Pattern recognition using spiking antiferromagnetic neurons

Pattern recognition using spiking antiferromagnetic neurons | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Pattern recognition using spiking antiferromagnetic neurons Hannah Bradley, Steven Louis, Andrei Slavin, Vasyl Tyberkevych This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4365235/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 10 You are reading this latest preprint version Abstract Spintronic devices offer a promising avenue for the development of nanoscale, energy-efficient artificial neurons for neuromorphic computing. It has previously been shown that with antiferromagnetic (AFM) oscillators, ultra-fast spiking artificial neurons can be made that mimic many unique features of biological neurons. In this work, we train an artificial neural network of AFM neurons to perform pattern recognition. A simple machine learning algorithm called spike pattern association neuron (SPAN), which relies on the temporal position of neuron spikes, is used during training. In under a microsecond of physical time, the AFM neural network is trained to recognize symbols composed from a grid by producing a spike within a specified time window. We further achieve multi-symbol recognition with the addition of an output layer to suppress undesirable spikes. Through the utilization of AFM neurons and the SPAN algorithm, we create a neural network capable of high-accuracy recognition with overall power consumption on the order of picojoules. Physical sciences/Physics/Applied physics Physical sciences/Materials science/Theory and computation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Introduction Despite the increase in the computational capability of typical von Neumann architecture, the human brain still outperforms modern computers at classification tasks with a fraction of power consumption [ 1 ], [ 2 ]. By mimicking brain-like behaviors through hardware-implemented artificial neural networks, neuromorphic chips perform pattern recognition with reduced power consumption and increased efficiency [ 3 ]. A biological neural network is comprised of two critical components: the individual processing units called neurons and the synapses that determine their connections. Current neuromorphic chips use silicon-based transistors to make up both of these components [ 4 ]. In spite of the fact that transistor-based neuromorphic computing is an improvement over Von Neumann architecture, a number of drawbacks still exist. Mainly, it requires multiple transistors to create one artificial neuron, thereby requiring a large amount of physical area and increasing power consumption. In recent years, there has been growing interest in the use of spintronic devices for neuromorphic computing [ 5 ], [ 6 ]. Spintronic artificial neurons offer numerous advantages over their transistor-based counterparts. Apart from possessing intrinsic non-linear dynamics, these neurons can be fabricated at the nanoscale, with each device serving as a single neuron. As a result, power and area requirements are significantly reduced. The creation of artificial neurons has been shown to be possible with domain wall motion [ 7 ]–[ 9 ], skyrmions [ 10 ]–[ 12 ], spin torque nano oscillators [ 13 ]–[ 15 ], and magnetic tunnel junctions [ 16 ], [ 17 ]. One of the prospective designs of artificial spintronic neurons is based on antiferromagnetic (AFM) spin-Hall oscillators operating in a subcritical regime [ 18 ]. These artificial “AFM neurons” generate voltage spikes that closely resemble the action potentials elicited by biological neurons, with properties that include response latency, bursting, and refraction. The main advantages of artificial AFM neurons are their nano-sized footprint, relatively low power consumption, and ultra-high operational speed, generating spikes with a duration on the order of 5 ps [ 19 ]. In light of the high speed and low power consumption of AFM neurons, it is important to consider AFM neurons as a possible candidate for post-silicon neuromorphic computer systems. Until now, the literature shows no attempt to develop a method to perform machine learning with AFM spiking neurons. While a simple neural network employing AFM neurons in memory loops was presented in Ref. [ 19 ], that neural network featured copper bridge synapses that carried spikes from neuron to neuron with constant coupling. Thus, it did not have the ability to demonstrate that more complex neural networks based on AFM neurons can be trained for cognitive tasks like pattern recognition and how efficient these networks are in terms of training, recognition time, and power consumption. In this work, we theoretically investigate the possibility of using AFM neurons combined with a supervised learning algorithm to create neural networks that recognize symbols generated from a grid of input neurons [ 20 ], [ 21 ]. We show that, due to the strongly nonlinear and inertial dynamics of AFM neurons, even a single AFM neuron is capable of successfully recognizing various symbols from a \(5\times 5\) input grid, which is enough to encode various printed symbols. Our simulations show that the total training time of an AFM SPAN neuron can be below \(1 {\mu }\text{s}\) , while the power consumption during the training is of the order of 30 pJ. This research provides the first demonstration of the ability of AFM neurons to perform learning tasks, thus making clear the potential for using artificial AFM neurons in machine learning applications. Methods Antiferromagnets (AFM) have two magnetic sublattices orientated in opposing directions. The direction of AFM magnetic sublattices relative to the crystal lattice can be manipulated using spin currents. Usually, this is achieved in spin Hall geometry, in which a layer of heavy metal covers an AFM element. When a DC electric current flows in the heavy metal layer, it induces a perpendicular spin current that penetrates into the AFM. The most interesting effect of spin current on the AFM dynamics happens when the spin polarization of the spin current is perpendicular to the easy plane of the AFM. In this case, spin-transfer torque induced by a sufficiently large spin current causes the AFM sublattices to rotate in the easy plane [ 22 ]. For AFM materials with bi-axial anisotropy, the rotation of the sublattices is not uniform with time. This results in a sequence of short spin-pumping spikes at a frequency that can reach the THz range. The threshold current needed to achieve this auto-oscillating regime depends on the easy-plane anisotropy of the AFM material and is of the order of \({10}^{8} \text{A}/{\text{c}\text{m}}^{2}\) for NiO AFM [ 22 ]. If the driving current is below the generation threshold, the AFM oscillator will not have enough energy to overcome the anisotropy, but the equilibrium orientation of the AFM sublattices will be moved towards the hard direction in the easy plane. With an additional impulse of current, the AFM magnetizations will surpass the anisotropy energy barrier and perform a single half-turn in the easy plane, which will cause a single spike of the spin-pumping voltage. This response of a sub-threshold AFM spin Hall oscillator is similar to the reaction of a biological neuron to an external stimulus [ 18 ]. The AFM neurons and their networks also have other properties that resemble biological neural systems, such as refraction and delayed response [ 19 ]. As it was shown in Ref. [ 22 ], the dynamics of an AFM neuron can be described by the in-plane angle \(\varphi\) that the Neel vector of the AFM makes with the easy axis of the AFM. Under rather general assumptions, the angle \(\varphi\) obeys the second-order dynamical equation, $$\frac{1}{{\omega }_{ex}}\ddot{\varphi }+\alpha \dot{\varphi }+\frac{{\omega }_{e}}{2}\text{sin}2\varphi =\sigma I$$ 1 , where \({{\omega }_{ex}=2\pi f}_{ex}\) is the exchange frequency of the AFM, \(\alpha\) is the effective Gilbert damping constant, \({\omega }_{e}=2\pi {f}_{e}\) is the easy axis anisotropy frequency, \(\sigma\) is the spin-torque