Spiking artificial neurons emulate the voltage spikes of biological neurons and constitute the building blocks of a new class of energy efficient, neuromorphic computing systems. Antiferromagnetic materials can, in theory, be used to construct spiking artificial neurons. When configured as a neuron, the magnetization in antiferromagnetic materials has an effective inertia that gives them intrinsic characteristics that closely resemble biological neurons, in contrast with conventional artificial spiking neurons. It is shown here that antiferromagnetic neurons have a spike duration on the order of picoseconds, a power consumption of about 10−3 pJ per synaptic operation, and built-in features that directly resemble biological neurons, including response latency, refraction, and inhibition. It is also demonstrated that antiferromagnetic neurons interconnected into physical neural networks can perform unidirectional data processing even for passive symmetrical interconnects. The flexibility of antiferromagnetic neurons is illustrated by simulations of simple neuromorphic circuits realizing Boolean logic gates and controllable memory loops.
I. INTRODUCTION
Artificial Intelligence (AI) and machine learning seek to replicate the cognitive functions of the human brain. AI is currently employed in many applications, for example, in image recognition, data processing, natural language processing, computer vision, and decision making. In the near future, AI will be of increasing importance for self-driving cars, medical diagnostics, and many other areas. Interestingly, computers using deep learning and other machine learning algorithms can identify handwritten numbers with an accuracy that surpasses the ability of the human brain.1,2
Machine learning algorithms typically use an artificial neural network (ANN) running on a conventional, silicon semiconductor based computer platform. Unfortunately, training ANNs can be quite computationally intensive and consequently consume a substantial amount of energy; for example, AlphaGo—the first AI program to defeat a professional human player in the game of Go—required more than 1 MW for operation.3,4 In another example, the energy cost to train an ANN for natural language processing is estimated to exceed ten million dollars, with significant negative environmental consequences.5,6 Computational intensity also limits the utility of AI for applications that require low power; for example, in mobile phones, at remote data sensors, or in other edge-computing applications that bring computational resources closer to data sources. Moreover, for applications that require processing and learning from vast quantities of data in real time, traditional silicon based computing platforms greatly limit the utility of AI.
In contrast, the human brain can perform these tasks in real time with a power requirement that is substantially less than 20 W.7 It is anticipated that the speed and power efficiency of AI algorithms will be greatly enhanced by the use of specialized brain-inspired, neuromorphic computing hardware.8–11 It has been proposed that neuromorphic computers that use spikes in a manner similar to biological neurons can greatly improve the computational speed and power consumption of spiking neural networks (SNNs), the power consumption of which will be substantially lower than that of conventional ANNs.12 For example, Intel recently developed a 2 × 109 transistor Loihi chip that has an architecture based on spiking neurons.13,14 Before that, IBM designed and fabricated TrueNorth, which has 1 × 106 digital neurons and a power consumption of less than 65 mW.15–17 Other prominent computing platforms that feature spiking neurons include SpiNNaker (University of Manchester),18 Braindrop (Stanford),19,20 and BrainChip.21 These systems all exhibit improved efficiency by employing a computer architecture that is primarily event driven, similar to the human brain. They employ spiking neurons that consist of multiple silicon based transistors per neuron, which have a behavior that differs substantially from biological neurons.22
Artificial neurons composed of antiferromagnetic (AFM) material, which we call “AFM neurons,” have characteristics that resemble biological neurons.23–25 A schematic diagram of a nanometer-sized, single element artificial AFM neuron is shown in Fig. 1(a). In this figure, the AFM material is shown in yellow, and it is in contact with a metallic electrical conductor, shown in blue. The metallic conductor acts as a terminal for the AFM neuron, with an input and an output. When an electrical current, with a DC bias current and an input current impulse, passes through the conductor [Fig. 1(b)], the AFM material will respond by generating a short-duration voltage spike, as shown in Fig. 1(c). This output spike is similar to an action potential that is generated by a biological neuron. It is evident from this figure that a characteristic output spike for an AFM neuron is quite fast, with a duration of ps. This high speed, which is faster than other artificial neurons, is a direct result of the intrinsic properties of the AFM material.
AFM neuron. (a) A cartoon of the AFM neuron, which consists of an AFM material and a metallic conductor. The input and output of the neuron are as labeled. (b) Simulated input current supplied to the AFM neuron. The bias current is A, with input impulses just before 100 and 300 ps. (c) Simulated output spikes from an AFM neuron in response to the input at 100 and 300 ps.
AFM neuron. (a) A cartoon of the AFM neuron, which consists of an AFM material and a metallic conductor. The input and output of the neuron are as labeled. (b) Simulated input current supplied to the AFM neuron. The bias current is A, with input impulses just before 100 and 300 ps. (c) Simulated output spikes from an AFM neuron in response to the input at 100 and 300 ps.
As will be shown in this paper, there are several distinct advantages of AFM neurons over the currently employed silicon based alternatives. First, AFM neurons exhibit biologically realistic characteristics including refraction, latency, bursting, and inhibition, all of which can be controlled dynamically. These are intrinsic physical characteristics of AFM neurons. Second, the duration of spikes is in the picosecond timescale, allowing for an operational speed that is substantially faster than that of conventional computers. Third, the power to operate this device is several orders of magnitude lower than the state of the art. Fourth, as this artificial neuron consists of a single element, it has the potential to greatly simplify the design and fabrication stage of manufacture. Moreover, the spatial dimension is on the nanometer scale so that they can, in principle, be included with integrated circuits that use complementary metal–oxide–semiconductor (CMOS) technology.26 Finally, AFM neurons operate at room temperature, and when interconnected, it will allow the development of physical spiking neural networks.
It is worth noting that AFM neurons belong to a broader class of technology known as spintronics, which has been implemented widely in data storage, among other novel applications. There is much research exploring spintronic devices for novel computing architectures.8,27–36
This paper is organized as follows: In Sec. II, the physics of the AFM neuron will be discussed in simple terms, and the differential equations governing their behavior will be introduced. Then, Sec. III demonstrates the operation of a single AFM neuron. After this, Sec. IV will demonstrate the interconnection of neurons and explains how these neurons could be employed in a spiking neural network (SNN). Finally, Sec. V demonstrates a few simple neuromorphic circuits that employ unique physical properties of AFM neurons. This will be followed by conclusion remarks.
II. PHYSICS OF AFM NEURON OPERATION
This section sets out to explain how AFM neurons function and the physical principles underlying their behavior. It begins by presenting a physical model of an AFM neuron, including equations describing its behavior. Then, realistic parameters for AFM neurons are presented, which will allow for simulation of AFM neurons and AFM neural networks. After this is a brief section with estimates on the speed of performance, size, power consumption, and feasibility of manufacture.
A. Antiferromagnetic neuron physics
Magnetic materials can be classified into different categories, including ferromagnetic (FM) materials and antiferromagnetic (AFM) materials.37,38 FM materials are familiar in everyday life as permanent magnets and have a single magnetic lattice with the magnetization M pointing in a certain direction.37,38
Antiferromagnetic materials are similar to FM materials, except they can have two or more intrinsic magnetic sublattices with different magnetizations. A visualization of a simple AFM material is shown in Fig. 2(a), with two intrinsic magnetizations M1 and M2. Due to a very strong exchange interaction, these two magnetizations are inclined to be anti-parallel. Thus, if the direction of M1 is changed, M2 will also be reoriented, and vice versa.37,38 Due to the antiparallel orientation of M1 and M2, the net magnetic moment of an AFM material vanishes, and thus, AFM materials do not create any stray magnetic fields. This allows for close packing of AFM elements in integrated circuits, thus greatly increasing the areal density compared to FM-based circuits.
AFM neuron configuration. (a) An antiferromagnetic material, NiO, with two anti-parallel magnetizations M1 and M2. At equilibrium, these two magnetizations are oriented along the easy axis be. Also shown, by a semi-transparent blue color, is the easy plane. (b) The AFM material together with a Pt film. The electric current is flowing through the Pt. When the NiO easy plane is oriented as shown in this figure, the electric current creates a torque on the AFM that causes M1 and M2 to rotate. The rotation in the easy plane is shown by a dashed red arc.
AFM neuron configuration. (a) An antiferromagnetic material, NiO, with two anti-parallel magnetizations M1 and M2. At equilibrium, these two magnetizations are oriented along the easy axis be. Also shown, by a semi-transparent blue color, is the easy plane. (b) The AFM material together with a Pt film. The electric current is flowing through the Pt. When the NiO easy plane is oriented as shown in this figure, the electric current creates a torque on the AFM that causes M1 and M2 to rotate. The rotation in the easy plane is shown by a dashed red arc.
AFM materials are much more common in nature than FM materials. The great variety of AFM materials allows one to choose, for each particular application, a material with optimal parameters. In this article, we consider AFM neurons based on nickel oxide (NiO)—a high-quality AFM dielectric (insulator), which is currently one of the most extensively studied materials in AFM spintronics.39–41
AFM materials can be isotropic or anisotropic. In isotropic AFM materials, the magnetizations M1 and M2 are free to rotate in any direction as long as they stay antiparallel to each other. In anisotropic AFM materials, on the other hand, there are more and less preferable directions of magnetization. Nickel oxide is a bi-axial anisotropic AFM. The anisotropy in NiO causes M1 and M2 to remain in a plane called the “easy plane,” as shown in Fig. 2(a).37,38 Deviations from the easy plane cost a lot of energy and typically can be ignored. Within the easy plane, additional in-plane anisotropy in NiO aligns M1 and M2 with be, the easy axis, which is also shown in Fig. 2(a). The in-plane anisotropy in NiO is relatively weak, so magnetization can rotate in the easy plane if disturbed by an external signal. This in-plane rotation of M1 and M2 is the basis of AFM neuron operation.
