arXiv subjects

Rodolphe Sepulchre

Publications and source records attributed to Rodolphe Sepulchre.

At least 19 recordsLinked to original sources

arXiv2026

State-space fading memory

The fading-memory (FM) property captures the progressive loss of influence of past inputs on a system's current output and has originally been formalized by Boyd and Chua in an operator-theoretic framework. Despite its importance for systems approximation, reservoir computing, and recurrent neural networks, its connection with state-space notions of nonlinear stability, especially incremental ones, remains understudied. This paper introduces a state-space definition of FM. In state-space, FM can be interpreted as an extension of incremental input-to-output stability ($\delta$IOS) that explicitly incorporates a memory kernel upper-bounding the decay of past input differences. It is also closely related to Boyd and Chua's FM definition, with the sole difference of requiring uniform, instead of general, continuity of the memory functional with respect to an input-fading norm. We demonstrate that incremental input-to-state stability ($\delta$ISS) implies FM semi-globally for time-invariant systems under an equibounded input assumption. Notably, Boyd and Chua's approximation theorems apply to $\delta$ISS state-space models. As a closing application, we show that, under mild assumptions, the state-space model of a current-driven memristive device possesses the FM property.

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arXiv2025

A Neuromodulable Current-Mode Silicon Neuron for Robust and Adaptive Neuromorphic Systems

Neuromorphic engineering makes use of mixed-signal analog and digital circuits to directly emulate the computational principles of biological brains. Such electronic systems offer a high degree of adaptability, robustness, and energy efficiency across a wide range of tasks, from edge computing to robotics. Within this context, we investigate a key feature of biological neurons: their ability to carry out robust and reliable computation by adapting their input responses and spiking patterns to context through neuromodulation. Achieving analogous levels of robustness and adaptation in neuromorphic circuits through modulatory mechanisms is a largely unexplored path. We present a novel current-mode neuron design that supports robust neuromodulation with minimal model complexity, compatible with standard CMOS technologies. We first introduce a mathematical model of the circuit and provide tools to analyze and tune the neuron behavior; we then demonstrate both theoretically and experimentally the biologically plausible neuromodulation adaptation capabilities of the circuit over a wide range of parameters. All theoretical predictions were verified in experiments on a low-power 180 nm CMOS implementation of the proposed neuron circuit. Due to the analog underlying feedback structure, the proposed adaptive neuromodulable neuron exhibits a high degree of robustness, flexibility, and scalability across operating ranges of currents and temperatures, making it a perfect candidate for real-world neuromorphic applications.

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arXiv2025

Describing Functions and Phase Response Curves of Excitable Systems

The describing function (DF) and phase response curve (PRC) are classical tools for the analysis of feedback oscillations and rhythmic behaviors, widely used across control engineering, biology, and neuroscience. These tools are known to have limitations in networks of relaxation oscillators and excitable systems. For this reason, the paper proposes a novel approach tailored to excitable systems. Our analysis focuses on the discrete-event operator mapping input trains of events to output trains of events. The methodology is illustrated on the excitability model of Hodgkin-Huxley. The proposed framework provides a basis for designing and analyzing central pattern generators in networks of excitable neurons, with direct relevance to neuromorphic control and neurophysiology.

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arXiv2025

On the Contraction of Excitable Systems

We study the contraction of Hodgkin-Huxley model and its role in the reliability of spike timings. Without input, the model is contractive in the region of physiological interest. With impulsive synaptic inputs, contraction is retained provided that the input events are sparse enough. Contraction is lost when the input firing rate is too high. Spike timings are shown to be reliable in the contracting regime.

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arXiv2025

Discrete Event System Modeling of Neuromorphic Circuits

Excitable neuromorphic circuits are physical models of event behaviors: their continuous-time trajectories consist of sequences of discrete events. This paper explores the possibility of extracting a discrete-event model out of the physical continuous-time model. We discuss the potential of this methodology for analysis and design of neuromorphic control systems.

