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...q(s) with s, using log-transformed data. The piecewise or segmented relationship between the mean response µ= E[Y ] and the variable X, for observation i= 1,2,...,n was modeled by adding the following terms in the linear predictor: β+β1Xi+β2(Xi−δ)+ (5) where (Xi−gδ)+= (Xi−gδ)×I (Xi>ψ) and δ is the fitted breakpoint or crossover point and I (·) is an indicator function that is equal to one when the statement is true and is equal to zero when the statement is false (Muggeo, 2003). Piecewise linear models were fitted using the segmented library (Muggeo, 2003) in the R program (R Development Core Team, 2014). If no crossovers were observed, then linear regression would be favored over a piecewise regression. To test this, the segmented library uses Davie’s test to test for a non-constant regression parameter in the linear predictor (Muggeo, 2003). Once the correct regression model is identified, the regression slopes provide the asymptotic estimates for the scaling exponents h(q). If no crossover is present, only one scaling exponent h(q) is obtained for every moment q. If a crossover point is detected, then two scaling exponents h(q) and h(q)are obtained for every moment q. For monofractal self-...

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... between 0 and 1 with a+b≥ 1. An additional advantage is that these functions also extend to negative q values, and thus allow estimation of the multifractal spectrum f (α) for these values as well (Koscielny-Bunde et al., 2006). Using the τ (q) spectra, we estimated the parameters a and b for Eq. (8), allowing us to obtain continuous τ (q) and f (α) spectra from the MMCM fits. To test whether observed long term correlation behaviour was different from a random expectation, we randomized all time series using an amplitude-adjusted Fourier transform algorithm (AAFT) (Schreiber & Schmitz, 1996; Schreiber & Schmitz, 2000). The scaling functions were calculated for all surrogate time series and the corresponding scaling exponents (e.g., β and αDFA for Fourier spectral density and DFA respectively) were calculated (Schreiber & Schmitz, 1996; Schreiber & Schmitz, 2000). Assessing the effect of temperature on multifractality of metabolic rate fluctuations As explained above, regular VO2 time series were obtained under temperature-controlled conditions (see ‘Methods’ sections for details). To assess the effect of Ta on long range and multifractal measures of r(VO2) fluctuations, we calculated the average fluctuatio...

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...mly shuffled series, andmay not be explained as an artifact of stochastic sampling of a linear frequency spectrum. These results show that metabolic rate dynamics in a well studied micro-endotherm are consistent with a highly non-linear feedback control system. Subjects Computational Biology, Ecology, Mathematical Biology, Statistics Keywords Physiological complexity, Metabolic rate fluctuations, Endothermy,Mus musculus, Long-range correlations, Multifractality, Metabolic rate INTRODUCTION Physiologic complexity is ubiquitous in all living organisms (Bassingthwaighte, Liebovitch & West, 1994; Glass, 2001; Goldberger et al., 2002; Burggren & Monticino, 2005). It emerges as the result of interactions among multiple structural units and regulatory feedback loops, all of which function over a wide range of temporal and spatial scales, allowing the organism to respond to the stresses and challenges of everyday life (Bassingthwaighte, Liebovitch & West, 1994; Goldberger et al., 2002). As a consequence of these intricate How to cite this article Labra et al. (2016), Nonlinear temperature effects on multifractal complexity of metabolic rate of mice. PeerJ 4:e2607; DOI 10.7717/peerj.2607 regulation fe...

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...identified below which multifractalitymay be observedwould indicate the onset of a highly nonlinear configuration of control processes acting in the regulation of body temperature. The alternative outcome would be that multifractal long-range correlations also hold true for larger endotherms. This alternative scenario would indicate that a more detailed model analysis is required to account for the processes affecting metabolic rate oscillations. GENERAL CONCLUSION While an increasing number of authors have pointed out the complex nature of physiological processes (Burggren & Monticino, 2005; Spicer & Gaston, 1999), an emerging research question is what are the consequences and implications of physiological complexity for the homeostatic adaptive capability of animals, particularly on a scenario of global climate change. In addition to considering the potential role of organism body size, it is important to determine whether the observed multifractal correlation structure is a general trait of all endotherm taxa, or if it is a characteristic trait of mammals as a lineage. Comparative experimental studies may help to untangle the relative importance of body size and taxonomic inertia in the emergence of ...

