7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 103 



variable about some line other than its steady state, for example x,- = i/ where v 

 may be a positive or a negative number. The appropriate modification is 



00 



- \x,\e-^^y^dy, 



Qi J 



z,.z. 



(52) 



Similarly for j, we have 



oj(yi-v) = f \yi\h{yi-v)e-^^dvi{ e-^^dv 





-j36a./-iog(i+i')] 



7 7 



CO 



r \y\e-^''^^dx (53) 



If we now take the ratio of the mean frequency of zeros about the line 

 Xi = f to that about x^ = 0, we get 



and similarly 



^relCy/) = e-^'"[^-l°S('+''l = (l+v)^''"^-^''''' 



For V j^O, these ratios are always less than 1, which shows us that the 

 variables x, and j,- cross the axis x,- = 0, >',■ = more frequently than any other 

 axis, A',- = V, yi = v. Furthermore, if /3 is very large, then these ratios decrease 

 extremely rapidly as v moves away from zero in either a positive or a negative 

 direction. That is to say, when 6 is very small the oscillations are small so that 

 the trajectories cross lines displaced from the steady state much less frequently 

 than they do the steady state axis. But when j8 is very small (9 large), then the 

 mean frequency of zeros drops off much less rapidly as one moves away from 

 the steady states, since the amplitudes of oscillation are large for large 6. 



We now turn to the problem of evaluating the mean frequency functions 

 explicitly. We will restrict our attention now to the yi variables. The calculation 

 depends upon finding the integral 



00 



-Pi 

 We must content ourselves with the evaluation of this expression in the limits 

 of large and small ^, since an exact solution valid over all values of ^ depends 



