### Abstract

- Top of page
- Abstract
- Introduction
- Data recording and preprocessing
- Determination of current sources on single neurons – an overview
- The new sCSD method
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

Traditional current source density (tCSD) calculation method calculates neural current source distribution of extracellular (EC) potential patterns, thus providing important neurophysiological information. While the tCSD method is based on physical principles, it adopts some assumptions, which can not hold for single-cell activity. Consequently, tCSD method gives false results for single-cell activity. A new, spike CSD (sCSD) method has been developed, specifically designed to reveal CSD distribution of single cells during action potential generation. This method is based on the inverse solution of the Poisson-equation. The efficiency of the method is tested and demonstrated with simulations, and showed, that the sCSD method reconstructed the original CSD more precisely than the tCSD. The sCSD method is applied to EC spatial potential patterns of spikes, measured in cat primary auditory cortex with a 16-channel chronically implanted linear probe *in vivo*. Using our method, the cell–electrode distances were estimated and the spatio-temporal CSD distributions were reconstructed. The results suggested, that the new method is potentially useful in determining fine details of the spatio-temporal dynamics of spikes.

### Introduction

- Top of page
- Abstract
- Introduction
- Data recording and preprocessing
- Determination of current sources on single neurons – an overview
- The new sCSD method
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

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).

### The new sCSD method

- Top of page
- Abstract
- Introduction
- Data recording and preprocessing
- Determination of current sources on single neurons – an overview
- The new sCSD method
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

The sCSD method requires a model with new assumptions, consistent with the properties of the source and which ensure the uniqueness of the inverse solution. It has been show, that current source distribution of a spiking neuron can be described by the counter current model (CCM) during the initial period of the spike (Somogyvári *et al.,* 2005). The sCSD method suggested here is built on the basis of the CCM.

The validity period of the CCM extends from the onset of the spike until the largest amplitude negative peak of an EC recorded spike. The CCM describes the instantaneous spatial CSD distribution of a spiking cell, as it is driven by a large amplitude and point-like inward current (sink), which is accompanied by a one-dimensional distribution of small-amplitude outward currents (sources) parallel to the electrode. This line source distribution simplifies the complex morphology of neurons, while preserving their basic geometrical properties with elongated shape and typical orientation parallel to the recording probe. The simplification in the assumption of the morphology implies numerical advantages – this model ensures the uniqueness of the inverse solution of the Poisson-equation, thus forming a good basis for a CSD calculation method. In CCM, the EC potential is generated by a discretized line source, parallel to the electrode, thus allowing the elements of the transfer matrix to be calculated as:

- (8)

where *d* is the distance between the electrode and the line source while *x*_{i} and *x*_{j} are the fixed positions of the measurements and the sources respectively. It was assumed that for every measurement position exists a point source with the same respective depth (*x* coordinate, Fig. 1A).

With fixed electrode and source geometry, *V* is only dependent on the *d* distance and the source itself:

- (9)

Let us assume the case *M*=*N*, the number of point sources equals the number of electrodes, where **T**(*d*) is a square matrix and invertible. Thus, if *V* is given, the inverse problem can be formulated as:

- (10)

Thus, the only source of ambiguity is the *d* distance between the line source and the electrode. There are infinitely many source distributions, which generate the same spatial potential pattern (*V*), but they can be distinguished by *d*. If *d* is known, then **T**^{−1}(*d*) can be calculated, and the original discretized *I* can be exactly determined up to the spatial resolution of the electrode array, based on the measured *V*. Thus, in this setup, *d* plays a key role – the task is to find the correct distance by using *a priori* knowledge about the *I*(*d*) distribution.

There are physiological constraints for the cell–electrode distance. Henze *et al.* (2000) showed that EC potential of a spike can be measured up to 200 μm laterally away from the cell. However, current spike identification and clustering algorithms require an appropriate signal to noise ratio, which can be achieved typically at 60 μm away at most. Our calculation will be performed under the assumption that neurons are within the 200 μm limit.

