No abstract

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.

We consider a heteroclinic network in the framework of winnerless competition, realized by generalized Lotka-Volterra equations. By an appropriate choice of predation rates we impose a structural hierarchy so that the network consists of a heteroclinic cycle of three heteroclinic cycles which connect saddles on the basic level. As we have demonstrated in previous work, the structural hierarchy can induce a hierarchy in time scales such that slow oscillations modulate fast oscillations of species concentrations. Here we derive a Poincaré map to determine analytically the number of revolutions of the trajectory within one heteroclinic cycle on the basic level, before it switches to the heteroclinic connection on the second level. This provides an understanding of which parameters control the separation of time scales and determine the decisions of the trajectory at branching points of this network.

We construct n levels of nested, self-similar winnerless competition dynamics of which we explicitly work out the first three levels in the framework of generalized Lotka-Volterra equations. We choose microscopic rules such that the competition in the form of rock-paper-scissors is played between metapopulations, populations, and individuals at the same time. The trajectory of individual activities moves through a hierarchically structured heteroclinic network in a desired way. The hierarchy in structure is able to induce a separation of timescales that translates into nested spirals if the heteroclinic networks are coupled via diffusion on a spatial grid. For sufficiently strong diffusion the dynamics of interacting heteroclinic networks gets synchronized between the sites, which amounts to a large dimensional reduction of phase space. Possible applications lie in ecology and in brain dynamics. Our model reproduces in particular chunking dynamics with slow oscillations modulating fast oscillations modulating faster ones as observed in brain dynamics.

Heteroclinic networks are structures in phase space that consist of multiple saddle fixed points as nodes, connected by heteroclinic orbits as edges. They provide a promising candidate attractor to generate reproducible sequential series of metastable states. While from an engineering point of view it is known how to construct heteroclinic networks to achieve certain dynamics, a data based approach for the inference of heteroclinic dynamics is still missing. Here, we present a method by which a template system dynamically learns to mimic an input sequence of metastable states. To this end, the template is unidirectionally, linearly coupled to the input in a master-slave fashion, so that it is forced to follow the same sequence. Simultaneously, its eigenvalues are adapted to minimize the difference of template dynamics and input sequence. Hence, after the learning procedure, the trained template constitutes a model with dynamics that are most similar to the training data. We demonstrate the performance of this method at various examples, including dynamics that differ from the template, as well as a regular and a random heteroclinic network. In all cases the topology of the heteroclinic network is recovered precisely, as are most eigenvalues. Our approach may thus be applied to infer the topology and the connection strength of a heteroclinic network from data in a dynamical fashion. Moreover, it may serve as a model for learning in systems of winnerless competition.