Adaptive spiking, itinerancy, and quantum effects in artificial neuron circuit hardware with niobium–hafnium oxide-niobium memristor devices inserted

Harnessing and merging the fields of neuromorphic computing and quantum information processing is a practical way to take steps toward designing and developing new hardware that operate with biological-inspired autonomous control.¹ Therefore, increasing our understanding and knowledge of neuron behavior and developing technologies that can mimic it are essential to move forward with this goal.² Currently, artificial neuron circuit designs are mostly aimed at matching a circuit response that can produce operations to mimic the known output response of biological neurons, as extracted either experimentally from neuroscience experiments or modeled with biological models. These biological neuron models range in complexity and biological inspiration. The current approaches typically emulate neuron models with digital and analog transistor circuits³ and have achieved impressive results; however, they may lack sufficient built-in memory, non-linear dynamical behavior, and quantum phenomenon in their operation to come close to biological functionality. Thus, integrating memristors and other novel memory devices in the design of artificial neuron hardware is an intense topic in the scientific community.⁴ Memristors provide a dynamic and non-linear operation and memory effect, where the most common mechanism is reversible ionic motion in a thin film. Reducing this film, to the point where quantum tunneling is the primary conduction mechanism provides for a faster and lower voltage operation and emergence of quantum effects. Recently, memristive circuits were demonstrated that output various spiking modes, but observation of hybrid non-chaotic/chaotic attractor modes and itinerancy, where time is spent in changing attractor modes, requires increased non-linearity and feedback.⁵ Theoretical proposals have also recently pointed to the possibility of harnessing a quantum mechanical phenomenon in the design of artificial neurons to take advantage of quantum information science protocols, but experimental demonstrations are limited.^6–8 

We designed and experimentally examined a high-speed artificial neuron circuit and simple networks with quantum tunneling memristor hardware inserted in the positive and negative feedback loops of an analog spiking oscillator, providing inhibitory and excitatory dynamics in a coupled non-linear dynamical system with built-in synaptic memory. We demonstrate the adaptive properties, hybrid non-chaotic/chaotic modes, and the emergence of strong itinerant behavior when networked. With the memories in the superconducting state, Josephson tunneling emerges and results in a new solid-state superconductor-ionic system, where signs of the influence of coherent quantum control are observed in accordance with the quantum master equation (QME) calculations of the expectation values and correlation functions using a calibrated time-dependent Hamiltonian under strong driving field conditions. As a utility of this hardware, we experimentally demonstrate a four-neuron ring network performing autonomous learning and the modulation of signals that exploit the biologically inspired functionality.

To provide further insights into the learning operations, larger scale circuit simulations (see methods) of a network with nine artificial neurons, as shown in Fig. 1(e), were also done with the model parameters presented in Table S1. Figure S2 shows several examples of various spiking modes that can be obtained for a single artificial neuron. Figures 1(f) and 1(g) shows the simulated output and the dynamics of the memristor dynamical state variables upon the application of a small sinusoidal stimulus signal. The output starts with adaptive training and finally to a steady state spiking output that represents the learned encoded information that was reinforced by the autonomous learning as determined from the memristors dynamical state variables. We note that such simulations are limited to a scale of around 40 artificial neurons on a digital computer, thus further motivating the development of dedicated AI hardware.

