Critical fluctuations in an optical parametric oscillator: when light behaves like magnetism

We study the nondegenerate optical parametric oscillator in a planar interferometer near threshold, where critical phenomena are expected. These phenomena are associated with nonequilibrium quantum dynamics that are known to lead to quadrature entanglement and squeezing in the oscillator field modes. We obtain a universal form for the equation describing this system, which allows a comparison with other phase transitions. We find that the unsqueezed quadratures of this system correspond to a two-dimensional XY-type model with a tricritical Lifshitz point. This leaves open the possibility of a controlled experimental investigation into this unusual class of statistical models. We evaluate the correlations of the unsqueezed quadrature using both an exact numerical simulation and a Gaussian approximation, and obtain an accurate numerical calculation of the non-Gaussian correlations.Comment: Title changed. New figures adde

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

Computational study of resting state network dynamics

Lo scopo di questa tesi è quello di mostrare, attraverso una simulazione con il software The Virtual Brain, le più importanti proprietà della dinamica cerebrale durante il resting state, ovvero quando non si è coinvolti in nessun compito preciso e non si è sottoposti a nessuno stimolo particolare. Si comincia con lo spiegare cos’è il resting state attraverso una breve revisione storica della sua scoperta, quindi si passano in rassegna alcuni metodi sperimentali utilizzati nell’analisi dell’attività cerebrale, per poi evidenziare la differenza tra connettività strutturale e funzionale. In seguito, si riassumono brevemente i concetti dei sistemi dinamici, teoria indispensabile per capire un sistema complesso come il cervello. Nel capitolo successivo, attraverso un approccio ‘bottom-up’, si illustrano sotto il profilo biologico le principali strutture del sistema nervoso, dal neurone alla corteccia cerebrale. Tutto ciò viene spiegato anche dal punto di vista dei sistemi dinamici, illustrando il pionieristico modello di Hodgkin-Huxley e poi il concetto di dinamica di popolazione. Dopo questa prima parte preliminare si entra nel dettaglio della simulazione. Prima di tutto si danno maggiori informazioni sul software The Virtual Brain, si definisce il modello di network del resting state utilizzato nella simulazione e si descrive il ‘connettoma’ adoperato. Successivamente vengono mostrati i risultati dell’analisi svolta sui dati ricavati, dai quali si mostra come la criticità e il rumore svolgano un ruolo chiave nell'emergenza di questa attività di fondo del cervello. Questi risultati vengono poi confrontati con le più importanti e recenti ricerche in questo ambito, le quali confermano i risultati del nostro lavoro. Infine, si riportano brevemente le conseguenze che porterebbe in campo medico e clinico una piena comprensione del fenomeno del resting state e la possibilità di virtualizzare l’attività cerebrale

Shaping bursting by electrical coupling and noise

Gap-junctional coupling is an important way of communication between neurons and other excitable cells. Strong electrical coupling synchronizes activity across cell ensembles. Surprisingly, in the presence of noise synchronous oscillations generated by an electrically coupled network may differ qualitatively from the oscillations produced by uncoupled individual cells forming the network. A prominent example of such behavior is the synchronized bursting in islets of Langerhans formed by pancreatic \beta-cells, which in isolation are known to exhibit irregular spiking. At the heart of this intriguing phenomenon lies denoising, a remarkable ability of electrical coupling to diminish the effects of noise acting on individual cells. In this paper, we derive quantitative estimates characterizing denoising in electrically coupled networks of conductance-based models of square wave bursting cells. Our analysis reveals the interplay of the intrinsic properties of the individual cells and network topology and their respective contributions to this important effect. In particular, we show that networks on graphs with large algebraic connectivity or small total effective resistance are better equipped for implementing denoising. As a by-product of the analysis of denoising, we analytically estimate the rate with which trajectories converge to the synchronization subspace and the stability of the latter to random perturbations. These estimates reveal the role of the network topology in synchronization. The analysis is complemented by numerical simulations of electrically coupled conductance-based networks. Taken together, these results explain the mechanisms underlying synchronization and denoising in an important class of biological models

Generative Models of Cortical Oscillations: Neurobiological Implications of the Kuramoto Model

Understanding the fundamental mechanisms governing fluctuating oscillations in large-scale cortical circuits is a crucial prelude to a proper knowledge of their role in both adaptive and pathological cortical processes. Neuroscience research in this area has much to gain from understanding the Kuramoto model, a mathematical model that speaks to the very nature of coupled oscillating processes, and which has elucidated the core mechanisms of a range of biological and physical phenomena. In this paper, we provide a brief introduction to the Kuramoto model in its original, rather abstract, form and then focus on modifications that increase its neurobiological plausibility by incorporating topological properties of local cortical connectivity. The extended model elicits elaborate spatial patterns of synchronous oscillations that exhibit persistent dynamical instabilities reminiscent of cortical activity. We review how the Kuramoto model may be recast from an ordinary differential equation to a population level description using the nonlinear Fokker–Planck equation. We argue that such formulations are able to provide a mechanistic and unifying explanation of oscillatory phenomena in the human cortex, such as fluctuating beta oscillations, and their relationship to basic computational processes including multistability, criticality, and information capacity