Synchronization in periodically driven and coupled stochastic systems-A discrete state approach

Wir untersuchen das Verhalten von stochastischen bistabilen und erregbaren Systemen auf der Basis einer Modellierung mit diskreten Zuständen. In Ergänzung zum bekannten Markovschen Zwei-Zustandsmodell bistabiler stochastischer Dynamik stellen wir ein nicht Markovsches Drei-Zustandsmodell für erregbare Systeme vor. Seine relative Einfachheit, verglichen mit stochastischen Modellen erregbarer Dynamik mit kontinuierlichem Phasenraum, ermöglicht eine teilweise analytische Auswertung in verschiedenen Zusammenhängen. Zunächst untersuchen wir den gemeinsamen Einfluß eines periodischen Treibens und Rauschens. Dieser wird entweder mit Hilfe spektraler Größen oder durch Synchronisation des Systems mit dem treibenden Signal charakterisiert. Wir leiten analytische Ausdrücke für die spektrale Leistungsverstärkung und das Signal-zu-Rauschen Verhältnis für periodisch getriebene Renewal-Prozesse her und wenden diese auf das diskrete Modell für erregbare Dynamik an. Stochastische Synchronization des Systems mit dem treibenden Signal wird auf der Basis der Diffusionseigenschaften der Übergangsereignisse zwischen den diskreten Zuständen untersucht. Wir leiten allgemeine Formeln her, um die mittlere Häufigkeit dieser Ereignisse sowie deren effektiven Diffusionskoeffizienten zu berechnen. Über die konkrete Anwendung auf die untersuchten diskreten Modelle hinaus stellen diese Ergebnisse ein neues Werkzeug für die Untersuchung periodischer Renewal-Prozesse dar. Schließlich betrachten wir noch das Verhalten global gekoppelter bistabiler und erregbarer Systeme. Im Gegensatz zu bistabilen System können erregbare Systeme synchronisiert werden und zeigen kohärente Oszillationen. Alle Untersuchungen des nicht Markovschen Drei-Zustandsmodells werden mit dem prototypischen Modell für erregbare Dynamik, dem FitzHugh-Nagumo System, verglichen und zeigen eine gute Übereinstimmung.We investigate the behavior of stochastic bistable and excitable dynamics based on a discrete state modeling. In addition to the well known Markovian two state model for bistable dynamics we introduce a non Markovian three state model for excitable systems. Its relative simplicity compared to stochastic models of excitable dynamics with continuous phase space allows to obtain analytical results in different contexts. First, we study the joint influence of periodic signals and noise, both based on a characterization in terms of spectral quantities and in terms of synchronization with the periodic driving. We present expressions for the spectral power amplification and signal to noise ratio for renewal processes driven by periodic signals and apply these results to the discrete model for excitable systems. Stochastic synchronization of the system to the driving signal is investigated based on diffusion properties of the transition events between the discrete states. We derive general results for the mean frequency and effective diffusion coefficient which, beyond the application to the discrete models considered in this work, provide a new tool in the study of periodically driven renewal processes. Finally the behavior of globally coupled excitable and bistable units is investigated based on the discrete state description. In contrast to the bistable systems, the excitable system exhibits synchronization and thus coherent oscillations. All investigations of the non Markovian three state model are compared with the prototypical continuous model for excitable dynamics, the FitzHugh-Nagumo system, revealing a good agreement between both models

Complex and Adaptive Dynamical Systems: A Primer

An thorough introduction is given at an introductory level to the field of quantitative complex system science, with special emphasis on emergence in dynamical systems based on network topologies. Subjects treated include graph theory and small-world networks, a generic introduction to the concepts of dynamical system theory, random Boolean networks, cellular automata and self-organized criticality, the statistical modeling of Darwinian evolution, synchronization phenomena and an introduction to the theory of cognitive systems. It inludes chapter on Graph Theory and Small-World Networks, Chaos, Bifurcations and Diffusion, Complexity and Information Theory, Random Boolean Networks, Cellular Automata and Self-Organized Criticality, Darwinian evolution, Hypercycles and Game Theory, Synchronization Phenomena and Elements of Cognitive System Theory.Comment: unformatted version of the textbook; published in Springer, Complexity Series (2008, second edition 2010