An AFM neuron can be formed if a metal film with a strong spin–orbit interaction, such as platinum (Pt), is grown onto the AFM material. Platinum is an electrical conductor. When an electric current flows through the Pt in the geometry depicted in Fig. 2(b), the AFM magnetization can be made to rotate in the easy plane due to the spin Hall effect and spin transfer torque (STT).42,43 As M1 rotates with respect to the easy plane, it has an azimuthal angle ϕ with the easy axis. The azimuthal angle ϕ is shown in the coordinate axis at the top right corner of Fig. 2. The dynamic change in ϕ can be modeled with the following differential equation:23,42,44
In this equation, ωex = 2πfex is the exchange frequency, α is the dimensionless effective damping parameter, ωe = 2πfe is the easy axis anisotropy frequency, σ is the spin-torque efficiency, and I is the input electric current for this system. Further details about the physics of AFM neurons and the derivation of Eq. (1) can be found in Refs. 42 and 44. In this paper, we refer to (1) as the artificial neuron equation.
When M1 and M2 have a non-zero angular velocity , an impulse of electric voltage is generated via the inverse spin Hall effect.44–46 This is the output signal of the AFM neuron. It is given by a simple equation,23,42
where v(t) is a time dependent voltage and β is the spin pumping efficiency.
B. Mechanical analog: simple pendulum
To develop an intuitive understanding of (1), this section investigates a mechanical system that is also described by the artificial neuron equation. The system is a simple pendulum with an applied torque.23 A schematic of the pendulum is shown in Fig. 3(a). In this schematic, an object with a mass m is situated at a distance l from its axis of rotation. It is attached to the axis of rotation by a thin rod, which is assumed to be both rigid and weightless. The mass is subject to a gravitational force with an acceleration g. The pendulum is displaced from the vertical axis by the angle ϕ, as depicted in Fig. 3(a). It is assumed that the mass is free to rotate continually about its axis, although the rotation is subject to viscous friction.
Simple pendulum as a mechanical analog of an AFM neuron. (a) The mass m is affixed at a distance l from an axis and is displaced vertically by an angle ϕ. It is subject to a torque from gravity τg, an external driving torque τd, and a frictional torque τf. (b) The driving torque has displaced the mass by ϕ0 = 155°. With a small push, it will pass the vertical line and revolve once about its axis, which is shown by a red arc.
Simple pendulum as a mechanical analog of an AFM neuron. (a) The mass m is affixed at a distance l from an axis and is displaced vertically by an angle ϕ. It is subject to a torque from gravity τg, an external driving torque τd, and a frictional torque τf. (b) The driving torque has displaced the mass by ϕ0 = 155°. With a small push, it will pass the vertical line and revolve once about its axis, which is shown by a red arc.
In this model, the mass is affected by three torques. First, the mass is subject to a gravitational force, so the torque due to gravity is given by τg = −mgl sin ϕ. Second, it is assumed that there is an external driving torque applied to the mass, which we define as τd. Third, there is a frictional torque, which is given by , where b is a damping constant and is the angular velocity of the mass. Thus, the damping torque τf is proportional to the angular velocity and acts against the motion of the pendulum.
Before deriving the equation of motion, it is useful to develop an intuitive feel for the motion of the pendulum. Three cases will be considered here. First, when there is no driving torque, τd = 0, the mass will be in equilibrium with an equilibrium angle ϕ0 that is zero (ϕ0 = 0). If the position of the pendulum is momentarily displaced, the mass will oscillate back and forth until friction returns the mass to its equilibrium position.
Consider a second situation where there is a very large driving torque. In this case, the mass will revolve repeatedly around the axis of rotation, with a velocity that varies depending on the instantaneous angle ϕ of the mass. Specifically, when ϕ > 180°, the angular velocity will increase under the influence of gravity, and when ϕ < 180°, gravity will act to slow the angular velocity of the pendulum. These changes in velocity will be approximately sinusoidal in form.
The two previous situations are relevant for understanding the mechanics of this system. The third situation is analogous to the functioning of the AFM neuron, as will be explained with an example that is depicted in Fig. 3(b). In this example, a moderate torque τd, which is not sufficient to cause permanent rotations, is applied. Instead, the torque will shift the equilibrium angle ϕ0 of the mass to a point where τd is compensated by the gravity torque, τg(ϕ0) = τd [in Fig. 3(b) ϕ0 = 155°]. If the mass is subject to a small perturbation, the pendulum will oscillate about ϕ0 until damping brings the mass back to its equilibrium position. However, if the perturbation is large enough that the mass passes the threshold angle ϕth = 180° (vertically above the axis of rotation), the mass will follow the path depicted by a solid red arc in Fig. 3(b). When ϕ > ϕth, τd and τg are in the same direction, and the mass will accelerate rapidly. This rapid acceleration continues until the mass passes ϕ = 0°. After this, when ϕ > 0°, τg works opposite to τd. For a moderate perturbation, the gravitational torque and friction will be sufficient to slow the mass, and the mass will return to ϕ0 = 155°. Of course, if the momentary perturbation is large enough, the pendulum may rotate two or more times.
It is clear from this mechanical analog that the closer the equilibrium angle ϕ0 is to the threshold angle ϕth, a smaller perturbation will cause a single pendulum rotation. Thus, by applying torque τd close to its threshold value, one can substantially increase the pendulum “sensitivity.” At the same time, the angular speed of the pendulum during the rotation is determined, mostly, by the gravity and bias torque and is almost independent of the amplitude of the initial “kick.” In this regime, the pendulum works as a threshold element, producing the same rotation every time input perturbations exceed a certain critical value and, in this sense, which is very similar to a biological neuron.
The three torques can be used to derive an equation of motion, which can be found from Newton’s second law for rotation, . Here, Im is the moment of inertia of the system, is the angular acceleration of the mass, and ∑τ is the sum of the torques applied to the system. Thus, we can explicitly write the equations of motion by substituting the torques in this equation,
By comparing this equation with (1), it is evident that this system is analogous to that of an AFM neuron. It is useful to describe each term in this equation individually in order to gain a better understanding of this system.
The first term on the left-hand side of (3), , contains the moment of inertia. The moment of inertia describes how difficult it is to change the angular velocity of an object rotating about an axis. When comparing the mechanical analog in (3) with the artificial neuron Eq. (1), it is evident that the first term on the left-hand side of (1) models the inertia of the system. This implies that (1/ωex) is analogous to the moment of inertia. Thus, the exchange interaction in the AFM material leads to the AFM neuron having an effective inertia. This effective inertia is one of the factors that make AFM neurons unique, giving AFM neurons properties such as response latency and finite refraction time.
The second term on the left-hand side of (3), , represents the damping due to friction experienced by the pendulum. Friction acts to slow any movement by the pendulum. The magnitude of b determines how much friction is present. In the pendulum model, we consider b to be related to air resistance. To decrease b, the air can be removed so that the pendulum moves with no friction in perfect vacuum. Likewise, b can be increased by immersing the pendulum in a viscous fluid such as water or honey. By comparison with the artificial neuron equation, the second term on the left-hand side of (1), , represents damping. When M1 has a non-zero angular velocity , damping will slow this rotation. The magnitude of α determines how much damping is present in this system. The damping acts like a frictional force to slow M1 and M2, similar to the damping in a pendulum. For the AFM neuron, α can be controlled in fabrication to a certain extent. Bulk NiO, for example, has an intrinsic effective damping of about 10−4; in a thin film, α will be larger, depending on fabrication characteristics.41 The effective damping will increase when NiO is fabricated as a thin film. It will also increase when covered by a conducting thin film due to spin pumping.51 Thus, how the AFM neuron is fabricated will impact the effective damping.
The third term on the left-hand side of (3) is mgl sin ϕ. This is the gravitational potential for the pendulum. This potential defines a preferable orientation (i.e., anisotropy) of the pendulum (ϕ = 0°) and governs how the velocity of the mass will change as it revolves around the axis. Likewise, the third term on the left-hand side of (1), (ωe/2) sin 2ϕ, models the anisotropy of the AFM material. This anisotropy governs how the velocity of M1 and M2 will change as it rotates. It is notable that ωe, the easy axis anisotropy frequency, is present in this term. When M1 and M2 are rotated under the action of the external spin current, the anisotropy of the AFM acts to realign M1 and M2 with the easy axis.
Finally, the term on the right-hand side of (3), τd, is the externally applied driving torque to the pendulum. When at equilibrium, this torque displaces the pendulum to a position of ϕ0. When in motion, this torque acts to increase the angular velocity. Likewise, in the model of the AFM neuron, the term on the right-hand side of (1) represents the spin transfer torque applied by the spin current to M1 and M2 in the AFM neuron. STT is analogous to the driving torque in the pendulum. The STT depends on the current I = Idc + ip(t) that flows through the Pt substrate. Here, Idc is a bias current that brings the angle of the magnetization to an initial angular displacement of ϕ0, and ip(t) represents a momentary perturbation that incites the magnetization to rotate.
The magnitude of the bias current determines the regime in which the AFM neuron is operating, that is, if the bias current exceeds some threshold current Ith, the system will behave as an auto-oscillator, with magnetizations M1 and M2 continually rotating in the easy plane. In a biaxial AFM material such as NiO, the transition to the auto-oscillatory regime occurs when the magnetization cross the threshold angle ϕth = 45°, making the threshold current23
Thus, for this system to be in the neuronal regime and thus exhibit behavior similar to that of a typical biological neuron, it is required that the bias current be sub-threshold, Idc < Ith. When Idc < Ith, the initial angular displacement of the magnetization will be ϕ0 = arcsin(I/Ith)/2.
