Artificial neurons made of active matter memristors

Abstract

In this study we propose a new class of artificial neurons and memristors made of active chiral particles. We formulate a single-particle model to simulate active chiral particle behavior in a two-terminal device, with resistance depending on the particle position. We create a dynamical phase map connecting particle trajectories and memristor electrical properties to applied voltage and particle's self-propulsion parameters. Analysis of spiking modes in artificial neurons, with and without noise, shows the memristor switches between high- and low-resistance states, exhibiting stable limit cycles in the position-voltage phase response.

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Introduction

Memristors, first proposed by Leon Chua in 1971,¹ have gained significant attention for their ability to mimic synaptic and spiking behavior in biological neurons.² As the fourth fundamental passive circuit element, memristors enable energy-efficient neuromorphic computing, replicating the efficiency of biological neural networks.^3–5 With ongoing advances in materials science and nanotechnology, memristors hold great promise for applications in artificial intelligence and advanced data storage systems.⁶

The resistance of a memristor depends on the internal parameters controlled by the history of applied electric pulses.^7,8 An artificial neuron, consisting of a memristor connected in parallel to a capacitor and in series with an external resistor, is locally electrically active.⁹ Such devices can emulate spiking dynamics and facilitate hardware-based learning.^10–14 Their ability to learn in situ and retain memory highlights their potential in neuromorphic engineering.¹⁵

The mechanisms behind memristor operation are diverse and can generally stem from a combination of ion migration, phase transitions, and charge trapping/detrapping processes.¹⁶ For instance, ion migration involves the movement of ions within a solid electrolyte, forming and dissolving conductive filaments that modulate the device's resistance state.¹⁷ Recently, a new type of volatile memristor, known as a diffusive memristor, has been developed.^18–21 In these devices, resistance switching occurs due to the diffusion of metallic Ag nanoclusters in SiO₂, which create a conductive bridge between terminals.^22,23 The interplay of nanomechanical, thermal, and electrical factors results in rich and complex dynamics. Moreover, diffusive memristors can emulate short- and long-term plasticity in biological synapses, offering capabilities that surpass those of existing memristive systems.²⁴ Understanding these mechanisms is essential for optimizing the performance and reliability of memristive devices, enabling advancements in artificial intelligence, the Internet of Things (IoT), and energy-efficient data processing systems.^25–27

Inspired by the switching mechanism of diffusive memristors, we propose and numerically investigate a novel memristor based on active particles. There have been efforts to develop intelligent active matter based on swarm behavior of colloids and other soft matter systems²⁸ or applications of machine learning,²⁹ and investigations of collective dynamics of intelligent agents.^30–33 Here, we consider active Brownian particles–either biological or synthetic–that self-propelling in an fluid medium, such as Janus particles,^34,35 which due to their functional asymmetry, can convert chemical energy from their surroundings into kinetic energy. Currently, various effective strategies exist for synthesizing and manipulating Janus particles with diverse material composition or structural configurations.³⁶ The self-propulsion of these active particles introduces an additional timescale into the model, which, when combined with the typical spiking timescale of artificial neurons, is expected to produce rich dynamics. Indeed, as shown below, artificial neurons based on this new type of memristor exhibit unconventional spiking regimes.

Model

The memristor under investigation is a two-terminal device where particles can diffuse between terminals, and whose distribution determines the device's resistance. In this work, we simulate an artificial neuron model based on the single active particle memristor illustrated in Fig. 1(a). Our model is formulated by means of the following equations governing the position r ≡ (x,y) and orientation e = (cos θ, sin θ) of the active particle, the medium's temperature T, and the voltage V across the memristor:

Fig. 1 (a) Schematic of the artificial neuron circuit. R_ext represents the external resistor, C_ext the external capacitor, and V_ext the external voltage. R(x) denotes the resistance of a memristor embedding one chiral active Brownian particle, with self-propulsion speed v₀, and where θ is the angle between the self-propulsion direction and the positive x-axis, and ω is the angular velocity of the particle. (b) Relationship between the angular velocity ω of the particles and the voltage V across the memristor, see eqn (6).