Stochasticity,complexity and synchronization in semiconductor lasers

The purpose of this Thesis is study the dynamical behavior of semiconductor lasers with optical feedback, as well as analyze the synchronization of this kind of systems under different coupling arquitectures. This study has been done from an experimental point of view, but in some cases we have used numerical models in order to verify and/or extend the experimental results. A semiconductor laser in absence of any optical feedback emits light at constant power. If one wants to induce dynamics in the laser, a good strategy is to introduce an external cavity able to reflect the emitted light back into the laser. Due to this feedback, the laser can show a large variety of dynamical behaviors. In this Thesis we will focus mainly in a dynamical regime known as low frequency fluctuations regime (LFF). The LFF regime takes place when the pump current of the laser is close to its threshold current and the feedback strength is sufficiently large, and it consists in sudden intensity dropouts arising at irregular times, followed by a gradual and stepwise recovery. During this Thesis, we have characterized in detail the dynamical behavior of the time between intensity dropouts for a semiconductor laser with feedback, by using different statistical techniques based on information theory concepts. We have quantified the probability of appearance of certain patterns within the temporal series, as well as its degree of complexity. As a result of these studies, we can conclude that the dynamics of a semiconductor laser with optical feedback is stochastic for pump current values close to the laser threshold. On the other hand, for larger pump currents the dynamics is basically deterministic (chaotic). Numerical simulations have shown a good qualitative and quantitive agreement with the experimental results. During this Thesis we have also studied the ability of semiconductor lasers to synchronize under different coupling architectures. First, we have characterized the leader-laggard dynamics showed by two semiconductor lasers bidirectionally coupled operating at the LFF regime, with a method that takes into account the number of forbidden patterns that appears in the temporal series. We have quantified the degree of stochasticity of the system as a function of the pump current of both lasers. A second coupling architecture studied here, consists in two lasers unidirectionally coupled via two paths. In this case, we have analyzed how the synchronization is affected under different values of the coupling strength of both paths, as well as the potential of this system (or rather, the lack thereof) to be used in chaotic communications. Finally we have characterized the synchronization at zero lag for two lasers coupled bidirectionally via a passive relay. In particular, we have studied the desynchronization events and their statistics for different pump currents. The experimental results obtained in this Thesis give a global perspective of the dynamical statistical properties of semiconductor laser dynamics, both isolated or coupled to other lasers, which contributes to a better understanding of this kind of dynamical systems.L’objectiu d’aquesta Tesi ´es l’estudi de la din`amica de l`asers de semiconductor amb retroalimentaci´o `optica, aix´ý com l’an`alisis de la sincronitzaci´o d’aquest tipus de sistemes sota diferents