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.
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).
Data recording and preprocessing
A 16-channel probe with 100 μm inter-electrode distances was implanted chronically into a cat's primary auditory cortex perpendicular to the surface. The depth of the first channel was 500 μm below the surface, near the border of the second and third cortical layer. Spontaneous and sound-evoked potential patterns were recorded from awake animals, with at a 20-kHz sampling rate, and were digitalized with 12-bit precision. Data were band-pass filtered from 100 Hz to 6 kHz (for more details about the recording system, see Ulbert et al., 2001). Spikes were collected and clustered with Spike-o-matic free software (Pouzat et al., 2002). Spikes presumably originating from each individual neurons were distinguished and clustered based on their waveshape similarity on all channels, 13 clusters were produced and used for further analysis (see Supporting Information). The samples of each cluster were then averaged to produce a spatio-temporal potential pattern with high signal to noise ratio. To uncover fine details at low amplitudes, the CSD of these clusters averaged was analysed. All the calculations and analyses were performed with Scilab free mathematical software (Gomez, 1999). Surgical procedures and use of animals were approved by the Institutional Committee for Care and Use of Laboratory Animals.
Determination of current sources on single neurons – an overview
The source of the EC potential
The aim of our work is to reconstruct the neural current sources of the EC potentials, thus first we have to clarify notation of CSD and its relation to the transmembrane currents.
The net membrane current I(r,t) flowing through the membrane of the neurons in a position r and moment t is also called ephaptic current by Holt & Koch (1999). It is described as the sum of two terms, the resistive [IR(r,t)] and capacitive membrane currents [IC(r,t)]:
While IR(r,t) denotes the resistive membrane current carried by ions moving through the membrane, IC(r,t) does not involve charge transfer through the membrane, but describes the accumulation of opposing charges on the two sides of the cell membrane, corresponding to the increase of potential across the membrane.
The spatial density distribution of I(r,t) is called CSD and serves as sources for the EC potential (Fig. 2B). Any CSD calculation method provides an estimation for the net membrane current I(r,t) or ephaptic current, which does not correspond to the resistive ionic membrane current IR(r,t) on the right side of the Hodgkin–Huxley equations, rather corresponds to the spatial derivative of the axial currents along the neural processes. The sum of CSD is zero through the whole cell in every moment, while this does not necessarily hold for the IR(r,t) ionic membrane current (Holt & Koch, 1999). As a consequence of this, an equipotential neuron does not generate any EC. Furthermore, in a voltage-clamped situation dVm/dt = 0, thus IC = 0 and IR can be directly measured by an IC electrode.
From the CSD to the EC potential and back – the forward and the inverse problem
Extracellular space was considered a homogeneous and isotropic electrolytic volume conductor with conductivity σe = 0.003 S/mm based on Varona et al. (2000). This low-frequency quasistatic approximation allows a spike to be represented by a series of consecutive independent spatial potential patterns, denoted by V. The relation between CSD distribution and the EC potential in a homogeneous electrolytic volume conductor is described by Poisson's equation (Malmivuo & Plonsey, 1995):
Here ∇2 is the sum of the second spatial derivatives in the 3D space, and σe is the conductivity of the neural tissue. The forward problem of the Poisson equation is to find the V(r,t) potential distribution if the I(r,t) source distribution is known, while the inverse problem is the reconstruction of the I(r,t) source distribution, if V(r,t) is known.
Equation (3) is analogous to the first law of Maxwell, describing the electrostatics, but the source of the field on the right side is the CSD instead of the charge density distribution. A well-known solution of the forward problem is the electric potential field of a monopole, decreasing with the inverse of the distance. In a homogeneous and isotropic medium, the forward problem could be solved in general case by using the linearity of the electric fields and discretizing the sources – the field generated by any arbitrary CSD distribution can be described as the sum of the fields of the point sources. With the general assumption, that the source consists of N point sources, the solution for the forward problem, the Vi potential on the ith electrode can be obtained as:
where ri and r′j are the position vectors of the electrodes and the sources, respectively.
