Learning Spatio-Temporally Encoded Pattern Transformations in - PowerPoint PPT Presentation

Introduction Background Our Approach Results Summary Learning Spatio-Temporally Encoded Pattern Transformations in Structured Spiking Neural Networks 12 Andr´ e Gr¨ uning, Brian Gardner and Ioana Sporea Department of Computer Science University of Surrey Guildford, UK 9th November 2015 1 http://dx.doi.org/10.6084/m9.figshare.1517779 2 $Id: multilayerspiker.txt 1827 2015-11-09 07:39:53Z ag0015 $

Introduction Background Our Approach Results Summary

Introduction Background Our Approach Results Summary

Introduction Background Our Approach Results Summary Introduction 1 Background 2 Our Approach 3 Results 4 Summary 5

Introduction Background Our Approach Results Summary What are we doing? What are we doing? Formulate a supervised learning rule for spiking neural networks that can train spiking networks containing a hidden layer of neurons, can map arbitrary spatio-temporal input into arbitrary output spike patterns, ie multiple spike trains. Why worthwhile? Understand how spike-pattern based information processing takes place in the brain. A learning rule for spiking neural networks with technical potential. Find a rule that is to spiking networks what is backprop to rate neuron networks. Human Brain Project The SPIKEFRAME project:

Introduction Background Our Approach Results Summary Scientific Area Where are we scientifically? In the middle of nowhere between: computational neuroscience cognitive science artificial intelligence / machine learning

Introduction Background Our Approach Results Summary

Introduction Background Our Approach Results Summary Introduction 1 Background 2 Our Approach 3 Results 4 Summary 5

Introduction Background Our Approach Results Summary Spiking Neurons (a) input spikes output spike (c) u output spike (b) input spikes Spiking neurons: real neurons communicate with each other via sequences of pulses – spikes . 1 Dendritic tree, axon and cell body of a neuron. 2 Top: Spikes arrive from other neurons and its membrane potential rises. Bottom: incoming spikes on various dendrites elicit timed spikes responses as the output. 3 response of the membrane potential to incoming spikes. If the threshold θ is crossed, the membrane potential is reset to a low value, and a spike fired. From Andre Gruning and Sander Bohte. Spiking neural networks: Principles and challenges. In Proceedings of the 22nd European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning – ESANN , Brugge, 2014. Invited Contribution.

Introduction Background Our Approach Results Summary Spiking Neurons Spiking Information Processing The precise timing of spikes generated by neurons conveys meaningful information. Synaptic plasticity forms the basis of learning. Changes in synaptic strength depend on relative pre- and postsynaptic spike times, and third signals . Challenge: to relate such localised plasticity changes to learning on the network level .

Introduction Background Our Approach Results Summary Learning for Spiking NN General Learning Algorithms for Spiking NN? There is no general-purpose algorithm for spiking neural networks. Challenge: discontinuous nature of spiking events. Various supervised learning algorithms exist, each with its own limitations eg: network topology, adaptability (e.g. reservoir computing), limited spike encoding (e.g. latency, or spike vs no spike). Most focus on classification rather than more challenging tasks like mapping from one spike train to another.

Introduction Background Our Approach Results Summary Some Learning Algorithms for Spiking NN SpikeProp 3 , ReSuMe 4 , Tempotron 5 , Chronotron 6 , SPAN 7 , Urbanczik and Senn 8 , Brea et al. 9 , Freimaux et al. 10 , . . . 3 S.M. Bohte, J.N. Kok, and H. La Poutr´ e. Spike-prop: error-backpropagation in multi-layer networks of spiking neurons. Neurocomputing , 48(1–4):17–37, 2002 4 Filip Ponulak and Andrzej Kasi´ nski. Supervised learning in spiking neural networks with ReSuMe: Sequence learning, classification and spike shifting. Neural Computation , 22:467–510, 2010 5 Robert G¨ utig and Haim Sompolinsky. The tempotron: a neuron that learns spike timing-based decisions. Nature Neuroscience , 9(3), 2006. doi: 10.1038/nn1643 6 R˘ azvan V Florian. The chronotron: A neuron that learns to fire temporally precise spike patterns. PLoS ONE , 7(8):e40233, 2012 7 A. Mohemmed, S. Schliebs, and N. Kasabov. SPAN: Spike pattern association neuron for learning spatio-temporal sequences. Int. J. Neural Systems , 2011 8 R. Urbanczik and W. Senn. A gradient learning rule for the tempotron. Neural Computation , 21:340–352, 2009 9 Johanni Brea, Walter Senn, and Jean-Pascal Pfister. Matching recall and storage in sequence learning with spiking neural networks. The Journal of Neuroscience , 33(23):9565–9575, 2013 10 Nicolas Fremaux, Henning Sprekeler, and Wulfram Gerstner. Functional requirements for reward-modulated spile-timing-dependent plasticity. The Journal of Neuroscience , 30(40):13326–13337, 10 2010