efficiency defined by Eq. ( 3 ) in Ref. [ 22 ], \(I\) is the driving electric current. Further details about the derivation of Eq. ( 1 ) can be found in [ 18 ], [ 22 ]. Note that the spin-pumping signal produced by the AFM is proportional to the angular velocity of the sublattice rotation \(\dot{\varphi }\) . Namely, the inverse spin Hall voltage produced by the AFM neuron can be found as $$V=\beta \dot{\varphi }$$ 2 , where the efficiency \(\beta =0.11x{10}^{-15} V\bullet s\) is defined by Eq. ( 2 ) in Ref. [ 19 ]. In this work, we study the dynamics of a network of interconnected AFM neurons. Each neuron is described by its own phase \({\varphi }_{i}\) and obeys an equation similar to Eq. ( 1 ) with additional terms describing synaptic connections between the neurons: $$\frac{1}{{\omega }_{ex}}{\ddot{\varphi }}_{i}+\alpha {\dot{\varphi }}_{i}+\frac{{\omega }_{e}}{2}\text{sin}2{\varphi }_{i}=\sigma I+\sum _{i\ne k}{\kappa }_{ik}{\dot{\varphi }}_{k}$$ 3 . Here, \(i\) and \(k\) are indices that represent the \(i\) -th and \(k\) -th neurons, and \({\kappa }_{ik}\) represents a matrix of coupling coefficients. Note that the coupling signal produced by the \(k\) -th neuron is proportional to \({\dot{\varphi }}_{k}\) , in agreement with Eq. ( 2 ). The coupling coefficients that constitute \({\kappa }_{ik}\) can behave as the synaptic weights in a machine learning system. To a large extent, the challenge of building a fast and efficient neuromorphic computing system depends on the efficient implementation of variable spintronic synapses capable of changing inter-neuron connectivity. The need for variable synapses dramatically increases the complexity of a neuromorphic neural network. This problem is even more serious for AFM neurons since, to fully employ ultra-fast AFM dynamics in neuromorphic hardware, the reaction times of artificial synapses should be on the timescale of AFM neuron dynamics. As AFM neurons spike with a duration that can be less than 5 ps, it should be noted that traditional CMOS technology would severely limit the capabilities of an AFM neural network. To our knowledge, no variable weight synapses have been developed that are suitable to work in conjunction with AFM neurons. As no CMOS or spintronic hardware is capable of being used as variable synapses for AFM neurons, circuit simulations such as SPICE simulations cannot be done. Therefore, in this paper, which primarily focuses on investigating the dynamics of AFM neurons, we did not assume any particular physical model of a synapse. Instead, the simulated synapses are considered to be “ideal” such that they can be adjusted instantaneously and to any value. Nevertheless, it is important to consider how the latency, or synaptic delay, would impact our model. In a previous work [ 19 ], copper bridges with fixed dimensions were used to provide constant weight synaptic coupling or fixed connections \({\kappa }_{ik}\) between AFM neurons. Copper is capable of carrying spin current from one neuron to the next, allowing the output of one neuron to act as the input for a second neuron. The synaptic delay of copper bridges can be found by solving the diffusion equation for spin accumulation in copper. By using standard diffusion parameters for copper [ 23 ] and AFM neuron dimensions found in Ref. [ 19 ], the synaptic delay for a copper bridge with a length of 100 nm can be found to be about 1.5 ps. We consider this delay to be short enough to have a negligible impact on our system. There is a remarkable similarity between the equation describing the AFM neuron and that describing the dynamics of a physical pendulum; therefore, each term in Eq. ( 3 ) can be characterized by its mechanical analog. As a result, the coefficient of the first term on the left-hand side of Eq. ( 3 ) defines an effective mass, indicating that the AFM neuron possesses an effective inertia due to AFM exchange. This inertia results in a delay between a neuron receiving an input and the resulting output, an effect not found in conventional artificial neurons. When AFM neurons are linked together, such that the output of one neuron acts as the input of the next, the delay is dependent on the coupling strength \({\kappa }_{ik}\) between the neurons. The delay caused by inertia decreases as the strength between neurons increases. Thus, the firing time of the neuron can be easily controlled. This means that the AFM neurons are well-suited for neuromorphic algorithms in which time encoding of neuron spikes is used. One such time-encoding approach, namely, spike pattern association neuron (SPAN) [ 20 ], is studied in this paper. The architecture of an AFM neural network realizing the SPAN algorithm is shown in Fig. 1 (a). It consists of one output “SPAN” neuron connected to many neurons of the input layer. In our simulations, the input layer consisted of 25 neurons and encoding input symbols drawn in a \(5\times 5\) binary grid. We used several shapes of the input symbols shown in Fig. 1 (b). A blackened pixel in the input symbol causes a spike in the corresponding input neuron, while a white pixel will have no spike. The SPAN neuron is trained to output a spike at a certain prescribed time if the input symbol matches the pattern to be recognized. To achieve this, synaptic connections between the input layer and the SPAN neuron are adjusted during the training, as explained below. We used parallel encoding of the input layer; namely, the input symbol triggers the input neurons to fire simultaneously. If the combined weights connected to the SPAN are strong enough, there will be an output spike. The goal of training is to move this output spike to the desired time for a chosen symbol. If the spike is produced earlier (later) than the target time, the weights connected to the SPAN should reduce (increase). In more detail, the SPAN training algorithm is based on the Widrow-Hoff rule, where the difference between the desired spike time \({t}_{d}\) and the actual spike time \({t}_{a}\) is used to update the synaptic weights. After some manipulation, shown in Ref. [ 24 ], the Widrow-Hoff rule is transformed to describe the change in weights during training: $${\Delta }\kappa = \lambda {\left(\frac{e}{2}\right)}^{2}\left[\left({t}_{d}-{t}_{i}+{\tau }\right){e}^{-\left({t}_{d}-{t}_{i}\right)/\tau }-\left({t}_{a}-{t}_{i}+{\tau }\right){e}^{-({t}_{a}-{t}_{i})/\tau }\right]$$ 4 , where \(\lambda\) is a positive and constant learning rate, \({t}_{i}\) is the timing of the input spike, \({t}_{d}\) is the desired timing of the output spike, \({t}_{a}\) is the actual timing of the output spike, and \(\tau\) is a time constant corresponding to the width of a spike. Due to the simplicity of the SPAN algorithm, it is only capable of training a neuron to a single symbol. Upon training, a SPAN should output its spike at the target time \({t}_{d}\) for the correct symbol and spike away from the target time for any other symbol. A library of 20 symbols is used to train the neural network. These symbols are all variations of the correct symbol chosen from one of the symbols shown in Fig. 1 (b). Variations include symbols with multiple additional or missing pixels. Initially initialized with random synaptic weights \kappa_{ik}, the neural network receives each symbol as an input during one training epoch. A symbol is associated with a target time corresponding to the image's difference from the correct symbol. Using this time and the actual timing of the output neuron, the SPAN algorithm determines how the weights should be changed in accordance with Eq. ( 4 ). The algorithm is modified to ensure that the weights cannot become negative, ensuring a more straightforward implementation in hardware. When all images have been processed, the weight changes resulting from each symbol are averaged, the neural network is updated, and the next epoch begins. Figure 2 shows the output spikes of a SPAN neural network after training. When the correct symbol is supplied as input, the SPAN spikes within a 10 ps time window of the target time; this implies that the neural