C. Physical implementation and simulation parameters
This subsection will discuss the physical implementation of an AFM neuron. It begins by presenting physical parameters that allow for the realistic simulation of AFM neuron behavior. After this, the feasibility of implementing an AFM neuron is discussed.
A summary of material constants related to Eqs. (1) and (2) is given in Table I. Many of the values in this table are the same as those presented in an earlier work42 and are specific for an AFM neuron composed of a NiO/Pt bilayer. It should be noted that Ith (4) is proportional to the AFM resonance frequency ωe and, thus, depends on the in-plane anisotropy of the AFM material. It is possible, in principle, to decrease Ith by reducing the anisotropy field by voltage-controlled magnetic anisotropy (VCMA) or by using magnetoelastic interactions. Similar effects might be achieved by the interfacial Dzyaloshinskii–Moriya field present at the AFM-heavy metal interface. Complete elimination of Ith, however, is not possible since the operation of AFM neurons relies on the anisotropic angular dependence of the AFM potential energy.
AFM material parameters, fundamental constants, and physical dimensions used in simulation.
Parameter . | Description . | Simulation value . |
---|---|---|
fex | Exchange frequency | 27.5 THz |
α | Effective damping | 0.001–0.1 |
fe | Easy axis anisotropy frequency | 1.75 GHz |
|γ|/2π | Gyromagnetic ratio | 28 GHz/T |
Ms | Saturation magnetization | 351 kA/m |
θSH | Spin Hall angle | 0.1 |
gr | Spin mixing conductance | 6.9 × 1018 m−2 |
e | Elementary charge | 1.6 × 10−19 C |
λ | Pt spin diffusion length | 7.3 nm |
ρ | Resistivity of Pt | 4.8 × 10−7 Ω m |
dAFM | NiO thickness | 5 nm |
wAFM | NiO/Pt interface width | 10 nm |
ℓAFM | NiO/Pt interface length | 40 nm |
dPt | Pt thickness | 20 nm |
η | ⋯ | 5.4 × 10−17 V s |
σ | Spin-torque efficiency | 27.1 × 1012 rad/A s |
β | Spin pumping efficiency | 0.11 × 10−15 V s |
Ith | Threshold current | 0.203 mA |
Parameter . | Description . | Simulation value . |
---|---|---|
fex | Exchange frequency | 27.5 THz |
α | Effective damping | 0.001–0.1 |
fe | Easy axis anisotropy frequency | 1.75 GHz |
|γ|/2π | Gyromagnetic ratio | 28 GHz/T |
Ms | Saturation magnetization | 351 kA/m |
θSH | Spin Hall angle | 0.1 |
gr | Spin mixing conductance | 6.9 × 1018 m−2 |
e | Elementary charge | 1.6 × 10−19 C |
λ | Pt spin diffusion length | 7.3 nm |
ρ | Resistivity of Pt | 4.8 × 10−7 Ω m |
dAFM | NiO thickness | 5 nm |
wAFM | NiO/Pt interface width | 10 nm |
ℓAFM | NiO/Pt interface length | 40 nm |
dPt | Pt thickness | 20 nm |
η | ⋯ | 5.4 × 10−17 V s |
σ | Spin-torque efficiency | 27.1 × 1012 rad/A s |
β | Spin pumping efficiency | 0.11 × 10−15 V s |
Ith | Threshold current | 0.203 mA |
The spin-torque efficiency σ and the voltage phase proportionality constant β are given by42
where γ is the gyromagnetic ratio, Ms is the saturation magnetization of one NiO sublattice, dAFM is the thickness of the NiO, wAFM is the width of the NiO/Pt interface, dPt is the thickness of the Pt, and ℓPt is the length of the NiO/Pt interface. These physical dimensions are shown in Fig. 2. The constant η is given by42
where θSH is the spin Hall angle, gr is the spin mixing conductance, e is the magnitude of the fundamental electric charge, λ is the spin-diffusion length in Pt, and ρ is the resistivity of thin film Pt.
It should be noted in Eqs. (1) and (2) that by increasing the values of η, σ, and β, the performance of the AFM neuron will be improved. Therefore, the performance characteristics of the AFM neuron depend on the physical dimensions of the AFM neuron, specifically dAFM, wAFM, and dPt. For example, in (5), the value of σ increases for small values of dAFM and wAFM. Therefore, we have chosen dAFM = 5 nm and wAFM = 10 nm as simulation parameters. These values are small yet reasonable for nano-fabrication.
Likewise, dPt appears in (5)–(7). To maximize η, a thickness with dPt > 2λ should be chosen. However, a thin dPt is beneficial to maximize σ and β. Taking this into consideration, dPt ∼ 20 nm was chosen as a simulation parameter. This thickness is reasonable for nano-fabrication.
The size of the entire structure must be considered for the choice of ℓAFM, the length of the NiO/Pt interface. The NiO material should be small enough that its entire volume is in a uniform ground state, that is, it should be a single crystal with a single domain and without any domain walls. In addition, the volume of the AFM material should be large enough that the magnetization will not spontaneously rotate in the easy plane due to thermal energy. The energy required for thermal noise to rotate the magnetization is equal to VAFMBeMs, where VAFM is the volume of the AFM material and Be = ωe/|γ| is the easy axis anisotropy field. The thermal energy is equal to kBT, where kB is the Boltzmann constant and the temperature is T = 300 K. Thus, a reasonable volume would be VAFM > 10(kBT/BeMs), which corresponds to an AFM material with a volume of VAFM ∼ 2000 nm.3 For our chosen dAFM, this corresponds to a NiO/Pt interface with a surface area of 400 nm2. This surface area is on par with the size of CMOS transistors. With this surface area and wAFM as defined above, it is appropriate to use a simulation parameter for the length of the NiO/Pt interface as ℓAFM = 40 nm.
With these size parameters, the spin torque efficiency is approximately σ = 27.1 × 1012 rad/A s, and the spin pumping efficiency is β ∼ 0.11 × 10−15 V s/rad. Considering that a typical value for is about 1 rad/ps, we can assume that altogether, voltage spikes produced by an AFM neuron will have a magnitude on the order of V, with the duration as short as 1 ps. This is demonstrated in Fig. 1. In addition, with this σ, the threshold current is Ith ∼ 0.2 mA.
The effective damping α for NiO is shown in Table I to vary between 0.001 and 0.1. Damping in NiO depends on the technology of film preparation, and therefore, there is some flexibility in how α can be varied in simulation. Smaller damping parameters lead to a shorter spike duration.
It is important to note that AFM neurons require a DC electric current Idc to be constantly flowing through the Pt substrate. The purpose of this current, as stated earlier, is to bias the magnetization so that ϕ0 is close to ϕth, so it can generate a spike with the receipt of a small current impulse ip(t). The energy consumption of an AFM neuron can be estimated by considering Idc. Previously, it was estimated that a current near the threshold bias current of Ith = 0.2 mA would be required to drive the rotation of M1 and M2. If the resistance of the platinum beneath the NiO is given by Rpt = ρℓAFM/dPtwAFM, then the power consumption of this structure can be estimated as μW. If the time per synaptic operation is conservatively estimated as 100 ps, the energy consumption of a single AFM neuron is pJ per synaptic operation. This power consumption can be compared with other spiking neuromorphic hardware.52 For example, the energy per synaptic operation for the Intel Loihi neuromorphic chip was reported to be 20 pJ.17 The efficiency of an AFM neuron can be evaluated in units of SOPS/W, where SOPS stands for synaptic operations per second.12 Using this measure, a single AFM neuron will have a performance of ten TSOPS and an efficiency of about 2500 TSOPS/W, which are far more efficient than those of comparable systems reported in Ref. 12. The high efficiency of AFM neurons is a direct result of its high speed of operation, which is a result of the high exchange frequency of NiO and other antiferromagnetic materials. Please note that these efficiency estimates are for the AFM neurons only and do not include the power consumed by support circuitry or by interconnections between the neurons.
It is worth noting that platinum was chosen for simulations because it is relatively straightforward to fabricate and has an acceptable spin Hall angle θSH ∼ 0.1. However, it may be possible to employ a topological insulator with a much higher spin Hall angle, for example, θSH ∼ 50 for BiSb.53 The use of a material with a higher spin Hall angle will greatly improve the power efficiency of this system. For example, the use of material with θSH ∼ 10 will decrease the power consumption to pJ per synaptic operation and, at the same time, increase the output spike amplitude to ∼10 mV.
At present, AFM neurons have not yet been fabricated. The reason relates to the difficulty in fabricating an insulating structure with nanometer dimensions. To function properly, the AFM structure should be fabricated with several important characteristics. As previously mentioned, the structure should be large enough so that the magnetization will not rotate spontaneously due to thermal noise but small enough that it is monocrystalline and of a single domain. In addition, the easy plane anisotropy in the AFM film must be oriented correctly with respect to the Pt film plane, and a clean interface between the substrate and the AFM material is required for an efficient interfacial effect. These technical challenges will need to be overcome prior to the experimental demonstration of an AFM neuron. However, we believe that the performance of this system justifies the expanded research in this area. In addition, it has been suggested that a neuron with properties similar to those of an AFM neuron can be constructed from a magnetic tunnel junction using available fabrication technology.54–56
Of course, the AFM neurons have not yet been realized experimentally. Thus, the configuration of support circuitry is yet to be determined. In addition, the presence of defects and other noise sources will require attention in the design of support circuitry to allow the propagation of spiking signals. Thus, details about integration with CMOS or other technologies will not be addressed in this paper.
