Source-linked AI summary
Supervised Learning in Multilayer Spiking Neural Networks
Ioana Sporea, André Grüning
TL;DR
Existing approaches face limitations in learning from precisely timed spikes, including methods restricted to single spikes. The paper presents multilayer ReSuMe, which trains hidden-layer spiking networks and learns the XOR problem while remaining robust to spike-timing variability.
Problem
Existing learning approaches are limited by ignoring subsequent spikes, despite evidence that neural systems use exact spike timing.
Method
The paper presents a new learning algorithm for feed-forward spiking neural networks, with multilayer ReSuMe enabling training networks with hidden layers.
Results
The additional hidden layer permits learning the XOR problem, while simulations show learning is robust to variability in spike timing.
Takeaways & Limitations
Multilayer ReSuMe supports supervised learning in spiking networks with hidden layers and accommodates variable spike-timing patterns.
Takeaways & Limitations
Existing methods such as SpikeProp are limited to neurons firing a single spike and may ignore subsequent spikes.
Abstract
from arXiv · showhide
The current article introduces a supervised learning algorithm for multilayer spiking neural networks. The algorithm presented here overcomes some limitations of existing learning algorithms as it can be applied to neurons firing multiple spikes and it can in principle be applied to any linearisable neuron model. The algorithm is applied successfully to various benchmarks, such as the XOR problem and the Iris data set, as well as complex classifications problems. The simulations also show the flexibility of this supervised learning algorithm which permits different encodings of the spike timing patterns, including precise spike trains encoding.
1 Introduction
Spiking neural networks are motivated by evidence that precise spike timing carries information, while existing supervised methods have important architectural and spike-count limitations. The paper introduces a multilayer learning algorithm intended to address these limitations and support flexible temporal encodings.
- Motivation: Experimental evidence indicates that single-spike timing and precise temporal firing patterns convey sensory information.Evidence includes visual, tactile, and grouped-neuron responses.
- Motivation: Spiking neurons offer computational models that can approximate continuous functions and may be more powerful than sigmoidal neurons.Their communication is also described as suitable for VLSI implementation with speed advantages.
- Contribution: The paper presents a supervised learning rule for feed-forward spiking neural networks with multiple layers.The rule extends ReSuMe to multiple layers through backpropagation of network error.
- Contribution: Unlike SpikeProp, the proposed approach can be applied to neurons firing multiple spikes in all layers.This supports multilayer networks in which multiple spikes are considered throughout the network.
- Contribution: The algorithm supports precise spike-time encoding for both network inputs and outputs.The paper presents this as a first learning algorithm with hidden layers and this encoding capability.
- Evaluation: Simulations showcase the flexibility of multilayer ReSuMe on nonlinear problems and classification tasks using spike-timing patterns.The patterns span timescales from tens to hundreds of milliseconds.
2 Background
Earlier supervised spiking-neuron algorithms differ in neuron-model assumptions, network depth, spike multiplicity, and output representation. The paper positions its method as combining multilayer learning with the flexibility of multiple spikes and different neuron models.
- Prior approaches: Existing supervised learning algorithms investigate how spiking neurons learn, but definite biological conclusions about their learning mechanisms remain unavailable.The background frames this as an open question concerning biological neurons.
- Prior approaches: SpikeProp trains feed-forward networks of single-spike neurons by minimizing the time difference between target and actual output spikes.Its derivation is based on explicit neuron-model dynamics.
- Limitations: Extensions of SpikeProp can allow multiple spikes in input and hidden layers, but output learning still considers only the first spike.The network error is represented by the first-spike time difference.
- Prior approaches: ReSuMe uses Hebbian learning and gradient descent, and its dependence on input and output spike trains allows use with various neuron models.However, the algorithm is limited to a single layer or to training readouts for reservoir networks.
- Prior approaches: A ReSuMe extension with one hidden layer can perform nonlinear logical operations, whereas networks without hidden layers cannot.Other layer-feedforward applications used a hidden layer as a frequency filter and fixed-rate input and target outputs.
- Proposed approach: The proposed algorithm combines SpikeProp’s multilayer quality with ReSuMe’s flexibility across multiple spikes and neuron models.This is the paper’s stated positioning of the new supervised learning algorithm.
3 Learning algorithm
The section derives a supervised learning rule for feed-forward multilayer spiking networks by extending ReSuMe with continuous-time backpropagation. The formulation supports multiple hidden layers and, in principle, linearisable neuron models.
