  The Hodgkin-Huxley model describes the behavior of the cell membrane in
neurons, treating each part of it as an electric circuit element, namely
capacitors, memristors, and voltage sources. We focus on the activation channel
of potassium ions, due to its simplicity, while keeping most of the features
displayed by the original model. This reduced version is essentially a
classical memristor, a resistor whose resistance depends on the history of
electric signals that have crossed it, coupled to a voltage source and a
capacitor. Here, we will consider a quantized Hodgkin-Huxley model based on a
quantum memristor formalism. We compare the behavior of the membrane voltage
and the potassium channel conductance, when the circuit is subjected to AC
sources, in both classical and quantum realms. Numerical simulations show an
expected adaptation of the considered channel conductance depending on the
signal history in all regimes. Remarkably, the computation of higher moments of
the voltage manifest purely quantum features related to the circuit zero-point
energy. Finally, we study the implementation of the Hodgkin-Huxley quantum
memristor as an asymmetric rf SQUID in superconducting circuits. This study may
allow the construction of quantum neuron networks inspired in the brain
function, as well as the design of neuromorphic quantum architectures for
quantum machine learning.