We experimentally examined this artificial neuron circuit using memristor/resistive switching (ReRAM) hardware that we built on silicon wafers with 4.2 nm of atomic layer deposited (ALD) ionically active hafnium-oxide (HfO_x) and sputtered niobium (Nb) electrodes, i.e., (Nb–HfO_x–Nb) produced with the process given in Ref. 9. First, memristors were independently characterized. Figures 2(a) and S3 show pulsed current/voltage (IV) measurements taken at room temperature sweeping forward and reverse from −2.0 to 2.0 V with 300/900 ns pulse widths and 60/100 ns rise/fall times. The characteristics are non-linear with a significant hysteresis and resistance switching of current of 4–5 orders of magnitude. The primary conduction mechanism for this voltage range and with 4.2 nm of HfO_x and a Nb–HfO_x barrier of 2.2 eV is by direct and field-effect quantum tunneling as determined from the dependency from a plot of ln(I/V²) vs 1/V, as shown in Fig. 2(b), where regimes of direct tunneling (DT) with a transition to field-effect tunneling is evident.^9,12 As shown in Fig. 2(c), we present a representative band diagram in accordance with the expected tunneling electron flow between normal metals at these warm temperatures, where T_Nb−HfOx is the transmission probability that can be understood from a Schrödinger solution and we consider the presence of the ionic states impacting it, and hence, $I=2πAℏ∫−∞+∞TNb−HfOx2NNbε−VNNbεfε−V−fεdε$⁠,¹³ where N_Nb are the density of electron states available for tunneling and f is the Fermi–Dirac function. The changes in the memory response with increasing pulsing speed point to a combination of effects. The hysteresis is consistent with volatile ionic changes/motion due to the creation and motion of ionic oxygen vacancy states and any residual thermal fluctuations.^14–17 The increasingly larger hysteresis of 0.5–0.7 V with 300 vs 60 ns pulsing and the more pronounced shoulder is indicative of negative differential resistance and the combined contributions of the partial ferro-electric effects and metal–insulator transitions due to the ionic composition and the Nb–HfO_x interfaces.^18,19

We measured the full circuit response with the memristors inserted. The experiments were done with the memristors probed on a chip while in a cryogenic probe station and wired up with discrete op-amps, resistors, capacitors, and diodes that resided on an electronics breadboard, Fig. S4. Figure 2(d) shows example experimental outputs with changing DC bias voltages. First, we isolated a spiking mode that is classical and stable in appearance to those produced with the Hodgkin–Huxley (HH) neuron model.^20,21 Next, we adjust the DC stimulus to ±2.4 and ±2.0 V to change the response to a chaotic mode and a bursting mode. The extracted spiking rates for these modes are greater than 200 kHz with the MHz spectrum. To examine the degree of spike rate adaption (SRA) and other adaptive behaviors, which are common attributes in biological neurons and key for training and learning, we also examined the addition of a small signal excitation. Figure 2(e) compares the outputs with just the DC voltage stimulus and with an added RF sinusoidal stimulus (0.2 V_pp/5 MHz), and Fig. 2(f) shows the extracted spiking rates vs time. For the DC input, this rate adapts starting from around 200.1 kHz and reduces over the course of around half a milli-second to 100.3 kHz with a dependence like that produced with the biologically inspired adaptive exponential model (AdEx).²² With the added RF analog stimulus, the spiking rates start at 201.3 kHz and reduce to 160.2 kHz with a more abrupt transition that effectively produced a new bursting spiking mode as the adaptive response. An example of the phase-plane trajectories of a single artificial neuron at room temperature that produces hybrid non-chaotic/chaotic attractor modes is shown in Fig. 2(g) and Video S1. With a single neuron, the attractors enter hybrid non-chaotic/chaotic modes with a distinct saddle point,^23,24 and as we add neurons, the complexity observably increases, consistent with the formation of a neural network, as shown in the examples shown in Figs. 2(h)–2(j) Ref. 25 and Videos S2–S4.

Next, we experimentally examined the effect of cryogenically cooling the memories into a superconducting state. Figure 3(a) show the measured data of a Nb–HfO_x–Nb device taken at 8.1 K sweeping from ±1.0 mV in the forward and reverse directions. A critical current, due to cooper pair tunneling, a sum-gap voltage Δ = 0.31 mV, and at increasing voltage a quasiparticle current. These characteristics have a hysteretic memory effect. With 0.24 T applied, a decreased tunneling current is apparent and consistent with a Josephson tunneling junction with Nb electrodes.¹² The characteristics are impacted by the excitation and decay of carriers into ionic centers where there is energy level splitting in the quantized environment. The fluctuations are attributed to residual noise with a thermal energy of 0.6 meV. The impacts of the cryogenic temperature and field effect further impacts the coherence length $ξn=DkBTℏ2π1/2$ through the diffusion constant D = (1/3)v_fl_n, with Fermi velocity v_f; l_n is the mean free path depending on the level of ionic states¹³ and ℏ is the reduced Planck’s constant resulting in observable hysteresis.

