Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

See globally, spike locally: oscillations in a retinal model encode large visual features

We show that coherent oscillations among neighboring ganglion cells in a retinal model encode global topological properties, such as size, that cannot be deduced unambiguously from their local, time-averaged firing rates. Whereas ganglion cells may fire similar numbers of spikes in response to both small and large spots, only large spots evoke coherent high frequency oscillations, potentially allowing downstream neurons to infer global stimulus properties from their local afferents. To determine whether such information might be extracted over physiologically realistic spatial and temporal scales, we analyzed artificial spike trains whose oscillatory correlations were similar to those measured experimentally. Oscillatory power in the upper gamma band, extracted on single-trials from multi-unit spike trains, supported good to excellent size discrimination between small and large spots, with performance improving as the number of cells and/or duration of the analysis window was increased. By using Poisson distributed spikes to normalize the firing rate across stimulus conditions, we further found that coincidence detection, or synchrony, yielded substantially poorer performance on identical size discrimination tasks. To determine whether size encoding depended on contiguity independent of object shape, we examined the total oscillatory activity across the entire model retina in response to random binary images. As the ON-pixel probability crossed the percolation threshold, which marks the sudden emergence of large connected clusters, the total gamma-band activity exhibited a sharp transition, a phenomena that may be experimentally observable. Finally, a reanalysis of previously published oscillatory responses from cat ganglion cells revealed size encoding consistent with that predicted by the retinal model.     

Empirical mode decomposition of field potentials from macaque V4 in visual spatial attention

Empirical mode decomposition (EMD) has recently been introduced as a local and fully data-driven technique for the analysis of non-stationary time-series. It allows the frequency and amplitude of a time-series to be evaluated with excellent time resolution. In this article we consider the application of EMD to the analysis of neuronal activity in visual cortical area V4 of a macaque monkey performing a visual spatial attention task. We show that, by virtue of EMD, field potentials can be resolved into a sum of intrinsic components with different degrees of oscillatory content. Low-frequency components in single-trial recordings contribute to the average visual evoked potential (AVEP), whereas high-frequency components do not, but are identified as gamma-band (30–90 Hz) oscillations. The magnitude of time-varying gamma activity is shown to be enhanced when the monkey attends to a visual stimulus as compared to when it is not attending to the same stimulus. Comparison with Fourier analysis shows that EMD may offer better temporal and frequency resolution. These results support the idea that the magnitude of gamma activity reflects the modulation of V4 neurons by visual spatial attention. EMD, coupled with instantaneous frequency analysis, is demonstrated to be a useful technique for the analysis of neurobiological time-series.     

Dynamic Processes in Regulation and Some Implications for Biofeedback and Biobehavioral Interventions

Systems theory has long been used in psychology, biology, and sociology. This paper applies newer methods of control systems modeling for assessing system stability in health and disease. Control systems can be characterized as open or closed systems with feedback loops. Feedback produces oscillatory activity, and the complexity of naturally occurring oscillatory patterns reflects the multiplicity of feedback mechanisms, such that many mechanisms operate simultaneously to control the system. Unstable systems, often associated with poor health, are characterized by absence of oscillation, random noise, or a very simple pattern of oscillation. This modeling approach can be applied to a diverse range of phenomena, including cardiovascular and brain activity, mood and thermal regulation, and social system stability. External system stressors such as disease, psychological stress, injury, or interpersonal conflict may perturb a system, yet simultaneously stimulate oscillatory processes and exercise control mechanisms. Resonance can occur in systems with negative feedback loops, causing high-amplitude oscillations at a single frequency. Resonance effects can be used to strengthen modulatory oscillations, but may obscure other information and control mechanisms, and weaken system stability. Positive as well as negative feedback loops are important for system function and stability. Examples are presented of oscillatory processes in heart rate variability, and regulation of autonomic, thermal, pancreatic and central nervous system processes, as well as in social/organizational systems such as marriages and business organizations. Resonance in negative feedback loops can help stimulate oscillations and exercise control reflexes, but also can deprive the system of important information. Empirical hypotheses derived from this approach are presented, including that moderate stress may enhance health and functioning.     

