Pathological conditions resulting from instabilities in physiological control systems

Strong delayed negative feedback

In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit.

Transcription factor competition facilitates self-sustained oscillations in single gene genetic circuits

Genetic feedback loops can be used by cells to regulate internal processes or to keep track of time. It is often thought that, for a genetic circuit to display self-sustained oscillations, a degree of cooperativity is needed in the binding and unbinding of actor species. This cooperativity is usually modeled using a Hill function, regardless of the actual promoter architecture. Furthermore, genetic circuits do not operate in isolation and often transcription factors are shared between different promoters. In this work we show how mathematical modelling of genetic feedback loops can be facilitated with a mechanistic fold-change function that takes into account the titration effect caused by competing binding sites for transcription factors. The model shows how the titration effect facilitates self-sustained oscillations in a minimal genetic feedback loop: a gene that produces its own repressor directly without cooperative transcription factor binding. The use of delay-differential equations leads to a stability contour that predicts whether a genetic feedback loop will show self-sustained oscillations, even when taking the bursty nature of transcription into account.

The time scales of irreversibility in spontaneous brain activity are altered in obsessive compulsive disorder

We study how obsessive-compulsive disorder (OCD) affects the complexity and time-reversal symmetry-breaking (irreversibility) of the brain resting-state activity as measured by magnetoencephalography (MEG). Comparing MEG recordings from OCD patients and age/sex matched control subjects, we find that irreversibility is more concentrated at faster time scales and more uniformly distributed across different channels of the same hemisphere in OCD patients than in control subjects. Furthermore, the interhemispheric asymmetry between homologous areas of OCD patients and controls is also markedly different. Some of these differences were reduced by 1-year of Kundalini Yoga meditation treatment. Taken together, these results suggest that OCD alters the dynamic attractor of the brain's resting state and hint at a possible novel neurophysiological characterization of this psychiatric disorder and how this therapy can possibly modulate brain function.

Ordinal pattern-based complexity analysis of high-dimensional chaotic time series

The ordinal pattern-based complexity-entropy plane is a popular tool in nonlinear dynamics for distinguishing stochastic signals (noise) from deterministic chaos. Its performance, however, has mainly been demonstrated for time series from low-dimensional discrete or continuous dynamical systems. In order to evaluate the usefulness and power of the complexity-entropy (CE) plane approach for data representing high-dimensional chaotic dynamics, we applied this method to time series generated by the Lorenz-96 system, the generalized Hénon map, the Mackey-Glass equation, the Kuramoto-Sivashinsky equation, and to phase-randomized surrogates of these data. We find that both the high-dimensional deterministic time series and the stochastic surrogate data may be located in the same region of the complexity-entropy plane, and their representations show very similar behavior with varying lag and pattern lengths. Therefore, the classification of these data by means of their position in the CE plane can be challenging or even misleading, while surrogate data tests based on (entropy, complexity) yield significant results in most cases.

Bifurcation analysis of motoneuronal excitability mechanisms under normal and ALS conditions

Introduction

Bifurcation analysis allows the examination of steady-state, non-linear dynamics of neurons and their effects on cell firing, yet its usage in neuroscience is limited to single-compartment models of highly reduced states. This is primarily due to the difficulty in developing high-fidelity neuronal models with 3D anatomy and multiple ion channels in XPPAUT, the primary bifurcation analysis software in neuroscience.

Methods

To facilitate bifurcation analysis of high-fidelity neuronal models under normal and disease conditions, we developed a multi-compartment model of a spinal motoneuron (MN) in XPPAUT and verified its firing accuracy against its original experimental data and against an anatomically detailed cell model that incorporates known MN non-linear firing mechanisms. We used the new model in XPPAUT to study the effects of somatic and dendritic ion channels on the MN bifurcation diagram under normal conditions and after amyotrophic lateral sclerosis (ALS) cellular changes.

