Understanding the Convergence in Balanced Resonate-and-Fire Neurons

Resonate-and-Fire (RF) neurons are an interesting complementary model for integrator neurons in spiking neural networks (SNNs). Due to their resonating membrane dynamics they can extract frequency patterns within the time domain. While established RF variants suffer from intrinsic shortcomings, the recently proposed balanced resonate-and-fire (BRF) neuron marked a significant methodological advance in terms of task performance, spiking and parameter efficiency, as well as, general stability and robustness, demonstrated for recurrent SNNs in various sequence learning tasks. One of the most intriguing result, however, was an immense improvement in training convergence speed and smoothness, overcoming the typical convergence dilemma in backprop-based SNN training. This paper aims at providing further intuitions about how and why these convergence advantages emerge. We show that BRF neurons, in contrast to well-established ALIF neurons, span a very clean and smooth - almost convex - error landscape. Furthermore, empirical results reveal that the convergence benefits are predominantly coupled with a divergence boundary-aware optimization, a major component of the BRF formulation that addresses the numerical stability of the time-discrete resonator approximation. These results are supported by a formal investigation of the membrane dynamics indicating that the gradient is transferred back through time without loss of magnitude.