## 4 Discussion and Examples
Hence, using a fractional-order generalized model of type (5), it is possible to study the dependence of the output variables (product image) on the input variables (the configuration parameters and element shapes) using the fractional-order differences between their current and previous values. Moreover, in all extended models, impulsive controllers are proposed at some instances, at which impulsive control is performed to avoid the unwanted behavior of the states. The fixed moments \(\eta_{I}\), \(l\in\mathbb{N}_{1}\) are the intervention times, and the functions \(V_{hl}\) determine the control actions. In addition, instead of considering learning mechanisms, as in [5; 6; 7; 8; 9; 10; 11], this paper investigates the specific stability behavior of the neuronal states and formulates rigorous results that can be applied to the concrete model. In fact, learning algorithms allow optimal solutions to be reached in different design tasks. For example, in [6], the goal of the created learning rule is the best-fitting shape. However, the issue of the stability of a trained neural network model is also very important [48; 49], but the investigations in this area are not sufficient. In this paper, the concept of the global practical Mittag-Leffler stability is introduced to the impulsive discrete fractional-order neural network model, which has not been considered for such systems. The introduced concept extends the Mittag-Leffler stability notion investigated in [28; 30]. The practical stability extensions [53] of classical stability concepts are more appropriate for applied problems [54; 55; 56; 57; 58] when the corresponding system is not mathematically Mittag-Leffler stable, but its behavior is acceptable. The comparison demonstrated in Table 1 displays the novelty offered in this paper. Table 1 also illustrates that the study of the fractional-order neural network modeling framework in engineering design is far from complete and needs further development. In Theorems 1 and 2, global practical Mittag-Leffler stability results are established using the fractional Lyapunov method. The obtained novel stability and impulsive control criteria provide new perspectives and tools that could be extremely useful, as they provide a framework enabling control to better design fractional-order neural network models that are resilient in the face of impulsive shocks. From the applied point of view, if the restrictions of the model's parameters satisfy conditions \(\mathcal{H}_{1}\), (9) and (18), and the impulsive controllers are such that condition \(\mathcal{H}_{2}\) is true, then the constructed fractional-order neural network model is globally practically Mittag-Leffler stable. The conditions of Theorem 2 are less restrictive. Having different stability criteria is very useful in applications, which will be demonstrated through examples in the next section. **Example 1**. : _Consider the impulsive discrete fractional-order neural network model (5) for \(n=3\), \(\alpha=0.96\), \(a_{h}=3.5\), \(h=1,2,3\), \(f_{j}(u_{j})=\frac{1}{2}(|u_{j}+1|-|u_{j}-1|)\), \(u_{j}\in\mathbb{R}\), \(j=1,2,3\), with the connection weights given as_  
\[W=\left[ \begin{array}{ccc}w_{11}&w_{12}&w_{13}\\ w_{21}&w_{22}&w_{23}\\ w_{31}&w_{32}&w_{33}\end{array} \right]=\left[ \begin{array}{ccc}1.2&-1.4&1.4 \\ -0.8&0.6&0.6\\ 0.3&0.3&-0.2\end{array} \right].\]  
_and the impulsive controllers defined as_  
\[V_{hl}(\mathcal{U}_{h}(\eta_{l}-1))=\left(1-\frac{1}{5h}\right)\mathcal{U}_{h }(\eta_{l}-1), \tag{27}\]  
_at the the impulsive instances \(\eta_{l}\) that satisfy (4)._  
_If_  
\[|\mathcal{U}_{h}(\eta)|\leq\gamma_{h},\;\gamma_{h}>0,\eta\in\mathbb{N}_{0},h=1,2,3\]  
_and_  
\[\sum_{h=1}^{n}\gamma_{h}||\Theta_{h}|<\sigma,\;\sigma>0,\;\eta\in\mathbb{N}_{0},\]  
_then, for the choice of the model's parameters, hypotheses \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) hold for \(\mathcal{F}_{j}=1,j=1,2,3\) and \(\rho_{hl}=\frac{1}{5h}\), \(h=1,2,3,l\in\mathbb{N}_{1}\).