### INTRODUCTION

^{1,}

^{2}and are, in common with other functional psychiatric disorders, defined by their clinical features. Finding objective, biological, differences between these disorders could give important insights in relation to pathophysiology and classification. Electroencephalography is one commonly used method for this purpose,

^{3}another is recording of motor activity. Alterations of motor activity are prominent features in several psychiatric disorders, such as ADHD,

^{4}mood disorders and schizophrenia,

^{5,}

^{6}with the most pronounced disturbances seen in the catatonia.

^{7}In a number of studies of patients with psychiatric disorders objective registrations of motor activity have demonstrated differences in activity patterns compared to normal controls and between disorders.

^{8,}

^{9,}

^{10,}

^{11}However, assessment of such changes has only to a limited extent been implemented in routine clinical practice.

^{5,}

^{6}

^{12}Such actions will follow a heavy-tail distribution, and on a log-log plot (probability vs. waiting time) this will appear as a straight line, suggestive of a power-law distribution.

^{13}and Sano et al.

^{14}have found that the distribution of active and inactive periods can be used to characterize the behavioural organization of patients with depression and schizophrenia, and that these distributions show differences compared to healthy controls. Furthermore, Nakamura et al.,

^{15}found that the organization of motor activity of mice displayed a similar pattern, such that the cumulative distribution of resting period durations seemed to obey a power-law distribution, and they also found that mice lacking clock genes differed from normal mice.

^{13}and Sano et al.

^{14}and 2) to look for possible correlations between these findings and the altered variability and complexity parameters found in short- and long-term recordings from the same patients.

### METHODS

### Ethics statement

### Subjects

^{5,}

^{6}The control group consisted of 18 women and 11 men, average age 37.8±13.3 years (mean±SD), range 21-66, medical students (n=5), patients without serious medical or psychiatric symptoms from a primary care office (n=4) and employees from Knappentunet (n=20). None of the control subjects had a history of affective or psychotic symptoms.

### Psychiatric assessment

^{16}for mood disorders. Depressive symptoms were assessed by the use of the Montgomery-Asberg Depression Rating Scale (MADRS).

^{17}

^{18}

### Recording of motor activity

^{19,}

^{20}Total activity counts were recorded for one minute intervals for a continuous period of at least 12 days for all participants. Patients were instructed to remove the actigraphs when taking a bath/shower, and to record these time intervals. The recorded activity counts (0) in these sequences were then replaced with the mean for the whole recording period.

### Mathematical analyses

^{13}and Sano et al.

^{14}we assumed we would find an approximately straight line on a log-log plot of A (minutes) versus P for inactive periods in both patient groups and controls, suggestive of a power law distribution, and a stretched exponential function for the active periods. However, in preliminary analyses of our data, we found that for low values of A (≤35 min for active sequences and ≤20 min for inactive sequences) it seemed that a straight line best fitted the data on log-log plots. We therefore decided to calculate the slope of the line (the scaling exponent of a corresponding power function) that best fitted the data using the least squares method, using these values of A. The slope is negative, but in the table absolute values are given. We also calculated the number of active periods (expressed as % of all the active periods) with a length (A) of 36 min or longer, and the number of inactive periods (% of all the inactive periods) with a length (A) of 21 min or longer.

### RESULTS

^{21}are used to analyse behavioural rhythms.

^{22}The IS quantifies the invariability between the days, the strength of coupling of the rhythm to environmental factors. The IV indicates fragmentation of the rhythm, that is, the transitions between rest and activity. In the patients with schizophrenia, IS is increased and IV is reduced, reflecting a more structured behavioural pattern.

^{5}The most noteworthy findings from Table 3 are the significant correlations of IS to all the six measures used in the present study, with positive correlations to the mean lengths of both active and inactive periods and to the amount of long active (≥36 min) and inactive (≥21 min) periods. In Table 4 are shown correlations between two measures used to characterize short term variability and complexity in recordings of 300 min durations; standard deviation (SD) and sample entropy, a measure of complexity of time series.

^{6}SD is negatively correlated to the length of active periods and to the amount of long active periods (≥36 min) and positively correlated to the scaling exponent of inactive periods. Sample entropy is also correlated to the same measures, but the correlation is positive to the length of active periods and to the amount of long active periods (≥36 min), and negative to the scaling exponent of inactive periods. Sample entropy is also positively correlated to the mean length of inactive periods and the amount of long inactive periods (≥21 min), while SD is negatively (but not significantly) correlated also to these two measures.

### DISCUSSION

^{13}found that when examining the cumulative probability distribution (P) of inactive periods, with lengths (A) from 2 to 200 min, the relation of P to A on a log-log plot followed a straight line, suggestive of a power-law distribution, and the patients had a lower mean value for the scaling exponent compared to controls. This is the opposite of what we found, namely a higher value for the scaling exponent, and we have no clear explanation of this discrepancy. In agreement with Nakamura et al.

^{13}we also found a straight line on a log-log plot of the relation P to A, but only for periods of A≤20 min, so obviously the distribution of the inactive periods, both in depressed patients and controls, is different in these two studies. The mean length of inactive periods of controls in the Nakamura study

^{13}(inactive periods defined as non-zero activity counts) is quite similar to what we found (7.7 min vs. 6.0 min), but the depressed patients in their study has a substantially higher value (15.6 min vs. 5.9 min). However, it is important to note that the periods we have used to calculate the scaling exponents (having lengths ≤20 min) comprise approximately 93% of all the inactive periods, and in studies from a wide range of different fields it not unusual that power-law behaviour only applies to a restricted range of parameter values.

^{23}We have not performed a more rigorous study of whether our distributions follows a strict power-law function,

^{23,}

^{24}but then our main purpose has not been to determine whether this is the case, but to look for differences between diagnostic groups.

^{14}with similar methods, for the cumulative probability distribution of inactive periods, schizophrenic patients were also found to have a significantly lower value for the scaling exponent compared to controls. We did not find any difference between schizophrenic patients and controls in our study, but in agreement with Sano et al.

^{14}we found that the schizophrenic patients had a higher value for the mean length of inactive periods compared to controls. In agreement with this the schizophrenic patients also had a higher value for longest inactive period compared to controls (135% higher). One difference between our patient groups is that we found a lower total activity count in the schizophrenic patients compared to controls, while Sano et al.

^{14}did not find any such difference.

^{13}and Sano et al.

^{14}did not find such a relationship, but instead a stretched exponential function, when using the whole range of lengths. We cannot therefore directly compare our results concerning scaling exponents, but while we found no significant differences between any of the patient groups and controls, we found that the depressed patients had a significantly higher value for the scaling exponent compared to the schizophrenic patients. Nakamura et al.

^{13}did not find any differences between depressed patients and controls concerning parameter values for the stretched exponential functions, but Sano et al.

^{14}found that the schizophrenic patients were different from controls. Concerning active period durations we did not find any difference between schizophrenic patients and controls, in agreement with Sano et al.,

^{14}but the depressed patients in our study had a significantly lower value, compared both to controls and to schizophrenic patients.

^{22}particularly interdaily stability (Table 3). The correlations are positive with mean lengths of active and inactive periods and number of long active and inactive periods, which is intuitively reasonable, since higher values of all these four measures presumably will tend to give a more stable rhythm. The correlations with the scaling exponents are negative, which again is reasonable given the results form Table 2, showing negative correlations between the scaling exponents and the other four measures.

^{6}The correlations between sample entropy and SD, taken from analyses of 300 min sequences in the same group of patients, and scaling exponents, duration of active and inactive periods and number of long active and inactive periods, show moderate correlations, which are not easy to interpret, but probably reflect common mechanisms in these aspects of motor activity regulation. However, it is interesting to note that when sample entropy shows a negative correlation, the correlation with SD is positive, and the other way around.

^{14}found that the parameter β of the stretched exponential function for active periods increased with age in schizophrenic patients. We did not find any correlations with age for any of our measures either in the whole study group or in the group of schizophrenic patients separately, neither did we find any correlation with scores on the Montgomery-Asberg Depression Rating Scale in the depressed patients or on the score on the Brief Psychiatric Rating Scale sin the schizophrenic patients (data not shown). However, we did not have patients with a wide range of severity, mainly moderately depressed outpatients and chronically ill, institutionalized schizophrenic patients. It would clearly have been an advantage to have a more heterogenous group in order to look at a possible relationship with severity. This also makes it difficult to know if our findings can be generalized to other groups of depressed and schizophrenic patients.

^{25}and it is possible that mathematical analyses of actigraph registrations from periods with continuous motor activity

^{6,}

^{26}combined with analyses of active and inactive periods during longer registration periods may give a biological "signature" that can be useful for diagnostic purposes. It is obviously important if clinical impression of altered behavior can be supplanted with objective registrations of motor behavior. It will also be interesting to see if any of these measures can be used to predict effect of treatment, or be altered as the result of treatment. In the present study we used 12 days of registration, in the study of Nakamura et al.

^{13}seven days were used, and in Sano et al.

^{14}five days. However, to make this more applicable to clinical practice it would be useful to see if analyses of active and inactive periods can be reliably applied to shorter to time periods, such as one or two days.

^{27}

^{5}but in the present study there were no significant differences between patients treated with clozapine and the other schizophrenic patients.

^{28}and schizophrenic patients.

^{29}This may have influenced our findings, but on the other hand it would be very difficult to disentangle such effects from other effects on rest and activity rhythms. The controls in our study were employed and working, or students, while the patients were not. This is a potential source of bias that is difficult to evaluate. It would also have been desirable to have larger study groups and longer observation time, but this was not possible to accomplish with the resources we had available.