Locomotion-related oscillatory body movements at 6-12 Hz modulate the hippocampal theta rhythm

The hippocampal theta rhythm is required for accurate navigation and spatial memory but its relation to the dynamics of locomotion is poorly understood. We used miniature accelerometers to quantify with high temporal and spatial resolution the oscillatory movements associated with running in rats. Simultaneously, we recorded local field potentials in the CA1 area of the hippocampus. We report that when rats run their heads display prominent vertical oscillations with frequencies in the same range as the hippocampal theta rhythm (i.e., 6-12 Hz). In our behavioral set-up, rats run mainly with speeds between 50 and 100 cm/s. In this range of speeds, both the amplitude and frequency of the "theta" head oscillations were increasing functions of running speed, demonstrating that the head oscillations are part of the locomotion dynamics. We found evidence that these rhythmical locomotor dynamics interact with the neuronal activity in the hippocampus. The amplitude of the hippocampal theta rhythm depended on the relative phase shift with the head oscillations, being maximal when the two signals were in phase. Despite similarity in frequency, the head movements and LFP oscillations only displayed weak phase and frequency locking. Our results are consistent with that neurons in the CA1 region receive inputs that are phase locked to the head acceleration signal and that these inputs are integrated with the ongoing theta rhythm.     

Ca2+ oscillations induced by histamine H1 receptor stimulation in HeLa cells: Fura-2 and patch clamp analysis

The response of HeLa cells to histamine H1 receptor stimulation is characterized by periodic increases in cytosolic free Ca2+ concentration. The mechanisms underlying this oscillatory behaviour are not well understood. Fura-2 and patch clamp experiments carried out on HeLa cells have previously shown: (a) that Ca2+ oscillations are not initially dependent on the presence of external Ca2+, that external Ca2+ is required to maintain the oscillatory activity; (b) that a depolarization of the cell membrane leads to an inhibition of Ca2+ oscillations during the external Ca2+ dependent phase of the process; and (c) that Ca2+ oscillations can be abolished during this latter phase by the exogenous addition of Ca2+ channel blocking agents, such as Co2+ or La3+. The contribution of the inositol phosphate pathway to Ca2+ oscillations was more recently investigated in whole cell experiments performed with patch pipettes containing IP3 or the non-hydrolysable GTP analogue GTP-gamma S. Clear periodic current fluctuations were recorded using both patch pipette solutions. Assuming that the intracellular IP3 level remained constant under these conditions, these findings provide direct evidence that the Ca2+ oscillations in HeLa cells do not arise from a periodic production of IP3. The effect of the internal and external cell pH on the oscillatory process was also investigated in Fura-2 and patch clamp experiments. It was found that an increase in intracellular pH from 7.4 to 7.7 during the external Ca2+ dependent phase of the histamine stimulation abolishes the appearance of Ca2+ spikes whereas, a cellular acidification to pH 7.2 maintains or stimulates the Ca2+ oscillatory activity. The former effect was observed in the absence of Ca2+ in the bathing medium, indicating that the inhibitory action of alkaline pH was not related to a reduced Ca2+ entry. An increase in extracellular pH from 7.3 to 9.0 in contrast elicited an intracellular Ca2+ accumulation which resulted in most cases in an inhibition of the oscillatory process. This effect was dependent on external Ca2+ and was observed in alkaline internal pH conditions (pH 7.7). These observations suggest: (a) that the net Ca2+ influx in HeLa cells is strongly dependent on the cell internal and external pH; and (b) that the magnitude of this Ca2+ influx controls to a large extent the oscillation frequency. Finally, an inhibition of the histamine induced Ca2+ oscillatory activity was observed following the addition of the Ca(2+)-induced Ca(2+)-release (CICR) inhibitor adenine to the external medium.(ABSTRACT TRUNCATED AT 250 WORDS)     

