The scale-free dynamics of eukaryotic cells

Temporal organization of biological processes requires massively parallel processing on a synchronized time-base. We analyzed time-series data obtained from the bioenergetic oscillatory outputs of Saccharomyces cerevisiae and isolated cardiomyocytes utilizing Relative Dispersional (RDA) and Power Spectral (PSA) analyses. These analyses revealed broad frequency distributions and evidence for long-term memory in the observed dynamics. Moreover RDA and PSA showed that the bioenergetic dynamics in both systems show fractal scaling over at least 3 orders of magnitude, and that this scaling obeys an inverse power law. Therefore we conclude that in S. cerevisiae and cardiomyocytes the dynamics are scale-free in vivo. Applying RDA and PSA to data generated from an in silico model of mitochondrial function indicated that in yeast and cardiomyocytes the underlying mechanisms regulating the scale-free behavior are similar. We validated this finding in vivo using single cells, and attenuating the activity of the mitochondrial inner membrane anion channel with 4-chlorodiazepam to show that the oscillation of NAD(P)H and reactive oxygen species (ROS) can be abated in these two evolutionarily distant species. Taken together these data strongly support our hypothesis that the generation of ROS, coupled to redox cycling, driven by cytoplasmic and mitochondrial processes, are at the core of the observed rhythmicity and scale-free dynamics. We argue that the operation of scale-free bioenergetic dynamics plays a fundamental role to integrate cellular function, while providing a framework for robust, yet flexible, responses to the environment.     

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.     

Mitochondrial criticality: a new concept at the turning point of life or death

A variety of stressors can cause the collapse of mitochondrial membrane potential (DeltaPsi(m)), but the events leading up to this catastrophic cellular event are not well understood at the mechanistic level. Based on our recent studies of oscillations in mitochondrial energetics, we have coined the term "mitochondrial criticality" to describe the state in which the mitochondrial network of cardiomyocytes becomes very sensitive to small perturbations in reactive oxygen species (ROS), resulting in the scaling of local mitochondrial uncoupling and DeltaPsi(m) loss to the whole cell, and the myocardial syncytium. At the point of criticality, the dynamics of the mitochondrial network bifurcate to oscillatory behavior. These energetic changes are translated into effects on the electrical excitability of the cell, inducing dramatic changes in the morphology and the threshold for activating an action potential. Emerging evidence suggests that this mechanism, by creating spatial and temporal heterogeneity of excitability in the heart during ischemia and reperfusion, underlies the genesis of potentially lethal cardiac arrhythmias.     

Oscillations in energy metabolism

Organisation of mitochondrial metabolism is a quintessential example of a complex dissipative system which can display dynamic instabilities. Several findings have indicated that the conditions inducing instabilities are within the physiological range and that mild perturbations could elicit oscillations. Different mathematical models have been put forth in order to explain the genesis of oscillations in energy metabolism. One model considers mitochondria as an organised network of oscillators and indicates that communication between mitochondria involves mitochondrial reactive oxygen species (ROS) production acting as synchronisers of the energy status of the whole population of mitochondria. An alternative model proposes that extramitochondrial pH variations could lead to mitochondrial oscillations. Oscillatory phenomena in energy metabolism have also been investigated in vivo on the basis of ³¹P magnetic resonance spectroscopy (MRS) measurements of phosphocreatine post-exercise recovery in human and animal skeletal muscle. The corresponding results provide experimental evidences about the role exerted by cytosolic pH on oscillations. Finally a new simple non-linear mathematical model describing the overall chemical reaction of phosphocreatine recovery predicting oscillatory recovery pattern under certain experimental conditions is presented and discussed in the light of the experimental results reported so far.     

Oscillations and waves in the models of interactive neural populations

The dynamics of activity in interactive neural populations is simulated by the networks of Wilson–Cowan oscillators. Two extreme cases of connection architectures in the networks are considered: (1) 1D and 2D regular and homogeneous grids with local connections and (2) sparse random coupling. Propagating waves in the network have been found under the stationary external input and the regime of partial synchronization has been obtained for the periodic input. It has been shown that in the case of random coupling about 60% of neural populations demonstrate oscillatory activity and some of these oscillations are synchronous. The role of different types of dynamics in information processing is discussed. In particular, we discuss the regime of partial synchronization in the context of cortical microcircuits.     