Introduction Background Our Approach Results Summary

Introduction Background Our Approach Results Summary Introduction 1 Background 2 Our Approach 3 Results 4 Summary 5

Introduction Background Our Approach Results Summary Our Approach MultilayerSpiker Generalise backpropagation to Spiking Neural Networks with hidden neurons. Use stochastic neuron model to connect smooth quantities (derivative exists) with discrete spike trains (no derivative)

Introduction Background Our Approach Results Summary Neuron model Membrane potential � t � t Z o ( t ′ ) κ ( t − t ′ ) d t ′ , (1) � u o ( t ) := Y h ( t ′ ) ǫ ( t − t ′ ) d t ′ + w oh 0 0 h o postsynaptic neurons, h presynaptic neuron u o membrane potential of o . w oh strength of synaptic connection from h to o . Y h ( t ) = � t h < t δ ( t − t h ) spike train of neuron h where t h are the firing times of h Z o ( t ) = � t o < t δ ( t − t o ) spike train of neuron o where t o are the firing times of o .

Introduction Background Our Approach Results Summary Neuron model Spike response kernel ǫ and reset kernel κ ǫ ( s ) = ǫ 0 [ e − s /τ m − e − s /τ s ] Θ( s ) κ ( s ) = κ 0 e − s /τ m Θ( s ) , (2) and spike response kernel ǫ 0 = 4 mV , reset kernel κ 0 = − 15 mV , membrane time constant τ m = 10 ms , the synaptic rise time τ s = 5 ms Heaviside step function Θ( s ).

Introduction Background Our Approach Results Summary Neuron model Stochastic Intensity (instantaneous firing rate) and Spikes � u ( t ) − ϑ � ρ ( t ) = ρ [ u ( t )] = ρ 0 exp (3) , ∆ u firing rate at threshold ρ 0 = 0 . 01 ms − 1 . “threshold” ϑ = 15 mV . smoothness of the threshold ∆ u o = 0 . 2 mV (output layer) or ∆ u h = 2 mV (hidden layer) Spikes are generated by a point process taking stochastic intensity ρ o ( t ). Ie in a small time interval [ t , t + δ t ) a spike is generated with probability ρ o ( t ) δ t .

Introduction Background Our Approach Results Summary Backpropagation Objective (“Error”) function � � � P ( z ref log ( ρ o ( t )) Z ref o | x ) = exp o ( t ) − ρ o ( t ) d t (4) , where Z ref f δ ( t − ˜ t f o ( t ) = � o ) is the target output spike train for input x . a a J. P. Pfister, T. Toyoizumi, K. Aihara, and W. Gerstner. Optimal spike-timing dependent plasticity for precise action potential firing in supervised learning. Neural Computation , 18(6):1309–1339, 2006 Backprop approach ∂ log P ( z ref | x ) ∆ w oh = η o (5) ∂ w oh

Introduction Background Our Approach Results Summary Backprop approach . . . and some ten slides later Lots of derivatives, indices, probabilities. Derivatives only possible due to smoothness of probability function. Relatively freely switching between expected values and their best estimates to be had when you only have single cast.

Introduction Background Our Approach Results Summary Backprop Weight Update Backpropagated Error Signal 1 � � Z ref δ o ( t ) := o ( t ) − ρ o ( t ) (6) , ∆ u o Hidden-to-Output Weights � T ∆ w oh = η o δ o ( t ) ( Y h ∗ ǫ )( t ) d t . (7) 0 Input-to-Hidden Weights � T η h � ∆ w hi = w oh δ o ( t )([ Y h ( X i ∗ ǫ )] ∗ ǫ )( t ) d t . (8) ∆ u h 0 o a a Brian Gardner, Ioana Sporea, and Andre Gruning. Learning spatio-temporally encoded pattern transformations in structured spiking neural networks. Neural Computation , To appear. 2015. Preprint available at http://arxiv.org/abs/1503.09129

Introduction Background Our Approach Results Summary

Introduction Background Our Approach Results Summary Introduction 1 Background 2 Our Approach 3 Results 4 Summary 5

Introduction Background Our Approach Results Summary Task Task Purpose: explore the properties of the new learning algorithm. Map an input (given as a set of spike trains) to an output (given again as a set of spike trains). Simulation details a . a Brian Gardner, Ioana Sporea, and Andre Gruning. Learning spatio-temporally encoded pattern transformations in structured spiking neural networks. Neural Computation , To appear. 2015. Preprint available at http://arxiv.org/abs/1503.09129