Leaders do not look back, or do they?

We study the effect of adding to a directed chain of interconnected systems a directed feedback from the last element in the chain to the first. The problem is closely related to the fundamental question of how a change in network topology may influence the behavior of coupled systems. We begin the analysis by investigating a simple linear system. The matrix that specifies the system dynamics is the transpose of the network Laplacian matrix, which codes the connectivity of the network. Our analysis shows that for any nonzero complex eigenvalue $\lambda$ of this matrix, the following inequality holds: $\frac{|\Im \lambda |}{|\Re \lambda |} \leq \cot\frac{\pi}{n}$. This bound is sharp, as it becomes an equality for an eigenvalue of a simple directed cycle with uniform interaction weights. The latter has the slowest decay of oscillations among all other network configurations with the same number of states. The result is generalized to directed rings and chains of identical nonlinear oscillators. For directed rings, a lower bound $\sigma_c$ for the connection strengths that guarantees asymptotic synchronization is found to follow a similar pattern: $\sigma_c=\frac{1}{1-\cos\left( 2\pi /n\right)}$. Numerical analysis revealed that, depending on the network size $n$, multiple dynamic regimes co-exist in the state space of the system. In addition to the fully synchronous state a rotating wave solution occurs. The effect is observed in networks exceeding a certain critical size. The emergence of a rotating wave highlights the importance of long chains and loops in networks of oscillators: the larger the size of chains and loops, the more sensitive the network dynamics becomes to removal or addition of a single connection

Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems

We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable subsystems with one-dimensional unstable dynamics or critically stable dynamics. Systems of this type arise in problems of nonlinear output regulation, parameter estimation and adaptive control. In addition to providing boundedness and convergence criteria the results allow to derive domains of initial conditions corresponding to solutions leaving a given neighborhood of the origin at least once. In contrast to other works addressing convergence issues in unstable systems, our results require neither input-output characterizations for the stable part nor estimates of convergence rates. The results are illustrated with examples, including the analysis of phase synchronization of neural oscillators with heterogenous coupling

Parental functioning during maintenance treatment for childhood acute lymphoblastic leukemia: Effects of treatment intensity and dexamethasone pulses

Background: During maintenance treatment, Dutch pediatric patients with medium-risk (MR) acute lymphoblastic leukemia (ALL) receive intravenous chemotherapy and cyclic dexamethasone. Dexamethasone affects child's sleep and behavior. Standard-risk (SR) patients only receive oral chemotherapy, without dexamethasone. Effects of stratified therapy on parents are not well known. This study compares parental sleep, distress and quality of life (QoL) with the general population, between MR and SR groups, and on- and off-dexamethasone (MR group). Procedure: One year after diagnosis, parents of MR patients completed the Medical Outcomes Study (MOS) sleep, distress thermometer for parents and Short Form-12 (SF-12) twice; once on-dexamethasone and once off-dexamethasone. SR parents completed one measurement. Sleep problems, distress and QoL scores (off-dexamethasone) were compared to reference values and between MR and SR. Score differences on- and off-dexamethasone were assessed by multilevel regression analysis. Results: Parents (80% mothers) of 121 patients (57% males; 75% MR, 25% SR) completed 191 measurements. Compared to reference values, parents reported more sleep disturbances, higher distress, and lower mental QoL. Additionally, MR parents reported clinical distress (score ≥ 4), whereas SR parents (on average) did not (mean 4.8 ± 2.4 vs 3.5 ± 2.4, P =.02). Within the MR group, outcomes did not significantly differ on- and off-dexamethasone. Conclusions: Parents of ALL patients report sleep problems, high distress, and QoL impairment. Within the MR group, parental functioning did not differ on- and off-dexamethasone. However, MR parents reported clinical distress more often than SR parents, possibly reflecting differences in prognostic estimates and treatment burden. This perhaps includes the overall strain of cyclic dexamethasone. This study highlights the need for psychosocial support throughout treatment, regardless of risk stratification

The impact of maintenance therapy on sleep-wake rhythms and cancer-related fatigue in pediatric acute lymphoblastic leukemia

Purpose: To assess the impact of maintenance therapy and the additional impact of dexamethasone treatment on cancer-related fatigue and sleep-wake rhythms in pediatric acute lymphoblastic leukemia (ALL) patients and to determine the association between these outcomes. Methods: A national cohort of pediatric ALL patients (≥ 2 years) was included (± 1 year post-diagnosis). Patients receiving dexamethasone were assessed twice (assessment with and without dexamethasone). Actigraphy assessments were used to calculate sleep-wake outcomes with nonparametric methods. Cancer-related fatigue was assessed with the PedsQL Multidimensional Fatigue Scale. Sleep-wake rhythms and cancer-related fatigue were compared between patients participating in the assessment without dexamethasone and healthy children (linear regression) and between assessments with and without dexamethasone (mixed models). Using linear regression, associations between sleep-wake outcomes and cancer-related fatigue were determined during assessments with and without dexamethasone. Results: Responses were collected for 125 patients (113 assessments with and 81 without dexamethasone). The sleep-wake rhythm was less stable (p = 0.03) and less robust (p = 0.01), with lower physical activity l