On 01-18-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
  
 WE>  WE> Given a function f measured at discrete times t1, t2, ...,  
 WE>  WE> ti, ... with ti < ti' where i' = i+1, plot the pairs <   
 WE>  WE> f(ti), f(ti') >.  So what's the spread function?  A measure   
 
 WE>  JB> The spread function is the thousand dollar question and
 WE>  JB> one I am still pondering.  Something, more mathematical,  
 WE>  JB> "what is the minimum domain of the representation function  
 WE>  JB> (f(ti),f(ti')) such that the heart shows sufficient chaotic  
 WE>  JB> behavior?"  
 
 WE> I would think the spread function would be a measure of 
 WE> distribution such as standard deviation of the distances  
 WE> between all pairs of points. 
    
  Standard deviation would be one way.  For a graphical standpoint, 
the domain, the set (x,y), for healthy chaotic heart beats is 
"spread out" and quite "neat" to look at. 
  For me, the most intuitive way to look at is the way I am 
doing. 
  
 WE> WE> of what?  The function of your example is 60/(Pulse 
 WE> WE> Interval).  Why the reciprocal?  
 
 WE> JB> In response to "Why the reciprocal?"  I assume by this 
 WE> JB> why plot (f(ti),f(ti'))?  It was arbitary and gleamed by  
 WE> JB> the work of some in "Scientific American" four years ago.  
 
 WE> But why plot 60/(Pulse Interval)?  The 60 is most likely because of 1 
 
  I am interested in the -momentary- changes of heart rate. 
The P-P internal is the time from the beginning of one P-wave, 
Pi, to the beginning to the next P-wave, Pi+1, representing the 
time period from one beat to the next.  In order to determine 
heart rate, HR, it is necessary to divide 60 by P-P interval, 
(60/P-P). 
  For example if the time for one beat to the next beat is 
.5 seconds, P-P interval is .5 seconds, the persons heart 
rate is 120 beats per minute, (60/.5). 
  
 WE> minuet 
 WE> = 60 seconds.  But why not (Pulse Interval) directly or  
 WE> (Pulse Interval)/60? 
   
  I am interested in Heart rate, however, one could just measure 
the P-P interval variability as well. 
  
 WE> JB> There are many other ways to represent the data. 
 WE> JB> However, it a reasonable way to represent the data.  
 WE> JB> If you look at a -simple- system of degree 2, pendulum  
 WE> JB> with moving pivot point, the representation of the  
 WE> JB> movement of pendulum as a function of time could be  
 WE> JB> done by (P(ti),P(ti')).  
 
 WE> (P(ti),P(ti')) and then (P(ti'),P(ti'')).  This switch of 
 WE> ti to ti' defies my ability to visualize just what is  
 WE> happening.  This glib switching the x and y axis indicates  
 
  "Scientific American" had some graphical representations 
about 4 to 5 years ago. I believe "a picture is worth a 
thousaund words." 
  IAE, make up some heart rates and plot it on a x-y 
graph, here are some realistic ones.  HR1=80, HR2=75, HR3=70, 
HR4=85, HR5=90, HR6=72, HR7=65, HR8=75, .... 
  Point 1 is (80,75), point 2 (75,70), point 3 (70,85), 
point 4 (85,90), etc. 
  
 WE> some conceptual symmetry I'm not grasping.  Why the two 
 WE> dimensional data display?  Why not just calculate delta- 
 
  I think they are pretty and graphical representation is 
easier to "see" for me.  
 
 WE> f(t) = f(t') - f(t) and take the standard deviation of the 
 WE> delta-f's?   
   
  One could.  For a second there, I thought I was seeing 
the Newton-Raphson method , now that brings back 
Numerical Analysis, and the zeros of polynominals .  
 
 WE> JB> It seems reasonable to continue the method of 
 WE> JB> representation with heart rate.    
 
 WE> Don't get the pendulum example, too complex.  How about a 
 
  Think of a simple pendulum with the pivot point fixed. 
Now move the pivot point horizontal say with a certain 
omega, frequency.  
  Does that help? 
  
 WE> drummer, thump, thump, thump or a typist, click, click, 
 WE> click. 
    
  Not really sure how to do this. I'll think about it. 
  
 WE> JB> Ah, bingo, in healthy hearts, the pulse interval doesn't 
 WE> JB> hold steady, but in fact is "choatic." Reason, the pulse  
 WE> JB> interval is a representation of a multiple of excitable  
 WE> JB> tissue or cells, in this case, the heart.  
 WE> JB> IOW, the less "choatic" the heart the more unhealthy  
 WE> JB> it is.   
 
 WE> Hm, what about fibrillation and other erratic heart 
 
  Hard one for me to answer, in particular. But I can 
answer ventricular fibrillantion, VF.   VF does not plot 
out "choatic", interesting no. 
  Of interesting note, if one looks at EEG, electro-encephalo 
graph, activity, brain activity, the plot becomes very 
non-chaotic just before a seizure, interesting no. 
  
 WE> mummers?  Aren't they chaotic?  A healthy person movements 
 WE> has a large plot domain while a dying person movements has 
 WE> a shrinking plot domain. 
    
  Yep. 
 
 WE>  WE> will all fall at about the same point.  A -small- spread 
 WE>  WE> but a likelihood of death?    
 WE>  JB> We find that as the plot domain diminishes, the less spread,  
 WE>  JB> the more unhealthy the heart.   
 
 WE> Plot domain?  The smallest rectangle with sides parallel to 
 WE> the axis that contains all the plotted points. 
   
  Yep, the plot, even though choatic, is constrained within 
limits on the x,y plane.     
 
 WE>  JB> Nope, the one I am currently reading, reentrant supraventricular 
 WE>  JB> tachycardia, fast heart beat.  
 WE>  JB> There doesn't seem to be much research in this area, perhaps,  
 WE>  JB> there doesn't seem to be an awareness or understanding of "fuzzy" 
 WE>  JB> or fractals.  
 
 WE> I've heard it rumored some years ago that medical students 
 WE> are mathematical shy.  Is that so?  Are you in medicine or  
 
  Some are, some are not.  I was once a medical student. 
  
 WE> biology?  Are you considering a fuzzy thesis or research 
 
  Medicine. 
  
 WE> proposal.  Wish I could help you, but my fuzzy background 
 
  Research while working leaving little time for research.  
  
 WE> is scant.  What you're considering would require 
 WE> understanding application examples.  BTW, do you have  
 WE> access to the SCIENCE fidonet echo? 
 
  I probably could get it, but right now, I only write 
in one or two echos, any more would be too much for me 
time wise.  
 
Take care, 
John 
 
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