# Identification of saturated networked systems

Adel Malik Annabi, Jean-Baptiste Pomet, Dario Prandi, Ludovic Sacchelli

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Adel Malik Annabi, Jean-Baptiste Pomet, Dario Prandi, Ludovic Sacchelli. Identification of saturated networked systems. 64th IEEE Conf. on Decision and Control (CDC), Dec 2025, Rio de Janeiro, Brazil. hal-05236385

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# Identification of saturated networked systems

A.M. Annabi, J.-B. Pomet, D. Prandi, L. Sacchelli

**Abstract**— We address the challenge of parameter identification in networks of neural mass models. We consider a system of $n$ interconnected populations, structured in successive layers, where measurements are available from only one. Traditional offline methods struggle with parameter ambiguity and time-dependent variability. We propose an online approach that exploits the system's nonlinear characteristics, particularly saturation functions, to enable effective parameter recovery. By designing specific control inputs applied to targeted nodes, it becomes possible to isolate and retrieve parameters effectively. Crucially, we show that this process must be conducted online to achieve full identification, as control inputs must adapt to previously unknown states and parameters.

## I. INTRODUCTION

The application of control theory techniques to neuroscience presents a rich and challenging problem. Its diverse applications range from brain health monitoring [17] to brain-computer interfacing [18]. For example, feedback control has been employed to stabilize brain signals and mitigate Parkinson's disease symptoms [12], [9].

These methods and techniques are subject to the choice of a relevant model. We focus here on neural mass models, particularly those based on the Wilson-Cowan framework [20]. They serve as powerful tools for studying the macroscopic dynamics of neuronal populations [11], [20]. See [6] for review on large scale models of the brain.

All dynamical models depend on a certain number of parameters whose values influence the behavior and the response to controls. It is therefore critical to know, exactly or approximately, the value of these parameters. Some of them may be known from physical considerations, but most of the parameters have to be estimated from indirect measurement via *parameter identification*. By identification we mean any method or process that uses measurements of various outputs of the system corresponding to some inputs (or excitation) to deduce some estimate of the parameters. The choice of the excitation, or experiment design, may be an important part of the process.

Parameter identification in neural mass models is a challenging problem due to the non-linear nature of the system. Offline methods struggle with issues such as parameter ambiguity, where different parameter sets can produce equally good representations of the data [4]. Moreover, this approach fail to account for time-dependent variations in physiological

parameters [18]. This underscores the necessity of developing tools that leverage online approaches.

Since we are dealing with a case where both the state is not fully measured and parameters are unknown, it is natural to resort to adaptive observers, see e.g. [13]. This is done for instance in [7], [8], even for a distributed state in the first case. This was done under various sorts of contraction and persistent excitation assumptions, that are not easy to check in general, and are very weakly satisfied in realistic neural systems where saturation effects induce non-monotonic behaviors, leading to disconnection between variables.

We want propose an identification method that includes experiment design, i.e. proposes explicit control that facilitates identification; see [3] for a discussion of this problem and [16], [19] for experiment design for systems with interconnections in a different context from ours.

In this work, we study a network of $n$ interconnected neural populations, structured in successive layers, governed by Wilson-Cowan-like equations, where measurements are available from only one population. We propose an identification approach that exploits the non-linear characteristics of the system, particularly the role of saturation functions that are common in these models [6], [20], to establish a one-to-one correspondence between the output of the system and its parameters. By designing specific control inputs applied to selected nodes, we ensure that all parameters can be retrieved effectively. Crucially, we show that the parameter estimation must be done online, as it is inherently recursive. Indeed, control inputs must account for previously unknown states or parameters to achieve full system identification. This highlights the necessity of online estimation techniques capable of adapting to unknown and time-varying dynamics. Our approach addresses key limitations of offline methods and introduces new possibilities for real-time neural modulation and feedback control.

### Notations

1. For any two integers $p < q$, we denote by $[[p, q]]$ the set of integers $[[p, q]] = \{i \in \mathbb{Z}, p \le i \le q\}$.

2. Given an element $V$ in $\mathbb{R}^{n_1} \times \dots \times \mathbb{R}^{n_p} = \mathbb{R}^{n_1+\dots+n_p}$ and $q$ in $[[1, p-1]]$, we let $V^q = (V_1, \dots, V_q)$.

3. Given a function $g: \mathbb{R} \to \mathbb{R}$ and a vector $z \in \mathbb{R}^n$, we denote by $g(z) \in \mathbb{R}^n$ the vector obtained by applying $g$ element-wise, i.e., $g(z) = (g(z_1), \dots, g(z_n))$.

