122 TEMPORAL ORGANIZATION IN CELLS 



The analogous expression for X2 is (for jS small, Q large) 



00 





(74) 



a2A:i2v'Wc2i/2/cii)>' 



Now we may expect that if a subharmonic resonance appears in the variable .Yj 

 as an oscillation of large amplitude, but not in a'2, as we suggested might occur 

 for small jS (large 6), small k2i, and large ki2, then the mean frequency of zeros 

 of xi should drop off more slowly than that for X2 as v moves away from the 

 steady state axis in a positive direction. This behaviour is shown by equations 

 (73) and (74) for these parametric values, (oj^rei tending to decrease more 

 rapidly than (co^Jrei ^s v increases, although their relative rates of decrease are 

 obviously dependent upon the other parameters as well. Thus, for example, 

 «! is an important parameter in determining how rapidly (wyrei decreases 

 as V increases, the rate of decrease being small when a^ is small. The same is 

 true for the effect of the parameter a2 on {co^xXeh but the interpretation of this 

 effect is not clear, and there is no direct evidence to connect these effects with 

 possible subharmonic phenomena. However, it would appear that for certain 

 values of the microscopic parameters and with 6 large so that the non-linearities 

 are marked, the variable Xi shows an oscillatory pattern with a large amplitude 

 which is not due to either of the autoperiodic oscillations in the system, and 

 must arise in consequence of their interaction. 



Entrainment 



Another phenomenon arising from the interaction of non-linear oscil- 

 lations is the occurrence of entrainment. Under certain conditions the auto- 

 periodic components of two coupled oscillators "lock" together to produce 

 a single system oscillation, so that the coupled pair appears to behave as a 

 single oscillator. It has been shown by studies on electrical and mechanical 

 systems (Appleton, 1922; van der Pol, 1922) that there is an asymmetry in the 

 interaction of such oscillators prior to entrainment, so that we may speak of 

 one of the oscillations "capturing" the other and forcing it to oscillate in 

 synchrony with it. Pringle (1951) has shown that the direction of this "prey- 

 predator" relationship is determined by the direction in which one oscillation 

 approaches another. Thus if one of the autoperiodic oscillations is stationary, 

 then it will be "captured" by an oscillator coupled to it if this second oscillator 

 approaches the frequency of the stationary oscillator through greater frequency 

 values. If, however, the approach by the second oscillator to the stationary 

 frequency of the first is through smaller frequencies, then the stationary 

 oscillator is the predator and "captures" the approaching "prey" oscillator. 

 Pringle has shown further that the distance between the frequencies when 

 entrainment occurs is different in these two cases, there being a greater jump 

 or discontinuity when the approach is through greater frequencies than when it 

 occurs through smaller frequencies. On the basis of these properties of inter- 

 acting non-linear oscillators, Pringle has constructed an extremely interesting 



