Classification of bursting patterns: A tale of two ducks

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple-timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram et al., and then by Golubitsky et al., which together with the Rinzel-Izhikevich proposals provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least two slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the two main families of folded-node bursters, depending upon the phase (active/spiking or silent/non-spiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast-subsystem approach

Spike-adding in a canonical three time scale model: superslow explosion & folded-saddle canards

International audienceWe examine the origin of complex bursting oscillations in a phenomenological ordinary differential equation model with three time scales. We show that bursting solutions in this model arise from a Hopf bifurcation followed by a sequence of spike-adding transitions, in a manner reminiscent of spike- adding transitions previously observed in systems with two time scales. However, the details of the process can be much more complex in this three-time-scale context than in two-time-scale systems. In particular, we find that spike-adding can involve canard explosions occurring on two different time scales and is associated with passage near a folded-saddle singularity. We show that the character of the bursting and the form of spike-adding transitions that occur depend on the geometry of certain singular limit systems, specifically the relative positions of the critical and superslow manifolds. We also show that, unlike the case of spike-adding in two-time-scale systems, the onset of a new spike in our model is not typically associated with a local maximum in the period of the bursting oscillation

Complex oscillations with multiple timescales - Application to neuronal dynamics

The results gathered in this thesis deal with multiple time scale dynamical systems near non-hyperbolic points, giving rise to canard-type solutions, in systems of dimension 2, 3 and 4. Bifurcation theory and numerical continuation methods adapted for such systems are used to analyse canard cycles as well as canard-induced complex oscillations in three-dimensional systems. Two families of such complex oscillations are considered: mixed-mode oscillations (MMOs) in systems with two slow variables, and bursting oscillations in systems with two fast variables. In the last chapter, we present recent results on systems with two slow and two fast variables, where both MMO-type dynamics and bursting-type dynamics can arise and where complex oscillations are also organised by canard solutions. The main application area that we consider here is that of neuroscience, more precisely low-dimensional point models of neurons displaying both sub-threshold and spiking behaviour. We focus on analysing how canard objects allow to control the oscillatory patterns observed in these neuron models, in particular the crossings of excitability thresholds

Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

International audienceIn this work, we analyze a four dimensional slow-fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse

Understanding spiking and bursting electrical activity through piece-wise linear systems

In recent years there has been an increased interest in working with piece-wise linear caricatures of nonlinear models. Such models are often preferred over more detailed conductance based models for their small number of parameters and low computational overhead. Moreover, their piece-wise linear (PWL) form, allow the construction of action potential shapes in closed form as well as the calculation of phase response curves (PRC). With the inclusion of PWL adaptive currents they can also support bursting behaviour, though remain amenable to mathematical analysis at both the single neuron and network level. In fact, PWL models caricaturing conductance based models such as that of Morris-Lecar or McKean have also been studied for some time now and are known to be mathematically tractable at the network level. In this work we proceed to analyse PWL neuron models of conductance type. In particular we focus on PWL models of the FitzHugh-Nagumo type and describe in detail the mechanism for a canard explosion. This model is further explored at the network level in the presence of gap junction coupling. The study moves to a different area where excitable cells (pancreatic beta-cells) are used to explain insulin secretion phenomena. Here, Ca2+ signals obtained from pancreatic beta-cells of mice are extracted from image data and analysed using signal processing techniques. Both synchrony and functional connectivity analyses are performed. As regards to PWL bursting models we focus on a variant of the adaptive absolute IF model that can support bursting. We investigate the bursting electrical activity of such models with an emphasis on pancreatic beta-cells

Analysis of Complex Bursting Patterns in Multiple Timescale Respiratory Neuron Models

Many physical systems feature interacting components that evolve on disparate timescales. Significant insights about the dynamics of such systems have resulted from grouping timescales into two classes and exploiting the timescale separation between classes through the use of geometric singular perturbation theory. It is natural to expect, however, that some dynamic phenomena cannot be captured by a two timescale decomposition. One example is the mixed burst firing mode, observed in both recordings and model pre-B\"{o}tzinger neurons, which appears to involve at least three timescales based on its time course. With this motivation, we construct a model system consisting of a pair of Morris-Lecar systems coupled so that there are three timescales in the full system. We demonstrate that the approach previously developed in the context of geometric singular perturbation theory for the analysis of two timescale systems extends naturally to the three timescale setting. To elucidate which characteristics truly represent three timescale features, we investigate certain reductions to two timescales and the parameter dependence of solution features in the three timescale framework. Furthermore, these analyses and methods are extended and applied to understand multiple timescale bursting dynamics in a realistic single pre-B\"{o}tzinger complex neuron and a heterogeneous population of these neurons, both of which can generate a novel mixed bursting (MB) solution, also observed in pre-B\"{o}tC neuron recordings. Rather surprisingly, we discover that a third timescale is not actually required to generate mixed bursting solution in the single neuron model, whereas at least three timescales should be involved in the latter model to yield a similar mixed bursting pattern. Through our analysis of timescales, we also elucidate how the single pre-B\"{o}tC neuron model can be tuned to improve the robustness of the MB solution

Canard-Mediated (De)Synchronization in Coupled Phantom Bursters

International audienceIn this work, we study canard-mediated transitions in mutually coupled phantom bursters. We extend a multiple-timescale model which provides a sequence of dynamic events, i.e., transition from a frequency modulated relaxation cycle to a quasi-steady-state and resumption of the relaxation regime through small amplitude oscillations. Folded singularities and associated canard solutions have a particular impact on the dynamics of the original system, which consists of two feedforward coupled FitzHugh--Nagumo oscillators, where the slow subsystem (regulator) controls the periodic behavior of the fast subsystem (secretor). We first investigate the variability in the dynamics depending on the canard mechanism that occurs near the folded singularities of the four-dimensional secretor-regulator configuration. Then, we introduce a second secretor and focus on the slow-fast transitions in the presence of a linear coupling between the secretors. In particular, we explore the impact of the relationship between the canard structures and the coupling on patterns of synchronization and desynchronization of the collective dynamics of the resulting six-dimensional system. We identify two different sources of desynchronization induced by canards, near a folded-saddle singularity and a folded-node singularity, respectively

Desarrollo de paradigmas de neuromodulación adaptativa para el tratamiento de trastornos motores

Actualmente, diversas técnicas de neuromodulación basadas en dispositivos implantables (e.g., open-loop DBS: estimulación cerebral profunda a lazo abierto) conforman un conjunto de terapias bien establecidas para el tratamiento de estados avanzados de trastornos motores (e.g., enfermedad de Parkinson y epilepsia). Las limitaciones inherentes a estos tratamientos han motivado el desarrollo de nuevos paradigmas de neuromodulación adaptativa. Este trabajo tiene como objetivo el estudio de diversos aspectos asociados al desarrollo de un esquema de neuromodulación adaptativa (closed-loop DBS) con el n de optimizar la eciencia de los dispositivos implantables destinados al tratamiento de trastornos motores. Se implementaron diversos modelos computacionales que capturan la dinámica de la red de ganglios basales-tálamocortical involucrada en la enfermedad de Parkinson. Mediante técnicas analíticas y de procesamiento de señales, se caracterizaron los mecanismos asociados a la aparición de diferentes biomarcadores electrofisiológicos (i.e., potencia en bandas de frecuencias específicas, acoplamientos inter-frecuencia, forma de onda) observables en pacientes y/o modelos animales parkinsonianos. Se encontró que estos biomarcadores emergen de bifurcaciones en la dinámica de circuitos neuronales biológicamente plausibles y pueden coexistir de diferentes maneras: 1) correlacionados debido a que uno es epifenómeno de otro ó 2) independientemente debido a diferentes mecanismos subyacentes. Este tipo de estudio permitió diseñar nuevos algoritmos especializados para identificar diferentes dinámicas oscilatorias que se han observado experimentalmente y que son indistinguibles para los algoritmos tradicionales utilizados en la cuantificación de acoplamientos inter-frecuencia. Por otro lado, se estudió la dinámica de la actividad ictal en pacientes con epilepsia focal fármaco-resistente. Aplicando los algoritmos desarrollados, se mostró que diferentes patrones de acoplamiento inter-frecuencia coexisten en la actividad ictal registrada en la zona de inicio de crisis. El estudio presentado constituye una herramienta capaz de asistir el análisis de los registros iEEG realizado por los epileptólogos y proveer información útil en diferentes aspectos: 1) definición de los electrodos involucrados en la zona de inicio de la actividad ictal e identificación del núcleo ictal e 2) interpretación apropiada de los mecanismos ictales asociados a la propagación de la actividad ictal. Basados en modelos computacionales, se identificaron posibles mecanismos de ac ción de la estimulación cerebral profunda sobre la disfunción de los ganglios basales. El principal mecanismo consiste en un efecto de resetting de la actividad provocado por la estimulación eléctrica y es consistente con observaciones en modelos de fisiopatología previamente reportados de la enfermedad de Parkinson (e.g., inecacia de los patrones de estimulación irregulares). Este enfoque permitió mostrar que el rango clínicamente relevante para la frecuencia y la intensidad del Patrón de estimulación eléctrica, es una propiedad emergente de la anatomía de la red de ganglios basales y puede entenderse sin tener en cuenta los detalles biofísicos de las estructuras relevantes. Finalmente, se propone un esquema closed-loop DBS basado en la teoría del aprendizaje por refuerzo para el diseño de un lazo de retroalimentación. Esta propuesta permite extender los controladores/enfoques de closed-loop DBS presentados hasta el momento. Se evaluó el esquema en ambientes simulados de la red de ganglios basales y los resultados permitieron demostrar la factibilidad y analizar el desempeño del paradigma de neuromodulación adaptativa basado en un algoritmo independiente del modelo y capaz de ser extensible a acciones continuas (i.e., cantidad/rango de parámetros a controlar) y multi-objetivos (i.e., un conjunto de biomarcadores)

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described