Numerical simulations of the spiking activity and the related first exit time of stochastic neural systems

The aim of this thesis was to study, using numerical simulation techniques, the possible effects of an additive noise on the firing properties of stochastic neural models, and the related first exit time problems. The research is divided into three main investigations. First, using SDELab, mathematical software for solving stochastic differential equations within MATLAB, we examine the influence of an additive noise on the output spike trains for the space-clamped Hodgkin Huxley (HH) model and the spatially-extended FitzHugh Nagumo (FHN) system. We find that a suitable amount of additive noise can enhance the regularity of the repetitive spiking of the space-clamped HH model. Meanwhile, we find the FHN system to be sensitive to noise, requiring that very small values of noise are chosen, in order to produce regular spikes. Second, under additive noise, we use fixed and exponential time-stepping Euler algorithms, with boundary tests, to calculate the mean first exit times (MFET) for one-dimensional neural diffusion models, represented by a stochastic space-clamped FHN system and the Ornstein-Uhlenbeck (OU) model. The strategies and theory behind these numerical methods and their convergence rates in the MFET are also considered. We find that, for different values of noise, these methods with boundary tests can improve the rate of convergence from order one half to order one, which coincides with previous studies. Finally, we look at spatially-extended systems, represented by the Barkley system with additive noise that is white in time and correlated in space, calculating mean nucleation times and mean lifetimes of traveling waves, using an efficient numerical simulation. A simple model of the dynamics of the underlying Barkley model is introduced, in order to compute the mean lifetimes, particulary for interacting waves. The reduced model is easy to use and allows us to explore the full dynamics of the kinks and antikinks, in particular over long periods. One application of the reduced model is to calculate the mean number of kinks at a given time and use this to obtain the probability that the system is excitable at a given position. With these three investigations into the effects of additive noise on stochastic neural models, we have demonstrated some of the interesting results that can be achieved using numerical techniques. We hope to extend this work, in the future, to include the effects of multiplicative noise.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

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 Dynamics in Dedicated / Multifunctional Neural Networks and Chaotic Nonlinear Systems

We study complex behaviors arising in neuroscience and other nonlinear systems by combining dynamical systems analysis with modern computational approaches including GPU parallelization and unsupervised machine learning. To gain insights into the behaviors of brain networks and complex central pattern generators (CPGs), it is important to understand the dynamical principles regulating individual neurons as well as the basic structural and functional building blocks of neural networks. In the first section, we discuss how symbolic methods can help us analyze neural dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations in various models of individual neurons, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits, such as network bursting from non-intrinsic bursters. The second section is focused on the origin and coexistence of multistable rhythms in oscillatory neural networks of inhibitory coupled cells. We discuss how network connectivity and intrinsic properties of the cells affect the dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. Our analyses can help generate verifiable hypotheses for neurophysiological experiments on central pattern generators. In the last section, we demonstrate the inter-disciplinary nature of this research through the applications of these techniques to identify the universal principles governing both simple and complex dynamics, and chaotic structure in diverse nonlinear systems. Using a classical example from nonlinear laser optics, we elaborate on the multiplicity and self-similarity of key organizing structures in 2D parameter space such as homoclinic and heteroclinic bifurcation curves, Bykov T-point spirals, and inclination flips. This is followed by detailed computational reconstructions of the spatial organization and 3D embedding of bifurcation surfaces, parametric saddles, and isolated closed curves (isolas). The generality of our modeling approaches could lead to novel methodologies and nonlinear science applications in biological, medical and engineering systems

Models for reinforcement learning and design of a soft robot inspired by Drosophila larvae

