BACKGROUND
brain size is a traditional metric of intelligence.
species with larger brain-body size ratios are often associated with greater intelligence, better tool-making skills, more social interactions, and other enhanced characteristics  <cit> .
although many correlations between large brains and complex behavioral measures have been posited over the past century, the driving forces and key factors underlying the development of larger brain-body size ratios in mammals and birds are still an open issue  <cit> .
the evolution of large brains in mammals and birds coincided with the development of endothermy  <cit>  at the end of the triassic period   <cit>  during which time the first mammals evolved from therapsids  and has continued in primates and humans, which have the largest brain-to-body weight ratios  <cit> , to the present day  <cit> .
in  <dig>  snell  <cit>  first described the allometric scaling relationship that emphasizes the dependence of brain size on body size.
the most striking observation is that mammals and birds have developed larger brains that are approximately 5– <dig> times heavier than those of the lower-level vertebrates  with similar body mass.
it is surprising that mammals and birds developed larger brains, because, brain tissue is extremely demanding in terms of metabolic consumption, because the neuronal electronic signaling that occurs within the brain may have a per-neuron energy cost as high as ten times that of other cells in the body.
this difference has attracted much attention to the search for links between brain size, body size and metabolic production.
martin  <cit>  proposed that brain size is linked to maternal metabolic turnover via a 3/ <dig> allometric scaling law  <cit> .
this work indicated that the total amount of energy available from the body to supply the brain may constrain the size of the brain; however, species with larger brains do not have basal metabolic rates significantly higher than those in species with smaller brains  <cit> .
there is also no significant difference in the percentage of energy supplied by the body to the brain between endotherms and ectotherms  <cit> .
given these similar metabolic processes and energy supply ratios, why do mammals and birds have larger brains than fish and reptiles of similar weight?
how do mammals and birds meet the energy demands of brains that are 5– <dig> times larger?
although much research has focused on the several-fold differences in the sizes of the brains of primates and those of low-level mammals by considering the improved energy source acquisition styles of primates, such as improvements in food hunting style due to the use of tools, the consumption of cooked food, and differences in metabolic allocation, few studies have focused on the proximate causes of the differences in brain size between lower-level animals  and higher-level animals .
what is the role of temperature and metabolism in promoting or constraining brain size?
what is the relative contribution of neuron and glial densities in increasing brain size?
in the present study, we re-examined the relationships between temperature, metabolism, neuronal density, energy cost of individual neurons, brain size and body size across of all the major types of vertebrates.
here we specifically focus on the critical effect of temperature and growth of glia cells in the development of large brain-body size ratio in mammals and birds and origin of allometric body-brain size scaling in vertebrates.
we then develop a general model to describe the intrinsic relationship among the examined factors.
RESULTS
body temperature, metabolism, external temperature and brain size
we first re-plotted the relationships between brain size and body size across different species .
in each type of species, the relationship between brain size and body size is highly ordered.
brain size increases with body size, and the relationship can be fit using the following allometric equation  <cit> :figure  <dig>
allometric relationships of the brain and body weights of vertebrate animals on a log-log plot.
a. endothermic animals included  <dig> mammal species  <cit>  ,  <dig> bird species  <cit>   and  <dig> insectivore species  <cit>  .
ectothermic animals included  <dig> fish species  <cit>  ,  <dig> reptilian species  <cit>  , and  <dig> amphibian species  <cit>  .
the fitting functions are w =  <dig> p
<dig>   for endotherms and w =  <dig> p
<dig>   for ectotherms.
all these datasets are provided in additional file 1: table s <dig>  b. the relationship of the brains and bodies of randomly selected fish.
data from  <dig> polar-water fish species ,  <dig> temperate species ,  <dig> tropical species  and  <dig> shark species that live in sub-tropical waters  were randomly selected from a fish database   <cit> .
the fitting functions are w =  <dig> p
<dig>   for polar-water fish, w =  <dig> p
<dig>   for temperate-water fish, and w =  <dig> p
<dig>   for tropical-water fish and the sharks.
all these datasets are provided in additional file 2: table s <dig>  c. brain size as a function of temperature for fixed body masses, i.e.,  <dig> grams and  <dig> grams respectively.
the open triangles represent the averaged data from  <dig> fish species with body masses of 5– <dig> grams, and the open squares represent the average of  <dig> species with body masses of 500– <dig> grams from three temperature conditions.
the data for 38°c are from  <dig> endotherms with body masses of 5– <dig> grams  and  <dig> endotherms with body masses of 500– <dig> grams .
the fitting equations are log10
w =  <dig> p
<dig>  for p =  <dig> g  and log10
w =  <dig> p
<dig>  for  <dig> g .
d. the brain size ratios as a function of body sizes: brainendotherm/brainectotherm , braintropic fish/brainpolar fish  and brainprimate/brainmammal .
<dig> w=cpα
where exponent α is the allometric slope, c is a constant, and w and p are brain and body weight, respectively.
analyses have shown that c =  <dig>  and α =  <dig>  for endotherms , and c =  <dig>  and α =  <dig>  for ectotherms .
the parameters c and α vary across species .
a closer examination of the distributions of body mass versus brain mass reveals that the warm-blooded, endothermic animals with relatively constant body temperatures of approximately 8°c  form one distinct distribution, and cold-blooded ectothermic animals with body temperatures that vary with the environment and average approximately 16°c  form a clear second distribution.
warm-blooded species have brains that are 4– <dig> times larger than those of ectothermic animals.
the fact that the most apparent difference between the two groups is temperature suggests that it is a key factor that correlates with brain size.
if temperature is important, we might predict that cold-blooded species living in disparate environmental temperatures might also have different brain sizes.
to examine whether the extrinsic, i.e., environmental, temperature also explains the variation in brain size, we used fish brain and body-size data from locations of different temperatures, obtained from a fish database , and plotted the body-brain mass relationships.
the fish and environments examined included the following: 1)  <dig> fish species living in polar conditions in which the average temperature was ~1°c, 2)  <dig> fish species living in temperate conditions with an average water temperature of ~15°c, 3)  <dig> species living in tropical conditions with an average water temperature of 25°c, and 4)  <dig> shark species living in sub-tropical waters that maintain their body temperatures at 20-30°c via steady contractile activity of muscles  <cit>  .
surprisingly, we found that the fish brain-to-body mass ratio data fell into  <dig> distinct groups .
fish living in warm tropical water had brains that were 2– <dig> times larger than those of the temperate-water fish and 3– <dig> times larger than those of the fish living in cold polar water.
the fit equation for the fish species living in tropical conditions had an allometric slope of  <dig>  and a constant of  <dig>  , the equation for fish living in the temperate waters had an allometric slope of  <dig>  and a constant of  <dig>  constant , and the fish living in the polar waters had the shallowest allometric slope  and a constant of  <dig> .
figure 1c directly quantifies the allometric relationship between the environmental temperature  of the fish and their average brain mass for two body masses .
the fit equations indicate that fish brain mass more than doubles for each 10°c increase in temperature.
as a comparison, the average mammalian brain sizes in animals weighing 10 g and  <dig>  g are plotted in this figure .
this figure shows a nice fit between body mass and brain temperature for three points, but not for the largest brain sizes and highest temperatures.
the data from the larger brain sizes/higher temperatures falls above the fit lines, indicating that temperature alone may not be the sole contributor to the development of larger brains.
other factors may contribute to the development of larger brains in mammals.
