Perturbations both trigger and delay seizures due to generic properties of slow-fast relaxation oscillators

The mechanisms underlying the emergence of seizures are one of the most important unresolved issues in epilepsy research. In this paper, we study how perturbations, exogenous or endogenous, may promote or delay seizure emergence. To this aim, due to the increasingly adopted view of epileptic dynamics in terms of slow-fast systems, we perform a theoretical analysis of the phase response of a generic relaxation oscillator. As relaxation oscillators are effectively bistable systems at the fast time scale, it is intuitive that perturbations of the non-seizing state with a suitable direction and amplitude may cause an immediate transition to seizure. By contrast, and perhaps less intuitively, smaller amplitude perturbations have been found to delay the spontaneous seizure initiation. By studying the isochrons of relaxation oscillators, we show that this is a generic phenomenon, with the size of such delay depending on the slow flow component. Therefore, depending on perturbation amplitudes, frequency and timing, a train of perturbations causes an occurrence increase, decrease or complete suppression of seizures. This dependence lends itself to analysis and mechanistic understanding through methods outlined in this paper. We illustrate this methodology by computing the isochrons, phase response curves and the response to perturbations in several epileptic models possessing different slow vector fields. While our theoretical results are applicable to any planar relaxation oscillator, in the motivating context of epilepsy they elucidate mechanisms of triggering and abating seizures, thus suggesting stimulation strategies with effects ranging from mere delaying to full suppression of seizures.     

Isochron-based phase response analysis of circadian rhythms

Circadian rhythms possess the ability to robustly entrain to the environmental cycles. This ability relies on the phase synchronization of circadian rhythm gene regulation to different environmental cues, of which light is the most obvious and important. The elucidation of the mechanism of circadian entrainment requires an understanding of circadian phase behavior. This article presents two phase analyses of oscillatory systems for infinitesimal and finite perturbations based on isochrons as a phase metric of a limit cycle. The phase response curve of circadian rhythm can be computed from the results of the analyses. The application to a mechanistic Drosophila circadian rhythm model gives experimentally testable hypotheses for the control mechanisms of circadian phase responses and evidence for the role of phase and period modulations in circadian photic entrainment.     

Phase computations and phase models for discrete molecular oscillators

Background

Biochemical oscillators perform crucial functions in cells, e.g., they set up circadian clocks. The dynamical behavior of oscillators is best described and analyzed in terms of the scalar quantity, phase. A rigorous and useful definition for phase is based on the so-called isochrons of oscillators. Phase computation techniques for continuous oscillators that are based on isochrons have been used for characterizing the behavior of various types of oscillators under the influence of perturbations such as noise.

Results

In this article, we extend the applicability of these phase computation methods to biochemical oscillators as discrete molecular systems, upon the information obtained from a continuous-state approximation of such oscillators. In particular, we describe techniques for computing the instantaneous phase of discrete, molecular oscillators for stochastic simulation algorithm generated sample paths. We comment on the accuracies and derive certain measures for assessing the feasibilities of the proposed phase computation methods. Phase computation experiments on the sample paths of well-known biological oscillators validate our analyses.

Conclusions

The impact of noise that arises from the discrete and random nature of the mechanisms that make up molecular oscillators can be characterized based on the phase computation techniques proposed in this article. The concept of isochrons is the natural choice upon which the phase notion of oscillators can be founded. The isochron-theoretic phase computation methods that we propose can be applied to discrete molecular oscillators of any dimension, provided that the oscillatory behavior observed in discrete-state does not vanish in a continuous-state approximation. Analysis of the full versatility of phase noise phenomena in molecular oscillators will be possible if a proper phase model theory is developed, without resorting to such approximations.     

Spectral methods for parametric sensitivity in stochastic dynamical systems

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

Signal acquisition and analysis for cortical control of neuroprosthetics

Work in cortically controlled neuroprosthetic systems has concentrated on decoding natural behaviors from neural activity, with the idea that if the behavior could be fully decoded it could be duplicated using an artificial system. Initial estimates from this approach suggested that a high-fidelity signal comprised of many hundreds of neurons would be required to control a neuroprosthetic system successfully. However, recent studies are showing hints that these systems can be controlled effectively using only a few tens of neurons. Attempting to decode the pre-existing relationship between neural activity and natural behavior is not nearly as important as choosing a decoding scheme that can be more readily deployed and trained to generate the desired actions of the artificial system. These artificial systems need not resemble or behave similarly to any natural biological system. Effective matching of discrete and continuous neural command signals to appropriately configured device functions will enable effective control of both natural and abstract artificial systems using compatible thought processes.     

