Ultralow Power Electronic Analog of a Biological Fitzhugh–Nagumo Neuron

Abstract

Here, we introduce an electronic circuit that mimics the functionality of a biological spiking neuron following the Fitzhugh–Nagumo (FN) model. The circuit consists of a tunnel diode that exhibits negative differential resistance (NDR) and an active inductive element implemented by a single MOSFET. The FN neuron converts a DC voltage excitation into voltage spikes analogous to biological action potentials. We predict an energy cost of 2 aJ/cycle through detailed simulation and modeling for these FN neurons. Such an FN neuron is CMOS compatible and enables ultralow power oscillatory and spiking neural network hardware. We demonstrate that FN neurons can be used for oscillator-based computing in a coupled oscillator network to form an oscillator Ising machine (OIM) that can solve computationally hard NP-complete max-cut problems while showing robustness toward process variations.

Introduction

Biological brains can perform computational tasks at an ∼100,000× efficiency compared to the digital computers.¹⁻⁶ A typical biological neuron has a surface area of ∼10 μm², spends ∼10 pJ energy to generate each spike, and operates at a frequency of ∼100 Hz, which translates to a power cost of ∼1 nW for biological systems.^3,4,6 The first set of efforts in emulating biological neurons dates back to 1960s following the FN model using voltage controlled NDR devices^7,8 paired with inductors to produce relaxation oscillations similar to neuronal spiking behavior.⁹⁻¹¹ The inductor element is the main scaling bottleneck of this circuit implementation of spiking neuron, as coil-based passive inductors are difficult to fabricate at nanoscale with the required inductance values. The emergence of current-controlled NDR devices featuring metal–insulator phase transition materials has enabled generation of relaxation oscillations using capacitors, leading to considerable progress in artificial spiking neurons.¹²⁻²³ There have been other approaches to producing NDR, such as band to band tunneling, resonant tunneling, Gunn effect, real space electron transfer in III–V heterostructures, body biasing of MOSFET, exploiting graphene’s unique dispersion relationship near its Dirac point, using trap-based recombination processes, redox behavior of molecular junctions, and multiple circuits.^7,8,24−42 Recently, we have shown that a graphene–silicon photodetector can show voltage-dependent NDR behavior under optical illumination while operating in the photovoltaic regime.⁴³ This photosensor coupled with an inductive circuit element generates optically driven voltage oscillations similar to those of ganglion cells in the retina, following the FN model of spiking neurons. Using a frequency multiplexed parallel computing approach, these oscillatory retinal neurons (ORNs) were then used to demonstrate ultralow-power in-sensor neuronal computing for machine vision. Purely electronic oscillatory neurons can extend the ability of FN neurons to different types of sensors and enable the construction of deep oscillator-based neural networks. In addition, electronic oscillatory neurons are instrumental to traditional oscillatory neural networks (ONNs), where a network of coupled oscillators can be used as associative memory⁴⁴⁻⁵⁸ or an engine for convolution-like operations.^{44,45,59−62} However, there are three important challenges to achieving ultralow power oscillator-based computing: (1) scalability, (2) energy consumption, and (3) variability of the oscillatory neurons. These neurons serve as the smallest unit of computation in oscillator networks similar to the different logic gates in digital computing systems. Therefore, a cohesive effort is necessary to improve the design of individual neuron circuits and the algorithms that would allow the employment of these neurons for the promised excellent performance of the brain-inspired computing architectures.

In this work, we have introduced an FN neuron implementation with a tunnel diode and an active inductor. The scaling bottleneck presented by passive inductors is removed in this implementation of the FN neuron by using the active inductor that can easily be fabricated at nanoscale by traditional CMOS processes. The active inductor emulates the behavior of a passive coil-based inductor by taking advantage of an active element: a single MOSFET, and a resistor. Through circuit simulations, we demonstrated the oscillation behavior of the FN neuron implemented using this active inductor and predicted the scalability limits. The key feature of this neuron is that it is expected to generate voltage spikes with ∼2 aJ/cycle energy cost, whereas the Mott transition-based neurons consume an electrical energy of ∼50 pJ/cycle, and the state-of-the-art CMOS neuron consumes ∼4 fJ/cycle.^{12,14,17−21,23} These FN neurons can generate firing patterns similar to biological neurons. We have then used FN neurons to form a coupled oscillator network onto which a computationally hard combinatorial problem such as max-cut problem can be mapped in an Ising machine-like fashion. This OIM can then minimize the Ising Hamiltonian associated with the max-cut problem and identify correct solutions while consuming a small electrical energy.