together, these data indicate that temperature may be a critical factor in the marked changes in brain size during evolution in different climate conditions.
specifically, for a given body mass, warmer living conditions should result in larger brains.
in addition to this temperature-driven brain enlargement, the difference in brain sizes of endotherms and ectotherms increases as body size increases .
for small body sizes , the brains of endotherms are atleast  <dig> times larger than those of ectotherms .
similarly, tropical-water fish have brains that are at least  <dig> times larger than those of cold-water fish .
in contrast to these effects in animals with small body sizes, for animals with body sizes over  <dig> kilograms, the brains of endotherms are approximately  <dig> to  <dig> times larger than those of ectotherms, and tropical-water fish have brains that are more than  <dig> to  <dig> times larger than those of cold-water fish.
in comparison, the brains of primates are, relatively consistently, 2– <dig> times larger than those of other mammals, and this difference is less dependent on body size differences .
the emergence of intrinsically constant warm body temperatures together with an increasing brain size, with increasing body mass may be related to a temperature-dependent regulation of body metabolism.
it is possible, that changes in body or environmental temperature affects the metabolic rate and consequently affected the evolution of large brains.
this effect may depend on body mass.
the basal metabolic rate of the body  has been found to be governed primarily by two interacting processes  <cit> : the boltzmann-arrhenius factor, which describes the temperature dependence of biochemical processes  <cit> , and an allometric relation that describes how biological rate processes scale with body size for all vertebrates  <cit> .
<dig> mbody=c1pγe−evkt
where p is body weight, c <dig> is a constant , γ is the allometric scaling factor, t is the absolute temperature, ev is the activation energy   <cit> ), and k is boltzmann’s constant .
earlier analyses have revealed that c <dig> =  <dig>  and γ =  <dig>  for endotherms , measured and scaled at 38°c, and c <dig> =  <dig>  and γ =  <dig>  for ectotherms when normalized to a temperature of 16°c ; these findings are consistent with those of recent reports  <cit> .
when the metabolic rates of the endotherms are normalized to a common temperature of 16°c using eq.
, the metabolic rates of the endotherms  remain several times higher than those of ectotherms , particularly for animals with small body sizes .
in large animals, the metabolic rates of the endotherms and ectotherms were similar .
these findings suggest that at similar metabolic rates, large endotherms should support brain masses several times greater than those of ectotherms , given that brain size differences increase with body size among endotherms and ectotherms .figure  <dig>
temperature-regulated basal metabolic rate   and brain size .
a. allometric relationships between basal metabolic rate   and body mass  for the endotherms   and the ectotherms  .
the bmr versus body mass data were well fit by the equation  <dig> × <dig>   for the endotherms and by  <dig> × <dig>   for the ectotherms.
the datasets are given in additional file 3: table s <dig>  all the data plotted here for ectotherms were normalized to the temperature of 16°c, and the data for the endotherms were normalized to 38°c based on eq.
to minimize the errors induced by measuring temperatures.
b. when the metabolic rate data for the endotherms were normalized to 16°c, they merged into the group of ectothermic metabolic data.
however, there some difference between these groups remained, particularly for the small animals.
c. the bmr ratios as a function of body size for the endotherms to ectotherms was bmrendotherm/bmrectotherm   and bmrendotherm/bmrectotherm  .
d. the brain masses  as a function of body bmrs  for the ectotherms  and endotherms  were based on the fit equations in figures 1a and 2a.
these analyses suggest that temperature-dependent body metabolism alone cannot explain the differences in brain size among endotherms than among ectotherms.
these results also indicate that other factors may play a role in explaining the differences in brain size between warm blooded endothermic animals and cold blooded exothermic animals.
it is possible that neural, glial density or the energy demands on these cells are dependent on body temperature.
a general model linking brain and body sizes to metabolism and temperature
in this section, we derive a general model to quantify the relationships between brain size, temperature-dependent metabolism, individual neuronal metabolism and body size to account for the origin of eq.
.
we began by considering the relationship between brain mass and the number of neurons and supporting cells in the brain.
brain mass  is determined by the total number of neurons and supporting cells; hence w = nw+ n1w <dig>  where n and w are the number and weight of brain neurons, respectively, and n <dig> and w <dig> are the corresponding quantities for the supporting cells.
if we assume that the weight and number of the supporting cells are proportional to those of the neurons, i.e., if n1= k1n, and w1= k2w, then the relationship between brain mass and neurons simplifies to w = c2n, where c <dig> = w. this theoretical picture in which w is proportional to n is supported by a recent analysis of mammalian and primate brains .
analyses of these data yielded.figure  <dig>
scaling relationship between the total number of neurons or supporting cells in the brain and brain mass.
a brain mass  compared with the total number of neurons in the brain for  <dig> low-level mammal species  <cit>  ,  <dig> primates  <cit>   and  <dig> ectotherms  <cit> .
the fit function  gives w =  <dig>  n
<dig>   for the endotherms  for the endotherms.
b brain mass in grams compared with the total number of non-neuron cells in the brain for  <dig> low-level mammal species  <cit>  ,  <dig> primates  <cit>   and  <dig> ectotherms  <cit> .
the fit functions  have pendotherm =  <dig>  n
<dig>   for the endotherms  as reported in recent studies  <cit> .
the numbers of supporting non-neuronal cells in endotherms are proportional to the numbers of neurons and are also strongly correlated with brain mass .
the supporting cells play an important role in providing mechanical and trophic support to neurons.
additionally, data from several ectothermic animals  <cit>  indicate that that c2 =  <dig>  and β =  <dig>  .
neurons are the brain’s major energy consumer, and most  of the brain’s energy is consumed by signaling processes within neurons  <cit> .
brains are energetically costly.
experimental investigations have shown that the brain’s metabolic consumption accounts for 1-10% of the total body metabolic rate  in most animal species, although brain mass accounts for no more than 1% of an animal’s body weight  <cit> .
hence, the total energy required to support the brain can be quantified as  <dig> mbrain=φmbody
where mbrain is the brain’s basal metabolic rate, mbody is the body’s basal metabolic rate as defined in eq.
, and φ is a variable that varies with species.
re-examination of the data reported by mink et al.
suggested that φ can be described by the following equation for both ectotherms and endotherms  <dig> φ=c3pζ
where p is the body mass, c3 =  <dig>  and ζ = − <dig>  for ectotherms, and c3 =  <dig>  and ζ = − <dig>  for endotherms .figure  <dig>
the dependence of brain metabolism on body size and invertebrate species.
a) the ratios of brain metabolism to resting body metabolism as a function of body mass  for  <dig> ectotherms  <cit>  were fit by y =  <dig> ×- <dig>  , and for  <dig> endotherms  <cit>  which were fit by y =  <dig> ×- <dig>  , b).
c. the average metabolic cost of an individual neuron for endotherms was  <dig>  ± <dig>  × 10− <dig> watts, and for ectotherms this cost was  <dig>  ± <dig>  × 10− <dig> watts  <cit> .
the datasets used for this figure are given in additional file 5: table s <dig>
the total amount of energy supplied to the brain  should be equal to the total energy consumption of the brain’s neurons.
note that here we make a simplifying assumption, by considering the metabolic costs of neurons only because neurons consume most of the energy while the supporting cells account for 10-20% of total brain energy.
if the total number of neurons in the brain is n and the metabolic consumption of each individual neuron is mneuron, we can describe the relationship as follows:  <dig> n=mbrainmneuron=ϕmbodymneuron
where the total number of neurons in the brain  contributes directly to the brain’s mass .
by inserting eqs.