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arXiv2025

A Neuromorphic Architecture for Scalable Event-Based Control

This paper introduces the ``rebound Winner-Take-All (RWTA)" motif as the basic element of a scalable neuromorphic control architecture. From the cellular level to the system level, the resulting architecture combines the reliability of discrete computation and the tunability of continuous regulation: it inherits the discrete computation capabilities of winner-take-all state machines and the continuous tuning capabilities of excitable biophysical circuits. The proposed event-based framework addresses continuous rhythmic generation and discrete decision-making in a unified physical modeling language. We illustrate the versatility, robustness, and modularity of the architecture through the nervous system design of a snake robot.

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arXiv2025

Passive and reciprocal linear time-and-space-invariant systems

Reciprocity is a fundamental symmetry property observed across many physical domains, including acoustics, elasticity, electromagnetics, and thermodynamics. In systems and control theory, it provides key insights into the internal structure of linear time-invariant (LTI) systems and is closely linked to properties such as passivity, relaxation, and time-reversibility. This paper extends the concept of reciprocity to linear time-and-space-invariant (LTSI) systems, a class of infinite-dimensional systems with spatio-temporal dynamics. It is suggested that, analogously to the LTI case, combining the internal properties of reciprocity and (impedance) passivity entails physical state-space realizations. This is of particular relevance for infinite-dimensional systems, where issues of unboundedness can be detrimental to the well-posedness of the system. The results are motivated and illustrated with a physical example.

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arXiv2025

Spiking control systems for soft robotics: a rhythmic case study in a soft robotic crawler

Inspired by spiking neural feedback, we propose a spiking controller to engineer the locomotion of a soft robotic crawler. Its bistability, akin to neural fast positive feedback, combined with a sensorimotor slow negative feedback loop, generates rhythmic spiking and self-sustained peristaltic locomotion. We analytically characterize the local equilibrium bifurcations induced by the sensorimotor gain and complement this analysis through numerical continuation of periodic orbits. The resulting bifurcation diagram reveals the emergence of qualitatively distinct solutions, including resting and crawling behaviors. For a representative regime with separated mechanical and electrical timescales, Geometric Singular Perturbation Theory reveals the geometry of the relaxation oscillations leading to endogenous crawling. Within this singularly perturbed regime, we formulate and analytically solve an optimization problem, proving that locomotion speed is maximized at mechanical resonance through a matching of neuromechanical scales. Given the importance and ubiquity of rhythms and waves in soft-bodied locomotion, we envision that spiking control systems could be utilized in a variety of soft-robotic morphologies and modular distributed architectures.

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arXiv2025

A memristive model of spatio-temporal excitability

This paper introduces a model of excitability that unifies the mechanism of an important neuronal property both in time and in space. As a starting point, we revisit both a key model of temporal excitability, proposed by Hodgkin and Huxley, and a key model of spatial excitability, proposed by Amari. We then propose a novel model that captures the temporal and spatial properties of both models. Our aim is to regard neuronal excitability as a property across scales, and to explore the benefits of modeling excitability with one and the same mechanism, whether at the cellular or the population level.

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arXiv2025

Operator-Splitting Methods for Neuromorphic Circuit Simulation

A novel splitting algorithm is proposed for the numerical simulation of neuromorphic circuits. The algorithm is grounded in the operator-theoretic concept of monotonicity, which bears both physical and algorithmic significance. The splitting exploits this correspondence to translate the circuit architecture into the algorithmic architecture. The paper illustrates the many advantages of the proposed operator-theoretic framework over conventional numerical integration for the simulation of multiscale hierarchical events that characterize neuromorphic behaviors.

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arXiv2025

Event disturbance rejection: a case study

This article introduces the problem of robust event disturbance rejection. Inspired by the design principle of linear output regulation, a control structure based on excitable systems is proposed. Unlike the linear case, contraction of the closed-loop system must be enforced through specific input signals. This induced contraction enables a steady-state analysis similar to the linear case. Thanks to the excitable nature of the systems, the focus shifts from precise trajectory tracking to the regulation of discrete events, such as spikes. The study emphasizes rejecting events rather than trajectories and demonstrates the robustness of the approach, even under mismatches between the controller and the exosystem. This work is a first step towards developing a design principle for event regulation.

