## Introduction

The use of silicone micro-electrode arrays (MEA) has become increasingly widespread in recent years (Kipke *et al.,* 2008). Besides the traditionally high temporal resolution of electric recording techniques, they provide increasing spatial resolution and well-defined geometry of recording sites. The increasing spatial extent and resolution provide higher number of simultaneously recorded cells, better spike separation and the possibility of monitoring local field potential and multi-unit activity in all layers of the cortex simultaneously. Single-cell analysis techniques, however, have not kept up with this development. The transformation of the spatio-temporal potential information into time series of spikes neglects the spatial information contained by the signal. While the information content, especially the spatial information of the measurements, increased drastically, the appropriate techniques for analysis are still missing. Our work aims to develop an analysis method to extract the spatial information of the extracellularly (EC) measured single-neuron action potentials by reconstructing the spatial distribution of single-cell current source density (CSD).

We will demonstrate the capabilities of the methods by focusing on the spatial aspects of action potential generation. The current optical imaging methods combined with voltage-sensitive dyes provide information on spatial distribution of instantaneous membrane potential, but *in vivo* recordings of multiple cells and resolution of spatio-temporal dynamics of action potentials on the full extent of individual neurons is still very challenging, especially in freely behaving animals (Scanziani & Häusser, 2009). The spatio-temporal aspects of action potential generation, such as initiation and back-propagation (BP), were examined by optical imaging techniques and intracellular electrodes (IC), but all these experiments were carried out on *in vitro* slices (Stuart *et al.,* 1997; Antic, 2003; Zhou *et al.,* 2008). In this paper, we will apply our new source reconstruction method to examine the spatio-temporal dynamics of action potentials *in vivo* conditions.

Extracellular potential measurements with MEAs provide information about spatio-temporal dynamics of action potentials, but the effects of the membrane current sources appear in an integrated form. The potential on each electrode is a weighted sum of the current source system on the whole cell. Thus, the key question, can we bridge the gap between EC potential and membrane current sources, and if yes, how can we do it?

The traditional CSD (tCSD) method (Nicholson & Freeman, 1975; Mitzdorf, 1985) provides a solution for this problem. It is based on the continuity of the current and calculates the CSD as the second spatial derivatives of the EC potential. Unfortunately, when it is applied onto one-dimensional data, the derivatives according to the orthogonal dimensions are neglected, due to the lack of information. This one-dimensional method uses an implicit assumption, that CSD changes can be neglected in these two dimensions on the spatial scale of the electrode. In other words, it assumes laminar source distribution, with infinitely large, homogeneous laminar sources. Considering the laminar organization of the cortex, this can be a good approximation in case of large population activities such as epilepsy or evoked potentials, but certainly not valid in case of single cells. Thus, one-dimensional tCSD method gives incorrect results for spatial potential patterns, originated from a single neuron. The solution of this problem required designing a new CSD method, which fits better to the properties of individual cellular sources, thus it is able reconstruct the cellular currents, based on EC potential measurements.

In the first part of this paper we review the problem of determination of CSD distributions on single neurons. Thus, the relation between CSD and the transmembrane currents is clarified, and the connection between the CSD and the EC potential is presented in terms of the forward and inverse problem of the Poisson equation. The central problem of this work, the non-unique solution of the inverse problem of the Poisson equation, is briefly described, and previous solutions and their applicability to the single-cell problem is reviewed. In the second part, a new method is introduced as a new inverse solution. Then, test results of the spike sCSD method are presented in comparison to the tCSD method on simulated data. Finally, the spatio-temporal dynamics of spikes, recorded *in vivo* by a 16-channel MEA in a cat's A1 cortex, is described with much more detail and completeness, than was possible before (Fig. 2A).