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

三次元心臓モデルと不均一振動モデルによる心臓活動電位のコンピュータシミュレーション

Tohoku University西條芳文論

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems: Proceedings

Proceedings of the 18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems, which took place in Dresden, Germany, 26 – 28 May 2010.:Welcome Address ........................ Page I Table of Contents ........................ Page III Symposium Committees .............. Page IV Special Thanks ............................. Page V Conference program (incl. page numbers of papers) ................... Page VI Conference papers Invited talks ................................ Page 1 Regular Papers ........................... Page 14 Wednesday, May 26th, 2010 ......... Page 15 Thursday, May 27th, 2010 .......... Page 110 Friday, May 28th, 2010 ............... Page 210 Author index ............................... Page XII

Physical grounds for causal perspectivalism

In this paper we ground the asymmetry of causal relations in the internal physical states of a special kind of open dissipative physical system, a causal agent. A causal agent is an autonomous physical system, maintained far from equilibrium by a low entropy source of energy, with accurate sensors and actuators. It has a memory to record sensor measurements and actuator operations. It contains a learning system that can access the sensor and actuator records to learn and represent the causal relations. We claim that causal relations are relations between the internal sensor and actuator records and the causal concept inherent in these correlations is then inscribed in the physical dynamics of the internal learning machine. The existence of contingent internal memory states means each causal agent is in a different physical state. We argue that it is in this sense that causal relations are perspectival. From the outside, averaging over internal states, the causal agents are identical thermodynamic systems.Comment: 25 pages. Revised list of reference

Multifractal Desynchronization of the Cardiac Excitable Cell Network During Atrial Fibrillation. II. Modeling

In a companion paper (I. Multifractal analysis of clinical data), we used a wavelet-based multiscale analysis to reveal and quantify the multifractal intermittent nature of the cardiac impulse energy in the low frequency range ≲ 2Hz during atrial fibrillation (AF). It demarcated two distinct areas within the coronary sinus (CS) with regionally stable multifractal spectra likely corresponding to different anatomical substrates. The electrical activity also showed no sign of the kind of temporal correlations typical of cascading processes across scales, thereby indicating that the multifractal scaling is carried by variations in the large amplitude oscillations of the recorded bipolar electric potential. In the present study, to account for these observations, we explore the role of the kinetics of gap junction channels (GJCs), in dynamically creating a new kind of imbalance between depolarizing and repolarizing currents. We propose a one-dimensional (1D) spatial model of a denervated myocardium, where the coupling of cardiac cells fails to synchronize the network of cardiac cells because of abnormal transjunctional capacitive charging of GJCs. We show that this non-ohmic nonlinear conduction 1D modeling accounts quantitatively well for the “multifractal random noise” dynamics of the electrical activity experimentally recorded in the left atrial posterior wall area. We further demonstrate that the multifractal properties of the numerical impulse energy are robust to changes in the model parameters

Target patterns and pacemakers in reaction-diffusion systems

Pattern formation in systems far from thermal equilibrium is a fascinating phenomenon. Reaction-diffusion systems are an important type of system where pattern formation is observed. The target pattern and the associated wave source called pacemaker are typical patterns in such systems. This thesis studies pacemakers and target patterns systematically by analytical and numerical means. The underlying dynamics of the system may be oscillatory or excitable and the pacemakers may either consist of spatial heterogeneities of the medium or be self-organized, i.e. result of intrinsic processes. The investigation of heterogeneous pacemakers in oscillatory systems in the framework of the complex Ginzburg-Landau equation focuses on two aspects. First, the conditions of the creation of pacemakers and extended target patterns versus the creation of wave sinks and localized target patterns are derived systematically. In particular, inward traveling target patterns and large heterogeneities are discussed. Then, pacemakers which emit target waves with high frequencies are considered. In this case, the waves become Eckhaus unstable, causing ring-shaped amplitude defects or other complex patterns. For even larger frequencies, the amplitude defects already take place at the boundary of the heterogeneity, giving rise to a localized desynchronization phenomenon. Moreover, wave sinks can have a significant impact on the spatio-temporal dynamics of the system by breaking the waves arriving from other wave sources. It is well known that oscillatory media close to a Hopf bifurcation are not able to give rise to stable self-organized pacemakers. Therefore, to model such pacemakers, a system close to a pitchfork-Hopf bifurcation is proposed. The normal form and amplitude equations of the pitchfork-Hopf bifurcation are derived. Such a system displays birhythmicity, i.e. bistability of limit cycles, and it is demonstrated analytically that stable self-organized pacemakers are possible. Simulations confirm the existence of stable self-organized pacemakers. In the presence of a parameter gradient, such patterns drift, as shown analytically and numerically. The interaction between pacemakers is studied numerically, giving rise either to coexisting pacemakers or to a new phenomenon called global inhibition: Established pacemakers suppress new cores or merge with them. When the frequencies of the limit cycles differ strongly, the waves may become Eckhaus unstable and the pacemaker may destabilize. Furthermore, kinetic instabilities of the pacemakers are possible, creating breathing and swinging pacemakers. Self-organized pacemakers in excitable media are usually unstable. In this thesis, a three-component activator-inhibitor system on the basis of the FitzHugh-Nagumo model is proposed that gives rise to stable self-organized pacemakers in the excitable regime. The formation of such patterns is demonstrated if several conditions are fulfilled: The system is close to relaxational oscillations, the additional component is strongly diffusive, and the additional component inhibits the inhibitor. Moreover, bistability of pulse solutions is observed in such a system. Different pulses can interact and may create pacemakers. Alternatively, other complex spatio-temporal dynamics is observed. If the diffusion of the activator vanishes, the waves emitted by the wave source are unstable and spatio-temporal chaos appears. Thus, this thesis presents new results on the dynamics of pacemakers with large frequencies and demonstrates for the first time the possibility of stable self-organized pacemakers in birhythmic and excitable systems

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

Emulation of Neural Dynamics in Neuromorphic Circuits Based on Memristive Devices

The most impressive properties of the human brain are widely acknowledged as being perception and consciousness. While the underlying mechanisms are not yet understood, it is very likely that neural dynamics, in connection with the topology of neural networks, may play a decisive role. Neuromorphic systems offer an interesting approach to emulate and model these processes, as they allow the complexity of neural networks to be mapped onto energy-efficient and real-time capable systems. For this purpose, analogue electrical circuits that are oriented as closely as possible to biological networks are investigated. Electronic devices are particularly important for this purpose, as they make it possible to emulate the mechanisms that are important to the learning and memory processes that occur at the connections of neurons in form of synapses. In this context, it has been shown that nano-ionic mechanisms, in socalled memristive devices, allow the emulation of synaptic plasticity on a descriptive level within a single device. Memristive devices are passive, non-volatile components whose resistance value depends on the applied electrical potentials. In recent years, the important plasticity mechanisms of synaptic information-processing have been emulated using memristive devices. The importance of memristive devices in terms of emulating dynamic processes within novel bio-inspired computing schemes attract worldwide interest and is the subject of this thesis