A Convergent Iterative Solution of the Quantum Double-well Potential

We present a new convergent iterative solution for the two lowest quantum wave functions $\psi_{ev}$ and $\psi_{od}$ of the Hamiltonian with a quartic double well potential $V$ in one dimension. By starting from a trial function, which is by itself the exact lowest even or odd eigenstate of a different Hamiltonian with a modified potential $V+\delta V$, we construct the Green's function for the modified potential. The true wave functions, $\psi_{ev}$ or $\psi_{od}$, then satisfies a linear inhomogeneous integral equation, in which the inhomogeneous term is the trial function, and the kernel is the product of the Green's function times the sum of $\delta V$, the potential difference, and the corresponding energy shift. By iterating this equation we obtain successive approximations to the true wave function; furthermore, the approximate energy shift is also adjusted at each iteration so that the approximate wave function is well behaved everywhere. We are able to prove that this iterative procedure converges for both the energy and the wave function at all $x$.Comment: 76 pages, Latex, no figure, 1 tabl

Bayesian analysis of trinomial data in behavioral experiments and its application to human studies of general anesthesia

Accurate quantification of loss of response to external stimuli is essential for understanding the mechanisms of loss of consciousness under general anesthesia. We present a new approach for quantifying three possible outcomes that are encountered in behavioral experiments during general anesthesia: correct responses, incorrect responses and no response. We use a state-space model with two state variables representing a probability of response and a conditional probability of correct response. We show applications of this approach to an example of responses to auditory stimuli at varying levels of propofol anesthesia ranging from light sedation to deep anesthesia in human subjects. The posterior probability densities of model parameters and the response probability are computed within a Bayesian framework using Markov Chain Monte Carlo methods.National Institutes of Health (U.S.) (Grant DP2-OD006454)National Institutes of Health (U.S.) (Grant K25-NS057580)National Institutes of Health (U.S.) (Grant DP1-OD003646)National Institutes of Health (U.S.) (Grant R01-EB006385)National Institutes of Health (U.S.) (Grant R01-MH071847

Studying the song development process rationale and methods

Current technology makes it possible to measure song development continuously throughout a vocal ontogeny. Here we briefly review some of the problems involved and describe experimental and analytic methods for automatic tracing of vocal changes. These techniques make it possible to characterize the specific methods the bird uses to imitate sounds: an automated song recognition procedure allows continuous song recording, followed by automated sound analysis that partition the song to syllables, extract acoustic features of each syllable, and summarize the entire song development process over time into a single database. The entire song development is then presentable in the form of images or movie clips. These Dynamic Vocal Development (DVD) maps show how each syllable type emerges, and how the bird manipulates syllable features to eventually approximate the model song. Most of the experimental and analytic methods described here have been organized into a software package, which also allows combined neural and sound recording to monitor changes in brain activity as vocal learning occurs. The software is available at http://ofer.sci.ccny.cuny.edu

Thalamocortical model for a propofol-induced alpha rhythm associated with loss of consciousness

Recent data reveal that the general anesthetic propofol gives rise to a frontal α-rhythm [alpha rhythm] at dose levels sufficient to induce loss of consciousness. In this work, a computational model is developed that suggests the network mechanisms responsible for such a rhythm. It is shown that propofol can alter the dynamics in thalamocortical loops, leading to persistent and synchronous α-activity [alpha activity]. The synchrony that forms in the cortex by virtue of the involvement of the thalamus may impede responsiveness to external stimuli, thus providing a correlate for the unconscious state.National Institutes of Health (U.S.) (Grant DP1-OD003646)National Institutes of Health (U.S.) (Grant K25-NS057580)National Institutes of Health (U.S.) (Grant DP2-OD006454)National Science Foundation (U.S.) (Grant DMS-0717670