- into eq.
, we obtained a general model of the relationships between brain mass , body mass  and temperature :  <dig> w=λpαe−tvt
where λ=c2c1c3mneuronβ, α = βγ + βξ, and tv=βevk, called effective temperature.
equation  links body temperature, body mass, brain mass and neuron metabolic rate , provides a general theoretical framework for quantifying changes in brain size during evolution, and provides an origin of the formula of eq.
that was derived from experimental data.
the temperature range used for eq.
is 0 ~ 42°c, corresponds to what is normally observed in polar fish to the body temperature of birds.
the equations are not valid for temperatures above or below these values, because all the scaling parameters have been derived from animal data acquired within this temperature range.
equation  links brain mass with the metabolic consumption rate of individual neurons .
recent studies suggest that the energy cost per neuron  in the brain is stable and nearly invariant across mammal species  <cit> .
calculations based on the available data revealed that mneuron is, on average,  <dig>  × 10− <dig> watts for endotherms and  <dig>  × 10− <dig> watts for ectotherms .
we inserted empirically determined parameters  into eq.
to determine how well this data-based theoretical model predicts actual animal brain size.
figure 5a plots the results of linerar regression analysis between log <dig>  and log <dig>  for endotherms  -- mammals, birds, primates and dolphins, primarily -- at 38°c and endothermic animals at 16°c .
the predicted parameters: for endotherms were λ =  <dig>  and α =  <dig> , and the analysis results were a correlation coefficient cc =  <dig> , an adjusted-r <dig> =  <dig> , and a mean squared error mse =  <dig> , p-value p <1×10−5).
for endothermic animals at 16°c, the predicted parameters were: λ =  <dig>  and α =  <dig> ) and the multiple linear regression analysis results were cc =  <dig> , adjusted-r <dig> =  <dig> , and a mse =  <dig>  .
in this analysis body mass p was the regressor variable and temperature was a constant.
the model predictions closely matched the real brain data, with data-derived parameters of c <dig> =  <dig> , γ =  <dig> , c <dig> =  <dig> , β =  <dig> , c <dig> =  <dig> , and ζ = − <dig>  for the endotherm-based data analysis that resulted in the equation w= <dig> p <dig> e <dig> evk1t+ <dig> −138+ <dig>   based on eq.
and c <dig> =  <dig> , γ =  <dig> , c <dig> =  <dig> , β =  <dig> , c <dig> =  <dig>  and ζ = − <dig>  for ectotherms, which led to the following equation: w= <dig> p <dig> e <dig> evk1t+ <dig> −116+ <dig> .
the difference in brain size of endotherms compared to ectotherm  also increased from  <dig>  to 36-fold with increasing body masses .
if we constrain all parameters in ectotherms, in the same fashion as we have done for endotherms with the exception of the temperature difference, the brain size prediction for the ectotherms is only  <dig> -fold lower than that for the endotherms .
furthermore the scaling parameter of α =  <dig>  from eq.
is close to the measured value of  <dig>  for endotherms, and α =  <dig>  is close to the measured value of  <dig>  for ectotherms .
thus, temperature-regulated metabolism can only partially explain the 4-40-fold brain size difference shown in figure 1a, and other factors, such as increases in non-neuron cell  numbers , may be essential factors in explaining brain size differences.
thus, contributions from metabolism, neural density and temperature together explain most of the differences in brain mass between endotherms and ectotherms shown in figure 1a.figure  <dig>
model prediction for allometric scaling between the brain and body weights of vertebrate animals on a log-log plot.
a. for the brain-body mass data listed in figure 1a, the model predictions based on eq.
for endotherms in 38°c  and for ectotherms in 16°c  are displayed.
multiple linear regression analysis  for the endothermic prediction yielded adjusted-r <dig> =  <dig> , p < 10− <dig>  and for ectothermic prediction, these results were adjusted- r <dig> =  <dig> , p < 10− <dig>  insert: ratios of predicted endothermic brain sizes to ectothermic brain sizes as a function of body mass .
for comparison, the ratios of predicted endothermic brain sizes at 38°c to the predicted ectothermic brain sizes at 16°c using all the endothermic parameters are presented .
b. for the fish brain-body mass data listed in figure 1b, the model predictions based on eq.
for fish in 1°c , 15°c  and 25°c  conditions are shown.
least-squares regression analysis of the model predictions and the actual data revealed that, for the tropical-water fish, n =  <dig>  cc =  <dig> , adjusted- r <dig> =  <dig> , p < 10− <dig>  for the temperate-water fish n =  <dig>  cc =  <dig> , adjusted- r <dig> =  <dig> , p < 10− <dig>  and for the polar-water fish n =  <dig>  cc =  <dig> , adjusted- r <dig> =  <dig> , p < 10− <dig>  c. for the brain-body mass data of the insectivores listed in figure 1a, the model predictions based on eq.
and the endothermic parameters at the temperature of 31°c  produced regression analysis results of cc =  <dig> , adjusted- r <dig> =  <dig> , p < 1×10− <dig>  d. for the fish data in figure 1c, the model predictions for the relationship between brain size and temperature for the two body sizes of  <dig> grams and  <dig> grams based on eq.
are given in black lines.
multiple linear regression analysis was used here  between log <dig> and log <dig>
note that model predictions for endothermic brain sizes are generally 2– <dig> times lower than actual brain masses of primates and dolphins, suggesting there other evolutionary factors mat contribute to the larger brain size in these species, as has been indicated in recent reports  <cit> .
the model in the present study does not make any attempt to capture these effects.
the model predictions for tropical-water fish, temperate-water fish, and cold-water fish are shown in figure 5b, which predicts that brain size at 25°c should be 6-fold higher than that in fish at 1°c and that at 15°c brain size should be 3-fold higher than at 1°c.
the model used actual data-derived parameters of c1 =  <dig> , γ =  <dig> , c2 =  <dig> , β =  <dig> , c3 =  <dig>  and ζ = − <dig> , which for fish results in the equation w= <dig> p <dig> e <dig> evk1t+ <dig> −116+ <dig>  based on eq.
.
note, data for fish brain, glial density data are currently not available, so the fitting is imperfect, and only includes the temperature- related differences.
the model prediction for insectivores  is perfectly aligned with the actual data with predicted parameters of λ =  <dig>  and α =  <dig>  .
figure 5d quantifies the allometric relationship between the temperatures that fish live at, and their average brain mass for two body masses .
the equations for the fish were given by w= <dig> p <dig> e <dig> evk1t+ <dig> −116+ <dig>  and for mammals were given by w= <dig> p <dig> e <dig> evk1t+ <dig> −138+ <dig> .
at a given body weight, as temperature increases, brain mass of fish increases, but the increase of brain mass with temperature is larger in mammals.
this relationship is well captured by the model predictions , with multiple linear regression analysis -- here temperature t is the regressor variable -- between log <dig>  and log <dig> for  <dig> g: cc =  <dig> , adjusted- r <dig> =  <dig> , mse =  <dig> , p <  <dig> ; and for  <dig> g: cc =  <dig> , adjusted- r <dig> =  <dig> , mse =  <dig> , p <  <dig> .
by applying the general model described by eq.
to all the major types of vertebrates , we found that the actual brain masses and model predictions were aligned well in a log-log plot .
