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Biological plausibility and stochasticity in scalable VO2 active memristor neurons

Wei Yi, Kenneth K. Tsang, Stephen K. Lam, Xiwei Bai, Jack A. Crowell, Elias A. Flores

arXiv:1809.07867v1cs.ETphysics.app-ph

TL;DR

Biologically competitive memristor neurons have a stated energy-efficiency threshold of >10^13 spikes/J, and VO2-neuron energy efficiency and area are evaluated against biological benchmarks. The paper studies VO2 active-memristor neurons and reports that they can surpass estimated human-brain energy efficiency at neuron sizes below 3 μm2.

Abstract

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Neuromorphic networks of artificial neurons and synapses can solve computational hard problems with energy efficiencies unattainable for von Neumann architectures. For image processing, silicon neuromorphic processors outperform graphic processing units (GPUs) in energy efficiency by a large margin, but they deliver much lower chip-scale throughput. The performance-efficiency dilemma for silicon processors may not be overcome by Moore's law scaling of complementary metal-oxide-semiconductor (CMOS) field-effect transistors. Scalable and biomimetic active memristor neurons and passive memristor synapses form a self-sufficient basis for a transistorless neural network. However, previous demonstrations of memristor neurons only showed simple integrate-and-fire (I&F) behaviors and did not reveal the rich dynamics and computational complexity of biological neurons. Here we show that neurons built with nanoscale vanadium dioxide active memristors possess all three classes of excitability and most of the known biological neuronal dynamics, and are intrinsically stochastic. With the favorable size and power scaling, there is a path toward an all-memristor neuromorphic cortical computer.

Locally-active memristors

This section examines VO₂ active memristors as compact neuron elements, covering device fabrication and performance, circuit mechanisms for biological spiking and excitability, and stochastic firing. It also considers energy scaling and circuit-level topics including synaptic plasticity, thermal management, and vertical stacking.

Locally-active memristors

  • A memristor operating in the current-controlled S-shaped NDR regime is locally active and can provide a.c. gain greater than 1 or excite oscillations in circuits with reactive elements.The work focuses on these devices because they are high-resistance when powered off, offering low standby power dissipation.Source: Locally-active memristors · p. 5
  • Electroform-free VO2 devices achieved >98 % typical yield, <13 % coefficient of variation in switching threshold voltage, and >26.6 million cycles without discernible I-V change.The nano-crossbar devices had critical dimensions from 50-600 nm.Source: Locally-active memristors · p. 6
  • Simulated VO2 Mott transition was 100 times faster than NbO2 and consumed about one-sixth (16 %) of its energy.The paper characterizes VO2 as superior to NbO2 in both switching speed and switching energy.Source: Locally-active memristors · p. 7