In our previous work, we assumed that *I*(*d*) satisfied the restrictions of the CCM and fit to the measured *V* best (Somogyvári *et al.,* 2005). This was achieved by defining a model, which consists of a negative monopole source and a chain of positive line source segments, and fitting the model parameters to minimize the squared distance between the model-generated and the measured potential. There were numerous free parameters of the model – the distance between the line source and the electrode, the position of the main monopole along the line and amplitudes of the monopole and the line; source segments. We were able to determine the CSD distributions underlying the observed spatial potential patterns by fitting this model, but only during its very short validity period, which was the main drawback of that method. The aim of the new sCSD method is the determination of the spatio-temporal CSD distributions during the whole length of the spikes. So we applied a different approach, to reach this extension – the potential measurement is used to constrain the possible solutions and a measure is constructed, which quantifies how much a given CSD distribution fulfills the assumptions of CCM. This measure is used as a regularization function, thus the maximum of it marks the best solution, which corresponds to finding the best-fitting *d* distance parameter in our case. Because the CCM describes a sharp peak surrounded by the smooth background of its counter currents, the constructed measure quantifies the sharpness of the peak in a given CSD distribution. To measure this spike-likeness the following regularization function was applied:

- (11)

Although *S*(*d*) depends on *I*(*d*) directly, under our assumptions, *I*(*d*) depends only on the distance so we simplified the notations to *S*(*d*). The most spike-like CSD distribution *I*_{opt} and the distance estimation *d*_{opt} could be found by maximizing *S*(*d*) for all *I*(*d*) CSD distributions, which generate the actual *V* from different distances.

This distance estimation method resembles the auto-focus methods applied in automated photo cameras. There the basic assumption is, that the object has sharp contours. The picture is transformed according to the laws of optic radiation, depending on the distance and the lens makes an inverse transformation, which also has a distance parameter, the focus distance. The autofocus system measures the sharpness of the picture and changes the focus until the sharpest picture is reached. Typically it gives only the sharp picture, but it also would be able to estimate the object distance *d*_{opt} that maximizes the sharpness.

The *V* is time dependent during the spike. Our *a priori* knowledge about spike-likeness is generally valid during the first negative deflection of the EC potential, which is brief in comparison to the duration of the action potential. However, the source can be determined over a longer period of time, using the distance determined by the *V* data inside the CCM validity period. For distance determination, an appropriate choice is the *V* at the end of the validity period with maximal negative amplitude, as it ensures the most dominant somatic sink and the highest signal to noise ratio. Once the cell–electrode distance has been determined on the maximal amplitude sample, the determined distance can be used during the whole spike to calculate the inverse solution. The steps of the sCSD algorithm are summarized in Box 1.

The schema of the sCSD method is represented in Fig. 2C. Here, and in all the graphs in this paper, the sink is shown upward, but considered to be negative. The potential was generated by a sharp CSD distribution, shown in Fig. 4, top left, at 100 μm distance. The potential is transformed into a series of normalized *I*(*d*) distributions, by using **T**^{−1}(*d*) for each distance. When the *V* was mapped at a smaller distance, the peak of the corresponding *I*(*d*) became wider, while at larger distances it became a zig-zag CSD pattern with a wide envelope. Mapping *V* back to the original distance resulted in the sharpest peak, with the highest *S* value.

### Discussion

- Top of page
- Abstract
- Introduction
- Data recording and preprocessing
- Determination of current sources on single neurons – an overview
- The new sCSD method
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

Reviewing the problem of determination of CSD distributions on single neurons and the existing solutions, we concluded, that a new method is necessary to match the specific properties of single neuron sources. Based on these specific properties of the single-cell CSD distributions (Somogyvári *et al.,* 2005), a new method, called sCSD, was introduced, tested and applied to *in vivo* experimental data.

Compared with our previous work (Somogyvári *et al.,* 2005), we modified the model behind the analysis, so its only unknown parameter, needed for the inversion, is the cell–electrode distance. The most important consequence is that the new method made possible the extension of the time period of analysis to the whole length of the action potentials, thus allowing the reconstruction of the spatio-temporal dynamics of spikes.

The main technical difference between the method of Somogyvári *et al.* (2005) and the one presented here is that in our previous work we used the assumptions of the CCM as constrains and fitted the source to the measured potential patterns. Here we used the measurements as constrain and fitted the source to the assumptions of CCM. As a consequence, the sCSD method allows sinks in the background line source, while the original CCM does not. This way, we also reached greater numerical stability and the new algorithm requires less computational power.