Results

Our results show that somatic small-conductance Ca2+-activated K (SK) channels and dendritic L-type Ca2+ channels have the strongest effects on the bifurcation diagram of MNs under normal conditions. Specifically, somatic SK channels extend the limit cycles and generate a subcritical Hopf bifurcation node in the V-I bifurcation diagram of the MN to replace a supercritical node Hopf node, whereas L-type Ca2+ channels shift the limit cycles to negative currents. In ALS, our results show that dendritic enlargement has opposing effects on MN excitability, has a greater overall impact than somatic enlargement, and dendritic overbranching offsets the dendritic enlargement hyperexcitability effects.

Discussion

Together, the new multi-compartment model developed in XPPAUT facilitates studying neuronal excitability in health and disease using bifurcation analysis.

Bifurcation analysis for a single population model with advection

In this paper, the dynamics of a single population model with a general growth function is investigated in an advective environment. We show the existence of a nonconstant positive steady state, and give sufficient conditions for the occurrence of a Hopf bifurcation at the positive steady state. Moreover, the theoretical results are applied to the diffusive Nicholson's blowflies and Mackey-Glass's models with advection and delay, respectively. We numerically show that the population density decreases as the increase of advection rate or death rate, and a delay-induced Hopf bifurcation is more likely to occur with small advection or low mortality rate.

Stability Analysis of fMRI BOLD Signals for Disease Diagnosis

Previous studies have demonstrated that the stability changes in physiological signals can reflect individuals' pathological conditions. Apart from this, according to system science theory, a large-scale system can generally be divided into many subsystems whose stability level govern its overall performance. Therefore, this study attempts to investigate the possibility of analyzing the stability of decomposed subsystems of resting-state fMRI (rs-fMRI) BOLD signals in order to assess the overall characteristic of the human brain and individuals' health conditions. We used attention deficit/hyperactive disorder (ADHD) as an example to illustrate our method. Rs-fMRI BOLD signals were first decomposed into dynamic modes (DMs) which can illuminate the patterns of brain subsystems. Each DM is associated with one eigenvalue that determines its stability as well as oscillation frequency. Accordingly, we divided the DMs within common BOLD frequency bands into stable and unstable DMs. Then, the features related to the stability of those DMs were extracted, and nine common classifiers were used to differentiate healthy controls from ADHD patients taken from ADHD-200, a well-known dataset. The results showed that almost all features were statistically significant. Additionally, our proposed approach outperforms all existing methods with the highest possible precision, recall, and area under the receiver operating characteristic curve of 100%. In sum, we are the first to evaluate the stability of BOLD signals and demonstrate its possibility for disease diagnosis. This method can unveil new mechanisms of brain function, and could be widely used in medicine and engineering.

Analysis of ECG Signals by Dynamic Mode Decomposition

OBJECTIVE

Based on cybernetics, a large system can be divided into subsystems, and the stability of each can determine the overall properties of the system. However, this stability analysis perspective has not yet been employed in electrocardiogram (ECG) signals. This is the first study to attempt to evaluate whether the stability of decomposed ECG subsystems can be analyzed in order to effectively investigate the overall performance of ECG signals, and aid in disease diagnosis.

METHODS

We used seven different cardiac pathologies (myocardial infarction, cardiomyopathy, bundle branch block, dysrhythmia, hypertrophy, myocarditis, and valvular heart disease) to illustrate our method. Dynamic mode decomposition (DMD) was first used to decompose ECG signals into dynamic modes (DMs) which can be regarded as ECG subsystems. Then, the features related to the DMs stabilities were extracted, and nine common classifiers were implemented for classification of these pathologies.

RESULTS

Most features were significant for differentiating the above-mentioned groups (value<0.05 after Bonferroni correction). In addition, our method outperformed all existing methods for cardiac pathology classification.

CONCLUSION

We have provided a new spatial and temporal decomposition method, namely DMD, to study ECG signals.

SIGNIFICANCE

Our method can reveal new cardiac mechanisms, which can contribute to the comprehensive understanding of its underlying mechanisms and disease diagnosis, and thus, can be widely used for ECG signal analysis in the future.

