Memory effects in disease modelling through kernel estimates with oscillatory time history.

We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR-model and on a susceptible-exposed-infected-recovered SEIR-model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e. locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero.

Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking.

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

Pattern formation in a 2-population homogenized neuronal network model.

We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of [Formula: see text]-stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

Polarization noise places severe constraints on coherence of all-normal dispersion femtosecond supercontinuum generation.

Supercontinuum (SC) generated with all-normal dispersion (ANDi) fibers has been of special interest in recent years due to its potentially superior coherence properties when compared to anomalous dispersion-pumped SC. However, care must be taken in the design of such sources since too long pump pulses and fiber length has been demonstrated to degrade the coherence. To assess the noise performance of ANDi fiber SC generation numerically, a scalar single-polarization model has so far been used, thereby excluding important sources of noise, such as polarization modulational instability (PMI). In this work we numerically study the influence of pump power, pulse length and fiber length on coherence and relative intensity noise (RIN), taking into account both polarization components in a standard ANDi fiber for SC generation pumped at 1064 nm. We demonstrate that the PMI introduces a power dependence not found in a scalar model, which means that even with short ~120 fs pump pulses the coherence of ANDi SC can be degraded at reasonable power levels above ~40 kW. We further demonstrate how the PMI significantly decreases the pump pulse length and fiber length at which the coherence of the ANDi SC is degraded. The numerical predictions are confirmed by RIN measurements of fs-pumped ANDi fiber SC.

A finite-element method model for droplets moving down a hydrophobic surface.

We set up a 2D computational Finite-Element Method (FEM) model describing the initial descent of a droplet down an inclined hydrophobic substrate. We solve the full Navier-Stokes equations inside the drop domain, and use the arbitrary Lagrangian-Eulerian method to keep track of the droplet surface. The contact angle is included by using the Frennet-Serret equations. We investigate the behaviour of the drop velocity as a function of the slip length and compare with experimental results. Furthermore, we quantify the energy associated with centre-of-mass translation and internal fluid motion, and we also compute the local dissipation of energy inside the drop. The model predicts trajectories for tracer particles deposited inside the drop, and satisfactorily describes the sliding motion of steadily accelerating droplets. The model can be used for determining a characteristic slip parameter, associated with slip lengths and drag reduction for hydrophobic surfaces.

Wave-block due to a threshold gradient underlies limited coordination in pancreatic islets.

Two models for coupled pancreatic beta cells are used to investigate excited wave propagation in spatially inhomogeneous islets of Langerhans. The application concerns spatial variation of glucose concentration across the islet. A comprehensive model of coupled cells shows that wave blocking occurs as the conductance of adenosine triphosphate regulated potassium channels increases, corresponding to spatially decreasing glucose concentration. A simplified model based on a perturbed version of Fisher's equation has been investigated using perturbation theory. We show that the perturbed Fisher's equation likewise can exhibit wave blocking.

Kink shape solutions of the Maxwell-Lorentz system.

In the limit of high amplitude oscillating electromagnetic fields, a sequence of kink antikink shaped optical waves has been found in the Maxwell's equations coupled to a single Lorentz oscillator and with Kerr nonlinearity. The individual kinks and antikinks result from a traveling wave assumption and their stability has been assessed by numerical simulations. For typical physical parameter values the kink width is of the order of tens of femtoseconds.

Escape angles in bulk chi((2)) soliton interactions.

We develop a theory for nonplanar interaction between two identical type I spatial solitons propagating at opposite, but arbitrary transverse angles in quadratic nonlinear (or so-called chi((2))) bulk media. We predict quantitatively the outwards escape angle, below which the solitons turn around and collide, and above which they continue to move-away from each other. For in-plane interaction, the theory allows prediction of the outcome of a collision through the inwards escape angle, i.e., whether the solitons fuse or cross. We find an analytical expression determining the inwards escape angle using Gaussian approximations for the solitons. The theory is verified numerically.