pith. sign in

arxiv: 2511.18015 · v2 · pith:JRLROSAOnew · submitted 2025-11-22 · 📡 eess.SY · cs.SY

On the stability of event-based control with neuronal dynamics

Luke Eilers , Jonas Stapmanns , Catarina Dias , Jean-Pascal Pfister This is my paper

Pith reviewed 2026-05-22 12:43 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords event-based controlimpulsive controlneuronal dynamicsleaky integrate-and-fireinput-to-state stabilityhybrid systemsneuromorphic controlglobal practical stability
0
0 comments X

The pith

An event-based impulsive controller from leaky integrate-and-fire neurons achieves global practical stability by linearly cancelling jumps with the plant state, matching the properties of a corresponding analogue controller.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a multivariable control-affine plant driven by multiple neuronal units that follow leaky integrate-and-fire dynamics and trigger impulses at discrete events. It observes that the discontinuous jumps appearing in the plant state and in the neuronal states cancel exactly when combined in a linear way. This cancellation produces an exact dynamical correspondence between the hybrid event-based closed loop and a continuous-time analogue controller. When the analogue system is input-to-state stable with respect to particular disturbances, the event-based system therefore inherits global practical asymptotic stability in the nonlinear case and global practical exponential stability in the linear case.

Core claim

The central claim is that the linear cancellation of discontinuities between the plant and the neuronal units creates a direct equivalence to an analogue controller; therefore the event-based impulsive system is globally practically asymptotically stable whenever the analogue system is input-to-state stable with respect to the relevant disturbances, and globally practically exponentially stable for linear plants that are stable in the analogue setting.

What carries the argument

Linear cancellation of state discontinuities between the plant and the neuronal units, which produces an exact dynamical correspondence to the analogue controller and transfers its stability properties.

If this is right

  • Global practical asymptotic stability holds for nonlinear plants whenever the analogue controller is input-to-state stable with respect to the disturbances induced by the event-based implementation.
  • Global practical exponential stability holds for linear plants that are stable in the analogue setting.
  • The inter-event times remain positive and the hybrid system remains well-defined under the same conditions that guarantee the analogue stability.
  • Numerical simulations of the closed-loop trajectories confirm that the practical stability bounds predicted by the analogue analysis are attained.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Control engineers could reuse existing input-to-state stability certificates for continuous systems rather than developing new hybrid Lyapunov functions for each neuromorphic design.
  • The same cancellation argument might apply to other spiking neuron models whose reset or threshold jumps can be expressed as linear corrections to the plant state.
  • Hardware implementations on neuromorphic chips could exploit this equivalence to guarantee stability without solving the full hybrid stability problem at runtime.

Load-bearing premise

The discontinuities that occur in the plant state and in the neuronal units must cancel each other when their states are added linearly.

What would settle it

A numerical integration or direct calculation at any event instant showing that the sum of the plant jump vector and the neuronal jump vectors is nonzero would break the claimed correspondence and invalidate the stability reduction.

Figures

Figures reproduced from arXiv: 2511.18015 by Catarina Dias, Jean-Pascal Pfister, Jonas Stapmanns, Luke Eilers.

Figure 1
Figure 1. Figure 1: The control loop between the controller, which receives the error [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A one-dimensional linear plant controlled by an event-based [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heatmap of the stability of the event-based impulsive control system [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A two-dimensional linear plant controlled by an event-based [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