Node differentiation dynamics along the route to synchronization in complex networks

Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper we show how in a network of identical dynamical systems, nodes belonging to the same degree class, differentiate in the same manner, visiting a sequence of states of diverse complexity along the route to synchronization independently on the global network structure. In particular, we observe, just after interaction starts pulling orbits from the initially uncoupled attractor, a general reduction of the complexity of the dynamics of all units being more pronounced in those with higher connectivity. In the weak-coupling regime, when synchronization starts to build up, there is an increase in the dynamical complexity, whose maximum is achieved, in general, first in the hubs due to their earlier synchronization with the mean field. For very strong coupling, just before complete synchronization, we found a hierarchical dynamical differentiation with lower degree nodes being the ones exhibiting the largest complexity departure. We unveil how this differentiation route holds for several models of nonlinear dynamics, including toroidal chaos and how it depends on the coupling function. This study provides insights to understand better strategies for network identification or to devise effective methods for network inference.

Introduction to Focus Issue: Dynamical disease: A translational approach

The concept of Dynamical Diseases provides a framework to understand physiological control systems in pathological states due to their operating in an abnormal range of control parameters: this allows for the possibility of a return to normal condition by a redress of the values of the governing parameters. The analogy with bifurcations in dynamical systems opens the possibility of mathematically modeling clinical conditions and investigating possible parameter changes that lead to avoidance of their pathological states. Since its introduction, this concept has been applied to a number of physiological systems, most notably cardiac, hematological, and neurological. A quarter century after the inaugural meeting on dynamical diseases held in Mont Tremblant, Québec [Bélair et al., Dynamical Diseases: Mathematical Analysis of Human Illness (American Institute of Physics, Woodbury, NY, 1995)], this Focus Issue offers an opportunity to reflect on the evolution of the field in traditional areas as well as contemporary data-based methods.

Some elements for a history of the dynamical systems theory

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.

Adaptive Learning for Robust Radial Basis Function Networks

This article addresses the robust estimation of the output layer linear parameters in a radial basis function network (RBFN). A prominent method used to estimate the output layer parameters in an RBFN with the predetermined hidden layer parameters is the least-squares estimation, which is the maximum-likelihood (ML) solution in the specific case of the Gaussian noise. We highlight the connection between the ML estimation and minimizing the Kullback-Leibler (KL) divergence between the actual noise distribution and the assumed Gaussian noise. Based on this connection, a method is proposed using a variant of a generalized KL divergence, which is known to be more robust to outliers in the pattern recognition and machine-learning problems. The proposed approach produces a surrogate-likelihood function, which is robust in the sense that it is adaptive to a broader class of noise distributions. Several signal processing experiments are conducted using artificially generated and real-world data. It is shown that in all cases, the proposed adaptive learning algorithm outperforms the standard approaches in terms of mean-squared error (MSE). Using the relative increase in the MSE for different noise conditions, we compare the robustness of our proposed algorithm with the existing methods for robust RBFN training and show that our method results in overall improvement in terms of absolute MSE values and consistency.

Brain Network Analysis by Stable and Unstable EEG Components

OBJECTIVE

Previous studies have already shown that electroencephalography (EEG) brain network (BN) can reflect the health status of individuals. However, novel methods are still needed for BN analysis. Therefore, in this study, BNs were constructed based on stable and unstable EEG components, and these may be implemented for disease diagnosis.

METHODS

Parkinson's disease (PD) was used as an example to illustrate this method. First, EEG signals were decomposed into dynamic modes (DMs). Each DM contains one eigenvalue that can determine not only the stability of that mode, but also its corresponding oscillatory frequency. Second, the stable and unstable components of EEG signals in each frequency band (delta, theta, alpha and beta) were calculated, which are based on the stable and unstable DMs within each respective frequency band. Third, newly developed BNs were constructed, including stable brain network (SBN), unstable brain network (UBN) and inter-connected brain network (IBN). Finally, their topological attributes were extracted in order to differentiate between PD patients and healthy controls (HC). Furthermore, topological attributes were also derived from traditional brain network (TBN) for comparison.

RESULTS

Most topological attributes of SBN, UBN and IBN can significantly differentiate between PD patients and HC ( p value 0.05). Furthermore, the area under the curve (AUC), precision and recall values of SBN analysis are all significantly higher than TBN.

CONCLUSION

We proposed a new perspective on EEG BN analysis.

SIGNIFICANCE

These newly developed BNs not only have biological significance, but also could be widely applied in most medical and engineering fields.

Describing the evolution of myeloid-derived leucocytes in treated B-lineage paediatric acute lymphoblastic leukaemia with a data-driven granulocyte-monocyte-blast model

Acute lymphoblastic leukaemia (ALL) is associated with a compromised myeloid system. Understanding the state of granulopoiesis in a patient during treatment, places the clinician in an advantageous position. Mathematical models are aids able to present the clinician with insight into the behaviour of myeloid-derived leucocytes. The main objective of this investigation was to determine whether a proposed model of ALL during induction therapy would be a usable descriptor of the system. The model assumes the co-occurrence of the independent leukaemic and normal marrow populations. It is comprised of four delay-differential equations, capturing the fundamental characteristics of the blood and bone marrow myeloid leucocytes and B-lineage lymphoblasts. The effect of treatment was presumed to amplify cell loss within both populations. Clinical data was used to inform the construction, calibration and examination of the model. The model is promising-presenting a good foundation for the development of a clinical supportive tool. The predicted parameters and forecasts aligned with clinical expectations. The starting assumptions were also found to be sound. A comparative investigation highlighted the differing responses of similarly diagnosed patients during treatment and further testing on patient data emphasized patient specificity. Model examination recommended the explicit consideration of the suppressive effects of treatment on the normal population production. Additionally, patient-related factors that could have resulted in such different responses between patients need to be considered. The parameter estimates require refinement to incorporate the action of treatment. Furthermore, the myeloid populations require separate consideration. Despite the model providing explanation, incorporating these recommendations would enhance both model usability and predictive capacity.

Translational approaches to treating dynamical diseases through in silico clinical trials

The primary goal of drug developers is to establish efficient and effective therapeutic protocols. Multifactorial pathologies, including dynamical diseases and complex disorders, can be difficult to treat, given the high degree of inter- and intra-patient variability and nonlinear physiological relationships. Quantitative approaches combining mechanistic disease modeling and computational strategies are increasingly leveraged to rationalize pre-clinical and clinical studies and to establish effective treatment strategies. The development of clinical trials has led to new computational methods that allow for large clinical data sets to be combined with pharmacokinetic and pharmacodynamic models of diseases. Here, we discuss recent progress using in silico clinical trials to explore treatments for a variety of complex diseases, ultimately demonstrating the immense utility of quantitative methods in drug development and medicine.

Characterizing Chemotherapy-Induced Neutropenia and Monocytopenia Through Mathematical Modelling

In spite of the recent focus on the development of novel targeted drugs to treat cancer, cytotoxic chemotherapy remains the standard treatment for the vast majority of patients. Unfortunately, chemotherapy is associated with high hematopoietic toxicity that may limit its efficacy. We have previously established potential strategies to mitigate chemotherapy-induced neutropenia (a lack of circulating neutrophils) using a mechanistic model of granulopoiesis to predict the interactions defining the neutrophil response to chemotherapy and to define optimal strategies for concurrent chemotherapy/prophylactic granulocyte colony-stimulating factor (G-CSF). Here, we extend our analyses to include monocyte production by constructing and parameterizing a model of monocytopoiesis. Using data for neutrophil and monocyte concentrations during chemotherapy in a large cohort of childhood acute lymphoblastic leukemia patients, we leveraged our model to determine the relationship between the monocyte and neutrophil nadirs during cyclic chemotherapy. We show that monocytopenia precedes neutropenia by 3 days, and rationalize the use of G-CSF during chemotherapy by establishing that the onset of monocytopenia can be used as a clinical marker for G-CSF dosing post-chemotherapy. This work therefore has important clinical applications as a comprehensive approach to understanding the relationship between monocyte and neutrophils after cyclic chemotherapy with or without G-CSF support.

