Nonlinear Dynamics, Synchronisation and Chaos in Coupled FHN Cardiac and Neural Cells

Physiological systems are amongst the most challenging systems to investigate from a mathematically based approach. The eld of mathematical biology is a relatively recent one when compared to physics. In this thesis I present an introduction to the physiological aspects needed to gain access to both cardiac and neural systems for a researcher trained in a mathematically based discipline. By using techniques from nonlinear dynamical systems theory I show a number of results that have implications for both neural and cardiac cells. Examining a reduced model of an excitable biological oscillator I show how rich the dynamical behaviour of such systems can be when coupled together. Quantifying the dynamics of coupled cells in terms of synchronisation measures is treated at length. Most notably it is shown that for cells that themselves cannot admit chaotic solutions, communication between cells be it through electrical coupling or synaptic like coupling, can lead to the emergence of chaotic behaviour. I also show that in the presence of emergent chaos one nds great variability in intervals of activity between the constituent cells. This implies that chaos in both cardiac and neural systems can be a direct result of interactions between the constituent cells rather than intrinsic to the cells themselves. Furthermore the ubiquity of chaotic solutions in the coupled systems may be a means of information production and signaling in neural systems

Synchronization in STDP-driven memristive neural networks with time-varying topology

Synchronization is a widespread phenomenon in the brain. Despite numerous studies, the specific parameter configurations of the synaptic network structure and learning rules needed to achieve robust and enduring synchronization in neurons driven by spike-timing-dependent plasticity (STDP) and temporal networks subject to homeostatic structural plasticity (HSP) rules remain unclear. Here, we bridge this gap by determining the configurations required to achieve high and stable degrees of complete synchronization (CS) and phase synchronization (PS) in time-varying small-world and random neural networks driven by STDP and HSP. In particular, we found that decreasing $P$ (which enhances the strengthening effect of STDP on the average synaptic weight) and increasing $F$ (which speeds up the swapping rate of synapses between neurons) always lead to higher and more stable degrees of CS and PS in small-world and random networks, provided that the network parameters such as the synaptic time delay $\tau_c$, the average degree $\langle k \rangle$, and the rewiring probability $\beta$ have some appropriate values. When $\tau_c$, $\langle k \rangle$, and $\beta$ are not fixed at these appropriate values, the degree and stability of CS and PS may increase or decrease when $F$ increases, depending on the network topology. It is also found that the time delay $\tau_c$ can induce intermittent CS and PS whose occurrence is independent $F$. Our results could have applications in designing neuromorphic circuits for optimal information processing and transmission via synchronization phenomena.Comment: 28 pages, 86 references, 8 figures, 2 Table

The constructive role of diversity in the global response of coupled neuron systems

We study the effect that the heterogeneity present among the elements of an ensemble of coupled excitable neurons has on the collective response of the system to an external signal. We consider two different interaction scenarios, one in which the neurons are diffusively coupled and another in which the neurons interact via pulse-like signals. We find that the type of interaction between the neurons has a crucial role in determining the response of the system to the external modulation. We develop a mean-field theory based on an order parameter expansion that quantitatively reproduces the numerical results in the case of diffusive coupling.We acknowledge financial support from the following organizations: National Science Foundation (grant DMR- 0702890); G. Harold and Leila Y. Mathers Foundation; European Commission Project GABA (FP6-NEST Contract 043309); EU NoE Biosim (LSHB-CT-2004-005137); and MEC (Spain) and Feder (project FIS2007-60327).Peer reviewe

Stochastic neural network dynamics: synchronisation and control

Biological brains exhibit many interesting and complex behaviours. Understanding of the mechanisms behind brain behaviours is critical for continuing advancement in fields of research such as artificial intelligence and medicine. In particular, synchronisation of neuronal firing is associated with both improvements to and degeneration of the brain’s performance; increased synchronisation can lead to enhanced information-processing or neurological disorders such as epilepsy and Parkinson’s disease. As a result, it is desirable to research under which conditions synchronisation arises in neural networks and the possibility of controlling its prevalence. Stochastic ensembles of FitzHugh-Nagumo elements are used to model neural networks for numerical simulations and bifurcation analysis. The FitzHugh-Nagumo model is employed because of its realistic representation of the flow of sodium and potassium ions in addition to its advantageous property of allowing phase plane dynamics to be observed. Network characteristics such as connectivity, configuration and size are explored to determine their influences on global synchronisation generation in their respective systems. Oscillations in the mean-field are used to detect the presence of synchronisation over a range of coupling strength values. To ensure simulation efficiency, coupling strengths between neurons that are identical and fixed with time are investigated initially. Such networks where the interaction strengths are fixed are referred to as homogeneously coupled. The capacity of controlling and altering behaviours produced by homogeneously coupled networks is assessed through the application of weak and strong delayed feedback independently with various time delays. To imitate learning, the coupling strengths later deviate from one another and evolve with time in networks that are referred to as heterogeneously coupled. The intensity of coupling strength fluctuations and the rate at which coupling strengths converge to a desired mean value are studied to determine their impact upon synchronisation performance. The stochastic delay differential equations governing the numerically simulated networks are then converted into a finite set of deterministic cumulant equations by virtue of the Gaussian approximation method. Cumulant equations for maximal and sub-maximal connectivity are used to generate two-parameter bifurcation diagrams on the noise intensity and coupling strength plane, which provides qualitative agreement with numerical simulations. Analysis of artificial brain networks, in respect to biological brain networks, are discussed in light of recent research in sleep theor

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

Dynamics and Synchronization in Neuronal Models

La tesis está principalmente dedicada al modelado y simulación de sistemas neuronales. Entre otros aspectos se investiga el papel del ruido cuando actua sobre neuronas. El fenómeno de resonancia estocástica es caracterizado tanto a nivel teórico como reportado experimentalmente en un conjunto de neuronas del sistema motor. También se estudia el papel que juega la heterogeneidad en un conjunto de neuronas acopladas demostrando que la heterogeneidad en algunos parámetros de las neuronas puede mejorar la respuesta del sistema a una modulación periódica externa. También estudiamos del efecto de la topología y el retraso en las conexiones en una red neuronal. Se explora como las propiedades topológicas y los retrasos en la conducción de diferentes clases de redes afectan la capacidad de las neuronas para establecer una relación temporal bien definida mediante sus potenciales de acción. En particular, el concepto de consistencia se introduce y estudia en una red neuronal cuando plasticidad neuronal es tenida en cuenta entre las conexiones de la re

Dynamical systems techniques in the analysis of neural systems

As we strive to understand the mechanisms underlying neural computation, mathematical models are increasingly being used as a counterpart to biological experimentation. Alongside building such models, there is a need for mathematical techniques to be developed to examine the often complex behaviour that can arise from even the simplest models. There are now a plethora of mathematical models to describe activity at the single neuron level, ranging from one-dimensional, phenomenological ones, to complex biophysical models with large numbers of state variables. Network models present even more of a challenge, as rich patterns of behaviour can arise due to the coupling alone. We first analyse a planar integrate-and-fire model in a piecewise-linear regime. We advocate using piecewise-linear models as caricatures of nonlinear models, owing to the fact that explicit solutions can be found in the former. Through the use of explicit solutions that are available to us, we categorise the model in terms of its bifurcation structure, noting that the non-smooth dynamics involving the reset mechanism give rise to mathematically interesting behaviour. We highlight the pitfalls in using techniques for smooth dynamical systems in the study of non-smooth models, and show how these can be overcome using non-smooth analysis. Following this, we shift our focus onto the use of phase reduction techniques in the analysis of neural oscillators. We begin by presenting concrete examples showcasing where these techniques fail to capture dynamics of the full system for both deterministic and stochastic forcing. To overcome these failures, we derive new coordinate systems which include some notion of distance from the underlying limit cycle. With these coordinates, we are able to capture the effect of phase space structures away from the limit cycle, and we go on to show how they can be used to explain complex behaviour in typical oscillatory neuron models

