118 



TEMPORAL ORGANIZATION IN CELLS 



but there is one observation which is of considerable interest and that is when 

 d IS large and ^21 is very small {A+)l, tends to be large. The condition of very 

 small ^21 means that the oscillating pair {X^, Y,), to which we will refer as O, 

 as If It were an independent oscillator, has very httle effect on the oscillator O, 

 (defined by the pair {X2, Y^), again looking upon it as an independent oscillator) 

 If we assume further that k,^ is large (but keeping /:„^22-^i2^2i > 0), then 

 what we have is an oscillator O^ which is asymmetrically coupled to a second 

 oscillator O2 m such a manner that O2 tends to "drive" O^. These are the 

 conditions which favour the occurrence of subharmonic resonance or frequency 

 demultiphcation in the system, so that we may expect to find a large oscillation 

 in Oi arising from the fundamental oscillation generated by O2 and transmitted 

 to Oi by strong coupling. 



12 3 4 5 



Time 

 (a) 



Time 

 (b) 



Figure 9. 



Actually the experimental studies which have been made on subharmonic 

 resonance in electrical and mechanical system were performed under conditions 

 where one oscillator drives another system which has no autoperiodic oscil- 

 lation at all; i.e. only one of the two coupled systems is an autonomous 

 oscillator, but the driven system has a non-linear restoring force which causes 

 It to return to its equilibrium position after a disturbance. The type of oscil- 

 latory behaviour which was observed by Ludeke (1946) in the driven component 

 of such a mechanical system is shown in Fig. 9. 



Figure 9a shows a subharmonic resonance of order |, while in 9b we see one 

 of order \. The characteristic feature of this phenomenon is that an oscilla- 

 tion appears in the driven component which has a considerably larger ampli- 

 tude and smaller frequency than that of the autonomous oscillator which 

 IS driving it. In the case of the subharmonic resonance of order \ it is possible 

 to see the driving oscillations as components of the complete wave-form, but 

 not in the other case where a greater fusion of oscillatory motion' has 

 occurred. 



This type of coupling cannot be studied analytically by means of the 

 statistical mechanics developed in connection with the present theory, because 

 the equations of an interacting pair of oscillators with completely asym- 

 metrical coupling (with A'2i = 0, e.g.) cannot be integrated. However, we can 

 approach this condition by taking A-j, very small and A-, 2 large in the coupled sys- 

 tem. There will still be an autoperiodic oscillation in Oj, so that a subharmonic 



