Delays in feedback loops tend to destabilize dynamical systems, inducing self-sustained oscillations or chaos. I will show some typical examples in my presentation. I will also show how one can reduce the study of periodic oscillations in systems with delay to low-dimensional smooth algebraic systems of equations. The approach works also when the delay depends on the state, a case in which it is not clear in general if the underlying differential equations are smooth dynamical systems.
In many biological models multiple time scale dynamics occurs due to the presence of variables and parameters of very different orders of magnitudes. Situations with a clear "global" separation into fast and slow variables governed by singularly perturbed ordinary differential equations in standard form have been investigated in great detail.
For multi-scale problems depending on several parameters it can already be a nontrivial task to identify meaningful scalings. Typically these scalings and the corresponding asymptotic regimes are valid only in certain regions in phase-space or parameter-space. Another issue is how to match these asymptotic regimes to understand the global dynamics. In this talk I will show in the context of examples from enzyme kinetics that geometric methods based on the blow-up method provide a systematic approach to problems of this type.
(Joint work with Ilona Kosiuk, MPI MIS Leipzig)
The propagation of waves of neural activity across the surface of the brain is known to subserve both natural and pathological neurobiological phenomena. An example of the former is spreading excitation associated with sensory processing, whilst waves in epilepsy are a classic example of the latter. There is now a long history of using integro-differential neural field models to understand the properties of such waves. For mathematical convenience these models are often assumed to be spatially translationally-invariant. However, it is hard even at a first approximation to view the brain as a homogeneous system and so there is a pressing need to develop a set of mathematical tools for the study of waves in heterogeneous media that can be used in brain modeling. Homogenization is one natural multi-scale approach that can be utilized in this regard, though as a perturbation technique it requires that modulation on the micro-scale be both small in amplitude and rapidly varying in space. In this talk I will present novel techniques that improve upon this standard approach and can further tackle cases where the inhomogeneous environment is modeled as a random process.
Dynamical systems with delayed feedback often exhibit complex oscillations not observed in analogous systems without delay. Stochastic effects can change the picture dramatically, particularly if multiple time scales are present. Then transients ignored in the deterministic system can dominate the long range behavior. This talk will contrast the effects of different noise sources in certain systems with delayed feedback. We show how ideas from canonical physical and mechanical systems can be applied in biological models for disease and balance. The approaches we consider capture the effects of noise and delay in the contexts of piecewise smooth systems, nonlinearities, and discontinuities.
I will trace the history of models for bursting, concentrating on square-wave bursters descended from the Chay-Keizer model for pancreatic beta cells. The model was originally developed on a biophysical and intutive basis but was put into a mathematical context by John Rinzel's fast-slow analysis. Rinzel also began the process of classifying bursting oscillations based on the bifurcations undergone by the fast subsystem, which led to important mathematical generalization by others. Further mathematical work, notably by Terman, Mosekilde and others, focused rather on bifurcations of the full bursting system, which showed a fundamental role for chaos in mediating transitions between bursting and spiking and between bursts with different numbers of spikes. The development of mathematical theory was in turn both a blessing and a curse for those interested in modeling the biological phenomena - having a template of what to expect made it easy to construct a plethora of models that were superficially different but mathematically redundant. This may also have steered modelers away from alternative ways of achieving bursting, but instructive examples exist in which unbiased adherence to the data led to discovery of new bursting patterns. Some of these had been anticipated by the general theory but not previously instantiated by Hodgkin-Huxley-based examples. A final level of generalization has been the addition of multiple slow variables. While often mathematically reducible to models with a one-variable slow subsystem, such models also exhibit novel resetting properties and enhanced dynamic range. Analysis of the dynamics of such models remains a current challenge for mathematicians.
The interchange between dynamical systems theory with biology has had lasting impact upon both. As biology becomes increasingly quantitative, this relationship is likely to strengthen still further. This lecture will review my experience as a mathematician working at the interface with biology, emphasizing the role of multiple time scales in biological models. It will also look discuss why the solution of outstanding mathematical questions is essential to progress within biology.
Central Pattern Generators (CPGs) are limited neural networks that drive rhythmic behaviors such as locomotion, respiration and mastication. We have been studying the structure, function, and modulation of CPGs, with an emphasis on neuronal and ionic mechanisms that allow flexibility in the output from an anatomically defined network. Both biological and modeling studies show that individual oscillatory neurons can be modulated to generate bursting activity by a variety of independent ionic mechanisms, allowing flexibility in the frequency and output properties of these important neurons. The phasing of neuronal activity in the rhythmic pattern is not determined only by the pattern of synaptic connections; the intrinsic electrophysiological properties of the neurons also play a major role. These points raise issues with regard to the appropriate level of complexity in models of neural networks. I will discuss these issues based on work done in collaboration with John Guckenheimer on the pyloric network in the crustacean stomatogastric ganglion and the rodent spinal locomotor CPG. Supported by NIH grants NS17323, NS050943 and NSF grant IOS-0749467
John Rinzel Lecture
In the first part of this talk I will briefly describe previous work on quadruped gaits (which distinguishing gaits by their spatio-temporal symmetries). In the second part, I will discuss how the application to gaits has led to results about phase-shift synchrony in periodic solutions of coupled systems of differential equations. This work is joint with David Romano, Yunjiao Wang, and Ian Stewart.
The Lorenz system is the classical example of a seemingly simple dynamical system that exhibits chaotic dynamics. In fact, there are numerous studies to characterize the complicated dynamics on the famous butterfly attractor. This talk addresses how the dynamics is organized more globally. An important role in this regard is played by the stable manifold of the origin, also known as the Lorenz manifold. In 1992 John Guckenheimer suggested this manifold as a bench-mark challenge for developing computational methods in dynamical systems. We show how the numerical continuation of orbit segments can be used to investigate and characterize the transition to chaos in the Lorenz system.
Joint work with Eusebius Doedel (Concordia University, Montreal) and Bernd Krauskopf (University of Bristol).
The Mackey-Glass equation is a seemingly simple delay differential equation (DDE) with one fixed delay which can exhibit the full gamut of dynamics from a trivial stable steady state to fully chaotic dynamics, and has inspired decades of mathematical research into DDEs. However, much of that research has focused on equations with fixed or prescribed delays, whereas many biological delays would be more naturally modelled as state-dependent delays. Before incorporating state-dependent delays in complex biochemical network models, it is desirable to understand the dynamics which result from including state-dependent delays in simpler model problems. Accordingly, in this talk we will consider a simple model problem with multiple state-dependent delays, and show that it can exhibit a wide range of dynamical behaviour, including stable periodic solutions and bi-stable periodic solutions, to stable tori, together with the associated bifurcation structures.
Many neuronal systems and models display so-called mixed-mode oscillations (MMOs) consisting of small-amplitude oscillations alternating with large-amplitude oscillations. Different mechanisms have been identified which may cause this type of behaviour. In this talk, we will focus on MMOs in a slow-fast dynamical system with one fast and two slow variables, containing a folded-node singularity. The main question we will address is whether and how noise may change the dynamics.
We will first outline a general approach to stochastic slow-fast systems which allows
1. to construct small sets in which the sample paths are typically concentrated, and
2. to give precise bounds on the exponentially small probability to observe atypical behaviour.
Applying this method to our model system shows the existence of a critical noise intensity beyond which the small-amplitude oscillations become hidden by noise. Furthermore, we will show that in the presence of noise sample paths are likely to jump away from so-called canard solutions earlier than the corresponding deterministic orbits. This early-jump mechanism can drastically change the mixed-mode patterns, even for rather small noise intensities.
Joint work with Nils Berglund (Orleans) and Christian Kuehn (Dresden).
Coherent neuronal activity is ubiquitous and presumably important in brain function. I will review my group's experimental studies of the mechanisms underlying coherent activity using dynamic clamp technology, which allows us to perform virtual-reality-inspired experiments in neurons in vitro. Using these techniques and mathematical tools from dynamical systems theory, we are trying to understand which factors give rise to stable neuronal synchronization in the presence of heterogeneity, noise, and conduction delays.
Sue Ann Campbell
We consider a network of inherently oscillatory neurons with time delayed connections. We reduce the system of delay differential equations to a phase model representation and show how the time delay enters into the reduced model. For the case of two neurons, we show how the time delay may affect the stability of the periodic solution leading to stability switching between synchronous and antiphase solutions as the delay is increased. Numerical bifurcation analysis of the full system of delay differential equations is used determine constraints on the coupling strength such that the phase model is valid. Both type I and type II oscillators are considered.
Ale Jan Homburg
Random dynamical systems with bounded noise can have multiple stationary measures with different supports. Under variation of a parameter, such as the amplitude of the noise, bifurcations of these measures may occur. We discuss such bifurcations both in a context of random diffeomorphisms and of random differential equations.
* A.J. Homburg, T. Young. Bifurcations for random differential equations with bounded noise on surfaces Topol. Methods Nonlinear Anal. 35 (2010), 77-98.
* H. Zmarrou, A.J. Homburg. Dynamics and bifurcations of random circle diffeomorphisms Discrete Contin. Dyn. Syst. Ser. B 10 (2008), 719-731.
* H. Zmarrou, A.J. Homburg. Bifurcations of stationary measures of random diffeomorphisms Ergod. Th. and Dynam. Sys. 27 (2007), 1651-1692.
The plan is to divide the talk in three distinct but related parts.
First, the question of asymptotic stability for equilibria of delay differential equations is addressed numerically. The proposed method, based on the discretization of the infinitesimal generator of the solution operator semigroup via pseudospectral differentiation, allows to approximate the stability determining eigenvalue with spectral accuracy. Hence it is fast and suitable for robust analysis.
Second, the numerical scheme is extended for investigating the stability of steady states of population dynamics, where the study of the associated transcendental characteristic equations is often too difficult to be approached analytically. The fruitful interplay between theoretical and numerical analysis is highlighted through examples taken from age- and physiologically-structured models, as well as delayed epidemics.
Third, recent advances in the numerical stability analysis of delay systems are illustrated, showing how equilibria (characteristic roots), periodic orbits (Floquet multipliers) and chaotic motion (Lyapunov exponents) can be faced under the same discretization framework. Examples arising in the populations context are discussed which demand for adapting such treatment.
Stochastic delay differential equations often arise in biosciences as models involving, e.g., negative feedback terms and intrinsic or extrinsic noise. Examples of applications range from stochastic models of human immune response systems, neural networks or neural fields to genetic regulatory systems. Stability theory for stochastic delay differential equations is quite well established and we will provide a brief review of available methods and results. Stochastic dynamical systems theory for stochastic delay differential equations beyond the stability analysis of equilibria is much less developed and we will report on some open problems in this area.