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Towards model-based control of active matter: active nematics and oscillator networks
The richness of active matter's spatiotemporal patterns continues to capture our imagination. Shaping these emergent dynamics into pre-determined forms of our choosing is a grand challenge in the field. To complicate matters, multiple dynamical attractors can coexist in such systems, leading to initial condition-dependent dynamics. Consequently, non-trivial spatiotemporal inputs are generally needed to access these states. Optimal control theory provides a general framework for identifying such inputs and represents a promising computational tool for guiding experiments and interacting with various systems in soft active matter and biology. As an exemplar, I first consider an extensile active nematic fluid confined to a disk. In the absence of control, the system produces two topological defects that perpetually circulate. Optimal control identifies a time-varying active stress field that restructures the director field, flipping the system to its other attractor that rotates in the opposite direction. As a second, analogous case, I examine a small network of coupled Belousov-Zhabotinsky chemical oscillators that possesses two dominant attractors, two wave states of opposing chirality. Optimal control similarly achieves the task of attractor switching. I conclude with a few forward-looking remarks on how the same model-based control approach might come to bear on problems in biology.
A robust neural integrator based on the interactions of three time scales
Neural integrators are circuits that are able to code analog information such as spatial location or amplitude. Storing amplitude requires the network to have a large number of attractors. In classic models with recurrent excitation, such networks require very careful tuning to behave as integrators and are not robust to small mistuning of the recurrent weights. In this talk, I introduce a circuit with recurrent connectivity that is subjected to a slow subthreshold oscillation (such as the theta rhythm in the hippocampus). I show that such a network can robustly maintain many discrete attracting states. Furthermore, the firing rates of the neurons in these attracting states are much closer to those seen in recordings of animals. I show the mechanism for this can be explained by the instability regions of the Mathieu equation. I then extend the model in various ways and, for example, show that in a spatially distributed network, it is possible to code location and amplitude simultaneously. I show that the resulting mean field equations are equivalent to a certain discontinuous differential equation.