network has recognized chosen symbol. Any other symbol should cause a spike outside the target time window, indicating that a different symbol was used as input. Results and Discussion Figure 3 (a) shows the error between the actual and desired spike time for the correct symbol throughout training, and Fig. 3 (b) shows the corresponding changes in each synaptic weight. In this case, the neural network is being trained to the "O" symbol. After an aggressive start, the change in weights is subtle for most of training. Due to the large number of inputs, each individual weight is relatively small, as only the total sum is relevant to the timing of the output spike. It should be noted that some weights continue to change throughout the whole of training. These weights, in particular, do not contribute significantly to any symbol in the training library and, therefore, have limited data when making weight adjustments. After about 10 epochs, the trained neural network will produce a spike within a 10ps widow of the target time when the symbol is recognized, as shown in Fig. 2 . Several examples of incorrect symbols serving as input and the resulting output spikes are shown in Fig. 4 . By spiking outside the target time window for any symbol other than the correct symbol, the neural network has high accuracy in recognizing the chosen symbol. Whether additional or missing pixels serve as the difference from the correct image does not matter in making a spike outside of the target time window. To gain a more complete understanding of how weights change when training with the "Z" symbol, a distribution of weights is plotted in Fig. 5 . Figure 5 (a) shows the random distribution of weight at the beginning of training, and Fig. 5 (c) shows the weights at the end of training, after 60 epochs. Figure 5 (b), in contrast, shows the training in the middle of training after 10 epochs. It is evident from this sequence that as training progresses, the weights are adjusted in such a way that the “Z” symbol is reflected in the weight distribution. As there is little difference between Fig. 5 (b) and Fig. 5 (c), it can be assumed that the most significant training happens in the first 10 epochs. Multiple AFM SPANs, trained to recognize different symbols, can be connected to the same layer of input neurons. This way, the SPAN trained to the input symbol will produce a spike within the target time window, while the others would spike outside it. With multiple SPANs all spiking at different times, the output can be unclear. Therefore, it would be convenient to clear the output by suppressing output spikes outside the target time window. This can be done by creating an additional output layer that consists of fixed synapses. The architecture of the neural network capable of suppressing unwanted outputs is shown in Fig. 6 . The input neurons are connected to three SPANs via trainable weights. These SPANs are each trained to recognize different symbols. The SPANs then serve as input to the output layer. The output layer’s synapses have weak coupling, such that a single pre-synaptic spike is insufficient to cause a spike in the post-synaptic neuron. Two spikes must happen simultaneously to produce a strong enough signal for a post-synaptic spike. The red neuron, shown in Fig. 6 , is a clock neuron spiking at the target time. The clock neuron receives an input independent of the input layer’s symbol. This independent input causes the clock neuron to generate a spike at the target time. Therefore, when a symbol is recognized, the SPAN will spike along with the clock neuron at the target time. These two signals are enough to overcome the weak coupling and cause the post-synaptic neuron to fire. The spikes from the SPANs that do not correspond to the input symbol would spike away from the target time, thus not combining with the clock neuron to cause an output spike. This output layer ensures that only the spike from the SPAN corresponding to the correct symbol is outputted. The output spiking signals of this neural network are shown in Fig. 7 . The blue, green, and magenta spikes correspond to three different SPANs trained to three different symbols, while the red spike is the clock neuron spiking at the target time. At this time, there are spikes of the clock neuron and a single SPAN corresponding to the inputted symbol. These two spikes combine to send a single spike to the output, indicating which symbol has been recognized. Therefore, this output layer creates an output that clearly identifies the recognized symbol. Conclusions Ultra-fast spiking artificial neurons built from AFM oscillators have a number of unique properties that can be harnessed to create simple neural networks with fixed synapses [ 19 ]. However, in order to use AFM neurons in neuromorphic computing, a trainable neural network with variable synapses is required. The physical implementation of variable synapses is much more complex than fixed ones; therefore, in this work, we limit the number of trainable weights to reduce the complexity of the neural network. Namely, we numerically investigated the performance of the SPAN algorithm for the recognition of symbols encoded in a \(5\times 5\) binary grid. The simulated AFM neural networks are capable of recognizing symbols by producing a spike within a target time window (10 ps). The training time of the AFM networks for such relatively small images is very short, about 10 epochs with a 20-symbol library. Due to the high speed of AFM neurons (200 ps between inputting a symbol and the output neuron firing), this training session may last for only ~ 40 ns of real-time. Multiple SPANs, trained to different symbols, can be connected to the same inputs, thus providing multi-symbol recognition capabilities. With the addition of a fixed output layer suppressing spikes outside the target time window, the neural network will produce a single spike corresponding to the recognized symbol in just a few hundred picoseconds. The energy consumption of a single AFM neuron, with dimensions described in Ref. [ 19 ] was calculated to be about \({10}^{-3}\) pJ per synaptic operation. This is a relatively low power consumption in comparison with other artificial spiking neurons. For example, a spin torque nano-oscillator-based neuron reports an energy consumption of 4.96 pJ per synaptic operation [ 25 ], and an all-CMOS neuron was reported to have a power consumption of 247 pJ per synaptic operation [ 26 ]. The low power consumption of AFM neurons is the result of the simple design of the AFM oscillator and the ultra-fast spikes that result from using AFM materials. To successfully train a single SPAN, the total energy consumption of the AFM neural network is 31.2 pJ. The use of the SPAN algorithm leads to a very simple, one-layer neural network. This neural network is limited in its ability to perform more complex tasks as a result of such simplifications. For example, the MNIST data set of handwritten digits, commonly used for neural network training, is encoded in \(28\times 28\) pixel grids. It is unlikely that a neural network such as the one studied here would be able to cope with inputs on this scale. The simple neural network used in this work demonstrates for the first time that AFM neurons are capable of being used for neuromorphic tasks such as pattern recognition. In order to advance the use of AFM neurons in neuromorphic computing, a more complex neural network and learning algorithm are required. However, even the simple network described in this work may find practical applications when high training, operation speed, and/or low consumed power are required. Declarations Author Contribution HB is responsible for numerical simulations, data analysis and writing the manuscript. VT contributed to formulating the program and provided overall supervision. All authors participated in reviewing and editing the manuscript. Acknowledgments This work was partially 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. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4365235","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":301233713,"identity":"f2a4317e-aa1f-42d7-a50a-ac3be7f9bbb9","order_by":0,"name":"Hannah Bradley","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA70lEQVRIiWNgGAWjYBACAwkgwVPAwMDPwMDM8ICBgbEBKEiEFqAayQaglgSStBgcIFaLuXTvww9vDGwSNx9vfmyQUHNHtoG9eZsEPi2Wc44bS84xSEvcduaYcULCsWfGDTzHyvBqMbiRxiDNY3A4cduNBOMDCWyHExskcswIaWH+zWPwP3HzjPTPBxL+AbXIvyGohQ1oy4HEDRI5xgmJbSBbeAhouXOMzXKOQbLxjDNnig0S+w4bt/GkFVvg1XK7jfnGmwo72f729s0SH74dlu1nP7zxBj4tMODYAGOxEaMcBOyJVTgKRsEoGAUjEAAAnjBP6VLqVdUAAAAASUVORK5CYII=","orcid":"","institution":"Oakland University","correspondingAuthor":true,"prefix":"","firstName":"Hannah","middleName":"","lastName":"Bradley","suffix":""},{"id":301233714,"identity":"c634020c-309a-4a28-8545-e8c8d03cacac","order_by":1,"name":"Steven Louis","email":"","orcid":"","institution":"Oakland University","correspondingAuthor":false,"prefix":"","firstName":"Steven","middleName":"","lastName":"Louis","suffix":""},{"id":301233718,"identity":"cf8910e0-ecb0-494e-b9b3-fc74fb422c46","order_by":2,"name":"Andrei Slavin","email":"","orcid":"","institution":"Oakland University","correspondingAuthor":false,"prefix":"","firstName":"Andrei","middleName":"","lastName":"Slavin","suffix":""},{"id":301233719,"identity":"21ab9f39-ff34-4a5d-8460-c76c6983b0e2","order_by":3,"name":"Vasyl Tyberkevych","email":"","orcid":"","institution":"Oakland University","correspondingAuthor":false,"prefix":"","firstName":"Vasyl","middleName":"","lastName":"Tyberkevych","suffix":""}],"badges":[],"createdAt":"2024-05-03 16:40:50","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4365235/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4365235/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":56399695,"identity":"af2adaca-00a1-433c-ba0c-d08aa2373fd4","added_by":"auto","created_at":"2024-05-13 16:13:49","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":224568,"visible":true,"origin":"","legend":"\u003cp\u003eSingle AFM SPAN neural network. (a) The architecture of AFM SPAN neural network. The input symbol is encoded in the spiking pattern of the input layer neurons. The synaptic weights between the input layer and the AFM SPAN neuron are adjusted during training. The temporal position of the SPAN’s output spike encodes the outcome of pattern recognition. (b) Examples of the different input symbols used in pattern recognition where each pixel cell represents a different input neuron that will fire if the pixel is black.\u003c/p\u003e","description":"","filename":"Fig1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/978d8e07d08f033017d88e80.jpg"},{"id":56399693,"identity":"efe3fbdf-c279-40ab-87ab-be9b39eda5ed","added_by":"auto","created_at":"2024-05-13 16:13:49","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":127576,"visible":true,"origin":"","legend":"\u003cp\u003eSimulation result of output signals generated by a trained AFM SPAN network. The red dashed line shows the target time for symbol recognition, with a 10 ps time window (red shading) encompassing the target time. The blue (green) line shows the simulated output spike of the AFM SPAN for the correct (incorrect) input symbol.\u003c/p\u003e","description":"","filename":"Fig2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/13033661837a536613b407cb.jpg"},{"id":56400213,"identity":"7da38166-8a0c-4798-86e6-ad7e52021737","added_by":"auto","created_at":"2024-05-13 16:21:52","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":365447,"visible":true,"origin":"","legend":"\u003cp\u003eSimulations depicting the training process of an AFM SPAN. (a) Time difference between the target and actual spike times of the AFM SPAN’s output over 60 epochs of training. (b) Each colored line illustrates the evolution of an individual synaptic weight from an input neuron to the AFM SPAN over 60 epochs of training.\u003c/p\u003e","description":"","filename":"Fig3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/0ee0dd84c188f19e6c476f6c.jpg"},{"id":56399700,"identity":"ef378095-0865-4df9-b772-2269a39ca020","added_by":"auto","created_at":"2024-05-13 16:13:49","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":221703,"visible":true,"origin":"","legend":"\u003cp\u003eSimulation result of the output spikes of a trained AFM SPAN for different incorrect symbols, shown by insets. The dashed line represents the target time, and the red shading illustrates a 10 ps time window surrounding the target time. The simulated response for each incorrect symbol is outside the target time window, indicating that the inputted symbol is recognized to be not the correct symbol. (a, b) Input symbols have additional pixels. (c, d) Input symbols have missing pixels.\u003c/p\u003e","description":"","filename":"Fig4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/6d568ec4407ca8fa4c2fef8a.jpg"},{"id":56400212,"identity":"75ad4980-f342-466b-b5a2-663df3c06c09","added_by":"auto","created_at":"2024-05-13 16:21:52","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":79649,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of weights connecting inputs to SPAN (a) at the beginning of training, (b) in the middle of training, (c) at the end of training.\u003c/p\u003e","description":"","filename":"Fig5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/649e8863b94976b826589330.jpg"},{"id":56399699,"identity":"aee50a71-29ed-4636-b857-e23d9fc049d7","added_by":"auto","created_at":"2024-05-13 16:13:49","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":252683,"visible":true,"origin":"","legend":"\u003cp\u003eArchitecture of a 3 AFM SPAN neural network. The input symbol is encoded in the spiking pattern of the input layer neurons. During training, the synaptic weights between the input neurons and each SPAN are adjusted individually as each SPAN is trained to recognize a different symbol. The weights of the output layer are weakened such that a single spike cannot propagate to the next neuron. The red neuron represents a clock neuron that spikes at the target time, combining with the SPAN corresponding to the recognized symbol to generate a sufficiently strong signal for the post-synaptic neuron to fire.\u003c/p\u003e","description":"","filename":"Fig6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/cb29c5dadc8ccc791a65b54b.jpg"},{"id":56399698,"identity":"e3794002-034f-4e29-8aa4-949e8fef0b3b","added_by":"auto","created_at":"2024-05-13 16:13:49","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":374610,"visible":true,"origin":"","legend":"\u003cp\u003eSimulated result of a 3 SPAN (pink, green, blue) neural network for 3 different input symbols (Z, O, X). The red shading represents a 10 ps time window surrounding the target time. The SPAN corresponding to the recognized symbol generates a spike within the target time window, along with the clock neuron (red dashed spike), resulting in a single output spike. The color of the output spike corresponds to the SPAN that was trained to recognize the inputted symbol.\u003c/p\u003e","description":"","filename":"Fig7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/97da73fe9ee763692b601be1.jpg"},{"id":56400222,"identity":"4cf7916b-ba26-4f51-abdc-34a12974859a","added_by":"auto","created_at":"2024-05-13 16:22:44","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1972520,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4365235/v1/aeab0cca-7cdc-4ef2-abe7-f5934259da5e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Pattern recognition using spiking antiferromagnetic neurons","fulltext":[{"header":"Introduction","content":"\u003cp\u003eDespite the increase in the computational capability of typical von Neumann architecture, the human brain still outperforms modern computers at classification tasks with a fraction of power consumption [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. By mimicking brain-like behaviors through hardware-implemented artificial neural networks, neuromorphic chips perform pattern recognition with reduced power consumption and increased efficiency [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eA biological neural network is comprised of two critical components: the individual processing units called neurons and the synapses that determine their connections. Current neuromorphic chips use silicon-based transistors to make up both of these components [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. In spite of the fact that transistor-based neuromorphic computing is an improvement over Von Neumann architecture, a number of drawbacks still exist. Mainly, it requires multiple transistors to create one artificial neuron, thereby requiring a large amount of physical area and increasing power consumption.