Dynamical systems techniques in the analysis of neural systems

As we strive to understand the mechanisms underlying neural computation, mathematical models are increasingly being used as a counterpart to biological experimentation. Alongside building such models, there is a need for mathematical techniques to be developed to examine the often complex behaviour that can arise from even the simplest models. There are now a plethora of mathematical models to describe activity at the single neuron level, ranging from one-dimensional, phenomenological ones, to complex biophysical models with large numbers of state variables. Network models present even more of a challenge, as rich patterns of behaviour can arise due to the coupling alone. We first analyse a planar integrate-and-fire model in a piecewise-linear regime. We advocate using piecewise-linear models as caricatures of nonlinear models, owing to the fact that explicit solutions can be found in the former. Through the use of explicit solutions that are available to us, we categorise the model in terms of its bifurcation structure, noting that the non-smooth dynamics involving the reset mechanism give rise to mathematically interesting behaviour. We highlight the pitfalls in using techniques for smooth dynamical systems in the study of non-smooth models, and show how these can be overcome using non-smooth analysis. Following this, we shift our focus onto the use of phase reduction techniques in the analysis of neural oscillators. We begin by presenting concrete examples showcasing where these techniques fail to capture dynamics of the full system for both deterministic and stochastic forcing. To overcome these failures, we derive new coordinate systems which include some notion of distance from the underlying limit cycle. With these coordinates, we are able to capture the effect of phase space structures away from the limit cycle, and we go on to show how they can be used to explain complex behaviour in typical oscillatory neuron models

Chimeras in physics and biology : Synchronization and desynchronization of rhythms

Rhythmen prägen unser Leben auf vielfältige Weise, z. B. durch Herzschlag und Atmung, oszillierende Gehirnströme, Lebenszyklen und Jahreszeiten, Uhren und Metronome, pulsierende Laser, Übertragung von Datenpaketen, und vieles andere. Die Physik komplexer nichtlinearer Systeme hat Methoden entwickelt, wie periodische Schwingungen und deren Synchronisation in komplexen Netzwerken, die aus vielen Bestandteilen zusammengesetzt sind, beschrieben und analysiert werden können. Synchronisierte Oszillationen, aber auch völlig desynchronisierte, chaotische Oszillationen spielen eine große Rolle in vielen Netzwerken in Natur und Technik. Beispielsweise ist das synchronisierte Feuern aller Neuronen im Gehirn ein pathologischer Zustand, etwa bei Epilepsie oder Parkinson-Erkrankung, und sollte unterdrückt werden, wie auch synchrone mechanische Schwingungen von Brücken. Andererseits ist die Synchronisation erwünscht beim stabilen Betrieb von Stromnetzen oder bei der verschlüsselten Kommunikation mit chaotischen Signalen. In Netzwerken aus identischen Komponenten können sich überraschenderweise auch spontan Hybrid-Zustände („Schimären“) bilden, die aus räumlich koexistierenden synchronisierten und desynchronisierten Bereichen bestehen, welche scheinbar nicht zusammen passen. Diese könnten relevant sein bei der Auslösung oder Beendigung epileptischer Anfälle, oder beim halbseitigen Schlaf einer Gehirnhälfte, der bei bestimmten Zugvögeln oder Säugetieren auftritt, oder beim kaskadenartigen Zusammenbruch des Stromnetzes.Rhythms influence our life in various ways, e.g., through heart beat and respiration, oscillating brain currents, life cycles and seasons, clocks and metronomes, pulsating lasers, transmission of data packets, and many others. The physics of complex nonlinear systems has developed methods to describe and analyze periodic oscillations and their synchronization in complex networks, which are composed of many components. Synchronized oscillations as well as completely asynchronous chaotic oscillations play a major role in many networks in nature and technology. For instance, the synchronous firing of all neurons in the brain represents a pathological state, like in epilepsy or Parkinson’s disease, and should be suppressed, as well as the synchronous mechanical vibration of bridges. On the other hand, synchronization is desirable for the stable operation of power grids or in encrypted communication with chaotic signals. In networks composed of identical components, intriguing hybrid states (“chimeras”) may form spontaneously, which consist of spatially coexisting synchronized and desynchronized domains, i.e., seemingly incongruous parts. This might be of relevance in inducing and terminating epileptic seizures, or in unihemispheric sleep which is found in certain migratory birds and mammals, or in cascading failures of the power grid.DFG, 163436311, Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und AnwendungskonzepteDFG, 308748074, DFG-RSF: Komplexe dynamische Netzwerke: Effekte von heterogenen, adaptiven und zeitverzögerten Kopplunge