Local Structural Identifiability in Dynamical Systems

• Local structural identifiability is the property where model parameters can be uniquely or finitely recovered in a neighborhood of the true value from ideal, noise-free data.
• Analytical methods such as differential algebra, Jacobian rank tests, and observability-based approaches are employed to assess identifiability in various model classes.
• This concept is crucial in fields like systems biology, control theory, and econometrics to design models that enable reliable parameter estimation.

Local structural identifiability addresses the fundamental question of whether, for a given mathematical model—most commonly formulated as a system of ordinary differential equations (ODEs), algebraic equations, or discrete-time dynamical systems—the parameters (or identifiable functions thereof) can be uniquely recovered from exact, noise-free input-output data in a neighborhood of the true parameter value. In contrast to global identifiability, which requires uniqueness over the entire parameter space, local identifiability only demands that the parameter map be injective (or finite-to-one) in a sufficiently small neighborhood. This concept is essential in mathematical modeling, systems biology, control theory, epidemiology, engineering, and econometrics, as it characterizes the inherent limitations of model-based parameter estimation, independent of data imperfections or practical measurement constraints.

1. Formal Definitions and Theoretical Criteria

Let $\mathcal{M}$ be a model characterized by unknown parameter(s) $\theta \in \Theta \subset \mathbb{R}^p$, state $x \in \mathbb{R}^n$, measurable output $y(t)\in\mathbb{R}^m$, and possibly exogenous input $u(t)$. The general mapping of interest is

$\theta \mapsto y(\cdot;\theta)$

where $y(\cdot;\theta)$ is the output trajectory induced by fixed initial conditions and input.

Local structural identifiability (LSI): $\theta_0$ is locally identifiable if, for all $\theta$ in some open neighborhood $U \ni \theta_0$, the implication $y(\cdot;\theta) = y(\cdot;\theta_0)$ implies $\theta = \theta_0$ (or, in the finite-to-one variant, that the fiber $\{\theta: y(\cdot;\theta) = y(\cdot;\theta_0)\}$ is finite).

In the context of rational ODE models,

$$x'(t) = f(x(t),u(t),\theta), \qquad y(t) = g(x(t),u(t),\theta), \qquad x(0) = x_0$$

local identifiability means that the map from $\theta$ to the collection of output coefficients (e.g., Taylor or jet coefficients, or input-output equation coefficients) is finite-to-one in a generic neighborhood. Jacobian-based criteria are central: if a local parameterization map $F(\theta)$ (e.g., the vector of input-output equation coefficients or output jets) satisfies $\operatorname{rank} \partial F/\partial\theta = p$ at a generic $\theta_0$, then $\theta_0$ is locally identifiable (Villaverde, 2024, Ilmer et al., 2021, Liyanage et al., 15 May 2025, Ovchinnikov et al., 2023).

For models with algebraic, stochastic, semi-parametric, or function-valued parameters, analogous definitions apply, now in infinite-dimensional or function spaces, using Fréchet derivatives and suitable injectivity or closed-range conditions (Chen et al., 2011, Shalgin, 26 Dec 2025, Browning et al., 25 Mar 2025).

2. Algebraic, Differential, and Graphical Methods

Differential algebraic approach: For ODE models with rational vector fields and outputs, local structural identifiability reduces to analysis of differential-algebraic elimination ideals. Given a system $\Sigma$, one constructs the differential ideal $I_\Sigma$ generated by the state equations and output constraints, saturates by denominators if rational, and eliminates the state variables to obtain input-output (IO) equations in $y,u$ and $\theta$. The field of IO-identifiable functions is then generated by the coefficients of a characteristic set of the eliminated ideal (Ovchinnikov et al., 2020, Ovchinnikov et al., 2023, Falkensteiner et al., 2024).

Rank (Jacobian) tests: Compute the Jacobian matrix of the mapping from $\theta$ to the vector of output jet coefficients (e.g., derivatives at $t_0$ or IO polynomial coefficients). If this matrix has full column rank at a generic point, then the parameters are locally identifiable. This criterion is both necessary and sufficient for generic local identifiability in continuous models (Villaverde, 2024, Liyanage et al., 15 May 2025).