Results and Discussion

The spiking behavior of biological neurons was first described by the Hodgkin–Huxley (HH) model. The HH model interprets the dynamics of ion flows across neuronal membrane and the resulting generation and propagation of action potentials through a set of coupled differential equations. The Fitzhugh–Nagumo (FN) model reduces the complexity of the HH model by using just two variables: the membrane potential and the recovery variable. Despite its simplicity, the FN model captures some of the essential features of the Hodgkin–Huxley model, such as the concept of a threshold for firing an action potential and the recovery of the membrane potential after firing. Moreover, the FN model can reproduce many of the behaviors observed in the Hodgkin–Huxley model such as the propagation of action potentials and the refractory period. Dynamics of Fitzhugh–Nagumo neurons can be described by the following coupled differential equations:⁶³

Here, v is reminiscent of the membrane potential of a neuron, which is excited by an external stimulus i_ext analogous to the ion pump current through a nonlinear medium f(v). f(v) can be simplified as a third-order polynomial. Such an excitation is then dissipated through a slower sink w. Figure S1a shows the nullclines and the trajectory of the system for for parameter values of (a, b, r, τ, i_ext) = (0.7, 0.8, 0.5, 12.5, 0.4). Figure S1b shows the waveforms obtained for the Fitzhugh–Nagumo neurons for different values of i_ext , showing that action potentials or spikes are not generated for under-excitation or over-excitation conditions. Figure S2 shows the nullclines and trajectories of the system under different parameter values.

Here, we develop a circuit to emulate FN neurons, with the equivalent circuit shown in Figure 1a, with the third-order nonlinear element f(v) approximated by a tunnel diode, a circuit element that exhibits negative differential resistance. The following coupled differential equations describe the dynamics of this circuit:

Figure 1.

Implementation of an FN neuron. (a) Equivalent circuit of an FN neuron using a tunnel diode. (b) Circuit schematic of an FN neuron using Hara inductor.

These equations are equivalent to those of the Fitzhugh–Nagumo neurons and differ only by a scaling factor. In the absence of the resistor R and the offset DC voltage V_applied, this circuit would simplify to a more traditional Van der Pol oscillator.⁶⁴⁻⁶⁶ While such a circuit accurately captures the description of a Fitzhugh–Nagumo neuron, the presence of an inductor in the circuit limits its implementation and scalability.

To overcome this challenge, we explore the effect of implementing this circuit with a Hara active inductor comprising a single FET and a resistor.⁶⁷⁻⁶⁹Figure 1b shows the circuit schematic for an FN neuron by using the Hara inductor and a tunnel diode. The parasitic capacitance of the tunnel diode (C) is implicit here. There are two sources of electrical power in this circuit: V_G and V_applied. However, impedance of the series R_G–C_gs branch is , which blocks any DC current flow, and therefore power consumption in the V_G source is zero. Therefore, all of the power in this neuron comes from the source, V_applied. This power can be calculated from

where T is the period of one fundamental oscillation. Here, the capacitive current integrates to zero over a full cycle of oscillation. The energy in each cycle can be calculated from . As a rule of thumb, the power consumptions in both the tunnel diode and active inductor are ∼I_peakV_peak, where I_peak is the peak current of the tunnel diode before the start of the NDR regime, and V_peak is the voltage at which I_peak is observed. Since I_peak decreases linearly with tunnel diode device area, any reduction in power and energy consumption requires us to scale down the diode to the smallest possible size.