Circuit topology and spiking behaviors of VO2 neurons

  • The prototype neuron uses two resistively coupled relaxation oscillators, with oppositely biased active memristors emulating voltage-gated Na+ and K+ channels and each paired with a membrane capacitor and load resistor.The grounded capacitors represent local membrane potentials.Source: Circuit topology and spiking behaviors… · p. 7
  • Under steady d.c. current input, the neuron circuit fires evenly spaced spike trains or periodic bursts, characteristic of tonically active neurons.The passages distinguish these tonic behaviors from phasic firing, in which a neuron fires only at stimulus onset.Source: Circuit topology and spiking behaviors… · p. 10
  • Capacitive coupling enables phasic firing, while placing a capacitor in parallel with the load resistor produces a phasic burst followed by tonic spiking under steady d.c. input.Phasic firing can be realized by replacing the load resistor with an input capacitor or inserting the capacitor before it.Source: Circuit topology and spiking behaviors… · p. 10
  • Twenty-three biological neuron spiking behaviors were experimentally demonstrated in single VO2 active memristor neurons, measured in tonic, phasic, or mixed-mode circuits containing two VO2 memristors and four or five passive R, C elements.The demonstrated set includes Class 1 and Class 2 excitability, spike-frequency adaptation, and refractory period.Source: Figure 3 · p. 14
  • Each demonstrated behavior was measured in a single neuron circuit containing only two VO2 active memristors and four or five passive R, C elements.Spiking heterogeneity is achieved through controllable circuit parameters, including R and C values, input-stage impedance, and the area ratio of the two VO2 devices.Source: Circuit topology and spiking behaviors… · pp. 12–13, Figure 3 · p. 14
  • Changing membrane capacitances switches tonic neurons between Class 1 and Class 2 excitability: C2 < C1 gives Class 1, while C2 > C1 gives Class 2.Class 1 produces tonic spiking when 0.35C1 < C2 < C1 and tonic bursting when C2 < 0.35C1; Class 2 spike frequency is relatively constant and insensitive to input strength.Source: Circuit topology and spiking behaviors… · p. 15, Figure 4 · p. 16
  • VO2 neurons exhibit input-noise-sensitive, stochastically phase-locked firing, or skipping, with higher noise producing irregular spikes and many dropouts.In a tonic neuron, low noise yielded regular spiking with a median ISI of 29.3 µs; higher noise spread the JISI pairs across return maps.Source: Circuit topology and spiking behaviors… · p. 17
  • As input noise increases, tonic firing skips cycles while the fundamental ISI remains robust, and missed spikes retain sawtooth subthreshold oscillations.The figure illustrates firing under 5, 15, 25, and 50 μA peak-to-peak white noise.Source: Circuit topology and spiking behaviors… · p. 18, Figure 5 · p. 19
  • Spike-amplitude irregularity and skewness increase with input noise; mean amplitude falls quickly, then partially recovers above ~20 µApp.The skewness follows a similar trend.Source: Circuit topology and spiking behaviors… · p. 18

Discussion

  • Simulated VO2 neurons show superior EE-area scaling to best-case simulated HH cells below 70 μm2 and can surpass estimated human-brain EE of 1.8x10^14 spike/J (or 5.6 fJ/spike) below 3 μm2.Source: Discussion · p. 20
  • Tuning passive R,C elements and circuit topology yields varied neuronal dynamics without changing VO2 parameters; memristors or memcapacitors42 could enable adaptive class transitions.The proposed adaptive elements would replace fixed R,C elements.Source: Discussion · p. 20
  • A simulated VO2-neuron/TaOx-synapse circuit continuously potentiated or depressed synaptic conductance without parameter adjustments, demonstrating feasibility of a precursor to STDP learning.The circuit modeled one neuron and one synapse.Source: Discussion · pp. 20–21
  • VO2’s Mott transition near 67 °C creates a thermal-management challenge, but Ge doping has raised the transition temperature to ~96 °C at 5.9 at.%.Thermal management is especially pertinent when a CMOS co-processor is involved.Source: Discussion · p. 21
  • Thin-film memristor neuron–synapse cores could be vertically stacked for cortical-layer mapping; their pseudo-three-dimensional connectivity is difficult to achieve with conventional CMOS.The proposed integrated neuron requires up to three interconnect-metal layers, with passive memristor synapse arrays stacked above.Source: Discussion · p. 21
  • VO2 devices were electroforming-free, yielded 98–100% across 576 devices, and endured >26.6 million pulsed switching cycles without discernible I–V change.Measured switching thresholds ranged from 0.4 V to 1.3 V.Source: Methods · p. 22

Supplementary Tables

The supplementary tables quantify the low transition-energy cost of VO2 and document the shared device assumptions used in SPICE simulations. They also report that chaos is the only compared biological-fidelity property of VO2 neurons that remains unconfirmed.

Supplementary Table 1

  • 6.1 times: at equal volume, the free-energy cost of the NbO2 phase transition is that of VO2 multiplied by 6.1; a VO2 nano-crossbar with r = 10 nm and L = 10 nm has a total volumetric enthalpy change of 1.15 fJ.The NbO2 values are from Table 1 in Ref. 16 and references therein.Source: Table 1 · p. 66

Supplementary Table 2

  • The SPICE simulations use one 56 nm-radius, 100 nm-long cylindrical VO2 channel model, vary only R and C, and include 150–500 Ω series resistance and 13–17 kΩ parallel leakage resistance.The model uses published VO2 material properties and is reported to reproduce experimental switching dynamics and neuron spiking behaviors in most cases.Source: Table 2 · p. 67