Adopting matrix formalism it can be read as
where I and V denote the vector of N source intensities and the vector of M measured potentials values, respectively. T denotes the M×N transfer matrix, consisting of the lead field vectors, which describe the potential patterns generated by the individual point sources Ij measured by the electrodes. All the geometry of the source and the electrode and the electric properties of the medium are comprised into the T.
If the three-dimensional (3D) potential distribution V(r,t) is known, the solution of the inverse problem is straightforward – performing the spatial derivations on the left side of Eqn (3) results in the source distribution. However, lacking of the full 3D data, the inverse solution is underdetermined, infinitely many source distributions I(r,t) can generate the same measured potential pattern V(r,t), as it is first demonstrated by Helmholtz (1853). The T matrix does not have proper inverse T−1 in these cases.
The key question of all inverse solution methods is how to choose the appropriate solution from the infinite multitude of the possible ones. Implicitly or explicitly all inverse solutions methods use assumptions on the source to include a priori knowledge into the choice and find a unique inverse solution for each measurement. From a theoretical point of view, any new CSD method gives a new solution for the inverse problem of the Poisson equation.
Applying these considerations to the case of one-dimensional MEA shows, that the lacking information lies in the unknown orthogonal derivatives. Simply neglecting these orthogonal derivatives results in one-dimensional tCSD method (Nicholson & Freeman, 1975). The tCSD directly follows the Poisson equation [Eqn (3)]. The derivatives orthogonal to the linear electrode array are neglected and the spatial derivative is discretized:
Where dx stands for the inter-electrode distance, which in our case was 100 μm.
The neglect of the orthogonal derivatives implies the implicit assumption, that the potential, as well as the CSD, is constant orthogonally to the electrode array. This assumption is violated and causes inaccuracies in case of spatially localized sources. This drawback of the tCSD method can be addressed by the model-based inverse solutions, where the finite extent of the source as well as any other a priori knowledge about the source can be incorporated into the source model.
Explicit model-based inverse solutions
While the 1D tCSD method uses only implicit assumptions on the geometry of the source, the explicit model-based inverse solutions, briefly reviewed in this section, use explicit source models and the inverse of the forward solution. In most cases, the discretized version of the forward problem [Eqn (5)] is invertible, when M = N, the number of measurements equals the number of sources. In this case the solution is given by the inverse of the transfer matrix:
If a higher number of sources than electrodes should be applied we have to face the problem of non-unique inverse solutions. In this case, where M < N, there are typically infinitely many solutions and a priori knowledge is required to choose among them.
The homogeneous laminar source assumption of the tCSD method ensures the uniqueness of the inverse solution in 1D cases. Pettersen et al. (2006) showed, that the larger the lateral extent of the laminar source distribution, the better the approximation is given by the tCSD method, as T−1 converges to the Laplacian matrix of the discrete spatial second derivative. However, it was shown by Pettersen et al. (2006), that the infinite laminar source assumption of the tCSD method can lead to inaccurate results even if the lateral extent of the source is in the range of a cortical column. Thus, the tCSD method does not provide the needed accuracy for local field potential analysis. To meet this demand, model-based inverse solutions are needed. Pettersen et al. (2006) provided inverse solutions suitable for reconstructing sources of the LFP with given extent of the cortical column. The method works under the assumption, that the geometry of the source is known and the electrode is positioned at the center (or other specific positions) of the cortical column. In each case, the number of free parameters of the source models was equal to the number of electrodes (M = N), so the models (T matrices) were invertible.
The infinite laminar source assumption of the tCSD method caused inaccuracies in case of cortical columns whose diameter is about 0.5 mm, and of course gave even worse results for single spiking neurons. Furthermore, the inverse CSD methods of Pettersen et al. (2006) cannot be applied directly to the EC potential of single neurons, due to the lack of knowledge on exact source geometry. While the shape and size of a neuron can be well approximated by simple geometrical forms, a method to determine the exact position and distance of the neuron relative to the electrode is still absent in the field.