## II. MODEL DISCUSSION

The $i^{th}$ population, $1 \le i \le p$, is described by the state variable $V_i \in \mathbb{R}^{n_i}$; the state of the network is $V =$

A.M. Annabi, J.-B. Pomet, L. Sacchelli are with Université Côte d'Azur, Inria, CNRS, LJAD, France (email: adel-malik.annabi@inria.fr, jean-baptiste.pomet@inria.fr, ludovic.sacchelli@inria.fr)

D. Prandi is with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif-sur-Yvette, France (email: dario.prandi@centralesupelec.fr)

This work has been partially supported by the ANR-20-CE48-0003.
$(V_1, \dots, V_p) \in \mathbb{R}^N = \mathbb{R}^{n_1} \times \dots \times \mathbb{R}^{n_p}$ with $n_1 \ge \dots \ge n_p$, $N_k = n_1 + \dots + n_k$, $N = N_p$.

The dynamics are given by

$$ \begin{cases} \dot{V} = f(V, u, \Theta), \\ y = V_1, \end{cases} \qquad (1) $$

where the output $y = V_1$ indicates that we entirely measure the first population, and this is the only measurement. The right-hand side $f = (f_1, \dots, f_p)$ has more structure: each $f_i$, $1 \le i \le p$ is given by

$$ f_i(V, u, \Theta) = -V_i + \sum_{j=1}^{\min\{p,i+1\}} W_{ij}S(V_j) + u_i. \quad (2) $$

**Assumption 1:** The function $S: \mathbb{R} \to \mathbb{R}$ is a known Lipschitz-continuous, piecewise $C^1$ function satisfying

$$ x \le R \Rightarrow S(x) = 0 \quad \text{and} \quad x \ge \bar{R} \Rightarrow S(x) = 1 \quad (3) $$

for known real numbers $\bar{R} < \bar{R}$, and increasing in $[\bar{R}, \bar{R}]$.

The function $S$ is modeled by a modified sigmoid-type function which attains its limits. A possible candidate will be the clipped ReLU function $S(\lambda) = (\lambda - \bar{R})/(\bar{R} - \bar{R})$, $\lambda \in [\bar{R}, \bar{R})$. This assumption is often made in the literature. (see, e.g, [10]).

Define, for convenience,

$$ \begin{aligned} \Gamma_k &:= \{(i, j) \in [1, k]^2, 1 \le j \le \min(i+1, k)\}, \\ \text{and } \Theta_k &:= (W_{ij})_{(i,j) \in \Gamma_k}, \quad 1 \le k \le p, \end{aligned} \quad (4) $$

and let $Q_k$ be the total number of entries in $\Theta_k$; one may consider that $\Theta_k$ is an element of $\mathbb{R}^{Q_k}$. The parameters of the vector field $f$ are the matrices $W_{i,j} \in \mathbb{R}^{n_i \times n_j}$, that can be lumped into $\Theta = \Theta_p \in \mathbb{R}^{Q_p}$. They are unknown but we assume the following.

**Assumption 2:** For some known number $\bar{W} > 0$, one has

$$ \|W_{i,j}\|_{\infty} \le \bar{W}, \quad (i, j) \in \Gamma_p, \quad (5) $$

$$ W_{k,k+1} \text{ is left-invertible, } 1 \le k \le p-1. \quad (6) $$

We call a system of type (1)-(2) a *p*-layer system, and we will sometimes call its *q*-layer subsystem ($q \le p$) the system with state $(V_1, \dots, V_q)$ obtained by discarding $V_{q+1}, \dots, V_p$ and removing the term $W_{q,q+1}S(V_{q+1})$ in $\dot{V}_q$.

In the rest of the article we will consider controls that are continuous by part. From Cauchy-Lipschitz theorem, solutions of (1) are well defined, global in time. Moreover, if $u_i$ are bounded, due to the contraction induced by the $-x$ term and the boundedness of the sigmoid $S$, there exist an invariant attracting bounded set $\mathcal{V}$ for the dynamic $f(v, u)$ which denotes the right hand side of (1). We assume from hereafter that all solutions of (1) are bounded and belong to the set $\mathcal{V}$.

### III. MAIN RESULTS

In this section we present the main contribution of this article in the form of the following theorem

**Theorem 3:** Assume hypotheses 1-2 hold. Let $T > 0, \rho > 0$. For $k \in \{1, \dots, p\}$ there exists

• a family of maps $U_t^k: \mathbb{R}^{Q_{k-1}} \times C([0, t]; \mathbb{R}^{n_1}) \to \mathbb{R}^{N_k}$,

• a map $\Phi_t^k: \mathbb{R}^{Q_{k-1}} \times C([0, t], \mathbb{R}^{n_1}) \to \mathbb{R}^{Q_k} \times C([0, t], \mathbb{R}^{N_k})$,

such that for any solutions of (1) with initial conditions in $B_N(0, \rho)$ and output feedback control

$$ u(t) = U_t^k(\Theta_{k-1}, y|_{[0,t]}), \quad (7) $$

we have that

$$ \phi_t^k(\Theta_{k-1}, y|_{[0,t]}) = (\Theta_k, V^k(t)), \quad \forall t \ge T. \quad (8) $$

In particular, there exist a map $\Psi: C([0, kT], \mathbb{R}^{n_1}) \to \mathbb{R}^Q$ such that

$$ \Psi(y|_{[0,kT]}) = \Theta. \quad (9) $$

The above theorem describe that from the knowledge of the output $V_1$ and the parameters $\Theta_k$, there exists a control for which the measurement is rich enough to let you obtain $\Theta_j$, for any degree of depth desired $j \ge k+1$. The specific statement of the theorem formalizes the idea that, through a sequence of finite open-loop experiments, it is possible to recover unknown parts of the parameter set. Once the parts of the parameters are established, a new control can then be designed to capture the remaining unknown parameters.