Supplementary Table 4

  • Chaos is the only compared biological-fidelity property of VO2 neurons that remains unconfirmed.The table compares biological fidelity and computational cost across neuron models; it defines FLOPS as the approximate operations needed to simulate 1 ms, but supplies no values in the passage.Source: Table 4 · p. 70

Supplementary Notes · Supplementary Note 1: VO2 active memristor relaxation oscillator

VO2 active memristors use a Joule-heating-driven Mott transition to enable relaxation oscillation, with simulations indicating faster, lower-energy switching than NbO2 and sub-picosecond operation at nanoscale channel radii.

Supplementary Note 1: VO2 active memristor relaxation oscillator

  • Joule heating drives VO2’s Mott insulator-to-metal transition, producing volatile hysteretic switching and negative differential resistance used in oscillator and neuron circuits.VO2 has a transition temperature near 67 °C.Source: Supplementary Note 1: VO2 active… · p. 71
  • Electroform-free VO2 nano-crossbars on CMOS-compatible silicon substrates show 98–100% typical yield and <13% switching-threshold-voltage coefficient of variation.The devices operate at low voltage, down to about 0.5 V, and have high endurance.Source: Supplementary Note 1: VO2 active… · p. 71
  • A Pearson–Anson relaxation oscillator models voltage-gated Na+ or K+ membrane channels, and coupled oscillators can generate an action potential.Other ion channels, including Cl− and Ca2+, can also be emulated with similar circuits.Source: Supplementary Note 1: VO2 active… · p. 71
  • Astable oscillation requires the Vdc-and-load-resistor load line to intersect the VO2 device’s V–I curve in its negative differential resistance regime.The actual VO2 switching time is faster than oscilloscope rise-time measurement; pump-probe measurements reported 100 fs to 5 ps.Source: Supplementary Note 1: VO2 active… · p. 71
  • At matched channel dimensions, simulated VO2 switching is 100 times faster than NbO2 and uses about one-sixth the energy.The comparison fixes channel length at 50 nm while varying channel radius; switching speed is material-dependent and unaffected by passive R and C values.Source: Supplementary Note 1: VO2 active… · p. 71
  • At 7–15 nm channel radius, VO2 can achieve <1 fJ switching energy and <1 ps switching speed.These channel dimensions are feasible for advanced-node lithography.Source: Supplementary Note 1: VO2 active… · p. 71

Supplementary Note 2: Action potential generation in a VO2 active memristor neuron

A VO2 active memristor neuron generates an action potential through sequential Na+ and K+ channel switching, followed by a slow refractory recovery. Its timing is governed by capacitor charging and discharging.

Supplementary Note 2: Action potential generation in a VO2 active memristor neuron

  • A resting potential of 0.2–0.3 V arises from leakage through the two oppositely biased VO2 devices while both channels are insulating.A suprathreshold stimulus switches X1 to its metallic state, pulling VNa toward –ENa and discharging C1; the ultrafast switching precedes the slower hyperpolarization set by C1.Source: Supplementary Note 2: Action potential… · p. 72
  • Na+ channel hyperpolarization pulls VK low enough to switch X2 metallic, triggering depolarization and pulling VK toward +EK as C2 discharges.The depolarization timescale is determined by the C2 discharge time constant.Source: Supplementary Note 2: Action potential… · p. 72
  • The spike ends with a refractory undershoot and recovery to rest, the longest phase because C1 and C2 charge slowly back to their resting states.During recovery, the neuron does not respond to the next stimulus; unlike biological Na+-K+ pump recovery, the device recovery is electrical.Source: Supplementary Note 2: Action potential… · p. 72

Supplementary Note 3: Device modeling of VO2 active memristors

The VO2 active-memristor dynamics are modeled with the analytical equations of Ref. 16, using the normalized metallic-channel radius as the state variable. The note also describes a compact SPICE implementation.

