Article

The Concept of Nonlinearity in Complex Systems

Authors:
  • Military Academic Hospital Berlin, Germany
  • Brandenburg Medical School- Theodor Fontane
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

Background: Nonlinear systems are found everywhere throughout the natural world. In these systems there exists no proportionality and no simple causality between the magnitude of responses and the strength of their stimuli: small changes can have striking and unanticipated effects, whereas great stimuli will not always lead to drastic changes in a system's behavior. Over the past few years, several groups have been interested in pursuing the relevance of nonlinear concepts to medicine. Although the initial focus was on cardiovascular and neurophysiologic dynamics, it soon became clear that the models they were using had more general applications in biology and medicine. In the field of traumatology, up to now the nonlinear dynamics of the innumerable reactions and feedback loops at many structural levels of the traumatized patient have not been analyzed. Method: For a better understanding of the concept of nonlinearity and its possible implications in the field of traumatology, three examples at the molecular, the cellular and the organic levels are presented. Results and Conclusions: Nonlinear behavior in principle is the rule in highly complex reactions. This nonlinearity exists also in traumatologically relevant systems. The theories of nonlinear dynamics offer new mathematical tools to quantify, model, predict or modulate the behavior of biological systems. It should be demonstrated that the traditional Newtonian linear approach and the new nonlinear approach are essential dual aspects of any system, both being essential for a better understanding of the pathophysiologic reactions in complex situations like trauma, shock, and sepsis.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... The natural behavior of physical systems modelled by linear models is quite predictable [14,20]. Contrary to that, many natural phenomena such as alcohol, heroin or drug transmission, epidemiological models, climate-vegetation, prey-predator and smoking cessation models are governed by non-linear systems and their behavior is often unpredictable. ...
... The corresponding characteristic polynomial of (19) is (20) (λ − λ 1 )(Âλ 2 +Bλ +Ĉ) = 0 ...
... On the other hand, λ 2 and λ 3 are the roots of the quadratic equation in (20). ...
Article
Full-text available
In this paper, the effect of e-cigarettes on smoking cessation is studied using the tools of dynamical systems theory. The purpose here is to examine this efficacy by representing and analyzing a non-linear ODE system modeling potential smokers, tobacco smokers, e-cigarette smokers and quitters. The transition from smoking class to e-cigarette smoking class is represented by the “peer pressure”. The model exhibits three possible equilibrium solutions which are the smoking-free equilibrium, e-cigarette smoking-free equilibrium and endemic equilibrium. It is shown that the smoking free equilibrium always exists. Moreover, this equilibrium is stable if the basic reproduction number R0 is less than unity. When R0>1, it is shown that the smoking free equilibrium is unstable, and of the other two equilibria is stable. The global stability analysis of the smoking-free equilibrium is also given. Some numerical simulations are plotted using the data obtained from the literature. The theoretical results are also confirmed by numerical results.
... The derived equation already revealed its non-linearity. To test the superposition principle, the polynomial which describes the behavior of root 2 is assumed to take the form of (4) and (5). ...
... (4) (5) In (6) is then obtained by summing up the two preceding equations. ...
Article
Full-text available
Precision agriculture aims to maximize yield with optimum resources. Vast majority of natural systems are acknowledged as complex and non-linear. However, prior to formulation of precise models, linearity tests are performed to validate plant behavior. This study has presented proof that the water uptake system in monopodial orchid is indeed non-linear. The change in physical growth of root and stem due to temperature and relative humidity factors are observed. The work focused on Ascocenda Fuchs Harvest Moon x (V. Chaophraya x Boots) orchid hybrid. Three complementary methods are presented: linearity tests through 1) regression fitting; 2) scatter plots; and 3) cross-correlation function tests. Root diameter, stem diameter, temperature, and relative humidity are logged at 15 minutes interval for a duration of 71 days. The polynomial equations derived for root diameter and stem diameter changes attained strong regression coefficients. The non-linear behavior is further confirmed by the scatter plots where no linear associations are present between the independent and dependent variables. Subsequently, the cross-correlation function tests conducted on temperature-root diameter, temperature-stem diameter, relative humidity-root diameter, and relative humidity-stem diameter combinations also revealed weak correlation. Despite using different techniques, the behavior of physical changes has been consistently proven to be non-linear.
... Whereas complicated systems-for example an advanced aircraft or super-computer-can be comprehensively described and understood through an observation and analysis of their components and how they work together to produce a specific effect, a system that is complex cannot be understood via an analysis of its constituent elements (Cilliers 1998). In contrast to linear complicated systems, a complex system output is more than just the sum of inputs (Willy et al. 2003). Non-linearity in relational mechanisms in complex systems means that small, localised disturbances can evolve into critical states that impact the entire system (Bak 1999). ...
... Non-linearity in relational mechanisms in complex systems means that small, localised disturbances can evolve into critical states that impact the entire system (Bak 1999). As such, the system will have properties, and exhibit behaviours or mechanisms that cannot be analysed or traced through an analysis of its elements (Willy et al. 2003). If an alien were to observe humans they may observe men, women and children, and some of their relationships, but they will not be able to easily identify the invisible emergent and self-organising cultural processes that organise them into families, clans and societies. ...
Chapter
Full-text available
Complexity science provides us with a theoretical framework for understanding how complex social systems lapse into violent conflict, and how they can prevent, or recover from conflict. For a peace process to become self-sustainable, resilient social institutions need to emerge from within, i.e. from the culture, history and socio-economic context of the relevant society. International actors can assist and facilitate this process, but if they interfere too much, they will undermine the self-organising processes necessary to sustain resilient social institutions. Adaptive Peacebuilding navigates this hybrid peacebuilding dilemma with an adaptive methodology where peacebuilders, together with the communities and people affected by the conflict, actively engage in a structured process to sustain peace and resolve conflicts by employing an iterative process of learning and adaptation. A complexity informed approach to hybrid peacebuilding aims to safeguard, stimulate, facilitate and create the space for societies to develop resilient capacities for self-organisation.
... The circumstances of these deaths are often obscured by complex sociopolitical factors, creating significant issues for investigators [11]. This unpredictability and uncertainty, combined with limited data (and hidden variables), can distort conclusions about these deaths and lead to erroneous investigation decision-making [12,13], underscoring the need for caution when interpreting these events retrospectively [14]. During times of political or social instability, resources for death investigations may be limited, suboptimal or absent, forcing improvised methods for managing the dead and the missing [7,15]. ...