Combination chemical genetics

Predicting the behavior of living organisms is an enormous challenge given their vast complexity. Efforts to model biological systems require large datasets generated by physical binding experiments and perturbation studies. Genetic perturbations have proven important and are greatly facilitated by the advent of comprehensive mutant libraries in model organisms. Small-molecule chemical perturbagens provide a complementary approach, especially for systems that lack mutant libraries, and can easily probe the function of essential genes. Though single chemical or genetic perturbations provide crucial information associating individual components (for example, genes, proteins or small molecules) with pathways or phenotypes, functional relationships between pathways and modules of components are most effectively obtained from combined perturbation experiments. Here we review the current state of and discuss some future directions for 'combination chemical genetics', the systematic application of multiple chemical or mixed chemical and genetic perturbations, both to gain insight into biological systems and to facilitate medical discoveries.     

What makes ecological systems reactive?

Although perturbations from a stable equilibrium must ultimately vanish, they can grow initially, and the maximum initial growth rate is called reactivity. Reactivity thus identifies systems that may undergo transient population surges or drops in response to perturbations; however, we lack biological and mathematical intuition about what makes a system reactive. This paper presents upper and lower bounds on reactivity for an arbitrary linearized model, explores their strictness, and discusses their biological implications. I find that less stable systems (i.e. systems with long transients) have a smaller possible range of reactivities for which no perturbations grow. Systems with more species have a higher capacity to be reactive, assuming species interactions do not weaken too rapidly as the number of species increases. Finally, I find that in discrete time, reactivity is determined largely by mean interaction strength and neither discrete nor continuous time reactivity are sensitive to food web topology.     

On the representation of multi-input systems: Computational properties of polynomial algorithms

This paper introduces a theoretical framework for characterizing and classifying simple parallel algorithms and systems with many inputs, for example an array of photoreceptors. The polynomial representation (Taylor series development) of a large class of operators is introduced and its range of validity discussed. The problems involved in the polynomial approximation of systems are also briefly reviewed. Symmetry properties of the input-output map and their implications for the system structure (i.e. its kernels) are studied. Finally, the computational properties of polynomial mappings are characterized.     

Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities.

The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.     

Optimizing the sensitivity of enzyme systems to perturbations from equilibrium

Algebraic derivations and numerical examples illustrate how metabolite pool sizes and enzyme rate constants influence the rate at which a multireactant enzyme system, initially poised in a near-equilibrium steady state, responds to small perturbations in the concentrations of the reactants. Certain enzymes, such as those employing the ordered bi bi catalytic mechanism, become relatively insensitive to perturbations when the reactants are all present at high concentrations. Other enzymes, such as those employing the ping-pong bi bi mechanism, are most sensitive to perturbations at high reactant concentrations. The ratio of the reactant concentrations to one another significantly alters sensitivity to perturbations; equations are presented for calculation of the reactant concentrations yielding maximal sensitivity to perturbations. Natural selection could choose metabolite pool sizes and enzyme rate constants which would optimize the performance of these systems, but changing metabolic loads (naturally or experimentally imposed) constantly alter the sensitivity of these systems to perturbations, changing the relative strengths of various connections in metabolic control networks.     

Smoothing molecular interactions: the "kinetic buffer" effect of intrinsically disordered proteins

Intrinsically disordered proteins (IDPs) widely participate in molecular recognition and signaling processes in cells by interacting with other molecules. Compared with ordered proteins, IDPs usually possess stronger intermolecular interactions in binding. As a result, the interface structure of IDPs in complexes is distinct from that of ordered-protein complexes, and this difference may have essential effect on the response to various perturbations in a cell. In this study, we examined the perturbations of intermolecular interactions and temperature on the coupled folding and binding processes of pKID to KIX domains by performing molecular dynamics simulations. By comparing a series of virtual pKID systems with various degree of disorder, we found that the complex stability and the binding kinetics of the disordered systems were less sensitive to the perturbations than the ordered systems. The origin of the lower response sensitivity of IDPs was attributed to their higher flexibility in the complex interface, which was further supported by an analysis on protein complex structures. On the basis of our simulations and results from the literature, we speculate IDPs may not only interact with their biological partners with high specificity and low affinity but also may be resistant to the perturbations in the environment and transmit signals fast and smooth. We proposed to name it the "kinetic buffer" effect.     