Designs for robots are often inspired by animals, as they are designed mimicking animals’ mechanics, motions, behaviours and learning. The Drosophila, known as the fruit fly, is a well-studied model animal. In this thesis, the Drosophila larva is studied and the results are applied to robots. More specifically: a part of the Drosophila larva’s neural circuit for operant learning is modelled, based on which a synaptic plasticity model and a neural circuit model for operant learning, as well as a dynamic neural network for robot reinforcement learning, are developed; then Drosophila larva’s motor system for locomotion is studied, and based on it a soft robot system is designed. Operant learning is a concept similar to reinforcement learning in computer science, i.e. learning by reward or punishment for behaviour. Experiments have shown that a wide range of animals is capable of operant learning, including animal with only a few neurons, such as Drosophila. The fact implies that operant learning can establish without a large number of neurons. With it as an assumption, the structure and dynamics of synapses are investigated, and a synaptic plasticity model is proposed. The model includes nonlinear dynamics of synapses, especially receptor trafficking which affects synaptic strength. Tests of this model show it can enable operant learning at the neuron level and apply to a broad range of NNs, including feedforward, recurrent and spiking NNs. The mushroom body is a learning centre of the insect brain known and modelled for associative learning, but not yet for operant learning. To investigate whether it participates in operant learning, Drosophila larvae are studied with a transgenic tool by my collaborators. Based on the experiment and the results, a mushroom body model capable of operant learning is modelled. The proposed neural circuit model can reproduce the operant learning of the turning behaviour of Drosophila larvae. Then the synaptic plasticity model is simplified for robot learning. With the simplified model, a recurrent neural network with internal neural dynamics can learn to control a planar bipedal robot in a benchmark reinforcement learning task which is called bipedal walker by OpenAI. Benefiting efficiency in parameter space exploration instead of action space exploration, it is the first known solution to the task with reinforcement learning approaches. Although existing pneumatic soft robots can have multiple muscles embedded in a component, it is far less than the muscles in the Drosophila larva, which are well-organised in a tiny space. A soft robot system is developed based on the muscle pattern of the Drosophila larva, to explore the possibility to embed a high density of muscles in a limited space. Three versions of the body wall with pneumatic muscles mimicking the muscle pattern are designed. A pneumatic control system and embedded control system are also developed for controlling the robot. With a bioinspired body wall will a large number of muscles, the robot performs lifelike motions in experiments

Deterministic and stochastic dynamics of multi-variable neuron models : resonance, filtered fluctuations and sodium-current inactivation

Neurons are the basic elements of the networks that constitute the computational units of the brain. They dynamically transform input information into sequences of electrical pulses. To conceive the complex function of the brain, it is crucial to understand this transformation and identify simple neuron models which accurately reproduce the known features of biological neurons. This thesis addresses three different features of neurons. We start by exploring the effect of subthreshold resonance on the response of a periodically forced neuron using a simple threshold model. The response is studied in terms of an implicit one-dimensional time map that corresponds to the Poincar´e map of the forced system. Qualitatively distinct responses are found, including mode locking and chaos. We analytically find the stability regions of mode-locking solutions, and identify the transition to chaos through period-adding bifurcations. We show that the response becomes chaotic when the forcing frequency is close to the resonant frequency. Then we will consider an experimentally verified model with realistic spikegenerating mechanism and study the effect of filtered synaptic fluctuations on the firing-rate response of the neuron. Using a population density method as well as an efficient numerical method, we find the steady-state firing rate in two limits of fast and slow synaptic inputs and present the linear response theory for the firing rate of the model in response to both time-dependent mean inputs and time-dependent noise intensity. Finally, a novel model is introduced that incorporates threshold variability of neurons. We determine the modulation of the input-output properties of the model due to oscillatory inputs and in the presence of filtered synaptic fluctuations.EThOS - Electronic Theses Online ServiceUniversity of WarwickOverseas Research Students Awards Scheme (ORSAS)GBUnited Kingdo

Advances in point process filters and their application to sympathetic neural activity

This thesis is concerned with the development of techniques for analyzing the sequences of stereotypical electrical impulses within neurons known as spikes. Sequences of spikes, also called spike trains, transmit neural information; decoding them often provides details about the physiological processes generating the neural activity. Here, the statistical theory of event arrivals, called point processes, is applied to human muscle sympathetic spike trains, a peripheral nerve signal responsible for cardiovascular regulation. A novel technique that uses observed spike trains to dynamically derive information about the physiological processes generating them is also introduced. Despite the emerging usage of individual spikes in the analysis of human muscle sympathetic nerve activity, the majority of studies in this field remain focused on bursts of activity at or below cardiac rhythm frequencies. Point process theory applied to multi-neuron spike trains captured both fast and slow spiking rhythms. First, analysis of high-frequency spiking patterns within cardiac cycles was performed and, surprisingly, revealed fibers with no cardiac rhythmicity. Modeling spikes as a function of average firing rates showed that individual nerves contribute substantially to the differences in the sympathetic stressor response across experimental conditions. Subsequent investigation of low-frequency spiking identified two physiologically relevant frequency bands, and modeling spike trains as a function of hemodynamic variables uncovered complex associations between spiking activity and biophysical covariates at these two frequencies. For example, exercise-induced neural activation enhances the relationship of spikes to respiration but does not affect the extremely precise alignment of spikes to diastolic blood pressure. Additionally, a novel method of utilizing point process observations to estimate an internal state process with partially linear dynamics was introduced. Separation of the linear components of the process model and reduction of the sampled space dimensionality improved the computational efficiency of the estimator. The method was tested on an established biophysical model by concurrently computing the dynamic electrical currents of a simulated neuron and estimating its conductance properties. Computational load reduction, improved accuracy, and applicability outside neuroscience establish the new technique as a valuable tool for decoding large dynamical systems with linear substructure and point process observations

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