Dynamic complexity inPhysarum polycephalum shuttle streaming

Summary The nature of the oscillator controlling shuttle streaming inPhysarum polycephalum is not well understood. To examine the possibility of complex behavior in shuttle streaming, the time between reversal of streaming direction was measured over several hours in an intact plasmodium to produce a time series. Time series data were then used to analyze shuttle streaming dynamics. Complexity in shuttle streaming is revealed by an inverse frequency (1/f) power spectrum where the amplitude of reversals is plotted against their frequency. The complex dynamics of shuttle streaming is also shown by a trajectory in phase space typical of a strange attractor. Finally, shuttle streaming time series data have a dominant Lyapunov exponent of approximately zero. Dynamic systems with a Lyapunov exponent of zero exist in a state at the edge of chaos. Systems at the edge exhibit self-organized criticality, which produces complex behavior in many physical and biological systems. We propose that complex dynamics inPhysarum shuttle streaming is an example of self-organized criticality in the cytoplasm. The complex behavior ofPhysarum is an emergent phenomenon that probably results from the interaction of actin filaments, myosin, ATP, and other components involved in cell motility.     

Inferring the Relative Resilience of Alternative States

Ecological systems may occur in alternative states that differ in ecological structures, functions and processes. Resilience is the measure of disturbance an ecological system can absorb before changing states. However, how the intrinsic structures and processes of systems that characterize their states affects their resilience remains unclear. We analyzed time series of phytoplankton communities at three sites in a floodplain in central Spain to assess the dominant frequencies or “temporal scales” in community dynamics and compared the patterns between a wet and a dry alternative state. The identified frequencies and cross-scale structures are expected to arise from positive feedbacks that are thought to reinforce processes in alternative states of ecological systems and regulate emergent phenomena such as resilience. Our analyses show a higher species richness and diversity but lower evenness in the dry state. Time series modeling revealed a decrease in the importance of short-term variability in the communities, suggesting that community dynamics slowed down in the dry relative to the wet state. The number of temporal scales at which community dynamics manifested, and the explanatory power of time series models, was lower in the dry state. The higher diversity, reduced number of temporal scales and the lower explanatory power of time series models suggest that species dynamics tended to be more stochastic in the dry state. From a resilience perspective our results highlight a paradox: increasing species richness may not necessarily enhance resilience. The loss of cross-scale structure (i.e. the lower number of temporal scales) in community dynamics across sites suggests that resilience erodes during drought. Phytoplankton communities in the dry state are therefore likely less resilient than in the wet state. Our case study demonstrates the potential of time series modeling to assess attributes that mediate resilience. The approach is useful for assessing resilience of alternative states across ecological and other complex systems.     

Self-organised criticality in a genetic model of species-species interaction

Summary Genetics incorporated into a two trophic level species-species interaction model allows the populations of species to evolve. The system enters a regime where the extinction of species follows an erratic pattern in time and appears chaotic to the eye. Quantitative measurements suggest that the dynamics of the model exhibits self organised criticality. Fourier Transform analysis of the time series and autocorrelation function for the population data, as well as the distribution of lineage sizes and longevity of lineages, all show power law behavior. The lineages are the analogue of avalanches in other models of self organised criticality     

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.     

The role of dynamic stimulation pattern in the analysis of bistable intracellular networks

Bistable systems play an important role in the functioning of living cells. Depending on the strength of the necessary positive feedback one can distinguish between (irreversible) “one-way switch” or (reversible) “toggle-switch” type behavior. Besides the well- established steady-state properties, some important characteristics of bistable systems arise from an analysis of their dynamics. We demonstrate that a supercritical stimulus amplitude is not sufficient to move the system from the lower (off-state) to the higher branch (on-state) for either a step or a pulse input. A switching surface is identified for the system as a function of the initial condition, input pulse amplitude and duration (a supercritical signal). We introduce the concept of bounded autonomy for single level systems with a pulse input. Towards this end, we investigate and characterize the role of the duration of the stimulus. Furthermore we show, that a minimal signal power is also necessary to change the steady state of the bistable system. This limiting signal power is independent of the applied stimulus and is determined only by systems parameters. These results are relevant for the design of experiments, where it is often difficult to create a defined pattern for the stimulus. Furthermore, intracellular processes, like receptor internalization, do manipulate the level of stimulus such that level and duration of the stimulus is conducive to characteristic behavior.     

