates

In situ measurements of extracellular pH by means of microelectrodes and in situ measurements of optical density were performed on aggregating cells of Dictyostelium discoideum. Early aggregation stage AX2 cells showed sinusoidal pH oscillations, which could be inhibited by the specific relay inhibitor caffeine, indicating that they were coupled to cAMP oscillations. Sometimes biphasic pH oscillations were found, which can be explained by the superposition of two harmonic pH oscillations. These harmonic oscillations might arise by gating of the cAMP signal; a part of the cells respond to every cAMP signal and another subpopulation to every second cAMP pulse. Late aggregation-stage cells showed complex changes of the extracellular pH, which could be inhibited by caffeine. Optical density measurements of wave propagation in aggregation streams of HG220 also revealed gating behavior. In addition to sinusoidal optical density oscillations, biphasic and still more complex oscillations were observed.     

Calcium oscillations in endothelial cells

Several different types of endothelial cells are now known to respond to agonist stimulation with oscillations of cytosolic free [Ca2+] ([Ca2+]i). The oscillations can be repetitive [Ca2+]i spikes or sinusoidal-like oscillations according to the type of endothelial cell. Several properties of these oscillations are described including the effect of removal of extracellular Ca2+ and of changes in membrane potential, and the spatial heterogeneity of the oscillations. Results obtained with human umbilical vein endothelial cells are assessed in relation to a model for [Ca2+]i oscillations that involves Ca(2+)-induced Ca2+ release. In some preparations the oscillations are synchronized in neighbouring cells, whereas in other preparations they are not. The degree of synchrony may have functional implications and this is discussed with respect to control of blood flow and transmural permeability. A third functional implication of oscillations, their possible effect on desensitization, is also discussed.     

Oscillations of Ca++ concentration during the cell differentiation of Dictyostelium discoideum: their relation to oscillations in cyclic AMP and other components

Periodic cyclic-AMP pulses control the cell aggregation and differentiation of Dictyostelium discoideum. Another component required for the aggregation and differentiation of these cells appears to be extracellular Ca+ +. Oscillations in extracellular Ca+ + concentration were investigated in suspensions of differentiating cells. We observed spike-shaped and sinusoidal Ca+ + oscillations. In the course of differentiation, spike-shaped Ca+ + oscillations preceded sinusoidal oscillations, and no phase change occurred at the transition from spike-shaped to sinusoidal Ca+ + oscillations. Spike-shaped and sinusoidal Ca+ + oscillations were related to oscillations in (1) the cyclic-AMP and cyclic-GMP content of cells, (2) the light-scattering properties of cells, and (3) the extracellular pH. Spikeshaped Ca+ + oscillations were observed together with cyclic-AMP oscillations. The minima of the extracellular Ca+ + concentration trailed the maxima of the cyclic-AMP concentration by about 30 s. Sinusoidal Ca+ + oscillations were not accompanied by measurable cyclic-AMP oscillations. The amplitudes of the sinusoidal Ca+ + oscillations were smaller than those of the spike-shaped Ca+ + oscillations. A Ca+ + oscillation of small amplitude (instead of a spike-shaped oscillation) was observed when one cyclic-AMP spike was skipped. Our results provide evidence for the existence of a sinusoidal cyclic-AMP-independent Ca+ + oscillation of small amplitude, and they also suggest that spike-shaped Ca+ + oscillations may be superimposed on such small-amplitude oscillations. When D. discoideum cells produce cyclic-AMP spikes, the uptake of additional Ca+ + is induced, resulting in Ca+ + oscillations of a large amplitude.     