The nonlinear element in the FN neuron does not necessarily have to be a tunnel diode, as there are many different CMOS compatible circuit topologies that allow NDR with a small footprint. However, a tunnel diode is also a CMOS compatible device that allows aggressive scaling to the nanometer scale. Figure S3a shows the experimentally measured J–V and C–V curves of a silicon tunnel diode.⁷⁰ We have considered a linear scaling behavior between diode current and physical diode area, which is a reasonable assumption for tunnel diodes as discussed in Section S2.1. This silicon tunnel diode is composed of a vertically stacked p⁺⁺ (∼10²⁰ cm^–3)/n⁺⁺ (∼10²⁰ cm^–3) silicon homojunction, completely compatible with CMOS fabrication processes. As discussed in Section S2.2, a large peak current density of 1.75 μA/μm² indicates that a tunnel diode can be scaled down to 100 nm × 100 nm size while keeping a ∼30 dB signal-to-noise ratio at 1 GHz bandwidth. In addition to tunnel diodes that achieve NDR behavior through band-to-band tunneling, resonant tunneling diodes (RTDs) utilizing the resonant tunneling phenomenon in multiple quantum well-like structures can also be used in this implementation of FN neurons. Due to the near unity transmission probability for carriers, Si/SiGe-based RTDs can achieve a much higher NDR peak current density (2.18 mA/μm²)⁴² compared to tunnel diodes while maintaining similar capacitance per area. As a result, an RTD-based oscillator can potentially reach a much higher oscillation frequency.^36,38,39,41 Recently, RTDs have therefore been explored for designing spiking neurons for applications in neuromorphic computing.^35,37,40 However, fabrication of RTDs is more complicated, since RTDs are heterojunction devices. It is also more difficult to achieve NDR behavior in RTDs at room temperature due to its inherent physics. On the other hand, the simplified fabrication process, increased variation tolerance, and comparatively smaller susceptibility to thermal noise of silicon tunnel diodes make them better candidates for practical implementation of FN neurons.

Figure 2a,b shows a small signal equivalent circuit for the Hara inductor. The small signal model gives us an equivalent inductance of , where R_G is the series resistance to the gate, C_gs is the gate to source capacitance of the FET, and g_m is the transconductance of the FET.^67,69 The DC voltage source in series to the gate determines the bias point and consequently g_m of the FET. However, it is important to note that the spiking neuron operation can lead to voltage swings of ∼100 mV, and the small signal model would not be applicable in that case. Nevertheless, it is a good starting point to design the neuron circuit regardless of the nonidealities introduced by the large signal voltage swings. Figure 2c shows L_eq as a function of operating frequency for a 65 nm CMOS process MOSFET with W/L = 1 at different bias voltages and R_G = 200 MΩ. Figure 2d shows the equivalent resistance seen at the input terminal. These results show that L_eq can be tuned by the bias voltage. However, an active inductor has a positive equivalent inductance only within a certain frequency range, which puts certain design constraints on the FN neuron.

Figure 2.

Small signal behavior of the Hara inductor. (a) Small signal model for Hara inductor and (b) equivalent circuit. (c) Equivalent inductance and (d) series resistance of Hara inductor using a 65 nm process MOSFET with W/L = 1 and R_G = 200 MΩ.

Using the established behavior of the individual elements, we combine them to form the FN neuron, as shown in Figure 1a. We have performed circuit simulations for a FN neuron using the I–V curve and device capacitance obtained from the experimental data of a silicon tunnel diode and an ideal inductor with no series resistance (ideal Van der Pol oscillator case). The experimentally obtained I–V curve was first mathematically fit and then interpolated to generate a lookup table. Using a lookup table allows us to speed up simulations by avoiding evaluation of the nonlinear function as we solve the system of ordinary differential equations at each time step. Figure 3a–d shows the colormaps for oscillation amplitude, frequency, power consumption, and energy consumption, respectively, for a diode of 1 μm² device area. We observe from these colormaps that oscillation is only possible within certain applied voltage ranges, and the oscillation amplitude is significant after a threshold inductance. These values are functions of the tunnel diode I–V curve as oscillations can only be sustained at the voltages for which NDR behavior exists. We have then introduced a normalized quantity analogous to the characteristic impedance of a transmission line. When the device area (A) decreases, both the current and capacitance of the diode decrease linearly. This quantity captures the effect of scaling both the device area and inductance. Figure 3e shows the fundamental oscillation frequency (f_peak,1) normalized to the resonant frequency of the LC circuit () and oscillation amplitude for V_applied = 0.4 V. We can observe a sharp increase in oscillation amplitude at clearly marking the onset of oscillation. It is noteworthy that such an oscillator approximates an LC oscillator for , where . Figure 3f shows some example oscillation waveforms for the FN neurons at different values of , where we can observe the emergence of sharp spikes as we keep increasing . As we scale the tunnel diode down to smaller sizes, both I and C will scale linearly, and we would need to increase the inductance L linearly to maintain the same oscillation threshold and frequency. For a tunnel diode with an area of 1 μm², C = 25 fF, we would need an inductance of ∼100 nH to start the oscillations.

