7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 121 



to the form 



1 



/ 



g-t'+h,,'/h„i''-^l 



(/3/(23)" = [-/7,+(/).2//(j2)l] 



SO that we get 



The ratio of this to the mean frequency of zeros about the line i' = is 



«)rel="^'^"f^ = ^-^^"'"*-^''^ J ^-'V// J .-'V/ (72) 



Regarded as a function of v, this expression takes a maximum at a root of the 

 equation 



dv 



Determining the partial derivative, this is 



= 



-(i-J"\.-..-.......--» J .-.v, 



0/l22)'/ = [-p2+(Al2/A22)l'] 



or 



00 



/;i2e-^''"f''«-(^3/''22)'']>2|//|[^]^'% f e-''dt = 



0/l22)l/2[-p2+(/ll2//l22)H 



The root of this expression is always some negative value of v. Therefore in 

 the coupled system the mean frequency of zeros is no longer a maximum about 

 the steady state axis, v = 0, as was the case for the uncoupled oscillators, but 

 about some axis displaced below the steady state {v < 0). This is true for all 

 values of /S, but as ^ decreases (6 increases) the root becomes increasingly 

 negative. 



For )8 very small, the relative mean frequency of zeros is approximately 



00 





0//l22)l''/ll2l' 



Substituting the original parameters in place of /ly, we get 



00 



aikiimnllkiiy^v 



