SpikeLM: A Second-Order Supervised Learning Algorithm for Training Spiking Neural Networks

1. Yongji Wang Jian Huang Huazhong University of Sci. & Tech. Wuhan, China SpikeLM: A Second-Order Supervised Learning Algorithm for Training Spiking Neural Networks

2. Overview • Introduction • Preliminaries • SpikeLM algorithm • Experimental validation • Conclusions

3. Introduction • Spiking neural networks get increased attention: • Biologically more plausible • Computational power not less than traditional ANN • Main problem: supervised learning algorithms, it is just in its infancy. • SpikeProp, a grads-descent supervising learning algorithm

4. Preliminaries • Model of SNN • Originally introduced by Natschläger andRuf • Every connection consists of several synaptic connections • Each terminal is associated with a different delay and weight

5. Preliminaries • Model of SNN (continued) • Notations: • : the set of spiking neurons for the r-th layer; • : the spike firing time from neuron to • : the weight of the m-th terminal between iand j; • : the delay of the m-th terminal; • : membrane potential of neuron i;

6. Preliminaries • Model of SNN (continued) • Spiking Response Model (SRM): • Where is the unweighted contribution of a single synaptic terminal from j to i.

7. t Preliminaries • Membrane potential and firing time:

8. SpikeLM algorithm • Training samples: • Note that all inputs and outputs are firing times. • We use and to describe the actual and desire firing time respectively.

9. SpikeLM algorithm • Dynamics equation of the three-layered SNN: • Vectorial form:

10. SpikeLM algorithm • The performance index: • To compute the Jacobian matrix, define

11. SpikeLM algorithm • The representation of Jacobian matrix: where the k,q,r,m,i,j can be easily obtained given h and l.

12. Sensitivities in SpikeLM SpikeLM algorithm • Computation of Jacobian matrix:

13. The same way as SpikeProp did SpikeLM algorithm • Computation of Jacobian matrix: • Sensitivities: (output layer)

14. SpikeLM algorithm • Computation of Jacobian matrix: • To form sensitivity matrix: (output layer) where

15. As SpikeProp did SpikeLM algorithm • Computation of Jacobian matrix: • Sensitivities: (hidden layer)

16. SpikeLM algorithm • Then the elements are given by

17. SpikeLM algorithm • Computation of Jacobian matrix: • To form sensitivity matrix: (hidden layer)

18. SpikeLM algorithm • Computation of Jacobian matrix: • The matrix form of computation: • Define and

19. SpikeLM algorithm

20. SpikeLM algorithm • Computation of Jacobian matrix: • The matrix form of computation:

21. SpikeLM algorithm • Adaptation of parameters:

22. SpikeLM algorithm • Summarize the SpikeLM algorithm: 1) Compute the performance index; 2) Compute Jacobian matrix via backpropagation method; 3) Solve 4) Recompute the performance index using . If the new index is smaller than that computed in step 1, then reduce by , go to 1). Otherwise, increase by and go back to 3). 5) If the convergent condition is met, then stop.

23. Experimental validation • XOR Problem: • We assume the same setup as Bothe did. a “late” and “early” firing time substitute 0 and 1. • Both SpikeProp and SpikeLM algorithm are applied to cope with this problem. The convergent rates are compared to illustrate the merit of the latter algorithm.

24. Experimental validation • 4 output spike time examples during learning

25. Experimental validation • Convergence comparison for SpikeProp and SpikeLM

26. Experimental validation • Nonlinear function approximation: • Select a nonlinear function • the values of F(x) are totally normalized into a interval from 10 to 22. • The approximation curve was obtained after about 200 epochs of learning.

27. Experimental validation • The function approximation by SpikeLM:

28. Conclusion • A second-order supervising learning rule is derived for feedforward spiking neural networks using temporal-coding scheme. • This procedure is represented by a fairly concise vectorial form, which can be easily implemented by any softwares. • Elementary tests show the great potential of this algorithm.