Geometric singular perturbation theory

Singular perturbation problems feature prominently both in the theory of differential equations and in their applications. Singularly perturbed systems are characterised by the presence of at least two fundamentally different scales, typically in space or time, and, hence, of one or more small perturbation parameters; perturbation theory aims to infer the system dynamics from the "singular problem" obtained by setting these parameters to zero.

Traditionally, singularly perturbed dynamics has been studied using a variety of (mostly formal) techniques, often on a case-by-case basis. More recently, a unified, geometric approach has been developed, which is based on dynamical systems theory and, in particular, on invariant manifold methods. While the approach is complete in the hyperbolic setting, it breaks down at non-hyperbolic points, which are frequently more relevant in applications. This loss of hyperbolicity can be remedied by geometric desingularisation, or "blow-up."

I have been significantly involved in the ongoing development of the blow-up technique. In my PhD thesis, I analysed the Lagerstrom equation, a model for viscous flow past a solid at low Reynolds number; I proved the existence and uniqueness of solutions, and systematically derived asymptotic solution expansions, in the framework of geometric singular perturbation theory and blow-up. I have since successfully applied geometric desingularisation, including in a multi-dimensional setting, thus obtaining a fairly complete picture of the global system dynamics in many cases. My research interests include applications in a number of fields which range from fluid mechanics to chemical kinetics and from neuronal modelling to front dynamics; a detailed description of some of my more recent work can be found below.

Traveling front propagation

Reaction-diffusion systems of partial differential equations have found widespread use as "minimal" model throughout the sciences. Propagating fronts are monotonic solutions to such equations that retain a fixed profile when studied in a co-moving frame; they are frequently relevant as asymptotic states to which more general solutions tend in the long-time limit. The resulting front dynamics is classifiable in terms of the reaction function: the propagation speed of "pulled" front solutions equals the (linear) spreading speed of perturbations, while "pushed" and "bistable" fronts propagate at a speed that is unrelated to this linearised speed; prototypical examples of these regimes are given by the Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation and the Nagumo equation, respectively.

I have been studying front propagation in scalar reaction-diffusion systems from a geometric point of view. In particular, I have analysed a very general family of such equations which contains both the FKPP equation and the (degenerate) Zeldovich equation. Perturbing off these two cases, I have proved the existence and uniqueness of (pushed) front solutions in the resulting perturbed equations; moreover, I have derived rigorous asymptotic expansions for the corresponding front propagation speeds, and I have explained the occurrence of fractional powers of the perturbation parameter in these expansions. More recently, I have obtained analogous results in the regime where the polynomial degree in the above family tends to infinity. Finally, I have shown that the asymptotic expansion for the corresponding front propagation speed has Gevrey properties, and I have co-developed a reliable numerical algorithm for evaluating the coefficients in that expansion to any desired order.

Mixed-mode dynamics

"Mixed-mode" dynamics is a type of complex behaviour which is frequently encountered in systems of ordinary differential equations that involve multiple scales. Mixed-mode oscillations are characterised by time series in which small-amplitude (sub-threshold) oscillations and large-amplitude (relaxation-type) excursions alternate. The vast difference in amplitude implies the existence of a small parameter in the model, thus allowing one to apply perturbation techniques.

I have been at the forefront of the development of geometric techniques for analysing mixed-mode dynamics in singularly perturbed neuronal models that are based on the classical Hodgkin-Huxley formalism. The irregular firing patterns observed in these models are often of mixed-mode type; the Wilson-Callaway model for the dopaminergic neuron in the mammalian brain is an example of such a system in which robust mixed-mode behaviour has been observed. I have characterised the admissible mixed-mode patterns in a simplified (three time-scale) normal form system which, though analytically simpler, still capture the essential dynamics of the full equations; in particular, I have explained the structure of these patterns by an underlying canard phenomenon. (Canards are solutions that can stay close to strongly repelling manifolds for substantial amounts of time.) Finally, I have obtained analogous results for the "full" Wilson-Callaway model.

In ongoing work, I am analysing a modi cation of the classical Hodgkin-Huxley equations in which the dynamics varies on three scales, as opposed to the conventional two. A short-term goal is to characterise in detail the mixed-mode dynamics of that physiologically more realistic system. In the longer term, my aim is to classify more generally the mixed-mode dynamics that can occur in three-dimensional Hodgkin-Huxley type models, as well as in systems that can be reduced to three dimensions. Other relevant questions in this context concern the (potentially chaotic) dynamics of the discrete maps induced by these continuous systems, the geometric characterisation of related firing patterns, such as bursting, and the efficient reduction of high-dimensional models to their low-dimensional normal forms.

Cut-off reaction-diffusion dynamics

Reaction-diffusion systems of partial differential equations have been widely applied in the continuum approximation of discrete, many-particle systems. However, the quality of this approximation deteriorates as the number of particles decreases; in particular, the front propagation speed is often misestimated, even when the front profile is accurately reproduced. These discrepancies are due to microscopic fluctuations which need to be taken into account for finite particle numbers. Recently, it has been suggested that the stochastic effects of this discreteness can be modelled by introducing a deterministic "cut-off" which deactivates the reaction terms whenever the particle concentration lies below some threshold. The effects of this cut-off approximation on the front propagation dynamics depend strongly on the type of reaction function: thus, the correction to the propagation speed that is due to the cut-off is negative, and logarithmic in the cut-off parameter, in the pulled propagation regime, whereas it is positive, and sublinear or superlinear, in the pushed and bistable regimes, respectively.

