Allan R. Willms
https://hdl.handle.net/10214/6586
Associate Professor, Dept. of Mathematics & Statistics2021-09-28T13:54:56ZUniform Sampling on the Standard Simplex
https://hdl.handle.net/10214/25386
Uniform Sampling on the Standard Simplex
Willms, Allan R.
Three methods for obtaining uniform sampling on the standard simplex are summarized and a new derivation of one method is provided, which only uses basic notions from calculus, difference equations, and differential equations.
2021-01-01T00:00:00ZHuygens' Clocks Revisited
https://hdl.handle.net/10214/11553
Huygens' Clocks Revisited
Willms, Allan R.; Kitanov, Petko M.; Langford, William F.
In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. In contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, it also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.
pre-print
2017-09-07T00:00:00ZBifurcation, Bursting, and Spike Frequency Adaptation
https://hdl.handle.net/10214/9513
Bifurcation, Bursting, and Spike Frequency Adaptation
Guckenheimer, John; Harris-Warrick, Ronald; Peck, Jack; Willms, Allan R.
Many neural systems display adaptive properties that occur on time scales that are slower than the time scales associated with repetitive firing of action potentials or bursting oscillations. Spike frequency adaptation is the name given to processes that reduce the frequency of rhythmic tonic firing of action potentials, sometimes leading to the termination of spiking and the cell becoming quiescent. This article examines these processes mathematically, within the context of singularly perturbed dynamical systems . We place emphasis on the lengths of successive interspike intervals during adaptation. Two different bifurcation mechanisms in singularly perturbed systems that correspond to the termination of firing are distinguished by the rate at which interspike intervals slow near the termination of firing. We compare theoretical predictions to measurement of spike frequency adaptation in a model of the LP cell of the lobster stomatogastric ganglion.
1997-01-01T00:00:00ZAsymptotic Analysis of Subcritical Hopf-homoclinic Bifurcation
https://hdl.handle.net/10214/9512
Asymptotic Analysis of Subcritical Hopf-homoclinic Bifurcation
Guckenheimer, John; Willms, Allan R.
This paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical Hopf bifurcation with a homoclinic orbit of the Hopf equilibrium. Our work is motivated by our previous analysis of a Hodgkin–Huxley neuron model which possesses a subcritical Hopf bifurcation (J. Guckenheimer, R. Harris-Warrick, J. Peck, A. Willms, J. Comput. Neurosci. 4 (1997) 257–277). In this model, the Hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium, that is, the Hopf bifurcation appears to be close to a homoclinic bifurcation as well. This model of the lateral pyloric (LP) cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike-frequency adaptation, in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent. The presence of a subcritical Hopf bifurcation in this model was the one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non-oscillatory solution. The analysis presented here explains the apparent proximity of homoclinic and Hopf bifurcations. We also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical Hopf bifurcation that may be close to a codimension two subcritical Hopf–homoclinic bifurcation.
2000-01-01T00:00:00Z