Eqn (1)–(3) describe the dynamics of a standard two-dimensional (2D) chiral active particle characterized by a constant self-propelling speed v₀, and angular velocity ω.³⁷ Under unconstrained conditions, it undergoes circular motion with a gyration radius r₀ = v₀/|ω|, while its spatial position and orientation diffuse due to thermal fluctuations ξ_x(t), ξ_y(t), and ξ_θ(t). These fluctuations are modeled as zero-mean, δ-correlated Gaussian noises, satisfying 〈ξ_α(t)〉 = 0 and 〈ξ_α(t)ξ_β(t′)〉 = δ_αβ(t − t′) for (α,β = x,y,θ), where δ_αβ is the Kronecker delta and δ(t − t′) is the Dirac delta distribution. The phenomenological parameter η relates the particle's translational and rotational diffusivities, both proportional to the temperature T. We consider a single active particle moving in the channel of width L = 2 between the memristor terminals located at x = ±1. Reflecting boundary conditions are imposed along the x direction; the channel is open in the y-direction. The particle is confined within the channel by electrostatic repulsion at the side walls. Specifically, we simulated a conducting active particle embed in an insulating liquid and assumed the channel walls metallic. During electron tunnelling, the particle accumulates charge of the same sign as the nearest electrode, resulting in repulsive forces that keep it centered. This geometry represents the idealized configuration of two infinitely extended, parallel electrode surfaces placed in close proximity. Moreover, the device may be fabricated in a circular geometry so that any off-axis deviation simply returns the particle to its original path after a full rotation, preventing escape in the y-direction.

Eqn (4) and (5) describe the heat and voltage dynamics in a standard diffusive memristor.¹⁸Eqn (4) represents Newton's law of cooling, modeling the heat exchange between the device and the substrate. In our system, electrons tunnelling into and out of the particle gain energy eV which is released as heat inside the particle. Due to the particle's small heat capacity, its temperature rises significantly. Heat then dissipates into the surrounding liquid and substrate, whose thermal capacitances are orders of magnitude larger and therefore experience negligible temperature change. The heat source arises from Joule dissipation, proportional to the square of the voltage V across the memristor, and inversely proportional to its resistance R(x) and heat capacity C. The heat sink is characterized by the heat transfer coefficient k and is proportional to the temperature difference between the memristor and the substrate, with the latter assumed to be negligible. Eqn (5) is derived from Kirchhoff's law for the circuit, shown in Fig. 1(a), where the memristor is connected in series with an external resistor R_ext, in parallel with a capacitor C_ext, and driven by a constant external voltage V_ext. This configuration defines the circuit's timescale as τ = R_extC_ext. The memristor's resistance depends on the position of the active particle. Assuming electrons tunnel from the left terminal to the right through the particle, the resistance can be approximated as R(x) = R₀ cosh(x/λ), where λ is the tunneling length (see, e.g., ref. 9), and R₀ represents the minimum memristor resistance. Accordingly, the conductivity, normalized to 1/R₀, is expressed as

The frequency ω is expected to depend on both the particle's location r, due to trajectory deformations near the memristor's terminals, and the voltage V across the memristor, due to variations in the particle's self-propulsion mechanism induced by the electric field. The equation for ω may depend on the specific realization of the active matter memristor. Here, for the sake of presentation, we model the dependence of chirality on the electric field³⁸ using a simple sigmoidal function, assuming that ω changes sign within a narrow interval, αV*, around the threshold voltage V*, as illustrated in Fig. 1(b).

In diffusive memristors, the electric field that drives particles between terminals also modulates the resistance R(x) by forming conductive filaments that bridge the electrodes. The switching behavior is primarily governed by the dynamics of a single particle in the filament's narrowest region,^18,21 justifying the single-particle approach adopted in prior studies^18–21 and employed here. In our current model, we neglect orientation-dependent conductance under the assumption that the particle's diameter is much smaller than both the inter-electrode gap and the electron tunneling length.

We numerically solve eqn (1)–(6) using a fourth-order stochastic Runge–Kutta algorithm. In the following sections, we analyze the memristor in two scenarios: without noise and with noise. For consistency, we fix the following parameters unless specified otherwise: η = 0.01, ω₀ = 1, v₀ = 1, V* = 1, α = 1, CR₀ = 1, λ = 0.12, k = 0.1, τ = 0.3, V_ext = 1, and R_ext/R₀ = 10. These parameter values are consistent with those observed in real experiments with Janus particles, provided that lengths are expressed in micrometers and times in seconds. In particular, the wide range of chiral frequencies, ω₀, accessible in current experimental setups, allows for the realization of small gyration radii, r₀ β L, even in the regimes of low (η = 0.01) or vanishing rotational noise (η = 0) discussed below. The initial conditions (x₀,y₀,θ₀) = (0,0,0) are used in all simulations. Note all our results have been obtained after removing transients.

Spiking modes with out noise (T = 0)

In the absence of noise (T ≡ 0), where eqn (4) is disregarded, the system exhibits a spiking behavior, as illustrated in Fig. 2. When a chiral particle is positioned near the center of the memristor channel, it facilitates electron transitions between electrodes, effectively reducing the memristor's resistance. This reduction manifests as a decrease in the capacitor's voltage. Conversely, when the chiral particle is located near the boundaries of the channel, the electron transfer becomes hindered, leading to an increase in resistance. Upon applying an external voltage V_ext, the memristor initially resides in a high-resistance state, and the current begins charging the capacitor. As the particle migrates toward the center of the channel, the resistance R(x) decreases to its minimum value. In this low-resistance state, the capacitor discharges through the memristor, causing the voltage V to drop and R(x) to rise again, thereby recharging the capacitor. Dynamically, this process implies the existence of stable limit-cycle loops in the (x,V) phase space.

Fig. 2 Temporal dependence of the particle position x (a) and voltage V (b) in the spiking mode. The simulation parameters are: v₀ = 1, V_ext = 1, ω₀ = 1, V* = 1, α = 1, λ = 0.12, τ = 0.3, and R_ext/R₀ = 10.

The phase diagram in Fig. 3 illustrates the relationship between V and x in panel (a), or between V̇ and in panel (b), as the self-propulsion speed, v₀, and external voltage, V_ext, are varied. Both diagrams are divided into three distinct regions: (i) a region enclosed by a red solid line with single loops, (ii) a region enclosed by a blue solid line with asymmetric double loops, and (iii) the remaining regions exhibiting symmetric double loops. As v₀ increases, the particle frequently crosses the channel and collides with the boundaries, inducing hysteresis and dual-loop structures in the phase space. It is noteworthy that these three distinct dynamical topologies are intrinsic to the device's response, as they emerge naturally within a higher-dimensional space.

Fig. 3 The phase diagrams for varying v₀ and V_ext. (a) V versus x phase diagrams: each panel shows the closed trajectories of the chiral active Brownian particle in (x,V) plane where x (space, abscissa) and V (voltage, ordinate). (b) V̇ versus phase diagrams: each panel shows the system dynamics behavior in plane where (abscissa) and V̇ (voltage derivative, ordinate). In both panels, the red (blue) solid line encircles single (asymmetric double) loops; the remaining plots are symmetric double loops. Other simulation parameters are: α = 1, R_ext/R₀ = 10, ω₀ = 1, V* = 1, λ = 0.12, and τ = 0.3.

Fig. 4 illustrates the relationship between particle trajectories in the (x,y) plane and the electric spiking dynamics represented by the function G(t), corresponding to the different limit cycles of the memristor observed in the V-x phase diagrams of Fig. 3.