Observability-based approach: Treat parameters as extended constant states, build the augmented observability matrix via successive Lie derivatives of the output, and check its rank. The parameter vector is locally identifiable if and only if this matrix has full (state+parameter) rank (Villaverde, 2024, Ilmer et al., 2021).

Finite-difference approach for discrete-space systems: In discrete settings, differentiability fails. Identifiability is tested using finite-difference Jacobians or, if possible, combinatorial/exhaustive injectivity checks on the IO mapping (Sarathchandra et al., 2024).

Graph-theoretic/combinatorial criteria: For certain graphical models (Gaussian graphical models with latent variables, linear compartmental models, discrete Lyapunov/VAR models), local identifiability reduces to rank conditions on structural (edge) incidence matrices, sometimes coupled to specific combinatorial constraints (e.g., presence of odd cycles, leak-interlacing) (Leung et al., 2015, Ahmed et al., 2024, Gerberding et al., 2019, Recke et al., 29 Jan 2026).

3. Symmetries, Ambiguity, and the Boundary with Global Identifiability

Discrete symmetries: A model is locally but not globally identifiable if and only if there exist finite (discrete) algebraic symmetries of the parameter space that preserve the model's input-output behavior. Each such symmetry maps a parameter vector $\theta$ to a distinct but observationally indistinguishable point, so the fiber cardinality equals the order of the symmetry group. Analysis via Thomas decomposition and solution of the finite determining system gives a complete characterization (Barreiro et al., 26 Jul 2025).

In contrast, continuous symmetries (Lie groups) lead to non-identifiability even locally, as there is then a continuum of indistinguishable parameter values (Barreiro et al., 26 Jul 2025).

Examples: In linear mammillary compartmental models, simultaneous permutation symmetries among identical peripheral compartments produce locally but not globally identifiable parameter sets of size equal to the permutation group order. In rational ODEs, only the combinations of parameters that are invariant under these symmetries are globally (else, locally) identifiable (Barreiro et al., 26 Jul 2025, Ovchinnikov et al., 2023).

4. Specialized Model Classes and Algorithmic Implementation

Linear compartmental models: Structural identifiability is determined by the rank of the coefficient map from parameters to the input–output equation coefficients. In catenary and cycle models, explicit combinatorial conditions (“leak-interlacing”, number of leaks, position of inputs/outputs) fully determine which parameter regimes are locally identifiable. Operations such as moving inputs/outputs or certain modifications of leaks preserve identifiability, while others (e.g., adding too many leaks in cycles) destroy it (Ahmed et al., 2024, Gerberding et al., 2019).

Latent-variable Gaussian graphical models: For models with a single latent source, local identifiability is equivalent to generic finite-to-oneness of a polynomial parameterization map. Sufficient and necessary conditions are formulated in terms of cycles in the undirected complement graph and certain derived covariance graphs; subgraph extension principles further allow efficient evaluation in high-dimensional instances (Leung et al., 2015).

Discrete Lyapunov (VAR) models: Local identifiability of the parameter matrix for non-Gaussian steady-state models is certified by the generic rank of the Jacobian from higher-order cumulants to parameters, conditioned by graph-theoretic properties such as presence of self-loops and connectivity. For DAGs with loops, identifiability is constructive, via rational inversion from cumulant data (Recke et al., 29 Jan 2026).

Nonparametric, semiparametric, and stochastic function models: Abstract models admit a generalized rank (injectivity) criterion via the Fréchet or Gâteaux derivative of the moment mapping. An additional remainder bound is necessary to control nonlinearity in infinite-dimensional settings. This framework explicitly bounds the neighborhood in which local identifiability holds (Chen et al., 2011, Shalgin, 26 Dec 2025, Browning et al., 25 Mar 2025).

Software & algorithmic advances: Symbolic computation tools (e.g., StructuralIdentifiability.jl, SIAN, web-based analyzers) implement differential-algebraic elimination, power-series, and coefficient-rank-based tests. Modern implementations optimize for performance, handle high-dimensional models, and provide workflow templates for identifiability checking (Liyanage et al., 15 May 2025, Ilmer et al., 2021, Ovchinnikov et al., 2023).

