The design of artificial neural systems, in robotics applications and others, often leads to the problem of constructing a recurrent neural network capable of producing a particular trajectory, in the state space of its visible units. Throughout evolution, biological neural systems, such as central pattern generators, have also been faced with similar challenges. A natural approach to tackle this problem is to try to "learn" the desired trajectory, for instance through a process of trial and error and subsequent optimization. Unfortunately, gradient descent learning of complex trajectories in amorphous networks is unsuccessful. Here, we suggest a possible approach where trajectories are realized, in a modular and hierarchical fashion, by combining simple oscillators. In particular, we show that banks of oscillators have universal approximation properties. To begin with, we can restrict ourselves to the simple case of a network with one!

Natural and artificial neural circuits must be capable of traversing specific state space trajectories. A natural approach to this problem is to learn the relevant trajectories from examples. Unfortunately, gradient descent learning of complex trajectories in amorphous networks is unsuccessful. We suggest a possible approach where trajectories are realized by combining simple oscillators, in various modular ways. We contrast two regimes of fast and slow oscillations. In all cases, we show that banks of oscillators with bounded frequencies have universal approximation properties. Open questions are also discussed briefly.

Natural and artificial neural circuits must be capable of traversing specificstate space trajectories. A natural approach to this problem is to learn the relevant trajectories from examples. Unfortunately, gradientdescent learning of complex trajectories in amorphous networks is unsuccessful. We suggest a possible approach wheretrajectories are realized by combining simple oscillators, in various modular ways. We contrast two regimes of fast and slow oscillations.

Natural and artificial neural circuits must be capable of traversing specific state space trajectories. A natural approach to this problem is to learn the relevant trajectories from examples. Unfortunately, gradient descent learning of complex trajectories in amorphous networks is unsuccessful. We suggest a possible approach where trajectories are realized by combining simple oscillators, in various modular ways. We contrast two regimes of fast and slow oscillations. In all cases, we show that banks of oscillators with bounded frequencies have universal approximation properties. Open questions are also discussed briefly.