Active research related to AFM materials is not limited to AFM neurons; at present, much research is ongoing for both fundamental science and applied technology related to AFM materials. There have been recent proposals to employ AFM materials in THz signal generators, detectors, and spectrum analysis, among other ideas.57–61 There has also been much research related to AFM based memory applications, which is relevant to the development of AFM neurons.40,62–70
III. SINGLE AFM NEURON BEHAVIOR
This section discusses the dynamic behavior of an AFM neuron that is modeled by Eq. (1). It will begin by demonstrating the spiking behavior of an AFM neuron by numerical simulation. After this, the refractory properties of AFM neurons are discussed, as will the response of AFM neurons to polarity changes. This section ends by demonstrating the utility of AFM neurons for Boolean logic.
A. AFM neuron spiking demonstration
Together, Eqs. (1) and (2) describe the input–output behavior of an AFM neuron; the input is the current I, and the output is the voltage v(t). In most cases, there is no general analytical solution; thus, (1) can only be solved numerically.
Numerical simulations of (1) and (2) were performed to demonstrate an AFM neuron generating spikes. Results of these simulations are shown in Fig. 4. The simulations were performed with a bias current of Idc = 198 μA, which is 5 μA below the threshold current Ith = 203 μA. As mentioned earlier, Ith is the threshold current that determines when the AFM neuron is in the neuronal regime (Idc < Ith) and when it behaves as an auto-oscillator (Idc > Ith).
Input current and simulated AFM neuron output. (a) The simulated input current. The blue curve represents the input current ip(t), the lower dashed pink line represents the threshold current ith, the center red dashed line represents the critical current, icr, and the top red dashed line represents the critical current for bursting behavior, icr2. (b) The simulated azimuthal angle ϕ of M1 in the easy plane. Green vertical dashed lines show the time of input perturbations. (c) Voltage output of the AFM neuron. The latency tℓ for two spikes is labeled. In this simulation, α = 0.009.
Input current and simulated AFM neuron output. (a) The simulated input current. The blue curve represents the input current ip(t), the lower dashed pink line represents the threshold current ith, the center red dashed line represents the critical current, icr, and the top red dashed line represents the critical current for bursting behavior, icr2. (b) The simulated azimuthal angle ϕ of M1 in the easy plane. Green vertical dashed lines show the time of input perturbations. (c) Voltage output of the AFM neuron. The latency tℓ for two spikes is labeled. In this simulation, α = 0.009.
The input to the AFM neuron, current impulse ip(t), is shown in Fig. 4(a). In this simulation, ip(t) has four peaks. The threshold current ith = Ith − Idc, which is 5 μA above the bias current, is represented in the figure by a red dashed line. Note that the threshold current ith is the input signal amplitude that leads to magnetization rotations in the DC regime, i.e., for a very long duration of the input signal. To induce rotation with a short input pulse, its amplitude should exceed a certain critical value icr, which depends on the input duration and for the parameters of Fig. 4 equals icr = 18 μA. For a further increase in the input current amplitude (ip > icr2 = 41 μA), one input pulse will generate burst spikes, as explained below. The four input pulses shown in Fig. 4(a) have different amplitudes: one is below the critical current (ith < ip < icr), two pulses have different amplitudes in the range icr < ip < icr2, and the last spike is in the burst range ip > icr2.
Figures 4(b) and 4(c) show the response of the AFM neuron to the current impulse. Figure 4(b) shows the azimuthal angle ϕ of M1, and Fig. 4(c) shows the output voltage according to Eq. (2). It is evident from this figure that for the first 50 ps, in the absence of a current impulse, that ϕ remains constant, and thus, . The output voltage, which is proportional to , remains at zero during this interval.
At t = 50 ps, there is a current impulse with an amplitude of 10 μA. This impulse is larger than ith but is less than icr. Because the impulse is less than the critical current, the magnetization does not rotate, and there is no output spike. This current impulse, however, does cause a small movement in M1, as can be seen in the small bump at t = 50 ps in Fig. 4(c).
Next, at t = 100 ps, there is a current impulse of 20 μA, which exceeds icr. The response to this input current impulse consists of, first, a small bump at t = 100 ps [see Fig. 4(c)]. Then, after a delay of tℓ ∼ 10 ps, the neuron fires. This can be seen at time t = 110 ps in the 180° rotation of ϕ in Fig. 4(b) and in the voltage spike in Fig. 4(c).
The behavior of the AFM neuron at t = 100 ps demonstrates two properties that closely resemble the behavior of biological neurons. First, AFM neurons follow the “all-or-nothing” law,71 that is, these neurons only fire when they receive sufficient stimulus. Biological neurons only elicit action potentials when the membrane potential rises above a threshold, which then depolarizes the neural membrane. In the AFM neuron, this is equivalent to the input current being momentarily larger than the critical current.
Second, concerning the delay tℓ, in biological neurons, this delay is called the neuronal response latency, which is the time between the input stimulus and the activation potential.72 In AFM neurons, the delay is due to the time it takes for the magnetization to rotate past the threshold angle. Visualized as a simple pendulum, the perturbation provides just enough energy for ϕ to pass ϕth. However, the velocity will be very slow as ϕ approaches the critical angle. Then, once ϕ is past ϕth, the mass will accelerate.
The duration of the response latency depends on the magnitude of the input impulse. This is demonstrated by the response of the AFM neuron at time t = 150 ps. At this time, the current impulse is 30 μA, which is larger than the previous spike. Once again, the AFM neuron magnetization rotates by 180°, as shown in Fig. 4(b), and the rotation brings about a voltage spike, as shown in Fig. 4(c). Please note that this voltage spike has the same amplitude and duration as the previous voltage spike. However, the response latency has a shorter duration because the AFM neuron received a larger input current impulse.
Finally, consider the behavior of the AFM neuron at t = 250 ps. At this time, the momentary increase in the instantaneous electric current is larger than icr2. When the electric current is larger than icr2, the AFM neuron will exhibit bursting behavior. This can be seen in Fig. 4(c), where the AFM neuron shows a double peak, and in Fig. 4(b), where M1 rotates by 360°, twice as much as for the previous two impulses. In biological neurons, this behavior is also known as bursting.73–77 This is analogous to the pendulum shown in Fig. 3 rotating twice about its axis.
For AFM neurons, icr ≠ ith; thus, it is expected that a change in the relative magnitudes of Idc and ip(t) will lead to different behaviors. This is demonstrated in Fig. 5, which was made via simulation with a sinusoidal input for ip(t) and a bias current Idc.23 The axes are scaled as a ratio of Idc/Ith and ip(t)/Ith. In this figure, there are three regions: single spike (red), bursting (blue), and no spikes (yellow). The boundary between the red and yellow regions represents the critical current icr, which is the minimum current impulse required to generate a spike. The boundary between the generation of single spiking signals and bursting signals, icr2, corresponds to a line between the red and blue regions. In addition, there is a dashed line in the yellow region, which represents the threshold current ip = ith = Ith − Idc for very long pulses. One noteworthy characteristic in this plot is that as Idc decreases, the separation between icr and ith increases, and there is an increased range where single spiking signals can occur.
AFM neuron spike characteristics. For the yellow region, the AFM neuron produces no spikes. In the red region, the AFM neuron generates a single spike. In the blue region, the AFM neuron exhibits bursting behavior. The behavior depends on the relative values of ip(t) and Idc. In this simulation, α = 0.01. Figure reproduced with permission from Khymyn et al., Sci. Rep. 8, 15727 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License.
AFM neuron spike characteristics. For the yellow region, the AFM neuron produces no spikes. In the red region, the AFM neuron generates a single spike. In the blue region, the AFM neuron exhibits bursting behavior. The behavior depends on the relative values of ip(t) and Idc. In this simulation, α = 0.01. Figure reproduced with permission from Khymyn et al., Sci. Rep. 8, 15727 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License.
It can also be seen from Fig. 5 that when Idc > Ith, the AFM magnetization will rotate more than once. In this regime, AFM neurons are capable of generating a continuous train of spikes that has a frequency that depends on the amplitude of Idc. Interestingly, biological neurons subject to sustained suprathreshold stimuli will also elicit a train of action potentials whose frequency depends on the strength of the stimulus.76,78 Biological neurons can exhibit adaptation in spike trains; they feature a change in the duration of interspike intervals. They can also exhibit stuttering, which are spike trains with inconsistent rhythms.76 AFM neurons can also exhibit adaptation by tuning the bias current and can exhibit stuttering with the introduction of noise to the bias current. For both types of neurons, suprathreshold spike generation characteristics are dependent on the size and variation of the stimulus.
B. AFM neuron refraction
Due to the effective inertia modeled by the first term in (1), AFM neurons intrinsically exhibit refraction, a property that will be explored in this subsection. This property, which bears close resemblance to the behavior of biological neurons, is a positive attribute for AFM neurons. Refraction is not intrinsically present in CMOS spiking neurons, although it can be emulated.52
In AFM neurons, there is a period of time when the AFM magnetization is in the act of rotation and the AFM neuron is unable to fire even if it receives an additional impulse ip(t). In biological systems, this is called the absolute refractory period, which is defined as the time interval after the neuron fires where it cannot fire again.71 There are also times where the AFM magnetization is in the act of recovery and the AFM neuron responds with modified behavior. In biological systems, this is generally called the relative refractory period.71 These two refractory properties of an AFM neuron are demonstrated in Fig. 6 and will be discussed below.