5. Extensions: Function-Valued Parameters and Discrete-Time/Space Models

Function-valued parameter identifiability: For models where an unknown function $p(t)$ enters the vector field (i.e., parametric nonautonomous systems), local identifiability is characterized by sensitivity matrices and their degeneracies at a discrete set of “observation” times. Sufficient conditions are given in terms of sensitivities and properties of finite-dimensional cones in parameter function space, allowing for identification of $p(\cdot)$ from finitely many solution snapshots (Shalgin, 26 Dec 2025).

Discrete-space/time and algebraic models: Notions of “local neighborhood” and injectivity must be adapted in discrete systems: identifiability can be neighborhood-radius-dependent, and standard infinitesimal arguments are replaced by finite difference and combinatorial enumeration. Despite these differences, the equivalence between algebraic and structural identifiability still holds in the discrete regime, under appropriate input-output solvability (Sarathchandra et al., 2024).

6. Implications, Limitations, and Practical Considerations

Practical identifiability is strictly weaker than structural (local) identifiability: even if the latter holds, finite data and experimental constraints may preclude reliable parameter estimation. However, failure of structural identifiability precludes any successful inference, regardless of data (Villaverde, 2024).

Modifications preserving/failing identifiability: Many graph and model modifications (adding a single leak, altering observable outputs, moving inputs) preserve local identifiability in linear compartmental and other structured models. Other modifications (overloading with leaks, inappropriate output choice, or increasing unmeasured state components) may introduce local or global non-identifiability (Ahmed et al., 2024, Gerberding et al., 2019).

Reparametrization and minimal models: For any ODE model, there exists a partial specialization (fixing some parameters as functions of others) such that the reparametrized model is locally identifiable but equivalent in input-output behavior; these reparametrizations preserve the algebraic "shape" of the system and are constructively computable (Ovchinnikov et al., 2023, Falkensteiner et al., 2024).

Algorithmic complexity: Although principal criteria are algebraic and constructive, symbolic elimination and Gröbner basis computations exhibit doubly-exponential complexity in the number of variables and equation degrees, limiting scalability in large, dense nonlinear models (Ilmer et al., 2021, Ovchinnikov et al., 2023). For linear and structured models, combinatorial and block-matrix methods mitigate some of these issues.

Summary Table: Key Criteria and Approaches

Model Type	Local Identifiability Test	Principal Reference
Rational ODE systems	Jacobian rank of output-jet map; IO elimination	(Liyanage et al., 15 May 2025, Ilmer et al., 2021, Ovchinnikov et al., 2023)
Observability-based (nonlinear systems)	Augmented observability matrix rank	(Villaverde, 2024)
Graphical Gaussian models (SEM/latent)	Graph-theoretic cycle/complement analysis	(Leung et al., 2015)
Linear Compartmental models	Coefficient map Jacobian rank; combinatorics	(Ahmed et al., 2024, Gerberding et al., 2019)
Discrete-time/density models	IO-equation injectivity, finite-difference Jacobian	(Sarathchandra et al., 2024, Recke et al., 29 Jan 2026)
Stochastic SDE models	Moment-recurrence, IO-equation analysis	(Browning et al., 25 Mar 2025)
Nonparametric/semi-parametric models	Fréchet-derivative injectivity, remainder bound	(Chen et al., 2011, Shalgin, 26 Dec 2025)

7. Broader Implications and Connections

Local structural identifiability is a mathematically rigorous, model-intrinsic precondition for parameter inference and model discrimination. Its analysis unifies differential algebra, algebraic geometry, combinatorics, control theory, and computational symbolic algebra. Advances in algorithmic reparametrization, sensitivity analysis, and symmetry detection are yielding increasingly general and scalable protocols for both theoretical and practical identifiability assessment. These developments inform not only the diagnosis of statistical estimability but also facilitate robust model design, minimal realization construction, and refinement of experimental protocols in systems biology, engineering, and data-driven science (Ovchinnikov et al., 2023, Falkensteiner et al., 2024, Liyanage et al., 15 May 2025, Leung et al., 2015).