Dissipation is introduced through the collapse operators C_n and $Cn†$⁠, and we adjusted their rate of decay to 0.15 σ_ge and simulated two conditions: (i) where the intensity of the driving field A = 10 and a Hilbert space dimension of 4 and (ii) A = 40 with a Hilbert space of 8 and plot the time evolution of ⟨a^†a⟩ and ⟨$σge†σge$⟩ and also plotted against each other to represent the experimental data collected in the phase space. Figures 3(g) and 3(h) show the calculations for scenario (i) and Figs. 3(i) and 3(j) for scenario (ii). The behavior in these simulations is in good qualitative agreement with the response observed experimentally. The larger Hilbert space required to capture the effects of the increasing driving field in the experiment points supports the notion that quantized energy level splitting is further enhanced and thus increases the dimensionality of the ionic states available for any coherent processes. To further evaluate, we calculated the first and second order correlation functions $g1(t)$ and $g2(t)$ as implemented in Ref. 25, which represent the level of quantum coherence and entanglement that could be supported:

$g1(t)=a†ta(0)a†ta(t)a†0a(0)$ and $g2(t)=a†0a†ta(t)a(0)a†0a(0)2$⁠. The calculated correlation functions are shown in Figs. 3(k) and 3(l), and their sustainment over the course of duration provides further evidence that quantum coherent control occurs under these conditions. We note that while this Lindblad formulation of the QME in Eq. (5) is appropriate for short-term memory effects, i.e., Markovian, our collected experimental data show some signs under network conditions^26,27 that point to a time-delay reversal phenomenon, followed by re-establishment that occurs in the dynamical behavior, as shown by the time evolution in Figs. 3(c)–3(f).

Finally, we created a neuron–neuron network where the memories of one artificial neuron are kept warm at room temperature and the second neuron with its memories operated in the quantum control regime, as shown in Fig. 4(a), and we optimized the circuit for MHz speeds. The goal was to provide a system where strong itinerant behavior becomes significantly pronounced by networking neurons with very different intrinsic attractor modes and regimes of bifurcation, as recently proposed in theoretical studies on designing artificial autonomous systems with chaotic itinerancy.^24,28 Figure 4(b) shows an example of spiking output when the memristors of one artificial neuron are cooled to 8.1 K. As shown in the three-dimensional plots in Figs. 4(c) and 4(d), as temperature is reduced, the two neuron networks’ collective behavior enters several regimes where the attractors show strong signs of itinerancy, i.e., behavior where time is spent in the trajectory of a dynamical attractor mode and then changes to several other modes (Videos S7–S9). Such behaviors are commonly seen in the collective effects of biological neurons,^29,30 and where, for example, such influence can result in spontaneous changes in motion and directionality of birds.

