On the Nonlearnability of a Single Spiking Neuron

• Published in: Neural computation Vol. 17; no. 12; pp. 2635 - 2647
• Main Authors: Šíma, Jiří, Sgall, Jiří
• Format: Journal Article
• Language: English
• Published: One Rogers Street, Cambridge, MA 02142-1209, USA MIT Press 01.12.2005; MIT Press Journals, The
• Subjects: Action Potentials - physiology; Algorithms; Applied sciences; Artificial intelligence; Biological and medical sciences; Computer science; control theory; systems; Computers; Connectionism. Neural networks; Cortical Synchronization - methods; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects. Models. Methods; Learning and adaptive systems; Models, Neurological; Neural Networks (Computer); Neurons; Neurons - physiology; Vertebrates: nervous system and sense organs; Learning; Input output; Spiking neuron; Neural computation; N variable function; NP hard problem; Boolean function; NP complete problem; Neural network; Synchronization; Computational complexity; Threshold
• ISSN: 0899-7667; 1530-888X
• DOI: 10.1162/089976605774320601

We study the computational complexity of training a single spiking neuron with binary coded inputs and output that, in addition to adaptive weights and a threshold, has adjustable synaptic delays. A synchronization technique is introduced so that the results concerning the nonlearn-ability of spiking neurons with binary delays are generalized to arbitrary real-valued delays. In particular, the consistency problem for with programmable weights, a threshold, and delays, and its approximation version are proven to be -complete. It follows that the spiking neurons with arbitrary synaptic delays are not properly PAC learnable and do not allow robust learning unless = . In addition, the representation problem for , a question whether an -variable Boolean function given in DNF (or as a disjunction of ( ) threshold gates) can be computed by a spiking neuron, is shown to be -hard.