On a macroscopic scale, the same inverse problem could be solved by the electroencephalogram (EEG) magnetoencephalography (MEG) imaging methods. There are two main families of methods that uniquely limit the continuum of the possible sources. A family of methods assumes smooth source distribution and approximates it with a large number of spatially fixed source voxels, leading to ill-posed linear problems identical to Eqn (5). The minimum-norm solution by Hämäläinen & Ilmoniemini (1984) and the LORETA method by Pascual-Marqui et al. (1994) belongs to this family. For a comprehensive review, see Grech et al. (2008).
The other family of inverse solutions assumes localized sources and uses a small number of point sources (dipoles) to approximate the source distributions of EEG and MEG (Schmidt,1986; Mosher & Leahy, 1999). In these methods, the spatial positions of the sources are free parameters to be determined. Because the positions appear in the denominator in Eqn (4) the problem is non-linear and the best fitting model can only be found by numerical fitting.
Despite the large number of existing macroscopical inverse solutions, none of them has been suitable to accurately fit to the unique combination of a localized, sharp peak and smooth background, which is a typical spatial current source distribution on a spiking neuron.
The new sCSD method
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:
where d is the distance between the electrode and the line source while xi and xj 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:
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:
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:
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 Iopt and the distance estimation dopt 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 dopt 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.
Test of the sCSD method on simulated data
To validate the results of the sCSD method, we tested them on simulated data. This procedure allowed us to determine the effects of the parameters affecting the precision of the source reconstruction. The quality of a source reconstruction depends on the shape of the source distribution itself. To test and compare the performance of the tCSD and sCSD methods, three different, one-dimensional CSD distribution patterns were used (Fig. 4, left column). Each of them consisted of 160 point sources, placed 10 μm apart. These test CSD distributions were chosen to fulfill the basic assumptions of CCM. All three distributions were dominated by a large, point-like sink, but their small spatially distributed current source patterns differed. The amplitude of the main sink was 1 for all three cases. The counter currents were symmetric and decayed exponentially as the distance from the soma in source 1 increased. The spatial decay constant was λ = 0.6 μm. The counter currents of source 2 were also exponential, but asymmetric λ1 = 0.9 μm and λ2 = 0.3 μm. In case of source 3, a Gaussian-shaped positive current was added to the first branch of source 2. The relative amplitude of these dendritic currents was as small as ±1−3% of the main sink, but in accordance with zero sum requirement, the sums of the currents along the line sources were set to zero in all the three cases. The purpose of the test was to compare the performance of the tCSD or sCSD methods to determine these currents, based on EC measurements.
The EC potential was calculated along a line, in 160 or 320 points parallel to the cell, every 1 μm from 10 to 200 μm away from the cell. We tested two different spatial resolutions – 100 and 50 μm, to simulate to two different linear microelectrode arrays MEA consisting of 16 and 32 channels with 100 and 50 μm inter-electrode distances (E100 and E50). Thus, to create 10 different simulated measurements, each consisting of 16 or 32 data points with 100 or 50 μm, sampling density, respectively, the calculated fine potential patterns were undersampled 10 times with different shifts. These different sampling phases correspond to different relative positions of the soma and the contact points of the probe. In addition, an appropriate level of noise was added to the calculated potentials.
Testing the precision of distance estimation
A key element of the sCSD method is the distance estimation of the cell, based on measured spatial potential pattern. The aim of the first test was to quantify the precision of the distance estimation method and the effects of each influencing factor. The precision of the distance estimation of each source depends on four identified factors – the spatial resolution of the MEA; the noise-to-signal ratio of the measurement; the cell–electrode distance; and the angle between the cell and the electrode. All of these factors have been studied by using simulations. To quantify the distance estimation errors, the systematic deviations [mean error (ME)] from the original distance and the root mean square errors (RMSE) of the estimations were calculated and averaged over the 10 different samplings. The patterns of estimation errors of the three source distributions (Fig. 4, left column) were found to be very similar, thus only the results of the estimation errors for source 1 are presented.