Simulation demonstrating the refractory period of the AFM neuron. For these simulations, inputs to the neuron are shown with red and blue curves, while the output of the neuron is shown with a green curve. (a) The red input spike leads to a green output spike with tℓ = 50 ps. The blue input spike occurs after an interval of 102.3 ps, during the absolute refractory period. The neuron does not spike a second time. (b) The blue input spike occurs after an interval of 102.4 ps, during the relative refractory period. The second output spike occurs after a long delay. (c) The blue input spike occurs after an interval of 185 ps. The second output spike has the same latency as the first spike because the AFM neuron is not in its refractory period. In these simulations, α = 0.1.
Simulation demonstrating the refractory period of the AFM neuron. For these simulations, inputs to the neuron are shown with red and blue curves, while the output of the neuron is shown with a green curve. (a) The red input spike leads to a green output spike with tℓ = 50 ps. The blue input spike occurs after an interval of 102.3 ps, during the absolute refractory period. The neuron does not spike a second time. (b) The blue input spike occurs after an interval of 102.4 ps, during the relative refractory period. The second output spike occurs after a long delay. (c) The blue input spike occurs after an interval of 185 ps. The second output spike has the same latency as the first spike because the AFM neuron is not in its refractory period. In these simulations, α = 0.1.
The absolute refractory period will be considered first in Fig. 6(a). In this simulation, the first spike arrives at an AFM neuron at t = 200 ps, as shown by a red curve. After a latency of about 50 ps, the neuron fires, as shown by the green curve. Then, as shown by a blue curve, a second input spike arrives at t = 302.3, 102.3 ps after the first spike. In this case, the second spike does not cause the AFM neuron to spike. This is because the AFM neuron is still within its absolute refractory period and it is unable to fire a second time.
The simulation results shown in Fig. 6(b) demonstrate the relative refractory period of AFM neurons. In this case, a second input arrives at t = 302.4 ps, which is 102.4 ps after the first spike and 0.1 ps later than above. In this case, the AFM neuron generates a second spike after a latency of about 400 ps. Thus, during the relative refractory period, the AFM neuron can generate a spike, but with an extended latency.
A simulation showing the response of an AFM neuron without refraction is considered in Fig. 6(c). Once again, the first spike arrives at t = 200 ps, and after a latency of about 50 ps, the neuron responds with a spike. The blue input then arrives at the AFM neuron at t = 385 ps, which is 185 ps after the initial input. Then, once again, after a latency of about 50 ps, the neuron fires, as is shown by a green curve. Please note that the latency responses for both inputs are the same. For the parameters simulated, 185 ps is the minimum time required for the AFM neuron to fire with the same latency.
The length of the refractory period of an AFM neuron can be dynamically controlled by changing the bias current Idc. The refractory period also depends on the effective damping, the anisotropy, and the exchange frequency of the AFM material. Please note that for an AFM neuron with a lower damping parameter, the refractory time will be substantially shorter.
C. AFM neuron polarity
Interestingly, AFM neurons can also reverse their polarity; specifically, they can spike with both positive and negative voltages. This can be achieved by biasing an AFM neuron with DC currents of different polarities, which induces magnetization rotation in the opposite direction.
AFM neuron behavior for different bias polarities is illustrated in Fig. 7. Here, red curves show responses of AFM neurons with positive current, and blue curves show the response of AFM neurons to negative current. It is evident from the figure that the polarity of AFM neuron spikes can be reversed in response to a change in the current direction.
Reversible polarity of a single AFM neuron. The red curve shows the response of the neuron to a positive current that surpasses icr, while the blue curve shows the response of the same AFM neuron to a negative current that surpasses −icr. In these simulations, α = 0.009.
Reversible polarity of a single AFM neuron. The red curve shows the response of the neuron to a positive current that surpasses icr, while the blue curve shows the response of the same AFM neuron to a negative current that surpasses −icr. In these simulations, α = 0.009.
Unlike AFM neurons, biological neurons are unable to produce negative action potentials in response to negative stimuli. However, biological neurons can be inhibited by hyper-polarization stimuli.79 A spike employing negative polarity to inhibit a neighboring neuron will be demonstrated in Sec. V.
The bi-polar character of AFM neuron dynamics may find useful applications in developing new types of SNNs consisting of two competing subnetworks (positively and negatively biased). The neurons in one subnetwork will stimulate neurons in the same network but inhibit neurons of the opposite network.
D. AFM neurons as gates for Boolean logic
Previous subsections demonstrated AFM neuron spiking, refraction, and polarity. This subsection will describe how AFM neurons can be used as logic gates to perform Boolean logic. Boolean logic, along with logic gates fabricated from silicon semiconductor transistors, forms the backbone for conventional computing architectures.80 While it is unlikely that AFM neurons will replace the CMOS, this section provides an application example that does not require an adaptive neural network.
The most quintessential Boolean logic gate might be the AND gate.80 The truth table for an AND gate is shown in Fig. 8(a). This table shows X and Y as inputs and Z1 = X ∪ Y as an output. In this table, 1 represents true, and 0 represents false. Figure 8(b) shows the neural network for an AND gate. As before, X and Y are inputs, and Z1 is an output. To configure this AFM neuron as an AND gate, the magnitude of each input is individually smaller than the critical current icr. However, when both X and Y spike at about the same time, the magnetization in the AFM material rotates, and the neuron generates a spike.
AND gate implemented with an AFM neuron. (a) Truth table for an AND gate. (b) Schematic of the AFM neuron for an AND gate. Arrows signify inputs and outputs. (c) Simulation with a logical inputs X = 1 and Y = 1. X is shown with a red curve, and Y is shown with a blue curve. The output Z1 = 1 is shown with a green curve. (d) Simulation with logical inputs X = 1 and Y = 0. X is shown with a red curve, and the Y = 0 curve is not visible. The Z1 = 0 curve does not show a spike but instead shows a small bump. In these simulations, α = 0.1.
AND gate implemented with an AFM neuron. (a) Truth table for an AND gate. (b) Schematic of the AFM neuron for an AND gate. Arrows signify inputs and outputs. (c) Simulation with a logical inputs X = 1 and Y = 1. X is shown with a red curve, and Y is shown with a blue curve. The output Z1 = 1 is shown with a green curve. (d) Simulation with logical inputs X = 1 and Y = 0. X is shown with a red curve, and the Y = 0 curve is not visible. The Z1 = 0 curve does not show a spike but instead shows a small bump. In these simulations, α = 0.1.
Numerical simulations of an AFM neuron configured as an AND gate were performed, with results shown in Figs. 8(c) and 8(d). The AND gate in Fig. 8(c) has inputs X = 1 and Y = 1. The X input is shown as a function of time with a red curve, and it has a spike at t = 100 ps. This spiking signal represents X = 1. Likewise, the Y input is shown as a function of time with a blue curve and has a spiking signal at t = 150 ps. This spike represents Y = 1. Together, these two inputs provide enough energy that the neuron is able to generate an output spike at t = 220 ps, which is represented by a green curve. This output spike represents Z1 = 1, thus demonstrating that this portion of the AND gate truth table can be realized by an AFM neuron.
Figure 8(d) shows the results of numerical simulation for an AND gate when the inputs are X = 1 and Y = 0. The X input is shown as a function of time with a red curve, and it has a spike at t = 100 ps. This red spiking signal represents an X = 1 input. In this plot, Y is represented by a blue curve, which does not spike at all, and hence, it properly represents Y = 0. As shown in Fig. 8(d), there is no green curve spike; only a small bump is seen at t = 100 ps, and thus, for this case, the output is Z1 = 0. As shown in Fig. 8(d), there is no green curve spike; only a small bump is seen at t = 100 ps, and thus, for this case, the output is Z1 = 0. This is as expected for these inputs. Additional simulations were performed to confirm that the same AFM neuron will fulfill lines 1 and 2 of the AND truth table, thus showing that a single AFM neuron can act as an AND gate.
An AFM neuron with a configuration similar to that shown in Fig. 1 can act as an OR gate. The only change required is to increase the amplitudes of the inputs signals so that the neuron will fire from one or more inputs.
A single AFM neuron can also be used to represent a majority gate. A majority gate is a Boolean circuit that outputs a true when more than half of its inputs are true. The truth table for a three-input majority gate is shown in Fig. 9(a), and the configuration for an AFM neuron implementation of this majority gate is shown in Fig. 9(b). For comparison, a three-input majority gate would typically require four NAND gates.80 To configure the AFM neuron as a majority gate, a single input would be smaller than icr, while two inputs together would cause the AFM neuron to generate a single spike. Likewise, three inputs together will also create a single spike and have a threshold current that does not induce bursting behavior.
Majority gate implemented with a single neuron. (a) Truth table for a majority gate. (b) Schematic of the AFM neuron for the majority gate, with inputs W, X and Y and output Z2. (c) Simulation of a majority gate with W = 1 in magenta, X = 1 in red, and Y = 1 in blue, as labeled. Because there are true inputs, Z2 = 1, as shown by the green spike. (d) Simulation of the majority gate for W = 0, X = 1, and Y = 1. Because inputs are true, the output Z2 is also true, as evidenced by the green curve. (e) Simulation of the majority gate for W = 0, X = 1, and Y = 0. Because inputs are true, Z2 = 0, and hence, there is no spike on the green curve. In these simulations, α = 0.1.