Event-based control, unlike analogue control, poses significant analytical challenges due to its hybrid dynamics. This work investigates the stability and inter-event time properties of a control-affine system under event-based impulsive control. The controller consists of multiple neuronal units with leaky integrate-and-fire dynamics acting on a time-invariant, multivariable plant in closed loop. Both the plant state and the neuronal units exhibit discontinuities that cancel if combined linearly, enabling a direct correspondence between the event-based impulsive controller and a corresponding analogue controller. Leveraging this observation, we prove global practical stability of the event-based impulsive control system. In the general nonlinear case, we show that the event-based impulsive controller ensures global practical asymptotic stability if the analogue system is input-to-state stable (ISS) with respect to specific disturbances. In the linear case, we further show global practical exponential stability if the analogue system is stable. We illustrate our results with numerical simulations. The findings reveal a fundamental link between analogue and event-based impulsive control, providing new insights for the design of neuromorphic controllers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes stability of a control-affine multivariable plant under event-based impulsive control realized by multiple leaky integrate-and-fire neuronal units. It identifies a linear cancellation between discontinuities in the plant state and the neuronal states that maps the hybrid closed-loop dynamics to an analogue controller plus bounded disturbances. Using this reduction, the authors prove global practical asymptotic stability when the analogue system is ISS with respect to the induced disturbances, and global practical exponential stability in the linear case when the analogue system is stable. The claims are supported by standard ISS/Lyapunov arguments and illustrated by numerical simulations.

Significance. If the exact linear cancellation identity holds globally, the work establishes a direct correspondence between event-based impulsive neuromorphic control and analogue control, which is a useful conceptual bridge for stability analysis and controller design in hybrid systems. The reliance on standard ISS assumptions rather than ad-hoc fitting is a methodological strength, and the provision of both nonlinear and linear results increases the scope. The numerical examples serve as illustration rather than proof.

major comments (2)
  1. [Abstract and Introduction] Abstract and §1 (linear cancellation paragraph): the claim that plant and neuronal discontinuities 'cancel if combined linearly' is load-bearing for the entire reduction. In the general nonlinear control-affine case, the vector field f(x,u) and the neuron reset maps must satisfy a precise algebraic identity at every firing instant; the manuscript should state the exact condition on the input matrix and weights that guarantees this identity holds without residual state-dependent jump terms.
  2. [Stability theorems] Nonlinear stability theorem (presumably §4): the global practical asymptotic stability conclusion assumes the analogue system is ISS w.r.t. 'specific disturbances' that absorb all hybrid effects. If the cancellation is only approximate for arbitrary f or unmatched neuronal weights, an additional state-dependent jump term remains; this term must be shown to lie in the ISS disturbance class or the theorem statement must be restricted to plants and weights satisfying the exact identity.
minor comments (2)
  1. [Numerical examples] The numerical simulations are described as illustration; adding a brief table of plant/neuron parameters and inter-event time statistics would improve reproducibility without altering the theoretical contribution.
  2. [Preliminaries] Notation for the combined state (plant + neuronal) and the precise definition of the 'specific disturbances' should be introduced earlier and used consistently in the ISS assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. We agree that explicit statement of the cancellation condition will strengthen the paper and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract and Introduction] Abstract and §1 (linear cancellation paragraph): the claim that plant and neuronal discontinuities 'cancel if combined linearly' is load-bearing for the entire reduction. In the general nonlinear control-affine case, the vector field f(x,u) and the neuron reset maps must satisfy a precise algebraic identity at every firing instant; the manuscript should state the exact condition on the input matrix and weights that guarantees this identity holds without residual state-dependent jump terms.

    Authors: We agree that the precise algebraic identity is essential for the reduction. The plant is control-affine with input matrix B and the controller applies impulses via weight matrix W applied to the LIF states v. At each firing, the neuron resets (v_i := 0) while the plant receives a matched impulse. The linear combination z = x - B W v is continuous because the jump satisfies Δx = B W v exactly, canceling the neuronal contribution independently of f(x) and the state. This holds globally under the condition B W = I (with normalized resets). We will revise the abstract and §1 to state this condition explicitly on B and W. revision: yes

  2. Referee: [Stability theorems] Nonlinear stability theorem (presumably §4): the global practical asymptotic stability conclusion assumes the analogue system is ISS w.r.t. 'specific disturbances' that absorb all hybrid effects. If the cancellation is only approximate for arbitrary f or unmatched neuronal weights, an additional state-dependent jump term remains; this term must be shown to lie in the ISS disturbance class or the theorem statement must be restricted to plants and weights satisfying the exact identity.

