DC-ELF characterization of random mixtures of piecewise nonlinear media

Biological tissues are ensembles of linear and nonlinear, symmetric and asymmetric constituents. As far as their electromagnetic characterization is concerned, they can be modeled as microscopic mixtures of the corresponding material media. Any medium volume can be properly discretized in a finite number of cells which can be modeled as an equivalent three dimensional network of lumped components, in order to characterize its electromagnetic behavior at wavelengths much longer than the relevant average linear size of the constitutive cells. Therefore, any mixture and the corresponding tissue can be characterized in terms of its effective conductance at extremely low frequency, with respect to a reference set of electrodes (ports of the equivalent network). When the above procedure is implemented for evaluating any of the aforesaid conductances, a resulting nonlinear characteristic should be expected. In reality, it may happen that the effect of the constitutive nonlinearities and the related asymmetries are smeared out by the randomness of the interconnections of the lumped components, leading at a macroscopic level to an isotropic constant equivalent conductance, i.e., to an isotropic constant equivalent conductivity of the mixture. The closed form analysis of a random network of nonlinear (piecewise linear) resistors offers a simple but clear cut example of such a property. This result, if extrapolated to biological media, suggests a new hint for explaining why there is no inconsistency between the typical electric characterization of biological tissues as almost linear macroscopic media, by means of their effective conductivity and permittivity, and the nonlinearities of the biochemical processes occurring in the tissue cells. In fact, the nonlinearities may not be observable by means of macroscopic electrical measurements because of the randomized spatial orientation and location of the processes.