Allele-Specific Behavior of Molecular Networks: Understanding Small-Molecule Drug Response in Yeast

The study of systems genetics is changing the way the genetic and molecular basis of phenotypic variation, such as disease susceptibility and drug response, is being analyzed. Moreover, systems genetics aids in the translation of insights from systems biology into genetics. The use of systems genetics enables greater attention to be focused on the potential impact of genetic perturbations on the molecular states of networks that in turn affects complex traits. In this study, we developed models to detect allele-specific perturbations on interactions, in which a genetic locus with alternative alleles exerted a differing influence on an interaction. We utilized the models to investigate the dynamic behavior of an integrated molecular network undergoing genetic perturbations in yeast. Our results revealed the complexity of regulatory relationships between genetic loci and networks, in which different genetic loci perturb specific network modules. In addition, significant within-module functional coherence was found. We then used the network perturbation model to elucidate the underlying molecular mechanisms of individual differences in response to 100 diverse small molecule drugs. As a result, we identified sub-networks in the integrated network that responded to variations in DNA associated with response to diverse compounds and were significantly enriched for known drug targets. Literature mining results provided strong independent evidence for the effectiveness of these genetic perturbing networks in the elucidation of small-molecule responses in yeast.     

Identifying and classifying biomedical perturbations in text

Molecular perturbations provide a powerful toolset for biomedical researchers to scrutinize the contributions of individual molecules in biological systems. Perturbations qualify the context of experimental results and, despite their diversity, share properties in different dimensions in ways that can be formalized. We propose a formal framework to describe and classify perturbations that allows accumulation of knowledge in order to inform the process of biomedical scientific experimentation and target analysis. We apply this framework to develop a novel algorithm for automatic detection and characterization of perturbations in text and show its relevance in the study of gene–phenotype associations and protein–protein interactions in diabetes and cancer. Analyzing perturbations introduces a novel view of the multivariate landscape of biological systems.     

Effect of perturbations on magnetic fields as a possible problem for implementing toroidal magnetic configurations with a zero rotational transform

Approved reliable methods and a unified quasi-resonant approach are used to reproduce the results characterizing the effect of dangerous perturbations on three particular types of toroidal magnetic configurations with a zero rotational transform (ι/2π = 0). An analysis is made of the perturbations whose geometry and amplitudes correspond to imperfections and defects in the assembly of the actual magnetic systems of conventional stellarators. The effect of perturbations on magnetic configurations with ι/2π = 0 is of the same quasi-resonant nature as that for systems with ι/2π ≠ 0 and a zero magnetic shear. However, in contrast to shearless configurations with a nonzero rotational transform, the most dangerous perturbations are those with n = 0 and m = 1, 2, 3, … (where m is the poloidal mode number and n is the toroidal mode number) and the effect of harmonics with the smallest poloidal and toroidal numbers is more destructive and nonlocal even for the amplitudes corresponding to imperfections and defects in the assembly of conventional stellarators. Such an extremely high sensitivity of magnetic systems with ι/2π = 0 to the perturbations under consideration may be a serious obstacle for their practical implementation.     

Deterministic characterization of phase noise in biomolecular oscillators

On top of the many external perturbations, cellular oscillators also face intrinsic perturbations due the randomness of chemical kinetics. Biomolecular oscillators, distinct in their parameter sets or distinct in their architecture, show different resilience with respect to such intrinsic perturbations. Assessing this resilience can be done by ensemble stochastic simulations. These are computationally costly and do not permit further insights into the mechanistic cause of the observed resilience. For reaction systems operating at a steady state, the linear noise approximation (LNA) can be used to determine the effect of molecular noise. Here we show that methods based on LNA fail for oscillatory systems and we propose an alternative ansatz. It yields an asymptotic expression for the phase diffusion coefficient of stochastic oscillators. Moreover, it allows us to single out the noise contribution of every reaction in an oscillatory system. We test the approach on the one-loop model of the Drosophila circadian clock. Our results are consistent with those obtained through stochastic simulations with a gain in computational efficiency of about three orders of magnitude.     