Article
Full-text available
Medicolegal systems investigate the cause and manner of death, particularly differentiating between unintentional and intentional deaths. The examination of remains from unlawfully killed individuals is critical in exposing human rights violations. However, forensic medical investigations of these human remains can face multiple challenges, especially in contexts marked by limited resources, political influence, and sub-optimal investigative procedures. When killings are state-sanctioned or facilitated by well-resourced non-state actors, the clandestine disposal of remains can create a culture of impunity, leaving affected families and communities without recourse or resolution. This study aggregates articles in English and Spanish, examining the current state of how forensic medical research on extrajudicial executions and forced disappearances informs practice. It highlights critical gaps in the empirical literature, particularly in the reporting of the scientific findings that impact the investigation of victims of these unlawful killings. These cases' inherently non-linear and unpredictable nature, often influenced by chaotic and unstable conditions, can create disproportionate challenges for forensics practitioners. To address these gaps, this review suggests leveraging epidemiological frameworks to track data trends in these unlawful killings, supporting public health initiatives in prevention and policy. It emphasises the need for comprehensive documentation, robust databases, and adaptive forensic methodologies to navigate uncertainties and systemic limitations inherent in this complex and unpredictable domain of medicolegal death investigation.
... COVID-19 infection, due to its complex dynamic nature, cannot be adequately described by common linear analysis methods [8]. Nonlinear mathematical approaches, involving interactions, feedback loops, or non-proportional relationships between variables, have been effectively used for the analysis, understanding, and predictions of the dynamics of this pandemic [9,10]. As among non-linear tools, dynamic system models provide valuable insights into the behavior and evolution of complex systems, enabling researchers and practitioners to make informed decisions, predict outcomes, and design interventions or control strategies over time [11], we operated this advanced data analysis method to predict the relationship between dietary mineral intake and COVID-19 incidence and hospitalization. ...
Article
Full-text available
Background The aim of this study was to determine the association between dietary mineral intake and Coronavirus-disease 2019 (COVID-19) infection and its associated hospitalization. Methods This cohort study utilized the MASHAD study population, which comprised individuals aged 35–65. Upon recruitment in 2007, dietary intake was documented using a validated 65-item food frequency questionnaire (FFQ). Data on COVID-19 PCR test results was collected from all relevant medical centers in Mashhad between February 2020 and June 2022. The regression model included dietary minerals and employed the backward variable selection method, along with advanced data analysis techniques. Results The final analysis involved 1957 participants, including 193 COVID-19-positive patients. The mean age was 49.71 and 50.28 years in the COVID-19-positive and negative groups, respectively (p = 0.12). Dietary intakes of magnesium, iron, and potassium were notably lower in COVID-19-positive patients (P < 0.05). Following adjustments for age and sex, dietary iron remained significantly associated with COVID-19 incidence (OR = 0.94, 95% CI: 0.90–0.98). Furthermore, a statistically significant relationship was observed between dietary zinc and hospitalization due to COVID-19 (OR = 0.69, 95% CI: 0.51–0.93). In dynamical system models, intakes of calcium, zinc, and iron below the cut-offs of 1138, 9.7, and 8.17 mg/day, respectively, were linked to an increased risk of COVID-19 incidence. Conclusion Higher dietary iron and zinc intake are associated with decreased risk of COVID-19 infection and hospitalization, respectively.
... Firstly, well-chosen state variables (often a single variable in one specie) as the metric for the characterization of states fail to capture and provide a systematic and comprehensive description of states associated with the integrity and the hierarchy of ecosystems, and application of these variables is and continues to be a great hindrance in the collection of reliable, long-term data (Lenton 2011;Scheffer et al. 2015). Secondly, the complexity of an ecosystem with strong nonlinearity and stochasticity can invalidate classic methods of assessing system equilibrium and transitions, due to the enormous analytical difficulties of treating the system behavior of nonlinearity and stochasticity (Willy et al. 2003). Furthermore, complex ecosystems comprise interactions of varied and complex physical, biological, and chemical processes. ...
Preprint
Complex ecosystems exhibit more nonlinearity and stochasticity than the simple ones, rendering timely and accurate detection regime shifts in complex dynamic ecosystems a challenge. To resolve this dilemma, one of the most critical steps is to determine and quantify the equilibrium states reached by complex ecosystems under a given disturbance. This study utilizes the energy-transfer-network equilibrium model based on Nash-equilibrium theory and the maximum power principle to quantify and predict the equilibrium state of a complex ecosystem with multiple trophic levels. The model successfully simulated ecosystem energy transfer under equilibrium and quantified ecosystem state. The application of the model to monitor the aboveground biomass of a long-term dataset of un-grazed steppe achieved the description and prediction of the regime shift. This approach can possibly be used not only to find the equilibrium state for complex and simple ecosystems but also to remove the limitations of current methods to determine the attraction domain or stable points through statistical or difference equations in regime shift studies.
... In nonlinear systems," there exists no proportionality and no simple causality between the magnitude of responses and the strength of their stimuli with small changes (or stressors) can have striking and unanticipated effects, whereas significant stimuli will not always lead to drastic changes in a system's behavior" (e.g. Willy et al., 2003). Both linear and nonlinear mechanisms exist within the dynamics dominating the field (Van Geert, 1998). ...
... In nonlinear systems," there exists no proportionality and no simple causality between the magnitude of responses and the strength of their stimuli with small changes (or stressors) can have striking and unanticipated effects, whereas significant stimuli will not always lead to drastic changes in a system's behavior" (e.g. Willy et al., 2003). Both linear and nonlinear mechanisms exist within the dynamics dominating the field (Van Geert, 1998). ...
Preprint
Full-text available
Objective The study aimed to clarify and refine the concepts of cumulative stressors and trauma (CST), the centrality of an event to an identity (COE), the existential annihilation anxieties (EAA), and psychopathology. The study aimed to propose and test a model in which CST affects psychopathology directly but mostly indirectly through COE and the four different types of identity-based EAA (personal/ psychic identity, collective identity, physical identity, and status identity EAA's). Further, the study aimed to replicate the previous finding that the non-linear model of CST's effects on internalizing, externalizing, and thought disorders (the psychopathology three major components) explains more variance than the linear model. Method Using path analysis, PROCESS mediation analysis, curve estimation regression, on a combined sample (N = 1566) from Egypt (N = 490), Turkey (N = 420), Kuwait (N = 300), Syria (N = 179), and the UK (N = 177), we tested the study assumptions. Results Status identity EAA and the other types of EAA related to different identities and COE mediated the major part of CST impact on psychopathology; with "status identity, EAA" had the strongest effect size. The non-linear model of the impact of CST's cumulative dynamics on psychopathology, internalizing, externalizing, thought disorders, and physical health accounted for much more variance than the linear model. Conclusions Results supported the proposed assumptions. The implications of these results for a paradigm shift in understanding stress and traumatization dynamics that go beyond the current linear approach with the sole focus on a single past stressor or traumatic stressor were discussed.