Dose and frequency in cancer therapy. Theoretical non-autonomous model of p53 network

A non-autonomous model for the regulation of apoptosis by p53 is proposed as a model for cancer chronotherapy. A perturbation is introduced that simulates damage caused to DNA in a periodic regime. To characterize the resulting system dynamics, the techniques used were stroboscopic analysis, Poincare section and power spectrum. The complexity of the time series was determined using the LZ index. Periodic and quasi-periodic dynamics were obtained as the control parameters were varied. The less complex states are those corresponding to higher values of the amplitude which indicates a strict control of the dose is required on periodic treatments.     

Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies

We study the dynamics of a system of coupled nonlinear oscillators that has been used to model coordinated human movement behavior. In contrast to earlier work we examine the case where the two component oscillators have different eigenfrequencies. Problems related to the decomposition of a time series (from an experiment) into amplitude and phase are discussed. We show that oscillations at multiples of the main frequency of the oscillator system may occur in the phase and amplitude due to the choice of a coordinate system and how these oscillations can be eliminated. We derive an explicit equation for the dynamics of the relative phase of the oscillator system in phase space that enables a direct comparison between theory and experiment.     

Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies

 We study the dynamics of a system of coupled nonlinear oscillators that has been used to model coordinated human movement behavior. In contrast to earlier work we examine the case where the two component oscillators have different eigenfrequencies. Problems related to the decomposition of a time series (from an experiment) into amplitude and phase are discussed. We show that oscillations at multiples of the main frequency of the oscillator system may occur in the phase and amplitude due to the choice of a coordinate system and how these oscillations can be eliminated. We derive an explicit equation for the dynamics of the relative phase of the oscillator system in phase space that enables a direct comparison between theory and experiment. Received: 30 December 1994/Accepted in revised form: 27 June 1995     

Local and global cycles in an acarine predator-prey system: a frequency domain analysis

Techniques of time series analysis are applied to the dynamics of the phytophagous miteTetranychus urticae Koch and its predatorTyphlodromus occidentalis Nesbitt in six experimental mini-orchards, sampled weekly during two years (van de Klashorst et al., 1992). Autocovariance and crosscovariance functions characterize local and global behaviour in the time domain. Spectral density function and cross amplitude spectrum provide important information of the system's behaviour in the frequency domain. Because of a strikingly different behaviour, the total record was cut into two distinct periods. During the first period, all local systems oscillate with a frequency of four cycles per year and all in phase, resulting in a strongly periodical global behaviour at a high mean level. Since the mean local power spectra coincide with the global power spectrum over a wide range of frequencies, it is concluded that the total system of six orchards is homogeneous. During the second period, all local power spectra are mutually different, more or less smoothed with no apparent peak pointing to a periodical component. The resulting global power spectrum is almost flat at a low mean level. The system is heterogeneous with a quasi-stable global behaviour.From the cross-amplitude spectrum it became clear that the dynamics of the two species within the orchards remained strongly coupled over a wide range of frequencies.Since the experimental circumstances had not been intentionally changed, the origin of the drastic change in behaviour could not be identified. If high rates of dispersal have been the synchronizer during the first period, it is not clear why asynchrony suddenly occured in the second period.A comparison of the overall record with the results of Nachman's (1987) stochastic simulation model suggests that the change could possibly be of stochastic origin.     

