3.3 Fractal Dynamics in HRV: Power Law and Pink Noise 

• If these two characteristics fractals and power law are related, and it can be seen in this equation alone, f over x f of x is equal x to the minus alpha. Let's just say you scale up by a factor of c and the CX enters here. And then this huge factor out the see from this parenthesis, and then you get this relationship here, because x minus a is this f of x. But again, you see that even at a larger scale, the f of x remains the same. And this explains the fractals. Basically, at different levels of scale, you have the same appearance of the function. And this parallel is quite ubiquitous. 
• An example of this is the population of United States cities. If you map out the distribution of the population of cities, and by frequencies, you again see that the bulk of distribution occurs for small sizes in small towns, whereas a small number of cities with much higher population produces the long tail to the right. And again, this follows a power law relationship. And in order to see if it's truly a parallel relationship, you take the log log of both, you take the log of the population log of the probability, and you should get a linear relationship with a slope of minus alpha. And this parallel is seen in many different situations, all across nature as well as complex social networks. And also even in books. The word occurrences in Moby Dick's is also follows a linear relationship, the number of citations to a scientific paper, number of calls magnitude of earthquakes, the diameters of craters and Moon, etc. So this power law is a ubiquitous law that's seen in complex systems. This complexity, or this fractal ality can also extend not only from a spatial situation, but also temporal situations by time. 
• Another example of this where you sort of zoom in see the similar patterns, despite differences in scales. In regards to time based data, the particular relationship that's most important is something called one over f power spectral density. It's one over f, because the power is inversely related to the frequencies, the higher frequencies tend to have higher lower power, whereas lower frequencies has higher power. And certain types of noises can be categorized according to this relationship. White Noise is something that has power spectral density is that uniform across frequencies, whereas the other extreme is brown noise, which has much more of a lower frequency component. And it's this halfway point between these two, which is called Peak noise, and is this peak noise, one over f, that is of great interest for us. Peak noise, just like the power law in spatial frequencies is quite ubiquitous. And you can see this in multiple artificial as well as natural settings. You can see this in vacuum tubes, semiconductors, and for natural settings, human hearts with squid axons, as well as FPGAs and EGS.
• The question is, how do these fractal in power laws properties arise? And so this is the million dollar question to which scientists are still trying to figure out we still don't have the answer. But there's some really critical ingredients for these nonlinear properties to occur. The first is the is a balance between order and disorder. criticality occurs when there's both the stabilizing force and destabilizing forces. And the balance of the two creates these really dynamic characteristics at criticality. The second thing and third thing that's involved is non linearity, there needs to be some kind of non linearity in these fractal systems, and particularly for this Ising model, it probably is found in the energy function itself. And the other thing is iterative feedback that you have for instance, one spin affect another spin, and then that spin can affect that other spin back again. So you have these iterative feedbacks, which can be positive or negative. So fractals and power law properties probably arise from a tug of war between stabilizing and destabilizing forces, non linearity, and feedback. So knowing this, which of these cases are healthy, you can see the time series or heart rates over time in for these examples. And for the ones that for this one that has absolutely no variability, it's congested firefighter. This one has a periodic variability is also congestive heart failure. And this has a lot of variability, almost like a randomness to it. And this is atrial fibrillation. And this is the healthy case where you have combination, basically, the balance between order and disorder. 
• You can take this a little further and say that if health is truly a balanced state, between order and disorder, then disease is got to be a pathological breakdown into order severe order, and then a severe disorder. We can ask, then how can you objectively quantify this healthy dynamic, and you can make this logical step first, if good health is truly a balance between order and disorder, this is just like a system that criticality. And if a systemic criticality demonstrates fractal properties, then truly Frank, good health must have fractal properties. And since this is a time series, what we're interested in is something called Pink Noise, which is a balance between the white noise and brown noise. And the nice balance here is similar to the pink signals that we see here. Whereas the breakdown into disorder is similar to white noise, and the breakdown into order is brown noise. Next part is to appraise the evidence for fractal based Heart Rate Variability measures. 
