INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 37 (2004) 5713–5727 PII: S0305-4470(04)72724-9 Decoding spikes in a spiking neuronal network Jianfeng Feng 1 and Mingzhou Ding 2 1 Department of Informatics, University of Sussex, Brighton BN1 9QH, UK 2 Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431, USA Received 27 November 2003 Published 18 May 2004 Online at stacks.iop.org/JPhysA/37/5713 DOI: 10.1088/0305-4470/37/22/001 Abstract We investigate how to reliably decode the input information from the output of a spiking neuronal network. A maximum likelihood estimator of the input signal, together with its Fisher information, is rigorously calculated. The advantage of the maximum likelihood estimation over the ‘brute-force rate coding’ estimate is clearly demonstrated. It is pointed out that the ergodic assumption in neuroscience, i.e. a temporal average is equivalent to an ensemble average, is in general not true. Averaging over an ensemble of neurons usually gives a biased estimate of the input information. A method on how to compensate for the bias is proposed. Reconstruction of dynamical input signals with a group of spiking neurons is extensively studied and our results show that less than a spike is sufficient to accurately decode dynamical inputs. PACS numbers: 89.70.+c, 87.19.La (Some figures in this article are in colour only in the electronic version) 1. Introduction In a spiking neuronal network, how to reliably decode its input information in terms of observed neuronal output activity? This is a long-standing and fundamental issue in (computational) neuroscience [11, 5]. Even in the simplest form of neuronal models, the integrate-and-fire model, the answer is not known [16]. The difficulty lies in the fact that we do not know the exact input and output relationship in any ‘realistic’ neuron (model) and hence a parametric estimate approach is hard to be implemented in practice. In the current paper, we tackle the issue in a network of integrate-and-fire models. We expect that our approach will open a pathway into the study of how the biological nervous system works. To implement a decoding scheme for a spiking network, we first need the exact relationship between the input and the output of the integrate-and-fire model. To this end, we begin by considering spiking models with exactly balanced inputs. By exactly balanced input we mean that the mean input equals the threshold. In fact, such a model has been extensively studied 0305-4470/04/225713+15$30.00 © 2004 IOP Publishing Ltd Printed in the UK 5713