Katherine A Newhall | Searcharxiv

arXiv2020

Network Mechanism for Insect Olfaction

Early olfactory pathway responses to the presentation of an odor exhibit remarkably similar dynamical behavior across phyla from insects to mammals, and frequently involve transitions among quiescence, collective network oscillations, and asynchronous firing. We hypothesize that the time scales of fast excitation and fast and slow inhibition present in these networks may be the essential element underlying this similar behavior, and design an idealized, conductance-based integrate-and-fire (I&F) model to verify this hypothesis via numerical simulations. To better understand the mathematical structure underlying the common dynamical behavior across species, we derive a firing-rate (FR) model and use it to extract a slow passage through a saddle-node-on-an-invariant-circle (SNIC) bifurcation structure. We expect this bifurcation structure to provide new insights into the understanding of the dynamical behavior of neuronal assemblies and that a similar structure can be found in other sensory systems.

q-bio.NC↗

arXiv2016

Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory

This paper is concerned with the long-time dynamics of the nonlinear wave equation in one-space dimension, $$ u_{tt} - δ^2 u_{xx} +V'(u) =0 \qquad x\in [0,1] $$ where $δ>0$ is a parameter and $V(u)$ is a potential bounded from below and growing at least like $u^2$ as $|u|\to\infty$. Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure and when the potential is double-welled, for example when $V(u) = \tfrac14(1-u^2)^2$, there is a regime such that two small disjoint sets in the system's phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here we quantify this phenomenon by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter $δ$ in small enough. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger $δ$, however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency, indicating that successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.

math.PR↗