# A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks

## TL;DR

This paper introduces real-part associative hypercomplex number systems and $$-projection activation functions to analyze and ensure the stability of a broad class of discrete-time hypercomplex-valued Hopfield neural networks, generalizing existing models.

## Contribution

It proposes novel hypercomplex number systems and activation functions, extending stability analysis to a wide class of hypercomplex-valued neural networks including Cayley-Dickson algebras.

## Key findings

- Confirmed stability of several existing hypercomplex neural networks
- Introduced a general stability framework for Cayley-Dickson algebra-based networks
- Extended stability analysis to new hypercomplex systems

## Abstract

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 real-part associative hypercomplex number systems. Real-part associative 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 $\mathcal{B}$-projection functions. Broadly speaking, a $\mathcal{B}$-projection 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.