Fig. 4 Particle's trajectory samples in the (x,y) plane (left) and G(t) curves (right) for α = 1 and R_ext/R₀ = 10. (a) and (d): v₀ = 5 and V_ext = 7; (b) and (e): v₀ = 0.5 and V_ext = 7; (c) and (f): v₀ = 5 and V_ext = 1. The remaining simulation parameters are as in Fig. 3.

For symmetric double loops (V_ext = 7, v₀ = 5), as shown in Fig. 4(a) and (d), the particle's trajectory is closed and center-symmetric, resulting in a regular, well-defined spiking regime. The interspike interval in this mode is determined by the time between consecutive collisions with the left and right terminals.

For the closed single loop (V_ext = 7, v₀ = 0.5), as shown in Fig. 4(b) and (e), the particle's trajectory is asymmetric with respect to x = 0. The particle repeatedly bounces off the right terminal, leading to an unbounded trajectory. Consequently, the spiking behavior transitions into a quasi-single, low-amplitude modulation of G(t), reflecting the asymmetric single closed loops in the V-x diagram of Fig. 3.

For asymmetric double loops (V_ext = 1, v₀ = 5), as shown in Fig. 4(c) and (f), the trajectories are also asymmetric with respect to x = 0 and unbounded. A high v₀ causes the particle to linger longer near the right boundary (high-resistance state), with rapid excursions across the center (low-resistance state) toward the left boundary (high-resistance state) and back. This behavior gives rise to the unusual double-peak spiking pattern displayed in Fig. 4(f), mimicking bursting activity observed in biological neurons.

Fig. 5 displays the spike rate, defined as the number of recorded spikes divided by the recording time,³⁹ plotted as a function of the external voltage V_ext for four self-propulsion speeds. At v₀ = 0.1 [Fig. 5(a)], the rate falls from an initial peak, bottoms near V_ext ≈ 5, then plateaus before rising past V_ext ∼ 30. At v₀ = 0.5 [Fig. 5(b)], this nonmonotonic shape persists but with higher minima and maxima. The observed maxima of the spike rate as a function of controlling parameter is reminiscent a well-known effect of selectivity of biological neurons,⁴⁰ which is key for brain computations.⁴¹ For v₀ = 1 [Fig. 5(c)], the low-voltage peak weakens and the rate declines steadily. At v₀ = 3 [Fig. 5(d)], the system rapidly reaches a constant high-frequency state after a brief low-voltage transient. Thus, increasing v₀ converts a nonmonotonic response into an almost flat, elevated rate curve.

Fig. 5 Spike rate versus V_ext for (a) v₀ = 0.1, (b) v₀ = 0.5, (c) v₀ = 1 and (d) v₀ = 3. The remaining simulation parameters are as in Fig. 3.

These characteristics highlight the adaptive nature of our memristor design: by adjusting v₀ and V_ext, we can engineer diverse spiking regimes akin to those found in the brain activity.

Spiking modes with noise (T≠0)

After investigating the system's behavior in the ideal case of negligible thermal fluctuations, we now consider the influence of noise by including a non-zero temperature T and eqn (4). As shown in Fig. 6, the dependence of position x, temperature T, and voltage V, on time t reveals intricate dynamics. The trajectories, x(t), exhibit a series of noisy oscillations. The temperature, T, shows a gradual increase followed by a plateau, which points to a transient, during which the memristor approaches a new stationary state. The voltage across it, V(t), displays a series of oscillations that correlate with the position x. Indeed, the voltage peaks correspond to the particle's transitions between high- and low-resistance states, as illustrated in Fig. 7.