Majority gate implemented with a single neuron. (a) Truth table for a majority gate. (b) Schematic of the AFM neuron for the majority gate, with inputs W, X and Y and output Z2. (c) Simulation of a majority gate with W = 1 in magenta, X = 1 in red, and Y = 1 in blue, as labeled. Because there are true inputs, Z2 = 1, as shown by the green spike. (d) Simulation of the majority gate for W = 0, X = 1, and Y = 1. Because inputs are true, the output Z2 is also true, as evidenced by the green curve. (e) Simulation of the majority gate for W = 0, X = 1, and Y = 0. Because inputs are true, Z2 = 0, and hence, there is no spike on the green curve. In these simulations, α = 0.1.
Numerical simulations of the majority gate are shown in Figs. 9(c)–9(e). Simulations for the last line of the truth table when W = 1, X = 1, and Y = 1 are shown in Fig. 9(c). These inputs are labeled on the graph. The output Z2 = 1 is on the same graph as labeled.
Figure 9(d) shows simulations for W = 0, X = 1, and Y = 1, and Fig. 9(e) shows simulations for W = 0, X = 1, and Y = 0. In both cases, Z2 responds as expected. It is evident that a single spike input is insufficient to generate an output spike, while two or more inputs spikes can generate an output spike, thus confirming that a single AFM neuron can act as a majority gate.
A previous work provides an in-depth analysis of performing Boolean logic with AFM neurons and includes a description of a Q-gate and a full-adder that consists of just three AFM neurons.24
IV. INTERCONNECTING AFM NEURONS
For AFM neurons to be useful, they should be interconnected into a neural network. The previous sections described the behavior of individual AFM neurons. This section will discuss how multiple AFM neurons can be interconnected to form a physical neural network.
A summary of interconnection schemes is shown in Fig. 10. In Fig. 10(a), neurons “A” and “B” are connected by a single piece of platinum. With this connection, both neurons are subject to the same bias current. Likewise, spikes generated by neuron “A” will travel downstream to induce a spike in neuron “B.”
AFM neuron interconnection. (a) Schematic of electrical interconnection. Here, neuron “A” is green, neuron “B” is magenta, and the two neurons are connected via the blue platinum substrate. (b) Schematic of spin current interconnection. Here, neurons “C” and “D” are biased independently by two Pt substrates, and they are interconnected via a copper coupling, shown in brown. (c) Schematic of a generic interconnection. Neuron “E” is represented by a green circle, and neuron “F” is represented by a magenta circle. These two neurons are interconnected by a generic synapse, which is represented by an arrow.
AFM neuron interconnection. (a) Schematic of electrical interconnection. Here, neuron “A” is green, neuron “B” is magenta, and the two neurons are connected via the blue platinum substrate. (b) Schematic of spin current interconnection. Here, neurons “C” and “D” are biased independently by two Pt substrates, and they are interconnected via a copper coupling, shown in brown. (c) Schematic of a generic interconnection. Neuron “E” is represented by a green circle, and neuron “F” is represented by a magenta circle. These two neurons are interconnected by a generic synapse, which is represented by an arrow.
A second simple method of interconnecting neurons is shown in Fig. 10(b). In this case, both AFM neurons are connected to separate Pt strips. This allows a bias current to be provided to each AFM neuron independently. In addition, a copper waveguide interconnects the neurons to transmit spin current between neurons “C” and “D.” By changing the width, thickness, and length of the copper waveguide, it is possible to change the coupling strength between the neurons.
Unfortunately, the strengths of the inter-neuron connections shown in Figs. 10(a) and 10(b) are fixed and cannot be adjusted. To use AFM neurons in neural networks for machine learning and AI, it is required that the connection between neurons be mediated by an adjustable synapse. This is depicted schematically in Fig. 10(c). In this figure, both neurons are assumed to be properly biased with a DC current. When a momentary impulse of sufficient amplitude is applied to neuron “E,” it will generate a spike. This spike will travel through a synapse, which is represented in this figure by an arrow, to synapse “F.” If the signal that arrives at neuron F is of sufficient amplitude, neuron F will also generate a spike.
All synapses simulated in this paper are assumed to be “ideal,” meaning that the synaptic weights can be adjusted instantaneously, and be of any value. This paper is focused primarily on presenting AFM neurons as realizable hardware; thus, we do not delve into the details of how synapses will be realized. It is worth mentioning that electrical synapses can be physically realized in different ways,22 for example, using memristors,81 multiferroic heterostructures,82 and Josephson junctions83 or by implementing in CMOSs.84
Artificial synapses can also be constructed using spintronic devices; i.e., devices that use magnetic elements and their dynamics. For example, spin torque nano-oscillators,85,86 domain walls,87–89 and skyrmions90–92 have all been shown to act like synapses to interconnect artificial neurons. Spintronic devices have many distinct advantages over their electrical counterparts, with an ability to closely resemble biological synapses and a low power consumption. Spintronic synapses have been shown to exhibit the potentiation and depression behaviors of biological synapses90,91 and also shown to exhibit spike time dependent plasticity, the biological process that adjusts the weights between neurons.87,88
A. Mathematical model for the AFM neural network
A mathematical model for a single AFM neuron was given in Eq. (1). In contrast, when multiple AFM neurons are interconnected into a neural network, the neurons will interact with each other via spiking output voltages. The interaction can be modeled by a system of differential equations,
In this equation, i and k are indices that represent the i-th and k-th neuron, and κik represents a matrix of coupling coefficients. Similar to all neural networks, we envision that the interaction between neurons will be mediated by synaptic weights. In this equation, synapses are represented by the coupling coefficients κik. In a few words, by simulating (8), one can simulate an entire spiking neural network of AFM neurons. It is hoped that, eventually, instead of simulations, fabricated AFM neurons will be available to perform computations at a picosecond timescale.
Equation (8) assumes that the timing delays related to the interconnections are negligible in comparison to the refractory period. Thus, will need to be modified to simulate a system with delays.
B. Simple neuron chain
Simulations were performed on a chain of five interconnected neurons, as presented in Fig. 11(a). In this figure, five AFM neurons are represented by circles, and electrical connections with synapses are represented by arrows.
Simulation results for five AFM neurons connected in a sequential chain. (a) The neural network for a five-neuron chain. Circles represent AFM neurons, and arrows represent synapses. (b) Voltage output with κ0 = 0.011. The color of each curve represents the voltage output of each neuron. (c) Voltage output with κ0 = 0.015. (d) The relationship between κ0 and response latency. Different colored lines show Idc/Ith values of 0.99 (green), 0.97 (orange), and 0.95 (blue). In these simulations, α = 0.1.
Simulation results for five AFM neurons connected in a sequential chain. (a) The neural network for a five-neuron chain. Circles represent AFM neurons, and arrows represent synapses. (b) Voltage output with κ0 = 0.011. The color of each curve represents the voltage output of each neuron. (c) Voltage output with κ0 = 0.015. (d) The relationship between κ0 and response latency. Different colored lines show Idc/Ith values of 0.99 (green), 0.97 (orange), and 0.95 (blue). In these simulations, α = 0.1.
The neuron chain is intended to function as follows: First, there is a current impulse that originates on the left that will cause the magnetization of the blue neuron to rotate and thus generate a voltage spike. The spike from the blue neuron then travels through a synapse to the black neuron, where it will have sufficient amplitude to cause the magnetization of black neuron to rotate, and thus generates a voltage spike. After this, the spike from the black neuron will flow to the next neuron, thus continuing down the chain.
Numerical simulations were run for this system according to Eq. (8). In this simulation, every neuron is considered to have a damping constant of α = 0.1 and a bias current of Idc = 198 μA, which is . Weights in the synaptic connections are assigned to the matrix κik, which in this case is a 5 × 5 coupling matrix given by
where κ0 is the coupling constant that determines the overall strength of neuronal interconnects.
Consider first the simulation results with κ0 = 0.011, shown in Fig. 11(b). For this simulation, a momentary stimulus is provided to the leftmost (blue) neuron, leading to a spike at time t = 100 ps, as can be seen from the blue curve in Fig. 11(b). The voltage produced by the blue neuron, which is about 10 μV, then causes the black AFM neuron to spike at t = 190 ps, as shown by the black curve. The spike from the black neuron, which has the same amplitude, continues toward the green neuron. This leads the green neuron to generate a spike, which then leads the purple and orange neurons to generate spikes in succession. Thus, a spiking signal can propagate through a chain of interconnected neurons.
Note that in Fig. 11(b), there is a uniform time delay of about 90 ps between two neighboring neuron spikes. This delay is the neuron response latency tℓ, which was discussed previously. As before, the duration of the latency can be adjusted by increasing the amplitude of the perturbation ip(t) that initiates the magnetization revolution in the AFM material. In a neural network, the amplitude of a spike incident on an AFM neuron can be changed by adjusting the value of the synaptic weight.
Thus, the duration of the response latency can be adjusted by changing the weights in the coupling matrix. Simulation results for a chain of neurons, with a larger synaptic weight, are demonstrated in Fig. 11(c). Here, the weights were changed to κ0 = 0.015, while the bias current, the effective damping, and the coupling matrix (9) remained unchanged. With the increased synaptic weights, the neurons fire as before, with a shortened latency of about 50 ps. The time between neuron spikes can thus be controlled by adjusting the synaptic weight. The relationship between synaptic weights and response latency may be useful in the implementation of machine learning algorithms for SNNs such as SpikeProp, where spike timing plays a critical role.93
Of course, the bias current also plays a role in determining the latency in a sequential chain of neurons. This is examined in Fig. 11(d). The figure shows how the latency and the coupling coefficient κ0 are related for three values of bias current Idc/Ith. Two trends can be ascertained from this graph. First, for all three bias currents, as κ0 increases, the latency decreases. Second, as the bias current increases, a smaller coupling constant is able to induce a shorter latency.