#### A. Structure of the proof

The remainder of this paper is structured to progressively develop the tools required to prove the main theorem. In Section IV we analyze the control of the nonlinearities in (1) through an open-loop strategy. We expend on this to show that we can “disconnect” lower layers from the network. We also define output feedback controls for identifying the terms $W_{kj}$ at each layer $k \in [1, p]$ for $j \in [1, k]$. In Section V we identify the interaction terms $W_{i,i+1}$ for $i \in [1, p-1]$. In Section VI we provide the tools to recover the state of successive layers up to order $k$ using measurements from the first layer $y = V_1$, when the parameters $\Theta_k$ are known up to a certain order $k$. Finally in Section VII we combine the results to prove Theorem 3.

We numerically illustrate the method in Section VIII, and recall some results on finite time observers in Appendix I.

### IV. 1-LAYER SYSTEM

In this section, we focus on the following 1-layer system

$$ \dot{x} = -x + \psi(x, t) + u, \quad (10) $$

where $x \in \mathbb{R}^n$, $\psi \in C_b(\mathbb{R}^n \times \mathbb{R}, \mathbb{R}^n)$ and $u \in C_b(\mathbb{R}, \mathbb{R}^n)$. Solutions of (13) are well defined and global in time. Indeed, due to the contraction induced by the $-x$ term, we have for any $R \ge R_0 := \|\psi\|_{\infty} + \|u\|_{\infty} > 0$ the ball $B_n(0, R)$ is an invariant set for (13).

#### A. Weak controllability of the non-linearities

In this section, we prove that we can choose constant controls steering $S(x)$ towards any vector whose components are either 0 or 1. This can be achieved in finite time and without a priori knowledge of $x$.
**Lemma 4:** Let $T > 0$, $\rho > 0$, $m > 0$, and $w \in A_n := \{0, 1\}^n \subset \mathbb{R}^n$. Consider the constant control $u^w$

$$u_j^w := \begin{cases} \mu, & \text{if } w_k = 1, \\ -\mu, & \text{if } w_k = 0, \end{cases} \quad (11)$$

with $\mu \ge \mu^*$, where

$$\mu^* := m + \frac{\rho + e^T \max\{|\bar{R}|, |\bar{R}|\}}{e^T - 1}. \quad (12)$$

Then, for any $\psi$ such that $\|\psi\|_{\infty} \le m$, solutions of

$$\dot{x} = -x + \psi(x, t) + u^w. \quad (13)$$

initialized in $B_n(0, \rho)$, satisfy $S(x(t)) = w$ when $t \ge T$. Here, $\bar{R}$ and $R$ are set in (3).

*Proof:* Since the components of $x$ are decoupled, without loss of generality, we can assume $n = 1$. Let us show that if $w = 1$, then $x(t) \ge \bar{R}$ for all times $t \ge T$. From the expression of $\mu^*$ in (12), we have $\dot{x} \ge -x + (\rho + \bar{R})/(e^T - 1)$. The claim follows by Grönwall's lemma and the fact that $x(0) \ge -\rho$. A similar argument shows that $x(t) \le \bar{R}$ if $w = 0$, completing the proof. ■

## B. Cutting Layers

Denote the restriction $f^q = (f_1, \dots, f_q)$. Recall that for all $i \in [1, p]$, $f_i(V, u, \Theta) = f_i(V^{i+1}, u_i, \Theta_{i+1})$.

From the above lemma, it follows that we can reduce $p$-layer systems to a $q$-layer system, $q < p$, without any knowledge of the parameters or the states of the removed layers. This is encapsulated in the following.

**Corollary 5:** Let $T > 0$, $\rho > 0$, and $q \in [1, p-1]$. Then, there exists constant controls $u_{q+1}, \dots, u_p$ such that for any solution $V$ of (1) with initial condition in $B_{n_1}(0, \rho) \times \dots \times B_{n_p}(0, \rho)$, the vector $V^q$ follows for $t > T$ the dynamics of a $q$-layer system (see (2)) with parameter $\Theta_q$. That is, in the dynamic $f_q$, for $t \ge T$,

$$f_q(V(t), u(t), \Theta) = -V_q(t) + \sum_{i=1}^{q} WS(V_i(t)) + u_q(t).$$

## C. Identification of the nonlinearities under structural assumptions

In this section, we show how to identify the non-linearity $\psi$ in the particular case where $\psi(x, t) = WS(x)$, for $W \in \mathbb{R}^{n \times n}$. We suppose here that we can estimate $S(x)$, it will be treated in the next section. We consider system (10) with output

$$\left\{ \begin{array}{l} \dot{x} = -x + WS(x) + u, \\ y = S(x). \end{array} \right. \quad (14)$$

To estimate the parameter $W$, we first construct a class of output feedback controls $u^*$. We show that under this class of functions, solutions of (14) initialized in a known bounded domain remain in the invertible region of $S$. Moreover, we demonstrate that this class of controls can make $S(x)$ a persistently exciting function, allowing the estimation of $W$ in finite time.