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...scielny-Bunde et al., 2006). Hence, we tested whether linear or piecewise linear regressions best fit the scaling relationship of Fq(s) with s, using log-transformed data. The piecewise or segmented relationship between the mean response µ= E[Y ] and the variable X, for observation i= 1,2,...,n was modeled by adding the following terms in the linear predictor: β+β1Xi+β2(Xi−δ)+ (5) where (Xi−gδ)+= (Xi−gδ)×I (Xi>ψ) and δ is the fitted breakpoint or crossover point and I (·) is an indicator function that is equal to one when the statement is true and is equal to zero when the statement is false (Muggeo, 2003). Piecewise linear models were fitted using the segmented library (Muggeo, 2003) in the R program (R Development Core Team, 2014). If no crossovers were observed, then linear regression would be favored over a piecewise regression. To test this, the segmented library uses Davie’s test to test for a non-constant regression parameter in the linear predictor (Muggeo, 2003). Once the correct regression model is identified, the regression slopes provide the asymptotic estimates for the scaling exponents h(q). If no crossover is present, only one scaling exponent h(q) is obtained for every moment q....

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...ctively. A smoothing procedure was applied, which consisted of averaging the spectra for consecutive overlapping segments of 256 data points. Fitted OLS scaling relationships are shown in dotted lines. (F) Detrended fluctuation analyses (DFA) for the two time series shown in (C) and (D). Fluctuation functions for original and shuffled time series in are shown in open and filled circles respectively. Fitted scaling relationships are shown in dashed lines. Note the change in exponent values above s= 100 for the original time series. anti-persistent, with the second scaling exponent αDFA2= 0.39 (Eke et al., 2000; Delignières et al., 2006; Delignières, Torre & Bernard, 2011). As mentioned above, in anti-persistent time series dynamics positive trends are usually followed by negative trends, thus showing a phenomenological signature of control or negative feedback over the rate of change of VO2 (Delignières, Torre & Bernard, 2011). Shuffling the data results in a loss of the observed crossover scaling behaviour, indicating this is property is not a result of randomness in the pattern of fluctuations (Fig. 1F). Thus, we find that r(VO2) fluctuations within the TNZ show non-trivial long-range correlation...

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...riables of healthy systems reveal complex variability associated with long-range (fractal) correlations, along with distinct classes of nonlinear interactions (Goldberger, 1996; Goldberger, Rigney & West, 1990; Goldberger et al., 2002). Over the last two decades, different studies have shown that the break down of this type ofmulti-scale, nonlinear complexity is a characteristic signature of disease and senescence, and as a result, the study of complexity in physiological variables has shown important promise in the efforts to understand and diagnose different pathologies (Costa et al., 2008; Delignières & Torre, 2009; Goldberger et al., 2002; Hausdorff et al., 2001; Hu et al., 2004; Ivanov et al., 2007; Lipsitz, 2004). While different quantitative approaches have been devised to measure the degree of complexity in physiological signals (e.g., Burggren & Monticino, 2005; Costa, Goldberger & Peng, 2002; Feldman & Crutchfield, 1998; Pincus, 1991; Rezek & Roberts, 1998; Richman & Moorman, 2000; Schaefer et al., 2014), most studies examining changes in physiological complexity as a result of pathological alterations have been conducted by examining either the change or loss of longrange correlations of physiol...

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... a and b, which take values between 0 and 1 with a+b≥ 1. An additional advantage is that these functions also extend to negative q values, and thus allow estimation of the multifractal spectrum f (α) for these values as well (Koscielny-Bunde et al., 2006). Using the τ (q) spectra, we estimated the parameters a and b for Eq. (8), allowing us to obtain continuous τ (q) and f (α) spectra from the MMCM fits. To test whether observed long term correlation behaviour was different from a random expectation, we randomized all time series using an amplitude-adjusted Fourier transform algorithm (AAFT) (Schreiber & Schmitz, 1996; Schreiber & Schmitz, 2000). The scaling functions were calculated for all surrogate time series and the corresponding scaling exponents (e.g., β and αDFA for Fourier spectral density and DFA respectively) were calculated (Schreiber & Schmitz, 1996; Schreiber & Schmitz, 2000). Assessing the effect of temperature on multifractality of metabolic rate fluctuations As explained above, regular VO2 time series were obtained under temperature-controlled conditions (see ‘Methods’ sections for details). To assess the effect of Ta on long range and multifractal measures of r(VO2) fluctuations, we calcu...

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...stence or predictability of the profile or cumulated time series (Kantelhardt, 2011). The exponent H may be studied by different methods including rescaled range analysis, fluctuation analysis, and detrended fluctuation analysis (Peng et al., 2002; Kantelhardt, 2011). In particular, Detrended fluctuation analysis (DFA) has been widely employed to reliably detect longrange autocorrelations in non-stationary time series, with a large number of studies using it to report long-range autocorrelations, although a few studies have reported anti-persistent anti correlations (e.g., Bahar et al., 2001; Delignières et al., 2006; Delignières, Torre & Bernard, 2011; Kantelhardt, 2011). The value of the Hurst exponent H may be approximated by the DFA, which calculates the scaling of mean-square fluctuations with time series scale, yielding the scaling exponent α (Feder, 1988; Hurst, 1951; Peng et al., 2002; Kantelhardt, 2011). When DFA scaling relationships are observed, the scaling exponent α≈H is related to the correlation exponent γ by the relationship α= 1−γ /2, with α= 0.5 being the threshold between anti persistence and persistence (Peng et al., 2002; Kantelhardt, 2011). Despite the increased interest to study fr...