Beginning with a study of the cut-off FKPP equation, I have been spearheading an ongoing programme to analyse systematically the dynamics of cut-off reaction-diffusion systems using geometric (dynamical systems) techniques. Thus, I proved the occurrence of logarithmic ("switchback") terms in the expansion for the front propagation speed in the cut-off Zeldovich equation; in a follow-up article, I studied geometrically the effects of a cut-off on propagating fronts in the bistable Nagumo equation. Finally, I considered the effects of a cut-off on stationary fronts in the Nagumo equation at a Maxwell point. In the process, the cut-off Nagumo equation has emerged as a suitable prototypical realisation: it realises all three propagation regimes (pushed, pulled, and bistable), while still allowing for a fairly explicit analysis. Correspondingly, I have classified these regimes, as well as the respective boundary cases, in that equation from a geometric point of view.

In ongoing work, I am investigating the stability and convergence of front solutions in the presence of a cut-off; the extension of results obtained in a scalar context to multiple-component systems in more than one spatial dimension; and the relationship between the geometric methodology and an alternative approach that is based on a variational principle. Other aspects that merit further investigation concern the numerical implementation of regimes where no exact (closed-form) solutions can be obtained and a systematic analysis of the relationship between deterministic cut-off dynamics and the underlying discrete systems that it approximates.

Stochastic gene expression

Gene expression -- transcription of mRNA from DNA and translation of protein from mRNA-- is a highly stochastic process, as the involved species of molecules are often only present in small numbers. Different hypotheses on the molecular details of the expression process have been proposed to account for the observed stochasticity in single-cell data: while transcriptional bursting -- i.e., rapid mRNA synthesis due to DNA switching between an active and inactive state -- has long been considered the most likely transcription mechanism, recent studies have rehabilitated a standard birth-death model of gene expression.

The accepted mathematical description of gene expression dynamics is the Chemical Master Equation (CME), a system of first-order ordinary differential equations whose solution yields the probability of observing certain numbers of DNA, mRNA, and protein in the system. While the CME can be solved exactly only in very special cases, solutions are needed to calculate efficiently the likelihood of competing gene expression models given observed molecular data; thus, different approximation techniques have been proposed. One class of such techniques is perturbative, and is based on the observation that the lifetimes of the involved species often differ by several orders of magnitude. The presence of a small perturbation parameter -- intuitively, the ratio of shortest versus longest lifetimes -- allows for a fast-slow decomposition and, hence, for the approximate solution of the CME in terms of an asymptotic series expansion.

In collaboration with Synthetic and Systems Biology Edinburgh (SynthSys), I have been developing perturbative solution techniques for the CME. Thus, I have co-developed a novel approach which draws heavily on graph theory, and which is based on an expansion of the transition state matrix in the CME in terms of a non-dimensional parameter that depends on the reaction rates and the reaction volume.

Moreover, I have contributed to a study of phenotypic switching in gene regulatory networks due to noise: specifically, we have shown that the inherent stochastic nature of gene expression can induce multimodality in the resulting probability distributions, and we have derived a general analytical framework for quantifying this phenomenon that relies on a fast-slow decomposition of the CME, and the formulation of an associated effective master equation.

In recent work, I have characterised completely the standard birth-death model for gene expression using geometric singular perturbation theory: I have derived closed-form asymptotic formulae for the propagator probability -- i.e., the probability of observing a particular number of proteins given an observation at some earlier point in time -- in that model to any order in the perturbation parameter. In particular, I have shown that the oftentimes neglected fast (transient) dynamics is vital in many biologically relevant parameter regimes.

Currently, I am investigating how the geometric framework established to date can be extended to models that include additional stages in the transcription process; moreover, in collaboration with the Helmholtz Zentrum Mü nchen (Germany), I am applying that framework to both simulated (in silico) and experimental (in vitro) time series data, with the aim of inferring the underlying parameter regimes and, ultimately, of selecting between competing models for gene expression.

Geometric theory for partial differential equations

I have been involved in work aimed at extending geometric singular perturbation theory to certain classes of partial differential equations. One recent project concerns the long-time behaviour of physical systems in which energy dissipation is accounted for by diffusive terms. Systems of parabolic type can be analysed by introducing similarity variables and weighted spaces. I have generalised this geometric approach to hyperbolic and hyperbolic-parabolic systems, proving the existence of invariant manifolds in these cases. In a related project, I have described canard solutions and bifurcation delay in a class of equations of reaction-diffusion type. In a joint study with colleagues at the School of Medicine (UoE) that is based on clinical data, I have illustrated loss of complexity in patients with chronic fatigue syndrome. Following that study, an experimental protocol is currently being developed to compare the complexity of movement patterns before and after treatment, and to test the hypothesis that complexity increases with successful treatment.