Delay-Differential Equations with Constant Lags

This article concerns delay-differential equations (DDEs) with constant lags. DDEs increasingly are being used to model various phenomena in mathematics and the physical sciences. For such equations the value of the derivative at any time depends on the solution at a previous lagged time. Although solving DDEs is similar in some respects to solving ordinary differential equations (ODEs), it differs in some rather significant ways. These differences are discussed briefly. The effect the differences can have on systems of ODEs and DDEs is illustrated. Popular approaches used in the development of numerical methods for solving DDEs are described. Available Matlab DDE solvers and a Fortran 90 solver based on these approaches are mentioned. Finally, some pointers to further resources available to interested readers are given

Multiadaptive Galerkin Methods for ODEs III: A Priori Error Estimates

The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution. In this paper, we prove general order a priori error estimates for the mcG(q) and mdG(q) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution

Does error control suppress spuriosity?

In the numerical solution of initial value ordinary differential equations, to what extent does local error control confer global properties? This work concentrates on global steady states or fixed points. It is shown that, for systems of equations, spurious fixed points generally cease to exist when local error control is used. For scalar problems, on the other hand, locally adaptive algorithms generally avoid spurious fixed points by an indirect method---the stepsize selection process causes spurious fixed points to be unstable. However, problem classes exist where, for arbitrarily small tolerances, stable spurious fixed points persist with significant basins of attraction. A technique is derived for generating such examples

Simultaneous quantification of GABAergic 3α,5α/3α,5β neuroactive steroids in human and rat serum

The 3α,5α- and 3α,5β-reduced derivatives of progesterone, deoxycorticosterone, dehydroepiandrosterone and testosterone enhance GABAergic neurotransmission and produce inhibitory neurobehavioral and anti-inflammatory effects. Despite substantial information on the progesterone derivative (3α,5α)-3-hydroxypregnan-20-one (3α,5α-THP, allopregnanolone), the physiological significance of the other endogenous GABAergic neuroactive steroids has remained elusive. Here, we describe the validation of a method using gas chromatography-mass spectrometry to simultaneously identify serum levels of the eight 3α,5α- and 3α,5β-reduced derivatives of progesterone, deoxycorticosterone, dehydroepiandrosterone and testosterone. The method shows specificity, sensitivity and enhanced throughput compared to other methods already available for neuroactive steroid quantification. Administration of pregnenolone to rats and progesterone to women produced selective effects on the 3α,5α- and 3α,5β-reduced neuroactive steroids, indicating differential regulation of their biosynthetic pathways. Pregnenolone administration increased serum levels of 3α,5α-THP (+1488%, p<0.001), (3α,5α)-3,21-dihydroxypregnan-20-one (3α,5α-THDOC, +205%, p<0.01), (3α,5α)-3-hydroxyandrostan-17-one (3α,5α-A, +216%, p<0.001), (3α,5α,17β)-androstane-3,17-diol (3α,5α-A-diol, +190%, p<0.01). (3α,5β)-3-hydroxypregnan-20-one (3α,5β-THP) and (3α,5β)-3-hydroxyandrostan-17-one (3α,5β-A) were not altered, while (3α,5β)-3,21-dihydroxypregnan-20-one (3α,5β-THDOC) and (3α,5β,17β)-androstane-3,17-diol (3α,5β-A-diol) were increased from undetectable levels to 271 ± 100 and 2.4 ± 0.9 pg ± SEM, respectively (5/8 rats). Progesterone administration increased serum levels of 3α,5α-THP (+1806%, p<0.0001), 3α,5β-THP (+575%, p<0.001), 3α,5α-THDOC (+309%, p<0.001). 3α,5β-THDOC levels were increased by 307%, although this increase was not significant because this steroid was detected only in 3/16 control subjects. Levels of 3α,5α-A, 3α,5β-A and pregnenolone were not altered. This method can be used to investigate the physiological and pathological role of neuroactive steroids and to develop biomarkers and new therapeutics for neurological and psychiatric disorders

Proof-of-Concept Trial with the Neurosteroid Pregnenolone Targeting Cognitive and Negative Symptoms in Schizophrenia

The neurosteroid pregnenolone and its sulfated derivative enhance learning and memory in rodents. Pregnenolone sulfate also positively modulates NMDA receptors and could thus ameliorate hypothesized NMDA receptor hypofunction in schizophrenia. Furthermore, clozapine increases pregnenolone in rodent hippocampus, possibly contributing to its superior efficacy. We therefore investigated adjunctive pregnenolone for cognitive and negative symptoms in patients with schizophrenia or schizoaffective disorder receiving stable doses of second-generation antipsychotics in a pilot randomized, placebo-controlled, double-blind trial. Following a 2-week single-blind placebo lead-in, patients were randomized to pregnenolone (fixed escalating doses to 500 mg/day) or placebo, for 8 weeks. Primary end points were changes in BACS and MCCB composite and total SANS scores. Of 21 patients randomized, 18 completed at least 4 weeks of treatment (n = 9/group). Pregnenolone was well tolerated. Patients receiving pregnenolone demonstrated significantly greater improvements in SANS scores (mean change = 10.38) compared with patients receiving placebo (mean change = 2.33), p = 0.048. Mean composite changes in BACS and MCCB scores were not significantly different in patients randomized to pregnenolone compared with placebo. However, serum pregnenolone increases predicted BACS composite scores at 8 weeks in the pregnenolone group (rs = 0.81, p = 0.022). Increases in allopregnanolone, a GABAergic pregnenolone metabolite, also predicted BACS composite scores (rs = 0.74, p = 0.046). In addition, baseline pregnenolone (rs = −0.76, p = 0.037), pregnenolone sulfate (rs = − 0.83, p = 0.015), and allopregnanolone levels (rs = −0.83, p = 0.015) were inversely correlated with improvements in MCCB composite scores, further supporting a possible role for neurosteroids in cognition. Mean BACS and MCCB composite scores were correlated (rs = 0.74, p <0.0001). Pregnenolone may be a promising therapeutic agent for negative symptoms and merits further investigation for cognitive symptoms in schizophrenia

Neurosteroids and Self-Reported Pain in Veterans Who Served in the U.S. Military after September 11, 2001

Nearly half of Operation Enduring Freedom / Operation Iraqi Freedom (OEF/OIF) veterans experience continued pain post-deployment. Several investigations report analgesic effects of allopregnanolone and other neurosteroids in animal models, but few data are currently available focusing on neurosteroids in clinical populations. Allopregnanolone positively modulates GABAA receptors and demonstrates pronounced analgesic and anxiolytic effects in rodents, yet studies examining the relationship between pain and allopregnanolone in humans are limited. We thus hypothesized that endogenous allopregnanolone and other neurosteroid levels may be negatively correlated with self-reported pain symptoms in humans

Asymptotic L2 Inequalities of Markoff type

The Markoff brothers have obtained bounds in the maximum norm for polynomials in terms of bounds for their derivatives. E. Schmidt has obtained asymptotic bounds for the analogous problem for weighted L[subscript 2] norms and the first derivative. This thesis extends Schmidt's work in one case to the second derivative. A new technique is applied which recovers the principal term of Schmidt's asymptotic result and furnishes information about the extremal polynomials. Exact bounds are derived in one case