Given $L > 0$, denote by $C^{1,L}$ the set of those $C^1$ trajectories whose derivative is bounded by $L$. Given $0 < \delta < 1/6$, $l > 0$, and $\gamma \in C^{1,L}(\mathbb{R}, (\bar{R} + \delta, \bar{R} - \delta)^n)$, we then define the following piecewise affine function

$$\eta(\sigma; l, \delta, \gamma) = \begin{cases} l & \text{if } \sigma \le \delta, \\ \alpha_1(\sigma; l, \delta) & \text{if } \delta \le \sigma \le 2\delta, \\ -l(S^{-1}(\sigma) - \gamma) & \text{if } 2\delta \le \sigma \le 1 - 2\delta, \\ \alpha_2(\sigma; l, \delta) & \text{if } 1 - 2\delta \le \sigma \le 1 - \delta, \\ -l & \text{if } \sigma \ge 1 - \delta. \end{cases}$$

Here, $\alpha_1$ and $\alpha_2$ are affine functions ensuring continuity of $\eta$, and $S^{-1}: [0, 1] \to [\bar{R}, \bar{R}]$ is the inverse of $S$ in the invertible region. Define now the output feedback control

$$u^*(y) = \eta(y; l, \delta, \gamma). \quad (15)$$

Define the following set

$$\Omega_{\delta} := (S^{-1}(3\delta), S^{-1}(1-3\delta)). \quad (16)$$

**Lemma 6:** Let $T, \rho, \bar{W}, L > 0$ and $0 < \delta < 1/6$. For any $\varepsilon > 0$ and any $\gamma \in C^{1,L}(\mathbb{R}, \Omega_{\delta}^n)$, there exist $l^* > 0$ such that for any $l > l^*$, and any $W \in \mathbb{R}^{n \times n}$ such that $\|W\| \le \bar{W}$, solutions of (14) initialized in $B_n(0, \rho)$ under the output feedback $u^* = \eta(S(x); l, \delta, \gamma)$ defined in (15) verify $x \in [\bar{R}, \bar{R}]$ and $|x - \gamma| \le \varepsilon$ when $t \ge T$.

Using the output feedback introduced above, for any solution $x$ initialized in $B_n(0, \rho)$, $x$ is ensured to enter the set $\Omega_{\delta}$ in finite time by taking $l > 0$ sufficiently large. Then, by analysis of the evolution of $e^2 := |x - \gamma|^2$ we are ensured that $x$ gets arbitrarily close to $\gamma$ when $l$ is sufficiently large.

We recall the following

**Definition 7:** A vector valued function $g \in C(\mathbb{R}, \mathbb{R}^n)$ verifies the Persistent Excitation condition (PE) if there exist $a, b, \tau > 0$ such that

$$aI_n \preceq \int_{t-\tau}^{t} g(s)g(s)^{\top}ds \preceq bI_n. \quad (17)$$

If there exists $U \subset \mathbb{R}^n$ such that $g \in C(\mathbb{R}, U)$, then we write $g \in PE(U)$.

The following corollary immediately follows from the previous lemma

**Corollary 8:** Let $T, \rho, \bar{W}, L > 0$ and $0 < \delta < 1/6$. Let $\gamma \in C^{1,L}(\mathbb{R}, \Omega_{\delta}^n)$ such that $S(\gamma) \in PE((0, 1)^n)$. There exists an application $\Phi_t^* : C([0, t], \mathbb{R}^n) \to \mathbb{R}^{n \times n} \times \mathbb{R}^n$ and $l^* > 0$ such that for any $l > l^*$, for any $W \in \mathbb{R}^{n \times n}$ such that $\|W\| \le \bar{W}$, for any solution of (14) with initial in $B_n(0, \rho)$ and output feedback $u^*$ in to the form of (15), we have

$$\Phi_t^*(y|_{[0,t]}) = (W, x(t)), \quad \forall t \ge T. \quad (18)$$

*Proof:* Let $0 < T_1 < T_2 < T$. First, from the second point of Lemma 6, under the output feedback $u^*$, we know that $x(t) \in (\bar{R}, \bar{R})$ for all times $t \ge T_1$ for a high enough $l > 0$. We have therefore $x = S^{-1}(y)$. Using now Proposition 14 we can retrieve $WS(x(t))$ for all time $t \ge T_2$ using the sliding mode observer for $z_1 = x, z_2 = WS(x)$. To ease the proof we consider it as a virtual output $y_1 = WS(x(t))$. From Lemma 6, choosing $\gamma$ such that $S(\gamma) \in PE((0, 1)^n)$, it
is immediate from the regularity on $S$ that for a high enough $l > 0$, $S(x(\cdot)) \in \mathcal{PE}((0,1)^n)$ after $t \ge T$. Therefore

$$W = \left( \int_{t-\tau}^{t} y_1(s)y(s)^\top \right) \left( \int_{t-\tau}^{t} y(s)y(s)^\top ds \right)^{-1}, \quad (19)$$

for all times $t \ge T$, with $\tau > 0$ small enough which can be tuned from the choice of $\gamma$.