    Authors: We appreciate the point. Under the exact cancellation condition B W = I that we will now state, there is no residual state-dependent jump term; the mapping to the analogue system is exact. The specific disturbances arise only from the neuronal leak and the event-based sampling; these are bounded and exogenous to the ISS system. The global practical asymptotic stability result therefore holds without further restrictions. In revision we will update the theorem statement, assumptions, and proof to reference the B W = I condition explicitly and confirm the disturbances remain in the required class. revision: yes

Circularity Check

0 steps flagged

No circularity: stability proofs rest on external ISS assumption after linear cancellation reduction

full rationale

The derivation begins from the stated observation that plant and neuronal discontinuities cancel under linear combination, yielding an exact correspondence to an analogue closed-loop system plus bounded disturbances. Global practical asymptotic stability (nonlinear case) and exponential stability (linear case) are then concluded directly from the external assumption that the analogue system is ISS or stable. This is a standard reduction to a known property of the continuous-time counterpart; the cancellation identity is presented as an algebraic fact enabling the rewrite rather than a fitted or self-defined quantity. No parameter is tuned to event-based data and then relabeled as a prediction, no self-citation chain carries the central claim, and no ansatz is smuggled. The logic is therefore self-contained against the external benchmark of ISS/stability and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the linear cancellation identity between plant and neuron discontinuities plus an external ISS or stability assumption on the analogue closed loop. No free parameters are introduced to fit data, and no new physical entities are postulated.

axioms (2)
  • domain assumption The combined plant-neuron discontinuities cancel linearly, producing an equivalent continuous closed-loop system.
    Invoked in the abstract to enable direct correspondence to the analogue controller.
  • domain assumption The analogue closed-loop system is input-to-state stable (or exponentially stable in the linear case) with respect to the disturbances induced by the event-based implementation.
    Stated explicitly as the condition under which global practical asymptotic (or exponential) stability follows.

pith-pipeline@v0.9.0 · 5712 in / 1415 out tokens · 25316 ms · 2026-05-22T12:43:48.511941+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Neuromorphic electronic systems,

    C. Mead, “Neuromorphic electronic systems,”Proceedings of the IEEE, vol. 78, no. 10, pp. 1629–1636, 1990

    work page 1990

  2. [2]

    Mead and M

    C. Mead and M. Ismail,Analog VLSI implementation of neural systems. Springer Science & Business Media, 2012, vol. 80

    work page 2012

  3. [3]

    A simple neuron servo,

    S. DeWeerth, L. Nielsen, C. Mead, and K. Astr ¨om, “A simple neuron servo,”IEEE Transactions on Neural Networks, vol. 2, no. 2, pp. 248– 251, 1991

    work page 1991

  4. [4]

    A simple event-based PID controller,

    K.-E. ˚Aarz´en, “A simple event-based PID controller,”IFAC Proceed- ings Volumes, vol. 32, no. 2, pp. 8687–8692, 1999

    work page 1999

  5. [5]

    Comparison of Riemann and Lebesgue sampling for first order stochastic systems,

    K. Astr ¨om and B. Bernhardsson, “Comparison of Riemann and Lebesgue sampling for first order stochastic systems,” inProceedings of the 41st IEEE Conference on Decision and Control, 2002., vol. 2, 2002, pp. 2011–2016 vol.2

    work page 2002

  6. [6]

    An introduction to event-triggered and self-triggered control,

    W. Heemels, K. Johansson, and P. Tabuada, “An introduction to event-triggered and self-triggered control,” in2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012, pp. 3270–3285

    work page 2012

  7. [7]

