Coulostatics in bioelectrochemistry: A physical interpretation of the electrode-tissue processes from the theory of fractional calculus

In this paper, we analyze the electrical response of an electrode-tissue-electrode system to the application of a dc current for a sufficiently short time in order to obtain coulostatic conditions: A finite amount of charge is “instantaneously” and efficiently transferred to the capacitors formed by biological membranes at the tissue level and the electrode biointerfacial regions. To allow a more realistic study, the capacitances formed by the electrode-tissue interfaces and those of the cell membranes were modeled using constant phase elements (CPEs). The mathematical expressions for the current, voltage, and charge of the CPEs are obtained in response to the sudden injection of the controlled electric charge. It is predicted theoretically how, under certain conditions, the current path could be restricted to flow through the capacitors formed by the electrode-tissue interfaces and those of the cell membranes, and thus, the total charge injected is practically transferred to both types of capacitance (i.e., a coulostatic charge injection). Finally, we study the influence of the pulse shape (retaining the coulostatic nature) on the technique, from the theoretical perspective of the fractional calculus. The shape of the excitation signal is shown to play a dominant role in the coulostatic relaxation processes, in sharp contrast to the conventional approach. This methodology could be extended to include the membranes of organelles and also to implement a coulostatic test method involving electrical characterizations of biological tissues.