Fast global oscillations in networks of integrate-and-fire neurons with low firing rates

We study analytically the dynamics of a network of sparsely connected inhibitory integrate-and-fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate. In the limit when the number of neurons N tends to infinity,the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized. The activity becomes oscillatory when the inhibitory feedback is strong enough. The period of the global oscillation is found to be mainly controlled by synaptic times, but depends also on the characteristics of the external input. In large but finite networks, the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold. Their characteristics are determined as functions of systems parameters, in these two different regimes. The results are found to be in good agreement with numerical simulations.Comment: 45 pages, 11 figures, to be published in Neural Computatio

Adaptive dynamical networks

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical 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

Collective phenomena in networks of spiking neurons with synaptic delays

A prominent feature of the dynamics of large neuronal networks are the synchrony- driven collective oscillations generated by the interplay between synaptic coupling and synaptic delays. This thesis investigates the emergence of delay-induced oscillations in networks of heterogeneous spiking neurons. Building on recent theoretical advances in exact mean field reductions for neuronal networks, this work explores the dynamics and bifurcations of an exact firing rate model with various forms of synaptic delays. In parallel, the results obtained using the novel firing rate model are compared with extensive numerical simulations of large networks of spiking neurons, which confirm the existence of numerous synchrony-based oscillatory states. Some of these states are novel and display complex forms of partial synchronization and collective chaos. Given the well-known limitation of traditional firing rate models to describe synchrony-based oscillations, previous studies greatly overlooked many of the oscillatory states found here. Therefore, this thesis provides a unique exploration of the oscillatory scenarios found in neuronal networks due to the presence of delays, and may substantially extend the mathematical tools available for modeling the plethora of oscillations detected in electrical recordings of brain activity

McKean-Vlasov limits, propagation of chaos and long-time behavior of some mean field interacting particle systems

In this thesis we study mean field interacting particle systems and their McKean-Vlasov limiting processes, in particular we focus on three different interaction mechanisms, mainly emerging from biological modelling. The first type of interaction is given by the so called simultaneous jumps. We consider a system of interacting jump-diffusion processes that interact by means of the discontinuous component: each particle performs a main jump and it simultaneously induces in all the other particles a simultaneous jump whose amplitude is rescaled with the size of the system. This peculiar interaction is motivated by recent neuroscience models and here we depict a general framework for this type of processes. We focus on the well-posedness of the McKean-Vlasov limits of these particle systems under different assumptions on the coefficients and we prove a pathwise propagation of chaos result. The second interaction we consider is an asymmetric one. We describe a system of biased random walks on the positive integers, reflected at zero, where each particle may perform a leftward jump with a rate proportional to the fraction of particles which are strictly at its left. We study the critical interaction strength able to ensure ergodicity to this system, that would be transient in absence of interaction. We compare this model with existing models of diffusions interacting through their CDF and we highlight their differences, mainly caused by the presence of clusters of particles in the discrete model. The third interaction we account for is based on a dynamical version of the generalized Curie-Weiss model. We modify a Langevin dynamics for this model with a dissipative evolution of the interaction component, breaking the reversibility of the system. We prove that, in the mean field limit, this gives rise to stable limit cycles, explaining self-sustained periodic behaviors. In particular, we build a flexible model in which a suitable change in the interaction function can result in a system which, in certain regimes of parameters, displays coexistence of stable periodic orbits