Chaos in the square billiard with a modified reflection law

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the standard reflection law.Comment: 17 figure

Repetitive Pertussis Toxin Promotes Development of Regulatory T Cells and Prevents Central Nervous System Autoimmune Disease

Bacterial and viral infections have long been implicated in pathogenesis and progression of multiple sclerosis (MS). Incidence and severity of its animal model experimental autoimmune encephalomyelitis (EAE) can be enhanced by concomitant administration of pertussis toxin (PTx), the major virulence factor of Bordetella pertussis. Its adjuvant effect at the time of immunization with myelin antigen is attributed to an unspecific activation and facilitated migration of immune cells across the blood brain barrier into the central nervous system (CNS). In order to evaluate whether recurring exposure to bacterial antigen may have a differential effect on development of CNS autoimmunity, we repetitively administered PTx prior to immunization. Mice weekly injected with PTx were largely protected from subsequent EAE induction which was reflected by a decreased proliferation and pro-inflammatory differentiation of myelin-reactive T cells. Splenocytes isolated from EAE-resistant mice predominantly produced IL-10 upon re-stimulation with PTx, while non-specific immune responses were unchanged. Longitudinal analyses revealed that repetitive exposure of mice to PTx gradually elevated serum levels for TGF-β and IL-10 which was associated with an expansion of peripheral CD4+CD25+FoxP3+ regulatory T cells (Treg). Increased frequency of Treg persisted upon immunization and thereafter. Collectively, these data suggest a scenario in which repetitive PTx treatment protects mice from development of CNS autoimmune disease through upregulation of regulatory cytokines and expansion of CD4+CD25+FoxP3+ Treg. Besides its therapeutic implication, this finding suggests that encounter of the immune system with microbial products may not only be part of CNS autoimmune disease pathogenesis but also of its regulation