\u003c/p\u003e \u003cp\u003eIn recent years, there has been growing interest in the use of spintronic devices for neuromorphic computing [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Spintronic artificial neurons offer numerous advantages over their transistor-based counterparts. Apart from possessing intrinsic non-linear dynamics, these neurons can be fabricated at the nanoscale, with each device serving as a single neuron. As a result, power and area requirements are significantly reduced. The creation of artificial neurons has been shown to be possible with domain wall motion [\u003cspan additionalcitationids=\"CR8\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]\u0026ndash;[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], skyrmions [\u003cspan additionalcitationids=\"CR11\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]\u0026ndash;[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], spin torque nano oscillators [\u003cspan additionalcitationids=\"CR14\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]\u0026ndash;[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], and magnetic tunnel junctions [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eOne of the prospective designs of artificial spintronic neurons is based on antiferromagnetic (AFM) spin-Hall oscillators operating in a subcritical regime [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. These artificial \u0026ldquo;AFM neurons\u0026rdquo; generate voltage spikes that closely resemble the action potentials elicited by biological neurons, with properties that include response latency, bursting, and refraction. The main advantages of artificial AFM neurons are their nano-sized footprint, relatively low power consumption, and ultra-high operational speed, generating spikes with a duration on the order of 5 ps [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. In light of the high speed and low power consumption of AFM neurons, it is important to consider AFM neurons as a possible candidate for post-silicon neuromorphic computer systems.\u003c/p\u003e \u003cp\u003eUntil now, the literature shows no attempt to develop a method to perform machine learning with AFM spiking neurons. While a simple neural network employing AFM neurons in memory loops was presented in Ref. [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], that neural network featured copper bridge synapses that carried spikes from neuron to neuron with constant coupling. Thus, it did not have the ability to demonstrate that more complex neural networks based on AFM neurons can be trained for cognitive tasks like pattern recognition and how efficient these networks are in terms of training, recognition time, and power consumption.\u003c/p\u003e \u003cp\u003eIn this work, we theoretically investigate the possibility of using AFM neurons combined with a supervised learning algorithm to create neural networks that recognize symbols generated from a grid of input neurons [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e], [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. We show that, due to the strongly nonlinear and inertial dynamics of AFM neurons, even a single AFM neuron is capable of successfully recognizing various symbols from a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(5\\times 5\\)\u003c/span\u003e\u003c/span\u003e input grid, which is enough to encode various printed symbols. Our simulations show that the total training time of an AFM SPAN neuron can be below \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(1 {\\mu }\\text{s}\\)\u003c/span\u003e\u003c/span\u003e, while the power consumption during the training is of the order of 30 pJ. This research provides the first demonstration of the ability of AFM neurons to perform learning tasks, thus making clear the potential for using artificial AFM neurons in machine learning applications.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eAntiferromagnets (AFM) have two magnetic sublattices orientated in opposing directions. The direction of AFM magnetic sublattices relative to the crystal lattice can be manipulated using spin currents. Usually, this is achieved in spin Hall geometry, in which a layer of heavy metal covers an AFM element. When a DC electric current flows in the heavy metal layer, it induces a perpendicular spin current that penetrates into the AFM.\u003c/p\u003e \u003cp\u003eThe most interesting effect of spin current on the AFM dynamics happens when the spin polarization of the spin current is perpendicular to the easy plane of the AFM. In this case, spin-transfer torque induced by a sufficiently large spin current causes the AFM sublattices to rotate in the easy plane [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. For AFM materials with bi-axial anisotropy, the rotation of the sublattices is not uniform with time. This results in a sequence of short spin-pumping spikes at a frequency that can reach the THz range. The threshold current needed to achieve this auto-oscillating regime depends on the easy-plane anisotropy of the AFM material and is of the order of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({10}^{8} \\text{A}/{\\text{c}\\text{m}}^{2}\\)\u003c/span\u003e\u003c/span\u003e for NiO AFM [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIf the driving current is below the generation threshold, the AFM oscillator will not have enough energy to overcome the anisotropy, but the equilibrium orientation of the AFM sublattices will be moved towards the hard direction in the easy plane. With an additional impulse of current, the AFM magnetizations will surpass the anisotropy energy barrier and perform a single half-turn in the easy plane, which will cause a single spike of the spin-pumping voltage. This response of a sub-threshold AFM spin Hall oscillator is similar to the reaction of a biological neuron to an external stimulus [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. The AFM neurons and their networks also have other properties that resemble biological neural systems, such as refraction and delayed response [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAs it was shown in Ref. [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e], the dynamics of an AFM neuron can be described by the in-plane angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varphi\\)\u003c/span\u003e\u003c/span\u003e that the Neel vector of the AFM makes with the easy axis of the AFM. Under rather general assumptions, the angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varphi\\)\u003c/span\u003e\u003c/span\u003e obeys the second-order dynamical equation,\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\frac{1}{{\\omega }_{ex}}\\ddot{\\varphi }+\\alpha \\dot{\\varphi }+\\frac{{\\omega }_{e}}{2}\\text{sin}2\\varphi =\\sigma I$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\omega }_{ex}=2\\pi f}_{ex}\\)\u003c/span\u003e\u003c/span\u003e is the exchange frequency of the AFM, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003e is the effective Gilbert damping constant, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega }_{e}=2\\pi {f}_{e}\\)\u003c/span\u003e\u003c/span\u003e is the easy axis anisotropy frequency, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sigma\\)\u003c/span\u003e\u003c/span\u003e is the spin-torque efficiency defined by Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) in