Interlinked mutual inhibitory positive feedbacks induce robust cellular memory effects

Mutual inhibitory positive feedback (MIPF), or double-negative feedback, is a key regulatory motif of cellular memory with the capability of maintaining switched states for transient stimuli. Such MIPFs are found in various biological systems where they are interlinked in many cases despite a single MIPF can still realize such a memory effect. An intriguing question then arises about the advantage of interlinking MIPFs instead of exploiting an isolated single MIPF to realize the memory effect. We have investigated the advantages of interlinked MIPF systems through mathematical modeling and computer simulations. Our results revealed that interlinking MIPFs expands the parameter range of achieving the memory effect, or the memory region, thereby making the system more robust to parameter perturbations. Moreover, the minimal duration and amplitude of an external stimulus required for off-to-on state transition are increased and, as a result, external noises can more effectively be filtered out. Hence, interlinked MIPF systems can realize more robust cellular memories with respect to both parameter perturbations and external noises. Our study suggests that interlinked MIPF systems might be an evolutionary consequence acquired for a more reliable memory effect by enhancing robustness against noisy cellular environments.     

Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations

Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.     

Phytoplankton responses to repeated pulse perturbations imposed on a trend of increasing eutrophication

While eutrophication remains one of the main pressures acting on freshwater ecosystems, the prevalence of anthropogenic and nature‐induced stochastic pulse perturbations is predicted to increase due to climate change. Despite all our knowledge on the effects of eutrophication and stochastic events operating in isolation, we know little about how eutrophication may affect the response and recovery of aquatic ecosystems to pulse perturbations. There are multiple ways in which eutrophication and pulse perturbations may interact to induce potentially synergic changes in the system, for instance, by increasing the amount of nutrients released after a pulse perturbation. Here, we performed a controlled press and pulse perturbation experiment using mesocosms filled with natural lake water to address how eutrophication modulates the phytoplankton response to sequential mortality pulse perturbations; and what is the combined effect of press and pulse perturbations on the resistance and resilience of the phytoplankton community. Our experiment showed that eutrophication increased the absolute scale of the chlorophyll‐a response to pulse perturbations but did not change the proportion of the response relative to its pre‐event condition (resistance). Moreover, the capacity of the community to recover from pulse perturbations was significantly affected by the cumulative effect of sequential pulse perturbations but not by eutrophication itself. By the end of the experiment, some mesocosms could not recover from pulse perturbations, irrespective of the trophic state induced by the press perturbation. While not resisting or recovering any less from pulse perturbations, phytoplankton communities from eutrophying systems showed chlorophyll‐a levels much higher than non‐eutrophying ones. This implies that the higher absolute response to stochastic pulse perturbations in a eutrophying system may increase the already significant risks for water quality (e.g., algal blooms in drinking water supplies), even if the relative scale of the response to pulse perturbations between eutrophying and non‐eutrophying systems remains the same.     

Transient Responses to Spatial Perturbations in Advective Systems

We study the transient dynamics, following a spatially-extended perturbation of models describing populations residing in advective media such as streams and rivers. Our analyses emphasize metrics that are independent of initial perturbations—resilience, reactivity, and the amplification envelope—and relate them to component spatial wavelengths of the perturbation using spatial Fourier transforms of the state variables. This approach offers a powerful way of understanding the influence of spatial scale on the initial dynamics of a population following a spatially variable environmental perturbation, an important property in determining the ecological implications of transient dynamics in advective systems. We find that asymptotically stable systems may exhibit transient amplification of perturbations (i.e., have positive reactivity) for some spatial wavelengths and not others. Furthermore, the degree and duration of amplification varies strongly with spatial wavelength. For two single-population models, there is a relationship between transient dynamics and the response length that characterizes the steady state response to spatial perturbations: a long response length implies that peak amplification of perturbations is small and occurs fast. This relationship holds less generally in a specialist consumer-resource model, likely due to the model’s tendency for flow-induced instabilities at an alternative characteristic spatial scale.     

The inherent multidimensionality of temporal variability: how common and rare species shape stability patterns

Empirical knowledge of diversity–stability relationships is mostly based on the analysis of temporal variability. Variability, however, often depends on external factors that act as disturbances, which makes comparisons across systems difficult to interpret. Here, we show how variability can reveal inherent stability properties of ecological communities. This requires that we abandon one‐dimensional representations, in which a single variability measurement is taken as a proxy for how stable a system is, and instead consider the whole set of variability values generated by all possible stochastic perturbations. Despite this complexity, in species‐rich systems, a generic pattern emerges from community assembly, relating variability to the abundance of perturbed species. Strikingly, the contrasting contributions of different species abundance classes to variability, driven by different types of perturbations, can lead to opposite diversity–stability patterns. We conclude that a multidimensional perspective on variability helps reveal the dynamical richness of ecological systems and the underlying meaning of their stability patterns.