As an application, we experimentally examined a four synthetic neuron ring feedback network proposed by the neuroscience community³¹ as a possible way that biological organisms contribute to implementing gait control or central pattern operations as connected, as shown in the photo of the breadboard Fig. S5 in a ring architecture, and we provide stimulus signals as a means of introducing data for the circuit and collected the outputs at all four nodes in real-time shown in Figs. 4(e) and 4(f). We isolated, as an example a mode, and as shown in Fig. S6(a) and (Video S10) and under close examination there are interesting changes to the level of phase shift happening as a function of time between the various nodes, which is the typical requirement for such a circuit to train and learn various gait or central control patterns consistent with the simulations done earlier in Figs. 1(f) and 1(g). With suitable driving of the circuit, we observe adaptive changes to the spiking modes because of the adaptive and itinerant properties shown in Figs. S6(a) and S6(b) even at the scale of networks with just four artificial neurons. As an additional near-term utility, we use the neuron output to modulate RF photonics communications signals. The output of an artificial neuron, as shown in Fig. S6(c), serves as the source of I/Q modulation of a higher frequency carrier signal at 20 MHz produced with a vector signal generator (VSG). This neuron modulated RF signal then drives an acousto-optic modulator (AOM) that modulates a near-infrared 975 nm fiber optic laser operating with a power of 40 mW. The modulated optical signal is then propagated in a single mode fiber and is detected by using an avalanche photodetector and the output is viewed on an oscilloscope. As observed, this protocol successfully results in a modulated output with a complex time-dependent pattern due to the hybrid non-chaotic/chaotic initial spiking mode with an increased spiking rate of 6–10 times, as shown in Fig. S6(d). Therefore, such a protocol using these synthetic neurons can be used to generate complex types of signal modulations.

An artificial neuron was designed and examined experimentally with quantum tunneling memristor hardware formed from a 4.2 nm of atomic layer deposited ionic hafnium oxide and niobium metal and inserted in the positive and negative feedback of an analog spiking oscillator. When operated at room/warm temperatures, these memories have memristive properties and enable the artificial neuron circuits to produce adaptive spiking behavior in accordance with biological models and a non-linear dynamical model we derived that supports the contributions of the inhibitory and excitatory feedback and the memory state non-volatilities. When networks were formed, pronounced hybrid chaotic/non-chaotic modes with increased complexity are observed and itinerant behavior emerges. We demonstrated a four-artificial neuron feedback ring network performing autonomous learning and the use of these artificial neurons for the modulation of RF photonics communications signals. The measurements of these artificial neurons when the Nb–HfO_x–Nb memories are cryogenically cooled into the superconducting Josephson tunneling regime at 8.1 K revealed that the influence of quantum control effects in this solid-state superconductor-ionic system and quantum mechanical calculations of the expectation values and correlation functions, with the introduction of a strong time-dependent Hamiltonian, are in good agreement with the acquired experimental spectrum. This study provides key steps forward in our understanding of the design and hardware implementation of artificial neuron and network hardware that can operate with adaptive-itinerant properties and quantum effects to help advance and improve the quality of biological inspired neuromorphic technologies.

The supplementary material PDF file includes the Materials and Methods, Full mathematical model derivation, Circuit simulations, Memory device measurements, Artificial neuron and networks measurements, Applications, Figs. S1–S6, Table. S1. The auxiliary supplementary material for this manuscript includes the following: Videos S1–S10.

We acknowledge the use of the UCSD Qualcomm Nano³ facility for nanofabrication. We also acknowledge the NIWC information technology and lab infrastructure. We thank to E. Bozeman for providing some of the discrete electronics. This manuscript is a tribute to the memory of Professor Judy L. Hoyt (MIT).

This research was funded by the Office of the Secretary of Defense (OSD) applied research for the advancement of priorities (ARAP) programs on quantum science and engineering (QSEP) and neuromorphic electronics (Neuro-pipe), the NIWC PAC Naval innovative science and engineering program (NISE), and the Office of Naval Research (ONR) Independent laboratory initiative for research program (ILIR). Distribution Statement A. Approved for public release: distribution is unlimited.

OMN declares US Patents (1) “Advanced process flow for quantum memory devices and Josephson junctions with heterogeneous integration,” US Patent 9 455 391 and (2) Quantum memory device and method, US Patent 9 385 293. HM, MK, and JM don't have competing interests.

Osama M. Nayfeh: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Haik Manukian: Investigation (supporting); Writing – review & editing (supporting). Matthew Kelly: Investigation (supporting); Writing – review & editing (supporting). Justin Mauger: Investigation (supporting); Writing – review & editing (supporting).

All data are available in the main text or the supplementary material. Several videos are included in the auxiliary supplementary material. Requests for additional data or questions should be made to the corresponding author.