## 1 Introduction
It is also well known that the use of new initial conditions in a trained NN model may lead to instability and \"instability is the Achilles' heel of modern AI and a paradox, with training algorithms finding unstable NNs despite the existence of stable ones\" [48]. Hence, stability is one of the most important issues if an ANN model is to be used in practical applications [49]. This has motivated numerous studies on the development of stable neural control strategies for different types of ANN models [38; 40; 41; 42; 44; 50; 51; 52], including fractional-order neural networks of a continuous and discrete nature [23; 24; 27; 28; 30; 43; 46; 47]. For discrete fractional-order neural network models, considerable investigations have been devoted to investigating the Mittag-Leffler stability properties [28; 30], which generalize the exponential stability ones. It is well known that, in many applications, mathematically ideal, stable behavior is impossible. In such cases, the concept of practical stability, defined in [53], is much more appropriate. In terms of practical stability, the behavior of a neuronal state is still acceptable but not necessarily mathematically stable when other states that are initially close to it oscillate close to it. Recently, theoretical studies of practical stability have become very mature and it is increasingly applied in engineering [54; 55; 56; 57; 58]. The practical stability notion has been also studied for linear fractional discrete-time systems [59; 60; 61]. Practical stability results for fractional nonlinear discrete NN models are reported only in the paper [31], without considering impulsive effects. Accordingly, there are no practical stability results on impulsive fractional discrete impulsive ANN models so far, which leads to the lack of a theoretical basis or performance analysis of such systems. Prompted by the above analysis, the purpose of this work is to design a discrete fractional-order neural network model that extends some existing models used in product form design. Appropriate impulsive controllers and practical stability concepts are also considered. The summary of the proposed contribution is as follows. 1. A new fractional nonlinear neural network modeling approach is applied to extend and generalize some existing ANN models applied in product form design analysis [5; 6; 7; 8; 9; 10; 11]. The proposed bio-inspired neural network approach is based on the use of discrete fractional calculus [62; 63]. Such ANN models have higher degrees of accuracy and freedom than existing integer-order models and can help engineering designers and researchers to better understand the behavior of ANN models, overcoming major difficulties in experimental modeling. 2. Compared with the discrete fractional-order ANN model introduced in [31], appropriate impulsive controllers are considered to control the qualitative behavior of the proposed model. The extended impulsive version will allow the implementation of impulsive control methods in the design process of the product form in cases of external and internal disturbing noise. Although impulsive control theory has gained great popularity in several scientific areas, such as biology, population dynamics, computer sciences and some others, it has not been applied to neural network models in product form design, and the goal of this research is to fill this gap. 3. Since mathematically ideal, stable behavior is impossible for many applied problems, including engineering design problems, the notion of practical stability, namely global practical Mittag-Leffler practical stability, is adopted for the introduced impulsive control fractional neural network model. Thus, the proposed research not only contributes to the development of new fractional-order neural network models in product form design, but also proposes the use of the formulated framework to gain a fundamental scientific understanding of the dynamical properties of such models. 4.