Figure 3.

Simulated oscillation behavior of a silicon tunnel-diode-based FN neuron. Colormaps of (a) fundamental oscillation frequency, (b) amplitude, (c) power, and (d) energy consumption in the FN neuron using passive inductors. (e) Normalized fundamental oscillation frequency and amplitude of the FN neuron. (f) Typical oscillation waveforms at different values.

In order to understand the scaling limits of this FN neuron, we have then simulated the Hara inductor-based neurons using the 65 nm CMOS process-based MOSFET.⁷¹ It is important to note that while a 65 nm MOSFET has been used here for these simulations, quantitatively similar results can be obtained by using any other FET such as 5 nm FINFETs. Since the inductive load depends on R_G, C_gs, and g_m (), FETs of different sizes and processes can achieve the same inductance by adjusting their transconductance (g_m) by applying an appropriate V_G so that ratio remains constant. In addition, the capacitive load seen by the oscillator is a parallel combination of the parasitic capacitance of the diode and the drain-to-source capacitance (C_ds) of the FET. Since C_ds is typically much smaller (∼100×) compared to diode capacitance, the operational frequency of the oscillator will not be affected significantly by using different FETs as long as C_diode ≫ C_ds. Figure 4a shows the oscillation frequency colormap for a FN neuron comprising a 1 μm² tunnel diode and R_G = 200 MΩ. As shown in Figure 4a, there is a particular range of V_G and V_applied values at which spiking neural behavior can be sustained. A decrease in V_G decreases the inductance and increases the series resistance. As a result, we see a larger oscillation frequency at a smaller V_G. However, the increasing series resistance requires us to increase V_applied to sustain oscillation. Figure 4b shows some oscillation waveforms for different V_G and V_applied. We then performed the same simulation for tunnel diodes of different device areas between 400 nm² and 1 μm². Figure 4c shows f_peak,1 as a function of diode area, showing an ∼80 MHz oscillation frequency for the smallest devices. Figure 4d,e shows the P_osc and E_osc for the FN neuron for a minimum oscillation amplitude of 50 mV. P_osc and E_osc are calculated using the following equations,

Figure 4.

Simulated oscillation behavior of FN neuron implemented with the Hara inductor. (a) Colormap of fundamental oscillation frequency. (b) Example oscillation waveforms at different conditions. (c) Fundamental oscillation frequency, (d) power, and (e) energy consumption as a function of diode area. (f) Energy cost comparison with other artificial spiking neuron technologies.

Here, T is the fundamental oscillation period. The decrease in P_osc and E_osc is linear to the neuron area until the MOSFET area becomes the limiting factor. For a 0.0046 μm² neuron area (20 nm × 20 nm diode, 65 nm × 65 nm MOSFET), we estimate a P_osc of 1.4 nW and a E_osc of 2 aJ, which is the lowest among reported energy per oscillation cycle for different electronic neurons.^{6,12−15,17,18,22,34,72−78} It is important to note that the spread in the data comes from all possible oscillation conditions at different V_G and V_applied. The current flow and operation voltage for this FN neuron implementation depend on the tunnel diode I–V characteristics. As discussed in more detail in Section S5, we do not expect any significant influence of the FET sizing or process on the oscillation frequency or amplitude and therefore also on power and energy consumption. Figure 4f shows the comparison between the energy consumption per cycle for different types of neurons found in the literature. There are mainly three other approaches to making an artificial neuron: (1) metal–insulator transition (MIT) memristors and a capacitor, (2) circuit-based techniques implemented with CMOS technology, and (3) ferroelectric FETs (FeFETs) with a regular FET. While the MIT neurons can be scaled down to ∼1000 nm², the minimum energy cost is ∼50 pJ for oscillation, which is significantly larger compared to that of biological neurons.^{6,12−15,17,18,22,72} CMOS-based neurons have a large distribution of power and energy costs depending on the circuit techniques used for implementing the spiking neural behavior.^34,73−78 Current state-of-the-art spiking neuron⁷⁵ has a 35 μm² area with a 4 fJ energy consumption, whereas a typical biological neuron of ∼10 μm² area has an energy consumption of ∼10 pJ. FeFET neurons also demonstrate promise in achieving oscillation with an energy cost of ∼1 pJ. In contrast to all these artificial neurons, our proposed FN neuron consumes an electrical energy of 2 aJ, which can further the state-of-the-art of this field.

