In this article, the authors investigate the global and exponential dissipativity of quaternion-valued inertial neural networks (QVINNs) with mixed time-varying delays, without utilizing order reduction of inertial neural networks (INNs) and quaternion separation methods. Using innovative Lyapunov functional and inequality techniques, several fruitful sufficient criteria with multi-parameters are derived for QVINNs to ensure global dissipativity (GD), which generalizes and refines existing results. This article estimates the attractive sets and exponentially attractive sets globally. Unlike previous studies in which quaternion-valued neural networks (QVNNs) are separated into real-valued neural networks (RVNNs) and INNs are reduced into first-order systems, the foundation of this article rests upon approaches that diverge from the traditional methods of separation and order reduction. Unlike existing results on the GD of traditional neural networks (NNs) with bounded discrete time delays, this article focuses on INNs with unbounded discrete time-varying delays, which is more realistic because neurons consider their entire past rather than partial history within bounded time delays. In general, in exponential stability, synchronization, and dissipativity results, researchers typically impose an upper bound on the rate of convergence (Formula presented.), but in the present article, the authors investigate dissipativity criteria without such a restriction on the convergence rate in global exponential dissipativity (GED). Finally, to demonstrate the efficiency of our theoretical work, three consecutive examples are proposed to illustrate the effectiveness of the obtained results. The first two examples verify the proposed results, and the third one, related to QVNNs, redemonstrates the efficiency of storing true color image patterns.

This work emphasizes the fixed-time synchronization (FTS) of a specific class of octonion-valued neural networks (OVNNs), which incorporate discrete and distributed time delays. This study also establishes several norm properties for the octonion domains and explores FTS and fixed-time projective synchronization (FTPS) in OVNNs having mixed time delays by a suitable choice of the Lyapunov function, controllers and the one norm property. Unlike previous research on the decomposition of neural networks with octonion-valued and quaternion-valued components, this study introduces an enhanced one-norm method based on the non-separation approach. It employs a direct analytical approach to address two synchronization challenges using several norm properties. The computational complexity is reduced to provide less conservative results for OVNNs. This article analyzes various properties of the one norm of octonion domains and introduces effective controllers for achieving FTS and FTPS between the drive and response systems of OVNNs. The present article also establishes results in a compact and more generalized form using one norm criteria, which are easily verifiable to ensure synchronization, even with mixed time delays, which can be achieved within fixed time intervals. The settling time in each case illustrates its effectiveness compared to the existing results in a more straightforward way through a unique analytical process and more versatile activation functions. Finally, the theoretical results are validated through two numerical examples, with the overall results presented and discussed. Additionally, OVNNs are employed to demonstrate their effectiveness in storing and retrieving true-color images. This application showcases the ability of OVNNs to handle high-dimensional data representations, particularly in contexts where color channels and spatial features are strongly interrelated. The results highlight the robustness and efficiency of the proposed OVNNs framework, confirming its potential for advanced multidimensional data processing tasks.