Necessary and sufficient condition for the global stability of a delayed discrete-time neuron model

Positron emission tomography (PET) studies acquired in list mode offer the opportunity to provide a cine loop showing the dynamics of 18F- PET uptake, giving a visualization of regional bone remodeling. The focus of this report is a group of patients treated with Taylor spatial frames (TSF). The studies were acquired for a period of 45 minutes and saved in list mode. The list was decoded and subsequently segmented into time intervals of one minute each. For each time interval a sinogram was generated from which volumes of one minute each were reconstructed. Slices projected from these volumes could then be displayed as a dynamic loop superimposed on the corresponding computed tomography (CT) slice in order to visualize the 18F- uptake insitu. It was indicated that this technique has the potential of becoming an additional technique to that of using static volumes and SUV values only. As the list mode data was decoded it also offered a method to evaluate the potential decrease in injected activity by eliminating every Nth event from the list before reconstructing the 45 minute volume. This was done and the indication was that the injected activity and hence the effective dose to the patient can be decreased. However, in this work, this was not proven clinically. The open source STIR software was used to reconstruct volumes from sinograms to enable an unlimited access to reconstructing volumes without disturbing the daily routine at the clinic. The data was acquired on a clinical Siemens Medical Solutions Biograph 64 TruePoint TrueV, PET/CT scanner situated at the Nuclear Medicine Department at the Karolinska University Hospital in Solna. This scanner was not supported by the STIR software, hence the data collected by the Siemens PET/CT scanner was translated so that 3D reconstructions could be computed using the STIR tools. The reconstructions made in STIR resulted in volumes of sufficient visual quality, but not as good as those reconstructed by the scanner itself. Further optimization in STIR was left for future work. According to the physicians who treat these patients, dynamic visualization was of sufficient interest to continue to develop and optimize this method. The cine loops that were presented to the physicians were made from JPEG slices produced from the one minute volumes and put together as GIF files. It was also possible to vary the reconstruction time (from uniformly one minute) as well as the presentation rate in the cine loop, but this was left for future work. Ultimately, the cine loop will be implemented in the locally developed software tool.Positronemissionstomografi (PET) studier som förvärvats i list mode erbjuder möjligheten att göra film slingor som visar dynamiken i 18F- PET upptag. Detta förväntas ge en dynamisk visualisering av var ben nybildas. Fokus i denna rapport är en grupp patienter som behandlats med Taylor spatial frames (TSF). Bilderna i studien förvärvades under en period av 45 minuter och sparades i list mode. List mode data avkodades och delades därefter upp i tidsintervaller på en minut vardera. För varje tidsintervall rekonstruerades ett sinogram för vilka bilder av volymen rekonstruerades. Skivor från dessa volymer kan sedan visas som en dynamisk slinga ovanpå motsvarande datortomografi (CT) skiva för att visualisera 18F- upptaget in situ. Denna teknik visade sig ha potential att bli en ytterligare teknik utöver de statiska volymer och SUV-värden som redan finns tillgängliga. Eftersom listmode data avkodats erbjuds också en metod för att utvärdera en potentiell minskning av den injicerade aktivitet genom att eliminera varje N: te händelse från listan innan volymen på 45 minuter rekonstrueras. Detta utfördes och det visade sig att den injicerade aktiviteten och därmed den effektiva dosen till patienten potentiellt kan minskas men i detta arbete har detta inte bevisats kliniskt. Mjukvaran STIR, som är tillgängligt för allmänheten, användes för att rekonstruera volymer från sinogram för att möjliggöra en obegränsad tillgång till att rekonstruera volymer utan att störa den dagliga rutinen på kliniken. Data förvärvades i en klinisk Siemens Medical Solutions Biograph 64 TruePoint TrueV, PET/CT-skanner som är placerad vid nuklearmedicinska avdelningen vid Karolinska Universitetssjukhuset i Solna. Skannern var inte kompatibel med STIR mjukvaran. Därför översattes det data som samlats in av Siemens PET/CT-skannern så att 3D-rekonstruktioner kan beräknas med hjälp av verktygen i STIR. De rekonstruktioner som gjorts i STIR gav upphov till volymer av tillräcklig visuell kvalitet för denna studie, men var inte lika bra som de som rekonstrueras genom själva skannern. Ytterligare optimering i STIR lämnades för framtida arbete. Enligt de läkare som behandlar dessa patienter var en dynamisk visualisering av tillräckligt intresse att fortsätta utveckla och optimera den här metoden. Film slingor som presenterades för läkarna gjordes från JPEG bilder tagna från skivor av volymer på en minut vardera som producerats och sattes ihop som GIF-filer. Det var också möjligt att variera återuppbyggnadstiden (från jämnt en minut) samt presentationshastigheten i film slingan, men detta lämnades till framtida arbetet. I slutändan kommer film slingan implementeras i ett lokalt utvecklat verktyg.QC 20151102</p