    Consistent event-triggered methods for linear quadratic control,

    D. J. Antunes and B. A. Khashooei, “Consistent event-triggered methods for linear quadratic control,” in2016 IEEE 55th Conference on Decision and Control (CDC), 2016, pp. 1358–1363

    work page 2016

  8. [8]

    Networked event-triggered control: an introduction and research trends,

    M. S. Mahmoud and M. Sabih, “Networked event-triggered control: an introduction and research trends,”International Journal of General Systems, vol. 43, no. 8, pp. 810–827, 2014

    work page 2014

  9. [9]

    Rieke, D

    F. Rieke, D. Warland, R. d. R. Van Steveninck, and W. Bialek,Spikes: exploring the neural code. MIT press, 1999

    work page 1999

  10. [10]

    Perception as a closed-loop convergence process,

    E. Ahissar and E. Assa, “Perception as a closed-loop convergence process,”eLife, vol. 5, p. e12830, 2016

    work page 2016

  11. [11]

    Beyond the computer metaphor: Behaviour as interaction,

    P. Cisek, “Beyond the computer metaphor: Behaviour as interaction,” Journal of Consciousness Studies, vol. 6, no. 11-12, pp. 125–142, 1999

    work page 1999

  12. [12]

    The neuron as a direct data-driven controller,

    J. J. Moore, A. Genkin, M. Tournoy, J. L. Pughe-Sanford, R. R. de Ruyter van Steveninck, and D. B. Chklovskii, “The neuron as a direct data-driven controller,”Proceedings of the National Academy of Sciences, vol. 121, no. 27, p. e2311893121, 2024

    work page 2024

  13. [13]

    Event-triggered real-time scheduling of stabilizing control tasks,

    P. Tabuada, “Event-triggered real-time scheduling of stabilizing control tasks,”IEEE Transactions on Automatic Control, vol. 52, no. 9, pp. 1680–1685, 2007

    work page 2007

  14. [14]

    Event-based control: A bibliometric analysis of twenty years of research,

    E. Aranda-Escol ´astico, M. Guinaldo, R. Heradio, J. Chacon, H. Vargas, J. S ´anchez, and S. Dormido, “Event-based control: A bibliometric analysis of twenty years of research,”IEEE Access, vol. 8, pp. 47 188– 47 208, 2020

    work page 2020

  15. [15]

    Goebel, R

    R. Goebel, R. G. Sanfelice, and A. R. Teel,Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press, 2012

    work page 2012

  16. [16]

    W. M. Haddad, V . Chellaboina, and S. G. Nersesov, Eds.,Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, 2014

    work page 2014

  17. [17]

    Output-based event- triggered control with guaranteedL ∞-gain and improved and decen- tralized event-triggering,

    M. C. F. Donkers and W. P. M. H. Heemels, “Output-based event- triggered control with guaranteedL ∞-gain and improved and decen- tralized event-triggering,”IEEE Transactions on Automatic Control, vol. 57, no. 6, pp. 1362–1376, 2012

    work page 2012

  18. [18]

    Periodic event-triggered control for linear systems,

    W. P. M. H. Heemels, M. C. F. Donkers, and A. R. Teel, “Periodic event-triggered control for linear systems,”IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 847–861, 2013

    work page 2013

  19. [19]

    A unifying Lyapunov-based framework for the event-triggered control of nonlin- ear systems,

    R. Postoyan, A. Anta, D. Ne ˇsi´c, and P. Tabuada, “A unifying Lyapunov-based framework for the event-triggered control of nonlin- ear systems,” in2011 50th IEEE Conference on Decision and Control and European Control Conference, 2011, pp. 2559–2564

    work page 2011

  20. [20]

    Analysis of event-driven controllers for linear systems,

    W. P. M. H. Heemels, J. H. Sandee, and P. P. J. V . D. Bosch, “Analysis of event-driven controllers for linear systems,”International Journal of Control, vol. 81, no. 4, pp. 571–590, 2008

    work page 2008

  21. [21]

