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Time series analysis, based on the idea that female reproductive endocrine physiology can be construed as a nonlinear dynamical system in a chaotic trajectory, is performed to measure the correlation dimension of the menstrual cycle data from subjects in two different age cohorts. The dimension is computed using a method proposed by Judd (Physica D, vol. 56, 1992, pp. 216-228) that does not assume the correlation dimension to be necessarily constant for all appropriate time scales of the system’s strange attractor. Significant time scale differences are found in the behavior of the dimension between the two age cohorts, but at the shortest time scales the correlation dimension converges to the same value, approximately 5.5, in both cases.

The typical conceptualization of the menstrual cycle is that it has a stable period of » 28 days throughout most of the reproductive years but becomes irregular and unstable during the perimenopause, i.e. the years preceding the end of menstruation (menopause) [

The application of concepts from nonlinear dynamics and chaos to physiological problems has now become well established [^{3} data points and none involving the reproductive system [9-11]. The novel approach that we have developed allows us to study the menstrual cycle and hence the reproductive endocrine system using time series analysis with ~10^{4} data points. In this paper, we employ these techniques, using a different methodological framework from our previous study [

The source of our data is the database maintained by the Tremin Research Program on Women’s Health, which contains the results of an ongoing longitudinal study begun in 1934 [_{i}), the other was a formal time series constructed using a novel procedure we have devised. The time series is defined such that f_{n}, the n^{th} term of the time series, is given by the difference between time t_{n} and the time at which the n^{th} menstrual cycle ends. The times t_{n} are equally spaced so that t_{n} = nt where t is the sampling time, giving this scheme the structure of a formal time series. More compactly,

where Dt_{i} is the time length of the i^{th} menstrual cycle in the sequence. Though this interevent time sequence itself (the set of Dt_{i}) does not meet the requirements of a formal time series, there is good reason to believe that it can be employed in the same manner to characterize chaotic trajectories [_{i} or using the f_{n} as input data to the time series analysis) makes different assumptions and approximations, so repeating the computations using both of these methods serves as a cross check on the validity of the results. In both cases, a phase space reconstruction (embedding) of the data must be implemented and repeated for a variety of embedding dimensions. Further details concerning this methodology, and a discussion of the issues that arise therefrom, can be found elsewhere [

A key concept in nonlinear dynamics and chaos is that the trajectory of the system in a multidimensional phase space of the relevant variables is described by a strange attractor in that space. One of the important quantities used to characterize a chaotic strange attractor is its fractal dimension. Experimentally, the quantity that is frequently calculated to obtain an approximation for this fractal dimension is the correlation dimension of the attractor, which can be computed using a time series of data measured for some system variable. This quantity is defined as

where C(e), the correlation sum, represents the number of interpoint distances in the time series data that are smaller than e. These interpoint distances are computed in an embedding space of dimensionality D greater than that of the correlation dimension itself, with vectors in the embedding space found from sets of points in the time series. A major conceptual problem with this formulation is the presence of the limit, because experimentally there are no data in that limit for a time series of finite resolution. In practice, one can instead use estimators of the correlation dimension, such the Takens estimator [

where d is the correlation dimension of the system. In essence, what Judd has shown is that the effect of not being in the asymptotic limit of e ® 0 is to modify the usual relationship with the multiplication by a polynomial in e. This method then allows us to experimentally fit data over a range of e values and opens the possibility to explore any length (i.e. time) scale dependence that may exist for the correlation dimension d itself.

In practice, the probability P(e), which basically corresponds to the correlation sum, is not used directly. Instead, the number of interpoint distances b_{i} within an interval De_{i} = e_{i} - e_{i+1} are counted; such intervals can be referred to as bins. The numbers b_{i} now correspond to the probability p_{i} = P_{i} - P_{i+1} and can be used to find the parameters of Equation (3) for all e smaller than some cutoff value e_{0}. As the cutoff e_{0} decreases, we approach closer to the asymptotic limit e ® 0, but the number of bin values b_{i} with which to fit the parameters decreases and the fits become problematic. The maximum value of e_{0} is determined by the fact that the theory is only valid on the decreasing tail of the p_{i} distribution. The correlation dimension d, in this formulation, is not required to be constant for all e_{0}, so in essence we are able to obtain information about how the dimension d varies with the characteristic length scale of the system’s strange attractor by using different values for the cutoff. Judd has shown that the b_{i} have a multinomial distribution, and he suggests fitting the parameters by a log-likelihood maximization of this probability function, but here we instead find the parameters by fitting the data to the probabilities entailed by Equation (3) directly, because we found this procedure to have some advantages in algorithmic speed and stability.