Control of glycolytic oscillations by temperature

External control of oscillatory glycolysis in yeast extract has been performed by application of either homogeneous temperature oscillations or stationary, spatial temperature gradients. Entrainment of the glycolytic oscillations by the 1/2- and 1/3-harmonic, as well as the fundamental input frequency, could be observed. From the phase response curve to a single temperature pulse, a distinct sensitivity of NADH-oxidizing processes, compared with NAD-reducing processes, is visible. Determination of glycolytic intermediates shows that the feedback-regulated phosphofructokinase as well as the glyceraldehyde-3-phosphate dehydrogenase are the most temperature-sensitive steps of glycolysis. We also find strong concentration changes in ATP and AMP at varying temperatures and, accordingly, in the energy charge. Construction of a feedback loop for spatial control of temperature by means of a Peltier element allowed us to apply a temperature gradient to the yeast extract. With this setup it is possible to initiate traveling waves and to control the wave velocity.     

Oscillations in plant membrane transport: model predictions, experimental validation, and physiological implications

Although oscillations in membrane-transport activity are ubiquitous in plants, the ionic mechanisms of ultradian oscillations in plant cells remain largely unknown, despite much phenomenological data. The physiological role of such oscillations is also the subject of much speculation. Over the last decade, much experimental evidence showing oscillations in net ion fluxes across the plasma membrane of plant cells has been accumulated using the non-invasive MIFE technique. In this study, a recently proposed feedback-controlled oscillatory model was used. The model adequately describes the observed ion flux oscillations within the minute range of periods and predicts: (i) strong dependence of the period of oscillations on the rate constants for the H+ pump; (ii) a substantial phase shift between oscillations in net H+ and K+ fluxes; (iii) cessation of oscillations when H+ pump activity is suppressed; (iv) the existence of some 'window' of external temperatures and ionic concentrations, where non-damped oscillations are observed: outside this range, even small changes in external parameters lead to progressive damping and aperiodic behaviour; (v) frequency encoding of environmental information by oscillatory patterns; and (vi) strong dependence of oscillatory characteristics on cell size. All these predictions were successfully confirmed by direct experimental observations, when net ion fluxes were measured from root and leaf tissues of various plant species, or from single cells. Because oscillatory behaviour is inherent in feedback control systems having phase shifts, it is argued from this model that suitable conditions will allow oscillations in any cell or tissue. The possible physiological role of such oscillations is discussed in the context of plant adaptive responses to salinity, temperature, osmotic, hypoxia, and pH stresses.     

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.     

Diurnal and ultradiurnal oscillations of growing organs within the framework of the information system of the plant

The analysis of growth and movements of seedling organs of kidney bean (Phaseolus vulgaris L.) provides a pattern of periodic phases of activity and relaxation. The existence of a central organ which would control the phase relationships, is not anticipated in the integrity of the plant. The cyclic activity of individual organs shows itself by growth associated with oscillation movements. One and the same organ may simultaneously accomplish oscillatory movements with a diurnal and ultradiurnal frequency. These rhythms originate during the organ development; the first pair of kidney bean leaves at first executes oscillation movements with a diurnal frequency and only after it is fully developed it exhibits a diurnal cycle with the photophil phase upwards and the scotophil downwards, the oscillations with an ultradiurnal oycle being maintained. The movements of the two leaves are synchronous, but there occur short sections with a desynchronous cycle. Simultaneously with these oscillations, in which the leaf petiole takes part, the adult leaf performs oscillatory movements perpendicular to the longitudinal leaf axis, the so-called side swings, controlled by periodical changes of the joint attaching the leaf blade. Their frequency is practically identical with that of the ultradiurnal cycle. Thus the periodic growth activity of the kidney bean results in growth oscillations passing in the diurnal cycle with a frequency of 0.043 rev.h^-1, their ascending and descending phases consisting of periodical ultradiurnal oscillations in cycles of 0.73–0.59 rev.h^-1. The epicotyl growth shows a similar pattern: into the basic diurnal nutation cycle with a frequency of 0.042 rev.h^-1 ultradiurnal oscillation cycles are incorporated having a similar frequency to that revealed in leaves (0.69–0.64 rev.h^-1). The diurnal oscillatory cycles belong to a system established on the basis of periodicity of day and night and other geophysical cycles. The ultradiurnal rhythmic oscillations are presumed to be an expression of the geocontrol system of root and shoot growth direction and orientation of the organ in space. The shape of their trajectories in bean leaves is contradictory to this; they are not spatial helices, as the kybernetic model would presuppose, but have a vertical, upwards and downwards course in one plane. Since these oscillatory movements with an ultradiurnal cycle cease after petiole excision from the stem and after shoot apex amputation, one may presume that they are coupled with the low-frequency oscillatory system of the epicotyl.     