- Learning algorithm: The algorithm extends ReSuMe to multilayer feed-forward spiking networks by backpropagating network error through hidden layers.The derivation starts with one hidden layer and generalises to multiple hidden layers.
- Neuron model: Spike trains represent neural inputs and outputs as sequences of impulses occurring at specific times.Spike trains are formalised as sums of Dirac delta functions.
- Neuron model: The derivation uses the linear Poisson neuron model and replaces instantaneous firing-rate expressions with spike-train estimators.The firing-rate formulation is used because of its smoothness before substitution by discontinuous spike-train estimates.
- Backpropagation of the network error: Network error is defined from differences between actual and target output firing rates, and weight updates follow functional-gradient backpropagation in continuous time.Output-layer and hidden-layer errors are propagated through the network to compute weight modifications.
- Weight modifications: ReSuMe weight changes replace problematic products of Dirac delta functions with STDP and anti-STDP processes.A non-Hebbian term helps maintain the correct update direction when actual and target spike counts differ.
- Simulation observations: Preliminary simulations report better convergence from synaptic scaling and a higher convergence rate as the hidden-layer neuron count increases.The supplied passages also state that scaling addresses weight initialisation and speeds learning.
- Generalisation: The rule can be generalised to multiple hidden layers and is intended to work with spiking neuron models that can be approximated by linear models.The paper connects this generalisation to ReSuMe’s applicability across neuron models.
4 Heuristic discussion of the learning rule
The heuristic discussion examines how the learning rule changes synaptic weights according to whether output spikes occur before or after their target times. These changes move output timing toward the desired timing.
- Assumptions: The analysis assumes single-spike trains, postsynaptic spikes after presynaptic spikes, and no synaptic scaling for simplicity.Cases with missing actual or desired spikes are represented using infinite spike times.
- Early output spikes: When the output spike occurs before its target time, output-to-hidden weight changes decrease excitatory strength or increase inhibitory strength.Both effects make the output neuron more likely to fire later.
- Hidden-layer synapses: For hidden-to-input synapses, the direction of hidden-layer weight changes depends on whether the output-to-hidden connection is excitatory or inhibitory.The resulting hidden-neuron timing changes are described as influencing the timing of the output neuron through the intervening connection.
- Late output spikes: When the output spike occurs after its target time, the update reverses sign and moves the output spike forward toward the target time.This is derived from the sign change when actual and target times are swapped.
- Special cases: The same reasoning handles cases with only an actual spike or only a desired spike by setting the corresponding absent spike time to infinity.An additional contribution arises from the factor associated with a sub-connection without delay.
5 Simulations
The simulations evaluate the supervised learning algorithm across benchmark classification tasks and analyze its convergence, encoding flexibility, and learning dynamics.
- Simulation scope: The experiments cover XOR, Iris, randomly generated classification patterns, and spike-timing tasks spanning 100 to 500 ms.
- Network and evaluation setup: The feed-forward networks use spiking neurons, including Spike Response Model neurons, with delayed sub-connections between layers.
- Learning-parameter analysis: 137 iterations yielded high convergence, averaging over 98% successful trials under the tested learning-parameter settings.
- Convergence behavior: Convergence decreases and iterations increase when A− reaches or exceeds A+, while more delayed sub-connections stabilize learning but slow it through additional weights.
- Learning dynamics: The learning rule modifies weights toward solutions that produce the correct number of output spikes, with weight vectors moving steadily despite irregular error trajectories.
5.2 The Iris benchmark
The Iris benchmark tests whether Multi-layer ReSuMe can classify three flower species from spike-time encodings. It learns the dataset with fewer than 200 average iterations and supports alternative temporal encodings.
- Encoding and task: Iris samples are represented by four measurements encoded as input spike timings, with the output class represented by the output neuron’s spike time.
- Encoding alternatives: The single-spike encoding uses multiple delayed sub-connections, whereas group spike-train encoding avoids both multiple delays and an additional reference neuron.
- Evaluation protocol: Training uses 75% of samples and testing uses 25% for cross-validation, with success requiring 95% training accuracy and an average error of 0.2.
- Results: Multi-layer ReSuMe learns the Iris dataset with straightforward encoding and averages fewer than 200 iterations, faster than SpikeProp.