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arXiv2025

Regulation without calibration

This article revisits the importance of the internal model principle in the literature of regulation and synchronization. Trajectory regulation, the task of regulating continuous-time signals generated by differential equations, is contrasted with event regulation, the task of only regulating discrete events associated with the trajectories. In trajectory regulation, the internal model principle requires an exact internal generator of the continuous-time trajectories, which translates into unrealistic calibration requirements. Event regulation is envisioned as a way to relieve calibration of the continuous behavior while ensuring reliability of the discrete events.

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arXiv2025

Graphical Dominance Analysis for Linear Systems: A Frequency-Domain Approach

We propose a frequency-domain approach to dominance analysis for multi-input multi-output (MIMO) linear time-invariant systems. The dominance of a MIMO system is defined to be the number of its poles in the open right half-plane. Our approach is graphical: we define a frequency-wise notion of the recently-introduced scaled graph of a MIMO system plotted in a complex plane. The scaled graph provides a bound of the eigenloci of the system, which can be viewed as a robust MIMO extension of the classical Nyquist plot. Our main results characterize sufficient conditions for quantifying the dominance of a closed-loop system based upon separation of scaled graphs of two open-loop systems in a frequency-wise manner. The results reconcile existing small gain, small phase and passivity theorems for feedback dominance analysis.

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arXiv2025

Soft and Hard Scaled Relative Graphs for Nonlinear Feedback Stability

This article presents input-output stability analysis of nonlinear feedback systems based on the notion of soft and hard scaled relative graphs (SRGs). The soft and hard SRGs acknowledge the distinction between incremental positivity and incremental passivity and reconcile them from a graphical perspective. The essence of our proposed analysis is that the separation of soft SRGs or hard SRGs of two open-loop systems on the complex plane guarantees closed-loop stability. The main results generalize an existing soft SRG separation theorem for bounded open-loop systems which was proved based on interconnection properties of soft SRGs under a chordal assumption. By comparison, our analysis does not require this chordal assumption and applies to possibly unbounded open-loop systems based on their hard SRGs.

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arXiv2025

A Winner-Takes-All Mechanism for Event Generation

We present a novel framework for central pattern generator design that leverages the intrinsic rebound excitability of neurons in combination with winner-takes-all computation. Our approach unifies decision-making and rhythmic pattern generation within a simple yet powerful network architecture that employs all-to-all inhibitory connections enhanced by designable excitatory interactions. This design offers significant advantages regarding ease of implementation, adaptability, and robustness. We demonstrate its efficacy through a ring oscillator model, which exhibits adaptive phase and frequency modulation, making the framework particularly promising for applications in neuromorphic systems and robotics.

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arXiv2025

Gradient modelling of memristive systems

We introduce a gradient modeling framework for memristive systems. Our focus is on memristive systems as they appear in neurophysiology and neuromorphic systems. Revisiting the original definition of Chua, we regard memristive elements as gradient operators of quadratic functionals with respect to a metric determined by the memristance. We explore the consequences of gradient properties for the analysis and design of neuromorphic circuits.

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arXiv2025

Variable Metric Splitting Methods for Neuromorphic Circuits Simulation

This paper proposes a variable metric splitting algorithm to solve the electrical behavior of neuromorphic circuits made of capacitors, memristive elements, and batteries. The gradient property of the memristive elements is exploited to split the current to voltage operator as the sum of the derivative operator, a Riemannian gradient operator, and a nonlinear residual operator that is linearized at each step of the algorithm. The diagonal structure of the three operators makes the variable metric forward-backward splitting algorithm scalable and amenable to the simulation of large-scale neuromorphic circuits.

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arXiv2025

Linear time-and-space-invariant relaxation systems

This paper generalizes the physical property of relaxation from linear time-invariant (LTI) to linear time-and-space-invariant (LTSI) systems. It is shown that the defining features of relaxation -- complete monotonicity, passivity, and memory-based storage -- carry over seamlessly to the spatio-temporal domain. An LTSI system is shown to be of relaxation type if and only if its associated spatio-temporal Hankel operator is cyclically monotone. This implies the existence of an intrinsic quadratic storage functional defined uniquely by past inputs, independently of any state-space realization. As in the LTI case, LTSI relaxation systems are shown to be those systems for which the state-space concept of storage coincides with the input-output concept of fading memory functional.

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