Ref. [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e], \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(I\\)\u003c/span\u003e\u003c/span\u003e is the driving electric current. Further details about the derivation of Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) can be found in [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e], [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. Note that the spin-pumping signal produced by the AFM is proportional to the angular velocity of the sublattice rotation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\dot{\\varphi }\\)\u003c/span\u003e\u003c/span\u003e. Namely, the inverse spin Hall voltage produced by the AFM neuron can be found as\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$V=\\beta \\dot{\\varphi }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere the efficiency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta =0.11x{10}^{-15} V\\bullet s\\)\u003c/span\u003e\u003c/span\u003e is defined by Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) in Ref. [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn this work, we study the dynamics of a network of interconnected AFM neurons. Each neuron is described by its own phase \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varphi }_{i}\\)\u003c/span\u003e\u003c/span\u003e and obeys an equation similar to Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) with additional terms describing synaptic connections between the neurons:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\frac{1}{{\\omega }_{ex}}{\\ddot{\\varphi }}_{i}+\\alpha {\\dot{\\varphi }}_{i}+\\frac{{\\omega }_{e}}{2}\\text{sin}2{\\varphi }_{i}=\\sigma I+\\sum _{i\\ne k}{\\kappa }_{ik}{\\dot{\\varphi }}_{k}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e.\u003c/p\u003e \u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k\\)\u003c/span\u003e\u003c/span\u003e are indices that represent the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003e-th and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k\\)\u003c/span\u003e\u003c/span\u003e-th neurons, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\kappa }_{ik}\\)\u003c/span\u003e\u003c/span\u003e represents a matrix of coupling coefficients. Note that the coupling signal produced by the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k\\)\u003c/span\u003e\u003c/span\u003e-th neuron is proportional to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\dot{\\varphi }}_{k}\\)\u003c/span\u003e\u003c/span\u003e, in agreement with Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe coupling coefficients that constitute \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\kappa }_{ik}\\)\u003c/span\u003e\u003c/span\u003e can behave as the synaptic weights in a machine learning system. To a large extent, the challenge of building a fast and efficient neuromorphic computing system depends on the efficient implementation of variable spintronic synapses capable of changing inter-neuron connectivity. The need for variable synapses dramatically increases the complexity of a neuromorphic neural network. This problem is even more serious for AFM neurons since, to fully employ ultra-fast AFM dynamics in neuromorphic hardware, the reaction times of artificial synapses should be on the timescale of AFM neuron dynamics. As AFM neurons spike with a duration that can be less than 5 ps, it should be noted that traditional CMOS technology would severely limit the capabilities of an AFM neural network. To our knowledge, no variable weight synapses have been developed that are suitable to work in conjunction with AFM neurons. As no CMOS or spintronic hardware is capable of being used as variable synapses for AFM neurons, circuit simulations such as SPICE simulations cannot be done. Therefore, in this paper, which primarily focuses on investigating the dynamics of AFM neurons, we did not assume any particular physical model of a synapse. Instead, the simulated synapses are considered to be \u0026ldquo;ideal\u0026rdquo; such that they can be adjusted instantaneously and to any value.\u003c/p\u003e \u003cp\u003eNevertheless, it is important to consider how the latency, or synaptic delay, would impact our model. In a previous work [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], copper bridges with fixed dimensions were used to provide constant weight synaptic coupling or fixed connections \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\kappa }_{ik}\\)\u003c/span\u003e\u003c/span\u003e between AFM neurons. Copper is capable of carrying spin current from one neuron to the next, allowing the output of one neuron to act as the input for a second neuron. The synaptic delay of copper bridges can be found by solving the diffusion equation for spin accumulation in copper. By using standard diffusion parameters for copper [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] and AFM neuron dimensions found in Ref. [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], the synaptic delay for a copper bridge with a length of 100 nm can be found to be about 1.5 ps. We consider this delay to be short enough to have a negligible impact on our system.\u003c/p\u003e \u003cp\u003eThere is a remarkable similarity between the equation describing the AFM neuron and that describing the dynamics of a physical pendulum; therefore, each term in Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) can be characterized by its mechanical analog. As a result, the coefficient of the first term on the left-hand side of Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) defines an effective mass, indicating that the AFM neuron possesses an effective inertia due to AFM exchange. This inertia results in a delay between a neuron receiving an input and the resulting output, an effect not found in conventional artificial neurons. When AFM neurons are linked together, such that the output of one neuron acts as the input of the next, the delay is dependent on the coupling strength \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\kappa }_{ik}\\)\u003c/span\u003e\u003c/span\u003e between the neurons. The delay caused by inertia decreases as the strength between neurons increases. Thus, the firing time of the neuron can be easily controlled. This means that the AFM neurons are well-suited for neuromorphic algorithms in which time encoding of neuron spikes is used.\u003c/p\u003e \u003cp\u003eOne such time-encoding approach, namely, spike pattern association neuron (SPAN) [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e], is studied in this paper. The architecture of an AFM neural network realizing the SPAN algorithm is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(a). It consists of one output \u0026ldquo;SPAN\u0026rdquo; neuron connected to many neurons of the input layer. In our simulations, the input layer consisted of 25 neurons and encoding input symbols drawn in a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(5\\times 5\\)\u003c/span\u003e\u003c/span\u003e binary grid. We used several shapes of the input symbols shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b). A blackened pixel in the input symbol causes a spike in the corresponding input neuron, while a white pixel will have no spike. The SPAN neuron is trained to output a spike at a certain prescribed time if the input symbol matches the pattern to be recognized. To achieve this, synaptic connections between the input layer and the SPAN neuron are adjusted during the training, as explained below.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWe used parallel encoding of the input layer; namely, the input symbol triggers the input neurons to fire simultaneously. If the combined weights connected to the SPAN are strong enough, there will be an output spike. The goal of training is to move this output spike to the desired time for a chosen symbol. If the spike is produced earlier (later) than the target time, the weights connected to the SPAN should reduce (increase).