Fig. 6 Temporal dependence of the particle position x (a), temperature T (b) and voltage V (c) in the spiking mode. Simulation parameters are: v₀ = 1, V_ext = 1, η = 0.01, CR₀ = 1, k = 0.1, ω₀ = 1, V* = 1, α = 1, λ = 0.12, τ = 0.3, and R_ext/R₀ = 10.

Fig. 7 (a) Heat maps of V (ordinate) versus x (abscissa) for varying v₀ and V_ext. P(x,V) represents the particle's probability density at the point (x,V). (b) Heat maps of V̇ (voltage derivative, ordinate) versus (abscissa) for varying v₀ and V_ext. represents the particle's probability density at the point . Other system parameters are: α = 1, R_ext/R₀ = 10, η = 0.01, CR₀ = 1, k = 0.1, ω₀ = 1, V* = 1, λ = 0.12, and τ = 0.3.

In Fig. 7(a) we compare heat maps in the (x,V) plane for different values of the control parameters, v₀ and V_ext. As the applied voltage increases, the distribution of the particle's (x,V) across the memristor changes significantly. At lower external voltages (V_ext ≤ 0.1), implying low temperatures, the heat maps show that the active particle exhibits similar trajectories to the T = 0 case but with some slight spread due to thermal fluctuations. This however means that even in the presence of noise the system keeps switching regularly between well-defined high- and low-resistance states. As V_ext increases, the density of the probability distribution (x,V) tends to spread out. The heat map also reveals that with increasing temperature, no distinct high- and low-resistance states are identifiable, so that the switching occurs between continuously distributed states with different conductivity. Upon further raising the applied voltage, the particle's probability distribution in the (x,V) plane grows even broader. Noise can indeed assist bifurcation in memristors.⁴² However, in most cases presented in this work, the spreading of trajectories and corresponding spiking can be considered as a crossover rather than a noise-induced bifurcation. In Fig. 7(b), we show heat maps of the system's state in the plane for varying self-propulsion speed v₀ and external voltage V_ext. It can be observed that, as the external voltage increases, the memristor's system dynamics evolves from asymmetric to symmetric behavior.

The transition from order to randomness is a key characteristic of the memristor's behavior under thermal fluctuations. This offers the possibility of realizing a variety of neuromorphic computation paradigms, from oscillator-based spiking computing to probabilistic dynamical algorithms.

Conclusions

In summary, the investigation of active particle-based memristors has revealed their distinct electrical characteristics and dynamic switching behaviors, setting them apart from conventional memristive systems. The asymmetric resistance distribution, controlled by the motion of chirality-driven particles, highlights a complex interplay between particle dynamics and the electrical properties of the device. Through single-particle modeling, this study offers insights into the possible mechanisms underlying resistance switching.

In the absence of noise, the memristor exhibits various stable limit cycles in the (x,V) phase space, depending on the control parameters. These include symmetric double loops, closed single loops, or asymmetric double loops. These patterns underscore a robust switching mechanism between high- and low-resistance states. Noise introduces additional complexity, affecting the stability and predictability of the switching process. Heat map analysis reveals the coexistence of multiple stable states, reinforcing the memristor's potential for reliable information storage and processing.

Overall, this study underscores the promise of active chiral particle-based memristors in advancing computational paradigms. Their nonlinear oscillatory behavior and tunable resistance states position these devices as key enablers for the next generation of neuromorphic systems. By leveraging their unique properties, these memristors offer exciting opportunities for the development of more efficient and adaptive technologies in memory and computing applications.⁴³

Author contributions

S. S. designed the research; J. L. performed the research; all authors contributed to the data analysis; J. L. wrote the manuscript; S. S., M. G. M., Y. L. and F. M. edited it.

Conflicts of interest

There are no conflicts to declare.

Data availability

The authors confirm that the data supporting the findings of this study are obtained from numerical simulations. These are available within the article.

Acknowledgements

Y. L. is supported by the NSF of China under Grant No. 12375037, F. M. and Y. L. are supported by the NSF of China under Grant and No. 12350710786.

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