Spiking neural networks (SNNs) have garnered a great amount of interest for supervised and unsupervised learning applications. This paper deals with the problem of training multilayer feedforward SNNs. The non-linear integrate-and-fire dynamics employed by spiking neurons make it difficult to train SNNs to output desired spike train in response to a given input. To tackle this, first the problem of training a multilayer SNN is formulated as an optimization problem such that its objective function is based on the deviation in membrane potential rather than the spike arrival instants. Then, an optimization method named Normalized Approximate Descent (NormAD), hand-crafted for such non-convex optimization problems, is employed to derive the iterative synaptic weight update rule. Next it is reformulated for a more efficient implementation, which can also be interpreted to be spatio-temporal error backpropagation. The learning rule is validated by employing it to solve generic spike based training problem as well as a spike based formulation of the XOR problem. Thus, the new algorithm is a key step towards building deep spiking neural networks capable of event-triggered learning.
One of the earliest attempts to demonstrate supervised learning with spiking neurons is the SpikeProp algorithm @cite_34 . However, it is restricted to single spike learning, thereby limiting its information representation capacity . SpikeProp was then extended in @cite_21 to neurons firing multiple spikes. In these studies, the training problem was formulated as an optimization problem with the objective function in terms of the difference between desired and observed spike arrival instants and gradient descent was used to adjust the weights. However, since spike arrival time is a discontinuous function of the synaptic strengths, the optimization problem is non-convex and gradient descent is prone to local minima.
@cite_34 @cite_21
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Abstract For a network of spiking neurons that encodes information in the timing of individual spike times, we derive a supervised learning rule, SpikeProp , akin to traditional error-backpropagation. With this algorithm, we demonstrate how networks of spiking neurons with biologically reasonable action potentials can perform complex non-linear classification in fast temporal coding just as well as rate-coded networks. We perform experiments for the classical XOR problem, when posed in a temporal setting, as well as for a number of other benchmark datasets. Comparing the (implicit) number of spiking neurons required for the encoding of the interpolated XOR problem, the trained networks demonstrate that temporal coding is a viable code for fast neural information processing, and as such requires less neurons than instantaneous rate-coding. Furthermore, we find that reliable temporal computation in the spiking networks was only accomplished when using spike response functions with a time constant longer than the coding interval, as has been predicted by theoretical considerations. A supervised learning rule for Spiking Neural Networks (SNNs) is presented that can cope with neurons that spike multiple times. The rule is developed by extending the existing SpikeProp algorithm which could only be used for one spike per neuron. The problem caused by the discontinuity in the spike process is counteracted with a simple but effective rule, which makes the learning process more efficient. Our learning rule is successfully tested on a classification task of Poisson spike trains. We also applied the algorithm on a temporal version of the XOR problem and show that it is possible to learn this classical problem using only one spiking neuron making use of a hair-trigger situation.