Nacer Eddine Boukacem

arXiv2026

Sandscapes: self-modifying energy landscapes with emergent branching and flips

Energy landscapes provide a common framework for describing learning, embryonic development, and collective dynamics. Although such landscapes may evolve over time, their dynamics are typically prescribed externally rather than generated by the system itself. Here we get inspiration from biology to introduce sandscapes : self-modifying landscapes in which the motions of interacting agents continuously reshape the landscape that governs their own trajectories. We derive sandscapes from a minimal model of interacting Hopfield units, where the basins of each attractor are modulated by their occupancies. Sandscapes spontaneously generate sequential symmetry breaking and differentiation trees, with local branching described by coupled Ising dynamics. We then drive the dynamics of sandscapes (using local proliferation common in biology) and leverage catastrophe theory to show that sandscapes self-organize toward flip bifurcations, suggesting a generic mechanism for the emergence of ubiquitous binary cell-fate decisions. We further demonstrate that sandscapes can act as generative models of developmental trajectories : starting from terminal states alone, we reconstruct realistic hematopoietic differentiation trees with multiple layers of intermediate progenitor states. More broadly, our results identify sandscapes as a general principle of adaptive dynamics, explaining how feedback between agents and landscapes produces branching, criticality, and self-organization across learning and biology.

q-bio.QM↗

arXiv2023

A Waddington landscape for prototype learning in generalized Hopfield networks

Networks in machine learning offer examples of complex high-dimensional dynamical systems reminiscent of biological systems. Here, we study the learning dynamics of Generalized Hopfield networks, which permit a visualization of internal memories. These networks have been shown to proceed through a 'feature-to-prototype' transition, as the strength of network nonlinearity is increased, wherein the learned, or terminal, states of internal memories transition from mixed to pure states. Focusing on the prototype learning dynamics of the internal memories we observe a strong resemblance to the canalized, or low-dimensional, dynamics of cells as they differentiate within a Waddingtonian landscape. Dynamically, we demonstrate that learning in a Generalized Hopfield Network proceeds through sequential 'splits' in memory space. Furthermore, order of splitting is interpretable and reproducible. The dynamics between the splits are canalized in the Waddington sense -- robust to variations in detailed aspects of the system. In attempting to make the analogy a rigorous equivalence, we study smaller subsystems that exhibit similar properties to the full system. We combine analytical calculations with numerical simulations to study the dynamical emergence of the feature-to-prototype transition, and the behaviour of splits in the landscape, saddles points, visited during learning. We exhibit regimes where saddles appear and disappear through saddle-node bifurcations, qualitatively changing the distribution of learned memories as the strength of the nonlinearity is varied -- allowing us to systematically investigate the mechanisms that underlie the emergence of Waddingtonian dynamics. Memories can thus differentiate in a predictive and controlled way, revealing new bridges between experimental biology, dynamical systems theory, and machine learning.

cond-mat.dis-nn↗