Abstract
Spiking Neural Networks (SNN) are mathematical models in neuroscience to describe the dynamics among a set of neurons that interact with each other by firing instantaneous signals, a.k.a., spikes. Interestingly, a recent advance in neuroscience [BarrettDenĂ¨veMachens, NIPS 2013] showed that the neurons' firing rate, i.e., the average number of spikes fired per unit of time, can be characterized by the optimal solution of a quadratic program defined by the parameters of the dynamics. This indicated that SNN potentially has the computational power to solve nontrivial quadratic programs. However, the results were justified empirically without rigorous analysis.
We put this into the context of natural algorithms and aim to investigate the algorithmic power of SNN. Especially, we emphasize on giving rigorous asymptotic analysis on the performance of SNN in solving optimization problems. To enforce a theoretical study, we first identify a simplified SNN model that is tractable for analysis. Next, we confirm the empirical observation in the work of Barrett et al. by giving an upper bound on the convergence rate of SNN in solving the quadratic program. Further, we observe that in the case where there are infinitely many optimal solutions, SNN tends to converge to the one with smaller l_1 norm. We give an affirmative answer to our finding by showing that SNN can solve the l_1 minimization problem under some regular conditions.
Our main technical insight is a dual view of the SNN dynamics, under which SNN can be viewed as a new natural primaldual algorithm for the l_1 minimization problem. We believe that the dual view is of independent interest and may potentially find interesting interpretation in neuroscience.
BibTeX  Entry
@InProceedings{chou_et_al:LIPIcs:2018:10119,
author = {ChiNing Chou and KaiMin Chung and ChiJen Lu},
title = {{On the Algorithmic Power of Spiking Neural Networks}},
booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
pages = {26:126:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770958},
ISSN = {18688969},
year = {2018},
volume = {124},
editor = {Avrim Blum},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10119},
URN = {urn:nbn:de:0030drops101191},
doi = {10.4230/LIPIcs.ITCS.2019.26},
annote = {Keywords: Spiking Neural Networks, Natural Algorithms, l_1 Minimization}
}
Keywords: 

Spiking Neural Networks, Natural Algorithms, l_1 Minimization 
Collection: 

10th Innovations in Theoretical Computer Science Conference (ITCS 2019) 
Issue Date: 

2018 
Date of publication: 

08.01.2019 