Oscillations and waves in the models of interactive neural populations

The dynamics of activity in interactive neural populations is simulated by the networks of Wilson-Cowan oscillators. Two extreme cases of connection architectures in the networks are considered: (1) 1D and 2D regular and homogeneous grids with local connections and (2) sparse random coupling. Propagating waves in the network have been found under the stationary external input and the regime of partial synchronization has been obtained for the periodic input. It has been shown that in the case of random coupling about 60% of neural populations demonstrate oscillatory activity and some of these oscillations are synchronous. The role of different types of dynamics in information processing is discussed. In particular, we discuss the regime of partial synchronization in the context of cortical microcircuits.     

From mitochondrial dynamics to arrhythmias

The reactive oxygen species (ROS)-dependent mitochondrial oscillator described in cardiac cells exhibits at least two modes of function under physiological conditions or in response to metabolic and oxidative stress. Both modes depend upon network behavior of mitochondria. Under physiological conditions cardiac mitochondria behave as a network of coupled oscillators with a broad range of frequencies. ROS weakly couples mitochondria under normal conditions but becomes a strong coupling messenger when, under oxidative stress, the mitochondrial network attains criticality. Mitochondrial criticality is achieved when a threshold of ROS is overcome and a certain density of mitochondria forms a cluster that spans the whole cell. Under these conditions, the slightest perturbation triggers a cell-wide collapse of the mitochondrial membrane potential, Δψ_m, visualized as a depolarization wave throughout the cell which is followed by whole cell synchronized oscillations in Δψ_m, NADH, ROS, and GSH. This dynamic behavior scales from the mitochondrion to the cell by driving cellular excitability and the whole heart into catastrophic arrhythmias. A network collapse of Δψ_m under criticality leads to: (i) energetic failure, (ii) temporal and regional alterations in action potential (AP), (iii) development of zones of impaired conduction in the myocardium, and, ultimately, (iv) a fatal ventricular arrhythmia.     

Rhythmic cortical neurons increase their oscillations and sculpt basal ganglia signaling during motor learning

The function and modulation of neural circuits underlying motor skill may involve rhythmic oscillations (Feller, 1999 ; Marder and Goaillard, 2006 ; Churchland et al., 2012 ). In the proposed pattern generator for birdsong, the cortical nucleus HVC, the frequency and power of oscillatory bursting during singing increases with development (Crandall et al., 2007 ; Day et al., 2009 ). We examined the maturation of cellular activity patterns that underlie these changes. Single unit ensemble recording combined with antidromic identification (Day et al., 2011 ) was used to study network development in anesthetized zebra finches. Autocovariance quantified oscillations within single units. A subset of neurons oscillated in the theta/alpha/mu/beta range (8–20 Hz), with greater power in adults compared to juveniles. Across the network, the normalized oscillatory power in the 8–20 Hz range was greater in adults than juveniles. In addition, the correlated activity between rhythmic neuron pairs increased with development. We next examined the functional impact of the oscillators on the output neurons of HVC. We found that the firing of oscillatory neurons negatively correlated with the activity of cortico‐basal ganglia neurons (HVC_Xs), which project to Area X (the song basal ganglia). If groups of oscillators work together to tonically inhibit and precisely control the spike timing of adult HVC_Xs with coordinated release from inhibition, then the activity of HVC_Xs in juveniles should be decreased relative to adults due to uncorrelated, tonic inhibition. Consistent with this hypothesis, HVC_Xs had lower activity in juveniles. These data reveal network changes that shape cortical‐to‐basal ganglia signaling during motor learning. © 2013 Wiley Periodicals, Inc. Develop Neurobiol 73: 754–768, 2013     

