Noise-constrained switching times for heteroclinic computing.

Heteroclinic computing offers a novel paradigm for universal computation by collective system dynamics. In such a paradigm, input signals are encoded as complex periodic orbits approaching specific sequences of saddle states. Without inputs, the relevant states together with the heteroclinic connections between them form a network of states-the heteroclinic network. Systems of pulse-coupled oscillators or spiking neurons naturally exhibit such heteroclinic networks of saddles, thereby providing a substrate for general analog computations. Several challenges need to be resolved before it becomes possible to effectively realize heteroclinic computing in hardware. The time scales on which computations are performed crucially depend on the switching times between saddles, which in turn are jointly controlled by the system's intrinsic dynamics and the level of external and measurement noise. The nonlinear dynamics of pulse-coupled systems often strongly deviate from that of time-continuously coupled (e.g., phase-coupled) systems. The factors impacting switching times in pulse-coupled systems are still not well understood. Here we systematically investigate switching times in dependence of the levels of noise and intrinsic dissipation in the system. We specifically reveal how local responses to pulses coact with external noise. Our findings confirm that, like in time-continuous phase-coupled systems, piecewise-continuous pulse-coupled systems exhibit switching times that transiently increase exponentially with the number of switches up to some order of magnitude set by the noise level. Complementarily, we show that switching times may constitute a good predictor for the computation reliability, indicating how often an input signal must be reiterated. By characterizing switching times between two saddles in conjunction with the reliability of a computation, our results provide a first step beyond the coding of input signal identities toward a complementary coding for the intensity of those signals. The results offer insights on how future heteroclinic computing systems may operate under natural, and thus noisy, conditions.