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.
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.
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.
Noise interacts with oscillators in often counterintuitive ways. In this talk, I will discuss the interactions of noise with coupling in systems of limit cycle oscillators.
Delayed coupling of oscillators can give rise to dynamical instabilities and the onset of characteristic synchronization phenomena. We show that two elements coupled with long delay will never synchronize isochronously. Nevertheless, isochronous synchronization of the chaotic dynamics can occur via coupling through a relay element. This has been demonstrated for lasers in experiments and modeling. This mechanisms has been extended to neuronal systems, where isochronously synchronized activity is assumed to underlie cognitive processes. Therefore, a major question is the stability of this state and the limits of its stability. In this talk we will present results from a stability analysis of delay-coupled lasers using either an additional laser or a semitransparent mirror as relay element. We discuss the occurrence of destabilizing mechanisms, including blow-out bifurcations and bubbling. Finally, we present that isochronous identical synchronization between distant elements can be even obtained when the relay element via which they are coupled exhibits uncorrelated dynamics. Even mutual information between the synchronized dynamics and the relay dynamics can vanish. We discuss the implications for neuronal networks.
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
Joint work with Nils Berglund (Orleans) and Christian Kuehn (Dresden).
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 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.
For the past 30 years I have been thinking on and off about neural circuits for generating rhythmic patterns in locomotion, and for the past 11 years, about brain circuits that make simple decisions in the laboratory, and perhaps, in life. In the former case, periodic oscillations are dominant and phase reduction and averaging methods afford useful simplifications of biophysically-based ion-channel models. In the latter, while some progress has been made in reducing pools of stochastic integrate-and-fire models to population models characterized by average firing rates and synaptic activities, the theory is not as well developed. I will sketch a recent attempt to reduce a spiking model with neuromodulation to a leaky accumulator model, and reveal some of the less pleasant features of the process.
I will also describe a vexing problem that is emerging from electropysiological, EEG, and imaging studies. Leaky competing accumulators and drift-diffusion models can reproduce reaction time distributions and other behavioral measures of decision making remarkably well, but they often fail to capture firing rate dynamics in relevant brain areas through the entire course of a trial. The lateral interparietal area (LIP), in particular, exhibits ramping firing rates that track evidence accumulation, but EEG and fMRI studies suggest that a distributed network of cortical areas is involved, and experiments on scanning and attending to multiple stimuli show that LIP reflects processes other than simple integration. How can mathematical models engage these data without degenerating into massive, incomprehensible simulations?
(Joint work with Philip Eckhoff, Sam Feng, Mike Schwemmer, KongFatt Wong-Lin, Pat Simen, Marieke van Vugt, Leigh Nystrom and Jonathan Cohen.)
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.
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.
Change in the concentration of free intracellular calcium is a crucial control mechanism in almost every cell type, with oscillations of calcium concentration being thought to play an important role in muscle contraction, secretion, cardiac electrophysiology and many other aspects of cell physiology. Experiments have been done in a number of different physiological settings to investigate intracellular calcium dynamics, with the results used to construct mathematical models of intracellular calcium dynamics. A main aim of experimental and modeling work is to identify the mechanisms underlying calcium oscillations.
In this talk, I will show how attempts to understand the oscillatory dynamics of calcium models has given rise to new results in bifurcation theory and geometric singular perturbation theory. I will also briefly outline some areas where current gaps in theory are delaying our understanding of the models.
Over the last decade, inspired by several key animal studies, my collaborators and I have extended the domain of dynamically dexterous legged robots to include running over rough natural terrain, quasi-static climbing of exterior vertical walls and trees, and more recently, dynamical ascents of more structured vertical surfaces. In so doing we have found it advantageous to engineer gait generators through the interconnection of internal coupled oscillators with the mechanical body and limbs through various proprioceptive and vestibular feedback channels. When construed in this manner as attracting limit cycles on the torus, the organization of these resulting gaits takes the form of a cell complex whose adjacency relations structure the design of robust steady behaviors and safe transitions between them.
Often, the imperatives of locomotion impose additional constraints (e.g., keep at least two limbs in contact with the wall at all times) manifest as combinatorial obstacles in the gait space, complicating any refinement of the gait complex that would effect their excision. We have been developing gait generators and gait transition mechanisms that respect these obstacles and I will give some examples of work in progress. The gait complex and its combinatorially punctured variants have characteristic topological signatures that constrain the manner in which the basins of distinct attracting limit cycles achieved by smooth controllers can fit together. Such constraints impact the engineering of dynamical gait controllers for robots and to the extent that animal motion controllers target smooth dynamical systems, they must impact the animals' designs as well. I will close with some speculation about the possibility for developing gait assays that probe those designs. If we hypothesize that biological preflexes (i.e., those animal motion controllers implemented by the tuned musculoskeletal system) must be smooth, then observations of animal gait transitions may help gain greater insight into the boundary between neural and mechanical control.
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.
Our recent research shows that a faulty or sub-optimally operating metabolic network can often be rescued by the targeted removal of enzyme-coding genes. Predictions go as far as to assert that certain gene knockouts can restore the growth of otherwise nonviable gene-deficient cells. In this talk, I will discuss how the theory of dynamical systems can be combined with network modeling to develop computational methods for the systematic identification of compensatory perturbations and rescue interactions in a range of biological contexts. The proposed problem is mathematically challenging and has the potential to illuminate biological and medical research.
Main references: A.E. Motter, Improved network performance via antagonism: From synthetic rescues to multi-drug combinations, BioEssays 32, 236 (2010); A.E. Motter, N. Gulbahce, E. Almaas, A.-L. Barabasi, Predicting synthetic rescues in metabolic networks, Molecular Systems Biology 4, 168 (2008).
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).
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.
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)
In this talk, I will describe our recent experimental and theoretical work on small synthetic gene networks exhibiting oscillatory behavior. Most living organisms use internal genetic "clocks" to govern fundamental cellular behavior. While the gene networks that produce oscillatory expression signals are typically quite complicated, certain recurring network motifs are often found at the core of these biological clocks. One common motif which may lead to oscillations is delayed auto-repression. We constructed a synthetic two-gene oscillator based on this design principle, and observed robust and tunable oscillations in bacteria. Computational and theoretical modeling suggests that the key mechanism of oscillations is a small time delay in the negative feedback loop. In a strongly nonlinear regime, this time delay can lead to long-period oscillations that can be characterized by "degrade and fire" dynamics. We also demonstrated synchronization of synthetic gene oscillators across cell population using a variant of the same design in which oscillators are synchronized by a chemical signal freely diffusing through cell membranes.
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
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.
I will discuss joint work with Aaditya Rangan in which we model a small patch of layer 2 of the primary visual cortex (V1) as a large network of point neurons. Network architecture is chosen to reflect a few coarse structures of V1. Our aim is to understand macroscopic observations from dynamics on the neuronal level. Using biological data to constrain parameters, we arrive at models which exhibit a number of empirically observed V1 phenomena (including e.g. localized receptive fields and spontaneous pattern formation in background). In this talk, I will discuss dynamical mechanisms behind a phenomenon called surround suppression.