Bifurcations and Intrinsic Chaotic and 1/f Dynamics in an Isolated Perfused Rat Heart

The application of the theory of chaotic dynamical systems has gradually evolved from computer simulations to assessment of erratic behavior of physical, chemical, and biological systems. Whereas physical and chemical systems lend themselves to fairly good experimental control, biologic systems, because of their inherent complexity, are limited in this respect. This has not, however, prevented a number of investigators from attempting to understand many biologic periodicities. This has been especially true regarding cardiac dynamics: the spontaneous beating of coupled and non-coupled cardiac pacemakers provides a convenient comparison to the dynamics of oscillating systems of the physical sciences. One potentially important hypothesis regarding cardiac dynamics put forth by Goldberger and colleagues, is that normal heart beat fluctuations are chaotic, and are characterized by a 1/f-like power spectrum. To evaluate these conjectures, we studied the heart beat intervals (R wave toR wave of the electocardiogram) of isolated, perfused rat hearts and their response to a variety of external perturbations. The results indicate bifurcations between complex patterns, states with positive dynamical entropies, and low values of fractal dimensions frequently seen in physical, chemical and cellular systems, as well as power law scaling of the spectrum. Additionally, these dynamics can be modeled by a simple, discrete map, which has been used to describe the dynamics of the Belousov-Zhabotinsky reaction.     

Behavior of Early Warnings near the Critical Temperature in the Two-Dimensional Ising Model

Among the properties that are common to complex systems, the presence of critical thresholds in the dynamics of the system is one of the most important. Recently, there has been interest in the universalities that occur in the behavior of systems near critical points. These universal properties make it possible to estimate how far a system is from a critical threshold. Several early-warning signals have been reported in time series representing systems near catastrophic shifts. The proper understanding of these early-warnings may allow the prediction and perhaps control of these dramatic shifts in a wide variety of systems. In this paper we analyze this universal behavior for a system that is a paradigm of phase transitions, the Ising model. We study the behavior of the early-warning signals and the way the temporal correlations of the system increase when the system is near the critical point.     

Systems biology towards life in silico: mathematics of the control of living cells

Systems Biology is the science that aims to understand how biological function absent from macromolecules in isolation, arises when they are components of their system. Dedicated to the memory of Reinhart Heinrich, this paper discusses the origin and evolution of the new part of systems biology that relates to metabolic and signal-transduction pathways and extends mathematical biology so as to address postgenomic experimental reality. Various approaches to modeling the dynamics generated by metabolic and signal-transduction pathways are compared. The silicon cell approach aims to describe the intracellular network of interest precisely, by numerically integrating the precise rate equations that characterize the ways macromolecules’ interact with each other. The non-equilibrium thermodynamic or ‘lin–log’ approach approximates the enzyme rate equations in terms of linear functions of the logarithms of the concentrations. Biochemical Systems Analysis approximates in terms of power laws. Importantly all these approaches link system behavior to molecular interaction properties. The latter two do this less precisely but enable analytical solutions. By limiting the questions asked, to optimal flux patterns, or to control of fluxes and concentrations around the (patho)physiological state, Flux Balance Analysis and Metabolic/Hierarchical Control Analysis again enable analytical solutions. Both the silicon cell approach and Metabolic/Hierarchical Control Analysis are able to highlight where and how system function derives from molecular interactions. The latter approach has also discovered a set of fundamental principles underlying the control of biological systems. The new law that relates concentration control to control by time is illustrated for an important signal transduction pathway, i.e. nuclear hormone receptor signaling such as relevant to bone formation. It is envisaged that there is much more Mathematical Biology to be discovered in the area between molecules and Life.     

Personality development in psychotherapy: a synergetic model of state-trait dynamics