Mitochondrial Ca2+ and the heart

It is now well established that mitochondria accumulate Ca(2+) ions during cytosolic Ca(2+) ([Ca(2+)](i)) elevations in a variety of cell types including cardiomyocytes. Elevations in intramitochondrial Ca(2+) ([Ca(2+)](m)) activate several key enzymes in the mitochondrial matrix to enhance ATP production, alter the spatial and temporal profile of intracellular Ca(2+) signaling, and play an important role in the initiation of cell death pathways. Moreover, mitochondrial Ca(2+) uptake stimulates nitric oxide (NO) production by mitochondria, which modulates oxygen consumption, ATP production, reactive oxygen species (ROS) generation, and in turn provides negative feedback for the regulation of mitochondrial Ca(2+) accumulation. Controversy remains, however, whether in cardiac myocytes mitochondrial Ca(2+) transport mechanisms allow beat-to-beat transmission of fast cytosolic [Ca(2+)](i) oscillations into oscillatory changes in mitochondrial matrix [Ca(2+)](m). This review critically summarizes the recent experimental work in this field.     

Dynamics and bifurcations of two coupled neural oscillators with different connection types

In this paper we present an oscillatory neural network composed of two coupled neural oscillators of the Wilson-Cowan type. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. The network serves as a model for several possible network architectures. We study how the type and the strength of the connections between the oscillators affect the dynamics of the neural network. We investigate, separately from each other, four possible connection types (excitatory→excitatory, excitatory→inhibitory, inhibitory→excitatory, and inhibitory→inhibitory) and compute the corresponding bifurcation diagrams. In case of weak connections (small strength), the connection of populations of different types lead to periodicin-phase oscillations, while the connection of populations of the same type lead to periodicanti-phase oscillations. For intermediate connection strengths, the networks can enter quasiperiodic or chaotic regimes, and can also exhibit multistability. More generally, our analysis highlights the great diversity of the response of neural networks to a change of the connection strength, for different connection architectures. In the discussion, we address in particular the problem of information coding in the brain using quasiperiodic and chaotic oscillations. In modeling low levels of information processing, we propose that feature binding should be sought as a temporally coherent phase-locking of neural activity. This phase-locking is provided by one or more interacting convergent zones and does not require a central “top level” subcortical circuit (e.g. the septo-hippocampal system). We build a two layer model to show that although the application of a complex stimulus usually leads to different convergent zones with high frequency oscillations, it is nevertheless possible to synchronize these oscillations at a lower frequency level using envelope oscillations. This is interpreted as a feature binding of a complex stimulus.     

Oscillatory states of mitochondria. Studies on the oscillatory mechanism of liver and heart mitochondria

Oscillatory movements of ions across the inner membrane of liver and heart mitochondria in vitro have been investigated. Our results indicate that the inverse of the square of oscillatory period is linear with respect to (a) the permeant cation concentration; (b) the inverse of the permeant weak acid anion concentration, and (c) the rate of energyproduction. It has been shown that various factors contribute to damping of the oscillations. These factors include: substrate utilization, mitochondrial deterioration, imperfect mitochondrial synchronization, and, possibly, an oscillatory mechanism which dictates damping. An increased period length and extensive damping of oscillations occurs at a critical mitochondrial protein concentration (less than 0.6 mg protein/ml). Such inhibition can be reversed by the addition of cytochrome c.     

Source Space Estimation of Oscillatory Power and Brain Connectivity in Tinnitus

Tinnitus is the perception of an internally generated sound that is postulated to emerge as a result of structural and functional changes in the brain. However, the precise pathophysiology of tinnitus remains unknown. Llinas’ thalamocortical dysrhythmia model suggests that neural deafferentation due to hearing loss causes a dysregulation of coherent activity between thalamus and auditory cortex. This leads to a pathological coupling of theta and gamma oscillatory activity in the resting state, localised to the auditory cortex where normally alpha oscillations should occur. Numerous studies also suggest that tinnitus perception relies on the interplay between auditory and non-auditory brain areas. According to the Global Brain Model, a network of global fronto—parietal—cingulate areas is important in the generation and maintenance of the conscious perception of tinnitus. Thus, the distress experienced by many individuals with tinnitus is related to the top—down influence of this global network on auditory areas. In this magnetoencephalographic study, we compare resting-state oscillatory activity of tinnitus participants and normal-hearing controls to examine effects on spectral power as well as functional and effective connectivity. The analysis is based on beamformer source projection and an atlas-based region-of-interest approach. We find increased functional connectivity within the auditory cortices in the alpha band. A significant increase is also found for the effective connectivity from a global brain network to the auditory cortices in the alpha and beta bands. We do not find evidence of effects on spectral power. Overall, our results provide only limited support for the thalamocortical dysrhythmia and Global Brain models of tinnitus.     

Mitochondrial Oscillations and Waves in Cardiac Myocytes: Insights from Computational Models

Periodic cellwide depolarizations of mitochondrial membrane potential (Ψ_M) which are triggered by reactive oxygen species (ROS) and propagated by ROS-induced ROS release (RIRR) have been postulated to contribute to cardiac arrhythmogenesis and injury during ischemia/reperfusion. Two different modes of RIRR have been described: Ψ_M oscillations involving ROS-sensitive mitochondrial inner membrane anion channels (IMAC), and slow depolarization waves related to mitochondrial permeability transition pore (MPTP) opening. In this study, we developed a computational model of mitochondria exhibiting both IMAC-mediated RIRR and MPTP-mediated RIRR, diffusively coupled in a spatially extended network, to study the spatiotemporal dynamics of RIRR on Ψ_M. Our major findings are: 1), as the rate of ROS production increases, mitochondria can exhibit either oscillatory dynamics facilitated by IMAC opening, or bistable dynamics facilitated by MPTP opening; 2), in a diffusively-coupled mitochondrial network, the oscillatory dynamics of IMAC-mediated RIRR results in rapidly propagating (∼25 μm/s) cellwide Ψ_M oscillations, whereas the bistable dynamics of MPTP-mediated RIRR results in slow (0.1-2 μm/s) Ψ_M depolarization waves; and 3), the slow velocity of the MPTP-mediated depolarization wave is related to competition between ROS scavenging systems and ROS diffusion. Our observations provide mechanistic insights into the spatiotemporal dynamics underlying RIRR-induced Ψ_M oscillations and waves observed experimentally in cardiac myocytes.     

Cardiac-specific mutation of Clock alters the quantitative measurements of physical activities without changing behavioral circadian rhythms

Even though peripheral circadian oscillators in the cardiovascular system are known to exist, the daily rhythms of the cardiovascular system are mainly attributed to autonomic or hormonal inputs under the control of the central oscillator, the suprachiasmatic nucleus (SCN). In order to examine the role of peripheral oscillators in the cardiovascular system, we used a transgenic mouse where the Clock gene is specifically disrupted in cardiomyocytes. In this cardiomyocyte-specific CLOCK mutant (CCM) mouse model, the circadian input from the SCN remains intact. Both CCM and wild-type (WT) littermates displayed circadian rhythms in wheel-running behavior. However, the overall wheel-running activities were significantly lower in CCM mice compared to WT over the course of 5 weeks, indicating that CCM mice either have lower baseline physical activities or they have lower physical adaptation abilities because daily wheel running, like routine exercise, induces physical adaptation over a period of time. Upon further biochemical analysis, it was revealed that the diurnal oscillations of phosphorylation states of several kinases and protein expression of the L-type voltage-gated calcium channel (L-VGCC) α1D subunit found in WT hearts were abolished in CCM hearts, indicating that in mammalian hearts, the daily oscillations of the activities of these kinases and L-VGCCs were downstream elements of the cardiac core oscillators. However, the phosphorylation of p38 MAPK exhibited robust diurnal rhythms in both WT and CCM hearts, indicating that cardiac p38 could be under the influence of the central clock through neurohormonal signals or be part of the circadian input pathway in cardiomyocytes. Taken together, these results indicate that the cardiac core oscillators have an impact in regulating circadian rhythmicities and cardiac function.     

The oscillatory behavior of pancreatic islets from mice with mitochondrial glycerol-3-phosphate dehydrogenase knockout

Glucose stimulation of pancreatic beta cells induces oscillations of the membrane potential, cytosolic Ca(2+) ([Ca(2+)](i)), and insulin secretion. Each of these events depends on glucose metabolism. Both intrinsic oscillations of metabolism and repetitive activation of mitochondrial dehydrogenases by Ca(2+) have been suggested to be decisive for this oscillatory behavior. Among these dehydrogenases, mitochondrial glycerol-3-phosphate dehydrogenase (mGPDH), the key enzyme of the glycerol phosphate NADH shuttle, is activated by cytosolic [Ca(2+)](i). In the present study, we compared different types of oscillations in beta cells from wild-type and mGPDH(-/-) mice. In clusters of 5-30 islet cells and in intact islets, 15 mM glucose induced an initial drop of [Ca(2+)](i), followed by an increase in three phases: a marked initial rise, a partial decrease with rapid oscillations and eventually large and slow oscillations. These changes, in particular the frequency of the oscillations and the magnitude of the [Ca(2+)] rise, were similar in wild-type and mGPDH(-/-) mice. Glucose-induced electrical activity (oscillations of the membrane potential with bursts of action potentials) was not altered in mGPDH(-/-) beta cells. In single islets from either type of mouse, insulin secretion strictly followed the changes in [Ca(2+)](i) during imposed oscillations induced by pulses of high K(+) or glucose and during the biphasic elevation induced by sustained stimulation with glucose. An imposed and controlled rise of [Ca(2+)](i) in beta cells similarly increased NAD(P)H fluorescence in control and mGDPH(-/-) islets. Inhibition of the malate-aspartate NADH shuttle with aminooxyacetate only had minor effects in control islets but abolished the electrical, [Ca(2+)](i) and secretory responses in mGPDH(-/-) islets. The results show that the two distinct NADH shuttles play an important but at least partially redundant role in glucose-induced insulin secretion. The oscillatory behavior of beta cells does not depend on the functioning of mGPDH and on metabolic oscillations that would be generated by cyclic activation of this enzyme by Ca(2+).     

Kinematic control of walking

The planar law of inter-segmental co-ordination we described may emerge from the coupling of neural oscillators between each other and with limb mechanical oscillators. Muscle contraction intervenes at variable times to re-excite the intrinsic oscillations of the system when energy is lost. The hypothesis that a law of coordinative control results from a minimal active tuning of the passive inertial and viscoelastic coupling among limb segments is congruent with the idea that movement has evolved according to minimum energy criteria (1, 8). It is known that multi-segment motion of mammals locomotion is controlled by a network of coupled oscillators (CPGs, see 18, 33, 37). Flexible combination of unit oscillators gives rise to different forms of locomotion. Inter-oscillator coupling can be modified by changing the synaptic strength (or polarity) of the relative spinal connections. As a result, unit oscillators can be coupled in phase, out of phase, or with a variable phase, giving rise to different behaviors, such as speed increments or reversal of gait direction (from forward to backward). Supra-spinal centers may drive or modulate functional sets of coordinating interneurons to generate different walking modes (or gaits). Although it is often assumed that CPGs control patterns of muscle activity, an equally plausible hypothesis is that they control patterns of limb segment motion instead (22). According to this kinematic view, each unit oscillator would directly control a limb segment, alternately generating forward and backward oscillations of the segment. Inter-segmental coordination would be achieved by coupling unit oscillators with a variable phase. Inter-segmental kinematic phase plays the role of global control variable previously postulated for the network of central oscillators. In fact, inter-segmental phase shifts systematically with increasing speed both in man (4) and cat (38). Because this phase-shift is correlated with the net mechanical power output over a gait cycle (3, 4), phase control could be used for limiting the overall energy expenditure with increasing speed (22). Adaptation to different walking conditions, such as changes in body posture, body weight unloading and backward walk, also involves inter-segmental phase tuning, as does the maturation of limb kinematics in toddlers.     