... These features are optimised according to the specific task for which the model has been trained, typically classification, regression, or recognition (LeCun et al., 2015). As biological systems are inherently non-linear, this ability to generate intricate nonlinear input-output mappings is of great benefit for resolving the heterogeneity and complexity contained within biological data (Willy et al., 2003;Janson, 2012). The features learned by ANNs can also be extracted as feature vectors from the intermediate layers of the trained model, and subsequently combined for downstream integrative analyses (Chen et al., 2020). ...
Article
Full-text available
Integrating single cell omics and single cell imaging allows for a more effective characterisation of the underlying mechanisms that drive a phenotype at the tissue level, creating a comprehensive profile at the cellular level. Although the use of imaging data is well established in biomedical research, its primary application has been to observe phenotypes at the tissue or organ level, often using medical imaging techniques such as MRI, CT, and PET. These imaging technologies complement omics-based data in biomedical research because they are helpful for identifying associations between genotype and phenotype, along with functional changes occurring at the tissue level. Single cell imaging can act as an intermediary between these levels. Meanwhile new technologies continue to arrive that can be used to interrogate the genome of single cells and its related omics datasets. As these two areas, single cell imaging and single cell omics, each advance independently with the development of novel techniques, the opportunity to integrate these data types becomes more and more attractive. This review outlines some of the technologies and methods currently available for generating, processing, and analysing single-cell omics- and imaging data, and how they could be integrated to further our understanding of complex biological phenomena like ageing. We include an emphasis on machine learning algorithms because of their ability to identify complex patterns in large multidimensional data.
... Complex systems exhibit non-linear interactions and feedback loops among their many elements (Willy, Neugebauer, & Gerngroß, 2003;Ottino, 2003;Batty & Torrens, 2005;Diez Roux, 2011;Holland, 2014;Sturmberg, 2018). This means that small causes can reverberate throughout the system leading to a disproportionately larger effect at a future point in time (i.e. ...
... It is known also as the small-world phenomenon that formalises Karinthy's idea that 'you are only ever six "degrees of separation" away from anybody else on the planet' (Watts, 2018, p. 4). 3 Interdependence becomes dynamic when over time two or more agents interact and produce changes based on their interaction. 4 And in the dynamic, complex networks the rule is nonlinear behaviour (Willy et al., 2003), where the law changes concerning the scale of variables (parameters) and as such is the distortion of the linear graph or the proportionality relation. (Yoshida, 2010) When people connect, the elements of complexity can be found in the nonlinearity, communication, emergence, diversity and prediction. ...
Article
Full-text available
The interactions, interdependence, dynamism, diversity, emergency and other elements of complexity should be embedded in legal rules to cope with the complex environment. If it is obvious that the latter is hard to manage with the classical forms of legal rules, this common-sense is tricking us into an insistence on such rules. The complex environment and the people are complex adaptive systems, and such should be also legal rules when applied in such an environment. Public systems should systemically address the environment because the latter is per se blind to rules. The aim of this paper is to give directions towards the use of complex adaptive rules with the enumeration of elements of complexity. Based on the elaborated and included elements of complexity the paper finds that collective decision-making, here named as synomy, presents the appropriate shift from experts to the people and database oracles. The possibility to store and process a large amount of data (with the better statistical prediction) gives collective wisdom preference over the people as individuals, over experts and the classical legal approaches. Based on this the paper presents different rules that are accustomed to different environments.
... In complex systems, small changes can have striking and unexpected effects; Thus, the system should be analysed as a whole rather than individual components (Willy et al., 2003). However, in healthcare, many projects either fail to take the systems approach or misapply it (Lau et al., 2015). ...
Article
The complex and dynamic features of neonatal intensive care units (NICUs) have made it necessary to think beyond traditional safety management approaches. The Functional Resonance Analysis Method (FRAM) was thus developed to explore how functional variability affects the overall system. This study performs the FRAM on the drug administration process in a NICU to understand performance variability as conditions change, as well as to understand how variability in functions influences the system in terms of both success and failure. A mixed methods approach was used, including observations, interviews and workshops. From the data obtained, we identified 21 foreground and 16 background functions and developed 58 scenarios in relation to the effects of potential variability on the system. This study shows that the FRAM can be used to determine how to respond to changing conditions, to anticipate how variability might lead to system success or failure, to understand how to monitor it and to learn from all these.
... A holistic and emergent perspective underpins complex systems theory and has already been noted as relevant to project management (Aritua, Smith & Bower 2009;Curlee 2011;Jaafari 2003;Morris 2013;Shenhar & Dvir 2007;Skyttner 2001). A central tenant of complex systems theory is that the whole cannot be understood by analysing its components in isolation (Whitty & Maylor 2009;Willy, Neugebauer & Gerngroß 2003). As posited in section 5.2 and as per Aritua, Smith and Bower (2009), to date, the reductionist paradigm has dominated project management theory. ...
Article
This paper proposes a tool that can be used by practitioners to identify and represent the enablers to, and constraints on, the progress of a specific project: the Project-space Model. The diagrammatic tool is a response to the limitations of universal “critical success factors” for projects, and the calls for a more tailored and contextualised approach to managing projects. The Project-space Model prototype presented in the article embeds concepts from Heideggerian thinking, complexity science, Gestalt theory, and Lewin's Force Field analysis and life-space model. The tool has a ‘current-space’ and a ‘forecast-space’ and information regarding the enabling and constraining factors is shown through colour, scale and placement of icons within the ‘spaces’. The model is currently being tested through an action research case study. It is anticipated that the model will enable stakeholders to identify where their attention and action is most required in a given project.
Article
The following article introduces complexity theory as an alternative for conceptualizing the dynamics of change in nonprofit health and human service organizations. We begin by reviewing theories most frequently used to frame change in nonprofits, identifying knowledge gaps that limit their explanatory capacity. The authors then introduce complexity theory as a lens for studying change as ongoing adaptation in dynamic systems; it emerges from the resolution of dualities and can appear unpredictable and nonlinear. Given the criticality of these organizations to public services, we consider new ways of analyzing nonprofit adaptation, and how understanding could lead to enhancing the sector’s capacity to respond to crises.
Chapter
The present chapter of this book delves into the exploration of metaheuristic algorithms as an avenue for solving optimization problems pertaining to engineering and intricate systems. Metaheuristics which encompass a diverse array of intelligent search and optimization techniques inspired by natural phenomena, have demonstrated their efficacy in addressing intricate, nonlinear, and multi-objective optimization challenges. Furthermore, a thorough and comprehensive overview of metaheuristic algorithms, including genetic algorithms, simulated annealing, particle swarm optimization, and ant colony optimization, among others, is provided. Additionally, the chapter delves into the synergistic potential of combining metaheuristics with other optimization techniques, as well as machine learning and data-driven approaches. Ultimately, this chapter culminates in serving as a valuable resource for researchers, practitioners, and students who possess an interest in employing metaheuristics for the optimization of engineering and complex systems.