the multiple linear regression results were cc =  <dig> , adjusted- r <dig> =  <dig> , mse =  <dig> , p < 10− <dig> for endotherms and cc =  <dig> , adjusted- r <dig> =  <dig> , mse =  <dig> , p < 10− <dig> for ectotherms.
figure 6b shows the residuals of the log brain sizes i.e., log10-log <dig>  for both the endotherms and ectotherms as a function of log body size with slopes close to  <dig>  indicating that the model performed well in predicting actual brain mass.
figure 6c shows that actual brain masses of fishes and model predictions were also aligned in a log-log plot .
the model predictions were underestimates compared to the actual brain masses for the tropical fish with larger body sizes, and were overestimates compared to the actual brain masses for smaller tropical fish.
the model underestimated the actual brain masses for fish from cold – i.e.
polar -- water , but for the temperate-water fish the model matched the real data well .
note that the model is missing parameters, it is worth noting that the model relies on actual data, and that for both tropical-water and polar-water fish there is a dearth of data for neural and supporting cell densities.
with adequate samples of all important measures, i.e.
the kinds of data we have for mammals and primates, shown in figure  <dig>  the model predictions would be improved.
figure 6d shows the residuals of the log brain sizes, i.e., log10-log <dig>  for three groups of fishes as a function of log body sizes.
this has a flat relationship with slopes close to  <dig>  indicating that the model performed well in predicting actual fish brain mass.figure  <dig>
regression residual analysis of model prediction and actual brain mass.
a. least-squares regression of the log <dig> versus log <dig> of the endotherms  , and ectotherms  .
b. least-squares regression of the residuals of the log <dig> = log10- log <dig> versus the log <dig> for the endotherms   and the ectotherms  .
c. least-squares regression of the log <dig> versus log <dig> for the tropical-water fish , temperate-water fish , and polar-water fish .
d. least-squares regression of the residuals of the log <dig> versus log <dig> for the tropical-water fish , temperate-water fish , and polar-water fish .
finally, we used eq.
to predict brain sizes of all the types of vertebrates  and  <dig> dinosaurs ).
top panel in figure 7a shows that actual brain masses, and the model predictions tracked each other well in a log-log plot .
compared to the ten- to hundred-fold variance in brain size for the same body mass shown in figure 1a, the model predictions and actual brain sizes exhibited a rather small variance, and only ~ 5-fold difference .
the predicted brain mass from the fitting equations in figure  <dig>  and the actual brain mass also tracked each other well, but the variance was a little bit higher, roughly  <dig> fold than in the analysis used in top panel.
the multiple linear regression analysis of these data show that: cc =  <dig> , adjusted- r <dig> =  <dig> ; mse =  <dig> ; p <  <dig> × 10− <dig>  note that the adjusted- r <dig> =  <dig>  is slightly less than the value  <dig>  in top panel using eq.
.
this suggests that eq.
fits the data better than do the fitting equations from figure 1a because that both body mass and temperature play important roles in determining the brain mass.figure  <dig>
matching analysis between model predictions and actual brain mass for all the major type of vertebrates.
a. the model ) predicted brain mass versus real brain mass for all the major types of vertebrates.
the correlation coefficient cc and mean squared error mse between model prediction and real brain mass are listed here: cc =  <dig>  and mse =  <dig>  for cold polar fishes; cc =  <dig>  and mse =  <dig>  for temperate fishes; cc =  <dig>  and mse =  <dig>  for tropical fishes; cc =  <dig>  and mse =  <dig>  for reptiles; cc =  <dig>  and mse =  <dig>  for amphibians; cc =  <dig>  and mse =  <dig>  for dinosaurs; cc =  <dig>  and mse =  <dig>  for mesozoic mammals; cc =  <dig>  and mse =  <dig>  for archaic mammals; cc =  <dig>  and mse =  <dig>  for insectivores; cc =  <dig>  and mse =  <dig>  for mammals; cc =  <dig>  and mse =  <dig>  for birds; cc =  <dig>  and mse =  <dig>  for primates; cc =  <dig>  and mse =  <dig>  for dolphins.
b. the ratios defined as / for the  <dig> animal species.
for most of animal species , model predictions matched real brain data well, with cc >  <dig> , mse =  <dig> , p < 10−5) with ratios of / close to  <dig>  while there are larger mean square errors for primates  and dolphins , and dinosaurs  with ratios of / either much less or much more than  <dig> .
the regression analysis gives cc =  <dig> , adjusted- r <dig> =  <dig> , mse =  <dig> , p <  <dig> × 10− <dig> for archiac mammals, and cc =  <dig> , adjusted- r <dig> =  <dig> , mse =  <dig> , p <  <dig> × 10− <dig> for mesozoic mammals.
the predictions of the model could be improved by more accurate measurements of the body temperature, body metabolism, neuronal metabolism and neuronal/glial densities of each group of species.
discussion and
CONCLUSIONS
there is increased interest in examining the key factors facilitating enlargement of the brain during evolution.
given that the energy cost of the brain’s neurons is more than ten times that of the cells of the body, the factors that regulate the brain’s energy supply and energy cost should be given thorough consideration.
in this study, we re-examined data on brain size, body size and metabolic rates for all the major types of vertebrates, developed a theoretical framework based on examination of the real data, and identified several critical factors that may trigger the dramatic enlargement of the brains of birds and mammals as compared with fish, reptiles and amphibians.
the most critical factor was temperature.
when the endothermic animals evolved from ectothermic animals, two unique features were bound together in endotherms: larger brains and higher body temperatures .
the warm temperature was critical in speeding the biochemical processes and making marked metabolic differences.
our analysis revealed that in small animals the basal metabolic rate of endotherms is  <dig> times greater than that of ectotherms, and in large animals it is ~  <dig> times greate .
however, the brain sizes of small endotherms are only approximately  <dig> times greater than that of small ectotherms but the brains of large endotherms are more than  <dig> times greater than the brains of large ectotherms.
clearly, in large animals, the same basal metabolic rate of endothermic animals should support larger brain tissue in ectothermic animals, than it actually does , suggesting that temperature was only one of the key factors in mediating the enlargement of the brains of endotherms.
note a similar analysis published recently points out that temperature correlates with brain size  <cit> .
other factors included in our analysis are brain metabolism and number of supporting glial neurons.
analysis of the brain metabolism data showed that that the average metabolic cost of individual neurons in ectothermic brains is ~  <dig>  × 10− <dig> watts, which is nearly  <dig> times lower than the average metabolic cost of endothermic brains .
the lower energy cost of individual neurons could be attributed to the effect of colder temperature on metabolism and the occurrence of fewer synapses in ectothermic brains  <cit> , or the lower energy cost of individual neurons could be associated to the smaller number of glia in ectotherms  <cit> .
glia are the major non-neuronal supporting cells that maintain homeostasis, form myelin, and provide mechanical and tropic support to neurons.
in the vertebrate cns, most glia originate from portions of the developing neural tube—the exception being olfactory ensheathing glia  <cit> —and the energy cost of glia is generally only a few percent of that of neurons  <cit> .
recent data suggest that the evolution of the nervous system was accompanied by increases in the number and size of glia  <cit> .
in vertebrates, the glia-to-neuron ratio in the cortex increases with the size of the brain  <cit> .
figure  <dig> and eq.