An FN neuron is a two-dimensional dynamical system capable of showing several biological neuron-like firing patterns, such as phasic and tonic spiking. Excitability of an FN neuron is dependent on the design parameters. Figure 5a shows three different operations of an FN neuron with a diode area of 0.02 μm², R_G = 110 MΩ, V_applied = 0.39 V, and three different values of V_G. At smaller V_G (0.1 V), transconductance of the FET is smaller, and the Hara inductor presents a large inductive impedance. As a result, the nonlinear term in the FN neuron model becomes smaller, and the FN neuron shows nearly harmonic oscillations. As V_G becomes larger (0.2 V), inductive impedance becomes smaller, and the oscillations become more nonlinear with sharper spiking nature. This is similar to tonic spiking in biological neurons. When V_G becomes even larger (0.35 V), the neuron shows a transient spike, followed by overdamped oscillations similar to phasic spiking patterns. Figure 5b shows the corresponding FFT spectra of the oscillation time series shown in Figure 5a. As expected, the nearly harmonic oscillations show much smaller high harmonic peaks compared to the fundamental frequency, while the tonic spiking pattern shows significantly larger high harmonic contents. The phasic spiking pattern does not show any significant frequency peak as the oscillation dies away with time. It is important to note that FN neuron model inherently lacks the capability to generate bursting patterns due to its limited dimensionality. However, it is possible to cascade two FN neurons, as shown in Figure 5c to increase the dimensionality and generate tonic bursting patterns (Figure 5d). Essentially, when the oscillations generated by one neuron are much slower than the other (R_G1 = 100R_G2 = 11 GΩ), it is possible to modulate the V_G2 input of the second neuron using the output of the first neuron (V_osc,1). As a result, the second neuron only fires when the first neuron provides the necessary voltage. Therefore, it is possible to generate different firing patterns by designing electronic circuits with multiple FN neurons.

Figure 5.

Biological neuron-like firing patterns. (a) Firing patterns of a single FN neuron showing tonic and phasic spiking and (b) corresponding FFT spectra. (c) Circuit schematic of two cascaded FN neurons. (d) Tonic bursting patterns generated by the cascaded circuit.

FN neurons can be used to create traditional spiking neural networks (SNNs) for performing tasks typical of artificial neural networks (ANNs). These traditional SNN architectures encode information in the timing (temporal encoding) or frequency (rate encoding) of spiking patterns and make use of all-or-nothing spiking dynamical behavior of neurons to process information.⁷⁹⁻⁸¹ These architectures typically mimic the behavior of ANNs while utilizing the low energy consumption behavior of spiking neurons. Performance of these networks has been further limited by their incompatibility with backpropagation methods of learning. The dimensionality of these networks is reduced due to their all-or-nothing spiking nature. On the other hand, oscillator-based computing approaches utilizing synchronization behavior of oscillator networks through interaction can potentially make use of a much richer computational space to perform more versatile tasks. A liquid state machine comprising a network of oscillators is theoretically capable of performing all tasks doable by a Turing machine without resorting to any learning mechanism.⁸²⁻⁸⁷ OIMs have been used to solve computationally hard combinatorial problems by exploiting the oscillation dynamics of a coupled oscillator network. In our previous work, we have shown that it is possible to perform massively parallel computing by multiplexing operations in the frequency domain. SNNs are therefore a subset of the massive computational space offered by computing systems by using oscillator dynamics.