Theoretical models of psychotherapy not only try to predict outcome but also intend to explain patterns of change. Studies showed that psychotherapeutic change processes are characterized by nonlinearity, complexity, and discontinuous transitions. By this, theoretical models of psychotherapy should be able to reproduce these dynamic features. Using time series derived from daily measures through internet-based real-time monitoring as empirical reference, we earlier presented a model of psychotherapy which includes five state variables and four trait variables. In mathematical terms, the traits modulate the shape of the functions which define the nonlinear interactions between the variables (states) of the model. The functions are integrated into five coupled nonlinear difference equations. In the present paper, we model how traits (dispositions or competencies of a person) can continuously be altered by new experiences and states (cognition, emotion, behavior). Adding equations that link states to traits, this model not only describes how therapeutic interventions modulate short-term change and fluctuations of psychological states, but also how these can influence traits. Speaking in terms of Synergetics (theory of self-organization in complex systems), the states correspond to the order parameters and the traits to the control parameters of the system. In terms of psychology, trait dynamics is driven by the states—i.e., by the concrete experiences of a client—and creates a process of personality development at a slower time scale than that of the state dynamics (separation of time scales between control and order parameter dynamics).     

Broadband Criticality of Human Brain Network Synchronization

Self-organized criticality is an attractive model for human brain dynamics, but there has been little direct evidence for its existence in large-scale systems measured by neuroimaging. In general, critical systems are associated with fractal or power law scaling, long-range correlations in space and time, and rapid reconfiguration in response to external inputs. Here, we consider two measures of phase synchronization: the phase-lock interval, or duration of coupling between a pair of (neurophysiological) processes, and the lability of global synchronization of a (brain functional) network. Using computational simulations of two mechanistically distinct systems displaying complex dynamics, the Ising model and the Kuramoto model, we show that both synchronization metrics have power law probability distributions specifically when these systems are in a critical state. We then demonstrate power law scaling of both pairwise and global synchronization metrics in functional MRI and magnetoencephalographic data recorded from normal volunteers under resting conditions. These results strongly suggest that human brain functional systems exist in an endogenous state of dynamical criticality, characterized by a greater than random probability of both prolonged periods of phase-locking and occurrence of large rapid changes in the state of global synchronization, analogous to the neuronal “avalanches” previously described in cellular systems. Moreover, evidence for critical dynamics was identified consistently in neurophysiological systems operating at frequency intervals ranging from 0.05–0.11 to 62.5–125 Hz, confirming that criticality is a property of human brain functional network organization at all frequency intervals in the brain's physiological bandwidth.     

A master relation defines the nonlinear viscoelasticity of single fibroblasts

Cell mechanical functions such as locomotion, contraction, and division are controlled by the cytoskeleton, a dynamic biopolymer network whose mechanical properties remain poorly understood. We perform single-cell uniaxial stretching experiments on 3T3 fibroblasts. By superimposing small amplitude oscillations on a mechanically prestressed cell, we find a transition from linear viscoelastic behavior to power law stress stiffening. Data from different cells over several stress decades can be uniquely scaled to obtain a master relation between the viscoelastic moduli and the average force. Remarkably, this relation holds independently of deformation history, adhesion biochemistry, and intensity of active contraction. In particular, it is irrelevant whether force is actively generated by the cell or externally imposed by stretching. We propose that the master relation reflects the mechanical behavior of the force-bearing actin cytoskeleton, in agreement with stress stiffening known from semiflexible filament networks.     

Do Cost Functions for Tracking Error Generalize across Tasks with Different Noise Levels?

Control of human-machine interfaces are well modeled by computational control models, which take into account the behavioral decisions people make in estimating task dynamics and state for a given control law. This control law is optimized according to a cost function, which for the sake of mathematical tractability is typically represented as a series of quadratic terms. Recent studies have found that people actually use cost functions for reaching tasks that are slightly different than a quadratic function, but it is unclear which of several cost functions best explain human behavior and if these cost functions generalize across tasks of similar nature but different scale. In this study, we used an inverse-decision-theory technique to reconstruct the cost function from empirical data collected on 24 able-bodied subjects controlling a myoelectric interface. Compared with previous studies, this experimental paradigm involved a different control source (myoelectric control, which has inherently large multiplicative noise), a different control interface (control signal was mapped to cursor velocity), and a different task (the tracking position dynamically moved on the screen throughout each trial). Several cost functions, including a linear-quadratic; an inverted Gaussian, and a power function, accurately described the behavior of subjects throughout this experiment better than a quadratic cost function or other explored candidate cost functions (p<0.05). Importantly, despite the differences in the experimental paradigm and a substantially larger scale of error, we found only one candidate cost function whose parameter was consistent with the previous studies: a power function (cost ∝ error^α) with a parameter value of α = 1.69 (1.53–1.78 interquartile range). This result suggests that a power-function is a representative function of user’s error cost over a range of noise amplitudes for pointing and tracking tasks.     