## V. IDENTIFICATION OF A 2-LAYER SYSTEM

In this section, we consider the identification problem for the following 2-level system

$$\begin{cases} \dot{x}_1 = -x_1 + \psi_1(x_1, t) + WS(x_2), \\ \dot{x}_2 = -x_2 + \psi_2(x_1, x_2, t) + u_2, \\ y = (y_1, y_2) = (x_1, \psi_1(x_1(t), t)), \end{cases} \quad (20)$$

where $(x_1, x_2) \in \mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$, $W \in \mathbb{R}^{d_1 \times d_2}$ is left invertible, $\psi_1 \in C_b(\mathbb{R}^{d_1} \times [0, +\infty), \mathbb{R}^{d_1})$, $\psi_2 \in C_b(\mathbb{R}^{d_2} \times \mathbb{R}^{d_1} \times [0, +\infty), \mathbb{R}^{d_2})$ and $u_2 \in C_b([0, +\infty), \mathbb{R}^{d_2})$. Solutions of (20) are global in time and for any $R \ge R_0 := \|\psi_1\|_{\infty} + \|\psi_2\|_{\infty} + \|u\|_{\infty} > 0$ the ball $B_{d_1+d_2}(0, R)$ is an invariant set for (20).

### A. Identification of interaction terms

We now want to retrieve from the output $y$ in equation (20) the matrix $W$. To this aim, given two times $t_2 > t_1 > 0$, we consider the map $\Psi_{t_1}^{t_2} : C_b([0, T], \mathbb{R}^{2d_1}) \to \mathbb{R}^{d_1}$ that associates to an output $y = (y_1, y_2)$, defined up to a time $T > t_2$, the vector

$$\Psi_{t_1}^{t_2}(y) := y_1(t_2) - e^{-t_2+t_1}y_1(t_1) - e^{-t_2} \int_{t_1}^{t_2} e^s y_2(s)ds. \quad (21)$$

From Lemma 4 we have the following.

**Lemma 9:** Let $t_2 > t_1 > 0$, $\rho > 0$, $m > 0$, and $w \in A_n := \{0, 1\}^n \subset \mathbb{R}^n$. Then, there exists a constant control $u_2 \in C_b([0, t_2], \mathbb{R}^{d_2})$, depending only on $(t_1, t_2, \rho, m, w)$, such that, for any $\psi_1$ and $\psi_2$ such that $\|\psi_2\|_{\infty} \le m$, for any solutions of (20) initialized in $B_{d_1}(0, \rho) \times B_{d_2}(0, \rho)$ we have that

$$\Psi_{t_1}^{t_2}(y) = (1 - e^{-(t_2-t_1)})Ww. \quad (22)$$

*Proof:* Using Lemma 4 for the chosen $w, t_1, \rho, m$, there exists $\mu > 0$, such that choosing $u_2$ to be a constant control as defined in (11), we have that for any solutions $x = (x_1, x_2)$ of (20) initialized in $B_{d_1}(0, \rho) \times B_{d_2}(0, \rho)$, $S(x_2(t)) = w$ for all times $t \ge t_1$. Using then Duhammel's formula for the equation on $x_1$, we obtain indeed that (21) is equal to (22).

Applying the previous lemma multiples times in a row, we have the following corollary.

**Corollary 10:** Let $T > 0$ and consider two sequences $t_1^k$ and $t_2^k$ such that

$$0 < \dots < t_1^k < t_2^k < t_1^{k+1} < \dots < t_2^{d_2} = T, \quad (23)$$

and $t_2^k - t_1^k = \alpha$ for some constant $\alpha > 0$, and any $k \in [1, d_2]$. Let $\rho > 0$, and $m > 0$. Then, there exists a control $u_2 \in C_b([0, T], \mathbb{R}^{d_2})$, depending only on $(T, (t_1^k)_k, (t_2^k)_k, \rho, m)$, such that, for any solution of (20)

initialized in $B_{d_1}(0, \rho) \times B_{d_2}(0, \rho)$, letting $\Psi_k := \Psi_{t_1^k}^{t_2^k}$, we have

$$W = \frac{1}{(1 - e^{-\alpha})} [\Psi_1(y) \ \Psi_2(y) \ \dots \ \Psi_{d_2}(y)].$$

### B. Estimation of $S(x_2)$

We consider system (20). At first, we want to estimate $S(x_2)$ from the knowledge of the output $y$ and the matrix $W$. In this case $\psi_2$ remains unknown.