While an oscillatory neuron is the building block for neuromorphic oscillatory computing, the actual computation emerges from the macroscopic behavior of a network of coupled oscillators. Harnessing these emergent computational behaviors is contingent on the construction of the coupling network. There have been prior works on associative memory property of coupled oscillators where the oscillator phases settle to certain values resulting from the minimization of the “energy” of the system.^{44,45,47,50,55,61,62} OIM is a special case of such a phase-based computation scheme, where a computationally hard combinatorial optimization problem can be mapped to the coupling impedances of the oscillator network. Under certain circumstances, the oscillators would then settle to a combination of binary phases (0 or π) that result in a minimization of the system energy that corresponds to the Ising Hamiltonian of the problem.⁵⁵⁻⁵⁸ Typically, the oscillator phases can take continuous values between 0 and π. However, if the oscillators are externally injected with a signal with twice their fundamental oscillation frequency, it is possible to force them to have binary phase differences of 0 or π because of subharmonic injection locking (SHIL).⁵⁵⁻⁵⁷ Detailed mathematical analysis on mapping NP-hard and NP-complete combinatorial problems to OIMs has been carried out elsewhere.^{44,55−58,62} This work focuses on how FN neurons can be utilized for oscillator-based computing, and we have chosen to demonstrate how an FN neuron-based OIM can be used to solve an NP-complete problem such as the “max-cut” problem. If we have a graph with N-nodes and the nodes have M-edges between them with different weights, then the max-cut problem aims to find a binary partition between the N-nodes that results in the maximum number of cuts to the edge weights. For the sake of simplicity, let us consider a graph with 2-nodes with a single edge with unity weight. Then, partitioning the two nodes into different groups results in the max-cut of 1 (Figure S5a). Figure S5b shows how this problem can be mapped on two FN neurons coupled with a resistor, R_coupling. When R_coupling is too small, the oscillators are always in phase, and there is no partition between the oscillators. On the other hand, when R_coupling is large, the oscillators are almost uncoupled, and therefore they do not maintain a specific phase difference between them. However, for intermediate values of R_coupling, they would always couple with a phase difference of π, resulting in a partition. We have performed simulations for X different random initial conditions for the FN neurons for a given R_coupling and then calculated the probability of finding correct solution as . Figure S5c shows P_solution as a function of R_coupling for V_inj = 0.5 V and R_SHIL = 50 MΩ (X = 1000). Figure S5d shows evolution of the oscillation waveform toward the correct solution, i.e., coupling antiphase. It is noteworthy that the symbol of oscillator includes the FN neuron as well as an externally applied SHIL signal, as shown in Figure S6a.