Thus, this section has established that AFM neurons can be interconnected and that spikes can be transmitted between neurons through these interconnections. It was also established that the weights can be used to change the response latency of neuron spikes.
C. Symmetric coupling
Interestingly, it is possible to transmit spin current directly between neurons with a non-magnetic metal such as copper. Pure spin current can be defined as the net flow of the spin angular momentum without the net flow of charge carriers.94 This idea is illustrated schematically in Fig. 10(b). In that figure, when neuron “C” spikes, it generates spin current that can flow into the copper connector and travel to neuron “D.” If the spin current is of sufficient magnitude, it can induce neuron D to generate a spike. The spike generated by neuron D can then re-enter the copper connecter and flow back to neuron C, that is, the coupling between these two neurons is bi-directional; hence, this type of coupling can be called symmetric coupling.
The concept of symmetric coupling in a system of bio-inspired neurons is novel; therefore, it will be considered in detail in this subsection. Unfortunately, machine algorithms that can use bi-directional signal propagation have not been developed. Therefore, it will be beneficial if AFM neurons could be interconnected via symmetric coupling yet transmit in just a single direction. This can be done by exploiting the refraction properties of AFM neurons to induce one-way spike transmission.
Figure 12 demonstrates how the refraction properties of AFM neurons can be used to send a signal unidirectionally through a chain of symmetrically coupled neurons. In Fig. 12(a), the signal will originate in the red neuron on the left, and in Fig. 12(b), the signal will originate in the purple neuron on the right. In this schematic, all four synapses are fully bidirectional; for example, a signal generated by the blue neuron will flow to both the red neuron and the green neuron. For this neuron chain, the coupling matrix is a symmetric matrix,
Despite the fact that all connections are bi-directional, simulations show that the signal will flow in only one direction.
Symmetric coupling and unidirectional signal propagation. (a) Simulation result for a symmetrically coupled neural network with an initial signal on the left and (b) with an initial signal on the right. In these simulations, α = 0.1 and κ0 = 0.011.
Symmetric coupling and unidirectional signal propagation. (a) Simulation result for a symmetrically coupled neural network with an initial signal on the left and (b) with an initial signal on the right. In these simulations, α = 0.1 and κ0 = 0.011.
First, consider a simulation performed where the first spike was initiated in the red neuron on the left. The simulation results are shown in Fig. 12(a). In this simulation, the red neuron generates a spike a t = 50 ps. This spike propagates to the blue neuron, which generates a spike at t = 120 ps. The spike from the blue neuron propagates symmetrically in two directions and arrives at both the red neuron and the green neuron at t = 120 ps. At t = 120 ps, the red neuron is in its absolute refractory period and will not generate a spike. In contrast, the green neuron will generate a spike as a result of the incoming spin current at t = 190 ps. After this, the green, black, and purple neurons spike in sequence. In each case, the signal will propagate symmetrically after a neuron spikes, but the refractory properties of AFM neurons prevent them from spiking, thus ensuring unidirectional signal propagation.
Consider now the network shown schematically in Fig. 12(b). This network is identical to that in Fig. 12(a), except that in this case the first spike is initiated in the purple neuron on the right. Results of simulating this neural network are also shown in Fig. 12(b), with the signal propagating from the purple neuron to the red neuron.
It is important to note that the neural networks in Figs. 12(a) and 12(b) are identical and unchanged; the only difference is where the initial spike was delivered. It should also be emphasized that spikes only propagate in one direction due to the refraction properties of the AFM neuron.
It is possible to form an isolator with a symmetrically connected neural network. An isolator serves a function similar to that of a diode or a check valve and restricts the signal flow to a single direction. A circuit that contains an isolator is shown in Fig. 13(a). The circuit consists of two sets of symmetrically coupled neurons, shown in red for neurons 1–5 and shown in blue for neurons 6–9. The portion of this circuit that functions as an isolator is neurons 4–6 and includes the two weak synaptic couplings that converge on neuron 6. In this configuration, signals can flow in just one direction, from neuron 1 to neuron 9, but cannot flow in the reverse direction. This will be demonstrated via simulation.
Isolator composed of symmetrically coupled neurons. (a) Schematic of the isolator circuit, with two sets of neurons, red and blue. Within these two sets, couplings are identical and fully symmetric. There is also a weak symmetric coupling between neuron 6 and neurons 4 and 5. (b) Simulation for a signal flowing from neuron 1 to neuron 9. The synaptic weights between 4, 5, and 6 are such that the combined spikes of 4 and 5 provide sufficient energy for neuron 6 to generate a spike. (c) Simulation for a signal flowing from neuron 9 to neuron 6. The signal is unable to continue flowing through the red chain due to the weak connections between neurons 6 and 4/5. In these simulations, α = 0.1.
Isolator composed of symmetrically coupled neurons. (a) Schematic of the isolator circuit, with two sets of neurons, red and blue. Within these two sets, couplings are identical and fully symmetric. There is also a weak symmetric coupling between neuron 6 and neurons 4 and 5. (b) Simulation for a signal flowing from neuron 1 to neuron 9. The synaptic weights between 4, 5, and 6 are such that the combined spikes of 4 and 5 provide sufficient energy for neuron 6 to generate a spike. (c) Simulation for a signal flowing from neuron 9 to neuron 6. The signal is unable to continue flowing through the red chain due to the weak connections between neurons 6 and 4/5. In these simulations, α = 0.1.
Results of the first simulations performed on this circuit are shown in Fig. 13(b). In this simulation, neuron 1 spikes at t ∼ 100 ps, and this signal propagates from neuron 1 toward neuron 5. At about t = 500 ps, neurons 4 and 5 generate spikes in rapid succession. Together, the two spikes generated by these two neurons have sufficient strength to cause neuron 6 to spike, despite the weak synaptic coupling between the two chains. After neuron 6 generates a spike, the signal continues down the chain toward neuron 9.
Simulations were performed on the same neural network with a signal originating at neuron 9. Results of this simulation are shown in Fig. 13(c). In this simulation, the signal at neuron 9 travels to neuron 6, which spikes at t = 500 ps. Due to the weak synaptic coupling, the spike generated by neuron 6 has insufficient strength to cause either neuron 4 or 5 to generate a spike. Thus, the signal is unable to travel from neuron 9 to neuron 1; spikes are prevented from propagating in the “backward” direction.
The design of the isolator can be extended into a combiner circuit, whose schematic is shown in Fig. 14(a). This circuit has three sets of neurons, shown in red, green, and blue. The red and green neurons represent input branches, and the blue neurons represent an output branch. With this circuit, signals can flow from the red to blue neurons or from the green to blue neurons. However, this configuration ensures that there is no signal leakage between the red and green branches.
Combiner composed of symmetrically coupled neurons. (a) Schematic of the combiner circuit, with three sets of neurons. Input neurons are red and green, while output neurons are blue. Arrow widths represent coupling strength. (b) Simulation result for a signal that begins in the red neurons. (c) Simulation results for a signal that begins in the green neurons. (d) Simulation result for a signal that begins in the blue neurons. In these simulations, α = 0.1.
Combiner composed of symmetrically coupled neurons. (a) Schematic of the combiner circuit, with three sets of neurons. Input neurons are red and green, while output neurons are blue. Arrow widths represent coupling strength. (b) Simulation result for a signal that begins in the red neurons. (c) Simulation results for a signal that begins in the green neurons. (d) Simulation result for a signal that begins in the blue neurons. In these simulations, α = 0.1.
Correct operation of this circuit is confirmed by simulation, with results shown in Figs. 14(b)–14(d). As shown in Fig. 14(b), a signal propagates from the red input through the red neurons to the magenta neuron and then to the blue output neurons. There is no leakage to the green neurons with this architecture. Likewise, as shown in Fig. 14(c), a signal propagates from the green input to green neurons and travels to the blue neurons, without leakage to the red neurons. Finally, as shown in Fig. 14(d), a signal that begins with the blue input will propagate through the blue neurons but will not propagate to the green or red neurons. Thus, this circuit can allow signals from the red and green branch to combine without crosstalk between branches and without signals flowing backward.
V. MEMORY LOOPS AND INHIBITION IN AFM NEURAL NETWORKS
This section will demonstrate neural networks that employ loops as memory and provide control with inhibition. This section begins with a simple loop and then continues with a demonstration of negative inhibition and positive inhibition. After this, the two concepts are combined to a controllable memory circuit.
Consider first the neural network shown in Fig. 15(a). Simulation results for this neural network are shown in Fig. 15(b). The simulation begins by first initiating a spike in the green neuron. This leads to a spike in neuron 1, which leads to a spike in neuron 2, which leads to a spike in neuron 3, which then continues around the loop back to neuron 1. As can be seen in the simulation results, the spike will continue around this loop indefinitely.
Simple memory loop. (a) Schematic of the neural network for the loop, with a green input neuron and four labeled blue loop neurons. (b) Simulation of the neural network, where the green curve represents a spike from the green input neuron and the blue curves represent spikes from the four loop neurons. In these simulations, α = 0.1.
Simple memory loop. (a) Schematic of the neural network for the loop, with a green input neuron and four labeled blue loop neurons. (b) Simulation of the neural network, where the green curve represents a spike from the green input neuron and the blue curves represent spikes from the four loop neurons. In these simulations, α = 0.1.
