Mathematical Medicine and Biology (2006) 23, 27−44 doi:10.1093/imammb/dql001 Advance Access publication on February 22, 2006 Modelling release of nitric oxide in a slice of rat’s brain: describing stimulated functional hyperemia with diffusion-reaction equations ALEXANDER I. OLEINICK,CHRISTIAN AMATORE†, MANON GUILLE AND STEPHANE ARBAULT Department de Chimie, Ecole Normale Superieure, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France AND OLEKSIY V. KLYMENKO AND I RINA SVIR‡ Mathematical and Computer Modelling Laboratory, Kharkov National University of Radioelectronics, 14 Lenin Avenue, 61166 Kharkov, Ukraine [Received on 22 August 2005; revised on 24 November 2005; accepted on 17 January 2006] The physicochemical process of nitric oxide (NO ◦ ) release from an active neuron is modelled based on the results obtained experimentally in independent series of experiments reported elsewhere in which the NO ◦ release elicited by patch-clamping a single neuron (stellate neuron from cerebellum area) is moni- tored by an ultramicroelectrode introduced into a slice of living rat’s brain. This process is believed to be central to brain behaviour by coupling neuronal activity with the blood supply to active areas of the living brain through precise control of NO ◦ -mediated dilatation of blood capillary vessels. This work, based on the conformal mapping approach initially proposed in a previous work, aims to model the overall physic- ochemical and diffusional processes giving rise to the release of NO ◦ by a neuron and during its collection at an electrode sensor. Fitting simulated currents to experimental ones published previously yields indeed the gross kinetic information which represents the overall neuron activation and defines the instant value of the concentration of NO ◦ at the neuron surface. This allows reconstructing the NO ◦ fluxes around the neuron body as they would have been in the absence of the electrode sensor. This permits one to appreci- ate how far NO ◦ is released by the neuron at concentrations which greatly exceed their basal values. The success of this procedure is exemplified using a set of three experimental data reported elsewhere. Keywords: brain slice; neuron; nitric oxide; conformal map; numerical simulation. 1. Introduction Delineating the intimate integrated mechanisms which control the local brain and cerebellum activ- ities are important challenges in modern medicine and biology and they have been within the scope of biochemical community for the last several decades. Understanding of such basic mechanisms is very important and provides new insights in how the brain works and therefore will have a lot of applications, especially in medicine, biochemistry, etc. In this connection, functional Magnetic Resonance Imaging (fMRI) provides important clues on how the brain performs its main function. It is the great advantage of fMRI to monitor efficiently and in-time in a perfectly non-invasive way the differential status (ca. a few percent at most) of blood circulation † Email: christian.amatore@ens.fr ‡ Email: svir@kture.kharkov.ua c  The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.