Impedance Spectroscopy Dynamics of Biological Neural Elements: From Memristors to Neurons and Synapses

Understanding the operation of neurons and synapses is essential to reproducing biological computation. Building artificial neuromorphic networks opens the door to a new generation of faster and low-energy-consuming electronic circuits for computation. The main candidates to imitate the natural biocomputation processes, such as the generation of action potentials and spiking, are memristors. Generally, the study of the performance of material neuromorphic elements is done by the analysis of time transient signals. Here, we present an analysis of neural systems in the frequency domain by small-amplitude ac impedance spectroscopy. We start from the constitutive equations for the conductance and memory effect, and we derive and classify the impedance spectroscopy spectra. We first provide a general analysis of a memristor and demonstrate that this element can be expressed as a combination of simple parts. In particular, we derive a basic equivalent circuit where the memory effect is represented by an RL branch. We show that this ac model is quite general and describes the inductive/negative capacitance response in many systems such as halide perovskites and organic LEDs. Thereafter, we derive the impedance response of the integrate-and-fire exponential adaptative neuron model that introduces a negative differential resistance and a richer set of spectra. On the basis of these insights, we provide an interpretation of the varied spectra that appear in the more general Hodgkin-Huxley neuron model. Our work provides important criteria to determine the properties that must be found in material realizations of neuronal elements. This approach has the great advantage that the analysis of highly complex phenomena can be based purely on the shape of experimental impedance spectra, avoiding the need for specific modeling of rather involved material processes that produce the required response.