## 2 The Proposed Model and Strategies
### Mittag-Leffler Practical Stability Concept  
Next, the Mittag-Leffler stability concept studied in [28,30] will be extended to the practical Mittag-Leffler stability case. The practical stability notions considered in [53, 54, 55, 56, 57, 58] will be adopted in system (5) via the next definition. **Definition 3**. : _The impulsive fractional discrete neural network model (5) is said to be globally practically Mittag-Leffler stable if, for any \(\mathcal{U}^{0}\in\mathbb{R}^{n}\), there exist constants \(\mathcal{L},\beta>\) and \(A\geq 0\) such that_  
\[|\mathcal{U}(\eta)||<\left[\mathcal{M}(\mathcal{U}^{0})\mathbb{E}_{\overline {\alpha}}(-\mathcal{L},\eta)\right]^{\beta}+A,\ \eta\in\mathbb{N}_{0},\]  
_where \(0<\alpha<1\), \(\mathcal{M}(0)=0\), \(\mathcal{M}(u)\geq 0\), \(\mathcal{M}(u)\) is locally Lipschitz for \(u\in\mathbb{R}^{n}\)._  
**Remark 5**. : _Definition 3 extends the Mittag-Leffler stability definition for the discrete fractional-order neural networks given in [28,30] to the practical stability case. When \(A=0\), Definition 3 is reduced to the global Mittag-Leffler stability of the zero solution of(5).The practical stability notion is a successful alternative when mathematical stability is impossible, insignificant or pointless in a set time. For practical stability, the corresponding state may not be mathematically stable, but its behavior may be acceptable from an applied point of view. Thus, practical stability concepts are more often studied for models used in applied sciences [54, 55, 56, 57, 58, 59, 60]._  
**Remark 6**. : _After appropriate substitutions, Definition 3 can be reformulated for the global practical Mittag-Leffler stability properties of non-zero states._  
**Remark 7**. : _Stability properties are directly related to synchronization issues. If the impulse-free system (3) is considered as a drive system and the impulsive control system (5) as a response system, then the synchronization error will be \(\mathcal{Y}(\eta)=\mathcal{U}(\eta)-\mathcal{Z}(\eta)\). In this case, the global practical Mittag-Leffler stability for the synchronization error system is_  
\[\left\{ \begin{array}{l}{}^{\mathrm{C}}\
abla^{\alpha}_{0}\mathcal{Y}_{h}( \eta)=-a_{h}\mathcal{Y}_{h}(\eta)+\sum_{j=1}^{n}w_{hj}f_{j}(\mathcal{Y}_{j}( \eta)),\ \eta\in\mathbb{N}_{0},\ \ \eta\
eq\eta_{l},\\ \\ \mathcal{Y}_{h}(\eta_{l})=V_{ht}(\mathcal{Y}_{h}(\eta_{l}-1)),\ l\in\mathbb{N}_ {1},\\ \\ \mathcal{Y}(0)=\mathcal{Y}^{0},\end{array} \right. \tag{6}\]  
_where \(f_{j}(\mathcal{V}_{j}(\eta))=f_{j}(\mathcal{U}_{j}(\eta))-f_{j}(\mathcal{Z}_{ j}(\eta))\) and \(\mathcal{Y}(0)=\mathcal{U}(0)-\mathcal{Z}(0)\) will lead to global practical Mittag-Leffler synchronization between the drive and the response systems [28]. Thus, the impulsive controllers will be used to globally practically Mittag-Leffler synchronize the states of the impulsive system (5) with those of system (3). Note that the impulsive synchronization mechanism has been applied to continuous fractional-order neural network models [41,69, 70, 71], but this control strategy has not been studied for the discrete case._  
**Remark 8**. : _One of the main goals in using the neural network approach in the design of industrial products is to achieve an optimal design solution by applying appropriate learning rules. However, the stable behavior of the optimal solution (equilibrium) of the neural network model is also essential for applications. In fact, the correct operation of a neural network model (even a trained one) depends essentially on its stability behavior [48]. Is the equilibrium stable or can small perturbations in the initial data lead to its instability and poor performance? For integer-order neural network models, the globally asymptotic stable equilibrium behavior is the most important stability behavior to guarantee convergence to a global optimal solution independently of the initial data, i.e., independently of spurious suboptimal responses [42]. The particular case of global exponential stability guarantees a fast convergence rate.