Flickering of cardiac state before the onset and termination of atrial fibrillation

Complex dynamical systems can shift abruptly from a stable state to an alternative stable state at a tipping point. Before the critical transition, the system either slows down in its recovery rate or flickers between the basins of attraction of the alternative stable states. Whether the heart critically slows down or flickers before it transitions into and out of paroxysmal atrial fibrillation (PAF) is still an open question. To address this issue, we propose a novel definition of cardiac states based on beat-to-beat (RR) interval fluctuations derived from electrocardiogram data. Our results show the cardiac state flickers before PAF onset and termination. Prior to onset, flickering is due to a "tug-of-war" between the sinus node (the natural pacemaker) and atrial ectopic focus/foci (abnormal pacemakers), or the pacing by the latter interspersed among the pacing by the former. It may also be due to an abnormal autonomic modulation of the sinus node. This abnormal modulation may be the sole cause of flickering prior to termination since atrial ectopic beats are absent. Flickering of the cardiac state could potentially be used as part of an early warning or screening system for PAF and guide the development of new methods to prevent or terminate PAF. The method we have developed to define system states and use them to detect flickering can be adapted to study critical transition in other complex systems.

Inferring the dynamics of oscillatory systems using recurrent neural networks

We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit.

Quantized Minimum Error Entropy Criterion

Comparing with traditional learning criteria, such as mean square error, the minimum error entropy (MEE) criterion is superior in nonlinear and non-Gaussian signal processing and machine learning. The argument of the logarithm in Renyi's entropy estimator, called information potential (IP), is a popular MEE cost in information theoretic learning. The computational complexity of IP is, however, quadratic in terms of sample number due to double summation. This creates the computational bottlenecks, especially for large-scale data sets. To address this problem, in this paper, we propose an efficient quantization approach to reduce the computational burden of IP, which decreases the complexity from O(N²) to O(MN) with M << N . The new learning criterion is called the quantized MEE (QMEE). Some basic properties of QMEE are presented. Illustrative examples with linear-in-parameter models are provided to verify the excellent performance of QMEE.

Connect and Conquer: Collectivized Behavior of Mitochondria and Bacteria

The connectedness of signaling components in network structures is a universal feature of biologic information processing. Such organization enables the transduction of complex input stimuli into coherent outputs and is essential in modulating activities as diverse as the cooperation of bacteria within populations and the dynamic organization of mitochondria within cells. Here, we highlight some common principles that underpin collectivization in bacteria and mitochondrial populations and the advantages conferred by such behavior. We discuss the concept that bacteria and mitochondria act as signal transducers of their localized metabolic environments to bring about energy-dependent clustering to modulate higher-order function across multiple scales.

From localization to anomalous diffusion in the dynamics of coupled kicked rotors

We study the effect of many-body quantum interference on the dynamics of coupled periodically kicked systems whose classical dynamics is chaotic and shows an unbounded energy increase. We specifically focus on an N-coupled kicked rotors model: We find that the interplay of quantumness and interactions dramatically modifies the system dynamics, inducing a transition between energy saturation and unbounded energy increase. We discuss this phenomenon both numerically and analytically through a mapping onto an N-dimensional Anderson model. The thermodynamic limit N→∞, in particular, always shows unbounded energy growth. This dynamical delocalization is genuinely quantum and very different from the classical one: Using a mean-field approximation, we see that the system self-organizes so that the energy per site increases in time as a power law with exponent smaller than 1. This wealth of phenomena is a genuine effect of quantum interference: The classical system for N≥2 always behaves ergodically with an energy per site linearly increasing in time. Our results show that quantum mechanics can deeply alter the regularity or ergodicity properties of a many-body-driven system.

Dynamical disease: Challenges for nonlinear dynamics and medicine

Dynamical disease refers to illnesses that are associated with striking changes in the dynamics of some bodily function. There is a large literature in mathematics and physics which proposes mathematical models for the physiological systems and carries out analyses of the properties of these models using nonlinear dynamics concepts involving analyses of the stability and bifurcations of attractors. This paper discusses how these concepts can be applied to medicine.

