arXiv subjects

Susanne Solem

Publications and source records attributed to Susanne Solem.

13 recordsLinked to original sources

arXiv2025

Estimating neural connection strengths from firing intervals

We propose and analyse a procedure for using a standard activity-based neuron network model and firing data to compute the effective connection strengths between neurons in a network. We assume a Heaviside response function, that the external inputs are given and that the initial state of the neural activity is known. The associated forward operator for this problem, which maps given connection strengths to the time intervals of firing, is highly nonlinear. Nevertheless, it turns out that the inverse problem of determining the connection strengths can be solved in a rather transparent manner, only employing standard mathematical tools. In fact, it is sufficient to solve a system of decoupled ODEs, which yields a linear system of algebraic equations for determining the connection strengths. The nature of the inverse problem is investigated by studying some mathematical properties of the aforementioned linear system and by a series of numerical experiments. Finally, under an assumption preventing the effective contribution of the network to each neuron from staying at zero, we prove that the involved forward operator is continuous. Sufficient criteria on the external input ensuring that the needed assumption holds are also provided.

math.DS↗

arXiv2024

A Convergent Numerical Algorithm for $α$-Dissipative Solutions of the Hunter-Saxton Equation

A convergent numerical method for $α$-dissipative solutions of the Hunter-Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step that introduces any approximation error. It is therefore crucial that its design ensures not only a good approximation of the initial data, but also that errors due to the energy dissipation at later times remain small. Furthermore, it is shown that the main quantity of interest, the wave profile, converges in $L^{\infty}$ for all $t \geq 0$, while a subsequence of the energy density converges weakly for almost every time.

math.NA↗

arXiv2023

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

math.AP↗

arXiv2023

Noise-driven bifurcations in a nonlinear Fokker-Planck system describing stochastic neural fields

The existence and characterisation of noise-driven bifurcations from the spatially homogeneous stationary states of a nonlinear, non-local Fokker--Planck type partial differential equation describing stochastic neural fields is established. The resulting theory is extended to a system of partial differential equations modelling noisy grid cells. It is shown that as the noise level decreases, multiple bifurcations from the homogeneous steady state occur. Furthermore, the shape of the branches at a bifurcation point is characterised locally. The theory is supported by a set of numerical illustrations of the condition leading to bifurcations, the patterns along the corresponding local bifurcation branches, and the stability of the homogeneous state and the most prevalent pattern: the hexagonal one.

math.AP↗

arXiv2023

The mean field limit of stochastic differential equation systems modelling grid cells

Several differential equation models have been proposed to explain the formation of patterns characteristic of the grid cell network. Understanding the robustness of these patterns with respect to noise is one of the key open questions in computational neuroscience. In the present work, we analyze a family of stochastic differential systems modelling grid cell networks. Furthermore, the well-posedness of the associated McKean--Vlasov and Fokker--Planck equations, describing the average behavior of the networks, is established. Finally, we rigorously prove the mean field limit of these systems and provide a sharp rate of convergence for their empirical measures.

math.PR↗

arXiv2022

Noise-driven bifurcations in a neural field system modelling networks of grid cells

The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by analysing the robustness of network activity patterns with respect to noise in an upscaled noisy grid cell model in the form of a system of partial differential equations. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We show that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value.

math.AP↗

arXiv2021

Analysis and simulation of a modified cardiac cell model givesaccurate predictions of the dynamics of the original one

The 19-dimensional TP06 cardiac muscle cell model is reduced to a 17-dimensional version, which satisfies the required conditions for performing an analysis of its dynamics by means of bifurcation theory. The reformulated model is shown to be a good approximation of the original one. As a consequence, one can extract fairly precise predictions of the behaviour of the original model from the bifurcation analysis of the modified model. Thus, the findings of bifurcations linked to complex dynamics in the modified model - like early afterdepolarisations (EADs), which can be precursors to cardiac death - predicts the occurrence of the same dynamics in the original model. It is shown that bifurcations linked to EADs in the modified model accurately predicts EADs in the original model at the single cell level. Finally, these bifurcations are linked to cardiac death at the tissue level by example.

math.DS↗

arXiv2021

Numerical conservative solutions of the Hunter--Saxton equation

In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws.

math.AP↗

arXiv2021

A second-order numerical method for the aggregation equations

Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, lambda-convex potentials with a possible Lipschitz singularity at the origin we prove that the method converges in the Monge--Kantorovich distance towards the unique gradient flow solution. Several numerical experiments are presented to validate the second-order convergence rate and to explore the performance of the scheme.

math.NA↗

arXiv2021

On complex dynamics in a Purkinje and a ventricular cardiac cell model

Cardiac muscle cells can exhibit complex patterns including irregular behaviour such as chaos or (chaotic) early afterdepolarisations (EADs), which can lead to sudden cardiac death. Suitable mathematical models and their analysis help to predict the occurrence of such phenomena and to decode their mechanisms. The focus of this paper is the investigation of dynamics of cardiac muscle cells described by systems of ordinary differential equations. This is generically performed by studying a Purkinje cell model and a modified ventricular cell model. We find chaotic dynamics with respect to the leak current in the Purkinje cell model, and EADs and chaos with respect to a reduced fast potassium current and an enhanced calcium current in the ventricular cell model -- features that have been experimentally observed and are known to exist in some models, but are new to the models under present consideration. We also investigate the related monodomain models of both systems to study synchronisation and the behaviour of the cells on macro-scale in connection with the discovered features. The models show qualitatively the same behaviour to what has been experimentally observed. However, for certain parameter settings the dynamics occur within a non-physiological range.

math.DS↗

arXiv2019

The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, $\Lip^+$-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of $\Lip^+$-unbounded initial data is worse than first-order.

math.NA↗

arXiv2018

Convergence rates of the front tracking method for conservation laws in the Wasserstein distances

We prove that front tracking approximations to entropy solutions of scalar conservation laws with convex fluxes converge at a rate of $Δx^2$ in the 1-Wasserstein distance $W_1$. Assuming positive initial data, we also show that the approximations converge at a rate of $Δx$ in the $\infty$-Wasserstein distance $W_\infty$. Moreover, from a simple interpolation inequality between $W_1$ and $W_\infty$ we obtain convergence rates in all the $p$-Wasserstein distances: $Δx^{1+1/p}$, $p \in [1,\infty]$.

math.NA↗

arXiv2016

Second-order convergence of monotone schemes for conservation laws

We prove that a class of monotone, \emph{$W_1$-contractive} schemes for scalar conservation laws converge at a rate of $Δx^2$ in the Wasserstein distance ($W_1$-distance), whenever the initial data is decreasing and consists of a finite number of piecewise constants. It is shown that the Lax--Friedrichs, Enquist--Osher and Godunov schemes are $W_1$-contractive. Numerical experiments are presented to illustrate the main result. To the best of our knowledge, this is the first proof of second-order convergence of any numerical method for discontinuous solutions of nonlinear conservation laws.

math.NA↗