Synchronization of Coupled Different Chaotic FitzHugh-Nagumo Neurons with Unknown Parameters under Communication-Direction-Dependent Coupling

This paper investigates the chaotic behavior and synchronization of two different coupled chaotic FitzHugh-Nagumo (FHN) neurons with unknown parameters under external electrical stimulation (EES). The coupled FHN neurons of different parameters admit unidirectional and bidirectional gap junctions in the medium between them. Dynamical properties, such as the increase in synchronization error as a consequence of the deviation of neuronal parameters for unlike neurons, the effect of difference in coupling strengths caused by the unidirectional gap junctions, and the impact of large time-delay due to separation of neurons, are studied in exploring the behavior of the coupled system. A novel integral-based nonlinear adaptive control scheme, to cope with the infeasibility of the recovery variable, for synchronization of two coupled delayed chaotic FHN neurons of different and unknown parameters under uncertain EES is derived. Further, to guarantee robust synchronization of different neurons against disturbances, the proposed control methodology is modified to achieve the uniformly ultimately bounded synchronization. The parametric estimation errors can be reduced by selecting suitable control parameters. The effectiveness of the proposed control scheme is illustrated via numerical simulations

Lyapunov-based synchronization of two coupled chaotic Hindmarsh-Rose neurons

This paper addresses the synchronization of coupled chaotic Hindmarsh-Rose neurons. A sufficient condition for self-synchronization is first attained by using Lyapunov method. Also, to achieve the synchronization between two coupled Hindmarsh-Rose neurons when the self-synchronization condition not satisfied, a Lyapunov-based nonlinear control law is proposed and its asymptotic stability is proved. To verify the effectiveness of the proposed method, numerical simulations are performed

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

Mathematical methods of factorization and a feedback approach for biological systems

The first part of the thesis is devoted to factorizations of linear and nonlinear differential equations leading to solutions of the kink type. The second part contains a study of the synchronization of the chaotic dynamics of two Hodgkin-Huxley neurons by means of the mathematical tools belonging to the geometrical control theory.Comment: Ph. D. Thesis at IPICyT, San Luis Potosi, Mexico, 102 pp, 40 figs. Supervisors: Dr. H.C. Rosu and Dr. R. Fema

Synchronous behavior in networks of coupled systems : with applications to neuronal dynamics

Synchronization in networks of interacting dynamical systems is an interesting phenomenon that arises in nature, science and engineering. Examples include the simultaneous flashing of thousands of fireflies, the synchronous firing of action potentials by groups of neurons, cooperative behavior of robots and synchronization of chaotic systems with applications to secure communication. How is it possible that systems in a network synchronize? A key ingredient is that the systems in the network "communicate" information about their state to the systems they are connected to. This exchange of information ultimately results in synchronization of the systems in the network. The question is how the systems in the network should be connected and respond to the received information to achieve synchronization. In other words, which network structures and what kind of coupling functions lead to synchronization of the systems? In addition, since the exchange of information is likely to take some time, can systems in networks show synchronous behavior in presence of time-delays? The first part of this thesis focusses on synchronization of identical systems that interact via diffusive coupling, that is a coupling defined through the weighted difference of the output signals of the systems. The coupling might contain timedelays. In particular, two types of diffusive time-delay coupling are considered: coupling type I is diffusive coupling in which only the transmitted signals contain a time-delay, and coupling type II is diffusive coupling in which every signal is timedelayed. It is proven that networks of diffusive time-delay coupled systems that satisfy a strict semipassivity property have solutions that are ultimately bounded. This means that the solutions of the interconnected systems always enter some compact set in finite time and remain in that set ever after. Moreover, it is proven that nonlinear minimum-phase strictly semipassive systems that interact via diffusive coupling always synchronize provided the interaction is sufficiently strong. If the coupling functions contain time-delays, then these systems synchronize if, in addition to the sufficiently strong interaction, the product of the time-delay and the coupling strength is sufficiently small. Next, the specific role of the topology of the network in relation to synchronization is discussed. First, using symmetries in the network, linear invariant manifolds for networks of the diffusively time-delayed coupled systems are identified. If such a linear invariant manifold is also attracting, then the network possibly shows partial synchronization. Partial synchronization is the phenomenon that some, at least two, systems in the network synchronize with each other but not with every system in the network. It is proven that a linear invariant manifold defined by a symmetry in a network of strictly semipassive systems is attracting if the coupling strength is sufficiently large and the product of the coupling strength and the time-delay is sufficiently small. The network shows partial synchronization if the values of the coupling strength and time-delay for which this manifold is attracting differ from those for which all systems in the network synchronize. Next, for systems that interact via symmetric coupling type II, it is shown that the values of the coupling strength and time-delay for which any network synchronizes can be determined from the structure of that network and the values of the coupling strength and time-delay for which two systems synchronize. In the second part of the thesis the theory presented in the first part is used to explain synchronization in networks of neurons that interact via electrical synapses. In particular, it is proven that four important models for neuronal activity, namely the Hodgkin-Huxley model, the Morris-Lecar model, the Hindmarsh-Rose model and the FitzHugh-Nagumo model, all have the semipassivity property. Since electrical synapses can be modeled by diffusive coupling, and all these neuronal models are nonlinear minimum-phase, synchronization in networks of these neurons happens if the interaction is sufficiently strong and the product of the time-delay and the coupling strength is sufficiently small. Numerical simulations with various networks of Hindmarsh-Rose neurons support this result. In addition to the results of numerical simulations, synchronization and partial synchronization is witnessed in an experimental setup with type II coupled electronic realizations of Hindmarsh-Rose neurons. These experimental results can be fully explained by the theoretical findings that are presented in the first part of the thesis. The thesis continues with a study of a network of pancreatic -cells. There is evidence that these beta-cells are diffusively coupled and that the synchronous bursting activity of the network is related to the secretion of insulin. However, if the network consists of active (oscillatory) beta-cells and inactive (dead) beta-cells, it might happen that, due to the interaction between the active and inactive cells, the activity of the network dies out which results in a inhibition of the insulin secretion. This problem is related to Diabetes Mellitus type 1. Whether the activity dies out or not depends on the number of cells that are active relative to the number of inactive cells. A bifurcation analysis gives estimates of the number of active cells relative to the number of inactive cells for which the network remains active. At last the controlled synchronization problem for all-to-all coupled strictly semipassive systems is considered. In particular, a systematic design procedure is presented which gives (nonlinear) coupling functions that guarantee synchronization of the systems. The coupling functions have the form of a definite integral of a scalar weight function on a interval defined by the outputs of the systems. The advantage of these coupling functions over linear diffusive coupling is that they provide high gain only when necessary, i.e. at those parts of the state space of the network where nonlinearities need to be suppressed. Numerical simulations in networks of Hindmarsh-Rose neurons support the theoretical results

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

Neuronal oscillations: from single-unit activity to emergent dynamics and back

L’objectiu principal d’aquesta tesi és avançar en la comprensió del processament d’informació en xarxes neuronals en presència d’oscil lacions subumbrals. La majoria de neurones propaguen la seva activitat elèctrica a través de sinapsis químiques que són activades, exclusivament, quan el corrent elèctric que les travessa supera un cert llindar. És per aquest motiu que les descàrregues ràpides i intenses produïdes al soma neuronal, els anomenats potencials d’acció, són considerades la unitat bàsica d’informació neuronal, és a dir, el senyal mínim i necessari per a iniciar la comunicació entre dues neurones. El codi neuronal és entès, doncs, com un llenguatge binari que expressa qualsevol missatge (estímul sensorial, memòries, etc.) en un tren de potencials d’acció. Tanmateix, cap funció cognitiva rau en la dinàmica d’una única neurona. Circuits de milers de neurones connectades entre sí donen lloc a determinats ritmes, palesos en registres d’activitat colectiva com els electroencefalogrames (EEG) o els potencials de camp local (LFP). Si els potencials d’acció de cada cèl lula, desencadenats per fluctuacions estocàstiques de les corrents sinàptiques, no assolissin un cert grau de sincronia, no apareixeria aquesta periodicitat a nivell de xarxa. Per tal de poder entendre si aquests ritmes intervenen en el codi neuronal hem estudiat tres situacions. Primer, en el Capítol 2, hem mostrat com una cadena oberta de neurones amb un potencial de membrana intrínsecament oscil latori filtra un senyal periòdic arribant per un dels extrems. La resposta de cada neurona (pulsar o no pulsar) depèn de la seva fase, de forma que cada una d’elles rep un missatge filtrat per la precedent. A més, cada potencial d’acció presinàptic provoca un canvi de fase en la neurona postsinàptica que depèn de la seva posició en l’espai de fases. Els períodes d’entrada capaços de sincronitzar les oscil lacions subumbrals són aquells que mantenen la fase d’arribada dels potencials d’acció fixa al llarg de la cadena. Per tal de què el missatge arribi intacte a la darrera neurona cal, a més a més, que aquesta fase permeti la descàrrega del voltatge transmembrana. En segon cas, hem estudiat una xarxa neuronal amb connexions tant a veïns propers com de llarg abast, on les oscil lacions subumbrals emergeixen de l’activitat col lectiva reflectida en els corrents sinàptics (o equivalentment en el LFP). Les neurones inhibidores aporten un ritme a l’excitabilitat de la xarxa, és a dir, que els episodis en què la inhibició és baixa, la probabilitat d’una descàrrega global de la població neuronal és alta. En el Capítol 3 mostrem com aquest ritme implica l’aparició d’una bretxa en la freqüència de descàrrega de les neurones: o bé polsen espaiadament en el temps o bé en ràfegues d’elevada intensitat. La fase del LFP determina l’estat de la xarxa neuronal codificant l’activitat de la població: els mínims indiquen la descàrrega simultània de moltes neurones que, ocasionalment, han superat el llindar d’excitabilitat degut a un decreixement global de la inhibició, mentre que els màxims indiquen la coexistència de ràfegues en diferents punts de la xarxa degut a decreixements locals de la inhibició en estats globals d’excitació. Aquesta dinàmica és possible gràcies al domini de la inhibició sobre l’excitació. En el Capítol 4 considerem acoblament entre dues xarxes neuronals per tal d’estudiar la interacció entre ritmes diferents. Les oscil lacions indiquen recurrència en la sincronització de l’activitat col lectiva, de manera que durant aquestes finestres temporals una població optimitza el seu impacte en una xarxa diana. Quan el ritme de la població receptora i el de l’emissora difereixen significativament, l’eficiència en la comunicació decreix, ja que les fases de màxima resposta de cada senyal LFP no mantenen una diferència constant entre elles. Finalment, en el Capítol 5 hem estudiat com les oscil lacions col lectives pròpies de l’estat de son donen lloc al fenomen de coherència estocàstica. Per a una intensitat òptima del soroll, modulat per l’excitabilitat de la xarxa, el LFP assoleix una regularitat màxima donant lloc a un període refractari de la població neuronal. En resum, aquesta Tesi mostra escenaris d’interacció