Preprint
Full-text available
This research studies the relation between money and prices and its practical implications analyzing quarterly data from United States (1959-2022), Canada (1961-2022), United Kingdom (1986-2022), and Brazil (1996-2022). The historical, logical, and econometric consistency of the logical core of the two main theories of money is analyzed using objective bayesian and frequentist machine learning models, bayesian regularized artificial neural networks, and ensemble learning. It is concluded that money is not neutral at any time horizon and that, despite money is ultimately subordinated to prices, there is a reciprocal influence over time between money and prices which constitute a complex system. Non-neutrality is transmitted through aggregate demand and is based on the exchange value of money as a monetary unit.
Chapter
In the last decades, despite considerable theoretical progress on the nature of chemical oscillations, the only known chemical oscillators show either biological origin, like the glycolysis.
Article
Full-text available
The role of public transportation has shifted over the last 2 decades as planners and policymakers increasingly integrate new transportation infrastructure as an economic growth tool that promotes density and desirability. This shift has also positioned new infrastructure as a driver for neighbourhood change and gentrification, leading to the evolution of literature that explores transit‐induced gentrification. As this scholarship grows however, research has become fragmented, as the political economy work, which frames much of gentrification, is antipathetic to the neoclassical perspective that frames transportation research. The resulting inconsistencies have left researchers calling for the integration of new and holistic approaches that can address growing gaps. With transit‐induced gentrification becoming more prevalent across large and mid‐sized cities, and research lacking methodological consistency, this review considers: Can a complex systems thinking framework be used to better understand and address the process of transit‐induced gentrification?
Chapter
In a present competitive world of globalization, activities of chemical industries specialists are striving to improve their processes and products for the dynamic and high expectations from consumers. However, the chemical businesses can move their attention to the cost challenges if they are able to fabricate innovative products that will carry a higher value to the customers, even if the cost of final products is higher. The most common challenges in almost all chemical engineering fields are manufacturing high quality products with the lowest cost and generating abundant profit from market sales.
Chapter
Full-text available
The halo effect can be colloquially defined as a tendency of individuals to extrapolate their impressions of an aspect of an object to other aspects of that same object...
Article
Full-text available
50 days' free access https://authors.elsevier.com/c/1b18naZ3EUxiR The halo effect is a cognitive bias whereby people form an opinion about a characteristic of an attribute of a product based on their predisposition (positive or negative) toward another attribute. No formal testing of this effect is available in the hospitality and tourism literature. Thus, this study fills this gap by analyzing a sample of 21,338 hotels. Results indicate that: i) the halo effect is supported (the “other” attributes explain nearly 50% of the focal attribute “location”); ii) asymmetric effects exist because negative variations have a stronger influence than positive variations (the halo effect actually becomes a crown of thorns); and iii) varying effects exist over the range of the dependent variable.
Article
Introduction: Translational systems biology approaches can be distinguished from mainstream systems biology in that their goal is to drive novel therapies and streamline clinical trials in critical illness. One systems biology approach, dynamic mathematical modeling (DMM), is increasingly used in dealing with the complexity of the inflammatory response and organ dysfunction. The use of DMM often requires a broadening of research methods and a multidisciplinary team approach that includes bioscientists, mathematicians, engineers, and computer scientists. However, the development of these groups must overcome domain-specific barriers to communication and understanding. Methods: We present 4 case studies of successful translational, interdisciplinary systems biology efforts, which differ by organizational level from an individual to an entire research community. Results: Case 1 is a single investigator involved in DMM of the acute inflammatory response at Cook County Hospital, in which extensive translational progress was made using agent-based models of inflammation and organ damage. Case 2 is a community-level effort from the University of Witten-Herdecke in Cologne, whose efforts have led to the formation of the Society for Complexity in Acute Illness. Case 3 is an institution-based group, the Biosystems Group at the University of California, San Francisco, whose work has included a focus on a common lexicon for DMM. Case 4 is an institution-based, transdisciplinary research group (the Center for Inflammation and Regenerative Modeling at the University of Pittsburgh), whose modeling work has led to internal education efforts, grant support, and commercialization. Conclusion: A transdisciplinary approach, which involves team interaction in an iterative fashion to address ambiguity and is supported by educational initiatives, is likely to be necessary for DMM in acute illness. Communitywide organizations such as the Society of Complexity in Acute Illness must strive to facilitate the implementation of DMM in sepsis/trauma research into the research community as a whole.
Article
Full-text available
An approach is presented for making short-term predictions about the trajectories of chaotic dynamical systems. The method is applied to data on measles, chickenpox, and marine phytoplankton populations, to show how apparent noise associated with deterministic chaos can be distinguished from sampling error and other sources of externally induced environmental noise.
Article
Full-text available
Ventricular fibrillation has been modeled as cardiac chaos occuring after a series of subharmonic bifurcations. However, previous experimental studies have suggested that fibrillatory oscillations have a relatively narrow-band frequency spectrum inconsistent with a turbulent process. Similarly, during the first minute of canine fibrillation we observed only a few localized frequency peaks from the epicardial and body surface electrocardiogram rather than a broadband type of spectrum as would be predicted for chaotic dynamics. Further narrowing of the frequency spectrum occured during the second minute of fibrillation. The frequency spectrum of ventricular fibrillation contrasts with scaled, broadband spectra observed in normal cardiac function. We suggest that ventricular fibrillation may serve as a general model for transitions from broadband stability to certain types of pathological periodicities in other physiological perturbations.
Article
Full-text available
The role of functional equations to describe the exact local structure of highly bifurcated attractors ofx n+1 = L[ y] = - a[ y( g( x \mathord/ \vphantom x a a ) ) + g¢( g( x \mathord/ \vphantom x a a ) )y( - x \mathord/ \vphantom - x a a ) ]\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right] We conjecture that possesses a unique eigenvalue in excess of 1, and show that this is the -convergence rate. The form (*) is then continued to all rather than just discrete r and bifurcation values r and dynamics at such is determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted 's spectral conjecture, the stability of theg r limit of highly iterated f's, thus establishing our theory in a local sense. We show in the course of this that highly iterated f's are conjugate tog r 's, thereby providing some elementary approximation schemes for obtaining r for a chosenf.
Article
Full-text available
We consider the notion of qualitative information and the practicalities of extracting it from experimental data. Our approach, based on a theorem of Takens, draws on ideas from the generalized theory of information known as singular system analysis due to Bertero, Pike and co-workers. We illustrate our technique with numerical data from the chaotic regime of the Lorenz model.
Article
Full-text available
Using both experimental and theoretical results, this Letter describes how low-energy, feedback control signals can be successfully utilized to suppress (laminarize) chaotic flow in a thermal convection loop.