To achieve this, it suffices to apply a finite time observer as the one introduced in Appendix I. This yields the following

**Proposition 11:** Let $T > 0$, $\rho > 0$. For any $m, \bar{u} > 0$ such that

$$\|\psi_2\|_{\infty} \le m, \quad \|u\|_{\infty} \le \bar{u}, \quad (24)$$

there exists $\Psi_1$, depending only on $(T, \rho, m, \bar{u})$, such that for any solutions of (20) initialized in $B_{d_1}(0, \rho) \times B_{d_2}(0, \rho)$ with output $y$, we have

$$\Psi_1(W, y|_{[0,t]}) = S(x_2(t)), \quad \forall t \ge T. \quad (25)$$

Combining all the previous results of this section, the following stands proved.

**Theorem 12:** Assume $\psi_2$ is Lipschitz continuous. Let $T, \rho > 0$ and $m > 0$ such that $\|\psi_2\|_{\infty} \le m$. Then, there exists a family of maps $U_t : C_b([0, t], \mathbb{R}^{d_1}) \to \mathbb{R}^{d_2}$ and map $\Phi_t : C_b([0, t], \mathbb{R}^{d_1}) \to \mathbb{R}^{d_1 \times d_2} \times \mathbb{R}^{d_2} \times \mathbb{R}^{d_2}$ such that for any solution of (20) with initial conditions in $B_{d_1+d_2}(0, \rho)$ and output feedback control $u_2(t) = U_t(y|_{[0,t]})$, we have

$$\Phi_t (y|_{[0,t]}) = (W, S(x_2(t))), \quad \forall t \ge T. \quad (26)$$

## VI. FINITE TIME STATE ESTIMATION WITH KNOWN PARAMETERS

In this section, we establish that the state of any $k$-layer subsystem $V^k = (V_1, \dots, V_k)$ of system (1) can be estimated in finite from $V_1$ when the parameters $\Theta_k$ are known. More precisely, we demonstrate that this can be achieved using output feedback controls of the form (15). This result is particularly relevant for subsequent developments, as the same class of controls will be employed for the identification process.

Recall the definition of $\Theta_k$ in (4) and $\Omega_\delta$ in (16). We have the following.

**Lemma 13:** Let $k \in [2, p]$. Let $T > 0$ and consider any sequence of times $0 = T_0 < T_1 < T_2 < \dots < T_k = T$. Let $\rho > 0$, $0 < \delta < 1/6$. For any sequence $\gamma_i \in C^{1,L}(\mathbb{R}, \Omega_\delta^{n_i})$ with $i \in [2, k]$, for any $\Theta_k \in \mathbb{R}^{Q_k}$, there exists a family of maps $O_t^i : \mathbb{R}^{Q_i} \times C([0, t], \mathbb{R}^{n_1}) \to \mathbb{R}^{n_1}$ and a map $O_t^* : \mathbb{R}^{Q_k} \times C([0, t], \mathbb{R}^{n_1}) \to \mathbb{R}^{N_k}$, and $l^* > 0$, such that for any $l > l^*$, for any $0 \le \alpha \le 1$, letting for $i \in [2, k]$

$$u_i = \eta(\alpha \mathbf{1}_{t \le T_i} + O_t^i \mathbf{1}_{t \ge T_i}; l, \delta, \gamma_i), \quad (27)$$

we have for any solution of (1) with starting in $B_N(0, \rho)$,

$$O_t^i(\Theta_i, y|_{[0,t]}) = S(V_i(t)), \quad \forall t \ge T_{i-1}, \quad (28)$$

$$O_t^*(\Theta_k, y|_{[0,t]}) = (V_1(t), \dots, V_k(t)), \quad t \ge T. \quad (29)$$

*Proof:* The proof relies on a cascade of sliding mode observers (32) as introduced in Appendix I, and follows a
recursive argument. We show here how to estimate $V_2$ from $V_1$, and $S(V_3)$ from $V_2$. The rest of the proof follows.

From the output $V_1$, using $z_1 = V_1$ and $z_2 = W_{12}S(V_2)$, we satisfy the conditions of Proposition 14 for a time $T_1 > 0$. For all $t \ge 0$, we obtain an estimate $\hat{z}_2(t)$ such that $\hat{z}_2(t) = W_{12}S(V_2(t))$ for all $t \ge T_1$. Since $W_{12}$ is known and left-invertible by Assumption 2, applying its left inverse $W_{11}^\dagger$ to the estimate allows us to construct the first mapping

$$O_t^2(\Theta_1, y|_{[0,t]}) = S(V_2(t)), \quad t \ge T_1. \quad (30)$$

From Lemma 6, we know that under such feedbacks, there exists a sufficiently large $l_1 > 0$ such that for a duration of $(T_2 - T_1)/2$, the trajectory $V_2$ remains in the invertible region of $S$ for all $t \ge T_1 + (T_2 - T_1)/2$. Setting $y_1(t) = S^{-1}(O_t^1(\Theta_1, y|_{[T_0,t]}))$, it follows that $y_1(t) = V_2(t)$ for all $t \ge T_1 + (T_2 - T_1)/2$. Following the same process, we can tune a sliding mode observer such that $\bar{O}_t^3$ and $O_t^3$ have

$$O_t^3(\Theta_2, y|_{[0,T]}) = \bar{O}_t^3(\Theta_2, y_1|_{[T_2,t]}) = S(V_3(t)), \quad t \ge T_2.$$