A mitochondrial oscillator dependent on reactive oxygen species

We describe a unique mitochondrial oscillator that depends on oxidative phosphorylation, reactive oxygen species (ROS), and mitochondrial inner membrane ion channels. Cell-wide synchronized oscillations in mitochondrial membrane potential (Delta Psi(m)), NADH, and ROS production have been recently described in isolated cardiomyocytes, and we have hypothesized that the balance between superoxide anion efflux through inner membrane anion channels and the intracellular ROS scavenging capacity play a key role in the oscillatory mechanism. Here, we formally test the hypothesis using a computational model of mitochondrial energetics and Ca(2+) handling including mitochondrial ROS production, cytoplasmic ROS scavenging, and ROS activation of inner membrane anion flux. The mathematical model reproduces the period and phase of the observed oscillations in Delta Psi(m), NADH, and ROS. Moreover, we experimentally verify model predictions that the period of the oscillator can be modulated by altering the concentration of ROS scavengers or the rate of oxidative phosphorylation, and that the redox state of the glutathione pool oscillates. In addition to its role in cellular dysfunction during metabolic stress, the period of the oscillator can be shown to span a wide range, from milliseconds to hours, suggesting that it may also be a mechanism for physiological timekeeping and/or redox signaling.     

Weakly Circadian Cells Improve Resynchrony

The mammalian suprachiasmatic nuclei (SCN) contain thousands of neurons capable of generating near 24-h rhythms. When isolated from their network, SCN neurons exhibit a range of oscillatory phenotypes: sustained or damping oscillations, or arrhythmic patterns. The implications of this variability are unknown. Experimentally, we found that cells within SCN explants recover from pharmacologically-induced desynchrony by re-establishing rhythmicity and synchrony in waves, independent of their intrinsic circadian period We therefore hypothesized that a cell''s location within the network may also critically determine its resynchronization. To test this, we employed a deterministic, mechanistic model of circadian oscillators where we could independently control cell-intrinsic and network-connectivity parameters. We found that small changes in key parameters produced the full range of oscillatory phenotypes seen in biological cells, including similar distributions of period, amplitude and ability to cycle. The model also predicted that weaker oscillators could adjust their phase more readily than stronger oscillators. Using these model cells we explored potential biological consequences of their number and placement within the network. We found that the population synchronized to a higher degree when weak oscillators were at highly connected nodes within the network. A mathematically independent phase-amplitude model reproduced these findings. Thus, small differences in cell-intrinsic parameters contribute to large changes in the oscillatory ability of a cell, but the location of weak oscillators within the network also critically shapes the degree of synchronization for the population.     

The redistribution of power: neurocardiac signaling, alcohol and gender

Human adaptability involves interconnected biological and psychological control processes that determine how successful we are in meeting internal and environmental challenges. Heart rate variability (HRV), the variability in consecutive R-wave to R-wave intervals (RRI) of the electrocardiogram, captures synergy between the brain and cardiovascular control systems that modulate adaptive responding. Here we introduce a qualitatively new dimension of adaptive change in HRV quantified as a redistribution of spectral power by applying the Wasserstein distance with exponent 1 metric (W(1)) to RRI spectral data. We further derived a new index, D, to specify the direction of spectral redistribution and clarify physiological interpretation. We examined gender differences in real time RRI spectral power response to alcohol, placebo and visual cue challenges. Adaptive changes were observed as changes in power of the various spectral frequency bands (i.e., standard frequency domain HRV indices) and, during both placebo and alcohol intoxication challenges, as changes in the structure (shape) of the RRI spectrum, with a redistribution towards lower frequency oscillations. The overall conclusions from the present study are that the RRI spectrum is capable of a fluid and highly flexible response, even when oscillations (and thus activity at the sinoatrial node) are pharmacologically suppressed, and that low frequency oscillations serve a crucial but less studied role in physical and mental health.