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Attention-deficit /hyperactivity disorder (ADHD) is a common neurodevelopmental syndrome characterized by age-inappropriate levels of motor activity, impulsivity and attention. The aim of the present study was to study diurnal variation of motor activity in adult ADHD patients, compared to healthy controls and clinical controls with mood and anxiety disorders. Wrist-worn actigraphs were used to record motor activity in a sample of 81 patients and 30 healthy controls. Time series from registrations in the morning and evening were analyzed using measures of variability, complexity and a newly developed method, the similarity algorithm, based on transforming time series into graphs. In healthy controls the evening registrations showed higher variability and lower complexity compared to morning registrations, however this was evident only in the female controls. In the two patient groups the same measures were not significantly different, with one exception, the graph measure bridges. This was the measure that most clearly separated morning and evening registrations and was significantly different both in healthy controls and in patients with a diagnosis of ADHD. These findings suggest that actigraph registrations, combined with mathematical methods based on graph theory, may be used to elucidate the mechanisms responsible for the diurnal regulation of motor activity.

Abstract. Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder Theorem yields a decomposition of GQC codes into a sum of concatenated codes. This decomposition leads to a trace formula, a minimum distance bound, and to a criteria for the GQC code to be self-dual or to be linear complementary dual (LCD). Explicit long GQC codes that are LCD, but not QC, are exhibited.

Changes in motor activity are core symptoms of mood episodes in bipolar disorder. The manic state is characterized by increased variance, augmented complexity and irregular circadian rhythmicity when compared to healthy controls. No previous studies have compared mania to euthymia intra-individually in motor activity. The aim of this study was to characterize differences in motor activity when comparing manic patients to their euthymic selves. Motor activity was collected from 16 bipolar inpatients in mania and remission. 24-h recordings and 2-h time series in the morning and evening were analyzed for mean activity, variability and complexity. Lastly, the recordings were analyzed with the similarity graph algorithm and graph theory concepts such as edges, bridges, connected components and cliques. The similarity graph measures fluctuations in activity reasonably comparable to both variability and complexity measures. However, direct comparisons are difficult as most graph measures reveal variability in constricted time windows. Compared to sample entropy, the similarity graph is less sensitive to outliers. The little-understood estimate Bridges is possibly revealing underlying dynamics in the time series. When compared to euthymia, over the duration of approximately one circadian cycle, the manic state presented reduced variability, displayed by decreased standard deviation (p = 0.013) and augmented complexity shown by increased sample entropy (p = 0.025). During mania there were also fewer edges (p = 0.039) and more bridges (p = 0.026). Similar significant changes in variability and complexity were observed in the 2-h morning and evening sequences, mainly in the estimates of the similarity graph algorithm. Finally, augmented complexity was present in morning samples during mania, displayed by increased sample entropy (p = 0.015). In conclusion, the motor activity of mania is characterized by altered complexity and variability when compared within-subject to euthymia.

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