\u003c/p\u003e \u003cp\u003eIn more detail, the SPAN training algorithm is based on the Widrow-Hoff rule, where the difference between the desired spike time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{d}\\)\u003c/span\u003e\u003c/span\u003e and the actual spike time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{a}\\)\u003c/span\u003e\u003c/span\u003e is used to update the synaptic weights. After some manipulation, shown in Ref. [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e], the Widrow-Hoff rule is transformed to describe the change in weights during training:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${\\Delta }\\kappa = \\lambda {\\left(\\frac{e}{2}\\right)}^{2}\\left[\\left({t}_{d}-{t}_{i}+{\\tau }\\right){e}^{-\\left({t}_{d}-{t}_{i}\\right)/\\tau }-\\left({t}_{a}-{t}_{i}+{\\tau }\\right){e}^{-({t}_{a}-{t}_{i})/\\tau }\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda\\)\u003c/span\u003e\u003c/span\u003e is a positive and constant learning rate, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the timing of the input spike, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{d}\\)\u003c/span\u003e\u003c/span\u003e is the desired timing of the output spike, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{a}\\)\u003c/span\u003e\u003c/span\u003e is the actual timing of the output spike, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\tau\\)\u003c/span\u003e\u003c/span\u003e is a time constant corresponding to the width of a spike. Due to the simplicity of the SPAN algorithm, it is only capable of training a neuron to a single symbol. Upon training, a SPAN should output its spike at the target time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{d}\\)\u003c/span\u003e\u003c/span\u003e for the correct symbol and spike away from the target time for any other symbol.\u003c/p\u003e \u003cp\u003eA library of 20 symbols is used to train the neural network. These symbols are all variations of the correct symbol chosen from one of the symbols shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b). Variations include symbols with multiple additional or missing pixels. Initially initialized with random synaptic weights \\kappa_{ik}, the neural network receives each symbol as an input during one training epoch. A symbol is associated with a target time corresponding to the image's difference from the correct symbol. Using this time and the actual timing of the output neuron, the SPAN algorithm determines how the weights should be changed in accordance with Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e). The algorithm is modified to ensure that the weights cannot become negative, ensuring a more straightforward implementation in hardware. When all images have been processed, the weight changes resulting from each symbol are averaged, the neural network is updated, and the next epoch begins.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the output spikes of a SPAN neural network after training. When the correct symbol is supplied as input, the SPAN spikes within a 10 ps time window of the target time; this implies that the neural network has recognized chosen symbol. Any other symbol should cause a spike outside the target time window, indicating that a different symbol was used as input.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"Results and Discussion","content":"\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e(a) shows the error between the actual and desired spike time for the correct symbol throughout training, and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e(b) shows the corresponding changes in each synaptic weight. In this case, the neural network is being trained to the \"O\" symbol. After an aggressive start, the change in weights is subtle for most of training. Due to the large number of inputs, each individual weight is relatively small, as only the total sum is relevant to the timing of the output spike. It should be noted that some weights continue to change throughout the whole of training. These weights, in particular, do not contribute significantly to any symbol in the training library and, therefore, have limited data when making weight adjustments.\u003c/p\u003e\n\u003cp\u003eAfter about 10 epochs, the trained neural network will produce a spike within a 10ps widow of the target time when the symbol is recognized, as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eSeveral examples of incorrect symbols serving as input and the resulting output spikes are shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. By spiking outside the target time window for any symbol other than the correct symbol, the neural network has high accuracy in recognizing the chosen symbol. Whether additional or missing pixels serve as the difference from the correct image does not matter in making a spike outside of the target time window.\u003c/p\u003e\n\u003cp\u003eTo gain a more complete understanding of how weights change when training with the \"Z\" symbol, a distribution of weights is plotted in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e(a) shows the random distribution of weight at the beginning of training, and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e(c) shows the weights at the end of training, after 60 epochs. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e(b), in contrast, shows the training in the middle of training after 10 epochs. It is evident from this sequence that as training progresses, the weights are adjusted in such a way that the \u0026ldquo;Z\u0026rdquo; symbol is reflected in the weight distribution. As there is little difference between Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e(b) and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e(c), it can be assumed that the most significant training happens in the first 10 epochs.\u003c/p\u003e\n\u003cp\u003eMultiple AFM SPANs, trained to recognize different symbols, can be connected to the same layer of input neurons. This way, the SPAN trained to the input symbol will produce a spike within the target time window, while the others would spike outside it. With multiple SPANs all spiking at different times, the output can be unclear. Therefore, it would be convenient to clear the output by suppressing output spikes outside the target time window. This can be done by creating an additional output layer that consists of fixed synapses. The architecture of the neural network capable of suppressing unwanted outputs is shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThe input neurons are connected to three SPANs via trainable weights. These SPANs are each trained to recognize different symbols. The SPANs then serve as input to the output layer. The output layer\u0026rsquo;s synapses have weak coupling, such that a single pre-synaptic spike is insufficient to cause a spike in the post-synaptic neuron. Two spikes must happen simultaneously to produce a strong enough signal for a post-synaptic spike.\u003c/p\u003e\n\u003cp\u003eThe red neuron, shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, is a clock neuron spiking at the target time. The clock neuron receives an input independent of the input layer\u0026rsquo;s symbol. This independent input causes the clock neuron to generate a spike at the target time. Therefore, when a symbol is recognized, the SPAN will spike along with the clock neuron at the target time. These two signals are enough to overcome the weak coupling and cause the post-synaptic neuron to fire. The spikes from the SPANs that do not correspond to the input symbol would spike away from the target time, thus not combining with the clock neuron to cause an output spike. This output layer ensures that only the spike from the SPAN corresponding to the correct symbol is outputted. The output spiking signals of this neural network are shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThe blue, green, and magenta spikes correspond to three different SPANs trained to three different symbols, while the red spike is the clock neuron spiking at the target time. At this time, there are spikes of the clock neuron and a single SPAN corresponding to the inputted symbol. These two spikes combine to send a single spike to the output, indicating which symbol has been recognized. Therefore, this output layer creates an output that clearly identifies the recognized symbol.