1 1 6 TEMPORAL ORGANIZATION IN CELLS 



There is nothing very perspicuous in these relations, and we see that the 

 equihbrium condition is consistent with a great variety of oscillatory patterns 

 between the constituent variables of the strongly coupled oscillators. 



It is of some interest to study now the function r+/rand A+ for the case of 

 strongly-interacting oscillators and compare the results with those obtained 

 in equations (45) and (48) for the uncoupled system. Using the superscript 

 to denote coupling and a subscript to refer to the variable involved we have 



w 



J J e-^'^^^^^.dx.dx^lZ,^,^ 



By using the same transformation as that used to reduce the phase integral, 

 defined by equations (59), it is easily verified that the numerator of this 

 expression reduces (for /3 very small) to the form 



=? tan 1 



vA"^^"- \ '~'^'^^^'^^' = 



hu 



^|//|i/2j -^^ j ^^ 2)S|7/|i^2 



(//.2/|//|'/=)f2 



(65) 



using equation (61). Since this expression is identical with Z/,,^, in the limit of 

 small /8 (large d), we see immediately that 



(t1 



1 as ^^00 (66) 



This is identical with the result (47), and so we see that the oscillations in 

 the .v-variables still show a strong asymmetry which increases with 6, the 

 variables spending more and more of their time at positive values so that the 

 A"s tend to be on the average greater than the steady state values. When ^ is 

 very large {6 small), we can use equation (60) to give us the result 



(t1 



tan -~,~^ ft|//|i'2 1 l//|i/2 



2iS|//|l'2 77 277 //12 ^ ^ 



The range of this quantity for variations in the parameters /j,y is to i, 

 so that for very small the oscillations in thecoupled system are not symmetrical 

 about the steady state axis as in the case of the single oscillators. Negative 

 values of x^ now predominate and {T-IT)l^ takes values between | and 1, 

 depending upon the parameters /?/;. 



The function defining the mean positive amplitude of the variable .Yj in 

 the coupled system is 





3D 00 



x^e-^^'^''-dx^dx2 