Coupled predator–prey oscillations in a chaotic food web

Coupling of several predator–prey oscillations can generate intriguing patterns of synchronization and chaos. Theory predicts that prey species will fluctuate in phase if predator–prey cycles are coupled through generalist predators, whereas they will fluctuate in anti-phase if predator–prey cycles are coupled through competition between prey species. Here, we investigate predator–prey oscillations in a long-term experiment with a marine plankton community. Wavelet analysis of the species fluctuations reveals two predator–prey cycles that fluctuate largely in anti-phase. The phase angles point at strong competition between the phytoplankton species, but relatively little prey overlap among the zooplankton species. This food web architecture is consistent with the size structure of the plankton community, and generates highly dynamic food webs. Continued alternations in species dominance enable coexistence of the prey species through a non-equilibrium 'killing-the-winner' mechanism, as the system shifts back and forth between the two predator–prey cycles in a chaotic fashion.     

Three-dimensional structure of an invertebrate rhodopsin and basis for ordered alignment in the photoreceptor membrane

Invertebrate rhodopsins activate a G-protein signalling pathway in microvillar photoreceptors. In contrast to the transducin-cyclic GMP phosphodiesterase pathway found in vertebrate rods and cones, visual transduction in cephalopod (squid, octopus, cuttlefish) invertebrates is signalled via Gq and phospholipase C. Squid rhodopsin contains the conserved residues of the G-protein coupled receptor (GPCR) family, but has only 35% identity with mammalian rhodopsins. Unlike vertebrate rhodopsins, cephalopod rhodopsin is arranged in an ordered lattice in the photoreceptor membranes. This organization confers sensitivity to the plane of polarized light and also provides the optimal orientation of the linear retinal chromophores in the cylindrical microvillar membranes for light capture. Two-dimensional crystals of squid rhodopsin show a rectilinear arrangement that is likely to be related to the alignment of rhodopsins in vivo.Here, we present a three-dimensional structure of squid rhodopsin determined by cryo-electron microscopy of two-dimensional crystals. Docking the atomic structure of bovine rhodopsin into the squid density map shows that the helix packing and extracellular plug structure are conserved. In addition, there are two novel structural features revealed by our map. The linear lattice contact appears to be made by the transverse C-terminal helix lying on the cytoplasmic surface of the membrane. Also at the cytoplasmic surface, additional density may correspond to a helix 5-6 loop insertion found in most GPCRs relative to vertebrate rhodopsins. The similarity supports the conservation in structure of rhodopsins (and other G-protein-coupled receptors) from phylogenetically distant organisms. The map provides the first indication of the structural basis for rhodopsin alignment in the microvillar membrane.     

Coupled map lattice approximations for spatially explicit individual-based models of ecology

Spatially explicit individual-based models are widely used in ecology but they are often difficult to treat analytically. Despite their intractability they often exhibit clear temporal and spatial patterning. We demonstrate how a spatially explicit individual-based model of scramble competition with local dispersal can be approximated by a stochastic coupled map lattice. The approximation disentangles the deterministic and stochastic element of local interaction and dispersal. We are thus able to understand the individual-based model through a simplified set of equations. In particular, we demonstrate that demographic noise leads to increased stability in the dynamics of locally dispersing single-species populations. The coupled map lattice approximation has general application to a range of spatially explicit individual-based models. It provides a new alternative to current approximation techniques, such as the method of moments and reaction-diffusion approximation, that captures both stochastic effects and large-scale patterning arising in individual-based models.     

Oscillatory network with self-organized dynamical connections for synchronization-based image segmentation

An oscillatory network of columnar architecture located in 3D spatial lattice was recently designed by the authors as oscillatory model of the brain visual cortex. Single network oscillator is a relaxational neural oscillator with internal dynamics tunable by visual image characteristics - local brightness and elementary bar orientation. It is able to demonstrate either activity state (stable undamped oscillations) or "silence" (quickly damped oscillations). Self-organized nonlocal dynamical connections of oscillators depend on oscillator activity levels and orientations of cortical receptive fields. Network performance consists in transfer into a state of clusterized synchronization. At current stage grey-level image segmentation tasks are carried out by 2D oscillatory network, obtained as a limit version of the source model. Due to supplemented network coupling strength control the 2D reduced network provides synchronization-based image segmentation. New results on segmentation of brightness and texture images presented in the paper demonstrate accurate network performance and informative visualization of segmentation results, inherent in the model.     