\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eUltra-fast spiking artificial neurons built from AFM oscillators have a number of unique properties that can be harnessed to create simple neural networks with fixed synapses [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHowever, in order to use AFM neurons in neuromorphic computing, a trainable neural network with variable synapses is required. The physical implementation of variable synapses is much more complex than fixed ones; therefore, in this work, we limit the number of trainable weights to reduce the complexity of the neural network. Namely, we numerically investigated the performance of the SPAN algorithm for the recognition of symbols encoded in a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(5\\times 5\\)\u003c/span\u003e\u003c/span\u003e binary grid.\u003c/p\u003e \u003cp\u003eThe simulated AFM neural networks are capable of recognizing symbols by producing a spike within a target time window (10 ps). The training time of the AFM networks for such relatively small images is very short, about 10 epochs with a 20-symbol library. Due to the high speed of AFM neurons (200 ps between inputting a symbol and the output neuron firing), this training session may last for only\u0026thinsp;~\u0026thinsp;40 ns of real-time.\u003c/p\u003e \u003cp\u003eMultiple SPANs, trained to different symbols, can be connected to the same inputs, thus providing multi-symbol recognition capabilities. With the addition of a fixed output layer suppressing spikes outside the target time window, the neural network will produce a single spike corresponding to the recognized symbol in just a few hundred picoseconds.\u003c/p\u003e \u003cp\u003eThe energy consumption of a single AFM neuron, with dimensions described in Ref. [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] was calculated to be about \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({10}^{-3}\\)\u003c/span\u003e\u003c/span\u003e pJ per synaptic operation. This is a relatively low power consumption in comparison with other artificial spiking neurons. For example, a spin torque nano-oscillator-based neuron reports an energy consumption of 4.96 pJ per synaptic operation [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e], and an all-CMOS neuron was reported to have a power consumption of 247 pJ per synaptic operation [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. The low power consumption of AFM neurons is the result of the simple design of the AFM oscillator and the ultra-fast spikes that result from using AFM materials. To successfully train a single SPAN, the total energy consumption of the AFM neural network is 31.2 pJ.\u003c/p\u003e \u003cp\u003eThe use of the SPAN algorithm leads to a very simple, one-layer neural network. This neural network is limited in its ability to perform more complex tasks as a result of such simplifications. For example, the MNIST data set of handwritten digits, commonly used for neural network training, is encoded in \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(28\\times 28\\)\u003c/span\u003e\u003c/span\u003e pixel grids. It is unlikely that a neural network such as the one studied here would be able to cope with inputs on this scale. The simple neural network used in this work demonstrates for the first time that AFM neurons are capable of being used for neuromorphic tasks such as pattern recognition. In order to advance the use of AFM neurons in neuromorphic computing, a more complex neural network and learning algorithm are required. However, even the simple network described in this work may find practical applications when high training, operation speed, and/or low consumed power are required.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\n\u003cp\u003eHB is responsible for numerical simulations, data analysis and writing the manuscript. VT contributed to formulating the program and provided overall supervision. All authors participated in reviewing and editing the manuscript.\u003c/p\u003e\n\u003ch2\u003eAcknowledgments\u003c/h2\u003e\n\u003cp\u003eThis work was partially 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.\u003c/p\u003e\n\u003ch2\u003eData Availability\u003c/h2\u003e\n\u003cp\u003eData may be made available at the request of the reader by contacting the authors at [email protected].\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eW. B. Levy and V. G. Calvert, \u0026ldquo;Communication consumes 35 times more energy than computation in the human cortex, but both costs are needed to predict synapse number,\u0026rdquo; \u003cem\u003eProc. Natl. Acad. Sci. U. S. 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Indiveri, \u0026ldquo;A current-mode conductance-based silicon neuron for address-event neuromorphic systems,\u0026rdquo; in \u003cem\u003e2009 IEEE International Symposium on Circuits and Systems\u003c/em\u003e, May 2009, pp. 2898\u0026ndash;2901. doi: 10.1109/ISCAS.2009.5118408.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-4365235/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4365235/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSpintronic devices offer a promising avenue for the development of nanoscale, energy-efficient artificial neurons for neuromorphic computing. It has previously been shown that with antiferromagnetic (AFM) oscillators, ultra-fast spiking artificial neurons can be made that mimic many unique features of biological neurons. In this work, we train an artificial neural network of AFM neurons to perform pattern recognition. A simple machine learning algorithm called spike pattern association neuron (SPAN), which relies on the temporal position of neuron spikes, is used during training. In under a microsecond of physical time, the AFM neural network is trained to recognize symbols composed from a grid by producing a spike within a specified time window. We further achieve multi-symbol recognition with the addition of an output layer to suppress undesirable spikes. Through the utilization of AFM neurons and the SPAN algorithm, we create a neural network capable of high-accuracy recognition with overall power consumption on the order of picojoules.\u003c/p\u003e","manuscriptTitle":"Pattern recognition using spiking antiferromagnetic neurons","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-05-13 16:13:45","doi":"10.21203/rs.3.rs-4365235/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-06-19T07:10:23+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-06-03T07:46:38+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-05-24T09:10:51+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"308193140172116493401079287039764602029","date":"2024-05-20T14:04:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"269839604569451923243270759496266833710","date":"2024-05-16T09:03:00+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-05-16T06:24:18+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-05-10T16:52:51+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-05-06T17:07:57+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-05-06T16:33:30+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-05-03T16:35:22+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"cb07213c-b233-426e-8457-f9cc266d0099","owner":[],"postedDate":"May 13th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":31787195,"name":"Physical sciences/Physics/Applied physics"},{"id":31787196,"name":"Physical sciences/Materials science/Theory and computation"}],"tags":[],"updatedAt":"2024-08-05T13:15:32+00:00","versionOfRecord":[],"versionCreatedAt":"2024-05-13 16:13:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4365235","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4365235","identity":"rs-4365235","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}