Francesco Bullo

In this paper, we provide a novel contraction-theoretic approach to analyze two-time scale systems, including those commonly encountered in online feedback optimization (OFO). Our framework endows these systems with several robustness properties, enabling a more comprehensive characterization of their behaviors. The primary assumptions are the contractivity of the fast subsystem and the reduced model, along with an explicit upper bound on the time-scale parameter. For two-time scale systems subject to disturbances, we show that the distance between solutions of the nominal system and solutions of its reduced model is uniformly upper bounded by a function of contraction rates, Lipschitz constants, the time-scale parameter, and the variability of the disturbances over time. Applying these general results to the OFO context, we establish new individual tracking error bounds, showing that solutions converge to their time-varying optimizer, provided the plant and steady-state feedback controller exhibit contractivity and the controller gain is suitably bounded. Finally, we explore two special cases: for autonomous nonlinear systems, we derive sharper bounds than those in the general results, and for linear time-invariant systems, we present novel bounds based on induced matrix norms and induced matrix log norms.

Chronic pain presents a leading challenge in the world today for both clinicians and researchers. Because chronic pain is difficult to explain and treat, it is often managed with opioids despite providing limited relief and contributing to dependence and misuse. Persistent pain can be maintained by altered central nervous system processing even in the absence of distinct tissue damage or disease, which may limit the efficacy of conventional pharmacological therapies that target nociceptive signal transmission rather than maladaptive central nervous system dynamics often present in those with chronic pain. Although neuroimaging studies have identified this shift from nociceptive to emotional circuits during pain chronification, a quantitative framework linking these neural changes to longitudinal pain trajectories or recovery is lacking. We present a parsimonious firing-rate model that can account for the development of and recovery from chronic pain, which is based on the theoretical framework established by Wilson and Cowan. The model provides a quantitative explanation of how sensitization, anxiety, and fear maintain pain even after an injury has healed, and how calming stimulus downregulates these processes to facilitate recovery. A study applying the same principles as the model produced an average pain decrease of 3.5 on the Visual Analog Scale (VAS), with all subjects experiencing a reduction in pain. These results, coupled with our model and findings in prior studies, suggest that increasing calming stimulus can reduce pain without necessitating pharmacological or invasive, resource-intensive interventions.

A recent breakthrough in biologically plausible normative frameworks for dimensionality reduction is based on the similarity matching cost function and the low-rank matrix approximation problem. Despite clear biological interpretation, successful application in several domains, and experimental validation, a formal complete convergence analysis remains elusive. Building on this framework, we consider and analyze a continuous-time neural network, the similarity matching network, for principal subspace projection. Derived from a min-max-min objective, this biologically plausible network consists of three coupled dynamics evolving at different timescales: neural dynamics, lateral synaptic dynamics, and feedforward synaptic dynamics at the fast, intermediate, and slow timescales, respectively. The feedforward and lateral synaptic dynamics consist of Hebbian and anti-Hebbian learning rules, respectively. By leveraging a multilevel optimization framework, we prove convergence of the dynamics in the offline setting. Specifically, at the first level (fast timescale), we show strong convexity of the cost function and global exponential convergence of the corresponding gradient-flow dynamics. At the second level (intermediate timescale), we prove strong concavity of the cost function and exponential convergence of the corresponding gradient-flow dynamics within the space of positive definite matrices. At the third and final level (slow timescale), we study a nonconvex and nonsmooth cost function, provide explicit expressions for its global minima, and prove almost sure convergence of the corresponding gradient-flow dynamics to the global minima. These results rely on two empirically motivated conjectures that are supported by thorough numerical experiments. Finally, we validate the effectiveness of our approach via a numerical example.

Recent studies on stability and contractivity have highlighted the importance of semi-inner products, which we refer to as “pairings,” associated with general norms. A pairing is a binary operation that relates the derivative of a curve’s norm to the radius vector of the curve and its tangent. This relationship, known as the curve norm derivative formula, is crucial when using the norm as a Lyapunov function. Another important property of the pairing, used in stability and contraction criteria, is the so-called Lumer inequality, which relates the pairing to the induced logarithmic norm. We prove that the curve norm derivative formula and Lumer’s inequality are, in fact, equivalent to each other and to several simpler properties. We then introduce and characterize regular pairings that satisfy all of these properties. Our results unify several independent theories of pairings (semi-inner products) developed in previous work on functional analysis and control theory. Additionally, we introduce the polyhedral max pairing and develop computational tools for polyhedral norms, advancing contraction theory in non-Euclidean spaces.

In competitive resource allocation formulations multiple agents compete over different contests by committing their limited resources in them. For these settings, contest games offer a game-theoretic foundation to analyze how players can efficiently invest their resources. In this class of games the resulting behavior can be affected by external interactions among the players. In particular, players may be able to make coalitions that allow transferring resources among them, seeking to improve their outcome. In this work, we study bilateral budgetary transfers in contest games played over networks. Particularly, we characterize the family of networks where there exist mutually beneficial bilateral transfers for some set of systems parameters. With this in mind, we provide sufficient conditions for the existence of mutually beneficial transfers. Moreover, we provide a constructive argument that guarantees that the benefit of making coalitions only depends on mild connectivity conditions of the graph structure. Lastly, we provide a characterization of the improvement of the utilities as a function of the transferred budget. Further, we demonstrate how gradient-based dynamics can be utilized to find desirable coalitional structures. Interestingly, our findings demonstrate that such collaborative opportunities extend well beyond the typical “enemy-of-my-enemy” alliances.

Firing rate models are dynamical systems widely used in applied and theoretical neuroscience to describe local cortical dynamics in neuronal populations. By providing a macroscopic perspective of neuronal activity, these models are essential for investigating oscillatory phenomena, chaotic behavior, and associative memory processes. Despite their widespread use, the application of firing rate models to associative memory networks has received limited mathematical exploration, and most existing studies are focused on specific models. Conversely, well-established associative memory designs, such as Hopfield networks, lack key biologically relevant features intrinsic to firing rate models, including positivity and interpretable synaptic matrices reflecting the action of long-term potentiation and long-term depression. To address this gap, we propose a general framework that ensures the emergence of rescaled memory patterns as stable equilibria in the firing rate dynamics. Furthermore, we analyze the conditions under which the memories are locally and globally asymptotically stable, providing insights into constructing biologically plausible and robust systems for associative memory retrieval.

We derive a new sufficient condition for 2-contraction of a feedback interconnection of two nonlinear dynamical systems. Our condition is based on, first, deriving new formulas for the 2-multiplicative [2-additive] compound of block matrices using block Kronecker products [sums], second, a hierarchical approach for proving standard contraction, and third, a network small-gain theorem for Metzler matrices. We demonstrate our results by deriving a simple sufficient condition for 2-contraction in a network of FitzHugh–Nagumo neurons.