The distance estimation method and the effect of spatial resolution on the estimation error was first tested in noise-free cases. The sCSD method estimated the cell–electrode distances with reasonable errors (Fig. 3A and B). In these noise-free situations, the estimation errors are solely the result of finite sampling rates, thus decrease with the increasing spatial resolution. The maximum RMSE of E100 and E50 were approximately 20 and 10 μm (Fig. 3D, thick solid and dashed lines).
During the simulations Gaussian noise was added to the calculated V patterns to test the effect of noise on the precision of the distance estimation. The tested noise levels were set at the observed noise levels of the experiments. The noise level of the V patterns of the cluster averages, were defined by the standard error of mean (SEM) divided by the spike amplitudes:
where the noise-to-signal ratio (Rs) of a cluster s was determined by three factors: the average standard deviations (Ds) of voltage signals; the number of spikes (ns); and the amplitude of spikes [As = −min(Vs)] in the cluster. The minimal observed noise-to-signal ratio of the clusters of measured spikes was 0.0015.
With the use of the simulations, it was found, that deteriorating effect of the noise was significant only in higher distances. If R ≤ 0.01, then the precision of distance estimation of E100 was not effected up to 140 μm, but RMSE increased rapidly for farther distances (Fig. 3D, thin solid line). We considered the range of applicability, where the RMSE did not exceed 20 μm. Figure (left, y-axis) shows, that the maximal distance of applicability decreases with the noise level. Based on this, clusters with R > 0.01 were excluded from the analysis. Increased spatial sampling rates of E50 were shown to be more sensitive to the noise. The range of applicability decreases more rapidly for E50 than for E100. Figure (right- y-axis) shows the noise dependence of the errors for E100 and E50, averaged over the physiologically recordable regime from 10 μm to 60 μm. It was found that up to R = 0.03, a higher spatial resolution resulted in lower errors, while the E100 was shown to be less sensitive for the noise and performed better with higher noise. The real distance is generally overestimated below 70 μm and underestimated above (Fig. 3F, solid lines). In the EC recordable regime, the average systematic error is about 7 μm.
Because, the inter-electrode distance of the probe was 100 μm and the noise levels were below 0.01, we can conclude, that the expected RMSE values are about 15 μm for our spike clusters, and that the distances are overestimated by 7 μm on average. These can cause large relative errors at smaller distances, but the results are surprisingly good, considering the 100 μm spatial resolution of the MEA.
Comparison of the CSD reconstruction
The original test CSD distributions consist of 160 point sources, but each E100 measurement uses only 16 data points from the potential, thus able to determine only 16 source amplitudes in case of sCSD and 14 with the tCSD method. To compare the original and the reconstructed distributions, 10 coarse grained CSD distributions were created from the original one, with different shifts according to the 10 sample potential. CSD distributions were coarse grained by averaging the closest 100 μm-long section of the original CSD distribution for each electrode position. For each simulated measurement, the coarse grained CSD represents the best possible results (Fig. 4, middle column, dashed lines).
The direct comparison between a line source and the result of tCSD is difficult, because they have different dimensions. The result of tCSD has dimension of A/m3, while a line source requires A/m units. Because the CSD is typically used qualitatively, normalized distributions were compared. Figure 4 shows the three test source distributions, and normalized results of the source reconstruction with the tCSD and sCSD methods. Comparisons were shown in six cases, with 20 and 100 μm distances for all three sources. In all cases the results of sCSD (thick line) were much closer to the coarse grained original (dashed line) than the results of the tCSD (thin line) were. The tCSD resulted in the greatest deviations on the sides of the dominant peak. It has the tendency to generate false local counter currents to compensate the dominant peak.
