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Dynamics of Excitatory-Inhibitory Coupling in Quantum Memristive LIF Neurons

This paper presents a minimal phenomenological model of two coupled quantum memristive leaky integrate-and-fire neurons that successfully replicates key excitatory-inhibitory dynamics, including six distinct firing regimes and a unique silent state, while highlighting the constraints of amplitude modulation imposed by Fock space truncation in quantum neuromorphic architectures.

Original authors: Marta Adamowska, Domenica Dibenedetto

Published 2026-09-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Marta Adamowska, Domenica Dibenedetto

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The human brain is a vast network of billions of tiny cells called neurons, communicating through a delicate balance of signals that either encourage activity or suppress it. This push-and-pull between excitement and inhibition is the fundamental rhythm of thought, allowing the brain to process information, learn, and adapt. For decades, scientists have tried to build computer models that mimic this biological dance, hoping to create machines that think more like living minds. Recently, a new frontier has opened where these brain-inspired models meet the strange, counterintuitive laws of quantum mechanics. In this emerging field, researchers are asking whether the complex, rhythmic behaviors seen in biological brains can survive when translated into the quantum realm, where particles can exist in multiple states at once and information is carried by light rather than electricity.

A team of researchers at Maastricht University has taken a significant step toward answering this question by simulating a tiny, two-neuron network using a new type of quantum model. They focused on a specific design called a "quantum memristive leaky integrate-and-fire" neuron. To understand this, imagine a simple electrical circuit that acts like a bucket: it slowly fills up with water (representing electrical charge) until it reaches a certain level, at which point it overflows, empties, and starts filling again. In the biological world, this overflow is a neuron firing a signal. The "memristive" part of the model adds a layer of memory, meaning the bucket's leakiness can change based on how much water has passed through it, mimicking how real neurons adapt. The "quantum" aspect replaces the simple bucket with a system that follows the rules of quantum physics, where the state of the neuron is described by probabilities and energy levels rather than just a simple voltage.

The researchers built a simulation of two of these quantum neurons connected in a loop. One neuron was designed to be excitatory, meaning it sends a signal that encourages the other to fire. The other was inhibitory, sending a signal that acts like a brake, making it harder for the first neuron to fire. They applied an external rhythm to the first neuron, similar to a steady heartbeat or a ticking clock, to see how the pair would behave together. The goal was to see if these quantum units could reproduce the rich, varied firing patterns observed in real biological networks, such as regular rhythms, bursts of activity, or periods of silence.

The simulation revealed that the quantum model successfully captured the essential roles of its biological counterparts. The excitatory connection worked as expected, pushing the receiving neuron toward firing, while the inhibitory connection acted as a suppressor, effectively increasing the "leak" in the system and preventing activity. By adjusting the strength of these connections and the rhythm of the external drive, the researchers were able to generate six distinct firing patterns. These included steady, regular firing; initial bursts of rapid activity that quickly died down; and irregular, chaotic spiking. This demonstrated that even a minimal quantum network could retain the complex dynamical features of classical neural systems.

However, the study also uncovered a unique behavior specific to this quantum model that does not appear in standard biological simulations. Under certain conditions, particularly when the inhibitory signal was strong, the network would fall into a permanent silent state. In this scenario, the first neuron would fire a few times, triggering the inhibitory neuron, which would then shut down the first neuron completely. Remarkably, even after the inhibitory signal faded away, the first neuron remained silent, unable to restart its firing cycle. The researchers found that this happened because the quantum model resets to a specific "ground" state after every spike. If the inhibitory signal prevents a spike from happening, the system never gets the chance to reset, and it becomes trapped just below the threshold needed to fire again. This suggests that while the model is powerful, it has specific structural constraints that must be managed if larger networks are to be built.

Another critical finding concerned how the neurons communicated the strength of their signals. In the model, the amount of information sent with each spike was supposed to vary based on the neuron's internal state. The researchers discovered that this communication was heavily influenced by a technical limitation in their simulation called "Fock space truncation." In quantum physics, calculations often require limiting the number of possible energy states to make the math manageable. The team found that the largest, most dramatic spikes in their simulation were not a natural result of the neuron's dynamics but were actually artifacts caused by hitting the artificial ceiling of this limit. When they increased the size of this limit, the maximum spike size grew, proving that the extreme values were numerical boundaries rather than physical realities. This implies that the current way the model handles signal strength is more of a binary switch—indicating whether the system is normal or hitting a limit—rather than a smooth, continuous channel for transmitting nuanced information.

Despite these limitations, the work provides a crucial proof of concept. It shows that the fundamental logic of excitatory and inhibitory coupling, which is so vital to brain function, can be preserved when moving from classical circuits to quantum ones. The model successfully reproduced the expected behaviors of a forced system, where the neurons locked their firing rhythms to the external drive, a phenomenon known as phase locking. This alignment suggests that the core nonlinear dynamics of the brain are robust enough to survive the transition into the quantum domain.

The study concludes that while this two-neuron network is a simplified testbed, it successfully captures the qualitative essence of coupled neural dynamics. It confirms that quantum neurons can interact in ways that mirror biological reality, generating complex patterns of activity and silence. At the same time, it highlights the specific challenges that remain, such as the silent trap and the artifacts of signal amplitude, which must be addressed before these models can scale up to larger, more complex architectures. By mapping out both the successes and the boundaries of this quantum approach, the research lays the groundwork for future developments in quantum neuromorphic computing, bringing the vision of brain-like quantum machines one step closer to reality.