arquitectures d’acoblament. Aquest estudi s’ha fet sempre des d’un punt de vista b`asicament experimental, tot i que en alguns casos hem utilitzat models num`erics per tal de verificar i/o ampliar els resultats experimentals. Un l`aser de semiconductor en abs`encia de retroalimentaci´o `optica o altres perturbacions externes, emet llum a una intensitat pr`acticament constant. Aix´ý doncs, si el que es vol ´es indu¨ýr din`amica en el l`aser, una bona estrat`egia ´es introdu¨ýr una cavitat externa capa¸c de reflexar la llum cap al l`aser. Un cop la llum ´es reinjectada, els l`asers de semiconductor poden mostrar una gran varietat de comportaments din`amics. En aquesta tesis ens centrarem principalment en un r`egim din`amic anomenat r`egim de fluctuacions de baixa frequ`encia (LFF en les seves sigles en angl`es). El r`egim d’LFF es d´ona quan el corrent d’injecci´o del l`aser es troba a prop del seu corrent llindar i la intensitat de la retroalimentaci´o ´es suficientment gran, i est`a caracteritzat per sobtades caigudes de la intensitat a temps irregulars, seguides per una recuperaci´o gradual i escalonada. Durant aquesta Tesi, hem caracteritzat de forma detallada el comportament din`amic de la distribuci´o dels temps entre les caigudes d’intensitat d’un l`aser de semiconductor amb retroalimentaci´o `optica, utilitzant diferents m`etodes estad´ýstics basats en conceptes de teoria de la informaci´o. En particular, hem quantificat la probabilitat d’aparici´o de certs patrons dins les s`eries temporals, aix´ý com el grau de complexitat d’aquestes. Durant aquest estudi hem observat que la din`amica d’un l`aser de semiconductor amb retroalimentaci´o es estoc`astica per valors del corrent d’injecci´o propers al corrent llindar del l`aser. D’altra banda, per a valors m´es grans del corrent d’injecci´o la din`amica ´es m´es determinista (ca`otica). Les simulacions num`eriques realitzades han mostrat un acord qualitatiu i quantitatiu amb els resultats experimentals. Durant aquesta Tesi tamb´e hem estudiat la sincronitzaci´o entre l`asers de semiconductor. Hem analitzat diferents arquitectures d’acoblament. Primer hem caracteritzat la din`amica leader-laggard que presenten dos l`asers de semiconductor acoblats bidireccionalment operant en r`egim de LFFs, amb un m`etode que t´e en compte el nombre de patrons prohibits que apareixen en la s`erie temporal. Hem quantificat el grau d’estocasticitat del sistema en funci´o del nivell de bombeig al qual est`an sotmesos els dos l`asers. La seg¨uent arquitectura d’acoblament que hem estudiat consisteix en dos l`asers acoblats unidireccionalment a trav´es de dos camins. En aquest cas hem analitzat com es veu afectada la sincronitzaci´o sota diferents valors de l’acoblament dels dos camins, aix´ý com el potencial d’aquest esquema experimental per realitzar comunicacions ca`otiques. Per ´ultim hem caracteritzat la sincronitzaci´o a retard zero per dos l`asers acoblats bidireccionalment, a on els dos l`asers tenen la seva pr`opia realimentaci´o `optica. En particular, hem estudiat els events de desincronitzaci´o i la seva estad´ýstica per a diferents corrents d’injecci´o. Els resultats experimentals obtinguts en aquesta Tesi, ofereixen una prespectiva global de les propietats estad´ýstiques de la din`amica de l`asers de semiconductor, tant a¨ýllats com acoblats a altres l`asers, que contribueixen a entendre millor aquests sistemes din`amics

