We use entropy to characterize intrinsic ageing properties of the human brain. Analysis of fMRI data from a large dataset of individuals, using resting state BOLD signals, demonstrated that a functional entropy associated with brain activity increases with age. During an average lifespan, the entropy, which was calculated from a population of individuals, increased by approximately 0.1 bits, due to correlations in BOLD activity becoming more widely distributed. We attribute this to the number of excitatory neurons and the excitatory conductance decreasing with age. Incorporating these properties into a computational model leads to quantitatively similar results to the fMRI data. Our dataset involved males and females and we found significant differences between them.
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The entropy of males at birth was lower than that of females. However, the entropies of the two sexes increase at different rates and intersect at approximately 50 years; after this age, males have a larger entropy. There is now a consensus that ageing is multifactorial; it is the joint outcome of genetics, the accumulation of random accidents and irreparable losses in molecular fidelity.
Use this site to download Maxent software for modeling species niches and distributions by applying a machine-learning technique called maximum entropy modeling. From a set of environmental (e.g., climatic) grids and georeferenced occurrence localities, the model expresses a probability distribution where each grid cell has a predicted. Figure 5 shows the entropy fluctuation in both traditional pairwise correlation and mutual information. Both diagrams contain time series for all of our stock market price. In each case, the entropy undergoes a sharp increase corresponding to the financial crises, which are associated with dramatic structural changes in the networks.
There is ample evidence that the genetic component alone plays a critical role in longevity determination. This is shown in regulatory and structural changes that occur with age in miRNA, mRNA, ncRNA, protein expression and functional MRI in many species. Intuitively, these changes could be expected to correspond to changes in the functioning of the brain. But in what precise sense? This is the question we address in this work.
The answer involves an explicitly quantitative way of characterizing the intrinsic ageing process of the human brain.We used entropy to quantify the functioning of the brain in individuals of different ages. Accordingly, we shall describe this as the functional entropy.Entropy characterizes the degree of underlying randomness of a random variable. Random variables with small entropies have a high level of predictability and hence a low level of randomness.
By contrast, large entropies correspond to low levels of predictability and high levels of randomness.As outlined below, we view the brain as being divided (parcellated) into a number of distinct regions. For each pair of distinct brain regions, we calculated the correlation coefficient of their neuronal activity; this characterizes the functional coupling of the two brain regions. The resulting set of correlation coefficients generates a frequency distribution. The correlation coefficient of a distinct pair of brain regions, that have been randomly selected, can be regarded as a random variable that follows this frequency distribution. We use the dispersion or variability of this random variable as a measure of the functional entropy (c.f., complexity) of the neuronal dynamics of the brain.
We investigate, in this work, how this measure of the functional entropy changes with age and in we illustrate the behaviors of the brain's dynamics that it captures. Illustration of the Functional Entropy of the Brain.In this figure, the brain is parcellated into a number of distinct regions. Different levels of brain activity (as measured by BOLD signals) are illustrated by different colors in the brain slices of the figure. We use two artificial data sets of BOLD signals to illustrate what the functional entropy captures about brain activity. Panels (a), (b) and (c) show brain activities of brain slices at three different times (T 1, T 2, T 3), when the correlation coefficients between all regions of the brain are unity. In this case, all regions of the brain have the same color since they are behaving synchronously.
Panel (d) shows the BOLD signals in different brain regions, for this case. Panel (e) shows the corresponding distribution of the correlation coefficients (a ‘spike’ located at a correlation coefficient of unity). The functional entropy for this case is zero (the minimum possible value). Panels (f), (g) and (h) show the brain activities at three different times (T 1, T 2, T 3), when all correlation coefficients are generally different (so all regions have different colors, indicating that all regions are behaving asynchronously. Panel (i) shows BOLD signals in different brain regions, for this case. Panel (j) shows the corresponding distribution of the correlation coefficients (a uniform distribution).
The functional entropy in this case is 4.32 bits (the maximum possible value). See for details. (top row) shows the situation where every brain region fluctuates over time, but is totally correlated with all other regions. In such a case, the functional entropy of correlation coefficients is zero; all correlation coefficients are unity and hence their distribution exhibits no randomness, just predictability. A case of non-zero functional entropy occurs when a range of different correlation coefficients are found between different pairs of brain regions. An example of this case is given by the second row in. See for details.