The majority of the sCSD reconstruction error is the consequence of the error in distance estimations, which is the consequence of finite spatial sampling rate and the unknown longitudinal position of the soma between two contact points. Thus, increasing the density of the electrodes improves not only the precision of the distance determination, but the quality of CSD reconstruction as well. However, to get relevant information, the spatial extent of the electrode is also important and should cover the whole spatial extent of the dendritic trees. Thus, with a given number of electrode channels, an optimum should be found in the spatial density. With higher spatial sampling rate, the algorithm is more sensitive to the noise, so care should be taken to increase size and the cleanness of the spike clusters.

Rall (1962) showed in detailed computational models and Somogyvári *et al.* (2005) in spikes measured *in vivo*, that monopole, dipole and quadrupole point sources, or a linear combination of these, are not satisfactory approximations of the CSD distributions on a spiking cell, in physiologically relevant distances. Thus, triangulation based on monopole model (Jog *et al.,* 2002; Chelaru & Jog, 2005) could not localize the spiking cells. Aur *et al.* (2005) and Aur & Jog (2006) extended this approach, using multiple monopoles for describing the CSD of a spike. This approach would require tests on simulated data to show, that it is able to approximate the real sources well.

Our method is similar to the multiple source localization method known as MUSIC (Schmidt, 1986; Mosher & Leahy, 1999). In this method, the EEG and MEG recorded on the scalp are interpreted by means of dipole sources. The method finds the most fitting dipole by calculating the principal angles between the high-dimensional signal subspace and the subspace of individual sources. Apart from many technical details, there is a conceptual difference also. In MUSIC algorithm the source is approximated by a few point sources and the uncovered background is considered as error. In the sCSD method, only one point source is applied in the description of the source and the residual line source contains important information.

Work of Gold *et al.* (2007) aims to reproduce the channel densities on the neuron, based on EC spatio-temporal potential patterns. Once the channel densities are reconstructed, the membrane currents can be reproduced. However, the drawback of his method is that it requires either numerical or manual searching in a large-dimensional parameter space, and also morphological reconstruction of the cell. These requirements limit the possible applications of their approach. Our method has the advantage that it can be applied off-line, and the only additional information required is the geometry of the recording MEA.

Based on tests on simulated data, the new sCSD method determines more accurately the spatial distribution of CSD on a spiking neuron than the tCSD method. Besides the greater precision, the sCSD method allows the estimation of the cell–electrode distance as well. As the simulations showed, the greater precision means that quantitative conclusions on spatio-temporal CSD distribution can be drawn more reliably based on the sCSD results. Comparisons of sCSD and tCSD maps (Supporting Information) show the tendency for tCSD to generate false side-currents on the neighboring channels affecting the tCSD results.

The tCSD method (Nicholson & Freeman, 1975; Mitzdorf, 1985) has rarely been applied to single-neuron potentials (Buzsáki & Kandel, 1998; Bereshpolova *et al.,* 2007). Both studies used 16-channel MEA with 100 μm inter-electrode distances and, in both studies, the somato-dendritic back-propagation (BP) into the apical direction was observed in mammalian neocortical neurons. Another common feature is that both papers focused on the analysis of the large neurons in layer V of the cortex, and both papers observed the initiation of the action potentials one channel below the soma in many cells at the presumptive position of the axon initial segment. Similar results were found in the hippocampus by Henze & Buzsáki (2003).

Buzsáki & Kandel (1998) showed in awake rat's sensory cortices, that the BP action potential could be followed up to 400 μm distance from the soma. The speed of propagation was found to be 0.67 m/s. Bereshpolova *et al.* (2007) recorded the EC potential of spikes in the primary visual cortices of awake rabbits. Both spontaneous and evoked spikes were studied, which were elicited by antidromical stimulation from the superior colliculus. Because special care was taken to align the electrode parallel to the cortical column and large number of spikes were averaged, the BP was observed up to 800 μm distance into the apical directions. The average propagation speed was found to be 0.78 m/s and largely independent from states of the cortex. They showed, that the BP was not observable in putative fast spiking interneurons. None of these works observed the BP into the basal dendrites or the FP of regenerative potentials, which were described by the *in vitro* optical imaging studies (Antic, 2003; Zhou *et al.,* 2008).

