INSTITUTE OF PHYSICS PUBLISHING NETWORK: COMPUTATION IN NEURAL SYSTEMS Network: Comput. Neural Syst. 14 (2003) 303–319 PII: S0954-898X(03)60761-X Optimal neural rate coding leads to bimodal firing rate distributions M Bethge, D Rotermund and K Pawelzik Institute of Theoretical Physics, University Bremen, Otto-Hahn-Alle, D-28334, Germany E-mail: mbethge@physik.uni-bremen.de Received 23 July 2002, in final form 4 February 2003 Published 8 April 2003 Online at stacks.iop.org/Network/14/303 Abstract Many experimental studies concerning the neuronal code are based on graded responses of neurons, given by the emitted number of spikes measured in a certain time window. Correspondingly, a large body of neural network theory deals with analogue neuron models and discusses their potential use for computation or function approximation. All physical signals, however, are of limited precision, and neuronal firing rates in cortex are relatively low. Here, we investigate the relevance of analogue signal processing with spikes in terms of optimal stimulus reconstruction and information theory. In particular, we derive optimal tuning functions taking the biological constraint of limited firing rates into account. It turns out that depending on the available decoding time T , optimal encoding undergoes a phase transition from discrete binary coding for small T towards analogue or quasi-analogue encoding for large T . The corresponding firing rate distributions are bimodal for all relevant T , in particular in the case of population coding. (Some figures in this article are in colour only in the electronic version) 1. Introduction Since the discovery by Adrian (1926) that action potentials are generated by sensory neurons with a frequency that is substantially determined by the stimulus, the idea of rate coding has become the prevalent paradigm in neuroscience (Perkel and Bullock 1968). In particular, today the coding properties of many neurons from various areas in the cortex have been characterized by tuning curves, which describe the average firing rate response as a function of certain stimulus parameters. Remarkably, almost all tuning functions measured in the mammalian cortex have a smooth bell-shaped form, which suggests an analogue neural code. On the other hand it is obvious that the maximum number of spikes k max that can be taken into account by subsequent neurons is limited by their integration time, so that a rate code actually constitutes a discrete code with k max + 1 different symbols. Therefore, the precision 0954-898X/03/020303+17$30.00 © 2003 IOP Publishing Ltd Printed in the UK 303