## 2 Problem Formulation  
In this section, we first consider a class of discrete measurement-saturated MNNs with mixed time delays in the following [20]:  
\[ \begin{cases}\tilde{\delta}_{u+1}=\tilde{A}(\tilde{\delta}_{u})\tilde{\delta }_{u}+\tilde{B}(\tilde{\delta}_{u})f(\tilde{\delta}_{u})+\tilde{D}(\tilde{ \delta}_{u})g(\tilde{\delta}_{u-\tau_{1}})\\ \qquad\qquad+\tilde{E}(\tilde{\delta}_{u})\sum_{i=1}^{\tau_{2}}\mu_{i}\tilde{ \delta}_{u-i}+\tilde{H}_{w}\tilde{\tilde{W}}_{u},\\ \tilde{q}_{u}=\Gamma(\tilde{\mathbb{C}}\tilde{\delta}_{u})+\tilde{H}_{v}\tilde {\
abla}_{u},\\ \tilde{\delta}_{u}=\tilde{\Psi}_{u},\ u\in[-\tau,0],\ \tau=\max\{\tau_{1},\tau_{2}\},\end{cases}  \tag{1}\]  
where \(\tilde{\delta}_{u}\)=\([\tilde{\delta}_{1,u},\ \tilde{\delta}_{2,u},\cdots,\tilde{\delta}_{n_{x},u}]^{ \mathrm{T}}\in\mathbb{R}^{n_{x}}\) is the state vector of \(n_{x}\) neurons with initial value \(\tilde{\Psi}_{u}\). \(\tilde{q}_{i}\)= \([\tilde{q}_{1,u},\ \tilde{q}_{2,u},\cdots,\tilde{q}_{n_{y},u}]^{\mathrm{T}}\in \mathbb{R}^{n_{y}}\) is the measurement output of the MNNs. \(\tilde{\
abla}_{u}\in\mathbb{R}^{n_{w}}\) and \(\tilde{\tilde{W}}_{u}\in\mathbb{R}^{n_{w}}\) represent the measurement noise and the process noise, respectively. \(f(\cdot)\), \(g(\cdot)\) and \(\Gamma(\cdot)\) denote the nonlinear functions. \(\tilde{A}(\tilde{\delta}_{u})=\mathrm{diag}_{n_{x}}\{a_{i}(\tilde{\delta}_{i,u})\}\) is the self-feedback matrix with suitable dimension. \(\tilde{B}(\tilde{\delta}_{u})=\left(b_{ij}(\tilde{\delta}_{i,u})\right)_{n_{ x}\times n_{x}}\) is the connection weight matrix with suitable dimension. \(\tilde{D}(\tilde{\delta}_{u})=\left(d_{ij}(\tilde{\delta}_{i,u})\right)_{n_{ x}\times n_{x}}\) and \(\tilde{E}(\tilde{\delta}_{u})=\left(h_{ij}(\tilde{\delta}_{i,u})\right)_{n_{ x}\times n_{x}}\) are the delayed connection weight matrices with suitable dimensions. \(\tilde{\mathbb{C}}\), \(\tilde{H}_{w}\) and \(\tilde{H}_{v}\) are the known real-valued matrices with proper dimensions. \(\tau_{1}\) represents the determined constant delay. \(\tau_{2}\) and \(\mu_{i}\) (\(i=1,\,2,\cdots,\tau_{2}\)) represent the distributed delay, which is caused by the delay accumulation influences of real circuit components and exists in MNNs widely. **Assumption 1**. : _The process noise \(\tilde{\Psi}_{u}\in R^{n_{w}}\) in the first formula in (1) and the measurement noise \(\tilde{\
abla}_{u}\in R^{n_{w}}\) in the second formula of (1) are UBB noises, which are restricted to the ellipsoidal sets specified below:_  
\[ \begin{cases}z_{w_{u}}=\{\tilde{\Psi}_{u}\in R^{n_{w}}:\,\tilde{\Psi}_{u}^{ \mathrm{T}}R_{w,u}^{-1}\tilde{\Psi}_{u}\leq 1\},\\ z_{v_{u}}=\{\tilde{\
abla}_{u}\in R^{n_{v}}:\tilde{\Psi}_{u}^{\mathrm{T}}R_{v,u }^{-1}\tilde{\
abla}_{u}\leq 1\},\end{cases}  \tag{2}\]  
_where \(R_{w,u}>0\) and \(R_{v,u}>0\) are known time-varying matrices with proper dimensions._  
The saturation function \(\Gamma(\cdot):\mathbb{R}^{n_{y}}\rightarrow\mathbb{R}^{n_{y}}\) is defined in the following:  
\[\Gamma(\tilde{\sigma})=\Big{(}\Gamma_{1}(\tilde{\sigma}_{1}),\ \Gamma_{2}(\tilde{\sigma}_{2}),\ \ \cdots,\ \Gamma_{n_{y}}(\tilde{\sigma}_{n_{y}})\Big{)}^{ \mathrm{T}}, \tag{3}\]  
with \(\Gamma_{t}(\tilde{\sigma}_{t})=\mathrm{sign}(\tilde{\sigma}_{t})\min\{\tilde{ \sigma}_{t,\max},|\tilde{\sigma}_{t}|\}\), where \(\tilde{\sigma}_{t,\max}\) denotes the t-th element of saturation level \(\tilde{\sigma}_{\max}\) and \(\mathrm{sign}\ (\ \cdot\ )\) represents the signum function. **Definition 1** ([61]).