Stochastic and complex dynamics in mesoscopic brain networks

The aim of this thesis is to deepen into the understanding of the mechanisms responsible for the generation of complex and stochastic dynamics, as well as emerging phenomena, in the human brain. We study typical features from the mesoscopic scale, i.e., the scale in which the dynamics is given by the activity of thousands or even millions of neurons. At this scale the synchronous activity of large neuronal populations gives rise to collective oscillations of the average voltage potential. These oscillations can easily be recorded using electroencephalography devices (EEG) or measuring the Local Field Potentials (LFPs). In Chapter 5 we show how the communication between two cortical columns (mesoscopic structures) can be mediated efficiently by a microscopic neural network. We use the synchronization of both cortical columns as a probe to ensure that an effective communication is established between the three neural structures. Our results indicate that there are certain dynamical regimes from the microscopic neural network that favor the correct communication between the cortical columns: therefore, if the LFP frequency of the neural network is of around 40Hz, the synchronization between the cortical columns is more robust compared to the situation in which the neural network oscillates at a lower frequency (10Hz). However, microscopic topological characteristics of the network also influence communication, being a small-world structure the one that best promotes the synchronization of the cortical columns. Finally, this Chapter shows how the mediation exerted by the neural network cannot be substituted by the average of its activity, that is, the dynamic properties of the microscopic neural network are essential for the proper transmission of information between all neural structures. The oscillatory brain electrical activity is largely dependent on the interplay between excitation and inhibition. In Chapter 6 we study how groups of cortical columns show complex patterns of cortical excitation and inhibition taking into account their topological features and the strength of their couplings. These cortical columns segregate between those dominated by excitation and those dominated by inhibition, affecting the synchronization properties of networks of cortical columns. In Chapter 7 we study a dynamic regime by which complex patterns of synchronization between chaotic oscillators appear spontaneously in a network. We show what conditions must a set of coupled dynamical systems fulfill in order to display heterogeneity in synchronization. Therefore, our results are related to the complex phenomenon of synchronization in the brain, which is a focus of study nowadays. Finally, in Chapter 8 we study the ability of the brain to compute and process information. The novelty here is our use of complex synchronization in the brain in order to implement basic elements of Boolean computation. In this way, we show that the partial synchronization of the oscillations in the brain establishes a code in terms of synchronization / non-synchronization (1/0, respectively), and thus all simple Boolean functions can be implemented (AND, OR, XOR, etc.). We also show that complex Boolean functions, such as a flip-flop memory, can be constructed in terms of states of dynamic synchronization of brain oscillations.L'objectiu d'aquesta Tesi és aprofundir en la comprensió dels mecanismes responsables de la generació de dinàmica complexa i estocàstica, així com de fenòmens emergents, en el cervell humà. Estudiem la fenomenologia característica de l'escala mesoscòpica, és a dir, aquella en la que la dinàmica característica ve donada per l'activitat de milers de neurones. En aquesta escala l'activitat síncrona de grans poblacions neuronals dóna lloc a un fenomen col·lectiu pel qual es produeixen oscil·lacions del seu potencial mitjà. Aquestes oscil·lacions poden ser fàcilment enregistrades mitjançant aparells d'electroencefalograma (EEG) o enregistradors de Potencials de Camp Local (LFP). En el Capítol 5 mostrem com la comunicació entre dos columnes corticals (estructures mesoscòpiques) pot ser conduïda de forma eficient per una xarxa neuronal microscòpica. De fet, emprem la sincronització de les dues columnes corticals per comprovar que s'ha establert una comunicació efectiva entre les tres estructures neuronals. Els resultats indiquen que hi ha règims dinàmics de la xarxa neuronal microscòpica que afavoreixen la correcta comunicació entre les columnes corticals: si la freqüència típica de LFP a la xarxa neuronal està al voltant dels 40Hz la sincronització entre les columnes corticals és més robusta que a una menor freqüència (10Hz). La topologia de la xarxa microscòpica també influeix en la comunicació, essent una estructura de tipus món petit (small-world) la que més afavoreix la sincronització. Finalment, la mediació de xarxa neuronal no pot ser substituïda per la mitjana de la seva activitat, és a dir, les propietats dinàmiques microscòpiques són imprescindibles per a la correcta transmissió d'informació entre totes les escales cerebrals. L'activitat elèctrica oscil·latòria cerebral ve donada en gran mesura per la interacció entre excitació i inhibició neuronal. En el Capítol 6 estudiem com grups de columnes corticals mostren patrons complexos d'excitació i inhibició segons quina sigui la seva topologia i d'acoblament. D'aquesta manera les columnes corticals se segreguen entre aquelles dominades per l'excitació i aquelles dominades per la inhibició, influint en les capacitats de sincronització de xarxes de columnes corticals. En el Capítol 7 estudiem un règim dinàmic segons el qual patrons complexos de sincronització apareixen espontàniament en xarxes d'oscil·ladors caòtics. Mostrem quines condicions s'han de donar en un conjunt de sistemes dinàmics acoblats per tal de mostrar heterogeneïtat en la sincronització, és a dir, coexistència de sincronitzacions. D'aquesta manera relacionem els nostres resultats amb el fenomen de sincronització complexa en el cervell. Finalment, en el Capítol 8 estudiem com el cervell computa i processa informació. La novetat aquí és l'ús que fem de la sincronització complexa de columnes corticals per tal d'implementar elements bàsics de computació Booleana. Mostrem com la sincronització parcial de les oscil·lacions cerebrals estableix un codi neuronal en termes de sincronització/no sincronització (1/0, respectivament) amb el qual totes les funcions Booleanes simples poden ésser implementades (AND, OR, XOR, etc). Mostrem, també, com emprant xarxes mesoscòpiques extenses les capacitats de computació creixen proporcionalment. Així funcions Booleanes complexes, com una memòria del tipus flip-flop, pot ésser construïda en termes d'estats de sincronització dinàmica d'oscil·lacions cerebrals.Postprint (published version

Mesoscopic dynamics of pitch processing in human auditory cortex.

