Matematiske aspekter ved lokalisert aktivitet i nevrofeltmodeller

Neural field models assume the form of integral and integro-differential equations, and describe non-linear interactions between neuron populations. Such models reduce the dimensionality and complexity of the microscopic neural-network dynamics and allow for mathematical treatment, efficient simulation and intuitive understanding. Since the seminal studies byWilson and Cowan (1973) and Amari (1977) neural field models have been used to describe phenomena like persistent neuronal activity, waves and pattern formation in the cortex. In the present thesis we focus on mathematical aspects of localized activity which is described by stationary solutions of a neural field model, so called bumps. While neural field models represent a considerable simplification of the neural dynamics in a large network, they are often studied under further simplifying assumptions, e.g., approximating the firing-rate function with a unit step function. In some cases these assumptions may not change essential features of the model, but in other cases they may cause some properties of the model to vary significantly or even break down. The work presented in the thesis aims at studying properties of bump solutions in one- and two-population models relaxing on the common simplifications. Numerical approaches used in mathematical neuroscience sometimes lack mathematical justification. This may lead to numerical instabilities, ill-conditioning or even divergence. Moreover, there are some methods which have not been used in neuroscience community but might be beneficial. We have initiated a work in this direction by studying advantages and disadvantages of a wavelet-Galerkin algorithm applied to a simplified framework of a one-population neural field model. We also focus on rigorous justification of iteration methods for constructing bumps. We use the theory of monotone operators in ordered Banach spaces, the theory of Sobolev spaces in unbounded domains, degree theory, and other functional analytical methods, which are still not very well developed in neuroscience, for analysis of the models.Nevrofeltmodeller formuleres som integral og integro-differensiallikninger. De beskriver ikke-lineære vekselvirkninger mellom populasjoner av nevroner. Slike modeller reduserer dimensjonalitet og kompleksitet til den mikroskopiske nevrale nettverksdynamikken og tillater matematisk behandling, effektiv simulering og intuitiv forståelse. Siden pionerarbeidene til Wilson og Cowan (1973) og Amari (1977), har nevrofeltmodeller blitt brukt til å beskrive fenomener som vedvarende nevroaktivitet, bølger og mønsterdannelse i hjernebarken. I denne avhandlingen vil vi fokusere på matematiske aspekter ved lokalisert aktivitet som beskrives ved stasjonære løsninger til nevrofeltmodeller, såkalte bumps. Mens nevrofeltmodeller innebærer en betydelig forenkling av den nevrale dynamikken i et større nettverk, så blir de ofte studert ved å gjøre forenklende tilleggsantakelser, som for eksempel å approksimere fyringratefunksjonen med en Heaviside-funksjon. I noen tilfeller vil disse forenklingene ikke endre vesentlige trekk ved modellen, mens i andre tilfeller kan de forårsake at modellegenskapene endres betydelig eller at de bryter sammen. Arbeidene presentert i denne avhandlingen har som mål å studere egenskapene til bump-løsninger i en- og to-populasjonsmodeller når en lemper på de vanlige antakelsene. Numeriske teknikker som brukes i matematisk nevrovitenskap mangler i noen tilfeller matematisk begrunnelse. Dette kan lede til numeriske instabiliteter, dårlig kondisjonering, og til og med divergens. I tillegg finnes det metoder som ikke er blitt brukt i nevrovitenskap, men som kunne være fordelaktige å bruke. Vi har startet et arbeid i denne retningen ved å studere fordeler og ulemper ved en wavelet-Galerkin algoritme anvendt på et forenklet rammeverk for en en-populasjons nevrofelt modell. Vi fokuserer også på rigorøs begrunnelse for iterasjonsmetoder for konstruksjon av bumps. Vi bruker teorien for monotone operatorer i ordnede Banachrom, teorien for Sobolevrom for ubegrensede domener, gradteori, og andre funksjonalanalytiske metoder, som for tiden ikke er vel utviklet i nevrovitenskap, for analyse av modellene

An intermediate value theorem in ordered Banach spaces

We consider a monotone increasing operator in an ordered Banach space having $u_-$ and $u_+$ as a strong super- and subsolution, respectively. In contrast with the well studied case $u_+ < u_-$, we suppose that $u_- < u_+$. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the ordered interval $[u_-,u_+].

Covering tour problem with varying coverage: Application to marine environmental monitoring

In this paper, we present a novel variant of the Covering Tour Problem (CTP), called the Covering Tour Problem with Varying Coverage (CTP-VC). We consider a simple graph = ( ,), with a measure of importance assigned to each node in . A vehicle with limited battery capacity visits the nodes of the graph and has the ability to stay in each node for a certain period of time, which determines the coverage radius at the node. We refer to this feature as stay-dependent varying coverage or, in short, varying coverage. The objective is to maximize a scalarization of the weighted coverage of the nodes and the negation of the cost of moving and staying at the nodes. This problem arises in the monitoring of marine environments, where pollutants can be measured at locations far from the source due to ocean currents. To solve the CTP-VC, we propose a mathematical formulation and a heuristic approach, given that the problem is NP-hard. Depending on the availability of solutions yielded by an exact solver, we evaluate our heuristic approach against the exact solver or a constructive heuristic on various instance sets and show how varying coverage improves performance. Additionally, we use an offshore CO2 storage site in the Gulf of Mexico as a case study to demonstrate the problem’s applicability. Our results demonstrate that the proposed heuristic approach is an efficient and practical solution to the problem of stay-dependent varying coverage. We conduct numerous experiments and provide managerial insights.publishedVersio

Semi-conditional variational auto-encoder for flow reconstruction and uncertainty quantification from limited observations

We present a new data-driven model to reconstruct nonlinear flow from spatially sparse observations. The proposed model is a version of a Conditional Variational Auto-Encoder (CVAE), which allows for probabilistic reconstruction and thus uncertainty quantification of the prediction. We show that in our model, conditioning on measurements from the complete flow data leads to a CVAE where only the decoder depends on the measurements. For this reason, we call the model semi-conditional variational autoencoder. The method, reconstructions, and associated uncertainty estimates are illustrated on the velocity data from simulations of 2D flow around a cylinder and bottom currents from a simulation of the southern North Sea by the Bergen Ocean Model. The reconstruction errors are compared to those of the Gappy proper orthogonal decomposition method.publishedVersio

Binary time series classification with Bayesian convolutional neural networks when monitoring for marine gas discharges

The world’s oceans are under stress from climate change, acidification and other human activities, and the UN has declared 2021–2030 as the decade for marine science. To monitor the marine waters, with the purpose of detecting discharges of tracers from unknown locations, large areas will need to be covered with limited resources. To increase the detectability of marine gas seepage we propose a deep probabilistic learning algorithm, a Bayesian Convolutional Neural Network (BCNN), to classify time series of measurements. The BCNN will classify time series to belong to a leak/no-leak situation, including classification uncertainty. The latter is important for decision makers who must decide to initiate costly confirmation surveys and, hence, would like to avoid false positives. Results from a transport model are used for the learning process of the BCNN and the task is to distinguish the signal from a leak hidden within the natural variability. We show that the BCNN classifies time series arising from leaks with high accuracy and estimates its associated uncertainty. We combine the output of the BCNN model, the posterior predictive distribution, with a Bayesian decision rule showcasing how the framework can be used in practice to make optimal decisions based on a given cost function.publishedVersio