show that brain mass increases much faster in endothermic brains than in ectothermic brains as the number of neurons increase, which is also suggestive of increases in the numbers of non-neuronal supporting cells.
glia contribute to the faster propagation of nerve signals and long-range communications, particularly via the evolution of myelinating glia in developed endotherms  <cit> .
these energetically inexpensive glial cells surround neurons and occupy a relatively large portion of the brain  <cit> , while also providing energy support to neurons and aiding efficient communication between cortical areas  <cit> .
thus, these cells may be one of the critical factors that promoted brain enlargement.
the evolution of the large brains of endotherms is associated with multiple factors.
to quantify the contribution of each factor to the brains of mammals and birds that are tens of times larger than those of fish, reptiles and amphibians, we built up a general theoretical model to examine the origin of body-brain size allometric scaling relationship by linking the key factors of temperature-dependent metabolism, neuron energetics, neuronal density and body size.
this basic framework suggests that temperature and the development of supporting non-neuronal cells may have been the critical factors that correlated with, or might even be causal, for the enlargement of the brains of endotherms .
this framework also predicts that the brains of cold-blooded animals living in warmer conditions should be several times larger than those of cold-blooded animals that live in colder conditions.
this prediction was confirmed by the analysis of the brain sizes of fish living in various temperatures .
indeed, owing to the absence of measurements of neuronal density  for the brains of fish that live in different temperatures, the model prediction in figure 5b did not capture the differences in the slopes between the tropical and polar fish that are shown in figure 1b and d. however, a prediction from the endothermic brain would be that tropical fishes have larger numbers of glia in their brains than do the fish that live in cold conditions.
these predictions can be tested by experimental investigations.

 brain size is a traditional metric of intelligence. species with larger brain-body size ratios are often associated with greater intelligence, better tool-making skills, more social interactions, and other enhanced characteristics. although many correlations between large brains and complex behavioral measures have been posited over the past century, the driving forces and key factors underlying the development of larger brain-body size ratios in mammals and birds are still an open issue. the evolution of large brains in mammals and birds coincided with the development of endothermy at the end of the triassic period during which time the first mammals evolved from therapsids and has continued in primates and humans, which have the largest brain-to-body weight ratios, to the present day. in snell first described the allometric scaling relationship that emphasizes the dependence of brain size on body size. the most striking observation is that mammals and birds have developed larger brains that are approximately 5– times heavier than those of the lower-level vertebrates with similar body mass. it is surprising that mammals and birds developed larger brains, because, brain tissue is extremely demanding in terms of metabolic consumption, because the neuronal electronic signaling that occurs within the brain may have a per-neuron energy cost as high as ten times that of other cells in the body. this difference has attracted much attention to the search for links between brain size, body size and metabolic production. martin proposed that brain size is linked to maternal metabolic turnover via a 3/ allometric scaling law. this work indicated that the total amount of energy available from the body to supply the brain may constrain the size of the brain; however, species with larger brains do not have basal metabolic rates significantly higher than those in species with smaller brains. there is also no significant difference in the percentage of energy supplied by the body to the brain between endotherms and ectotherms. given these similar metabolic processes and energy supply ratios, why do mammals and birds have larger brains than fish and reptiles of similar weight? how do mammals and birds meet the energy demands of brains that are 5– times larger? although much research has focused on the several-fold differences in the sizes of the brains of primates and those of low-level mammals by considering the improved energy source acquisition styles of primates, such as improvements in food hunting style due to the use of tools, the consumption of cooked food, and differences in metabolic allocation, few studies have focused on the proximate causes of the differences in brain size between lower-level animals and higher-level animals. what is the role of temperature and metabolism in promoting or constraining brain size? what is the relative contribution of neuron and glial densities in increasing brain size? in the present study, we re-examined the relationships between temperature, metabolism, neuronal density, energy cost of individual neurons, brain size and body size across of all the major types of vertebrates. here we specifically focus on the critical effect of temperature and growth of glia cells in the development of large brain-body size ratio in mammals and birds and origin of allometric body-brain size scaling in vertebrates. we then develop a general model to describe the intrinsic relationship among the examined factors.  body temperature, metabolism, external temperature and brain size we first re-plotted the relationships between brain size and body size across different species. in each type of species, the relationship between brain size and body size is highly ordered. brain size increases with body size, and the relationship can be fit using the following allometric equation:figure allometric relationships of the brain and body weights of vertebrate animals on a log-log plot. a. endothermic animals included mammal species, bird species and insectivore species. ectothermic animals included fish species, reptilian species, and amphibian species. the fitting functions are w = p for endotherms and w = p for ectotherms. all these datasets are provided in additional file 1: table s b. the relationship of the brains and bodies of randomly selected fish. data from polar-water fish species, temperate species, tropical species and shark species that live in sub-tropical waters were randomly selected from a fish database. the fitting functions are w = p for polar-water fish, w = p for temperate-water fish, and w = p for tropical-water fish and the sharks. all these datasets are provided in additional file 2: table s c. brain size as a function of temperature for fixed body masses, i.e., grams and grams respectively. the open triangles represent the averaged data from fish species with body masses of 5– grams, and the open squares represent the average of species with body masses of 500– grams from three temperature conditions. the data for 38°c are from endotherms with body masses of 5– grams and endotherms with body masses of 500– grams. the fitting equations are log10 w = p for p = g and log10 w = p for g. d. the brain size ratios as a function of body sizes: brainendotherm/brainectotherm, braintropic fish/brainpolar fish and brainprimate/brainmammal. w=cpα where exponent α is the allometric slope, c is a constant, and w and p are brain and body weight, respectively. analyses have shown that c =  and α =  for endotherms, and c =  and α =  for ectotherms. the parameters c and α vary across species. a closer examination of the distributions of body mass versus brain mass reveals that the warm-blooded, endothermic animals with relatively constant body temperatures of approximately 8°c form one distinct distribution, and cold-blooded ectothermic animals with body temperatures that vary with the environment and average approximately 16°c form a clear second distribution. warm-blooded species have brains that are 4– times larger than those of ectothermic animals. the fact that the most apparent difference between the two groups is temperature suggests that it is a key factor that correlates with brain size. if temperature is important, we might predict that cold-blooded species living in disparate environmental temperatures might also have different brain sizes. to examine whether the extrinsic, i.e., environmental, temperature also explains the variation in brain size, we used fish brain and body-size data from locations of different temperatures, obtained from a fish database, and plotted the body-brain mass relationships. the fish and environments examined included the following: 1) fish species living in polar conditions in which the average temperature was ~1°c, 2) fish species living in temperate conditions with an average water temperature of ~15°c, 3) species living in tropical conditions with an average water temperature of 25°c, and 4) shark species living in sub-tropical waters that maintain their body temperatures at 20-30°c via steady contractile activity of muscles. surprisingly, we found that the fish brain-to-body mass ratio data fell into distinct groups. fish living in warm tropical water had brains that were 2– times larger than those of the temperate-water fish and 3– times larger than those of the fish living in cold polar water. the fit equation