A new framework for consensus for discrete-time directed networks of multi-agents with distributed delays

Copyright @ 2012 Taylor & FrancisIn this article, the distributed consensus problem is considered for discrete-time delayed networks of dynamic agents with fixed topologies, where the networks under investigation are directed and the time-delays involved are distributed time delays including a single or multiple time delay(s) as special cases. By using the invariance principle of delay difference systems, a new unified framework is established to deal with the consensus for the discrete-time delayed multi-agent system. It is shown that the addressed discrete-time network with arbitrary distributed time delays reaches consensus provided that it is strongly connected. A numerical example is presented to illustrate the proposed methods.This work was supported in part by City University of Hong Kong under Grant 7008114, the Royal Society of the UK, the National Natural Science Foundation of China under Grants 60774073 and 61074129, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2010313

Local/global analysis of the stationary solutions of some neural field equations

Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integro-differential equations. The solutions of these equations represent the state of activity of these populations when submitted to inputs from neighbouring brain areas. Understanding the properties of these solutions is essential in advancing our understanding of the brain. In this paper we study the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree theory and bifurcation theory in the context of functional, in particular infinite dimensional, spaces. The joint use of these two theories allows us to make new detailed predictions about the global and local behaviours of the solutions. We also provide a generic finite dimensional approximation of these equations which allows us to study in great details two models. The first model is a neural mass model of a cortical hypercolumn of orientation sensitive neurons, the ring model. The second model is a general neural field model where the spatial connectivity isdescribed by heterogeneous Gaussian-like functions.Comment: 38 pages, 9 figure

Mean field modelling of human EEG: application to epilepsy

Aggregated electrical activity from brain regions recorded via an electroencephalogram (EEG), reveal that the brain is never at rest, producing a spectrum of ongoing oscillations that change as a result of different behavioural states and neurological conditions. In particular, this thesis focusses on pathological oscillations associated with absence seizures that typically affect 2–16 year old children. Investigation of the cellular and network mechanisms for absence seizures studies have implicated an abnormality in the cortical and thalamic activity in the generation of absence seizures, which have provided much insight to the potential cause of this disease. A number of competing hypotheses have been suggested, however the precise cause has yet to be determined. This work attempts to provide an explanation of these abnormal rhythms by considering a physiologically based, macroscopic continuum mean-field model of the brain's electrical activity. The methodology taken in this thesis is to assume that many of the physiological details of the involved brain structures can be aggregated into continuum state variables and parameters. The methodology has the advantage to indirectly encapsulate into state variables and parameters, many known physiological mechanisms underlying the genesis of epilepsy, which permits a reduction of the complexity of the problem. That is, a macroscopic description of the involved brain structures involved in epilepsy is taken and then by scanning the parameters of the model, identification of state changes in the system are made possible. Thus, this work demonstrates how changes in brain state as determined in EEG can be understood via dynamical state changes in the model providing an explanation of absence seizures. Furthermore, key observations from both the model and EEG data motivates a number of model reductions. These reductions provide approximate solutions of seizure oscillations and a better understanding of periodic oscillations arising from the involved brain regions. Local analysis of oscillations are performed by employing dynamical systems theory which provide necessary and sufficient conditions for their appearance. Finally local and global stability is then proved for the reduced model, for a reduced region in the parameter space. The results obtained in this thesis can be extended and suggestions are provided for future progress in this area

Data-Driven Modeling and Forecasting of Chaotic Dynamics on Inertial Manifolds Constructed as Spectral Submanifolds

We present a data-driven and interpretable approach for reducing the dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating from fixed points or periodic orbits, these SSMs are low-dimensional inertial manifolds containing the chaotic attractor of the underlying high-dimensional system. The reduced dynamics on the SSMs turn out to predict chaotic dynamics accurately over a few Lyapunov times and also reproduce long-term statistical features, such as the largest Lyapunov exponents and probability distributions, of the chaotic attractor. We illustrate this methodology on numerical data sets including a delay-embedded Lorenz attractor, a nine-dimensional Lorenz model, and a Duffing oscillator chain. We also demonstrate the predictive power of our approach by constructing an SSM-reduced model from unforced trajectories of a buckling beam, and then predicting its periodically forced chaotic response without using data from the forced beam.Comment: Submitted to Chao

Monotone and near-monotone biochemical networks

Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a “small” number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion