INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 35 (2002) 2379–2394 PII: S0305-4470(02)28638-2 Training the integrate-and-fire model with the informax principle: I Jianfeng Feng 1, 3 , Hilary Buxton 1 , and Yingchun Deng 2 1 COGS, Sussex University, Brighton BN1 9QH, UK 2 Department of Mathematics, Hunan Normal University, Changsha 410081, People’s Republic of China Received 7 September 2001, in final form 31 December 2001 Published 1 March 2002 Online at stacks.iop.org/JPhysA/35/2379 Abstract In terms of the informax principle, and the input–output relationship of the integrate-and-fire (IF) model, IF neuron learning rules are developed. For supervised learning and with uniform weight of synapses (the theoretically tractable case), we show that the derived learning rule is stable and the stable state is unique. For unsupervised learning, within physiologically reasonable parameter regions, both long-term potentiation (LTP) and long-term depression (LTD) could happen when the inhibitory input is weak, but LTD cannot be observed when inhibitory input is strong enough. When both LTP and LTD occur, LTD is observable when the output of the postsynaptic neuron is faster than pre-synaptic inputs, otherwise LTP is observable, as observed in recent experiments. Learning rules of general cases are also studied and numerical examples show that the derived learning rule tends to equalize the contribution of different inputs to the output firing rates. PACS numbers: 87.18.Sn, 87.19.La, 05.10.Gg, 05.40.-a 1. Introduction Learning or synaptic plasticity is of vital importance for biological systems [1]. In the present paper, we develop a learning rule, which is applicable to solving engineering problems [12] and is based upon (biophysical) models of a cell. The learning rule is derived under the principle of the maximization of the mutual information of input–output, which has been proposed and widely used in artificial neuron networks [2, 4, 18]. Due to recent developments in modelling single neurons, we know exactly the input–output relationship of some neuron models such as the integrate-and-fire (IF) model [27] and IF-FHN model [10] etc. Combining these two approaches together, we are able to develop learning rules relying on the input–output relationship of a neuron. We first consider an ideal case where all synaptic strengths are identical. For supervised learning, by which we mean that the input and output firing rates of a neuron are fixed, 3 http://www.cogs.susx.ac.uk/users/jianfeng 0305-4470/02/102379+16$30.00 © 2002 IOP Publishing Ltd Printed in the UK 2379