for the fish species living in tropical conditions had an allometric slope of and a constant of, the equation for fish living in the temperate waters had an allometric slope of and a constant of constant, and the fish living in the polar waters had the shallowest allometric slope and a constant of. figure 1c directly quantifies the allometric relationship between the environmental temperature of the fish and their average brain mass for two body masses. the fit equations indicate that fish brain mass more than doubles for each 10°c increase in temperature. as a comparison, the average mammalian brain sizes in animals weighing 10 g and  g are plotted in this figure. this figure shows a nice fit between body mass and brain temperature for three points, but not for the largest brain sizes and highest temperatures. the data from the larger brain sizes/higher temperatures falls above the fit lines, indicating that temperature alone may not be the sole contributor to the development of larger brains. other factors may contribute to the development of larger brains in mammals. together, these data indicate that temperature may be a critical factor in the marked changes in brain size during evolution in different climate conditions. specifically, for a given body mass, warmer living conditions should result in larger brains. in addition to this temperature-driven brain enlargement, the difference in brain sizes of endotherms and ectotherms increases as body size increases. for small body sizes, the brains of endotherms are atleast times larger than those of ectotherms. similarly, tropical-water fish have brains that are at least times larger than those of cold-water fish. in contrast to these effects in animals with small body sizes, for animals with body sizes over kilograms, the brains of endotherms are approximately to times larger than those of ectotherms, and tropical-water fish have brains that are more than to times larger than those of cold-water fish. in comparison, the brains of primates are, relatively consistently, 2– times larger than those of other mammals, and this difference is less dependent on body size differences. the emergence of intrinsically constant warm body temperatures together with an increasing brain size, with increasing body mass may be related to a temperature-dependent regulation of body metabolism. it is possible, that changes in body or environmental temperature affects the metabolic rate and consequently affected the evolution of large brains. this effect may depend on body mass. the basal metabolic rate of the body has been found to be governed primarily by two interacting processes: the boltzmann-arrhenius factor, which describes the temperature dependence of biochemical processes, and an allometric relation that describes how biological rate processes scale with body size for all vertebrates. mbody=c1pγe−evkt where p is body weight, c is a constant, γ is the allometric scaling factor, t is the absolute temperature, ev is the activation energy ), and k is boltzmann’s constant. earlier analyses have revealed that c = and γ = for endotherms, measured and scaled at 38°c, and c = and γ = for ectotherms when normalized to a temperature of 16°c; these findings are consistent with those of recent reports. when the metabolic rates of the endotherms are normalized to a common temperature of 16°c using eq., the metabolic rates of the endotherms remain several times higher than those of ectotherms, particularly for animals with small body sizes. in large animals, the metabolic rates of the endotherms and ectotherms were similar. these findings suggest that at similar metabolic rates, large endotherms should support brain masses several times greater than those of ectotherms, given that brain size differences increase with body size among endotherms and ectotherms.figure temperature-regulated basal metabolic rate and brain size. a. allometric relationships between basal metabolic rate and body mass for the endotherms and the ectotherms. the bmr versus body mass data were well fit by the equation × for the endotherms and by × for the ectotherms. the datasets are given in additional file 3: table s all the data plotted here for ectotherms were normalized to the temperature of 16°c, and the data for the endotherms were normalized to 38°c based on eq. to minimize the errors induced by measuring temperatures. b. when the metabolic rate data for the endotherms were normalized to 16°c, they merged into the group of ectothermic metabolic data. however, there some difference between these groups remained, particularly for the small animals. c. the bmr ratios as a function of body size for the endotherms to ectotherms was bmrendotherm/bmrectotherm and bmrendotherm/bmrectotherm. d. the brain masses as a function of body bmrs for the ectotherms and endotherms were based on the fit equations in figures 1a and 2a. these analyses suggest that temperature-dependent body metabolism alone cannot explain the differences in brain size among endotherms than among ectotherms. these results also indicate that other factors may play a role in explaining the differences in brain size between warm blooded endothermic animals and cold blooded exothermic animals. it is possible that neural, glial density or the energy demands on these cells are dependent on body temperature. a general model linking brain and body sizes to metabolism and temperature in this section, we derive a general model to quantify the relationships between brain size, temperature-dependent metabolism, individual neuronal metabolism and body size to account for the origin of eq.. we began by considering the relationship between brain mass and the number of neurons and supporting cells in the brain. brain mass is determined by the total number of neurons and supporting cells; hence w = nw+ n1w where n and w are the number and weight of brain neurons, respectively, and n and w are the corresponding quantities for the supporting cells. if we assume that the weight and number of the supporting cells are proportional to those of the neurons, i.e., if n1= k1n, and w1= k2w, then the relationship between brain mass and neurons simplifies to w = c2n, where c = w. this theoretical picture in which w is proportional to n is supported by a recent analysis of mammalian and primate brains. analyses of these data yielded.figure scaling relationship between the total number of neurons or supporting cells in the brain and brain mass. a brain mass compared with the total number of neurons in the brain for low-level mammal species, primates and ectotherms. the fit function gives w = n for the endotherms for the endotherms. b brain mass in grams compared with the total number of non-neuron cells in the brain for low-level mammal species, primates and ectotherms. the fit functions have pendotherm = n for the endotherms as reported in recent studies. the numbers of supporting non-neuronal cells in endotherms are proportional to the numbers of neurons and are also strongly correlated with brain mass. the supporting cells play an important role in providing mechanical and trophic support to neurons. additionally, data from several ectothermic animals indicate that that c2 =  and β = . neurons are the brain’s major energy consumer, and most of the brain’s energy is consumed by signaling processes within neurons. brains are energetically costly. experimental investigations have shown that the brain’s metabolic consumption accounts for 1-10% of the total body metabolic rate in most animal species, although brain mass accounts for no more than 1% of an animal’s body weight. hence, the total energy required to support the brain can be quantified as mbrain=φmbody where mbrain is the brain’s basal metabolic rate, mbody is the body’s basal metabolic rate as defined in eq., and φ is a variable that varies with species. re-examination of the data reported by mink et al. suggested that φ can be described by the following equation for both ectotherms and endotherms φ=c3pζ where p is the body mass, c3 =  and ζ = − for ectotherms, and c3 =  and ζ = − for endotherms.figure the dependence of brain metabolism on body size and invertebrate species. a) the ratios of brain metabolism to resting body metabolism as a function of body mass for ectotherms were fit by y = ×-, and for endotherms which were fit by y = ×-, b). c. the average metabolic cost of an individual neuron for endotherms was ± × 10− watts, and for ectotherms this cost was ± × 10− watts. the datasets used for this figure are given in additional file 5: table s the total amount of energy supplied to the brain should be equal to the total energy consumption of the brain’s neurons. note that here we make a simplifying assumption, by considering the metabolic costs of neurons only because neurons consume most of the energy while the supporting cells account for 10-20% of total brain energy. if the total number of neurons in the brain is n and the metabolic consumption of each individual neuron is mneuron, we can describe the relationship as follows: n=mbrainmneuron=ϕmbodymneuron where the total number of neurons in the brain contributes directly to the brain’s mass. by inserting eqs. - into eq., we obtained a general model of the relationships between brain mass, body mass and temperature: w=λpαe−tvt where λ=c2c1c3mneuronβ, α = βγ + βξ, and tv=βevk, called effective temperature. equation links body temperature, body mass, brain mass and neuron metabolic rate, provides a general theoretical framework for quantifying changes in brain size during evolution, and