- Practical constraints: More weights and incoming spikes slow learning, while early-versus-late encoding requires an additional reference neuron because otherwise patterns can become indistinguishable.
- Network topology: Increasing hidden-layer size improves convergence by distributing patterns across wider spiking activity, but more than 100 units slows learning without improving convergence.
5.5 Learning to Generalise
The experiments test generalization from clean or noisy training patterns to temporally distorted spike trains. Feed-forward networks classify patterns across hundreds of milliseconds and remain above random performance under substantial jitter.
- Generalization setup: Networks are trained on temporal spike patterns and evaluated on noisy versions to test classification under spike-time variation.
- Architecture: The feed-forward network performs these temporal classification tasks without requiring larger reservoir-style networks.
- Temporal scale and topology: The experiments also use patterns spanning 100 or 500 ms, with hidden layers of 210 neurons and outputs classified by a single neuron.
- Noise model: Training noise shifts spikes with Gaussian standard deviations from 1 to 4 ms, while target-pattern jitter remains fixed at 1 ms.
- Training robustness: Training with low input noise performs slightly better than training without noise, and learning remains possible with 3 or 4 ms spike distortion.
6 Discussion
Multilayer ReSuMe extends supervised spike-timing learning to hidden-layer networks, multiple spikes, and flexible temporal encodings. Simulations report successful nonlinear classification, robustness to timing noise, faster learning than SpikeProp in selected comparisons, and synaptic-scaling benefits.
- Multilayer ReSuMe trains spiking neural networks with hidden layers, extending supervised learning beyond single-layer settings.
- Unlike SpikeProp, the learning rule matches subsequent output spikes and the exact number of output spikes, not only the first target spike.
- The algorithm learns XOR patterns faster than SpikeProp in the reported direct comparison while matching precise target timing and spike counts.
- The method supports different spike-pattern encodings, including a four-neuron Iris encoding that learns in 5 times less iterations than SpikeProp’s population encoding.
- The reported advantages are balanced by encoding and convergence considerations, including extra sub-connections, additional training, and increased convergence rate with larger hidden layers.
- Spike-train encoding can successfully represent non-linear pattern transformations without an additional reference-start neuron, although distributed inputs may contain redundant information.
- Training is robust to timing variability: recognition exceeds 50% at variance up to 4 ms for clean-trained networks and 10 ms for noisy-trained networks.
- Synaptic scaling addresses weight-initialisation range and maintains hidden- and output-neuron activity within an optimal firing range during learning.
7 Conclusion
The paper concludes that multilayer ReSuMe provides supervised learning for feed-forward spiking networks with hidden layers and multiple spikes throughout the network. It supports nonlinear and complex classification tasks while remaining applicable in principle across linearisable neuron models.
- The paper introduces a supervised learning algorithm for feed-forward spiking neural networks.
- The method is presented as the first fully supervised algorithm to consider multiple spikes in all network layers.
- Hidden layers permit the network to learn nonlinear problems and complex classification tasks without many spiking neurons or input neurons.
- Because the weight updates depend on input, output, and target spike trains rather than specific neuron dynamics, the rule can in principle apply to any neuron model.
Neuron model
The simulations use feed-forward spiking neurons described by the Spike Response Model, in which membrane potential integrates weighted presynaptic responses and spikes occur at threshold crossing.
- The feed-forward network uses the Spike Response Model, representing each neuron with a membrane potential and threshold-triggered spike emission.
- A neuron fires when its membrane potential u(t) reaches threshold ϑ at firing time t^f.
- The membrane potential is formed from weighted presynaptic spike-response functions, with w_ji denoting the connection weight from neuron i to neuron j.
- The response kernel η(t) models the action potential and after-potential, while ε(t) describes a standard postsynaptic potential with a membrane time constant.
Network error
Network error is measured by the van Rossum distance between actual and target spike trains. The same distance determines the output pattern by selecting the closest target pattern.
- The error for one pattern is defined using the van Rossum distance between each output spike train and its target spike train.
- Filtering uses an exponential function associated with spike times, with τ_c chosen according to the output neurons’ interspike interval.
- In the simulations, output neurons fire approximately one spike in 10 ms, so τ_c = 10 ms.
- The distance compares filtered spike trains through squared Euclidean distance over a time domain [0, T] covering all system spikes.
- During learning and testing, the predicted output pattern is the target pattern with the smallest van Rossum distance.