Dynamics of electrostatic fluctuations in the edge plasma in the U-3M torsatron

Results are presented from experimental and theoretical investigations of oscillatory and wave phenomena observed in the edge region in the U-3M torsatron during plasma creation and heating by an RF discharge in the ICR frequency range, accompanied by a transition to improved confinement. The main results are reported of diagnostic measurements of the spectral composition of oscillations, as well as of how the phase and amplitude relationships depend on time and on the RF power during its injection into the plasma. The measurements were carried out with electrostatic probes positioned at the edge of the plasma confinement region. The experimental results are interpreted using the kinetic theory of the electron-ion parametric instability of a plasma in the ion cyclotron frequency range and are compared with the results of numerical simulations.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.     

The fundamental organization of cardiac mitochondria as a network of coupled oscillators

Mitochondria can behave as individual oscillators whose dynamics may obey collective, network properties. We have shown that cardiomyocytes exhibit high-amplitude, self-sustained, and synchronous oscillations of bioenergetic parameters when the mitochondrial network is stressed to a critical state. Computational studies suggested that additional low-amplitude, high-frequency oscillations were also possible. Herein, employing power spectral analysis, we show that the temporal behavior of mitochondrial membrane potential (DeltaPsi(m)) in cardiomyocytes under physiological conditions is oscillatory and characterized by a broad frequency distribution that obeys a homogeneous power law (1/f(beta)) with a spectral exponent, beta = 1.74. Additionally, relative dispersional analysis shows that mitochondrial oscillatory dynamics exhibits long-term memory, characterized by an inverse power law that scales with a fractal dimension (D(f)) of 1.008, distinct from random behavior (D(f) = 1.5), over at least three orders of magnitude. Analysis of a computational model of the mitochondrial oscillator suggests that the mechanistic origin of the power law behavior is based on the inverse dependence of amplitude versus frequency of oscillation related to the balance between reactive oxygen species production and scavenging. The results demonstrate that cardiac mitochondria behave as a network of coupled oscillators under both physiological and pathophysiological conditions.     

A Canonical Circuit for Generating Phase-Amplitude Coupling

‘Phase amplitude coupling’ (PAC) in oscillatory neural activity describes a phenomenon whereby the amplitude of higher frequency activity is modulated by the phase of lower frequency activity. Such coupled oscillatory activity – also referred to as ‘cross-frequency coupling’ or ‘nested rhythms’ – has been shown to occur in a number of brain regions and at behaviorally relevant time points during cognitive tasks; this suggests functional relevance, but the circuit mechanisms of PAC generation remain unclear. In this paper we present a model of a canonical circuit for generating PAC activity, showing how interconnected excitatory and inhibitory neural populations can be periodically shifted in to and out of oscillatory firing patterns by afferent drive, hence generating higher frequency oscillations phase-locked to a lower frequency, oscillating input signal. Since many brain regions contain mutually connected excitatory-inhibitory populations receiving oscillatory input, the simplicity of the mechanism generating PAC in such networks may explain the ubiquity of PAC across diverse neural systems and behaviors. Analytic treatment of this circuit as a nonlinear dynamical system demonstrates how connection strengths and inputs to the populations can be varied in order to change the extent and nature of PAC activity, importantly which phase of the lower frequency rhythm the higher frequency activity is locked to. Consequently, this model can inform attempts to associate distinct types of PAC with different network topologies and physiologies in real data.     

Control and variation ofin vivo nitrate reductase activity inLolium perenne L. cv. S24

Nitrate reductase activity in leaf material ofLolium perenne L. cv. S24 was estimated using anin vivo assay method such that activity could be estimated at intervals of 5 minutes for up to two hours. The pattern of nitrate reductase activity, as estimated by nitrite accumulation, showed pronounced oscillatory behaviour with frequency of approximately 4 cycles per hour; at certain seasons however oscillatory activity was not shown. The phase of the oscillations observed in different experiments was not co-incident with respect to time of day.     