Supplementary Note 3: Device modeling of VO2 active memristors

  • The model reuses the analytical equations developed by the authors of Ref. 16 to describe VO2 active-memristor dynamics.Its parameters are summarized in Supplementary Table 2.Source: Supplementary Note 3: Device modeling… · p. 73
  • The state variable u is the normalized radius of the metallic conducting channel heated above the Mott-transition temperature, and it determines channel resistance and state dynamics.The model specifies instantaneous current–voltage behavior, a first-order state equation, and state-dependent resistance, thermal conductance, and enthalpy change.Source: Supplementary Note 3: Device modeling… · p. 73
  • The VO2 compact model is constructed in SPICE following the approach of Ref. 16, and simulations were run on a personal computer with LTspice IV.Source: Supplementary Note 3: Device modeling… · p. 73

Supplementary Note 4: Dynamics equations of an active memristor neuron circuit

The note derives four coupled first-order ODEs for a tonic active-memristor neuron circuit and recasts them in terms of experimentally probed Na+ and K+ membrane potentials. The derivation assumes current clamp with no external output load; phasic equations are omitted.

Supplementary Note 4: Dynamics equations of an active memristor neuron circuit

  • Under current clamp, KCL gives input current I = IC1 + i1 + IC2 + i2 at the junction of RL1, C1, X1, and RL2.Source: Supplementary Note 4: Dynamics… · p. 74
  • Four coupled first-order ODEs in (u1, v1, u2, v2) govern the tonic active-memristor neuron, derived from device equations using KVL and KCL.The derivation assumes no external output load and current clamp; phasic-circuit equations are not included.Source: Supplementary Note 4: Dynamics… · pp. 74, 76
  • The equations are recast using experimentally probed membrane potentials VNa = v1 − E1 and VK = v2 + E2.Source: Supplementary Note 4: Dynamics… · p. 76
  • Although Eqs. (S17) and (S18) appeared in Ref. 32, that work used hard switching between Ron and Roff for reduced-dimension VNa–VK nullcline analysis, missing nonlinear dynamics of Eqs. (S7) and (S9).Source: Supplementary Note 4: Dynamics… · p. 76

Supplementary Note 5: Dynamic and static power scaling of VO2 neurons

VO2 neuron power scaling depends on membrane capacitance and firing rate: dynamic spike energy rises nearly linearly with capacitance, while static power can dominate at low firing rates.

Supplementary Note 5: Dynamic and static power scaling of VO2 neurons

  • Dynamic spiking energy scaled nearly linearly with membrane capacitance (slopes 0.96 at r/L = 10/10 nm and 0.924 at 36/50 nm), while an 18-fold VO2 volume reduction cut spike energy by ~24%.Dynamic spike energy was calculated by summing power from d.c. voltage sources and integrating it over a spike; at r/L = 10/10 nm, VO2 switching energy was 1.15 fJ/device. Neuron area also scales linearly with membrane capacitance, and no minimum membrane capacitance is required for operation.Source: Supplementary Note 5: Dynamic and… · p. 77
  • At 1 fF/μm2, VO2 neurons exceeded best-case HH-cell EE below 70 μm2 and estimated human-brain EE of 1.8u10^14 spike/J below 3 μm2.The estimated human-brain EE corresponds to 5.6 fJ/spike energy use.Source: Supplementary Note 5: Dynamic and… · p. 77
  • At fixed neuron area, lower capacitance density reduces dynamic spiking energy and improves EE; high-κ capacitors can enable <0.1 pJ/spike at ~1 μm2 area using 20 fF capacitors.The cited record-high high-κ MIM capacitance density is 43 fF/μm2, versus a typical 15–20 fF/μm2; boosting capacitance density is not an effective route to higher EE.Source: Supplementary Note 5: Dynamic and… · p. 77
  • Static power falls below 10% of total above 100 MHz at Vth bias and 400 MHz at Vth/2, with overall energy below 0.11 pJ/spike.Because static power is constant while dynamic power scales with firing rate, static power may dominate at low rates; estimates assume finite insulating-phase resistivity and do not consider improving it. Network power also depends on firing rate, synapse resistance, and the synapse/neuron ratio.Source: Supplementary Note 5: Dynamic and… · p. 77
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