Event-based backpropagation can compute exact gradients for spiking neural networks

Abstract

Spiking neural networks combine analog computation with event-based communication using discrete spikes. While the impressive advances of deep learning are enabled by training non-spiking artificial neural networks using the backpropagation algorithm, applying this algorithm to spiking networks was previously hindered by the existence of discrete spike events and discontinuities. For the first time, this work derives the backpropagation algorithm for a continuous-time spiking neural network and a general loss function by applying the adjoint method together with the proper partial derivative jumps, allowing for backpropagation through discrete spike events without approximations. This algorithm, EventProp, backpropagates errors at spike times in order to compute the exact gradient in an event-based, temporally and spatially sparse fashion. We use gradients computed via EventProp to train networks on the Yin-Yang and MNIST datasets using either a spike time or voltage based loss function and report competitive performance. Our work supports the rigorous study of gradient-based learning algorithms in spiking neural networks and provides insights toward their implementation in novel brain-inspired hardware.

The BrainScaleS-2 Accelerated Neuromorphic System With Hybrid Plasticity

“…In contrast to the time-to-first spike approach (Göltz et al, 2021), this does not require explicit or analytical knowledge of the function t ⋆ (x, p) and is also applicable to more complex neuron models. In the context of spiking neural networks, this was recognized by Wunderlich and Pehle (2020) and elaborated in full generality by Pehle (2021). Concurrent work also introduced this technique to the wider machine learning community (Chen et al, 2021).…”

“…A similar argument to the one made above for the parameter derivative of the transition times allows one to then relate the adjoint state variables after λ + to the adjoint state variables before the transition λ − and yields an event-based rule for gradient accumulation of the parameters that only enter the transition equations (in particular the synaptic weights). This is elaborated more explicitly in Wunderlich and Pehle ( 2020 ) and Pehle ( 2021 ). The event-based nature of the gradient accumulation and the sparse propagation of error information has immediate consequences for neuromorphic hardware.…”

mlGeNN: accelerating SNN inference using GPU-enabled neural networks

“…Numerous techniques for directly training SNNs have been developed. Notably, back-propagation can be applied to exact spike times [14][15][16] or the non-differentiable transfer function of each spiking neuron can be replaced with a 'surrogate gradient' function, allowing back-propagation through time [17,18] or more biologically-plausible learning rules [19][20][21] to be applied to spiking models. However, none of these techniques are yet capable of training deep feedforward SNNs.…”

Spiking neural networks trained via proxy

“…In the third approach, known as latency learning, the idea is to define the neuron activity as a function of its firing time [19,20,21,22,23,24,25]. In other words, neurons fire at most once and stronger outputs correspond to shorter spike delays.…”