Complex physiological dynamics have been argued to be a signature of healthy physiological function. Here we test whether the complexity of metabolic rate fluctuations in small endotherms decreases with lower environmental temperatures. To do so, we examine the multifractal temporal scaling properties of the rate of change in oxygen consumption _{2}), in the laboratory mouse _{2})^{2} s), either monofractal or weak multifractal dynamics are observed depending on whether _{a}_{a}_{2})_{2})

Physiologic complexity is ubiquitous in all living organisms (

While different quantitative approaches have been devised to measure the degree of complexity in physiological signals (e.g.,

Long-range correlated processes of this type are often referred to as 1∕^{β} processes or noises, and are characterized by a unique value of the scaling exponent _{i} = ∑_{2}, ^{H} in order to obtain a signal that is statistically self-similar to the original one (^{H}

Despite the increased interest to study fractal or long-range correlated dynamics across many systems, in some highly nonlinear complex systems, the resulting time series presents a scaling autocorrelation function and frequency power spectrum which may be better described by a large number of scaling exponents rather than by a single scaling exponent value (

In endotherms, metabolic rate (_{2}) is a global emergent property that reflects the sum of the energetic costs required to maintain homeostasis allowing body temperature (_{2} is expected to show minimal variation, and hence it is named the thermoneutral zone (_{2} signals is that, even within the _{2} dynamics within the ^{β} scaling exponent in the Fourier frequency spectrum (_{2} shows complex dynamics that are consistent with a dynamical system under non-linear control (_{2}) as a measure of the fluctuations of _{2}. It is defined as _{2}) = log10[_{2}(_{2}(_{2}) time series, or if similar variability is observed throughout. In addition, the calculation of _{2}) allows the de-trending of the data, yielding a much more stationary time series. Examination of _{2}) time series for different species of small mammals, birds and reptiles have shown that this variable has a symmetric power law probability distribution, centered in _{2}) = 0, with a universal triangular shape that does not change across different species (_{2}) has not been examined. In a similar fashion to other complex non-linear time series, long-term correlations in _{2}) 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 (_{2} would increase, showing a persistent trend. For _{2} to show homeostatic regulation; however, its fluctuations would be expected to show anti-persistence over at least at some scales, so that large _{2}) increases may be followed by large _{2}) decreases, ensuring that overall average _{2} values remain under homeostatic control. Thus, the presence of anti-persistent correlations may be expected for _{2}) time series, particularly if there are strong control feedback loops regulating total energy expenditure in an organism. This suggests that examination of the type of autocorrelations present in _{2}) time series, as well as the range of time scales involved may provide insight on the regulation feedback that may be acting on metabolic rate at the level of the organism. To gain some understanding of how this may be so, we examine the relationship between thermal stress and _{2} fluctuations.

In endotherms, _{2} fluctuations are expected to be proportional to the environmental thermal challenges measured as changes in the difference (_{2} and presumably _{2}) as well. In the case of small endotherms, their body size leads to higher challenges associated to the loss of temperature resulting from the large body surface through radiation (_{2} shows any changes as a result of decreasing environmental temperatures In this regard, a working hypothesis is that for _{2}) signal should show a more complex pattern of long-range correlations, resulting in a broader range of autocorrelation scaling exponents, as expected for multifractal signals. These changes should come about as a result of the activation of internal feedback mechanisms to regulate _{2} in animals exposed to _{2} and _{2}) complexity levels may also increase linearly. An alternative outcome may be the gradual decrease and eventual loss of complexity, due to a drop in the efficiency of the thermoregulatory feedback control at lower temperatures (_{2}) measurements n laboratory mice (_{2}) values exhibit either monofractal or multifractal long-term correlations under different environmental temperatures. We do this by testing whether metabolic rate fluctuations show any longrange correlations and, if so, testing whether there may be described either by a single scaling exponent or if multiple scaling exponents are required, using the multifractal detrended fluctuation analysis (MF-DFA) method. We then assess how these quantitative descriptors of longrange correlations vary with environmental temperature, assessing how they change with decreasing values of

Empirical _{2} time series were determined by measuring metabolic rate in wild-type male white laboratory mice. Mice were transferred to the laboratory and housed individually with sawdust bedding. Mice were provided with water and fed with food pellets _{b}) was recorded at the end of each measurement using a Digi-Sense copper-constant thermocouple to evaluate a possible torpor condition at the end of the experiment. In each experimental record _{2} was measured in a computerized open-flow respirometry system (Sable Systems, Las Vegas, Nevada). The metabolic chamber received dried air at a rate of 800 ml/min from mass flow-controllers (Sierra Instruments™, Monterey, California), which ensured adequate mixing in the chamber. Air passed through CO_{2} and H_{2}O absorbent granules of Baralyme™ and Drierite™ respectively before and after passing through the chamber and was monitored every 1 s. This allowed us to obtain time series of oxygen consumption recorded at periodic intervals of _{2}) time series were registered, they were then analysed by calculating the corresponding _{2}) time series.

To determine the presence of long-term correlations in the _{2}) time series, we examined the power spectral density ^{2}, where _{2}) data observations measured under experimental conditions (_{i}) evaluated at frequencies _{i} = ∑_{2}, _{s} non-overlapping segments of scale s. For every segment ^{2}(^{2} between the local trend and the profile in each segment ^{2} is then calculated by:

Examination of how _{2}(_{DFA}, which is often referred to as the global Hurst exponent _{DFA} = 0.5 (^{β} noise, where _{DFA} exhibits values of equal to 1.0. For values of _{DFA} below 0.5, the series is said to be anti-persistent, with positive trends being associated with negative trends (

To determine the presence of multifractality in the fluctuations of metabolic rate we applied multifractal detrended fluctuation analysis (MF-DFA) (_{2}) data measured under experimental conditions. This method yields similar results to other existing methods of multifractal analysis in time series (_{i} = ∑_{2}, _{s} non-overlapping segments of scale ^{2}(^{q∕2} between the local trend and the profile in each segment _{q}(

In general, _{q}(_{q}(^{h(q)}, which allows the estimation of a set of exponents _{q}(_{q}(_{i} − g_{i} − g_{i} >

For monofractal self-affine time series, _{DFA} ≈

For monofractal records,

To assess multifractality in _{2}) time series, we calculated the fluctuation function _{q}(_{2}) time series measured under controlled conditions. Following recent studies, we fit both the

The modified multiplicative cascade model functions (MMCM) allows the description of multifractal spectra with only two parameters,

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) (_{DFA} for Fourier spectral density and DFA respectively) were calculated (

As explained above, regular _{2} time series were obtained under temperature-controlled conditions (see ‘Methods’ sections for details). To assess the effect of _{2}) fluctuations, we calculated the average fluctuation function _{q}(_{max}, which indicates which is the dominant scaling exponent, or the one which shows greater support on average across the time series. We then summarized the various spectra across the experimental temperature treatments, allowing us to examine their response to temperature. To test whether observed multifractal behaviour was different from a random expectation, we randomized all time series using an amplitude-adjusted Fourier transform algorithm (AAFT) (_{q}(

As described in the physiological literature for endotherms, average _{2} values in the lab mouse show a marked thermal response below _{2} values that increase away from basal metabolic rate (_{2} measurements, and observed _{2} time series exhibit irregular non-stationary fluctuations (_{2}) yields a de-trended time series, which reveals abrupt changes in _{2}, with clusters of large fluctuations separated from clusters of smaller fluctuations(_{2}) fluctuations may be associated with the autocorrelation structure of the time series (_{2}) time series do not exhibit obvious trends in the mean, they do show changes in variability through time, and as a result may not meet the statistical assumptions of spectral frequency estimation (_{2}(_{2}) fluctuations (_{DFA1}, indicates the presence of persistent, long-range correlated fluctuations (_{DFA1} = 0.91) (_{DFA2} = 0.39 (_{2} (_{2}) fluctuations within the TNZ show non-trivial long-range correlations, in agreement with previous observations for _{2} in small endotherms (

(A) Average metabolic rates (_{2}) measured at different ambient temperatures. Average values ± standard errors are shown with open circles and error bars. Straight line shows calculated thermal conductance, while the humped curve corresponds to a fitted three parameter Gaussian function (^{2})). (B) Metabolic rate (_{2}) time series shown for a representative individual measured at 30 °C for 1 3/4 h at 1 (s) intervals. Note the irregular, nonstationary dynamics, despite thermo neutral ambient temperature. (C) Observed _{2} fluctuations _{2}(_{2}(_{2}) values, showing the loss of the clustering of fluctuations. (E) Fourier power spectra for time series in (C) and (D) shown by blue and red lines respectively. 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

The figure shows the average _{2}(_{2}) time series, while figures (E)–(H) show average DFA functions calculated for the AAFT shuffled data. All figures show the DFA root-mean-square fluctuation functions obtained using three different orders of detrending polynomials: linear (open circles), quadratic (open squares) and cubic functions (open triangles). Two scaling regimes can be observed across all temperatures and for all polynomial detrending orders. The first scaling regime spans scales between 8 and 100 s, while the second one spans scales from 100 to 1,024 s. All curves have been shifted vertically for clarity. Please note that while only four experimental temperatures are shown, the remaining three temperatures show similar patterns.

When we examined the DFA scaling functions for _{2}) fluctuations both within and outside the _{2}(_{DFA1} < 1.0) (_{DFA2} < 0.5) (_{2}(_{DFA1} values becoming smaller (_{DFA2} values do not show large changes for shuffled data (_{2}) may be interpreted as evidence that two dominant scaling exponents may suffice to account for the correlation structure of the _{2}) time series. An alternative possibility may be that a continuous spectrum of scaling exponents are required in order to account for the observed pattern of long-term correlations in _{2} fluctuations. If the latter were the case, local scaling exponents would show a large number of possible values.

The figure shows the average DFA scaling exponent _{DFA} calculated as a function of experimental temperature. Average scaling exponents corresponding to exponent for raw

To visualize whether a sample _{2}) time series is consistent with a multifractal process, we examined the changes in the value of local DFA scaling exponent _{DFA} through time in the time series shown in _{DFA} as for a moving window placed along the time series. We calculated _{DFA} values using moving windows of 128, 256 and 512 s (_{DFA} exponent values change through time for all window sizes used, forming an irregular pattern (_{DFA} values range broadly between 0.5 and 1.5, as shown by the blue lines in _{DFA} scaling exponent may take values below 0.5, corresponding to anti-persistent fluctuations (_{DFA}, with all exponent values clustering around 0.5, as shown by the red lines in _{2}) fluctuations cannot be characterized by a single scaling exponent, and hence may be multifractal.

The figure shows the value of local DFA scaling exponents _{DFA} for the time series in _{DFA}, which shows a complex structure in time as opposed to the simpler and more restricted changes in the shuffled time series.

Figure shows log–log plots of the average generalized fluctuation function _{q}(_{2}) time series. Figures (A)–(D) show the average _{q}(_{2}) time series measured at 30 °C, 20 °C, 10 °C and 0 °C respectively. Figures (E)–(F) show average _{q}(_{2}) time series measured at 30 °C, 20 °C, 10 °C and 0 °C respectively. Open circles in all figures show the observed _{q}(_{q}(

To determine whether this is the case, we examined whether the MF-DFA formalism can describe _{2} fluctuations across different environmental temperatures. _{q}(_{q}(_{q}(_{2}) time series rejected the hypothesis of the presence of trends, and we observed that the ADF test yields _{2}) time series does not completely remove the crossover scales

When we examined average Hurst (_{2}) fluctuations, as indicated by the dependence of _{2}) time series show different scaling behaviour, similar to what has been observed other complex systems (_{1}(_{2}(_{2}(_{2} fluctuations are characterized by larger scaling exponents _{2}(_{2} fluctuations present smaller _{2}(_{2}) values are balanced by large negative values. On the other hand, for this range of scales, small _{2}) values are persistent, such that small positive increases are followed by similarly valued changes, resulting in gradual positive trends in _{2}. A similar pattern occurs for negative rates of change, which leads to gradual negative trends in _{2}. Shuffling the _{2}) time series results in markedly lower values of

The figure shows the results of the multifractal scaling analysis for all mice studied. The results for the generalized Hurst exponent spectra (_{a} < 10 °C the time scales in the 8 <

Observed differences in the range of ^{2} values for the nonlinear fitting procedure being close to 1.0 in all cases (see

Examination of the average singularity spectra

The figure shows the average widths Δ_{a}. Figures (A)–(C ) show the average widths Δ_{2}) time series with linear, quadratic and cubic polynomial detrending respectively. Figures (D)–(F) show the average widths Δ

On the other hand, when we examine the exponent _{max} of the singularity spectra, we see that the first scaling regime is characterized by much stronger singularities, with _{max} taking values closer to 1.5, being slightly larger for 15 °C and 20 °C (_{max} below 0.5 (see _{max} as a function of temperature for the first scaling regime indicates that the value of _{max} has significant increases with temperature only for the linear and cubic cases (linear de-trending: _{max}, coherent with the persistent, Brownian motion-like values of _{max} does not show significant changes with temperature for any de-trending order (_{max} = 0.9, and shuffled data for the second scaling regime clustering around values close to _{max} = 0.3 (_{max} in these two scaling regimes cannot be attributed to random fluctuations.

The figure shows the average dominant fractal exponent _{max}, for the different the _{a}. Figures (A)–(C ) show the average _{max} values calculated for the raw _{2}) time series using linear, quadratic and cubic polynomial detrending respectively. Figures (D)–(F) show the average _{max} values calculated for the AAFT shuffled time series using linear, quadratic and cubic polynomial detrending respectively.

Physiological systems, and their state variables and signals, have been recognized as complex (_{2}),under different _{2}) time series show two distinct scaling regimes in the fluctuation functions _{q}(^{2} s. Examination of the generalized _{a} is decreased below the

The first aspect we discuss is the robustness of the rather complex long-correlation structure observed for our data. While previous analysis of _{2} have reported long-range persistent 1∕^{β} fluctuations, described by a single dominant monofractal scaling exponent (_{2} fluctuations of different magnitudes are clustered throughout the experimental time series with varying types of long-range correlation, depending on the time scale analyzed. Thus, _{2}) is a multifractal self-affine signal. This suggests that the feedback control mechanisms underlying rapid changes in energy consumption involve strongly non-linear dynamic processes. Both the observed multifractal exponent spectra and the scaling crossover differ from those observed under a random linear transformation in the frequency domain (_{2}) is a robust property of metabolic rate. The existence of this long-range correlation structure indicates the potential for plastic dynamic responses to thermal stress (_{2} and _{2}). As we have shown for data within the _{2} time series may show periods of higher energy consumption interspersed with periods of lower energy use (_{2} changes, which are reflected in the pattern of _{2}) fluctuations. Thus, higher average energy uses (larger mean _{2} values) are associated with less variable values of _{2}), in agreement with observed results for inter-specific scaling of _{2}) across different vertebrate species (_{2}) data using different approaches Fourier power spectra, DFA and MF-DFA reveal that small-scale and larger scales present different scaling relationships. The first two methods agree qualitatively with the pattern shown by the MF-DFA _{q}(_{max} ≈ 0.5 for first de-trending order MF-DFA, or 0.5 > _{max} > 1.0 for 2nd and 3rd de-trending order MF-DFA.

The second aspect we discuss is the possible explanations for the qualitative changes observed in the long-range correlation structure in the vicinity of 15 °C, as well as their potential significance. Metabolic rate changes are central for the control of _{2}, which become nearly twice the BMR. These additional homeostatic requirements may be offset with different thermoregulation strategies that include behavioral, postural and physiological adjustments, 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 (_{2} 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-body metabolic rate is an emergent phenomenon, resulting from microscopic interactions with a large number of degrees of freedom and a complex set of opposing feedback mechanisms acting at different time scales (_{2}) are completely different from simple linear random fluctuations. This opens an interesting scenario regarding the potential use of multifractal properties as either a diagnostic tool or as baseline to determine animal response to environmental stress. This improved characterization may also eventually allow the modeling the dynamics and projection of the likelihood of extreme events or prediction of future behavior (_{2} registered in a small section of the time series under specific environmental conditions (_{2} under the maximum sustainable rate of exercise (i.e., maximal metabolic rate) have been shown to be mostly a function of aerobic capacity of the muscle mass (_{2} fluctuations under conditions of maximum sustainable exercise would also show multifractal long-term correlations as well as power law distributed fluctuations.

In addition to the physiological significance of long-range multifractal correlations of _{2}), a related aspect pertains the taxonomic and systemic generality and significance of our results. It is relevant to discuss whether these observed patterns are expected to hold true for all endothermic species. While previous work on _{2}) has reported a universal probability distribution function across different vertebrate species (^{β} persistent oscillations of _{2}, even within the _{2} are not only long-range correlated, but that have a complex multifractal structure, which indicates that the model of _{2}) oscillations for larger endotherms, with multifractal dynamics being found only in micro-endotherms, regardless of whether they are mammals or birds. Whether a threshold body size may be identified below which multifractality may be observed would 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.

While an increasing number of authors have pointed out the complex nature of physiological processes (

Our results show that the dynamic response of the metabolic machinery in a model mammal species facing thermal challenge do not reduce themselves to the linear variance response expected, evidencing in addition that this response is regulated by environmental history experienced of individual. In this regards, the humped shape observed from the relationship between complexity level of _{2} and decrease of temperature agree with a limit at the physiological capability to control of body temperature. Future work in this area may focus on experimental explorations of the physiological basis of long-term correlations and multifractality of _{2} fluctuations. For example, such work may examine the relative importance of different control mechanisms regulating the rate of oxygen uptake as part of a hierarchical cascade of feedback loops that lead to multifractality.

The data shows individual time series for 18 mice assigned at random to seven different temperature treatments. Given that experimental records differed in total length, data in the matrix have been padded with “nan” codes. For every time series, we provide the original experimental code, individual body mass measured after metabolic rate recording, and environmental temperature at which the metabolic rate signal was registered.

The figure shows the average DFA scaling exponent _{DFA} calculated as a function of experimental temperature for (a) Linear DFA de-trending, (b) quadratic polynomial de-trending and (c) cubic de-trending. Average scaling exponents corresponding to exponent for raw

The figure shows the average generalized fluctuation function _{q}(_{a} = 0°C when data are detrended using (a) a linear function, (b) a quadratic polynomial and (c) a cubic polynomial. The bottom row shows the results for the shuffled time series when the data are detrended using (d) a linear function, (e) a quadratic polynomial and (f) a cubic polynomial. Open circles show the observed _{q}(_{q}(

The figure shows the average generalized fluctuation function _{q}(_{a} = 5°C when data are detrended using (a) a linear function, (b) a quadratic polynomial and (c) a cubic polynomial. The bottom row shows the results for the shuffled time series when the data are detrended using (d) a linear function, (e) a quadratic polynomial and (f) a cubic polynomial. Open circles show the observed _{q}(_{q}(

The figure shows the average generalized fluctuation function _{q}(_{a} = 10°C when data are detrended using (a) a linear function, (b) a quadratic polynomial and (c) a cubic polynomial. The bottom row shows the results for the shuffled time series when the data are detrended using (d) a linear function, (e) a quadratic polynomial and (f) a cubic polynomial. Open circles show the observed _{q}(_{q}(

The figure shows the average generalized fluctuation function _{q}(_{a} = 15°C when data are detrended using (a) a linear function, (b) a quadratic polynomial and (c) a cubic polynomial. The bottom row shows the results for the shuffled time series when the data are detrended using (d) a linear function, (e) a quadratic polynomial and (f) a cubic polynomial. Open circles show the observed _{q}(_{q}(

The figure shows the average generalized fluctuation function _{q}(_{a} = 20°C when data are detrended using (a) a linear function, (b) a quadratic polynomial and (c) a cubic polynomial. The bottom row shows the results for the shuffled time series when the data are detrended using (d) a linear function, (e) a quadratic polynomial and (f) a cubic polynomial. Open circles show the observed _{q}(_{q}(

The figure shows the average generalized fluctuation function _{q}(_{a} = 25°C when data are detrended using (a) a linear function, (b) a quadratic polynomial and (c) a cubic polynomial. The bottom row shows the results for the shuffled time series when the data are detrended using (d) a linear function, (e) a quadratic polynomial and (f) a cubic polynomial. Open circles show the observed _{q}(_{q}(

The figure shows the average generalized fluctuation function _{q}(_{a} = 30°C when data are detrended using (a) a linear function, (b) a quadratic polynomial and (c) a cubic polynomial. The bottom row shows the results for the shuffled time series when the data are detrended using (d) a linear function, (e) a quadratic polynomial and (f) a cubic polynomial. Open circles show the observed _{q}(_{q}(

Top to bottom rows show the results for _{a} = 0°C to _{a} = 30°C respectively. Left hand, central and right hand columns show the results for linear, quadratic and cubic de-trending polynomials respectively. In all figures, black lines show the smoothed conditional mean estimate of the breakpoint, while red lines show the smoothed conditional mean estimates for shuffled data.

The figure shows the average coeficient of determination (^{2}) for the fit of Renyi exponent spectra (^{2} value in raw ^{2} value in AAFT shuffled ^{2} values for ^{2} values for

The figure shows the effects of quadratic de-trending on the multifractal scaling analysis for all mice studied. Left, central and right hand column show the results for the generalized Hurst exponent spectra (_{a} < 10°C the time scales in the 8 <

The figure shows the effects of cubic de-trending on the multifractal scaling analysis for all mice studied. Left, central and right hand column show the results for the generalized Hurst exponent spectra (

We thank F Boher and S Clavijo for their assistance during the development of these experiments. FAL thanks C Huerta and E Labra for their continued support.

The authors declare there are no competing interests.

The following information was supplied relating to ethical approvals (i.e., approving body and any reference numbers):

Care of experimental animals was in accordance with institutional guidelines, and experimental protocols followed were approved by the following Review Boards:

Bioethics committee, Universidad Santo Tomás.

Bioethics committee, Pontificia Universidad Católica de Chile.

Bioethics committe, Chilean National Committee of Science and Technology (CONICYT).

All three review boards issued an approval letter, indicating the endorsement of animal care and experimental protocols.

The following information was supplied regarding data availability:

The raw data has been supplied as a

^{β}processes with applications to the analysis of stride interval time series