Le funzioni wavelet nelle applicazioni scientifiche

Le wavelet sono una nuova famiglia di funzioni matematiche che permettono di decomporre una data funzione nelle sue diverse componenti in frequenza. Esse combinano le proprietà dell’ortogonalità, il supporto compatto, la localizzazione in tempo e frequenza e algoritmi veloci. Sono considerate, perciò, uno strumento versatile sia per il contenuto matematico, sia per le applicazioni. Nell’ultimo decennio si sono diffuse e imposte come uno degli strumenti migliori nell’analisi dei segnali, a fianco, o addirittura come sostitute, dei metodi di Fourier. Si parte dalla nascita di esse (1807) attribuita a J. Fourier, si considera la wavelet di A. Haar (1909) per poi incentrare l’attenzione sugli anni ’80, in cui J. Morlet e A. Grossmann definiscono compiutamente le wavelet nel campo della fisica quantistica. Altri matematici e scienziati, nel corso del Novecento, danno il loro contributo a questo tipo di funzioni matematiche. Tra tutti emerge il lavoro (1987) della matematica e fisica belga, I. Daubechies, che propone le wavelet a supporto compatto, considerate la pietra miliare delle applicazioni wavelet moderne. Dopo una trattazione matematica delle wavalet, dei relativi algoritmi e del confronto con il metodo di Fourier, si passano in rassegna le principali applicazioni di esse nei vari campi: compressione delle impronte digitali, compressione delle immagini, medicina, finanza, astonomia, ecc. . . . Si riserva maggiore attenzione ed approfondimento alle applicazioni delle wavelet in campo sonoro, relativamente alla compressione audio, alla rimozione del rumore e alle tecniche di rappresentazione del segnale. In conclusione si accenna ai possibili sviluppi e impieghi delle wavelet nel futuro.

Sonno e funzioni cognitive: ruolo della microstruttura del sonno NREM

STUDY OBJECTIVE: Cyclic Alternating Pattern (CAP) is a fluctuation of the arousal level during NREM sleep and consists of the alternation between two phases: phase A (divided into three subtypes A1, A2, and A3) and phase B. A1 is thought to be generated by the frontal cortex and is characterized by the presence of K complexes or delta bursts; additionally, CAP A1 seems to have a role in the involvement of sleep slow wave activity in cognitive processing. Our hypothesis was that an overall CAP rate would have a negative influence on cognitive performance due to excessive fluctuation of the arousal level during NREM sleep. However, we also predicted that CAP A1 would be positively correlated with cognitive functions, especially those related to frontal lobe functioning. For this reason, the objective of our study was to correlate objective sleep parameters with cognitive behavioral measures in normal healthy adults. METHODS: 8 subjects (4 males; 4 females; mean age 27.75 years, range 2334) were recruited for this study. Two nocturnal polysomnography (night 2 and 3 = N2 and N3) were carried out after a night of adaptation. A series of neuropsychological tests were performed by the subjects in the morning and afternoon of the second day (D2am; D2pm) and in the morning of the third day (D3am). Raw scores from the neuropsychological tests were used as dependent variables in the statistical analysis of the results. RESULTS: We computed a series of partial correlations between sleep microstructure parameters (CAP, A1, A2 and A3 rate) and a number of indices of cognitive functioning. CAP rate was positively correlated with visuospatial working memory (Corsi block test), Trial Making Test Part A (planning and motor sequencing) and the retention of words from the Hopkins Verbal Learning Test (HVLT). Conversely, CAP was negatively correlated with visuospatial fluency (Ruff Figure Fluency Test). CAP A1 were correlated with many of the tests of neuropsychological functioning, such as verbal fluency (as measured by the COWAT), working memory (as measured by the Digit Span – Backward test), and both delay recall and retention of the words from the HVLT. The same parameters were found to be negatively correlated with CAP A2 subtypes. CAP 3 were negatively correlated with the Trial Making Test Parts A and B. DISCUSSION: To our knowledge this is the first study indicating a role of CAP A1 and A2 on behavioral cognitive performance of healthy adults. The results suggest that high rate of CAP A1 might be related to an improvement whereas high rate of CAP A2 to a decline of cognitive functions. Further studies need to be done to better determine the role of the overall CAP rate and CAP A3 on cognitive behavioral performances.