Cortical Resonance Frequencies Emerge from Network Size and Connectivity

Neural oscillations occur within a wide frequency range with different brain regions exhibiting resonance-like characteristics at specific points in the spectrum. At the microscopic scale, single neurons possess intrinsic oscillatory properties, such that is not yet known whether cortical resonance is consequential to neural oscillations or an emergent property of the networks that interconnect them. Using a network model of loosely-coupled Wilson-Cowan oscillators to simulate a patch of cortical sheet, we demonstrate that the size of the activated network is inversely related to its resonance frequency. Further analysis of the parameter space indicated that the number of excitatory and inhibitory connections, as well as the average transmission delay between units, determined the resonance frequency. The model predicted that if an activated network within the visual cortex increased in size, the resonance frequency of the network would decrease. We tested this prediction experimentally using the steady-state visual evoked potential where we stimulated the visual cortex with different size stimuli at a range of driving frequencies. We demonstrate that the frequency corresponding to peak steady-state response inversely correlated with the size of the network. We conclude that although individual neurons possess resonance properties, oscillatory activity at the macroscopic level is strongly influenced by network interactions, and that the steady-state response can be used to investigate functional networks.     

Functional dissociation of ongoing oscillatory brain states

The state of a neural assembly preceding an incoming stimulus is assumed to modulate the processing of subsequently presented stimuli. The nature of this state can differ with respect to the frequency of ongoing oscillatory activity. Oscillatory brain activity of specific frequency range such as alpha (8-12 Hz) and gamma (above 30 Hz) band oscillations are hypothesized to play a functional role in cognitive processing. Therefore, a selective modulation of this prestimulus activity could clarify the functional role of these prestimulus fluctuations. For this purpose, we adopted a novel non-invasive brain-computer-interface (BCI) strategy to selectively increase alpha or gamma band activity in the occipital cortex combined with an adaptive presentation of visual stimuli within specific brain states. During training, oscillatory brain activity was estimated online and fed back to the participants to enable a deliberate modulation of alpha or gamma band oscillations. Results revealed that volunteers selectively increased alpha and gamma frequency oscillations with a high level of specificity regarding frequency range and localization. At testing, alpha or gamma band activity was classified online and at defined levels of activity, visual objects embedded in noise were presented instantly and had to be detected by the volunteer. In experiment I, the effect of two levels of prestimulus gamma band activity on visual processing was examined. During phases of increased gamma band activity significantly more visual objects were detected. In experiment II, the effect was compared against increased levels of alpha band activity. An improvement of visual processing was only observed for enhanced gamma band activity. Both experiments demonstrate the specific functional role of prestimulus gamma band oscillations for perceptual processing. We propose that the BCI method permits the selective modulation of oscillatory activity and the direct assessment of behavioral consequences to test for functional dissociations of different oscillatory brain states.     

Cracking the code of oscillatory activity

Neural oscillations are ubiquitous measurements of cognitive processes and dynamic routing and gating of information. The fundamental and so far unresolved problem for neuroscience remains to understand how oscillatory activity in the brain codes information for human cognition. In a biologically relevant cognitive task, we instructed six human observers to categorize facial expressions of emotion while we measured the observers' EEG. We combined state-of-the-art stimulus control with statistical information theory analysis to quantify how the three parameters of oscillations (i.e., power, phase, and frequency) code the visual information relevant for behavior in a cognitive task. We make three points: First, we demonstrate that phase codes considerably more information (2.4 times) relating to the cognitive task than power. Second, we show that the conjunction of power and phase coding reflects detailed visual features relevant for behavioral response--that is, features of facial expressions predicted by behavior. Third, we demonstrate, in analogy to communication technology, that oscillatory frequencies in the brain multiplex the coding of visual features, increasing coding capacity. Together, our findings about the fundamental coding properties of neural oscillations will redirect the research agenda in neuroscience by establishing the differential role of frequency, phase, and amplitude in coding behaviorally relevant information in the brain.