From nonlinear dynamics to proof of concept for Heteroclinic Computing

• Applicant: Professor Dr. Marc Timme
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 419424741

Heteroclinic dynamics naturally emerges in coupled nonlinear dynamical systems, in particular in networks of phase-coupled and pulse-coupled oscillators. As a dynamical phenomenon it is reasonably well understood mathematically. Heteroclinic networks, a collection of saddle states linked via heteroclinic connections, form the skeleton determining much of the collective dynamics in a range of systems. Dynamics near heteroclinic networks has been proposed to provide computing mechanisms in biological and bio-inspired systems and to offer highly effective, universal computational features. Heteroclinic networks also enable a new computing framework that is independent of specific implementations, such as networks of phase-coupled or pulse-coupled oscillators or competitive Lotka-Volterra-like systems. The core computational principles of encoding are reasonably understood. In particular, the direction of the driving signal vector acting on a state near a saddle determines the direction a trajectory leaves that saddle and thus the next saddle approached by the trajectory. On longer time scales, the driving signal thereby determines the sequence of saddles and reversely, the sequence of saddles reveals information about specific properties of the driving signal, i.e. a computation has been performed on that input signal. Here, signals with components in the same partial rank order are associated with each other and yield the same computational result, i.e. the computation is robust in this sense. Recent works also demonstrated how noise affects reliable switching in both phase-coupled and pulse-coupled oscillator systems, thereby offering hints into computational reliability and providing first insights about how heteroclinic computing systems may actually behave under noisy, real world conditions. Heteroclinic computing, however, has so far not been demonstrated in any real device and it remains an open question how exactly to realize it.In the proposed project, we plan to address three remaining key questions. One about efficient decoding and its interplay with encoding, one about how to realize intrinsic, self-organized memory in suitable coupled oscillator systems, and one about identifying potential substrates, architectures and implementation techniques to provide a proof of concept for heteroclinic computers in hardware. We will combine known properties about coupled oscillator networks, especially the theory of pulse-coupled oscillators and generally the theory of coupled dynamical systems, with ideas from neural network theory to address these questions. A successful study would not only yield new insights into theoretical aspects of the nonlinear dynamics of the heteroclinic computing paradigm but also provide crucial steps towards a new form of robust analogue computers, creating a bridge from the mathematical level of nonlinear dynamics and the conceptual level of the computing idea towards options for the technological level.

• DFG Programme: Research Grants