Inhibition functions are demonstrated in Fig. 16. As shown in Fig. 16(a), this neural network consists of five neurons, including input (green) and output (magenta) neurons. A signal can be carried from the input neuron to the output neuron through two intermediate neurons (blue). Please note that the synapses are adjusted so that the signal from the input neuron is strong enough to initiate a spike in both intermediate neurons. However, for the magenta neuron to fire, both blue intermediate neurons must spike at nearly the same time.
Negative inhibition and positive inhibition. (a) The neural network for demonstrating inhibition, with the input neuron (green), intermediate neurons (blue), output neuron (magenta), and inhibition neuron (red). The synapses are weighted such that solid arrows represent a full connection, while dashed arrows represent a half connection. (b) Simulation of uninhibited network. The input neuron, with a green curve, fires. This leads the two intermediate neurons, with blue curves, to fire. Together, the spikes from the intermediate neurons provide enough current for the magenta output neuron to fire. (c) Simulation demonstrating negative inhibition. The inhibitor neuron generates a spike with negative polarity, which prevents the lower intermediate neuron from spiking. (d) Simulation demonstrating positive inhibition. The red and green neurons fire, leading the lower intermediate neuron to quickly fire. After a delay, the upper intermediate neuron fires. Because the intermediate neurons fire at different times, the output neuron does not fire. In these simulations, α = 0.1.
Negative inhibition and positive inhibition. (a) The neural network for demonstrating inhibition, with the input neuron (green), intermediate neurons (blue), output neuron (magenta), and inhibition neuron (red). The synapses are weighted such that solid arrows represent a full connection, while dashed arrows represent a half connection. (b) Simulation of uninhibited network. The input neuron, with a green curve, fires. This leads the two intermediate neurons, with blue curves, to fire. Together, the spikes from the intermediate neurons provide enough current for the magenta output neuron to fire. (c) Simulation demonstrating negative inhibition. The inhibitor neuron generates a spike with negative polarity, which prevents the lower intermediate neuron from spiking. (d) Simulation demonstrating positive inhibition. The red and green neurons fire, leading the lower intermediate neuron to quickly fire. After a delay, the upper intermediate neuron fires. Because the intermediate neurons fire at different times, the output neuron does not fire. In these simulations, α = 0.1.
A simulation of the signal being carried from the input neuron to the output neuron is shown in Fig. 16(b). First, the green input neuron generates a spike. Then, the two intermediate neurons spike, which together deliver enough energy to allow the output neuron to generate a spike.
The diagram in Fig. 16(a) also shows a red “inhibitor” neuron. The utility of the inhibitor neuron is demonstrated by simulation in Fig. 16(c). In this simulation, the red inhibitor neuron fires at t = 90 ps, as shown by the red dashed line. As can be seen in the figure, the spike generated by the inhibitor neuron has a negative polarity, as was described in Sec. III C. When the green input neuron fires at t = 100 ps, only the upper blue intermediate neuron fires in response, which does not provide sufficient energy to cause the magenta neuron to fire. The lower blue neuron does not fire as the inhibitory spike with negative polarity suppressed the rotation of its M1 magnetization. Essentially, when the negative inhibitory signal arrives at the lower blue intermediate neuron, it effectively cancels the positive signal and prevents the AFM magnetization from rotating in that neuron. The inhibition demonstrated in Fig. 16(c) can be called negative inhibition. This form of inhibition resembles hyper-polarizing stimuli in biological neurons.79
A different form of inhibition, which can be called positive inhibition, is demonstrated by simulation in Fig. 16(d). Once again, in this simulation, the red inhibitor neuron fires at t = 90 ps, as shown by the red dashed line. Here, the spike generated by the inhibitor neuron has a positive polarity. After the inhibitor neuron fires, the green neuron generates a spike at t = 100 ps. The combined spikes from the red and green neurons cause the lower intermediate neuron to fire at a different time from the previous simulation. Because the lower intermediate neuron fires at a different time from the upper intermediate neuron, there is insufficient energy arriving at the magenta output neuron, which does not fire. Thus, a positive signal is able to inhibit an output. It is notable that positive inhibition plays an important role in biological neurons and that this feature of AFM neurons may be employed in future AI tasks.
This inhibitor functionality can have application in a memory cell, which is shown in Fig. 17(a). The memory cell consists of an input neuron, an inhibitor neuron, and five intermediate neurons, as shown in the figure. Operation of the memory cell will be demonstrated by simulation. Initially, the entire system is at rest. Then, at t = 100 ps, the green input neuron generates a spike. This is shown by a green curve in Fig. 17(b). This signal from the input neuron then enters the blue neuron loop at the neuron labeled “1,” as shown in Fig. 17(b). After this, the spiking signal will travel through the neuron loop, from neuron “1” to neuron “2,” then to neurons “3a” and “3b,” and then to neuron “4.” These neurons, spiking in succession, are shown by multiple blue curves in Fig. 17(c). Please note that the synapses have been adjusted so that both neurons 3a and 3b must spike at nearly the same time in order for neuron 4 to generate a spike.
AFM neuron memory cell. (a) The memory cell neural network. The input neuron is colored green, the inhibitor neuron is colored red, and the memory neurons are colored blue. (b) Simulated input and inhibition signals. (c) Simulated response to the input. The input signal at t = 100 ps initiates the signal to flow from 1 to 4 and cyclically repeats until t = 500 ps when the inhibitor signal stops the cycle. The cycle is restarted by an input spike at t = 800 ps. In these simulations, α = 0.1.
AFM neuron memory cell. (a) The memory cell neural network. The input neuron is colored green, the inhibitor neuron is colored red, and the memory neurons are colored blue. (b) Simulated input and inhibition signals. (c) Simulated response to the input. The input signal at t = 100 ps initiates the signal to flow from 1 to 4 and cyclically repeats until t = 500 ps when the inhibitor signal stops the cycle. The cycle is restarted by an input spike at t = 800 ps. In these simulations, α = 0.1.
The inhibitor neuron fires at t = 500 ps, as shown by a red curve in Fig. 17(b). The signal from the inhibitor neuron causes neuron 3b to fire at an earlier time than neuron 3a. Due to this variance in timing and the weak synaptic connection between neurons 3 and 4, there is insufficient energy for neuron 4 to generate a spike, which stops the signal from circulating. Thus, the properties of positive inhibition were used to control a spiking signal propagating in a memory loop. After this, the input signal fires once more at t = 800 ps, and the memory signal circulates once again, as shown in the figure.
VI. COMPARISON OF AFM NEURONS AND BIOLOGICAL NEURONS
In previous sections, the characteristics that the AFM neuron and biological neurons shared were presented. This section will provide a summary of these shared characteristics.
First, AFM neurons generate spiking voltages that closely resemble the action potentials of biological neurons. Furthermore, AFM neurons follow the “all-or-nothing” law, that is, AFM neurons only generate spikes when their input surpasses a critical current, very much in the same way that an action potential is elicited from biological neurons. These two properties were demonstrated in Sec. III A.
Second, AFM neurons have a response latency, as was also demonstrated in Sec. III A. Basically, in both biological neurons and AFM neurons, after a sufficient stimulus has been delivered to the neuron, there is a time delay before the spike occurs. Latency of a stimulated biological neuron varies, depending on both the frequency and the intensity of its stimulus. With AFM neurons, the duration of the response latency can be controlled dynamically by adjusting the amplitude of the stimulus ip(t) and by adjusting the magnitude of the bias current Idc. The latency period also depends on the effective damping, the anisotropy, and the exchange frequency of the AFM material.
Third, both AFM neurons and biological neurons have a refractory period. Specifically, both types of neurons have an absolute refractory period and a relative refractory period. This was discussed in Sec. III B.
Fourth, both AFM neurons and biological neurons have variable spiking modes, that is, AFM neurons can generate single spikes, bursting spikes, or spike trains. They can also exhibit adaptation and stuttering. This was discussed in Sec. III B.
Fifth, the generation of action potentials in both AFM neurons and biological neurons can be suppressed through inhibition. This was demonstrated in Sec. V. In AFM neurons, both negative inhibition and positive inhibition can be used to suppress neuronal activity. The duration of inhibition for AFM neurons depends on the characteristics of their refractory period.
Finally, AFM neurons can be interconnected into a neural network where connections between neurons are mediated by synapses. Two methods of interconnecting neurons, via electrical current and via spin current, along with examples of spintronic synapses, were discussed in Sec. IV.
VII. CONCLUSION
AFM neurons are nanometer scale electronic elements that exhibit behavior that resembles that of a biological neuron. Specifically, they have several properties that resemble the characteristics of biological neurons, including a finite refraction time, response latency, and bursting behavior. In addition, AFM neurons operate much faster than the state of the art, with a ps spike time, and with an ultra-low energy consumption of about 10−3 pJ per synaptic operation. In this paper, it was demonstrated by simulation that AFM neurons can be interconnected into a neural network, with response latency, refraction, and inhibition as tools to be potentially used in machine learning algorithms. It was also demonstrated that AFM neurons can perform Boolean operations, have a polarity that can be reversed, can exhibit inhibition, and can perform memory functions. In addition, a one-way signal transmission can be implemented with symmetric coupling, along with an isolator and a combiner.
ACKNOWLEDGMENTS
This work was supported by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary Research Program of the University Research Initiative (MURI), under Grant No. FA9550-19-1-0307.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
H. Bradley: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). S. Louis: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). C. Trevillian: Conceptualization (equal); Software (equal); Validation (equal); Writing – review & editing (equal). L. Quach: Investigation (equal); Writing – review & editing (equal). E. Bankowski: Validation (equal); Visualization (equal); Writing – review & editing (equal). A. Slavin: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Writing – review & editing (equal). V. Tyberkevych: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.