Article
Full-text available
The Hill length-tension curve of a muscle fibre is not linear. This had led us to investigate the behaviour of a model muscle sarcomere. The muscle is modelled as an oscillator of mass m subject to a nonlinear restoring force, a periodic external driving force and friction. We have found that this simple, idealized model exhibits spontaneous symmetry breaking and transition to chaos via a period doubling sequence. This result suggests that the nonlinearity in the length-tension curve of a muscle fibre could be responsible for irregular temporal behaviour and muscle tremor.
Article
Full-text available
The extreme sensitivity to initial conditions that chaotic systems display makes them unstable and unpredictable. Yet that same sensitivity also makes them highly susceptible to control, provided that the developing chaos can be analyzed in real time and that analysis is then used to make small control interventions. This strategy has been used here to stabilize cardiac arrhythmias induced by the drug ouabain in rabbit ventricle. By administering electrical stimuli to the heart at irregular times determined by chaos theory, the arrhythmia was converted to periodic beating.
Article
Full-text available
This review describes approaches to the analysis of fractal properties of physiological observations. Fractals are useful to describe the natural irregularity of physiological systems because their irregularity is not truly random and can be demonstrated to have spatial or temporal correlation. The concepts of fractal analysis are introduced from intuitive, visual, and mathematical perspectives. The regional heterogeneities of pulmonary and myocardial flows are discussed as applications of spatial fractal analysis, and methods for estimating a fractal dimension from physiological data are presented. Although the methods used for fractal analyses of physiological data are still under development and will require additional validation, they appear to have great potential for the study of physiology at scales of resolution ranging from the microcirculation to the intact organism.
Article
Full-text available
Nonlinear dynamics, a branch of the basic sciences that studies complex physical systems, offers novel approaches to long-standing problems of physiological form and function. The nonlinear concept of fractals, introduced and developed over the last decade, provides insights into the organization of complex structures such as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes. Images FIG. 2 FIG. 4
Article
Full-text available
After photodissociation of carbon monoxide bound to myoglobin, the protein relaxes to the deoxy equilibrium structure in a quake-like motion. Investigation of the proteinquake and of related intramolecular equilibrium motions shows that states and motions have a hierarchical glass-like structure.
Article
Full-text available
The concepts of chaos and its control are reviewed. Both are discussed from an experimental as well as a theoretical viewpoint. A detailed exposition of the mathematics of chaos control is presented, with an eye toward implementation in computer-controlled experiments.
Article
A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.
Article
The problems of absorption and elimination by an organism through its natural processes are in reality quite complicated, and can be analysed more easily in terms of general compartment theory. In the present study a six-compartment model is considered in order to study the kinetics of a 14C-labelled glucose. For identification purposes an optimisation method isu sed based on the properties of an Archimedes spiral, to transform the unknown parameters involved in d, the deviation of the drug concentrations from the observed concentration, into a single variable. The solution thus requires the global minimum of a function of a single variable. The results obtained are compared with those obtained with the generalised least squares method.
Article
A simple proportional‐feedback algorithm for controlling chaos is presented. The scheme is a map‐based variation of a method recently proposed by Ott, Grebogi, and Yorke [Phys. Rev. Lett. 64, 1196 (1990)] in which unstable periodic orbits embedded within a strange attractor are stabilized through deliberate perturbations of a system constraint. The simplified method offers advantages for control of systems in which more complicated algorithms might not be feasible due to short time scales or limited computational resources. Applications to chemical and biological models are presented to demonstrate the utility and limitations of the method. Low‐dimensional chaos can usually be stabilized through proportional feedback of one parameter; in some cases, however, a linear combination of several parameters must be utilized.
Article
Some time ago Jerne proposed a new theory to explain the basis of the behaviour of the immune system. He suggested the existence of a functional connected network, based on pattern recognition of the idiotypes carried by the lymphocytes, which is responsible for the self regulation of the immune system. Only 15–20% of the lymphocytes available in the immune repertoire will participate in this functional network, while the rest of the lymphocytes will be free to respond to any foreign antigen. Each individual immune repertoire will be different depending on the lymphocytes that participate in the connected network.Using a very simple cellular automata model of the immune repertoire dynamics we show that, although the usual regimes (stable and chaotic) attained by this automata, are not interesting from the biological point of view, the transition region, at the edge of chaos, is very appropriate to describe such dynamics. In this region we have obtained a functional connected network involving 10–20% of the lymphocytes available in the repertoire, as suggested by Jerne and others. The model also reproduces the immune system signature, the ensemble of different lymphocytes that each individual expresses in his immune repertoire, which varies from one individual to another. We show how the immune memory comes out as a consequence of the dynamics of the system. From our results we confirm and present evidence that the chaotic regime corresponds to a sort of non-healthy state, as has been suggested previously.
Article
A principle of self-organization toward a critical state is proposed as a metadynamics of evolutionary processes. When the propagation velocity of information is slow as in living systems, discrepancy occurs between the virtual process and the actual one. The degree of discrepancy is defined for discrete dynamical systems on the scheme of perpetual disequilibration (PD), proposed by Gunji and others. It is supposed that adaptable systems tend to evolve so that the discrepancy may be minimized. A principle of the minimum degree of PD is applied to cellular automata and Boolean networks. These complex systems have the minimum degree of PD at the border between order and chaos, and thus are expected to evolve to the critical state at the edge of chaos. This self-organized criticality is a generalized form of Bak's self-organized criticality.
Article
A semipopular account of the universal scaling theory for the period doubling route to chaos is presented.
Article
The purpose of this review article is to demonstrate via a few simple models the mechanism for a very general, universal instability - the Arnold diffusion—which occurs in the oscillating systems having more than two degrees of freedom. A peculiar feature of this instability results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations. The instability takes place generally for very special initial conditions (inside the so-called stochastic layers) which are, however, everywhere dense in the phase space of the systsm.The basic and simplest one of the models considered is that of a pendulum under an external periodic perturbation. This model represents the behavior of nonlinear oscillations near a resonance, including the phenomenon of the stochastic instability within the stochastic layer of resonance. All models are treated both analytically and numerically. Some general regulations concerning the stochastic instability are presented, including a general, semi-quantitative method-the overlap criterion—to estimate the conditions for this stochastic instability as well as its main characteristics.
Article
Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into consider­ably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic. The feasibility of very-long-range weather prediction is examined in the light of these results.
