Canonical Quantization of a Memristive Leaky Integrate-and-Fire Neuron Circuit

We present a theoretical framework for a quantized memristive Leaky Integrate-and-Fire (LIF) neuron, uniting principles from neuromorphic engineering and open quantum systems. Starting from a classical memristive LIF circuit, we apply canonical quantization techniques to derive a quantum model grounded in circuit quantum electrodynamics. Numerical simulations demonstrate key dynamical features of the quantized memristor and LIF neuron in the weak-coupling and adiabatic regime, including memory effects and spiking behavior. Applications of this model to a sound localization benchmark show that it outperforms a phenomenological quantum LIF model as well as a classical LIF. This work establishes a foundational model for quantum neuromorphic computing, offering a pathway towards biologically inspired quantum spiking neural networks and new paradigms in quantum machine learning.

💡 Research Summary

The paper presents a comprehensive theoretical framework for a quantized memristive Leaky Integrate‑and‑Fire (LIF) neuron, bridging neuromorphic engineering and open‑quantum‑systems theory. Starting from the classical LIF circuit—an RC low‑pass filter that integrates input current and leaks through a resistor—the authors replace the fixed resistor with a memristor, whose resistance depends on the total charge that has passed, thereby endowing the neuron with adaptive leak conductance and memory. The resulting classical equations (Cₘ dV/dt = −V M(q) + I_in, with dq/dt = I) capture both spiking dynamics and history‑dependent behavior.

Quantizing such a circuit is non‑trivial because a memristor is intrinsically dissipative, conflicting with the unitary evolution required in quantum mechanics. To resolve this, the authors model the memristive leak as a semi‑infinite, lossless transmission line (TL) composed of coupled LC oscillators, following the Caldeira‑Leggett approach. The TL provides an Ohmic admittance Z₀(t) at the node, and by adiabatically modulating Z₀(t) they mimic the memristor’s state‑dependent resistance M(q). A small coupling capacitor C_C links the membrane node (flux ϕ₀) to the TL, ensuring weak, perturbative interaction.

The Lagrangian of the combined system is written, the continuum limit taken, and the wave equation for the TL field ϕ(x,t) derived. Boundary conditions at x = 0 yield the equation Cₘ ¨ϕ₀ + (1/Z₀) · ˙ϕ₀ = I_in(t), which is precisely the classical memristive LIF equation when recognizing ˙ϕ₀ = V(t). Canonical quantization proceeds by promoting ϕ₀ and its conjugate charge Q₀ to operators with