The NMDA receptor as a target for cognitive enhancement.

We prove an analog of Bottcher’s theorem for transcendental entire functions in the Eremenko–Lyubich class $ \mathcal{B} $
 . More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are quasiconformally equivalent in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that the conjugacy is essentially unique. In particular, we show that a function $ f \in \mathcal{B} $
 has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions $ f,g \in \mathcal{B} $
 that belong to the same parameter space are conjugate on their sets of escaping points.

Rigidity of escaping dynamics for transcendental entire functions

By rapidly modulating neuronal excitability, neurosteroids regulate physiological processes, such as responses to stress and development. Excessive stress affects their biosynthesis and causes an imbalance in cognition and emotions. The progesterone derivative, allopregnanolone (Allo) enhances extrasynaptic and postsynaptic inhibition by directly binding at GABAA receptors, and thus, positively and allosterically modulates the function of GABA. Allo levels are decreased in stress-induced psychiatric disorders, including depression and post-traumatic stress disorder (PTSD), and elevating Allo levels may be a valid therapeutic approach to counteract behavioural dysfunction. While benzodiazepines are inefficient, selective serotonin reuptake inhibitors (SSRIs) represent the first choice treatment for depression and PTSD. Their mechanisms to improve behaviour in preclinical studies include neurosteroidogenic effects at low non-serotonergic doses. Unfortunately, half of PTSD and depressed patients are resistant to current prescribed 'high' dosage of these drugs that engage serotonergic mechanisms. Unveiling novel biomarkers to develop more efficient treatment strategies is in high demand. Stress-induced down-regulation of neurosteroid biosynthesis and changes in GABAA receptor subunit expression offer a putative biomarker axis to develop new PTSD treatments. The advantage of stimulating Allo biosynthesis relies on the variety of neurosteroidogenic receptors to be targeted, including TSPO and endocannabinoid receptors. Furthermore, stress favours a GABAA receptor subunit composition with higher sensitivity for Allo. The use of synthetic analogues of Allo is a valuable alternative. Pregnenolone or drugs that stimulate its levels increase Allo but also sulphated steroids, including pregnanolone sulphate which, by inhibiting NMDA tonic neurotransmission, provides neuroprotection and cognitive benefits. In this review, we describe current knowledge on the effects of stress on neurosteroid biosynthesis and GABAA receptor neurotransmission and summarize available pharmacological strategies that by enhancing neurosteroidogenesis are relevant for the treatment of SSRI-resistant patients. Linked Articles This article is part of a themed section on Pharmacology of Cognition: a Panacea for Neuropsychiatric Disease? To view the other articles in this section visit http://onlinelibrary.wiley.com/doi/10.1111/bph.v174.19/issuetoc.

https://bpspubs.onlinelibrary.wiley.com/doi/pdf/10.1111/bph.13843

Neurosteroid biosynthesis down-regulation and changes in GABAA receptor subunit composition: a biomarker axis in stress-induced cognitive and emotional impairment.

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. 
We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. 
Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.

Memory dysfunction is a symptomatic feature of many neurologic and neuropsychiatric disorders; however, the basic underlying mechanisms of memory and altered states of circuitry function associated with disorders of memory remain a vast unexplored territory. The initial discovery of endogenous neurosteroids triggered a quest to elucidate their role as neuromodulators in normal and diseased brain function. In this review, based on the perspective of our own research, the advances leading to the discovery of positive and negative neurosteroid allosteric modulators of GABA type-A (GABAA), NMDA, and non-NMDA type glutamate receptors are brought together in a historical and conceptual framework. We extend the analysis toward a state-of-the art view of how neurosteroid modulation of neural circuitry function may affect memory and memory deficits. By aggregating the results from multiple laboratories using both animal models for disease and human clinical research on neuropsychiatric and age-related neurodegenerative disorders, elements of a circuitry level view begins to emerge. Lastly, the effects of both endogenously active and exogenously administered neurosteroids on neural networks across the life span of women and men point to a possible underlying pharmacological connectome by which these neuromodulators might act to modulate memory across diverse altered states of mind.

/pdf/neurosteroid-actions-in-memory-and-neurologic-3ivs6rqaqh.pdf

Neurosteroid Actions in Memory and Neurologic/Neuropsychiatric Disorders

http://math.bu.edu/people/bob/papers/msmap.pdf

Rational maps with generalized Sierpinski gasket Julia sets

In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions Eλ(z) = λez where λ ∈ ℂ in the special case when λ is a Misiurewicz parameter, so that the Julia set of these maps is the entire complex plane. These invariant sets consist of points that share the same itinerary under iteration of Eλ. Previously, the only known types of such invariant sets were either simple hairs that extend from a definite endpoint to ∞ in the right half plane or else indecomposable continua for which a single hair accumulates everywhere upon itself. One new type of invariant set that we construct in this paper is an indecomposable continuum in which a pair of hairs accumulate upon each other, rather than a single hair having this property. The second type consists of an indecomposable continuum together with a completely separate hair that accumulates on this continuum.

/pdf/indecomposable-continua-and-misiurewicz-points-in-2m1ze86e59.pdf

Indecomposable continua and misiurewicz points in exponential dynamics

Sierpinski Carpets and Gaskets as Julia sets of Rational Maps

Neurosteroid binding to the amino terminal and glutamate binding domains of ionotropic glutamate receptors.

In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p/q-bulbs. The reason for this is that the largest component of the interior of these bulbs consists of c-values for which the quadratic function Qc(z) = z2 + c admits an attracting cycle with rotation number p/q. In this paper, we give a geometric method to read off p/q from the geometry of the antenna attached to the bulb.

Mónica Moreno Rocha

Papers

Rational maps with generalized Sierpinski gasket Julia sets

Indecomposable continua and misiurewicz points in exponential dynamics

Sierpinski Carpets and Gaskets as Julia sets of Rational Maps

Neurosteroid binding to the amino terminal and glutamate binding domains of ionotropic glutamate receptors.

Geometry of the antennas in the mandelbrot set