Pitch is a perceptual correlate of sound periodicity elicited by vibrating bodies; it plays a crucial role in music and speech. Although perceptual phenomenology of pitch has been studied for centuries, a detailed understanding of its underlying neural mechanisms is still lacking. Early theories suggesting that pitch is decoded in the peripheral auditory system fail to explain the perception of complex stimuli. More recent mechanistic models, focused on how subcortical structures process periodic discharges of the auditory nerve activity, are unable to explain fully key aspects of the processing dynamics observed during electrophysiological recordings. In this thesis, we propose a novel theory describing how subcortical representations of pitch-related information are integrated in cortex and how this integratory process gives rise to the dynamics observed in magnetoencephalographic (MEG) experimental recordings. Auditory evoked fields recorded with MEG reveal a systematic deflection around 100 ms after stimulus’ onset known as the N100m. This deflection consists of several components reflecting the onset of different perceptual dimensions of auditory stimuli such as pitch, timbre and loudness. The exact latency of the component elicited by pitch onset, known as the pitch onset response (POR), shows a strong linear relationship with the pitch of the stimulus. Our theory links the POR latency with processing time and explains, in a quantitative manner, the substrate of the relationship between processing time and pitch. Cortical integration is described using a model of neural ensembles located in two adjacent areas, putatively located along the lateral portion of Heschl’s Gyrus in human auditory cortex. Cortical areas are hierarchically structured and communicate with each other in a top-down fashion. Pitch processing is modelled as a multi-attractor system whose dynamics are driven by subcortical input. After tone onset, the system evolves from an initial equilibrium position to a new equilibrium state that represents the pitch elicited by the tone. A computational implementation of the model shows that: 1) the transient dynamics between equilibrium points explains the POR; 2) the latency of the transient is directly linked with the time required to reach the new equilibrium state; and 3) that such processing time depends linearly on the pitch of the stimuli. Our theory also addresses the problem of how tones with several simultaneous pitch values are processed in cortex. In Western music, dyads comprising tones with different pitch values are judged as more consonant or more dissonant depending on the ratio of the periods of the involved sounds. The latency of the POR evoked by such dyads also presents a strong correlation with the perceived consonance: dissonant dyads generate later PORs than consonant dyads. Our theory of pitch processing describes consonance (dissonance) as a direct effect of harmonic collaboration (competition) during the cortical integration process: the cortical model shows that harmonic collaboration facilitates convergence, explaining why dissonant dyads require longer processing times and evoke later PORs than consonant dyads

Integrated information theory in complex neural systems

This thesis concerns Integrated Information Theory (IIT), a branch of information theory aimed at providing a fundamental theory of consciousness. At its core, lie two powerful intuitions: • That a system that is somehow more than the sum of its parts has non-zero integrated information, Φ; and • That a system with non-zero integrated information is conscious. The audacity of IIT’s claims about consciousness has (understandably) sparked vigorous criticism, and experimental evidence for IIT as a theory of consciousness remains scarce and indirect. Nevertheless, I argue that IIT still has merits as a theory of informational complexity within complexity science, leaving aside all claims about consciousness. In my work I follow this broad line of reasoning: showcasing applications where IIT yields rich analyses of complex systems, while critically examining its merits and limitations as a theory of consciousness. This thesis is divided in three parts. First, I describe three example applications of IIT to complex systems from the computational neuroscience literature (coupled oscillators, spiking neurons, and cellular automata), and develop novel Φ estimators to extend IIT’s range of applicability. Second, I show two important limitations of current IIT: that its axiomatic foundation is not specific enough to determine a unique measure of integrated information; and that available measures do not behave as predicted by the theory when applied to neurophysiological data. Finally, I present new theoretical developments aimed at alleviating some of IIT’s flaws. These are based on the concepts of partial information decomposition and lead to a unification of both theories, Integrated Information Decomposition, or ΦID. The thesis concludes with two experimental studies on M/EEG data, showing that a much simpler informational theory of consciousness – the entropic brain hypothesis – can yield valuable insight without the mathematical challenges brought by IIT.Open Acces