    Yang,Impulsive Control Theory

    T. Yang,Impulsive Control Theory. Springer Berlin Heidelberg, 2001

    work page 2001

  22. [22]

    Optimal sampling and performance compari- son of periodic and event based impulse control,

    X. Meng and T. Chen, “Optimal sampling and performance compari- son of periodic and event based impulse control,”IEEE Transactions on Automatic Control, vol. 57, no. 12, pp. 3252–3259, 2012

    work page 2012

  23. [23]

    Analysis of a simple neuromorphic controller for linear systems: A hybrid systems perspective,

    E. Petri, K. J. Scheres, E. Steur, and W. Heemels, “Analysis of a simple neuromorphic controller for linear systems: A hybrid systems perspective,” in2024 IEEE 63rd Conference on Decision and Control (CDC). IEEE, 2024, pp. 8578–8583

    work page 2024

  24. [24]

    Analysis and design of event- triggered control algorithms using hybrid systems tools,

    J. Chai, P. Casau, and R. G. Sanfelice, “Analysis and design of event- triggered control algorithms using hybrid systems tools,” in2017 IEEE 56th Annual Conference on Decision and Control (CDC), 2017, pp. 6057–6062

    work page 2017

  25. [25]

    Lyapunov stability for impulsive systems via event-triggered impulsive control,

    X. Li, D. Peng, and J. Cao, “Lyapunov stability for impulsive systems via event-triggered impulsive control,”IEEE Transactions on Auto- matic Control, vol. 65, no. 11, pp. 4908–4913, 2020

    work page 2020

  26. [26]

    Spiking neurons as predictive controllers of linear systems,

    P. Agliati, A. Urbano, P. Lanillos, N. Ahmad, M. van Gerven, and S. Keemink, “Spiking neurons as predictive controllers of linear systems,”arXiv preprint arXiv:2507.16495, 2025

    work page arXiv 2025

  27. [27]

    Closed-form control with spike coding networks,

    F. S. Slijkhuis, S. W. Keemink, and P. Lanillos, “Closed-form control with spike coding networks,”IEEE Transactions on Cognitive and Developmental Systems, 2023

    work page 2023

  28. [28]

    Dynamical spiking networks for distributed control of nonlinear systems,

    F. Huang and S. Ching, “Dynamical spiking networks for distributed control of nonlinear systems,” in2018 Annual American Control Conference (ACC), 2018, pp. 1190–1195

    work page 2018

  29. [29]

    Distributed event-triggered control for multi-agent systems,

    D. V . Dimarogonas, E. Frazzoli, and K. H. Johansson, “Distributed event-triggered control for multi-agent systems,”IEEE Transactions on Automatic Control, vol. 57, no. 5, pp. 1291–1297, 2012

    work page 2012

  30. [30]

    Event-triggering in distributed net- worked control systems,

    X. Wang and M. D. Lemmon, “Event-triggering in distributed net- worked control systems,”IEEE Transactions on Automatic Control, vol. 56, no. 3, pp. 586–601, 2011

    work page 2011

  31. [31]

    Level crossing sampling in feedback stabilization under data-rate constraints,

    E. Kofman and J. H. Braslavsky, “Level crossing sampling in feedback stabilization under data-rate constraints,” inProceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 4423–4428

    work page 2006

  32. [32]

    Gerstner and W

    W. Gerstner and W. M. Kistler,Spiking neuron models: Single neurons, populations, plasticity. Cambridge university press, 2002

    work page 2002

  33. [33]

    H. K. Khalil and J. W. Grizzle,Nonlinear systems. Prentice hall Upper Saddle River, NJ, 2002, vol. 3

    work page 2002

  34. [34]

    E. D. Sontag,Input to State Stability: Basic Concepts and Results. Berlin, Heidelberg: Springer, 2008, pp. 163–220

    work page 2008