A growing issue within conservation bioacoustics is the task of analysing the vast amount of data generated from the use of passive acoustic monitoring devices. In this paper, we present an alternative AI model which has the potential to help alleviate this problem. Our model formulation addresses the key issues encountered when using current AI models for bioacoustic analysis, namely the: limited training data available; environmental impact, particularly in energy consumption and carbon footprint of training and implementing these models; and associated hardware requirements. The model developed in this work uses associative memory via a transparent, explainable Hopfield neural network to store signals and detect similar signals which can then be used to classify species. Training is rapid ($3$\,ms), as only one representative signal is required for each target sound within a dataset. The model is fast, taking only $5.4$\,s to pre-process and classify all $10384$ publicly available bat recordings, on a standard Apple MacBook Air. The model is also lightweight with a small memory footprint of $144.09$\,MB of RAM usage. Hence, the low computational demands make the model ideal for use on a variety of standard personal devices with potential for deployment in the field via edge-processing devices. It is also competitively accurate, with up to $86\%$ precision on the dataset used to evaluate the model. In fact, we could not find a single case of disagreement between model and manual identification via expert field guides. Although a dataset of bat echolocation calls was chosen to demo this first-of-its-kind AI model, trained on only two representative calls, the model is not species specific. In conclusion, we propose an equitable AI model that has the potential to be a game changer for fast, lightweight, sustainable, transparent, explainable and accurate bioacoustic analysis.

This study presents a hybrid model for classifying handwritten digits in the MNIST dataset, combining convolutional neural networks (CNNs) with a multi-well Hopfield network. The approach employs a CNN to extract high-dimensional features from input images, which are then clustered into class-specific prototypes using k-means clustering. These prototypes serve as attractors in a multi-well energy landscape, where a Hopfield network performs classification by minimizing an energy function that balances feature similarity and class assignment.The model's design enables robust handling of intraclass variability, such as diverse handwriting styles, while providing an interpretable framework through its energy-based decision process. Through systematic optimization of the CNN architecture and the number of wells, the model achieves a high test accuracy of 99.2% on 10,000 MNIST images, demonstrating its effectiveness for image classification tasks. The findings highlight the critical role of deep feature extraction and sufficient prototype coverage in achieving high performance, with potential for broader applications in pattern recognition.

--This research paper introduces two novel complex-valued Hopfield neural networks (CvHNNs) that incorporate phase and magnitude quantization. The first CvHNN employs a ceiling-type activation function that operates on the rectangular coordinate representation of the complex net contribution. The second CvHNN similarly incorporates phase and magnitude quantization but utilizes a ceiling-type activation function based on the polar coordinate representation of the complex net contribution. The proposed CvHNNs, with their phase and magnitude quantization, significantly increase the number of states compared to existing models in the literature, thereby expanding the range of potential applications for CvHNNs. Real-valued neural networks are primarily based on the McCulloch-Pitts model of neurons [1], [2].

Brain-inspired computing aims to mimic cognitive functions like associative memory, the ability to recall complete patterns from partial cues. Memristor technology offers promising hardware for such neuromorphic systems due to its potential for efficient in-memory analog computing. Hopfield Neural Networks (HNNs) are a classic model for associative memory, but implementations on conventional hardware suffer from efficiency bottlenecks, while prior memristor-based HNNs faced challenges with vulnerability to hardware defects due to offline training, limited storage capacity, and difficulty processing analog patterns. Here we introduce and experimentally demonstrate on integrated memristor hardware a new hardware-adaptive learning algorithm for associative memories that significantly improves defect tolerance and capacity, and naturally extends to scalable multilayer architectures capable of handling both binary and continuous patterns. Our approach achieves 3x effective capacity under 50% device faults compared to state-of-the-art methods. Furthermore, its extension to multilayer architectures enables superlinear capacity scaling (\(\propto N^{1.49}\ for binary patterns) and effective recalling of continuous patterns (\propto N^{1.74}\ scaling), as compared to linear capacity scaling for previous HNNs. It also provides flexibility to adjust capacity by tuning hidden neurons for the same-sized patterns. By leveraging the massive parallelism of the hardware enabled by synchronous updates, it reduces energy by 8.8x and latency by 99.7% for 64-dimensional patterns over asynchronous schemes, with greater improvements at scale. This promises the development of more reliable memristor-based associative memory systems and enables new applications research due to the significantly improved capacity, efficiency, and flexibility.