Computational modeling of neurostimulation in brain diseases

Neurostimulation as a therapeutic tool has been developed and used for a range of different diseases such as Parkinson's disease, epilepsy, and migraine. However, it is not known why the efficacy of the stimulation varies dramatically across patients or why some patients suffer from severe side effects. This is largely due to the lack of mechanistic understanding of neurostimulation. Hence, theoretical computational approaches to address this issue are in demand. This chapter provides a review of mechanistic computational modeling of brain stimulation. In particular, we will focus on brain diseases, where mechanistic models (e.g., neural population models or detailed neuronal models) have been used to bridge the gap between cellular-level processes of affected neural circuits and the symptomatic expression of disease dynamics. We show how such models have been, and can be, used to investigate the effects of neurostimulation in the diseased brain. We argue that these models are crucial for the mechanistic understanding of the effect of stimulation, allowing for a rational design of stimulation protocols. Based on mechanistic models, we argue that the development of closed-loop stimulation is essential in order to avoid inference with healthy ongoing brain activity. Furthermore, patient-specific data, such as neuroanatomic information and connectivity profiles obtainable from neuroimaging, can be readily incorporated to address the clinical issue of variability in efficacy between subjects. We conclude that mechanistic computational models can and should play a key role in the rational design of effective, fully integrated, patient-specific therapeutic brain stimulation.

Fractal gait patterns are retained after entrainment to a fractal stimulus

Previous work has shown that fractal patterns in gait can be altered by entraining to a fractal stimulus. However, little is understood about how long those patterns are retained or which factors may influence stronger entrainment or retention. In experiment one, participants walked on a treadmill for 45 continuous minutes, which was separated into three phases. The first 15 minutes (pre-synchronization phase) consisted of walking without a fractal stimulus, the second 15 minutes consisted of walking while entraining to a fractal visual stimulus (synchronization phase), and the last 15 minutes (post-synchronization phase) consisted of walking without the stimulus to determine if the patterns adopted from the stimulus were retained. Fractal gait patterns were strengthened during the synchronization phase and were retained in the post-synchronization phase. In experiment two, similar methods were used to compare a continuous fractal stimulus to a discrete fractal stimulus to determine which stimulus type led to more persistent fractal gait patterns in the synchronization and post-synchronization (i.e., retention) phases. Both stimulus types led to equally persistent patterns in the synchronization phase, but only the discrete fractal stimulus led to retention of the patterns. The results add to the growing body of literature showing that fractal gait patterns can be manipulated in a predictable manner. Further, our results add to the literature by showing that the newly adopted gait patterns are retained for up to 15 minutes after entrainment and showed that a discrete visual stimulus is a better method to influence retention.

Dynamic mechanisms of neocortical focal seizure onset

Recent experimental and clinical studies have provided diverse insight into the mechanisms of human focal seizure initiation and propagation. Often these findings exist at different scales of observation, and are not reconciled into a common understanding. Here we develop a new, multiscale mathematical model of cortical electric activity with realistic mesoscopic connectivity. Relating the model dynamics to experimental and clinical findings leads us to propose three classes of dynamical mechanisms for the onset of focal seizures in a unified framework. These three classes are: (i) globally induced focal seizures; (ii) globally supported focal seizures; (iii) locally induced focal seizures. Using model simulations we illustrate these onset mechanisms and show how the three classes can be distinguished. Specifically, we find that although all focal seizures typically appear to arise from localised tissue, the mechanisms of onset could be due to either localised processes or processes on a larger spatial scale. We conclude that although focal seizures might have different patient-specific aetiologies and electrographic signatures, our model suggests that dynamically they can still be classified in a clinically useful way. Additionally, this novel classification according to the dynamical mechanisms is able to resolve some of the previously conflicting experimental and clinical findings.

According to the WHO fact sheet, epilepsy is a neurological disorder affecting about 50 million people worldwide. Even today 30% of epilepsy patients do not respond well to drug therapies. Neocortical focal epilepsy is a particular type of epilepsy in which drug treatments fail and surgical success rate is low. Hence, research is essential to improve the treatment of this type of epilepsy. Recent advances in brain recording methods have led to new observations regarding the nature of neocortical focal epilepsy. However, some of the observations appear to be contradictory. Here, we develop a computational modelling framework that can explain the different observations as different aspects of possible mechanisms that can all lead to seizure onset. Specifically, we classify three main conditions under which focal seizure onset can happen. This classification is clinically important, as our model predicts different treatment strategies for each class. We conclude that focal seizures are diverse, not only in their electrographic appearance and aetiology, but also in their onset mechanism. Combined multiscale recordings as well as stimulation studies are required to elucidate the onset mechanism in each patient. Our work provides the first classification of possible onset mechanism.

Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations

In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.

A network-oriented perspective on cardiac calcium signaling

The normal contractile, electrical, and energetic function of the heart depends on the synchronization of biological oscillators and signal integrators that make up cellular signaling networks. In this review we interpret experimental data from molecular, cellular, and transgenic models of cardiac signaling behavior in the context of established concepts in cell network architecture and organization. Focusing on the cellular Ca(2+) handling machinery, we describe how the plasticity and adaptability of normal Ca(2+) signaling is dependent on dynamic network configurations that operate across a wide range of functional states. We consider how (mal)adaptive changes in signaling pathways restrict the dynamic range of the network such that it cannot respond appropriately to physiologic stimuli or perturbation. Based on these concepts, a model is proposed in which pathologic abnormalities in cardiac rhythm and contractility (e.g., arrhythmias and heart failure) arise as a consequence of progressive desynchronization and reduction in the dynamic range of the Ca(2+) signaling network. We discuss how a systems-level understanding of the network organization, cellular noise, and chaotic behavior may inform the design of new therapeutic modalities that prevent or reverse the disease-linked unraveling of the Ca(2+) signaling network.

Complexity analysis of spontaneous brain activity: effects of depression and antidepressant treatment

Magnetoencephalography (MEG) allows the real-time recording of neural activity and oscillatory activity in distributed neural networks. We applied a non-linear complexity analysis to resting-state neural activity as measured using whole-head MEG. Recordings were obtained from 20 unmedicated patients with major depressive disorder and 19 matched healthy controls. Subsequently, after 6 months of pharmacological treatment with the antidepressant mirtazapine 30 mg/day, patients received a second MEG scan. A measure of the complexity of neural signals, the Lempel-Ziv Complexity (LZC), was derived from the MEG time series. We found that depressed patients showed higher pre-treatment complexity values compared with controls, and that complexity values decreased after 6 months of effective pharmacological treatment, although this effect was statistically significant only in younger patients. The main treatment effect was to recover the tendency observed in controls of a positive correlation between age and complexity values. Importantly, the reduction of complexity with treatment correlated with the degree of clinical symptom remission. We suggest that LZC, a formal measure of neural activity complexity, is sensitive to the dynamic physiological changes observed in depression and may potentially offer an objective marker of depression and its remission after treatment.

Inverse problems from biomedicine: inference of putative disease mechanisms and robust therapeutic strategies

Many complex diseases that are difficult to treat cannot be mapped onto a single cause, but arise from the interplay of multiple contributing factors. In the study of such diseases, it is becoming apparent that therapeutic strategies targeting a single protein or metabolite are often not efficacious. Rather, a systems perspective describing the interaction of physiological components is needed. In this paper, we demonstrate via examples of disease models the kind of inverse problems that arise from the need to infer disease mechanisms and/or therapeutic strategies. We identify the challenges that arise, in particular the need to devise strategies that are robust against variable physiological states and parametric uncertainties.

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.

Vulnerability to paroxysmal oscillations in delayed neural networks: a basis for nocturnal frontal lobe epilepsy?

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ∼τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t → 2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.

PUPILLARY HIPPUS: AN UNRECOGNIZED EXAMPLE OF BIOLOGIC CHAOS

We analyzed the pupil size vs time from six subjects using pupilography and nonlinear techniques. The correlation dimensions ranged from 4.08 to 5.7. The Hurst exponents ranged from 0.132 to 0.546. All data sets contained at least one positive Lyapunov exponent. The use of surrogate yielded statistically significant differences for the correlation dimension. Phase space analysis yields a definite flow, and in subject two, period-doubling is evident. The accumulated evidence supports the notion that dynamics of pupil size are governed by deterministic chaos rather than a stochastic or linear process. This implies that one might discern between well and disease states using pupillography and that the dynamics can be mechanistically modeled.

