Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model

We study the stochastic FitzHugh-Nagumo equations, modelling the dynamics of neuronal action potentials, in parameter regimes characterised by mixed-mode oscillations. The interspike time interval is related to the random number of small-amplitude oscillations separating consecutive spikes. We prove that this number has an asymptotically geometric distribution, whose parameter is related to the principal eigenvalue of a substochastic Markov chain. We provide rigorous bounds on this eigenvalue in the small-noise regime, and derive an approximation of its dependence on the system's parameters for a large range of noise intensities. This yields a precise description of the probability distribution of observed mixed-mode patterns and interspike intervals.Comment: 36 page

Fibonacci facts and formulas

We investigate several methods of computing Fibonacci numbers quickly and generalize some properties of the Fibonacci numbers to degree r Fi­bonacci (R-nacci) numbers. Sections 2 and 3 present several algorithms for computing the traditional, degree two, Fibonacci numbers quickly. Sections 4 and 5 investigate the structure of the binary representation of the Fibonacci numbers. Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. Finally, in Sec­tion 9 the asymptotic proportion of 1's in the Zekendorf representation of integers is computed and an easy to compute closed formula is given

Creating a multi-center rare disease consortium - the Consortium of Eosinophilic Gastrointestinal Disease Researchers (CEGIR).

 Eosinophilic gastrointestinal disorders (EGIDs) affect various segments of the gastrointestinal tract. Since these disorders are rare, collaboration is essential to enroll subjects in clinical studies and study the broader population. The Rare Diseases Clinical Research Network (RDCRN), a program of the National Center for Advancing Translational Sciences (NCATS), funded the Consortium of Eosinophilic Gastrointestinal Disease Researchers (CEGIR) in 2014 to advance the field of EGIDs. CEGIR facilitates collaboration among various centers, subspecialties, patients, professional organizations and patient-advocacy groups and includes 14 clinical sites. It has successfully initiated two large multi-center clinical studies looking to refine EGID diagnoses and management. Several pilot studies are underway that focus on various aspects of EGIDs including novel therapeutic interventions, diagnostic and monitoring methods, and the role of the microbiome in pathogenesis. CEGIR currently nurtures five physician-scholars through a career training development program and has published more than 40 manuscripts since its inception. This review focuses on CEGIR's operating model and progress and how it facilitates a framework for exchange of ideas and stimulates research and innovation. This consortium provides a model for progress on other potential clinical areas

Treatment of persistent organic pollutants in wastewater using hydrodynamic cavitation in synergy with advanced oxidation process

Persistent organic pollutants (POPs) are very tenacious wastewater contaminants. The consequences of their existence have been acknowledged for negatively affecting the ecosystem with specific impact upon endocrine disruption and hormonal diseases in humans. Their recalcitrance and circumvention of nearly all the known wastewater treatment procedures are also well documented. The reported successes of POPs treatment using various advanced technologies are not without setbacks such as low degradation efficiency, generation of toxic intermediates, massive sludge production, and high energy expenditure and operational cost. However, advanced oxidation processes (AOPs) have recently recorded successes in the treatment of POPs in wastewater. AOPs are technologies which involve the generation of OH radicals for the purpose of oxidising recalcitrant organic contaminants to their inert end products. This review provides information on the existence of POPs and their effects on humans. Besides, the merits and demerits of various advanced treatment technologies as well as the synergistic efficiency of combined AOPs in the treatment of wastewater containing POPs was reported. A concise review of recently published studies on successful treatment of POPs in wastewater using hydrodynamic cavitation technology in combination with other advanced oxidation processes is presented with the highlight of direction for future research focus

A cybernetic approach to population dynamics modeling

Some problems arising when modeling population dynamics by means of stochastic difference and differential equations are discussed. For a particular parametrization of logistic growth equations limit diffusion processes are constructed and interpreted in the light of the Ito-Stratonovich controversy. An indirect confirmation of the validity of May's conjecture on the persistence of a population in a randomly varying environment is also obtained. An extension of these results to a wider class of growth equations is finally provided

A continuous Markovian model for neuronal activity

A diffusion model for the description of neurons' membrane potential fluctuations is proposed. Though retaining the well known feature consisting of the spontaneous exponential decay of the membrane potential to its resting value, the model discussed differs substantially from the ones in the current literature. Moreover, the Fokker-Planck equation now describing the membrane potential fluctuations is singular. The neuron's firing times probability density function is calculated in closed form as in a first passage time problem, and its expectation value and variance are evaluated. A detailed study of the mode of the firing times probability density function as related to the noise's intensity is performed. Some other auxiliary results are also obtained