On the scaling properties of oscillatory modes with balanced energy

Animal bodies maintain themselves with the help of networks of physiological processes operating over a wide range of timescales. Many physiological signals are characterized by 1/f scaling where the amplitude is inversely proportional to frequency, presumably reflecting the multi-scale nature of the underlying network. Although there are many general theories of such scaling, it is less clear how they are grounded on the specific constraints faced by biological systems. To help understand the nature of this phenomenon, we propose to pay attention not only to the geometry of scaling processes but also to their energy. The first key assumption is that physiological action modes constitute thermodynamic work cycles. This is formalized in terms of a theoretically defined oscillator with dissipation and energy-pumping terms. The second assumption is that the energy levels of the physiological action modes are balanced on average to enable flexible switching among them. These ideas were addressed with a modelling study. An ensemble of dissipative oscillators exhibited inverse scaling of amplitude and frequency when the individual oscillators’ energies are held equal. Furthermore, such ensembles behaved like the Weierstrass function and reproduced the scaling phenomenon. Finally, the question is raised whether this kind of constraint applies both to broadband aperiodic signals and periodic, narrow-band oscillations such as those found in electrical cortical activity.