Figure 6a shows the graph for an 8-node max-cut problem and the solution that yields maximum cuts of 10. Figure S6b shows the circuit schematic implementing this max-cut problem. Here, we consider R_ij = R_coupling for all values of (i,j). Figure 6b shows the evolution of oscillator phases toward the partition that gives us maximum cuts. This result was achieved for R_coupling = 150 MΩ, R_SHIL = 50 MΩ, and V_inj = 0.5 V for f_SHIL = 150 MHz. It is important to note that P_solution is heavily dependent on the parameters of the coupling network. The correct partition between oscillators depends critically on two factors: (1) the ability to achieve phase bipartition and (2) the ability to achieve the correct order of phase bipartition. While phase bipartition ability comes explicitly from the SHIL signal and therefore depends on the strength of V_inj, the correct order of phase bipartition comes from the coupling between different oscillators and hence on R_coupling. Figure 6c shows the colormap of P_solution as a function of R_coupling and V_inj for R_SHIL = 50 MΩ (X = 768). As shown in Figure 6c, high values of P_solution can be achieved only when the combination of R_coupling and V_inj is within a certain range. However, one of the key issues with implementing OIMs is the variation between the oscillators. Since the phase bipartition is very sensitive to the oscillation dynamics, variations in oscillators can lead to significant degradation of P_solution. For nanoscale devices, these variations are more pronounced due to random dopant fluctuation, lithography-based size variations, etc. Random dopant variation is most pronounced since ion implantation in CMOS processes is inherently a stochastic process, and such variation leads to significant spread in threshold voltages across different MOSFETs. In addition, the tunnel diodes also feature heavily doped p⁺⁺ (>10¹⁹ cm^–3) and n⁺⁺ regions (>10¹⁹ cm^–3). Therefore, we can also expect some variations in the tunnel diode current levels. However, when there is current flow in a diode, its capacitance is dominated by the charge transport capacitance or diffusion capacitance. This capacitance is directly proportional to current, and therefore, such current variations also lead to linear variations in device capacitance. As discussed in Section S6, when there is a variation in both device current and capacitance, the differential equation governing the oscillation dynamics remains immune to these fluctuations when the fluctuations are small. We have then introduced a Gaussian variation in both device current and capacitance of the tunnel diodes for the 8-node max-cut circuit and simulated the OIM oscillation dynamics. Figure 6d shows P_solution as a function of standard deviation (σ_current) in tunnel diode peak current, showing the insensitivity of P_solution as σ_current increases. When we consider variations in V_th ( = 20 mV), as experimentally observed in a previous study on 65 nm MOSFETs,⁸⁸ there is a slight degradation in P_solution as shown in Figure 5d. These results clearly show the promise of FN neurons in oscillator-based computing. While CMOS compatibility and scalability would allow realization of large scale coupled oscillator networks, electrical energy consumption of such networks can also be minimized because of the superior energy efficiency of FN neurons.

Figure 6.

Implementation of an Ising machine with FN neurons. (a) A 8-node graph for max-cut problem and its solution. (b) Evolution of oscillation waveforms toward correct solution from random initial conditions. (c) Colormap of solution probability as a function of R_coupling and V_inj without considering any variation. (f) Robustness of P_solution to variations in tunnel diode current.

Conclusion

In conclusion, we have demonstrated an active inductor-based implementation for an FN neuron and predicted the scaling behavior of this neuron through simulations. Our simulations show that it is possible to generate the oscillation behavior in these neurons at an extremely low energy cost of 2 aJ per cycle. These FN neurons can generate firing patterns similar to biological neurons, such as tonic and phasic spiking. Firing patterns unachievable by a single FN neuron, such as bursting patterns, can also be generated by designing appropriate circuits with these neurons. FN neurons can be used in a coupled oscillator network to form an Ising machine and solve NP-complete combinatorial problems such as the max-cut problem while being robust to process variations. This work shows a roadmap to designing and implementing FN neurons through voltage controlled NDR devices, replacing the real inductor with the MOSFET-based inductor, generation of oscillations with ultralow electrical energy, and possible application in large scale variation insensitive oscillator-based computing systems.

Data Availability Statement

The data that support the plots within this paper and other findings of this study are publicly available at https://doi.org/10.6084/m9.figshare.25405207.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.3c09936.

• Additional details about theoretical modeling of FN neuron and Ising machine; nullclines and trajectory of a FN neuron (Figure S1); FN neuron nullclines and trajectories for different values of I_ext (Figure S2); experimentally measured J–V and C–V curves of a silicon tunnel diode, peak current density of silicon homojunction, and GaSb/InAs heterojunction tunnel diodes of different diode areas (Figure S3); oscillation, frequency, amplitude, power consumption, and energy consumption for different NMOS areas (Figure S4); simple 2-node max-cut problem and mapping 2-node max-cut problem to FN neurons (Figure S5); symbol for oscillator circuit and the actual circuit (Figure S6) (PDF)

Author Contributions

R.A. and R.K. conceived the project. R.A., Z.W., and S.A.A.J performed the simulations. All authors contributed to analyzing the data. R.A. and R.K. wrote the paper while all the authors provided feedback.

1. Department of Energy Grant No. DE-SC0022248. 2. National Science Foundation Award No. 2004791. 3. Office of Naval Research Grant No. N00014–21–1–2634. 4. Air Force Office of Scientific Research Grant No. FA9550–21–1–0305.

The authors declare no competing financial interest.