We didn't tell it about the two wings of the butterfly.
It found them anyway.
I-Field RNN trained on Lorenz reconstruction , the learned phase space spontaneously mirrors the attractor's bistable topology. And the I-field drops before every regime switch. Critical slowing down. Emergent.
Preprint: https://doi.org/10.5281/zenodo.20767066
This is what happens when your recurrent unit has a phase space.
#DynamicalSystems #MachineLearning #Chaos #Neuroscience #IField #Bifurcation
Most RNNs are black boxes. This one isn't.
I built a recurrent unit with a defined phase space — bistability, hysteresis, and limit cycles emerge naturally from the equations. No gates. No black magic. Just nonlinear dynamics.
The I‑Field RNN: memory of its own activity.
📄 https://doi.org/10.5281/zenodo.20767066
#MachineLearning #DynamicalSystems #Neuroscience #RNN #Bifurcation
Most recurrent neural networks rely on high‑dimensional hidden states that lack an explicit dynamical interpretation and cannot autonomously generate multistability, hysteresis, or sustained oscillations. We introduce the I‑Field RNN, a low‑dimensional recurrent unit derived from a biophysical conductance‑based neuron model through adiabatic elimination. The unit is governed by two interpretable variables: a fast synaptic current and a slow internal field with learnable refractory feedback. This design yields a well‑defined phase space with several emergent properties. For a range of the refractory coefficient, the system undergoes a saddle‑node bifurcation, creating bistability and hysteresis; a brief input pulse permanently switches the unit between attractors, producing a transparent memory switch. With inhibitory feedback, a supercritical Hopf bifurcation ($l_1 = -7.38 < 0$) generates sustained limit cycles. We validate the model on synthetic binary event streams with a known refractory period, using a context window shorter than the refractory period to force reliance on the internal field. The field decay rate $\alpha_I$ shows a sharp two‑regime adaptation: for short refractory periods it increases significantly to accelerate the field decay, while for long periods it remains at its biophysically motivated slow initialisation ($p = 0.0013$, t‑test). The refractory coefficient $\rho$ provides complementary modulation. These results confirm that the unit actively uses its structural memory rather than relying solely on opaque recurrent weights. The I‑Field RNN bridges dynamical systems theory and differentiable computation, offering a transparent, multistable, and oscillatory alternative to black‑box recurrent architectures.
Many thanks to the Julius Maximilians Universität Würzburg and Leon Bungert for hosting #MSCA researcher Albert Alcalde for two months. It lead to another #ModConFlex paper:
Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime
by Albert Alcalde, Leon Bungert, Konstantin Riedl, Tim Roith
➡️ https://arxiv.org/abs/2605.10931
RE: https://mathstodon.xyz/@DurstewitzLab/116549716016889895
🧠 New preprint by Brändle et al./ @DurstewitzLab: Continuous-Time Piecewise-Linear #RecurrentNeuralNetworks introduces continuous-time #PLRNNs for #DynamicalSystems reconstruction.
The model combines interpretability and analytical tractability of pw-linear #RNN with cont.-time dynamics, allowing semi-analytic analysis of equilibria and limit cycles while handling irregularly sampled data better than standard Neural #ODEs.
#ModConFlex researcher Mario Sinano co-authored
"Data-Driven Parametric Aeroelastic Modeling of the Pazy Wing", AIAA SciTech 2026.
developing physics-informed machine learning models that learn accurate, simplified representations of highly flexible aircraft wings from very small amounts of simulation data, capturing how system trajectories and stability evolve across different operating conditions.
https://arc.aiaa.org/doi/10.2514/6.2026-0187
#physics_informed #machineLearning #dynamicalsystems #aerospace
New paper, with Zerong Guo, Zonghua Liu, Shuguang Guan, Stefano Boccaletti and Jie Zhou.
Everyone knows about chimera states. We show a new mechanism for the emergence of chimeras that is specific of higher-order interactions. Interestingly, this mechanism is somewhat analogous to what happens in some proteins with intrinsic disorder (which we showed a couple of years ago), so we called it intrinsic frustration.
#mathematics #physics #science #HigherOrderNetworks #synchronization #DynamicalSystems #ChimeraStates #chimeras #frustration #IntrinsicDisorder
New paper, with Kirill Kovalenko, Gonzalo Contreras, Stefano Boccaletti and Rubén Sánchez.
People have noticed that, in higher-order networks, synchronization is often explosive, and that cluster synchronization happens very rarely, if ever. We explain why, by showing that symultaneous dynamical equitability across layers or interaction orders is necessary and sufficient for cluster synchronization, except if the coupling functions depend linearly on each other. Since the probability of randomly satisfying this condition is exceedingly low, cluster synchronization is precluded in such networks.
https://www.nature.com/articles/s42005-026-02543-5
#mathematics #physics #science #networks #complexity #HigherOrderNetworks #MultiplexNetworks #synchronization #dynamicalsystems #graphs #graphtheory #equitability #ClusterSynchronization #ExplosiveSynchronization
This document presents a fully specified, pre-registered empirical validation pipeline for testing the CRTI (Compression–Response/Resonance Thermodynamic Index) framework on a canonical ecological regime-shift dataset: the Peter Lake whole-ecosystem manipulation experiment (Carpenter et al., 2011). The pipeline defines a reproducible workflow for constructing the state variables R (adaptive response capacity) and \Phi (structural compression) from multivariate time-series data, and for computing the composite index T = R/\Phi. All preprocessing steps, parameter choices, windowing strategies, and statistical tests are fixed ex ante and may not be modified post hoc. The design is explicitly falsification-first. Primary and secondary hypotheses, as well as detailed failure criteria, are pre-specified and reported with equal prominence to positive outcomes. The document does not claim empirical validation of the CRTI framework; it defines a transparent and reproducible protocol for testing whether T carries early-warning information prior to a documented regime shift. This pipeline provides a methodological foundation for fair comparison between CRTI-based metrics and classical early-warning signals under identical conditions. early warning signals, regime shifts, ecological data, Peter Lake, pre-registration, reproducibility, falsification, time series analysis, dynamical systems, complex adaptive systems, structural compression, adaptive capacity, Kendall tau, covariance analysis, CRTI, critical transitions
This paper introduces a structural class of two-dimensional competitive adaptive systems (CRTI-class systems) defined by competitive coupling, self-limitation, and a compact invariant domain. Using Poincaré index theory and the Stable Manifold Theorem, we show that systems in this class admitting two boundary attractors necessarily contain an interior saddle point whose stable manifold partitions state space into two qualitatively distinct basins corresponding to adaptive and compression-dominated outcomes. We further define a scalar index T = R/\Phi, where R denotes adaptive response capacity and \Phi denotes structural compression, and prove that T acts as a local first-order basin classifier in a neighbourhood of the saddle, without constituting a geometric distance to the separatrix. Operational estimators for R and \Phi are derived from linear response theory and spectral covariance analysis, enabling empirical application to multivariate time series. High-dimensional reduction, thermodynamic interpretation, and cross-domain universality are explicitly identified as open problems. The framework provides a mathematically grounded substrate for regime-shift analysis while maintaining clear limits of validity. bistability, basin of attraction, separatrix, competitive dynamical systems, regime shifts, early warning signals, complex adaptive systems, structural compression, adaptive capacity, Poincaré index, stable manifold, critical transitions, dynamical systems theory, CRTI
MathiasTCK.bsky.social
FundamentalTime
ModConFlex
Fabrizio Musacchio
Charo del Genio
Bernd von Mallinckrodt 🦋