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...orrelation properties of VO2 fluctuations under different strategies such as torpor or group huddling, in order to determine whether the degree of multifractality decreases below that observed at 0 ◦C, giving rise either to monofractal scaling or to the loss of fractal autocorrelations. A third point we discuss is the biological significance of these results. As mentioned earlier, whole-bodymetabolic rate is an emergent phenomenon, resulting frommicroscopic interactions with a large number of degrees of freedom and a complex set of opposing feedback mechanisms acting at different time scales (Bozinovic, 1992; Chaui-Berlinck et al., 2005). In this regard, the multifractal nature of metabolic rate highlights the complex and non-linear nature of the multiple feedback loops involved in the maintenance of physiological homeostasis (Chaui-Berlinck et al., 2005; Darveau et al., 2002; Hochachka et Labra et al. (2016), PeerJ, DOI 10.7717/peerj.2607 21/29 al., 2003). The existence of multifractality in metabolic rate fluctuations has several interesting implications, particularly regarding the sensitivity to initial conditions. In general, multifractal dynamics are generated by non-linear recursive process...

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...itions, the state variables of healthy systems reveal complex variability associated with long-range (fractal) correlations, along with distinct classes of nonlinear interactions (Goldberger, 1996; Goldberger, Rigney & West, 1990; Goldberger et al., 2002). Over the last two decades, different studies have shown that the break down of this type ofmulti-scale, nonlinear complexity is a characteristic signature of disease and senescence, and as a result, the study of complexity in physiological variables has shown important promise in the efforts to understand and diagnose different pathologies (Costa et al., 2008; Delignières & Torre, 2009; Goldberger et al., 2002; Hausdorff et al., 2001; Hu et al., 2004; Ivanov et al., 2007; Lipsitz, 2004). While different quantitative approaches have been devised to measure the degree of complexity in physiological signals (e.g., Burggren & Monticino, 2005; Costa, Goldberger & Peng, 2002; Feldman & Crutchfield, 1998; Pincus, 1991; Rezek & Roberts, 1998; Richman & Moorman, 2000; Schaefer et al., 2014), most studies examining changes in physiological complexity as a result of pathological alterations have been conducted by examining either the change or loss of longra...

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...a single statistical distribution despite differences in cardiovascular and respiratory designs, with distribution width scaling inversely with individual body size (Labra, Marquet & Bozinovic, 2007). However, to date, the correlation Labra et al. (2016), PeerJ, DOI 10.7717/peerj.2607 4/29 structure in r(VO2) has not been examined. In a similar fashion to other complex non-linear time series, long-term correlations in r(VO2) would mean that large fluctuations are more likely to be followed by another large oscillation, while a small oscillation is likely to be followed by a small oscillation (Ashkenazy et al., 2003; Bunde & Lennartz, 2012). If this were the case, the expected average value of VO2 would increase, showing a persistent trend. For VO2 to show homeostatic regulation; however, its fluctuations would be expected to show anti-persistence over at least at some scales, so that large r(VO2) increases may be followed by large r(VO2) decreases, ensuring that overall average VO2 values remain under homeostatic control. Thus, the presence of anti-persistent correlations may be expected for r(VO2) time series, particularly if there are strong control feedback loops regulating total energy expenditure i...

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...ustments, all of which carry with them increased energetic costs. Over longer periods of time, these energetic requirements may not be met without resorting to alternative physiological strategies such as torpor (Gordon, 2012). Interestingly, individuals in our measurements did not reach the torpor stage, resorting only to individual huddling within the measurement chamber. Studies on thermoregulatory behavior have shown that small mammals such as lab mice form groups by huddling together as a behavioral thermoregulatory response to temperature challenges (Canals, Rosenmann & Bozinovic, 1997; Canals et al., 1998). Interestingly, this behavioral response behaves as a system with a continuous (second-order) phase transition, with a critical environmental temperature value found between 16 ◦C and 20 ◦C (Canals & Bozinovic, 2011). For low temperatures, individuals spontaneously aggregate, forming groups with a higher fractal dimension and a lower mass-specific metabolic rate. This change in behavior occurs in the same temperature range where we have observed maximal values for the degree of multifractality, supporting the idea that different physiological regimes may occur above and below this temperature...

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...anti-persistent, with positive trends being associated with negative trends (Delignières et al., 2006; Delignières, Torre & Bernard, 2011). Assessing multifractality of metabolic rate To determine the presence of multifractality in the fluctuations of metabolic rate we applied multifractal detrended fluctuation analysis (MF-DFA) (Kantelhardt, 2011; Kantelhardt et al., 2002) to r(VO2) data measured under experimental conditions. This method yields similar results to other existing methods of multifractal analysis in time series (Ivanov et al., 2007; Kantelhardt, 2011; Kantelhardt et al., 2002; Ludescher et al., 2011; Oswiecimka, Kwapien & Drozdz, 2006), but is considerably easier to implement, being based on an extension of DFA (Kantelhardt, 2011; Kantelhardt et al., 2002; Ludescher et al., 2011). Briefly, MF-DFA analyses a profile or accumulated data series zi = ∑ r(VO2,i), for all samples i= 1 to N . The profile is divided into Ns non-overlapping segments of scale s. For every segment ν, the local trend is fit by a polynomial of a given order o, where o= 1, 2 or 3. The resulting variance is then raised to the q/2th power [σ 2(v,s)]q/2 between the local trend and the profile in each segment ν is calcula...

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...senescence, and as a result, the study of complexity in physiological variables has shown important promise in the efforts to understand and diagnose different pathologies (Costa et al., 2008; Delignières & Torre, 2009; Goldberger et al., 2002; Hausdorff et al., 2001; Hu et al., 2004; Ivanov et al., 2007; Lipsitz, 2004). While different quantitative approaches have been devised to measure the degree of complexity in physiological signals (e.g., Burggren & Monticino, 2005; Costa, Goldberger & Peng, 2002; Feldman & Crutchfield, 1998; Pincus, 1991; Rezek & Roberts, 1998; Richman & Moorman, 2000; Schaefer et al., 2014), most studies examining changes in physiological complexity as a result of pathological alterations have been conducted by examining either the change or loss of longrange correlations of physiologic signals (e.g., Costa et al., 2008; Delignières & Torre, 2009; Goldberger et al., 2002; Hausdorff et al., 2001; Hu et al., 2004; Ivanov et al., 2007; Lipsitz, 2004). Long-range correlated time series typically exhibit slowly decaying autocorrelation functions C(s) across different time scales s, which are characterized by power law decay: C(s)∝ s−γ (1) with scaling exponent taking values in the ra...

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...ctrum which may be better described by a large number of scaling exponents rather than by a single scaling exponent value (Kantelhardt, 2011). Thus, onemay distinguish betweenmonofractal andmultifractal Labra et al. (2016), PeerJ, DOI 10.7717/peerj.2607 3/29 signals.Monofractal signals present a long-range correlation structure where a single scaling exponent suffices to describe the correlation scaling. On the other hand, multifractal signals require an infinite spectrum of scaling exponents to describe their correlation structure (Humeau et al., 2009; Ivanov et al., 1999; Kantelhardt, 2011; Suki et al., 2003;West & Scafetta, 2003). Thus, multifractal time series are heterogeneous, showing a given value of the self-affinity exponent only in local ranges of the signal structure, such that their self-affinity exponent varies in time. Hence, multifractal signals may be characterized by a set of local fractal sets that represent the support for each Hurst exponent value (Bassingthwaighte & Raymond, 1994; Ivanov et al., 1999; Kantelhardt, 2011). In this regard, multifractal time series are more complex thanmonofractal ones, and determining whether a given complex physiologic system presents monofractal...

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...tribution despite differences in cardiovascular and respiratory designs, with distribution width scaling inversely with individual body size (Labra, Marquet & Bozinovic, 2007). However, to date, the correlation Labra et al. (2016), PeerJ, DOI 10.7717/peerj.2607 4/29 structure in r(VO2) has not been examined. In a similar fashion to other complex non-linear time series, long-term correlations in r(VO2) would mean that large fluctuations are more likely to be followed by another large oscillation, while a small oscillation is likely to be followed by a small oscillation (Ashkenazy et al., 2003; Bunde & Lennartz, 2012). If this were the case, the expected average value of VO2 would increase, showing a persistent trend. For VO2 to show homeostatic regulation; however, its fluctuations would be expected to show anti-persistence over at least at some scales, so that large r(VO2) increases may be followed by large r(VO2) decreases, ensuring that overall average VO2 values remain under homeostatic control. Thus, the presence of anti-persistent correlations may be expected for r(VO2) time series, particularly if there are strong control feedback loops regulating total energy expenditure in an organism. This sugge...