Unexpected course of nonlinear cardiac interbeat interval dynamics during childhood and adolescence

The fluctuations of the cardiac interbeat series contain rich information because they reflect variations of other functions on different time scales (e.g., respiration or blood pressure control). Nonlinear measures such as complexity and fractal scaling properties derived from 24 h heart rate dynamics of healthy subjects vary from childhood to old age. In this study, the age-related variations during childhood and adolescence were addressed. In particular, the cardiac interbeat interval series was quantified with respect to complexity and fractal scaling properties. The R-R interval series of 409 healthy children and adolescents (age range: 1 to 22 years, 220 females) was analyzed with respect to complexity (Approximate Entropy, ApEn) and fractal scaling properties on three time scales: long-term (slope β of the power spectrum, log power vs. log frequency, in the frequency range 10(-4) to 10(-2) Hz) intermediate-term (DFA, detrended fluctuation analysis, α(2)) and short-term (DFA α(1)). Unexpectedly, during age 7 to 13 years β and ApEn were higher compared to the age <7 years and age >13 years (β: -1.06 vs. -1.21; ApEn: 0.88 vs. 0.74). Hence, the heart rate dynamics were closer to a 1/f power law and most complex between 7 and 13 years. However, DFA α(1) and α(2) increased with progressing age similar to measures reflecting linear properties. In conclusion, the course of long-term fractal scaling properties and complexity of heart rate dynamics during childhood and adolescence indicates that these measures reflect complex changes possibly linked to hormonal changes during pre-puberty and puberty.     

Spectral methods for parametric sensitivity in stochastic dynamical systems

Stochastic dynamical systems governed by the chemical master equation find use in the modeling of biological phenomena in cells, where they provide more accurate representations than their deterministic counterparts, particularly when the levels of molecular population are small. The analysis of parametric sensitivity in such systems requires appropriate methods to capture the sensitivity of the system dynamics with respect to variations of the parameters amid the noise from inherent internal stochastic effects. We use spectral polynomial chaos expansions to represent statistics of the system dynamics as polynomial functions of the model parameters. These expansions capture the nonlinear behavior of the system statistics as a result of finite-sized parametric perturbations. We obtain the normalized sensitivity coefficients by taking the derivative of this functional representation with respect to the parameters. We apply this method in two stochastic dynamical systems exhibiting bimodal behavior, including a biologically relevant viral infection model.     

Osmotic stress decreases complexity underlying the electrophysiological dynamic in soybean

• Studies on plant electrophysiology are mostly focused on specific traits of single cells. Inspired by the complexity of the signalling network in plants, and by analogy with neurons in human brains, we sought evidence of high complexity in the electrical dynamics of plant signalling and a likely relationship with environmental cues.
• An EEG‐like standard protocol was adopted for high‐resolution measurements of the electrical signal in Glycine max seedlings. The signals were continuously recorded in the same plants before and after osmotic stimuli with a ?2 MPa mannitol solution. Non‐linear time series analyses methods were used as follows: auto‐correlation and cross‐correlation function, power spectra density function, and complexity of the time series estimated as Approximate Entropy (ApEn).
• Using Approximate Entropy analysis we found that the level of temporal complexity of the electrical signals was affected by the environmental conditions, decreasing when the plant was subjected to a low osmotic potential. Electrical spikes observed only after stimuli followed a power law distribution, which is indicative of scale invariance.
• Our results suggest that changes in complexity of the electrical signals could be associated with water stress conditions in plants. We hypothesised that the power law distribution of the spikes could be explained by a self‐organised critical state (SOC) after osmotic stress.