126 



TEMPORAL ORGANIZATION IN CELLS 



imposed by equation (79). Without going through the transformation pro- 

 cedure to reduce equation (81) to a recognizable integral, let us record only 

 the expression obtained, which is (for very small /S) 



Rl 



00 00 N 



v. - 00 



(82) 



The inner integral is the Fourier transform or characteristic function of a 

 stable probability distribution with exponent a = 1 +^bi of the type discussed 

 by Paul Levy (1948) in probability theory. This transform is known, but the 

 subsequent infinite integral in the variable s is rather difficult to obtain and 

 does not lead to a result which can be used to study the roots of the equation 

 (79). Looking at the expression in (82), we see that equation (79) imposes 

 a constraint on the parameters Z?, as well as on the a,'s and ki/s (which enters 

 the expressions by way of the y,'s). Thus it is certainly not sufficient for 

 entrainment that the coupling parameters only have certain values. Rather 

 the whole of the coupled system must be in a particular parametric state. 



There is one final set of conditions which we shall consider in relation to 

 the question of entrainment in the strongly-coupled oscillators. Observe that 

 if the variables x^ and Xi are oscillating in synchrony, thus behaving identically 

 in all respects, they should be indistinguishable from one another. In parti- 

 cular it should be true that the time averages of the product, a'iA'2, should be 

 equal to the time averages of a? or x^. In the statistical mechanics we replace 

 time averages by phase averages, so the following relation should be true: 



2 2 



(83) 



Now from the definition of the phase integral for Xi and a:2, 



^P:P.= J / e-^^''"^^'+^''"''''^«+''"-'»'^^A-it/x2 



— Pi — Pa 



we have the identities 



-5_ i aiogZp,p, 

 1 aiogz 



•^1-^2 — 



PiPt 



2j3 a/?i2 



(84) 



