A State-Dependent Delay Equation with Negative Feedback and ‘Mildly Unstable’ Rapidly Oscillating Periodic Solutions

We consider state-dependent delay equations of the form x′(t)=f(x(t−d(x(t)))) where d is smooth and f is smooth, bounded, nonincreasing, and satisfies the negative feedback condition xf(x)x≠0. We identify a special family of such equations each of which has a ``rapidly oscillating periodic solution p. The initial segment p0 of p is the fixed point of a return map R that is differentiable in an appropriate setting. We show that, although all the periodic solutions p we consider are unstable, the stability can be made arbitrarily mild in the sense that, given ϵ\u3e0, we can choose f and d such that the spectral radius of the derivative of R at p0 is less than 1+ϵ. The spectral radii are computed via a semiconjugacy of R with a finite-dimensional map

Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback

We study the scalar difference equation $$x(k+1) = x(k) + \frac{f(x(k-N))}{N},$$ where $f$ is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation $$x'(t) = f(x(t-1)).$$ We examine explicit families of such equations for which we can find, for infinitely many values of $ and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle

A state-dependent delay equation with chaotic solutions

We exhibit a scalar-valued state-dependent delay differential equation x 0 (t) = f(x(t − d(xt))) that has a chaotic solution. This equation has continuous (semi-strictly) monotonic negative feedback, and the quantity t − d(xt) is strictly increasing along solutions

Corrigendum to "A state-dependent delay equation with chaotic solutions" - Electron. J. Qual. Theory Differ. Equ. 2019, No. 22, 1-20

We correct an error in “A state-dependent delay equation with chaotic solutions” [Electron. J. Qual. Theory Differ. Equ. 2019, No. 22, 1–20]

Molecular and behavioral changes associated with adult hippocampus-specific SynGAP1 knockout

The synaptic Ras/Rap-GTPase-activating protein (SynGAP1) plays a unique role in regulating specific downstream intracellular events in response to N-methyl-D-aspartate receptor (NMDAR) activation. Constitutive heterozygous loss of SynGAP1 disrupts NMDAR-mediated physiological and behavioral processes, but the disruptions might be of developmental origin. Therefore, the precise role of SynGAP1 in the adult brain, including its relative functional significance within specific brain regions, remains unexplored. The present study constitutes the first attempt in achieving adult hippocampal-specific SynGAP1 knockout using the Cre/loxP approach. Here, we report that this manipulation led to a significant numerical increase in both small and large GluA1 and NR1 immunoreactive clusters, many of which were non-opposed to presynaptic terminals. In parallel, the observed marked decline in the amplitude of spontaneous excitatory currents (sEPSCs) and inter-event intervals supported the impression that SynGAP1 loss might facilitate the accumulation of extrasynaptic glutamatergic receptors. In addition, SynGAP1-mediated signaling appears to be critical for the proper integration and survival of newborn neurons. The manipulation impaired reversal learning in the probe test of the water maze and induced a delay-dependent impairment in spatial recognition memory. It did not significantly affect anxiety or reference memory acquisition but induced a substantial elevation in spontaneous locomotor activity in the open field test. Thus, the present study demonstrates the functional significance of SynGAP1 signaling in the adult brain by capturing several changes that are dependent on NMDAR and hippocampal integrity

Variational Approach to the Modulational Instability

We study the modulational stability of the nonlinear Schr\"odinger equation (NLS) using a time-dependent variational approach. Within this framework, we derive ordinary differential equations (ODEs) for the time evolution of the amplitude and phase of modulational perturbations. Analyzing the ensuing ODEs, we re-derive the classical modulational instability criterion. The case (relevant to applications in optics and Bose-Einstein condensation) where the coefficients of the equation are time-dependent, is also examined

The Third wave in globalization theory

This essay examines a proposition made in the literature that there are three waves in globalization theory—the globalist, skeptical, and postskeptical or transformational waves—and argues that this division requires a new look. The essay is a critique of the third of these waves and its relationship with the second wave. Contributors to the third wave not only defend the idea of globalization from criticism by the skeptics but also try to construct a more complex and qualified theory of globalization than provided by first-wave accounts. The argument made here is that third-wave authors come to conclusions that try to defend globalization yet include qualifications that in practice reaffirm skeptical claims. This feature of the literature has been overlooked in debates and the aim of this essay is to revisit the literature and identify as well as discuss this problem. Such a presentation has political implications. Third wavers propose globalist cosmopolitan democracy when the substance of their arguments does more in practice to bolster the skeptical view of politics based on inequality and conflict, nation-states and regional blocs, and alliances of common interest or ideology rather than cosmopolitan global structures

Content analysis, cultural grammars, and computers

This paper describes current theoretical work in a cognitive anthropological approach to the analysis of oral literature. A theory of narrative grammar is presented along with the description of a computer program, SAGE, that is currently under development. SAGE facilitates the labelling of clauses and sentences by different types of semantic features. It maps and counts the occurrences of these in the stories analyzed. © 1991 Human Sciences Press, Inc

Rescue of behavioral and electrophysiological phenotypes in a Pitt-Hopkins syndrome mouse model by genetic restoration of Tcf4 expression

Pitt-Hopkins syndrome (PTHS) is a neurodevelopmental disorder caused by monoallelic mutation or deletion in the transcription factor 4 (TCF4) gene. Individuals with PTHS typically present in the first year of life with developmental delay and exhibit intellectual disability, lack of speech, and motor incoordination. There are no effective treatments available for PTHS, but the root cause of the disorder, TCF4 haploinsufficiency, suggests that it could be treated by normalizing TCF4 gene expression. Here, we performed proof-of-concept viral gene therapy experiments using a conditional Tcf4 mouse model of PTHS and found that postnatally reinstating Tcf4 expression in neurons improved anxiety-like behavior, activity levels, innate behaviors, and memory. Postnatal reinstatement also partially corrected EEG abnormalities, which we characterized here for the first time, and the expression of key TCF4-regulated genes. Our results support a genetic normalization approach as a treatment strategy for PTHS, and possibly other TCF4-linked disorders