Neural Models of Subcortical Auditory Processing

An important feature of the auditory system is its ability to distinguish many simultaneous sound sources. The primary goal of this work was to understand how a robust, preattentive analysis of the auditory scene is accomplished by the subcortical auditory system. Reasonably accurate modelling of the morphology and organisation of the relevant auditory nuclei, was seen as being of great importance. The formulation of plausible models and their subsequent simulation was found to be invaluable in elucidating biological processes and in highlighting areas of uncertainty. In the thesis, a review of important aspects of mammalian auditory processing is presented and used as a basis for the subsequent modelling work. For each aspect of auditory processing modelled, psychophysical results are described and existing models reviewed, before the models used here are described and simulated. Auditory processes which are modelled include the peripheral system, and the production of tonotopic maps of the spectral content of complex acoustic stimuli, and of modulation frequency or periodicity. A model of the formation of sequential associations between successive sounds is described, and the model is shown to be capable of emulating a wide range of psychophysical behaviour. The grouping of related spectral components and the development of pitch perception is also investigated. Finally a critical assessment of the work and ideas for future developments are presented. The principal contributions of this work are the further development of a model for pitch perception and the development of a novel architecture for the sequential association of those groups. In the process of developing these ideas, further insights into subcortical auditory processing were gained, and explanations for a number of puzzling psychophysical characteristics suggested.Royal Naval Engineering College, Manadon, Plymout

Neural dynamics of social behavior : An evolutionary and mechanistic perspective on communication, cooperation, and competition among situated agents

Social behavior can be found on almost every level of life, ranging from microorganisms to human societies. However, explaining the evolutionary emergence of cooperation, communication, or competition still challenges modern biology. The most common approaches to this problem are based on game-theoretic models. The problem is that these models often assume fixed and limited rules and actions that individual agents can choose from, which excludes the dynamical nature of the mechanisms that underlie the behavior of living systems. So far, there exists a lack of convincing modeling approaches to investigate the emergence of social behavior from a mechanistic and evolutionary perspective. Instead of studying animals, the methodology employed in this thesis combines several aspects from alternative approaches to study behavior in a rather novel way. Robotic models are considered as individual agents which are controlled by recurrent neural networks representing non-linear dynamical system. The topology and parameters of these networks are evolved following an open-ended evolution approach, that is, individuals are not evaluated on high-level goals or optimized for specific functions. Instead, agents compete for limited resources to enhance their chance of survival. Further, there is no restriction with respect to how individuals interact with their environment or with each other. As its main objective, this thesis aims at a complementary approach for studying not only the evolution, but also the mechanisms of basic forms of communication. For this purpose it can be shown that a robot does not necessarily have to be as complex as a human, not even as complex as a bacterium. The strength of this approach is that it deals with rather simple, yet complete and situated systems, facing similar real world problems as animals do, such as sensory noise or dynamically changing environments. The experimental part of this thesis is substantiated in a five-part examination. First, self-organized aggregation patterns are discussed. Second, the advantages of evolving decentralized control with respect to behavioral robustness and flexibility is demonstrated. Third, it is shown that only minimalistic local acoustic communication is required to coordinate the behavior of large groups. This is followed by investigations of the evolutionary emergence of communication. Finally, it is shown how already evolved communicative behavior changes during further evolution when a population is confronted with competition about limited environmental resources. All presented experiments entail thorough analysis of the dynamical mechanisms that underlie evolved communication systems, which has not been done so far in the context of cooperative behavior. This framework leads to a better understanding of the relation between intrinsic neurodynamics and observable agent-environment interactions. The results discussed here provide a new perspective on the evolution of cooperation because they deal with aspects largely neglected in traditional approaches, aspects such as embodiment, situatedness, and the dynamical nature of the mechanisms that underlie behavior. For the first time, it can be demonstrated how noise influences specific signaling strategies and that versatile dynamics of very small-scale neural networks embedded in sensory-motor feedback loops give rise to sophisticated forms of communication such as signal coordination, cooperative intraspecific communication, and, most intriguingly, aggressive interspecific signaling. Further, the results demonstrate the development of counteractive niche construction based on a modification of communication strategies which generates an evolutionary feedback resulting in an active reduction of selection pressure, which has not been shown so far. Thus, the novel findings presented here strongly support the complementary nature of robotic experiments to study the evolution and mechanisms of communication and cooperation.</p