In the opposite case of completely independent or incoherent activity in all regions, the correlation coefficients will all be zero and their dispersion (functional entropy) will again be zero. This means our entropy measure is sensitive to co-ordinated activity that is most interesting, namely activity that is intermediate between fully synchronised and fully incoherent brain-region dynamics.The functional entropy effectively measures the dispersion (or spread) of functional connectivities that exist within the brain. We initiated this study under the assumption that the dispersion of functional connectivities is related to the age of the brain.For the current paper, we collected fMRI data from 1248 individuals, ranging from 6 to 76 years of age.
This provided a unique opportunity to characterize the ageing process of the human brain.In the analysis of our fMRI dataset of differently aged individuals, we found that, at the population level, the functional entropy of the human brain (as calculated below) has a definite tendency to increase over time. This can be viewed as there being a higher level of randomness in the way different brain-regions functionally interact with one another.Beyond showing that the functional entropy of the brain has the tendency to increase with age, we find quantitative differences between the entropies of males and females. In newborn males the functional entropy has a mean value of 3.536 bits; it is approximately 0.06% larger in newborn females, with a value of 3.555 bits.
However, there are is also a difference in the rate at which the functional entropies change in the two sexes. In males, the functional entropy increases at a mean rate of approximately 0.0015 bits/year but in females it increases at the slower mean rate of 0.0011 bits/year. The different values of the functional entropy in newborns and the different rates of increase in the two sexes, lead to the entropies of the two sexes approaching one another and then crossing. This crossover in entropies occurs at approximately 50 years of age. Beyond this age, the pattern of entropies exhibited at birth is reversed, with males then having the larger functional entropy.Given that the current world life expectancy is 65.59 years in males and 69.73 years in females, we estimate that at these ages, the functional entropy of the brain will be 3.633 bits in males and 3.647 bits in females.
Thus the mean functional entropy change, from birth to life expectancy, is 0.097 bits in males and 0.092 bits in females, even though female life expectancy is higher.In addition to determining the functional entropy of the whole brain, we have determined the functional entropy of different regions of the brain, again using correlations between the neuronal activity of different brain regions. We find that different brain regions have different entropic characteristics.
Typically, the observed changes are monotonic, but not all brain regions have increasing entropies. There are some regions where the functional entropy increases, others where it decreases and a third set where the functional entropy remains almost constant. With L and R denoting left and right brain regions, we find that the brain regions with the fastest rate of functional entropy increase are: the L and R paracentral lobules, the R olfactory cortex, the L middle frontal area, the L and R hippocampi and the L and R parahippocampal gyrus. We note that the hippocampus is well known to be associated with both short and long term memory formation. By contrast, the L and R insulars represent regions whose entropies most rapidly decrease with the age of an individual. Clear changes in regions of the brain, with age, are also found in other studies, INS, PCL, OLF, MFG, HIP and PHG.
We note that the functional entropy of the whole brain is not simply an average over entropies of individual brain regions. Accordingly, our findings, that the functional entropy of the whole brain increases with age while some regions of the brain exhibit decreasing entropies, are compatible.We have used a computational model based on diffusion tensor imaging (DTI) data to investigate the origins of the relationship between functional entropy and age. Extensive experimental data indicates that there is significant loss of neuron number with age and this is accompanied by the excitatory receptor number (especially NMDA) decreasing with age. Our computational model (see for details) yields a brain entropy that decreases when the excitatory connection strength and neuron number in each brain region are simultaneously reduced.To motivate the definition of functional entropy that we use in this work, let us consider the following example of an analysis we carried out.We parcellated the whole brain of three individuals into 90 regions, based on the AAL atlas. These were healthy males aged 24, 49 and 69 years, which we describe as ‘young’, ‘middle-aged’ and ‘elderly’. For these, we calculated the correlation coefficient between the BOLD signal of the thalamus in the right hemisphere and each of 45 brain regions in the left hemisphere (see ).
These signals are represented in the left-hand of panels (a), (b) and (c) of. Differences between the three individuals show up which are found in more extensive analyses. The Origin of the Functional Entropy.This figure presents time series from BOLD signals.