Technical Summary: Dynamics of Excitatory–Inhibitory Coupling in Quantum Memristive LIF Neurons

Problem Statement
Quantum neuromorphic computing aims to integrate brain-inspired architectures with quantum-mechanical principles to explore novel information processing paradigms. While individual quantum neuron models, such as the canonically quantized memristive leaky integrate-and-fire (QMLIF) neuron, have been established to reproduce quantum memristor characteristics and threshold-driven spiking, the collective dynamics of coupled QMLIF neurons remain largely unexplored. Specifically, it is unknown whether networks of these quantized units can retain the qualitative dynamical features associated with excitatory–inhibitory interactions, which are central to biological neural computation and neuromorphic system design. This work addresses the gap in understanding how QMLIF neurons interact through synaptic couplings, a necessary step toward scalable quantum neuromorphic architectures.

Methodology
The study introduces a minimal phenomenological model of two coupled QMLIF neurons (Neuron A and Neuron B) arranged in a closed loop. Neuron A receives an external drive and projects an excitatory synapse to Neuron B, which in turn projects an inhibitory synapse back to Neuron A.

Neuron Model: The QMLIF neuron is derived from a memristive LIF unit where the dissipative leak resistor is replaced by a semi-infinite, lossless transmission line bath (following the Caldeira–Leggett model). The neuron exhibits spiking behavior analogous to classical LIF models but resets to a vacuum state after firing, entering a refractory period.
Synaptic Coupling: Interactions are modeled using alpha-shaped postsynaptic kernels.
- Excitation: Modeled as an additional driving current ( $I_{exc}$ ) added to the Hamiltonian, promoting postsynaptic firing.
- Inhibition: Modeled as an enhanced effective leak ( $\gamma_{inh}$ ) in the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation, suppressing membrane depolarization.
- State-Dependence: Synaptic strength is modulated by a per-spike amplitude $\mu$ , defined as the expected photon number ( $\langle \hat{n} \rangle$ ) immediately before the reset to the vacuum state.
Numerical Implementation: Simulations utilize reduced units with fixed neuron parameters (adapted from Brand et al.) and a tunable Fock space dimension ( $N$ ) to assess truncation effects. Metrics include spike counts, inter-spike intervals (ISI), adaptation indices, and phase-locking values (PLV) to characterize firing patterns and synchrony.

Key Contributions
The paper makes four primary contributions:

Validation of Synaptic Roles: It demonstrates that in the quantum regime, excitatory synapses promote postsynaptic firing while inhibitory synapses suppress activity via enhanced effective leak, preserving the expected qualitative roles found in classical networks.
Generation of Firing Regimes: By tuning external drive and synaptic parameters, the network generates six distinct firing-pattern regimes (tonic spiking, initial/regular/delayed bursting, transient spiking, and irregular spiking) known from classical integrate-and-fire models.
Characterization of Phase-Locking: The study characterizes phase-locking under periodic forcing, showing that the coupled system exhibits entrainment phenomena consistent with periodically forced classical integrate-and-fire systems, including 1:1 and higher-order $n:m$ locking.
Identification of Model-Specific Constraints: The work identifies a distinctive "silent regime" where recurrent inhibition interrupts the reset-driven firing cycle, trapping the driven neuron below threshold. It further analyzes the per-spike amplitude, revealing its sensitivity to Fock space truncation.

Results

Firing Patterns: The network successfully reproduced six of eight canonical firing patterns. Two patterns, spike-frequency adaptation and delayed acceleration, were not reproduced. The authors attribute this to structural limitations: the feedback loop is purely inhibitory, lacking the slow state variables or positive subthreshold feedback required for those specific dynamics.
The Silent Regime: A significant finding is the emergence of a permanent silent state. When inhibition is sufficiently strong, it suppresses spikes during a drive cycle. Because the QMLIF model resets the density matrix to the ground state only upon firing, the absence of a spike prevents the reset. Consequently, the neuron settles just below the threshold and remains trapped there, even after the inhibitory leak decays. This is a structural feature of the formulation, not a generic network effect.
Phase-Locking: The driven neuron exhibits nonlinear entrainment behavior. High phase-locking values (PLV) correlate with sustained firing. Low 1:1 PLV regions at low frequencies were found to correspond to higher-order $n:m$ locking rather than weak synchronization. The locking maps for open-loop and closed-loop systems were nearly identical, suggesting recurrent inhibition has minimal effect on the locked region itself, though it introduces additional silent points at low drive amplitudes.
Synaptic Amplitude and Truncation: Analysis of the per-spike amplitude $\mu$ revealed that large-amplitude spikes are artifacts of Fock space truncation. The maximum amplitude scales linearly with the truncation dimension $N$ . Under constant negative current (which silences classical LIF neurons), the QMLIF neuron fires tonically with large amplitudes because the negative drive pushes the oscillator toward higher Fock states until the truncated basis saturates, forcing a rebound above threshold. This indicates that $\mu$ acts primarily as a binary marker of saturation rather than a continuous channel for graded quantum information.

Significance and Claims
The paper claims that even a minimal two-neuron QMLIF network can capture key phenomenological features of coupled neural dynamics, specifically the balance between excitation and inhibition and the nonlinear entrainment to external drives. These results suggest that the canonical quantization of memristive LIF neurons preserves the essential qualitative dynamics of their classical counterparts.

However, the authors are modest regarding the model's completeness. They explicitly state that the current amplitude-modulated synapse should be interpreted as a phenomenological coupling rule rather than a fully resolved channel for graded quantum information transfer, due to the dominance of truncation artifacts in the amplitude distribution. Furthermore, the identification of the silent regime and the inability to reproduce certain firing patterns highlight specific constraints of the current formulation. The authors conclude that these findings provide a controlled testbed for assessing coupled QMLIF dynamics but that future work must address internal dynamical mechanisms and truncation artifacts to enable scalable, robust quantum neuromorphic architectures.

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