Chapter
This book presents the concepts needed to deal with self-organizing complex systems from a unifying point of view that uses macroscopic data. The various meanings of the concept "information" are discussed and a general formulation of the maximum information (entropy) principle is used. With the aid of results from synergetics, adequate objective constraints for a large class of self-organizing systems are formulated and examples are given from physics, life and computer science. The relationship to chaos theory is examined and it is further shown that, based on possibly scarce and noisy data, unbiased guesses about processes of complex systems can be made and the underlying deterministic and random forces determined. This allows for probabilistic predictions of processes, with applications to numerous fields in science, technology, medicine and economics. The extensions of the third edition are essentially devoted to an introduction to the meaning of information in the quantum context. Indeed, quantum information science and technology is presently one of the most active fields of research at the interface of physics, technology and information sciences and has already established itself as one of the major future technologies for processing and communicating information on any scale. This book addresses graduate students and nonspecialist researchers wishing to get acquainted with the concept of information from a scientific perspective in more depth. It is suitable as a textbook for advanced courses or for self-study.
Article
Biological signals like arterial blood pressure (ABP) and electrocardiograms are usually displayed in a linear fashion. The often very complex structure may, however, be better described by phase space plots and time-delayed vectors, enabling an advantageous display of the dynamics contained in the signal. The potentials of such a display were investigated during elective aortic aneurysm repair, where profound haemodynamic changes frequently occur. The peripheral volume pulse was recorded at a digit using noninvasive near infrared photoplethysmography (NIRP). All patients (n = 20, mean age 72.8 years) were invasively monitored using arterial and Swan Ganz catheters. The ABP signal was continuously recorded with a computer (sample rate 128 Hz). Two different phase space plots, [x(t), y(t + 8/128 s) and x(t), d(x(t + 8/128 s) - x(t))/dt] were calculated for the NIRP and the ABP signals and continuously displayed. The stability was subjectively assessed and the fractal dimension calculated using the 'Hausdorff dimension'. The correlation between stability, fractal dimension and frequently used parameters of patient monitoring were investigated. All patients included in the study had an uncomplicated operation. Cardiac index (CI) and oxygen delivery (DO2) increased, and systemic vascular resistance (SVR) decreased following declamping of the aorta. The ABP signal was generally more stable. After declamping of the aorta, 14 of 16 NIRP signals became unstable, and 9 of 14 ABP signals destabilised. The time required for stabilisation of the signal varied between the individual patients. Thirty minutes after declamping, 11 of 12 ABP signals were stable, whereas 3 out of 9 NIRP signals still revealed an unstable pattern. A fractal dimension was calculated by box counting, which revealed a linear regression over two orders of magnitude in a log-log plot (Hausdorff dimension between 1.19 and 1.71). The mean fractal dimension for NIRP was significantly higher than that of the ABP signal. On clamping and declamping of the aorta, a trend to a higher fractal dimension (p = 0.08) was observed for both signals analysed. No correlation was observed between the fractal dimension and ABP, SVR index, CI, DO2 index and oxygen consumption. The dynamic changes of the signals were emphasised when they were displayed as phase space plots calculated by time-delayed vectors. The time series of the signal revealed a fractal dimension, and the observed increase at the critical time points of the operation, where the need for cardiovascular regulation is most pronounced, support the contention that a physiological system based on non-linear behaviour may enable a rapid response to haemodynamic challenges. An on-line display of phase space plots calculated by time-delayed vectors may in future provide a valuable method of monitoring for high-risk patients.
Article
Synergetic concepts allow to identify emergent coordination phenomena between interacting physiological systems, for example between the cutaneous microcirculation, the sympathetic nervous system and the cardiac and pulmonary systems. The temporal patterns (oscillations of various frequencies) that are found in the data obtained with laser-Doppler anemometers (LDA; e.g. Periflux 2 used in the study) can be investigated by simultaneous recording of photoplethysmographic data obtained in the identical region of interest, as well as in cutaneous regions treated with vasoparalytic procedures which permit to record the dynamics of the arterial system. These strategies were applied to studies in the cutaneous microcirculation (volar side of the index fingers) as well as to mucosal microcirculation (maxillar gingiva) in healthy subjects and in patients suffering from autonomic dysfunction (cutaneous microcirculation) or gingivitis. By this procedure, it could be corroborated that - contrary to popular notions - the temporal fluctuations in the LDA records do not necessarily reflect myogenic vasomotion, but can have multiple causes. In a confirming recent study [Schmid-Schönbein et al., J Auton Nerv Syst, 57, 136-140, 1996], we have demonstrated that the LDA fluctuations under conditions of normal ambient temperature and hand position most likely reflect neurogenic vasoconstriction. Under exceptional conditions, different patterns emerge. Prolonged exposure to ambient temperature (18 degrees C) leads to marked vasoconstriction, with occasional vasodilator escape ('miniature hunting reaction'). Normal subjects under gravitational load and in warm environment (28 degrees C ambient) silence their neurogenetic vasoconstriction reactions, which allows sinusoidal vasomotion to dominate. A similar phenomenon is seen in neuropathic patients at 21-24 degrees C (presumably due to structural defects). Fluctuations in LDA signal taken from the healthy gingiva are entrained to arterial, those taken from inflamed gingiva to respiratory activity. The theory and practice of nonlinear analysis is discussed, and data compression procedures allowing to portray characteristic temporal patterns for future diagnostic procedures are presented.
Article
This is a lucid introduction to some of the mathematical ideas which are useful to biologists. Professor Maynard Smith introduces the reader to the ways in which biological problems can be expressed mathematically, and shows how the mathematical equations which arise in biological work can be solved. Each chapter has a number of examples which present further points of biological and mathematical interest. interest. Professor Maynard Smith's book is written for all biologists, from undergraduate level upwards, who need mathematical tools. Only an elementary knowledge of mathematics is assumed. Since there are already a number of books dealing with statistics for biologists, this book is particularly concerned with non-statistical topics.
Article
Halobacteria spontaneously reverse their swimming direction about every 10-15 s. They respond to light stimuli by a transient perturbation of this rhythm. During periodic stimulation the system shows features that are known from nonlinear oscillators. Increasing stimulation frequencies cause the following phenomena: (i) the frequency of reversals follows the stimulation frequency, (ii) transition to a state where a long and a short interval occur alternatingly and further transition to four interval lengths, (iii) appearance of irregular interval sequences, which, in a two-dimensional plot of successive intervals, reveal clearly discernible structures and suggest chaotic motion. A similar series of events can be induced in the absence of periodic stimulation, when a control parameter is changed to various constant levels. The data suggest that the system is governed by deterministic dynamical laws.
Article
A mathematical model has been formulated to analyze the effect of nonequilibrium kinetics on oxygen delivery to tissue. The model takes into account molecular diffusion, facilitated diffusion in the capillary blood, convection, chemical kinetics of O2 with hemoglobin, and the rate of metabolic consumption. A line iterative technique is described to solve numerically the resulting coupled system of nonlinear partial differential equations with physiologically relevant boundary and entrance conditions. With nonequilibrium kinetics the end-capillary PO2 is found to be lower than that in the venous blood. The effect is more pronounced during hypoxia and anemia. It is found that the tissue PO2 at the lethal corner decreases with the decrease in blood velocity, arterial PO2, hemoglobin concentration, P50, and increase in COHb concentration or metabolic rate, while the difference between end-capillary PO2 and venous PO2 increases, which reflects the effect of nonequilibrium kinetics on the delivery of O2 to tissue. Thus, the consideration of venous PO2 as an indicator of tissue PO2 in clinical and experimental studies may be questionable.
Article
A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.
Article
Over the years, there has been much discussion about the relative importance of environmental and biological factors in regulating natural populations. Often it is thought that environmental factors are associated with stochastic fluctuations in population density, and biological ones with deterministic regulation. We revisit these ideas in the light of recent work on chaos and nonlinear systems. We show that completely deterministic regulatory factors can lead to apparently random fluctuations in population density, and we then develop a new method (that can be applied to limited data sets) to make practical distinctions between apparently noisy dynamics produced by low-dimensional chaos and population variation that in fact derives from random (high-dimensional) noise, such as environmental stochasticity or sampling error. To show its practical use, the method is first applied to models where the dynamics are known. We then apply the method to several sets of real data, including newly analysed data on the incidence of measles in the United Kingdom. Here the additional problems of secular trends and spatial effects are explored. In particular, we find that on a city-by-city scale measles exhibits low-dimensional chaos (as has previously been found for measles in New York City), whereas on a larger, country-wide scale the dynamics appear as a noisy two-year cycle. In addition to shedding light on the basic dynamics of some nonlinear biological systems, this work dramatizes how the scale on which data is collected and analysed can affect the conclusions drawn.
Article
A reduced standard deviation of RR intervals (SDRR) predicts increased mortality in groups of survivors of myocardial infarction. Like SDRR, the correlation dimension (D2) describes variation within a sampled time series, but uniquely it reveals 1) the epoch's geometric structure and 2) the degrees of freedom of the generator. These unique features may be more sensitive predictors of mortality than SDRR. We developed a new algorithm for estimating D2 (i.e., the "point-D2"), tested it with known data, and found that it had greater accuracy for finite data than other published algorithms. Analysis of RR intervals from eight conscious pigs undergoing acute occlusion of the left anterior descending coronary artery revealed a drop in the point-D2 from a control mean and standard deviation of 2.50 +/- 0.81 to 1.58 +/- 0.64 during the first minute of ischemia (p less than 0.01) and to 1.07 +/- 0.18 during the last minute preceding ventricular fibrillation (p less than 0.01). Partial occlusions (50-90% reduction of coronary blood flow) evoked point-D2 reductions only 25-30% of control (p less than 0.01). The point-D2 means were correlated between pigs with the magnitude of the respiratory sinus arrhythmia (p less than 0.01), but during ischemia this correlation was replaced by one between the standard deviation of the point-D2s and SDRRs. Because the simultaneous reduction in the mean point-D2 and its standard deviation to 1.07 +/- 0.18 occurred in every case, was unique to the few minutes preceding ventricular fibrillation, and never reached these low values during other conditions in which it was reduced, we conclude that the point-D2 may be an accurate prospective predictor of mortality within the individual subject.
Article
Expert systems are computer applications with a built-in knowledge of a special field to solve problems like a human expert on this subject. An expert system consists of an inference component for problem specification and solving, a knowledge base for storage of data, facts, rules and heuristics, an interface for knowledge acquisition, another for the interaction with the user, and a component for reasoning. Pulmonary expert systems are used for automatic interpretations of lung function data which are measured on-line and directly computed. There are also some expert systems for consultation in difficult pulmonary cases and for protection against overlooking rare diseases. Perhaps the most successful applications of pulmonary expert systems are for educational purposes and a large variety of teachware is available. Pulmonary expert systems are presently limited to small applications; often it is necessary to type in large amounts of data and to follow a deeply structured cumbersome dialog. Also, the problem of responsibility for computer decisions has to be mentioned. But pulmonary expert systems as integrated supplements of normal pulmonary measuring devices, as decision support, and as parts of pulmonary teachware will be helpful tools for the pneumological physician.
Article
Markov models with discrete states, such as closed in equilibrium with closed in equilibrium with open have been widely used to model the kinetics of ion channels in the cell membrane. In these models the transition probabilities per unit time (the kinetic rate constants) are independent of the time scale on which they are measured. However, in many physical systems, a property, L, depends on the scale, epsilon, at which it is measured such that L(epsilon) alpha epsilon 1-D where D is the fractal dimension. Such systems are said to be 'fractal'. Based on the assumption that the kinetic rates are given by k(t) alpha t1-D we derive a fractal model of ion-channel kinetics. This fractal model has fewer adjustable parameters, is more consistent with the dynamics of protein conformations, and fits the single-channel recordings from the corneal endothelium better than the discrete-state Markov model.
Article
The dependence of respiratory flux via the alternative pathway on the redox poise of the ubiquinone (Q) pool was investigated in soybean cotyledon mitochondria. A marked nonlinear relationship was observed between Q-pool reduction level and O2 uptake via the alternative oxidase. Significant engagement of the alternative pathway was not apparent until Q-pool reduction level reached 35-40% but increased disproportionately on further reduction. Similar results were obtained with electron donation from either Complex 1 or Complex 2. Close agreement was obtained over a range of experimental conditions between the estimated contribution of the alternative pathway to total respiratory flux, as measured with salicylhydroxamic acid, and that predicted from the redox poise of the Q-pool. These results are discussed in terms of existing models of the regulation of respiratory flux via the alternative pathway.
Article
Linear mathematical models of the kinetics of blood coagulation have previously been presented (Levine, 1966, Science, N.Y. 152, 651; Martorana & Moro, 1974, Math. Biosci. 21, 77). In this paper a non-linear mathematical model of the extrinsic pathway of blood coagulation is presented to take into account a positive feedback. The feedback is due to factor Va as a co-factor involved in thrombin formation. The extrinsic pathway is shown to function as an amplifier cascade if a vessel wall injury exceeds a threshold value. For sub-threshold stimulation, the extrinsic pathway does not function.
Article
With the help of several independent methods of nonlinear dynamics, the electrocardiograms (ECG) of four normal human hearts are studied qualitatively and quantitatively. A total of 36 leads were tested. The power spectrum, the autocorrelation function, the phase portrait, the Poincaré section, the correlation dimension, the Lyapunov exponent and the Kolmogorov entropy all point to the fact that the normal heart is not a perfect oscillator. The cardiac activity stems from deterministic dynamics of chaotic nature characterized by correlation dimensions D2 ranging from 3.6 to 5.2. Two different phase spaces are constructed for the evaluation of D2: the introduction of time lags and the direct use of space vectors give similar results. It is shown that the variabilities in interbeat intervals are not random but exhibit short range correlations governed by deterministic laws. These correlations may be related to the accelerating and decelerating physiological processes. This new approach to the cardiac activity may be used in clinical diagnosis. Also they are valuable tools for the evaluation of mathematical models which describe cardiac activity in terms of evolution equations.
Article
(1) Nonlinear mechanisms may apply both to the understanding of SA-AV node interactions and to bifurcations leading to certain types of AV block. (2) The fractal His-Purkinje system serves as the structural substrate for the generation of the broadband, inverse power-law spectrum of the stable ventricular depolarization (QRS) waveform. (3) Fractal anatomy is also seen in multiple other systems: pulmonary, hepatobiliary, renal, etc. Fractal morphogenesis may reflect a type of critical phenomenon that results in the generation of these irregular, but self-similar structures. (4) Self-similar (fractal) scaling may underlie the 1/f-like spectra seen in multiple systems (e.g., interbeat interval variability, daily neutrophil fluctuations). This fractal scaling may provide a mechanism for the "constrained randomness" that appears to underlie physiological variability and adaptability. (5) Behavior consistent with subharmonic bifurcations is seen in cardiac electrophysiology (e.g., sick sinus syndrome) and hemodynamic perturbations (e.g., swinging heart phenomenon in pericardial tamponade). (6) Ventricular tachyarrhythmias associated with sudden cardiac death (e.g., torsades de pointes, ventricular fibrillation) appear to reflect relatively periodic, not chaotic (turbulent) processes resulting from disruption of the physiologic fractal depolarization sequence. (7) Spectral analysis of Holter monitor data may help in the detection of patients at high risk for sudden death.
Article
The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negative and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.
Article
Knowledge-based expert systems for medical applications have received considerable attention in recent years. In this review, fundamental terms and notions of artificial intelligence techniques as applied to expert systems are introduced. The most well-known and influential medical expert systems are discussed in detail, and newer efforts are surveyed. A critical comparison of strengths and weaknesses of the systems is made, discussing depth and complexity of knowledge, acquisition of knowledge, user interaction and explanations, knowledge engineering tools, system evaluations, and user resistance. Long- and short-term trends are appraised.
Article
Electrical activation of the ventricles via the His-Purkinje system is represented on the body surface by a waveform with a broad range of frequency components. We speculate that this process is mediated by current flow through a fractal-like conduction network and therefore that the broadband spectrum of the depolarization waveform should be scaled as a power-law distribution. The prediction is confirmed by Fourier analysis of electrocardiographic data from healthy men. This observation suggests a new dynamical link between nonlinear (fractal) structure and nonlinear function in a stable physiologic system.
Article
To investigate how arterial baroreceptors affect the dynamic properties of short-term blood pressure control, we determined Lyapunov exponents and correlation dimensions of blood pressure. Two groups of conscious dogs were studied: a control group (n = 7) and a group subjected to total sinoaortic and cardiopulmonary baroreceptor denervation (n = 7). As a measure of variability, standard deviation was determined and power spectra were calculated. In the lower frequency range (f < 0.1 Hz) power density was inversely related to frequency in both groups, indicating "1/f noise." Estimating the correlation dimension via the Grassberger-Procaccia algorithm as a quantification of complexity revealed a decrease after baroreceptor denervation (1.74 +/- 0.2 vs. 3.05 +/- 0.23 control; P < 0.05). Determination of the largest Lyapunov exponents lambda 1, which indicates the sensitive dependence on initial conditions, a hallmark of chaos, also yielded a diminution after denervation (lambda 1 = 0.74 +/- 0.08 vs. 1.85 +/- 0.18, P < 0.01). The results were cross-checked with surrogate data statistics. The null hypothesis, that there is no nonlinear structure in arterial blood pressure time series, was rejected. This shows that after baroreceptor denervation, blood pressure control is less complex and less sensitive to initial conditions ("chaos"). In contrast, variability (standard deviation) is increased (22.2 +/- 3.1 denervation vs. 8.3 +/- 1.4 control; P < 0.05). It is concluded that under physiological conditions, arterial and cardiopulmonary baroreceptors reduce variability of blood pressure, however, at the cost of blood pressure being less predictable. Thus the regulation is more sensitive depending on initial conditions.(ABSTRACT TRUNCATED AT 250 WORDS)
Article
A new model for intracellular Ca2+ oscillations is presented. The new model reinterprets two previous models, the ICC and CICR mechanisms, and incorporates the bell-shaped dependence of Ca2+ release on cytosolic [Ca2+]. Complex oscillations and chaos are found with this new model, confirming experimental observations of complex oscillations. A rich bifurcation sequence is found for the model as the stimulation due to agonist (R) is varied, including a period doubling route to chaos and a period-adding sequence of mixed-mode states.
Article
Circulating growth hormone (GH) levels in normal persons fluctuate widely because of pulsatile GH secretion. It is not known whether this pulsatile nature and rhythmicity exist in severe injury. These data become necessary to decide the timing of supplementary GH administration for its optimal utilization. The purpose of this study was to investigate the GH circadian variation with respect to that of insulin-like growth factor-1 (IGF-1), insulin, C-peptide, and cortisol in the early flow phase of injury. Plasma GH, IGF-1, insulin, C-peptide, and cortisol levels were measured at 1-hour intervals during 24 hours (8 AM to 8 AM) in 10 severely injured adults with multiple trauma during the early catabolic flow phase 24 to 48 hours after injury, when patients received maintenance fluids without calories or nitrogen. The 24-hour integrated GH concentration is not different from either 12-hour mean diurnal or 12-hour mean nocturnal or mean 8 AM GH concentration. Pulsatile GH bursts persist in injured patients during both day and night. Pulsatile bursts do not exist for IGF-1, insulin, and C-peptide. The plasma levels of cortisol show time-dependent daily maximum and minimum levels. Pulsatile GH bursts persist in injured patients but less frequently than seen in normal persons. The time of bolus administration of GH to augment the anabolic GH action in patients with trauma does not matter; however, for convenience morning administration may be preferable for patients in the intensive care unit.
Article
Two computer programs are described for evaluating the evidence for chaos and nonlinearity in time series data. "bx" is an efficient algorithm for computing the correlation integral (from which correlation dimension can be estimated); and "surrogat" is a Fourier-transform-based algorithm for generating surrogate data consistent with a null hypothesis that the data arise as a result of a linear stochastic process.
Article
When a person attempts to produce from memory a given spatial or temporal interval, there is inevitably some error associated with the estimate. The time course of this error was measured in a series of experiments where subjects repeatedly attempted to replicate given target intervals. Sequences of the errors in both spatial and temporal replications were found to fluctuate as 1/f noises. 1/f noise is encountered in a wide variety of physical systems and is theorized to be a characteristic signature of complexity.