In this paper, we explore the dynamics of structured complex-valued Hopfield neural networks (CvHNNs), which arise when the synaptic weight matrix possesses specific structural properties. We begin by analyzing CvHNNs with a Hermitian synaptic weight matrix and establish the existence of four-cycle dynamics in CvHNNs with skew-Hermitian weight matrices operating synchronously. Furthermore, we introduce two new classes of complex-valued matrices: braided Hermitian and braided skew-Hermitian matrices. We demonstrate that CvHNNs utilizing these matrix types exhibit cycles of length eight when operating in full parallel update mode. Finally, we conduct extensive computational experiments on synchronous CvHNNs, exploring other synaptic weight matrix structures. This work was supported in part by the National Council for Scientific and Technological Development (CNPq) under grant no 315820/2021-7, the S ao Paulo Research Foundation (FAPESP) under grant no 2023/03368-0, and the Postdoctoral Researcher Program (PPPD) at the Universidade Estadual de Campinas (UNICAMP). Keywords-- Hopfield neural network, complex-valued neural network, associative memory, braided Hermitian matrix. 1 Introduction Artificial neural networks have been conceived as emulators of the biological neural network synapse process. Their processing units, the artificial neurons, usually act based on input signals received from other neurons or cells. Like a biological neuron firing an electric impulse in the presence of specific chemical components in appropriate concentrations, an artificial neuron fires when certain mathematical conditions are satisfied.

Recurrent correlation neural networks (RCNNs), introduced by Chiueh and Goodman as an improved version of the bipolar correlation-based Hopfield neural network, can be used to implement high-capacity associative memories. In this paper, we extend the bipolar RCNNs for processing hypercomplex-valued data. Precisely, we present the mathematical background for a broad class of hypercomplex-valued RCNNs. Then, we provide the necessary conditions which ensure that a hypercomplex-valued RCNN always settles at an equilibrium using either synchronous or asynchronous update modes. Examples with bipolar, complex, hyperbolic, quaternion, and octonion-valued RCNNs are given to illustrate the theoretical results. Finally, computational experiments confirm the potential application of hypercomplex-valued RCNNs as associative memories designed for the storage and recall of gray-scale images.

Hypercomplex-valued neural networks, including quaternion-valued neural networks, can treat multidimensional data as a single entity. In this paper, we present the quaternion-valued recurrent projection neural networks (QRPNNs). Briefly, QRPNNs are obtained by combining the non-local projection learning with the quaternion-valued recurrent correlation neural network (QRCNNs). We show that QRPNNs overcome the crosstalk problem of QRCNNs. Thus, they are appropriate to implement associative memories. Furthermore, computational experiments reveal that QRPNNs exhibit greater storage capacity and noise tolerance than their corresponding QRCNNs. Introduction The Hopfield neural network, developed in the early 1980s, is an important and widely-known recurrent neural network which can be used to implement associative memories [1, 2]. Successful applications of the Hopfield network include control [3, 4], computer vision and image processing [5, 6], classification [7, 8], and optimization [2, 9, 10]. Despite its many successful applications, the Hopfield network may suffer from a very low storage capacity when used to implement associative memories. Precisely, due to crosstalk between the stored items, the Hebbian learning adopted by Hopfield in his original work allows for the storage of approximately n/(2 ln n) items, where n denotes the length of the stored vectors [11]. For example, Personnaz et al. [12] as well as Kanter and Sompolinsky [13] proposed the projection rule to determine the synaptic weights of the Hopfield networks. The projection rule increases the storage capacity of the Hopfield network to n 1 items. Another simple but effective improvement on the storage capacity of the original Hopfield networks was achieved by Chiueh and Goodman's recurrent correlation neural networks (RCNNs) [14, 15]. Briefly, an RCNN is obtained by decomposing the Hopfield network with Hebbian learning into a two layer recurrent neural network.

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In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as Hopfield-type hypercomplex number systems. Hopfield-type hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called Hopfield-type activation functions. Broadly speaking, a Hopfield-type activation function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras.