Bilinear dynamical systems

In this paper, we propose the use of bilinear dynamical systems (BDS)s for model-based deconvolution of fMRI time-series. The importance of this work lies in being able to deconvolve haemodynamic time-series, in an informed way, to disclose the underlying neuronal activity. Being able to estimate neuronal responses in a particular brain region is fundamental for many models of functional integration and connectivity in the brain. BDSs comprise a stochastic bilinear neurodynamical model specified in discrete time, and a set of linear convolution kernels for the haemodynamics. We derive an expectation-maximization (EM) algorithm for parameter estimation, in which fMRI time-series are deconvolved in an E-step and model parameters are updated in an M-Step. We report preliminary results that focus on the assumed stochastic nature of the neurodynamic model and compare the method to Wiener deconvolution.     

On existence and uniqueness of the carrying simplex for competitive dynamical systems

Certain dynamical models of competition are shown to have a unique invariant hypersurface Σ, having simple geometry and topology, such that every non-zero tractory is asymptotic to a trajectory in Σ.     

A dynamical systems hypothesis of schizophrenia

We propose a top-down approach to the symptoms of schizophrenia based on a statistical dynamical framework. We show that a reduced depth in the basins of attraction of cortical attractor states destabilizes the activity at the network level due to the constant statistical fluctuations caused by the stochastic spiking of neurons. In integrate-and-fire network simulations, a decrease in the NMDA receptor conductances, which reduces the depth of the attractor basins, decreases the stability of short-term memory states and increases distractibility. The cognitive symptoms of schizophrenia such as distractibility, working memory deficits, or poor attention could be caused by this instability of attractor states in prefrontal cortical networks. Lower firing rates are also produced, and in the orbitofrontal and anterior cingulate cortex could account for the negative symptoms, including a reduction of emotions. Decreasing the GABA as well as the NMDA conductances produces not only switches between the attractor states, but also jumps from spontaneous activity into one of the attractors. We relate this to the positive symptoms of schizophrenia, including delusions, paranoia, and hallucinations, which may arise because the basins of attraction are shallow and there is instability in temporal lobe semantic memory networks, leading thoughts to move too freely round the attractor energy landscape.     

Consistency principle in biological dynamical systems

We propose a principle of consistency between different hierarchical levels of biological systems. Given a consistency between molecule replication and cell reproduction, universal statistical laws on cellular chemical abundances are derived and confirmed experimentally. They include a power law distribution of gene expressions, a lognormal distribution of cellular chemical abundances over cells, and embedding of the power law into the network connectivity distribution. Second, given a consistency between genotype and phenotype, a general relationship between phenotype fluctuations by genetic variation and isogenic phenotypic fluctuation by developmental noise is derived. Third, we discuss the chaos mechanism for stem cell differentiation with autonomous regulation, resulting from a consistency between cell reproduction and growth of the cell ensemble.

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On the functional diversity of dynamical behaviour in genetic and metabolic feedback systems

Background  

Feedback regulation plays crucial roles in the robust control and maintenance of many cellular systems. Negative feedbacks are found to underline both stable and unstable, often oscillatory, behaviours. We explore the dynamical characteristics of systems with single as well as coupled negative feedback loops using a combined approach of analytical and numerical techniques. Particularly, we emphasise how the loop's characterising factors (strength and cooperativity levels) affect system dynamics and how individual loops interact in the coupled-loop systems.     

A dynamical systems perspective on vegetation theory

A dynamical systems perspective is employed to develop a simple conceptual model of vegetation and environment as coupled dynamical systems. The conceptual model characterizes the influence of environment on vegetation, the effect of vegetation on the environment, and the subsequent response of vegetation to the modified environment. Vegetation and environmental dynamics are modeled as trajectories in complementary state spaces, with the trajectories jointly determined by the position of a given site in both spaces. The vegetation and environment state spaces are coupled by the physiological requirements of the component species and the modification of environment by vegetation. From a dynamical systems perspective, current vegetation theory and analyses overemphasize environmental determination of vegetation composition and neglect the effects of vegetation on environment. A dynamical systems perspective is capable of synthesizing previous concepts of vegetation; the continuum and community type concepts are possible consequences of site specific differences in vegetation metabolism and environmental plasticity.I would like to acknowledge the helpful comments and criticisms of Drs G. Cottam, J. D. Aber, T. F. H. Allen, R. P. McIntosh, C. G. Lorimer, and anonymous reviewers.I would like to thank the University of Wisconsin, Madison for a fellowship which supported this research. This paper is dedicated with great respect and gratitude to Professor G. Cottam on the occasion of his retirement from the University of Wisconsin.     

The extinction of slowly evolving dynamical systems

The time evolution of slowly evolving discrete dynamical systems x _{i + 1} = T(r _i ,x _i ), defined on an interval [0, L], where a parameter r _ichanges slowly with respect to i is considered. For certain transformations T, once r _i reaches a critical value the system faces a non-zero probability of extinction because some x _j [0, L]. Recent ergodic theory results of Ruelle, Pianigiani, and Lasota and Yorke are used to derive a simple expression for the probability of survival of these systems. The extinction process is illustrated with two examples. One is the quadratic map, T(r, x) = rx(1 – x), and the second is a simple model for the growth of a cellular population. The survival statistics for chronic myelogenous leukemia patients are discussed in light of these extinction processes. Two other dynamical processes of biological importance, to which our results are applicable, are mentioned.     

Ventilatory support: a dynamical systems approach

Misunderstanding of the dynamical behavior of the ventilatory system, especially under assisted ventilation, may explain the problems encountered in ventilatory support monitoring. Proportional assist ventilation (PAV) that theoretically gives a breath by breath assistance presents instability with high levels of assistance. We have constructed a mathematical model of interactions between three objects: the central respiratory pattern generator modelled by a modified Van der Pol oscillator, the mechanical respiratory system which is the passive part of the system and a controlled ventilator that follows its own law. The dynamical study of our model shows the existence of two crucial behaviors, i.e. oscillations and damping, depending on only two parameters, namely the time constant of the mechanical respiratory system and a cumulative interaction index. The same result is observed in simulations of spontaneous breathing as well as of PAV. In this last case, increasing assistance leads first to an increase of the tidal volume (V_T), a further increase in assistance inducing a decrease in V_T, ending in damping of the whole system to an attractive fixed point. We conclude that instabilities observed in PAV may be explained by the different possible dynamical behaviors of the system rather than changes in mechanical characteristics of the respiratory system.     

Modeling of dynamical systems through deep learning

This review presents a modern perspective on dynamical systems in the context of current goals and open challenges. In particular, our review focuses on the key challenges of discovering dynamics from data and finding data-driven representations that make nonlinear systems amenable to linear analysis. We explore various challenges in modern dynamical systems, along with emerging techniques in data science and machine learning to tackle them. The two chief challenges are (1) nonlinear dynamics and (2) unknown or partially known dynamics. Machine learning is providing new and powerful techniques for both challenges. Dimensionality reduction methods are used for projecting dynamical methods in reduced form, and these methods perform computational efficiency on real-world data. Data-driven models drive to discover the governing equations and give laws of physics. The identification of dynamical systems through deep learning techniques succeeds in inferring physical systems. Machine learning provides advanced new and powerful algorithms for nonlinear dynamics. Advanced deep learning methods like autoencoders, recurrent neural networks, convolutional neural networks, and reinforcement learning are used in modeling of dynamical systems.     

An optimization framework of biological dynamical systems

Different biological dynamics are often described by different mathematical equations. On the other hand, some mathematical models describe many biological dynamics universally. Here, we focus on three biological dynamics: the Lotka-Volterra equation, the Hopfield neural networks, and the replicator equation. We describe these three dynamical models using a single optimization framework, which is constructed with employing the Riemannian geometry. Then, we show that the optimization structures of these dynamics are identical, and the differences among the three dynamics are only in the constraints of the optimization. From this perspective, we discuss the unified view for biological dynamics. We also discuss the plausible categorizations, the fundamental nature, and the efficient modeling of the biological dynamics, which arise from the optimization perspective of the dynamical systems.     

Bilinear state space systems for nonlinear dynamical modelling

Summary We discuss the identification of multiple input, multiple output, discrete-time bilinear state space systems. We consider two identification problems. In the first case, the input to the system is a measurable white noise sequence. We show that it is possible to identify the system by solving a nonlinear optimization problem. The number of parameters in this optimization problem can be reduced by exploiting the principle of separable least squares. A subspace-based algorithm can be used to generate initial estimates for this nonlinear identification procedure. In the second case, the input to the system is not measurable. This makes it a much more difficult identification problem than the case with known inputs. At present, we can only solve this problem for a certain class of single input, single output bilinear state space systems, namely bilinear systems in phase variable form.     

Modelling biological systems from molecules to dynamical networks

A report of the 5^th IEEE International Conference on Systems Biology (IEEE ISB2011), 2-4 September 2011, Zhuhai, China.     

Absolute stability and dynamical stabilisation in predator-prey systems

Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised.     

Theoretical approaches for dynamical ordering of biomolecular systems

Background

Living systems are characterized by the dynamic assembly and disassembly of biomolecules. The dynamical ordering mechanism of these biomolecules has been investigated both experimentally and theoretically. The main theoretical approaches include quantum mechanical (QM) calculation, all-atom (AA) modeling, and coarse-grained (CG) modeling. The selected approach depends on the size of the target system (which differs among electrons, atoms, molecules, and molecular assemblies). These hierarchal approaches can be combined with molecular dynamics (MD) simulation and/or integral equation theories for liquids, which cover all size hierarchies.

Solution scattering approaches to dynamical ordering in biomolecular systems

Clarification of solution structure and its modulation in proteins and protein complexes is crucially important to understand dynamical ordering in macromolecular systems. Small-angle x-ray scattering (SAXS) and small-angle neutron scattering (SANS) are among the most powerful techniques to derive structural information. Recent progress in sample preparation, instruments and software analysis is opening up a new era for small-angle scattering. In this review, recent progress and trends of SAXS and SANS are introduced from the point of view of instrumentation and analysis, touching on general features and standard methods of small-angle scattering. This article is part of a Special Issue entitled “Biophysical Exploration of Dynamical Ordering of Biomolecular Systems” edited by Dr. Koichi Kato.