Excitable neurons, firing threshold manifolds and canards

We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674-1697, 2012). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model

Mixed mode oscillations in a conceptual climate model

Much work has been done on relaxation oscillations and other simple oscillators in conceptual climate models. However, the oscillatory patterns in climate data are often more complicated than what can be described by such mechanisms. This paper examines complex oscillatory behavior in climate data through the lens of mixed-mode oscillations. As a case study, a conceptual climate model with governing equations for global mean temperature, atmospheric carbon, and oceanic carbon is analyzed. The nondimensionalized model is a fast/slow system with one fast variable (corresponding to ice volume) and two slow variables (corresponding to the two carbon stores). Geometric singular perturbation theory is used to demonstrate the existence of a folded node singularity. A parameter regime is found in which (singular) trajectories that pass through the folded node are returned to the singular funnel in the limiting case where $\epsilon = 0$. In this parameter regime, the model has a stable periodic orbit of type $1^s$ for some $s>0$. To our knowledge, it is the first conceptual climate model demonstrated to have the capability to produce an MMO pattern.Comment: 28 pages, 11 figure

Multiple timescales and the parametrisation method in geometric singular perturbation theory

We present a novel method for computing slow manifolds and their fast fibre bundles in geometric singular perturbation problems. This coordinate-independent method simultaneously computes parametrisations of these objects and the dynamics on them. It does so by iteratively solving a so-called conjugacy equation, yielding approximations with arbitrarily high degrees of accuracy. We show the power of this top-down method for the study of systems with multiple (i.e., three or more) timescales. In particular, we highlight the emergence of hidden timescales and show how our method can uncover these surprising multiple timescale structures. We also apply our parametrisation method to several reaction network problems.Comment: 33 pages, 5 figure

Singularly Perturbed Boundary-Focus Bifurcations

We consider smooth systems limiting as $\epsilon \to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < \epsilon \ll 1$ using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an $\epsilon-$dependent domain which shrinks to zero as $\epsilon \to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the $\epsilon-$dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction

On the stability of isothermal shocks in black hole accretion disks

Most black holes possess accretion disks. Models of such disks inform observations and constrain the properties of the black holes and their surrounding medium. Here, we study isothermal shocks in a thin black hole accretion flow. Modelling infinitesimal molecular viscosity allows the use of multiple-scales matched asymptotic methods. We thus derive the first explicit calculations of isothermal shock stability. We find that the inner shock is always unstable, and the outer shock is always stable. The growth/decay rates of perturbations depend only on an effective potential and the incoming--outgoing flow difference at the shock location. We give a prescription of accretion regimes in terms of angular momentum and black hole radius. Accounting for angular momentum dissipation implies unstable outer shocks in much of parameter space, even for realistic viscous Reynolds numbers of the order $\approx 10^{20}$.Comment: 26 page