Many fine details of the spatio-temporal dynamics of spikes were observable on the sCSD maps; however, the underlying mechanisms need further clarification. As far as we know neither the action potential BP into the basal dendrites, nor FP regenerative potentials were observed in EC potentials, although these phenomena were demonstrated by measurements and optical imaging experiments with voltage-sensitive dye (Antic, 2003; Zhou *et al.,* 2008), and some indirect evidence for dendritic spike generation was found by Kamondi *et al.* (1998), and Henze & Buzsáki (2003). The advantage of our method compared with the IC and optical imaging methods is that chronically implanted EC MEA can easily be applied to monitor single-cell activity in freely behaving animals. The observation of axon terminals in EC potential is not unique, Leutgeb *et al.* (2007) argued that the short spikes showing grid cell-like spatial tuning, that they observed in the perforant path termination area of the dentate gyrus were the signs of axon terminals. However, the EC potential of the whole neuron has never been measured together with its Ranvier-nodes or axon terminals, to the best of our knowledge.

We emphasize that the FP sink-waves we observed in the majority of cells, are not necessarily the signs of dendritic spikes, nor do they mean the dendritic initiation of the action potential. These waves rather can be considered as parts of the not fully understood current system of the cell leading to the action potential initiation.

There are many possible directions to proceed further. Technically, different *a priori* assumptions, thus different regularization functions, could be applied, to substitute the actually applied function *S*(*I*). Source models, which allow more source elements than electrode contacts, would be able to decrease the distance and generally the position estimation error of the method. However, appropriate regularization function should be found to solve the underdetermined inverse problem in this case.

So far we have applied the sCSD method on 1D MEA measurements, which resulted in 2D localization of the neurons. A 2D MEA measurement would allow the full 3D localization of the neurons.

By applying the sCSD method, we are able to determine the net membrane current (*I*) on the cells, which is the sum of the capacitive and the ionic membrane current [Eqn (1)]. However, to reconstruct the membrane potential based on the EC measurements, the ionic membrane current *I*_{R} should be determined separately. In perspective, this new method raises the possibility of identifying synaptic inputs, which causes a cell fire.

### Supporting Information

- Top of page
- Abstract
- Introduction
- Data recording and preprocessing
- Determination of current sources on single neurons – an overview
- The new sCSD method
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

**Fig. S1.** Spike A. First row: The first 100 sample spikes from the filtered potential data on one channel only, the cluster average for all channels and the calculated sCSD average for all channels. Second row: Spatio-temporal maps of average potential, spike CSD (sCSD) and traditional CSD (tCSD) on linear color scale. Third row: Same as in the second row, but on arcus-tangent color scale. Fourth row: Same as in the third row, but only those amplitudes are colored, which differs from the 0 amplitude on significance level *P* = 0:99. In all maps, hot colors correspond to negative potentials and current sinks, cold colors correspond to positive potentials and current sources and zero is green. Fifth row: Amplitude attenuation vs. the distance from the soma, during action potential back-propagation (BP) or amplitude increase during forward propagation (FP) in case of EC potential, sCSD and tCSD. Black line stands for the apical BP, red marks the basal BP, blue stands for the apical FP and green stands for the basal FP.

**Fig. S2.** Spike B. Notations same as in Fig. 1.

**Fig. S3.** Spike C. Notations same as in Fig. 1.

**Fig. S4.** Spike D. Notations same as in Fig. 1.

**Fig. S5.** Spike E. Notations same as in Fig. 1.

**Fig. S6.** Spike F. Notations same as in Fig. 1.

**Fig. S7.** Spike G. Notations same as in Fig. 1.

**Fig. S8.** Spike H. Notations same as in Fig. 1.

**Fig. S9.** Spike I. Notations same as in Fig. 1.

**Fig. S10.** Spike J. Notations same as in Fig. 1.

**Fig. S11.** Spike K. Notations same as in Fig. 1.

**Fig. S12.** Spike L. Notations same as in Fig. 1.

**Fig. S13.** Spike M. Notations same as in Fig. 1.

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