Error-backpropagation in temporally encoded networks of spiking neurons

For a network of spiking neurons that encodes information in the timing of individual spike-times, we derive a supervised learning rule, emph{SpikeProp, akin to traditional error-backpropagation and show how to overcome the discontinuities introduced by thresholding. 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, it is demonstrated that temporal coding requires significantly less neurons than instantaneous rate-coding

SuperSpike: Supervised learning in multi-layer spiking neural networks

A vast majority of computation in the brain is performed by spiking neural networks. Despite the ubiquity of such spiking, we currently lack an understanding of how biological spiking neural circuits learn and compute in-vivo, as well as how we can instantiate such capabilities in artificial spiking circuits in-silico. Here we revisit the problem of supervised learning in temporally coding multi-layer spiking neural networks. First, by using a surrogate gradient approach, we derive SuperSpike, a nonlinear voltage-based three factor learning rule capable of training multi-layer networks of deterministic integrate-and-fire neurons to perform nonlinear computations on spatiotemporal spike patterns. Second, inspired by recent results on feedback alignment, we compare the performance of our learning rule under different credit assignment strategies for propagating output errors to hidden units. Specifically, we test uniform, symmetric and random feedback, finding that simpler tasks can be solved with any type of feedback, while more complex tasks require symmetric feedback. In summary, our results open the door to obtaining a better scientific understanding of learning and computation in spiking neural networks by advancing our ability to train them to solve nonlinear problems involving transformations between different spatiotemporal spike-time patterns

Training Spiking Neural Networks Using Lessons From Deep Learning

The brain is the perfect place to look for inspiration to develop more efficient neural networks. The inner workings of our synapses and neurons provide a glimpse at what the future of deep learning might look like. This paper serves as a tutorial and perspective showing how to apply the lessons learnt from several decades of research in deep learning, gradient descent, backpropagation and neuroscience to biologically plausible spiking neural neural networks. We also explore the delicate interplay between encoding data as spikes and the learning process; the challenges and solutions of applying gradient-based learning to spiking neural networks; the subtle link between temporal backpropagation and spike timing dependent plasticity, and how deep learning might move towards biologically plausible online learning. Some ideas are well accepted and commonly used amongst the neuromorphic engineering community, while others are presented or justified for the first time here. A series of companion interactive tutorials complementary to this paper using our Python package, snnTorch, are also made available: https://snntorch.readthedocs.io/en/latest/tutorials/index.htm

