Febuary 10 Daniel Rothman
Title: Carbon-Cycle Catastrophes: A Dynamical-Systems Perspective
The history of Earth's carbon cycle is punctuated by enigmatic
transient changes in the ocean's store of carbon. Mass extinction is
always accompanied by such a disruption, but most disruptions are
relatively benign. The less calamitous group exhibits a
characteristic carbon flux, while mass extinctions are associated with
greater fluxes, possibly from massive volcanism. Analysis of a
two-component dynamical system suggests that characteristic
disruptions are initiated by perturbation of a permanently stable
steady state. Surges are excited when CO2 enters the ocean at a
sufficiently fast rate; the threshold scales inversely with the
duration of injection. Similar excitations follow regardless of the
injection timescale. Consequently the unusually strong but
geologically brief duration of modern anthropogenic oceanic CO2 uptake
is roughly equivalent, in terms of its potential to excite a major
disruption, to relatively weak but longer-lived perturbations
associated with massive volcanism in the geologic past.
Febuary 24 Paul Carter
Title: Slow absolute spectrum and pulse replication in the FitzHugh--Nagumo system
Abstract: The FitzHugh--Nagumo system is a simplified model of nerve impulse propagation, which is known to admit stable traveling pulse solutions. I will present existence and stability results for (multi)pulse solutions, and I will describe a phenomenon whereby a single pulse can be continuously deformed into a double pulse by parameter continuation. Along this transition, eigenvalues accumulate on the positive real axis due to the fact that the pulse solutions spend long times near a slow manifold which exhibits absolute spectrum.
March 2 Elena Quierolo
Title: Oscillations in biological systems
Abstract: In cell biology, piecewise linear models are proposed to describe biological systems exhibiting periodic behaviour. They are interpreted as limits of smooth systems with high-dimensional parameter space. Such limit systems allow us to determine parameter regimes in which periodicity can occur. Once the parameter regions are found for piecewise linear models, we want to smoothen the vector field, introducing a new parameter n, and we want to find the lowest value of n for which oscillations occur. We expect such values to lie on a high-dimensional manifold of Hopf bifurcations that we want to study. The implications would permit us to determine which models describe most faithfully the behaviour we detect in experiments.
Youngmin Park (Brandeis)
Title: Scalar Reduction of a Neural Field Model with Spike Frequency Adaptation
Abstract: We study a deterministic version of a one- and two-dimensional attractor neural network model of hippocampal activity first studied by Itskov et al 2011. We analyze the dynamics of the system on the ring and torus domain with an even periodized weight matrix, assuming weak and slow spike frequency adaptation and a weak stationary input current. On these domains, we find transitions from spatially localized stationary solutions (“bumps”) to (periodically modulated) solutions (“sloshers”), as well as constant and non-constant velocity traveling bumps depending on the relative strength of external input current and adaptation. The weak and slow adaptation allows for a reduction of the system from a distributed partial integro-differential equation to a system of scalar Volterra integro-differential equations describing the movement of the centroid of the bump solution. Using this reduction, we show that on both domains, sloshing solutions arise through an Andronov-Hopf bifurcation and derive a normal form for the Hopf bifurcation on the ring. We also show existence and stability of constant velocity solutions on both domains using Evans functions. In contrast to existing studies, we assume a general weight matrix of Mexican-hat type in addition to a smooth firing rate function.