Epilepsy as a dynamic disease: a tutorial of the past with an eye to the future

How can clinical epileptologists and computational neuroscientists learn to function together within the confines of interdisciplinary teams to develop new and more effective treatment strategies for epilepsy? Here we introduce epileptologists to the way modelers think about epilepsy as a dynamic disease. Not only is there terminology to be learned, but also it is necessary to identify those areas where clinical input might be expected to have the greatest impact. It is concluded that both groups have major roles to play in educating, evaluating, and shaping the direction of the efforts of each other.

How brains make chaos in order to make sense of the world

Recent “connectionist” models provide a new explanatory alternative to the digital computer as a model for brain function. Evidence from our EEG research on the olfactory bulb suggests that the brain may indeed use computational mechanisms like those found in connectionist models. In the present paper we discuss our data and develop a model to describe the neural dynamics responsible for odor recognition and discrimination. The results indicate the existence of sensory- and motor-specific information in the spatial dimension of EEG activity and call for new physiological metaphors and techniques of analysis. Special emphasis is placed in our model on chaotic neural activity. We hypothesize that chaotic behavior serves as the essential ground state for the neural perceptual apparatus, and we propose a mechanism for acquiring new forms of patterned activity corresponding to new learned odors. Finally, some of the implications of our neural model for behavioral theories are briefly discussed. Our research, in concert with the connectionist work, encourages a reevaluation of explanatory models that are based only on the digital computer metaphor.

Is the normal heart rate "chaotic" due to respiration?

The incidence of cardiovascular diseases increases with the growth of the human population and an aging society, leading to very high expenses in the public health system. Therefore, it is challenging to develop sophisticated methods in order to improve medical diagnostics. The question whether the normal heart rate is chaotic or not is an attempt to elucidate the underlying mechanisms of cardiovascular dynamics and therefore a highly controversial topical challenge. In this contribution we demonstrate that linear and nonlinear parameters allow us to separate completely the data sets of the three groups provided for this controversial topic in nonlinear dynamics. The question whether these time series are chaotic or not cannot be answered satisfactorily without investigating the underlying mechanisms leading to them. We give an example of the dominant influence of respiration on heart beat dynamics, which shows that observed fluctuations can be mostly explained by respiratory modulations of heart rate and blood pressure (coefficient of determination: 96%). Therefore, we recommend reformulating the following initial question: "Is the normal heart rate chaotic?" We rather ask the following: "Is the normal heart rate 'chaotic' due to respiration?"

Design principles of biochemical oscillators

Cellular rhythms are generated by complex interactions among genes, proteins and metabolites. They are used to control every aspect of cell physiology, from signalling, motility and development to growth, division and death. We consider specific examples of oscillatory processes and discuss four general requirements for biochemical oscillations: negative feedback, time delay, sufficient 'nonlinearity' of the reaction kinetics and proper balancing of the timescales of opposing chemical reactions. Positive feedback is one mechanism to delay the negative-feedback signal. Biological oscillators can be classified according to the topology of the positive- and negative-feedback loops in the underlying regulatory mechanism.

Properties of strange attractors in yeast glycolysis

The properties of periodic and aperiodic glycolytic oscillations observed in yeast extracts under sinusoidal glucose input were analyzed by the following methods. (1) Spectral analysis, rendering sharp peaks for periodic responses and enhanced broad-band noise for aperiodic oscillations. (2) Phase plane analysis, leading to closed and to open trajectories for periodic and aperiodic oscillations, respectively. (3) Rotation of a phase plane proportionally to time, revealing strange attractors associated with the aperiodic oscillations. (4) Stroboscopic plot on the phase plane, showing that the strange attractors follow a stretch-fold-press process, if the stroboscoping phase is varied. (5) Stroboscopic transfer plot, admitting a period of three transfer processes and thus implying chaos according to the Li-Yorke theorem. (6) Determination of the rate of information production by differentiation of the transfer plot, yielding approx. 0.21 bits per min for the chaotically glycolyzing yeast extract.