entre els potencials d’acció, característics de la dinàmica de neurones individuals, i les oscil lacions subumbrals, fruit de l’acoblament entre les cèl lules i ubiqües en la dinàmica de poblacions neuronals. Els resultats obtinguts aporten funcionalitat a aquests ritmes emergents, agents sincronitzadors i moduladors de les descàrregues neuronals i reguladors de la comunicació entre xarxes neuronals.The main objective of this thesis is to better understand information processing in neuronal networks in the presence of subthreshold oscillations. Most neurons propagate their electrical activity via chemical synapses, which are only activated when the electric current that passes through them surpasses a certain threshold. Therefore, fast and intense discharges produced at the neuronal soma (the action potentials or spikes) are considered the basic unit of neuronal information. The neuronal code is understood, then, as a binary language that expresses any message (sensory stimulus, memories, etc.) in a train of action potentials. Circuits of thousands of interconnected neurons give rise to certain rhythms, revealed in collective activity measures such as electroencephalograms (EEG) and local field potentials (LFP). Synchronization of action potentials of each cell, triggered by stochastic fluctuations of the synaptic currents, cause this periodicity at the network level.To understand whether these rhythms are involved in the neuronal code we studied three situations. First, in Chapter 2, we showed how an open chain of neurons with an intrinsically oscillatory membrane potential filters a periodic signal coming from one of its ends. The response of each neuron (to spike or not) depends on its phase, so that each cell receives a message filtered by the preceding one. Each presynaptic action potential causes a phase change in the postsynaptic neuron, which depends on its position in the phase space. Those incoming periods that are able to synchronize the subthreshold oscillations, keep the phase of arrival of action potentials fixed along the chain. The original message reaches intact the last neuron provided that this phase allows the discharge of the transmembrane voltage.I the second case, we studied a neuronal network with connections to both long range and close neighbors, in which the subthreshold oscillations emerge from the collective activity apparent in the synaptic currents. The inhibitory neurons provide a rhythm to the excitability of the network. When inhibition is low, the likelihood of a global discharge of the neuronal population is high. In Chapter 3 we show how this rhythm causes a gap in the discharge frequency of neurons: either they pulse single spikes or they fire bursts of high intensity. The LFP phase determines the state of the neuronal network, coding the activity of the population: its minima indicate the simultaneous discharge of many neurons, while its maxima indicate the coexistence of bursts due to local decreases of inhibition at global states of excitation. In Chapter 4 we consider coupling between two neural networks in order to study the interaction between different rhythms. The oscillations indicate recurrence in the synchronization of collective activity, so that during these time windows a population optimizes its impact on a target network. When the rhythm of the emitter and receiver population differ significantly, the communication efficiency decreases as the phases of maximum response of each LFP signal do not maintain a constant difference between them.Finally, in Chapter 5 we studied how oscillations typical of the collective sleep state give rise to stochastic coherence. For an optimal noise intensity, modulated by the excitability of the network, the LFP reaches a maximal regularity leading to a refractory period of the neuronal population.In summary, this Thesis shows scenarios of interaction between action potentials, characteristics of the dynamics of individual neurons, and the subthreshold oscillations, outcome of the coupling between the cells and ubiquitous in the dynamics of neuronal populations . The results obtained provide functionality to these emerging rhythms, triggers of synchronization and modulator agents of the neuronal discharges and regulators of the communication between neuronal networks

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