The left-hand sides of panels (a), (b) and (c) contain 45 time series from brain regions of the left hemisphere. The vertical location of a time series, from a given brain region, is given by value of the correlation coefficient of that region with the right thalamus. Panel (a) is from a healthy young male (age 24 years), panel (b) is from a healthy middle-aged male (age 49 years) and panel (c) is from a healthy elderly male (age 69 years). The two horizontal red lines in panels (a), (b) and (c) give, separately, the mean over either positive or negative correlation coefficients. Thus the separation of these lines is a measure of the width of the distribution. In the right halves of panels (a), (b) and (c), the time series of the right thalamus is plotted (using a different vertical scale).
Panel (d) gives a histogram of the correlation coefficients of the young male (in blue), the middle-aged male (in white) and the elderly male (in red). The distribution of the correlation coefficients of the elderly male (red histogram in ) is more widely spread than that of the young male (blue histogram) and middle-aged male (white histogram). This leads to the elderly male having a larger functional entropy than that of the middle aged male, who has a yet larger functional entropy than that of the young male (see ). This implies that the dispersion of correlations, between the right thalamus and a region of the brain in the left hemisphere, is typically an increasing function of age. This conclusion is found to hold in a full analysis, where a pairwise comparison of all regions in the brain is used, rather than just comparing regions in the left hemisphere with the right hemispheric thalamus. We have carried out a full analysis to determine the functional entropy in all individuals in our data set. Shows the functional entropy of the full data set as a function of age, without taking into account gender differences.
In we present a running average of the functional entropy for males and females, with an averaging window of 25 years, whose choice is a compromise between stability and being substantially smaller than the maximum age; choosing windows from 19 to 30 years does not significantly affect any conclusions we draw (see ). In, we give the results for the functional entropy in males and females separately, which are different at birth (3.5336 bits in males, 3.5547 bits in females) and which have different rates of change (0.0015 bits/year in males, 0.0011 bits/year in females). The Pearson correlation between entropy and age is strongly significant (r = 0.23, N = 610, p = 5.6 × 10 −9 for males and r = 0.15, N = 634, p = 1.51 × 10 −4 for females) These lead to the crossover that can be seen in, at an approximate age of 50 years and will be considered in the Discussion. Functional Entropy vs.
Age.Panel (a) is a plot of the functional entropy of individuals versus their age (pooling results from males and females). A mean rate of increase of the entropy of 0.0013 bits/year was found from the data.
Panel (b) contains a plot of the running average of the entropy, versus age, with a window of width of 25 years adopted. There is a crossover in the male/female entropies in the vicinity of 50 years of age. Panels (c) and (d) plot the entropy versus age of males and females.
Males have a lower initial value of the entropy than females, but a faster mean rate of increase. The linear correlation between entropy and age is strongly significant ( p = 5.6 × 10 −9 for males and p = 1.51 × 10 −4 for females, r equals 0.23 for males and 0.15 for females, degree of freedom: 610 and 634, respectively.). For each brain region considered in this study, we have also determined trends in their individual entropies over time. The functional entropy of a given region is determined from the set of correlation coefficients between that region and the other 89 regions in the remainder of the brain. In we have used color to show the trend: the warmer (redder) the color, the more positive the rate of increase. Shows that the frontal area is more likely to have a higher rate of increase in functional entropy with age than an average region. By contrast, the occipital area exhibits an apparent resistance to functional entropy change and remains largely unaltered over time.
This pattern is considered in detail in the Discussion. Functional Entropy of Brain Regions.Panel a shows the trend exhibited by the functional entropy of each brain region with age. The warmer (redder) the color, the larger the rate of change of the entropy. Panels (b), (c) and (d) give the coronal, sagittal and axial views of the brain and show the brain regions with the most rapidly changing entropies. Red ones show the brain regions with the most rapidly increasing entropy, blue ones show the most rapidly decreasing regions.
Panel (e) contains the trends of all the brain regions with significant ones being in the brightest colors. In panel (e), bright blue lines show the behavior of the insula (L, R); bright dotted red lines indicate the paracentral lobule (L, R); solid red lines indicate the hippocampus (L, R); dashed red lines indicate the parahippocampal gyrus (L, R); red squares represent the olfactory cortex (R); diamonds represent the middle frontal gyrus (L). The brain regions that exhibit the most significant (p.
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