• The first study that looked into fractal relationships in disease was bigger in 1996, which was published in circulation. He had three groups, one with a recent acute MI, he had 715 subjects in this group, healthy group, which had 274 participants, heart transplant individuals, and there was 19. In this, he took a 24 hour continuous ECG, and evaluate the power loss slope, which is essentially the power spectral density versus frequency but paid specific attention from 10 to the minus four to 10 minus two hertz. And the reason for that is that if you take a look at the relation spectral density relationship of the heart rate, you see a linear relationship or linearity that exists pretty much from the low, very low frequencies, all the way to maybe about 10 to the minus two, and then you have a little, you have these squiggly parts that correlate with the higher frequencies. Bigger at I'll show three examples from each group. This is a healthy individual, this is an myocardial infarction individual and heart transplant person. And this person who is healthy had a slope that was close to minus one again, following the one over f Pink Noise characteristic. However, the myocardial infarction individual has a steeper slope at minus 1.2 in the heartland transplant individual at minus 2.2. 
• The other thing to note is that the total power was significantly different. The one with a healthy individual had higher power compared to the Mi in the heart transplant patients. If you take the average for all the individuals that were within this group, you find that these differences in slope and the total power are statistically significant. And again, the healthy individual has a slope that's nearly close to minus one, which is which mimics or resembles pink noise. And the same applies to the total power. And this is a representation of one of the summary of the the parallel relationships across the three groups, we can see that the healthy group has an overall higher power compared to the myocardial infarction and the transplanted patients.
• You can see this from the perspective of air low frequency and ultra low frequency again, healthy it has a higher fair low frequency compared to MI group and the transplant patients. This graph also indicates how the total power differentiates itself from the fractal slope or whatever f power law relationship, the power law relationship more looks at the distribution of the energy of the heart rate variability across the multiple frequencies. And it's not necessarily relating at the absolute value, but more so how these values are distributed across the frequencies, with the transplanted patients having much more of the power it dedicated to the very low frequencies ultra low frequencies compared to the very low frequencies. And again, the healthy individuals and partly the myocardial infarction, individuals had much more of a pink noise relationship, whereas the transplanted patients had much more of a brown noise type of pattern. And this makes sense because transplanted individuals tend not to have too much heart rate variability, they tend to have a smoother graph like and resembling brown noise. 
• This is a graph that came off again from that same publication by vigor at all in 1996. This is a survival graph for the airline group, which number seven 115 And we see here that for those individuals that had both high heart rate variability power and a normal slope, which resembled peak noise, they had higher survival overall and lower mortality. Whereas those individuals with low heart rate variability power had worse mortality. And the same also applied to those with a slope, parallel slope greater than 1.37. To a combination of both these factors, total power and parallel slope had much worse mortality overall. And this indicates to me that absolute total power, and one over f slope likely contains different information. And the combination of both may be clinically useful. This is a forest plot obtained from sent at all in 2018, which summarizes the application of parallel one over f slope to multiple conditions, and my left ventricular dysfunction noncardiac conditions. And we see here generally that the individuals who had died had a higher slope or more of a negative slope compared to those who remained a life. And this relationship was, you know, significant if you combined all the factors all together. These are figures that were obtained from the studies that listed in the meta analysis. And you can see here again, steeper slope, indicating a slope less than minus 1.5 indicates higher rates of mortality. And this occurred in post myocardial infarction, low ejection fraction groups, elderly individuals without past medical history of cardiovascular disease, and even post stroke. 
• The power law one over f slow power has certain limitations. First off, is that it does not assess short term heart rate complexity. If you recall of this graph, the linearity the linear relationship between frequency and spectral density occurred only from the very low frequencies down to about 10 to the minus 10. to hertz, it's the shorter frequencies, the short term frequencies, that really were not included in this power law relationship. And because we focused on the very low frequencies, it requires a significant amount of data acquisition and it requires data more than 24 hours. And the other thing is that power law slope requires the fast Fourier transform and the fast Fourier transform is not ideal for non stationary data. And what do I mean by that? non stationary data has the mean or standard deviation that changes with respect to time. So you can see for instance, if you take a segment here, the mean and the standard deviation is very different from here where you have you know, low RR interval average, but and also low standard deviation is just simply one of the weaknesses of Fast Fourier Transform

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