Using this argument recursively yields the desired result. The map $O_t^*$ is obtained by combining all the $O_t^i$ and $S^{-1}$. ■

## VII. PROOF OF THEOREM 3

Let $0 < T_1 < T_2 < T_3 < T$. Let $L > 0$. From the structure of (1), the parameters $\Theta_{k+1}$ appear in the dynamic of $V^{k+1} = (V_1, \dots, V_{k+1})$. Corollary 5 ensures that there exist constant controls $u_{q+i}, \dots, u_p$, such that after time $T_1$, all lower layers $q \ge k+2$ will be functionally disconnected. The state $V^{k+1}$ will follow a $k+1$-layers system.

Lemma 13 ensure that for $\rho > 0$ high enough, $0 < \delta < 1/6$, $\gamma_i \in C^{1,L}(\mathbb{R}, \Omega^{n_i})$ and for the time $T_2 - T_1 > 0$, placing feedback controls $u_i$ of type (27), there exist $l_1 > 0$ such that for any $l \ge l_1$ we can recover the values of the vector $V^k = (V_1, \dots, V_k)$ for all times after $t \ge T_2$. We denote for simplicity of proof, $y_1(t) := V^k(t)$.

We now wish to retrieve $W_{k,k+1}$. We consider the dynamics of $V^k$ and $V_{k+1}$ as a two layer system (20) with

$$\psi_1(t) = \sum_{j=1}^{n_k} W_{k,j} S(V_j(t)), \quad \psi_2(t) = \sum_{j=1}^{n_{k+1}} W_{k+1,j} S(V_j(t)).$$

From assumption, the matrices $W_{k,j}$ are known and from the virtual output $y_1$ we indeed have the same form of output as in (20). Moreover, we also verify the condition of $\psi_2$ unknown with known bound $(k+1)\bar{W}$.

We now are in condition to use Theorem 12 for a time $T_3 - T_2 > 0$, $\rho_2 > 0$ large enough and $m = k\bar{W}$. The theorem ensures that there exist an adequate piecewise continuous control $u_{k+1} : [T_2, T_3] \to \mathbb{R}^{n_k}$ such that we can recover the values of $W_{k,k+1}$ and $S(V_{k+1}(t))$ for all times $t \ge T_3$. We consider, once again for simplicity, $y_2(t) = S(V_{k+1}(t))$ as a virtual output our system after time $T_2$. Recall, from Lemma 13, that estimation of $(y_1, y_2)$ is maintained thanks to the output feedback controls $u_i$ of class (27), for $i \in [1, k]$, and the fact that the values of $S(V_{k+1}(t))$ can be recovered through a sliding mode observer independently of the control $u_{k+1}$ once $W_{k,k+1}$ is known.

Let now $\gamma = (\gamma_1, \gamma_2, \dots, \gamma_{k+1})$, with $\gamma_i \in C^{1,L}(\mathbb{R}, \Omega^{n_i})$ be such that $S(\gamma) \in \text{PE}((0,1)^n)$. We are in condition to apply Corollary 8 for a time $T - T_3 > 0$, $\rho_3$ high enough and the chosen $\gamma$. The theorem ensures that we recover the block matrix $W$ after time $T > 0$. In particular, we can recover its last row $W_{k+1,j}$ for $j \in [1, k+1]$. Moreover, the corollary also yields that we can recover $V^k(t)$ after time $T$. Theorem 3 therefore stands proved. ■

## VIII. ILLUSTRATION

In this section, we provide an illustration of the method's principle by highlighting how saturations are exploited to reveal the weights. We consider a toy example of a two-layer system with $n_1 = 2$, $n_2 = 1$. We pick arbitrary matrices

$$\begin{aligned}
W_{11} &= \begin{pmatrix} 0.6 & -0.2 \\ 0.4 & -0.7 \end{pmatrix}, & W_{12} &= \begin{pmatrix} 0.3 \\ -0.9 \end{pmatrix}, \\
W_{21} &= \begin{pmatrix} -0.4 & -0.1 \end{pmatrix}, & W_{22} &= \begin{pmatrix} -0.9 \end{pmatrix}.
\end{aligned}$$

Regarding the saturation function $S$, we pick $\bar{R} = 0$, $\bar{R} = 1$ and $S(x) = x$ on $[0, 1]$. Then we define the dates $T_1 = 1$, $T_2 = 3$, $T_3 = 5$, $T_4 = 8$. We set the controls to be

$$(u_1^1, u_1^2, u_2) = \begin{cases} (0, 0, -3) & \text{on } [0, T_1), \\ (3, -3, -3) & \text{on } [T_1, T_2), \\ (-3, 3, -3) & \text{on } [T_2, T_3), \\ (-3, -3, 3) & \text{on } [T_3, T_4). \end{cases}$$