provides an origin of the formula of eq. that was derived from experimental data. the temperature range used for eq. is 0 ~ 42°c, corresponds to what is normally observed in polar fish to the body temperature of birds. the equations are not valid for temperatures above or below these values, because all the scaling parameters have been derived from animal data acquired within this temperature range. equation links brain mass with the metabolic consumption rate of individual neurons. recent studies suggest that the energy cost per neuron in the brain is stable and nearly invariant across mammal species. calculations based on the available data revealed that mneuron is, on average, × 10− watts for endotherms and × 10− watts for ectotherms. we inserted empirically determined parameters into eq. to determine how well this data-based theoretical model predicts actual animal brain size. figure 5a plots the results of linerar regression analysis between log and log for endotherms -- mammals, birds, primates and dolphins, primarily -- at 38°c and endothermic animals at 16°c. the predicted parameters: for endotherms were λ = and α =, and the analysis results were a correlation coefficient cc =, an adjusted-r =, and a mean squared error mse =, p-value p <1×10−5). for endothermic animals at 16°c, the predicted parameters were: λ = and α = ) and the multiple linear regression analysis results were cc =, adjusted-r =, and a mse =. in this analysis body mass p was the regressor variable and temperature was a constant. the model predictions closely matched the real brain data, with data-derived parameters of c =, γ =, c =, β =, c =, and ζ = − for the endotherm-based data analysis that resulted in the equation w= p e evk1t+ −138+ based on eq. and c =, γ =, c =, β =, c = and ζ = − for ectotherms, which led to the following equation: w= p e evk1t+ −116+. the difference in brain size of endotherms compared to ectotherm also increased from to 36-fold with increasing body masses. if we constrain all parameters in ectotherms, in the same fashion as we have done for endotherms with the exception of the temperature difference, the brain size prediction for the ectotherms is only -fold lower than that for the endotherms. furthermore the scaling parameter of α = from eq. is close to the measured value of for endotherms, and α = is close to the measured value of for ectotherms. thus, temperature-regulated metabolism can only partially explain the 4-40-fold brain size difference shown in figure 1a, and other factors, such as increases in non-neuron cell numbers, may be essential factors in explaining brain size differences. thus, contributions from metabolism, neural density and temperature together explain most of the differences in brain mass between endotherms and ectotherms shown in figure 1a.figure model prediction for allometric scaling between the brain and body weights of vertebrate animals on a log-log plot. a. for the brain-body mass data listed in figure 1a, the model predictions based on eq. for endotherms in 38°c and for ectotherms in 16°c are displayed. multiple linear regression analysis for the endothermic prediction yielded adjusted-r =, p < 10− and for ectothermic prediction, these results were adjusted- r =, p < 10− insert: ratios of predicted endothermic brain sizes to ectothermic brain sizes as a function of body mass. for comparison, the ratios of predicted endothermic brain sizes at 38°c to the predicted ectothermic brain sizes at 16°c using all the endothermic parameters are presented. b. for the fish brain-body mass data listed in figure 1b, the model predictions based on eq. for fish in 1°c, 15°c and 25°c conditions are shown. least-squares regression analysis of the model predictions and the actual data revealed that, for the tropical-water fish, n = cc =, adjusted- r =, p < 10− for the temperate-water fish n = cc =, adjusted- r =, p < 10− and for the polar-water fish n = cc =, adjusted- r =, p < 10− c. for the brain-body mass data of the insectivores listed in figure 1a, the model predictions based on eq. and the endothermic parameters at the temperature of 31°c produced regression analysis results of cc =, adjusted- r =, p < 1×10− d. for the fish data in figure 1c, the model predictions for the relationship between brain size and temperature for the two body sizes of grams and grams based on eq. are given in black lines. multiple linear regression analysis was used here between log and log note that model predictions for endothermic brain sizes are generally 2– times lower than actual brain masses of primates and dolphins, suggesting there other evolutionary factors mat contribute to the larger brain size in these species, as has been indicated in recent reports. the model in the present study does not make any attempt to capture these effects. the model predictions for tropical-water fish, temperate-water fish, and cold-water fish are shown in figure 5b, which predicts that brain size at 25°c should be 6-fold higher than that in fish at 1°c and that at 15°c brain size should be 3-fold higher than at 1°c. the model used actual data-derived parameters of c1 = , γ =, c2 = , β = , c3 =  and ζ = −, which for fish results in the equation w= p e evk1t+ −116+ based on eq.. note, data for fish brain, glial density data are currently not available, so the fitting is imperfect, and only includes the temperature- related differences. the model prediction for insectivores is perfectly aligned with the actual data with predicted parameters of λ = and α =. figure 5d quantifies the allometric relationship between the temperatures that fish live at, and their average brain mass for two body masses. the equations for the fish were given by w= p e evk1t+ −116+ and for mammals were given by w= p e evk1t+ −138+. at a given body weight, as temperature increases, brain mass of fish increases, but the increase of brain mass with temperature is larger in mammals. this relationship is well captured by the model predictions, with multiple linear regression analysis -- here temperature t is the regressor variable -- between log and log for g: cc =, adjusted- r =, mse =, p <; and for g: cc =, adjusted- r =, mse =, p <. by applying the general model described by eq. to all the major types of vertebrates, we found that the actual brain masses and model predictions were aligned well in a log-log plot. the multiple linear regression results were cc =, adjusted- r =, mse =, p < 10− for endotherms and cc =, adjusted- r =, mse =, p < 10− for ectotherms. figure 6b shows the residuals of the log brain sizes i.e., log10-log for both the endotherms and ectotherms as a function of log body size with slopes close to indicating that the model performed well in predicting actual brain mass. figure 6c shows that actual brain masses of fishes and model predictions were also aligned in a log-log plot. the model predictions were underestimates compared to the actual brain masses for the tropical fish with larger body sizes, and were overestimates compared to the actual brain masses for smaller tropical fish. the model underestimated the actual brain masses for fish from cold – i.e. polar -- water, but for the temperate-water fish the model matched the real data well. note that the model is missing parameters, it is worth noting that the model relies on actual data, and that for both tropical-water and polar-water fish there is a dearth of data for neural and supporting cell densities. with adequate samples of all important measures, i.e. the kinds of data we have for mammals and primates, shown in figure  the model predictions would be improved. figure 6d shows the residuals of the log brain sizes, i.e., log10-log for three groups of fishes as a function of log body sizes. this has a flat relationship with slopes close to indicating that the model performed well in predicting actual fish brain mass.figure regression residual analysis of model prediction and actual brain mass. a. least-squares regression of the log versus log of the endotherms, and ectotherms. b. least-squares regression of the residuals of the log = log10- log versus the log for the endotherms and the ectotherms. c. least-squares regression of the log versus log for the tropical-water fish, temperate-water fish, and polar-water fish. d. least-squares regression of the residuals of the log versus log for the tropical-water fish, temperate-water fish, and polar-water fish. finally, we used eq. to predict brain sizes of all the types of vertebrates and dinosaurs ). top panel in figure 7a shows that actual brain masses, and the model predictions tracked each other well in a log-log plot. compared to the ten- to hundred-fold variance in brain size for the same body mass shown in figure 1a, the model predictions and actual brain sizes exhibited a rather small variance, and only ~ 5-fold difference. the predicted brain mass from the fitting equations in figure  and the actual brain mass also tracked each other well, but the variance was a little bit higher, roughly fold than in the analysis used in top panel. the multiple linear regression analysis of these data show that: cc =, adjusted- r =; mse =; p < × 10− note that the adjusted- r = is slightly less than the value in top panel using eq.. this suggests that eq. fits the data better than do the fitting equations from figure 1a because that both body mass and temperature play important roles in determining the brain mass.figure matching analysis between model predictions and actual brain mass for all the major type of vertebrates. a. the model ) predicted brain mass versus