Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation

Following the paper of K. Shimizu et al. (2011) we consider the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using the geometrical theory of singular perturbations. The considered model can be re-written as a 4-dimensional (locally 3-dimensional) autonomous system, which under certain conditions has a folded saddle-node singularity and additionally can be treated as a three time scale one

Sector-Delayed-Hopf-Type Mixed-Mode Oscillations in a Prototypical Three-Time-Scale Model

We consider a three-dimensional three-time-scale system that was first proposed by Krupa et al. (2008) under the additional assumption that two singular perturbation parameters are present in the equations. While the presence of three scales was shown to give use to canard induced periodic mixed mode oscillations (MMOs) (Desroches et al., 2012) in the parameter regime studied by Krupa et al. (2008,) we additionally observe mixed mode patterns that display delayed-Hopf-type behaviour (Neishtadt, 1987). We present analytical and numerical evidence for the occurrence of stable periodic dynamics that realises both mechanisms, and we discuss the transition between them. To the best of our knowledge, the resulting mixed sector-delayed-Hopf-type MMO trajectories represent a novel class of mixed-mode dynamics in singularly perturbed systems of ordinary differential equations. (C) 2015 Elsevier Inc. All rights reserved.The authors' research was supported by the Research Foundation Flanders (FWO) under Grant number G.0939.10N. Moreover, E.K. acknowledges support from the Polish Ministry of Science and Higher Education. Finally, the authors thank three anonymous reviewers whose comments and suggestions greatly improved the original manuscript.mixed-mode oscillations; delayed Hopf bifurcation; canards; neuronal modelling; fast-slow dynamics; singular perturbation

Three Time-Scales In An Extended Bonhoeffer-Van Der Pol Oscillator

We consider an extended three-dimensional Bonhoeffer-van der Pol oscillator which generalises the planar FitzHugh-Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. and by Freire and Gallas. Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. as a canonical form for such systems. Based on previously obtained results in, we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer-van der Pol oscillator from the point of view of geometric singular perturbation theory, thus complementing the findings reported in Sekikawa et al. In particular, we specify in detail the mixed-mode patterns that are observed upon variation of a bifurcation parameter which is naturally obtained by combining two of the original parameters in the system, and we derive asymptotic estimates for the corresponding parameter intervals. We thereby also disprove a conjecture of Tu (1989), where it was postulated that no stable periodic orbits of mixed-mode type can be observed in an equivalent extension of the Bonhoeffer-van der Pol equations.The authors' research was supported by the Research Foundation Flanders (FWO) under grant number G.0939.10N. Moreover, E. K. acknowledges support from the Polish Ministry of Science and Higher Education.Bonhoeffer-van der Pol oscillator; mixed-mode oscillations; canards; geometric singular perturbation theory; blow-up techniqu

Оценка физического развития и питания детей в возрасте 1–3 лет, проживающих в г. Москве

Irrational feeding can lead to a child’s physical and intellectual development disorders, a decrease in the body’s ability to resist aggressive environmental factors.Aim: to evaluate the physical development and nutrition of children aged 1–3 years living in Moscow.Methods: 106 children aged 1–3 years were examined: group 1 (n = 59) was comprised of children aged 1–2 years, group 2 — children aged 2–3 years (n = 47). Anthropometric data was evaluated using AntroPlus (WHO software). The following Z-score figures were calculated: WAZ (body mass for age), HAZ (height for age) and BAZ (body mass index for age). Nutrition was evaluated by reproducing a 3-day food allowance (actual nutrition) using the Dietplan 6 software. Figures analyzed: the volume of consumed food, daily caloricity, the amount of consumed proteins, fats and carbohydrates.Results: for the majority (76.4%) of examined children BAZ was between -2 and +1. 20% of children had an excessive body mass and obesity (BAZ > 1). Children with an excessive body mass demonstrated exceeding volumes of food consumption (by about 200–300g. per day, p < 0.001), protein consumption by 47.5%, fat consumption — 36.7% and caloricity by 21.3% (p < 0.001) as compared with the recommended consumption norms.Conclusion: the revealed nutrition disorders in children aged 1–3 years (overeating and unbalanced diet) lead to an increased body mass and obesity. Children with high body mass indexes at birth and Z-score and BAZ at the time of the study can be attributed to the obesity risk group. The BAZ index is the most informative one in terms of evaluating the child’s nutritive status.Нерациональное питание может привести к нарушениям физического и интеллектуального развития ребенка, снижению сопротивляемости организма к агрессивным факторам внешней среды.Цель исследования: оценить физическое развитие и питание детей в возрасте 1–3 лет, проживающих в г. Москве.Методы: обследовано 106 детей в возрасте от 1 до 3 лет: I группу (n = 59) составили дети 1–2 лет, II группу — дети 2–3 лет (n = 47). Антропометрические данные оценивались с использованием программы WHO AnthroPlus: расчитывались показатели Z-score — массы тела для возраста (WAZ), длины тела для возраста (HAZ), а также индекс массы тела/возраст (BAZ). Оценка питания проводилась методом воспроизведения 3-дневного рациона (фактического питания) с использованием программы Dietplan 6. Анализировались объем съеденной пищи, суточная калорийность питания, количество потребляемых белков, жиров, углеводов.Результаты: для большинства (76,4%) обследованных детей были характерны средневозрастные показатели физического развития и нутритивного статуса (BAZ от -2 до +1). У 1/5 детей были выявлены избыточная масса тела и ожирение (BAZ > 1). При избыточной массе тела, особенно у детей 1–2 лет, было установлено превышение потребляемых объемов пищи на 200–300 г/сут, белка — на 47,5%, жира — на 36,7% и энергетической ценности — на 21,3% (p < 0,001) по сравнению с рекомендуемыми нормами потребления.Заключение: выявленные нарушения питания у детей в возрасте 1–3 лет (переедание и разбалансированный рацион) приводят к развитию у них избыточной массы тела вплоть до ожирения. Детей с высокими показателями веса при рождении и Z-score BAZ на момент исследования можно отнести к группе риска по развитию ожирения. Наиболее информативным для оценки нутритивного статуса ребенка является показатель BAZ