Simulating this system yields Figure 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Trajectory of the toy system. In red and blue are represented $(x_1^1, x_1^2)$ of the first layer, while in green is represented $x_2$ of the second layer. Dashed are the trajectories saturated at 1.</figcaption></figure>

On $[0, T_3]$, $u_2 = -3$ in order for $S(x_2)$ to be 0 on $[T_1, T_3]$. On $[T_1, T_2]$ and $[T_2, T_3]$, controls are picked so that $S(x_1) = (1, 0)$ and $S(x_1) = (0, 1)$ on $[T_2 - 1, T_2]$ and $[T_3 - 1, T_3]$ respectively. Likewise, on $[T_3, T_4]$, controls are picked so that $S(x_1) = (0, 0)$ and $S(x_2) = 1$ on $[T_4 - 1, T_4]$. As a result, the virtual measurement $z = \dot{x}_1 + x_1 - u_1$, that we can estimate (following from Section V and Appendix I), occupies on $[T_k - 1, T_k)$, $k = 2, 3, 4$, $(W_{11}^{11}, W_{11}^{21})$, $(W_{11}^{12}, W_{11}^{22})$ and $W_{12}$, respectively. See Figure 2.

## IX. CONCLUSION

In this paper, we established novel identification methods for systems governed by networked Wilson-Cowan-like equation. By leveraging the properties of saturations, we designed open-loop controls that allow extraction of the
<figure><img src="image_3.png" /><figcaption>Fig. 2. Values of the virtual measurement $z = \dot{x}_1 + x_1 - u_1$. In red and blue are represented ($z^1, z^2$), respectively. Dashed lines are the constant weights in $W_{11}, W_{12}$. For $k = 2, 3, 4$, $z$ and coincides with the weights over $[T_k - 1, T_k)$, which allows through estimation of $z$ to recover them.</figcaption></figure>

weight parameters in finite time. In single-layer systems, self-referential weights can be identified by steering the system into specific configurations. For two-layer systems, where the top layer is measured, similar principles enable the recovery of layer-interaction weights. Furthermore, we demonstrated how well-designed feedback mechanisms can facilitate a descent into the lower layer, effectively allowing us to treat it as a virtual measurement. Extending this approach, we recursively generalized our results to systems with an arbitrary number of layers.

Additional parameters are identifiable within this framework, but fall beyond the scope of this paper. These ideas warrant further exploration, particularly with a view to relaxing certain assumptions on the structure of the system.

In the realm of artificial neural networks, this type of scheme would be well fitted to identify weights resulting from learning, as proposed in [2]. By performing identification on a neural network with substantially fewer nodes than the one providing the measure, this scheme could be used to produce a distillation of the initial one (see, e.g., [15]).

# APPENDIX I

## FINITE-TIME OBSERVERS

Let us recall some well known results on homogeneous/sliding mode [14], [1]. Consider the following system with $z = (z_1, z_2) \in \mathbb{R}^n \times \mathbb{R}^n$,

$$ egin{cases} \dot{z}_1 = g_1(z_1, t) + z_2, \\ \dot{z}_2 = g_2(z_1, z_2, t), \\ y = z_1. \end{cases} \quad (31) $$

with $g_i$ continuous maps. Sliding mode observers allow to recover the state of the system in finite time, without explicit knowledge of $g_2$ other than a bound. More precisely, consider the following observer, depending on $L, K_1, K_2 > 0$:

$$ \begin{cases} \dot{z}_1 = g_1(y, t) + \hat{z}_2 + LK_1[y_1 - \hat{z}_1]^2 \\ \dot{z}_2 \in L^2 K_2 \text{sign}(y_1 - \hat{z}_1). \end{cases} \quad (32) $$

Here, for $\sigma \in \mathbb{R}$, we let $[\sigma]^{\frac{1}{2}} := \text{sign}(\sigma) \sqrt{|\sigma|}$, with $\text{sign}(\sigma)$ the set valued mapping

$$ \text{sign}(\sigma) := \begin{cases} \{1\} & \text{if } \sigma > 0, \\ [-1, 1] & \text{if } \sigma = 0, \\ \{-1\} & \text{if } \sigma < 0. \end{cases} $$

We give the following proposition, adapted to our context from [5, Proposition 4].

**Proposition 14:** Assume that $g_1$ and $g_2$ piecewise continuously differentiable and bounded, and that $z_1 \mapsto g_1(z_1, t)$ is $\frac{1}{2}$-Hölder continuous uniformly w.r.t. $t$. Let $m > 0$ be such that $\|g_2\|_{\infty} < m$. Then, there exist $K_1, K_2 \in \mathbb{R}$ such that for any time $T > 0$ and $\rho > 0$ there exist a gain $L^* > 0$ such that for any $L > L^*$ and for any $\hat{z}(0), z(0) \in B_{2n}(0, \rho)$ we have,

$$ \hat{z}(t) = z(t), \quad \forall t \ge T. \quad (33) $$

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