real brain mass for all the major types of vertebrates. the correlation coefficient cc and mean squared error mse between model prediction and real brain mass are listed here: cc = and mse = for cold polar fishes; cc = and mse = for temperate fishes; cc = and mse = for tropical fishes; cc = and mse = for reptiles; cc = and mse = for amphibians; cc = and mse = for dinosaurs; cc = and mse = for mesozoic mammals; cc = and mse = for archaic mammals; cc = and mse = for insectivores; cc = and mse = for mammals; cc = and mse = for birds; cc = and mse = for primates; cc = and mse = for dolphins. b. the ratios defined as / for the animal species. for most of animal species, model predictions matched real brain data well, with cc >, mse =, p < 10−5) with ratios of / close to while there are larger mean square errors for primates and dolphins, and dinosaurs with ratios of / either much less or much more than. the regression analysis gives cc =, adjusted- r =, mse =, p < × 10− for archiac mammals, and cc =, adjusted- r =, mse =, p < × 10− for mesozoic mammals. the predictions of the model could be improved by more accurate measurements of the body temperature, body metabolism, neuronal metabolism and neuronal/glial densities of each group of species. discussion and  there is increased interest in examining the key factors facilitating enlargement of the brain during evolution. given that the energy cost of the brain’s neurons is more than ten times that of the cells of the body, the factors that regulate the brain’s energy supply and energy cost should be given thorough consideration. in this study, we re-examined data on brain size, body size and metabolic rates for all the major types of vertebrates, developed a theoretical framework based on examination of the real data, and identified several critical factors that may trigger the dramatic enlargement of the brains of birds and mammals as compared with fish, reptiles and amphibians. the most critical factor was temperature. when the endothermic animals evolved from ectothermic animals, two unique features were bound together in endotherms: larger brains and higher body temperatures. the warm temperature was critical in speeding the biochemical processes and making marked metabolic differences. our analysis revealed that in small animals the basal metabolic rate of endotherms is times greater than that of ectotherms, and in large animals it is ~  times greate. however, the brain sizes of small endotherms are only approximately times greater than that of small ectotherms but the brains of large endotherms are more than times greater than the brains of large ectotherms. clearly, in large animals, the same basal metabolic rate of endothermic animals should support larger brain tissue in ectothermic animals, than it actually does, suggesting that temperature was only one of the key factors in mediating the enlargement of the brains of endotherms. note a similar analysis published recently points out that temperature correlates with brain size. other factors included in our analysis are brain metabolism and number of supporting glial neurons. analysis of the brain metabolism data showed that that the average metabolic cost of individual neurons in ectothermic brains is ~  × 10− watts, which is nearly times lower than the average metabolic cost of endothermic brains. the lower energy cost of individual neurons could be attributed to the effect of colder temperature on metabolism and the occurrence of fewer synapses in ectothermic brains, or the lower energy cost of individual neurons could be associated to the smaller number of glia in ectotherms. glia are the major non-neuronal supporting cells that maintain homeostasis, form myelin, and provide mechanical and tropic support to neurons. in the vertebrate cns, most glia originate from portions of the developing neural tube—the exception being olfactory ensheathing glia —and the energy cost of glia is generally only a few percent of that of neurons. recent data suggest that the evolution of the nervous system was accompanied by increases in the number and size of glia. in vertebrates, the glia-to-neuron ratio in the cortex increases with the size of the brain. figure  and eq. show that brain mass increases much faster in endothermic brains than in ectothermic brains as the number of neurons increase, which is also suggestive of increases in the numbers of non-neuronal supporting cells. glia contribute to the faster propagation of nerve signals and long-range communications, particularly via the evolution of myelinating glia in developed endotherms. these energetically inexpensive glial cells surround neurons and occupy a relatively large portion of the brain, while also providing energy support to neurons and aiding efficient communication between cortical areas. thus, these cells may be one of the critical factors that promoted brain enlargement. the evolution of the large brains of endotherms is associated with multiple factors. to quantify the contribution of each factor to the brains of mammals and birds that are tens of times larger than those of fish, reptiles and amphibians, we built up a general theoretical model to examine the origin of body-brain size allometric scaling relationship by linking the key factors of temperature-dependent metabolism, neuron energetics, neuronal density and body size. this basic framework suggests that temperature and the development of supporting non-neuronal cells may have been the critical factors that correlated with, or might even be causal, for the enlargement of the brains of endotherms. this framework also predicts that the brains of cold-blooded animals living in warmer conditions should be several times larger than those of cold-blooded animals that live in colder conditions. this prediction was confirmed by the analysis of the brain sizes of fish living in various temperatures. indeed, owing to the absence of measurements of neuronal density for the brains of fish that live in different temperatures, the model prediction in figure 5b did not capture the differences in the slopes between the tropical and polar fish that are shown in figure 1b and d. however, a prediction from the endothermic brain would be that tropical fishes have larger numbers of glia in their brains than do the fish that live in cold conditions. these predictions can be tested by experimental investigations. 
this study thus provides a quantitative theory that predicts the brain sizes of all the major types of animals and quantifies the contributions of temperature-dependent metabolism, body size and neuronal density. brain-body sizeallometric scalinggliabody temperaturebrain enlargementmetabolismissue-copyright-statement© the author 2014 the cost of maintaining and supporting energetically demanding neurons is the key constraint on brain size. the dramatic increase in brain size among mammals and birds cannot be understood without solving this conundrum: larger brains, with more neurons, consume more energy. here we examined the intrinsic relationships between metabolism, body-brain size ratios and neuronal densities of both endothermic and ectothermic animals. we formulated a general model to elucidate the key factors that correlate with brain enlargement, and the origin of allometric body-brain size scaling. our framework predicts that ectothermic animals living in tropical climates should have brain sizes that are several times larger than those of ectothermic animals living in cold climates. our framework also suggests that a rapid increase in the number of less energy-demanding glial cells may be another important factor contributing to the ten-fold increase in the brain sizes of endotherms compared with ectotherms. this framework identified temperature as a critical factor in brain enlargement via temperature-regulated changes in metabolism.
 brain signaling requires energy. the cost of maintaining and supporting energetically demanding neurons is the key constraint on brain size. the dramatic increase in brain size among mammals and birds cannot be understood without solving this conundrum: larger brains, with more neurons, consume more energy.  here we examined the intrinsic relationships between metabolism, body-brain size ratios and neuronal densities of both endothermic and ectothermic animals. we formulated a general model to elucidate the key factors that correlate with brain enlargement, and the origin of allometric body-brain size scaling. this framework identified temperature as a critical factor in brain enlargement via temperature-regulated changes in metabolism. our framework predicts that ectothermic animals living in tropical climates should have brain sizes that are several times larger than those of ectothermic animals living in cold climates. this prediction was confirmed by data from experiments in fish brains. our framework also suggests that a rapid increase in the number of less energy-demanding glial cells may be another important factor contributing to the ten-fold increase in the brain sizes of endotherms compared with ectotherms.  this study thus provides a quantitative theory that predicts the brain sizes of all the major types of animals and quantifies the contributions of temperature-dependent metabolism, body size and neuronal density. electronic supplementary material the online version of this article contains supplementary material, which is available to authorized users. keywords brain-body sizeallometric scalinggliabody temperaturebrain enlargementmetabolismissue-copyright-statement© the author 2014 