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.
Recently, an algorithm to learn precisely timed spikes using a leaky integrate-and-fire neuron was presented in @cite_14 . The algorithm converges only when a synaptic weight configuration to the given training problem exists, and can not provide a close approximation, if the exact solution does not exist. To overcome this limitation, another algorithm to learn spike sequences with finite precision is also presented in the same paper. It allows a window of width @math around the desired spike instant within which the output spike could arrive and performs training only on the first deviation from such desired behavior. While it mitigates the non-linear accumulation of error due to interaction between output spikes, it also restricts the training to just one discrepancy per iteration. Backpropagation for training deep networks of LIF neurons has been presented in @cite_33 , derived assuming an impulse-shaped post-synaptic current kernel and treating the discontinuities at spike events as noise. It presents remarkable results on MNIST and N-MNIST benchmarks using rate coded outputs, while in the present work we are interested in training multilayer SNNs with temporally encoded outputs i.e., representing information in the timing of spikes.
@cite_14 @cite_33
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Summary To signal the onset of salient sensory features or execute well-timed motor sequences, neuronal circuits must transform streams of incoming spike trains into precisely timed firing. To address the efficiency and fidelity with which neurons can perform such computations, we developed a theory to characterize the capacity of feedforward networks to generate desired spike sequences. We find the maximum number of desired output spikes a neuron can implement to be 0.1–0.3 per synapse. We further present a biologically plausible learning rule that allows feedforward and recurrent networks to learn multiple mappings between inputs and desired spike sequences. We apply this framework to reconstruct synaptic weights from spiking activity and study the precision with which the temporal structure of ongoing behavior can be inferred from the spiking of premotor neurons. This work provides a powerful approach for characterizing the computational and learning capacities of single neurons and neuronal circuits. Recent advances in neurosciences have revealed that neural information in the brain is encoded through precisely timed spike trains, not only through the neural firing rate. This paper presents a new supervised, multi-spike learning algorithm for multilayer spiking neural networks, which can implement the complex spatio-temporal pattern learning of spike trains. The proposed algorithm firstly defines inner product operators to mathematically describe and manipulate spike trains, and then solves the problems of error function construction and backpropagation among multiple output spikes during learning. The algorithm is successfully applied to different temporal tasks, such as learning sequences of spikes and nonlinear pattern classification problems. The experimental results show that the proposed algorithm has higher learning accuracy and efficiency than the Multi-ReSuMe learning algorithm. It is effective for solving complex spatio-temporal pattern learning problems. Abstract Previous studies have shown that spike-timing-dependent plasticity (STDP) can be used in spiking neural networks (SNN) to extract visual features of low or intermediate complexity in an unsupervised manner. These studies, however, used relatively shallow architectures, and only one layer was trainable. Another line of research has demonstrated – using rate-based neural networks trained with back-propagation – that having many layers increases the recognition robustness, an approach known as deep learning. We thus designed a deep SNN, comprising several convolutional (trainable with STDP) and pooling layers. We used a temporal coding scheme where the most strongly activated neurons fire first, and less activated neurons fire later or not at all. The network was exposed to natural images. Thanks to STDP, neurons progressively learned features corresponding to prototypical patterns that were both salient and frequent. Only a few tens of examples per category were required and no label was needed. After learning, the complexity of the extracted features increased along the hierarchy, from edge detectors in the first layer to object prototypes in the last layer. Coding was very sparse, with only a few thousands spikes per image, and in some cases the object category could be reasonably well inferred from the activity of a single higher-order neuron. More generally, the activity of a few hundreds of such neurons contained robust category information, as demonstrated using a classifier on Caltech 101, ETH-80, and MNIST databases. We also demonstrate the superiority of STDP over other unsupervised techniques such as random crops (HMAX) or auto-encoders. Taken together, our results suggest that the combination of STDP with latency coding may be a key to understanding the way that the primate visual system learns, its remarkable processing speed and its low energy consumption. These mechanisms are also interesting for artificial vision systems, particularly for hardware solutions. The vast majority of natural sensory data is temporally redundant. Video frames or audio samples which are sampled at nearby points in time tend to have similar values. Typically, deep learning algorithms take no advantage of this redundancy to reduce computation. This can be an obscene waste of energy. We present a variant on backpropagation for neural networks in which computation scales with the rate of change of the data - not the rate at which we process the data. We do this by having neurons communicate a combination of their state, and their temporal change in state. Intriguingly, this simple communication rule give rise to units that resemble biologically-inspired leaky integrate-and-fire neurons, and to a weight-update rule that is equivalent to a form of Spike-Timing Dependent Plasticity (STDP), a synaptic learning rule observed in the brain. We demonstrate that on MNIST and a temporal variant of MNIST, our algorithm performs about as well as a Multilayer Perceptron trained with backpropagation, despite only communicating discrete values between layers.