Nouvelle approche analytique pour l'apprentissage du quantron

RÉSUMÉ : Le quantron est un neurone artificiel inspiré d’un modèle stochastique de la diffusion synaptique. Ce type de neurone biologiquement réaliste a le potentiel d’améliorer les capacités de classification des réseaux de neurones utilisés en reconnaissance de formes. Cependant, le quantron présente des difficultés pour l’implémentation d’un algorithme d’apprentissage efficace. Ceci est dû à la présence de discontinuités dans la fonction de réponse qui caractérise l’émission ou l’absence d’émission de neurotransmetteurs en réaction à la stimulation des synapses d’entrée. Ces discontinuités nuisent à l’apprentissage par modification itérative des paramètres du neurone. Ainsi, nous adoptons une approche analytique pour contourner ces difficultés et développer de nouveaux algorithmes d’apprentissage pour entraîner un quantron ou un réseau de quantrons. D’abord, nous nous intéressons au maximum de la fonction représentant le potentiel électrique du quantron, appelée fonction d’activation. Par comparaison à un seuil d’excitabilité, ce maximum détermine l’état d’activité du quantron, qui est alors utilisé comme classificateur. En utilisant des potentiels post-synaptiques ayant un profil rectangulaire, nous obtenons une approximation du maximum en substituant des fonctions quadratiques aux signaux provenant des synapses d’entrée. Avec cette approximation analytique, nous démontrons expérimentalement la possibilité d’entraîner le quantron en minimisant une surface d’erreur par descente du gradient. De plus, pour certains problèmes, nous observons une amélioration des résultats d’un algorithme de recherche directe. Ensuite, en utilisant une configuration particulière du quantron, nous trouvons une forme analytique simple pour la fonction d’activation dans le cas où les potentiels post-synaptiques possèdent un profil rectangulaire ou en rampe. Cette expression permet de lier les paramètres du quantron aux caractéristiques géométriques de sa frontière de décision. En se basant sur ces résultats, nous développons deux algorithmes d’apprentissage distincts, l’un procédant par l’analyse des configurations de la frontière de décision, et l’autre par l’inversion directe d’un système d’équations. Ces algorithmes permettent une résolution efficace de problèmes de classification pour lesquels le quantron admet une représentation sans erreur. Enfin, nous portons attention au problème de l’apprentissage d’un réseau de quantrons. Dans le cas de potentiels post-synaptique avec un potentiel triangulaire, nous proposons une approximation analytique du temps où s’active le quantron, qui est déterminé par le premier instant où la fonction d’activation atteint le seuil d’excitabilité. L’expression mathématique résultante, utilisée comme valeur de réponse du neurone, permet d’adapter l’algorithme de rétropropagation de l’erreur au réseau. Nous montrons qu’il devient alors possible d’entraîner des neurones qui autrement resteraient inactifs lors de l’apprentissage. De plus, nous illustrons la capacité des réseaux de quantrons à résoudre certains problèmes de classification en nécessitant moins de paramètres que des réseaux de neurones impulsionnels ou des réseaux de perceptrons. Les trois aspects du quantron étudiés dans cette thèse mènent à des algorithmes qui se distinguent des approches antérieures utilisées pour l’apprentissage des réseaux de neurones impulsionnels. En effet, notre approche analytique permet d’éviter les discontinuités qui perturbent le processus d’apprentissage grâce au lissage résultant de l’approximation analytique du maximum de la fonction d’activation et du temps d’activation. De plus, l’analyse géométrique de la frontière de décision est rendue possible par l’expression analytique de la fonction d’activation. Le résultat le plus probant est la tentative fructueuse de résolution du problème associé à l’entraînement des neurones inactifs, appelé problème des neurones silencieux. Par notre approche analytique de l’apprentissage du quantron, nous proposons donc des algorithmes originaux et innovateurs qui contribuent à une meilleure compréhension de l’apprentissage dans les réseaux de neurones biologiquement réalistes.---------- ABSTRACT : The quantron is an artificial neuron inspired by a stochastic model of synaptic diffusion. This type of biologically realistic neuron can improve the classification capacity of neural networks used in pattern recognition. However, the implementation of an efficient learning algorithm for the quantron proves to be challenging. This is due to the presence of discontinuities in the output function which characterizes the emission of neurotransmitters, or lack thereof, as a reaction to the stimulus applied to synaptic inputs. These discontinuities disrupt the iterative training of the neuron’s parameters. Thus, in this work, we follow an analytical approach to avoid these difficulties and develop new learning algorithms adapted to the quantron and to networks of quantrons. First, we study the maximum of the function representing the electrical potential of the quantron, called the activation function. By comparing this function to an excitability threshold, this maximum determines the activity state of the neuron, which can be used as a classifier. Using post-synaptic potentials with a rectangular profile, we obtain an analytical approximation of the maximum by substituting quadratic functions for the signals stemming from the synaptic inputs. With this analytical approximation, we provide an experimental demonstration of the quantron being trained by minimizing an error surface via gradient search. Also, for certain problems, we observe an improvement of the results obtained by using a direct search algorithm. Second, using a specific configuration of the quantron, we find a simple analytical form for the activation function when the post-synaptic potentials have a rectangular or ramp profile. This expression links the parameters of the quantron to the geometrical characteristics of its decision boundary. Building upon these results, we obtain two distinct learning algorithms, one proceeding by analyzing the configurations of the decision boundary, and the other by solving directly a system of equations. These algorithms are able to solve efficiently classification problems for which the quantron admits an errorless representation. Third, we focus on the problem of training a network of quantrons. For post-synaptic potentials having a triangular profile, we propose an analytical approximation of the time when the quantron’s activation function reaches the excitability threshold. The resulting mathematical expression, used as the neuron’s output, enables the adaptation of the error backpropagation algorithm to the network. We show that it is then possible to modify the parameters of neurons which would otherwise remain inactive during training. Furthermore, we show that networks of quantrons can solve particular classification problems using fewer parameters than networks of spiking neurons or networks of perceptrons. The three aspects of the quantron studied in this thesis yield algorithms which differ from previous attempts to train spiking neural networks. Indeed, we avoid the discontinuities that disturb the training process due to the smoothing effect of the analytical approximation of the activation function’s maximum and of the activation time. Also, the geometrical analysis of the decision boundary is made possible by the analytical expression of the activation function. The most important result is the successful attempt to solve the problem of training inactive neurons, called the silent neuron problem. By following an analytical approach in the study of the quantron, we propose original and innovative algorithms which contribute to a better understanding of the learning process in networks of biologically realistic neurons

Spiking Neural Networks for Inference and Learning: A Memristor-based Design Perspective

On metrics of density and power efficiency, neuromorphic technologies have the potential to surpass mainstream computing technologies in tasks where real-time functionality, adaptability, and autonomy are essential. While algorithmic advances in neuromorphic computing are proceeding successfully, the potential of memristors to improve neuromorphic computing have not yet born fruit, primarily because they are often used as a drop-in replacement to conventional memory. However, interdisciplinary approaches anchored in machine learning theory suggest that multifactor plasticity rules matching neural and synaptic dynamics to the device capabilities can take better advantage of memristor dynamics and its stochasticity. Furthermore, such plasticity rules generally show much higher performance than that of classical Spike Time Dependent Plasticity (STDP) rules. This chapter reviews the recent development in learning with spiking neural network models and their possible implementation with memristor-based hardware