Membrane potential and Na(+)-K+ pump activity modulate resting and bradykinin-stimulated changes in cytosolic free calcium in cultured endothelial cells from bovine atria

The effects of membrane potential on resting and bradykinin-stimulated changes in [Ca2+]i were measured in fura-2 loaded cultured endothelial cells from bovine atria by spectrofluorimetry. The basal and bradykinin-stimulated release of endothelium-derived relaxing factor, monitored by bioassay methods, were dependent on extracellular Ca2+. Similarly, the plateau phase of the biphasic [Ca2+]i response to bradykinin stimulation exhibited a dependence on extracellular Ca2+, whereas the initial transient [Ca2+]i peak was refractory to the removal of extracellular Ca2+. The effect of membrane depolarization on the plateau phase of the bradykinin-induced change in [Ca2+]i was determined by varying [K+]o. The resting membrane potential measured under current clamp conditions was positively correlated with the extracellular [K+] (52 mV change/10-fold change in [K+]o). The observed decrease in resting and bradykinin-stimulated changes in [Ca2+]i upon depolarization is consistent with an ion transport mechanism where the influx is linearly related to the electrochemical gradient for Ca2+ entry (Em - ECa). The inhibition of bradykinin-stimulated Ca2+ entry by isotonic K+ was not due to the absence of extracellular Na+ since Li+ substitution did not inhibit the agonist-induced Ca2+ entry. In K(+)-free solutions and in the presence of ouabain, bradykinin evoked synchronized oscillations in [Ca2+]i in confluent endothelial cell monolayers. These [Ca2+]i oscillations between the plateau and resting [Ca2+]i levels were dependent on extracellular Ca2+ and K+ concentrations. Although the mechanism(s) underlying [Ca2+]i oscillations in vascular endothelial cells is unclear, these results suggest a role of the membrane conductance.     

Hydrodynamics and cell volume oscillations in the pollen tube apical region are integral components of the biomechanics of Nicotiana tabacum pollen tube growth

Pollen tube growth is localized at the apex and displays oscillatory dynamics. It is thought that a balance between intracellular turgor pressure (hydrostatic pressure, reflected by the cell volume) and cell wall loosening is a critical factor driving pollen tube growth. We previously demonstrated that water flows freely into and out of the pollen tube apical region dependent on the extracellular osmotic potential, that cell volume changes reflect changes in the intracellular pressure, and that cell volume changes differentially induce, increases or decreases in specific phospholipid signals. This article shows that manipulation of the extracellular osmotic potential rapidly induces modulations in pollen tube growth rate frequencies, demonstrating that changes in the intracellular pressure are sufficient to reset the pollen tube growth oscillator. This indicates a direct link between intracellular hydrostatic pressure and pollen tube growth. Altering hydrodynamic flow through the pollen tube by replacing extracellular H₂O with ²H₂O adversely affects both cell volume and growth rate oscillations and induces aberrant morphologies. Normal growth and cell morphology are rescued by replacing ²H₂O with H₂O. Further studies revealed that the cell volume oscillates in the pollen tube apical region. These cell volume oscillations were not from changes in cell shape at the tip and were detectable up to 30 μm distal to the tip (the longest length measured). Cell volume in the apical region oscillates with the same frequency as growth rate oscillations but surprisingly the cycles are phase-shifted by 180°. Raman microscopy yields evidence that hydrodynamic flow out of the apex may be part of the biomechanics that drive cellular expansion. The combined results suggest that hydrodynamic loading/unloading in the apical region induces cell volume oscillations and has a role in driving cell elongation and pollen tube growth.     

Density-dependent birth rate, birth pulses and their population dynamic consequences

 In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Received: 13 June 2001 / Revised version: 7 September 2001 / Published online: 8 February 2002