For each of the four cases described above (i.e. younger women using f_{n}; older women using f_{n}; younger women using Dt_{i}; older women using Dt_{i}), we have computed the correlation dimension for the entire range of accessible e_{0} cutoffs in embedding dimensions ranging from D = 8 to D = 12. (Computations were also performed in lower embedding dimensions, but these are uninteresting since they merely fill the embedding space; results for D ³ 8 are reported since these values of D should be high enough to obtain valid results for d.) We did not use all of the interpoint distances computable for the N data points available, because that offered no means to check reproducibility, it might bias the results due to time correlations, and it would be computationally inefficient. Instead, for every correlation dimension computed, we sampled random pairs from the population N and used the interpoint distances for those pairs to fill the bins. Using just a few percent of the ~N^{2} possible pairs provided ample amounts of data in the bins with minimal bias, and this procedure could then be repeated to find out how the random selection affected the consistency of the results. Every case (i.e. selection of D and e_{0} values) was redone for distributions from three random samplings.

An example of the results for a single sampling at a particular value of D and e_{0} is shown in _{0}. The best-fit parameters (the correlation dimension d and the three polynomial coefficients in Equation (3) were found by minimizing the sum of the squared differences between the theoretical probability p_{i} and the bin value b_{i} for the set of available e_{i} in each case considered. The case illustrated in _{0} = 64.2 days, employing 21 e_{i} and b_{i} values in the fit. (Again, note that fewer values become available as e_{0} decreases, making reliable fits more difficult in the interesting asymptotic regime.) For the case shown in

results of a fitting process like that illustrated in

fore they reached menopause are shown in

_{0} tend toward roughly the same values as in Figures 2 and 3. Although the d values at low e_{0} in _{0} in

The important conclusion that we may infer from these results is that although the dynamics of the reproductive physiological system appear to be chaotic over the entire age range, significant changes occur in the strange attractor governing the system during the later reproductive years of the lifespan. The observed increase in the characteristic time scale involved is intuitively sensible, given the well-known increase in menstrual cycle variability during the perimenopause, but we should note that this is not merely an artifact of raw variability in the input. The standard deviations of the actual values of the f_{n} points comprising the two different time series are approximately equal, so the difference in characteristic time scales seen in the horizontal axes of Figures 2 and 3 reflects some deeper aspect of the dynamical behavior of the system. The specific nature of these age-related changes, however, is not well understood at this time. The different behaviors of the correlation dimension with e_{0} are difficult to interpret without further information, and speculation about the physiological causes underlying these differences is premature at this stage. We are presently engaged in modeling the system to obtain further insight into these questions. Previous mathematical models of the menstrual cycle offer little insight, because they have assumed either periodic solutions or stochastic processes [18-20], which are inconsistent with empirical data and with the results presented here.

A potentially exciting implication of these results is that nonlinear dynamics may open new avenues to categorize and explore the lifespan development of the reproductive system. There is at present no agreement concerning even such basic issues as the definition of perimenopause or markers for when it begins [4-6]. We plan to use the methodology employed here to test some of these proposed definitions. Finally, we would suggest that these results call into question the assumption that menopause is merely the final outcome of a process of senescence, given the presence of a chaotic trajectory in both age cohorts, a presumably lawful process by which the trajectory changes, and the convergence at low e_{0} to similar values of the attractor’s correlation dimension in all cases.

We would like to thank Phyllis Mansfield and the Tremin Research Program on Women’s Health for permission to download and utilize their database of women’s menstrual histories. We would also like to thank Kevin Judd for several personal communications that clarified our understanding of his work.