Quantized Single-Ion-Channel Hodgkin-Huxley Model for Quantum Neurons

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. We 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 the classical realm and the quantum realm. 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 shows 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 superconducting quantum-interference device in superconducting circuits. This study may allow the construction of quantum neuron networks inspired by the brain function, as well as the design of neuromorphic quantum architectures for quantum machine learning.