Feedback-induced self-oscillations in large interacting systems subjected to phase transitions

Abstract

In this article it is shown that large systems with many interacting units endowing multiple phases display self-oscillations in the presence of linear feedback between the control and order parameters, where an Andronov–Hopf bifurcation takes over the phase transition. This is simply illustrated through the mean field Landau theory whose feedback dynamics turn out to be described by the Van der Pol equation and it is then validated for the fully connected Ising model following heat bath dynamics. Despite its simplicity, this theory accounts potentially for a rich range of phenomena: here it is applied to describe in a stylized way (i) excess demand-price cycles due to strong herding in a simple agent-based market model; (ii) congestion waves in queuing networks triggered by user feedback to delays in overloaded conditions; and (iii) metabolic network oscillations resulting from cell growth control in a bistable phenotypic landscape.

Statistical modeling of adaptive neural networks explains coexistence of avalanches and oscillations in resting human brain

“…First, as in the standard Ising magnet, the interactions J can be restricted to simulate local excitatory connectivity, e.g., to nearest-neighbors on a 2D lattice. Second, feedback h i to neuron i could be derived from a local magnetization in a neighborhood around neuron i instead of the global magnetization; in the interesting limiting case where ḣi = −cs i , each neuron would feed back on its own past spiking history only, and the model would reduce to a set of coupled "binary oscillators" [27]. Irrespective of the exact setting, the model's mathematical attractiveness stems from its tractable interpolation between stochastic (spiking of excitatory units) and deterministic (feedback) elements.…”

“…Here we propose a minimal, microscopic, and analytically tractable model class that can capture a wide spectrum of emergent phenomena in brain dynamics, including neural oscillations, extreme event statistics, and scale-free neuronal avalanches [11]. These models are non-equilibrium extensions of the Ising model of statistical physics with an extra feedback loop which enables self-adaptation [27]. As a consequence of feedback, neuronal dynamics is driven by the ongoing network activity, generating a rich repertoire of dynamical behaviors.…”

Oscillations in feedback-driven systems: Thermodynamics and noise

“…3, the number of coherent oscillations depends on the system size and becomes indefinite in the thermodynamic limit. This feature has been demonstrated analytically for a model with continuous feedback [31].…”

“…Second, we consider the case β > ∼ 1. In the thermodynamic limit the dynamics of the system can be mapped into the equation for the Van der Pol oscillator [31] m + (β − 1 − m 2 )ṁ + √ cm = 0.…”

“…We then proceed to study a more complex system with several degrees of freedom, a fully interacting Ising model driven by a feedback scheme, which is discrete in time, between the external field and the magnetization. This is a generalization of the model with continuous feedback introduced in [31], which has been shown to display self-oscillations that persist in the thermodynamic limit below the static critical temperature.…”