We examined if it was possible to draw quantitative conclusions on the spatial distribution of the small-amplitude counter currents based on the CSD analysis. To do this, the spatial constants of the exponential decaying counter currents in source 1 and 2 in Fig. 4 were estimated based on the CSD estimations in 20 μm and in 100 μm cell-to-electrode distances by using 100 μm sampling rate (E100). The decay exponents were estimated by least-square linear fitting of the log[abs(Ij)] between 200 and 700 μm distance from the soma along the line source in both directions. The mean and the standard deviation of the estimated exponents over the 10 spatially shifted samplings are given in Table 1. These results show that all the exponents were precisely estimated by using the sCSD reconstruction in 20 μm distance and less accurately in 100 μm distance. However, the tCSD method does not make possible even approximate estimation of the spatial decay constants. Consequently, sCSD reconstruction allowed much more precise estimation of the cell parameters than the traditional method did.
Table 1. Test of spatial decay constant estimations based on CSD spike current source density (sCSD) reconstructions. The spatial decay constants were estimated for source 1 and 2 in Fig. 4, based on 100 μm spatial sampling (E100) and given in μm units with SD over the 10 spatially shifted samples. All the different spatial decay constants could be precisely estimated based on sCSD reconstructions at 20-μm cell–electrode distance and with moderate precision in 100 μm distance. However, decay parameter estimations were far from their original value in the case of the traditional (tCSD) method
λ1 = 600 μm
593 ± 116
818 ± 291
194 ± 83
184 ± 22
λ2 = 600 μm
589 ± 91
841 ± 342
207 ± 96
189 ± 23
λ1 = 900 μm
867 ± 201
1452 ± 816
306 ± 217
233 ± 40
λ2 = 300 μm
289 ± 37
371 ± 89
82 ± 29
123 ± 5
To quantify the difference between the coarse grained original and the reconstructed CSD distributions, calculated with the sCSD and tCSD methods, RMSE was calculated for 16 and 14 points, respectively. The distance dependence of errors was examined in distances from 10 to 200 μm with two different spatial sampling rates (E100 and E50), and with and without the presence of noise. Because all three source distributions resulted in very similar patterns of error, results only for source 1 are presented here (Fig. 5). The RMSE values clearly show that the sCSD method provides significantly smaller error, thus more faithful reconstruction, than tCSD in every distance for both sampling rates in the noise free-cases (Fig. 5A and C). Figure 5B and D show, that the significant difference between the precisions remained unaffected by the noise below 70 μm in E100 case and below 120 μm in E50 case. Considering, that the regime available for physiological recording is below 60 μm we concluded that sCSD method provides significantly better reconstruction than the traditional method even in the presence of the noise with R = 0.01.
The sCSD analysis assumed the parallel placement of the electrode and the cell. Thus, we tested the sensitivity of the results on the deviation from this ideal setup (Fig. 1B). Tilt is described by two angles: the deviation from the parallel denoted by α; and the direction of the tilt, denoted by β. Tilt towards the electrode within a common plane is denoted by β = 0° and the tilt perpendicular to the cell–electrode direction is denoted by β = 90°.
Figure 6 shows how the precision of the CSD reconstruction and the distance estimation depends on the relative angle of cell and the electrode. It was found, that the smaller details of the original CSD diminished on the reconstruction with the increasing tilt, but no spurious sources appeared on the sides of the main sink, which was typical for the tCSD (Fig. 6, top row). Parallel, the reconstruction error increased slightly for the sCSD with the increasing tilt, but the RMSE remained lower for sCSD than for tCSD for all α, β and distance combinations (Fig. 6, middle row). As the tilt increased, the originally equal distances between different parts of the cell and the line of the electrode became different. This effect is most pronounced in case of β = 0°, thus the reconstruction deteriorates most rapidly in this direction.
In these non-parallel cases, we defined the cell–electrode distance as the distance between the soma and the probe, and compared the sCSD distance estimations to it (Fig. 6, bottom row). The distance estimation was relatively insensitive to the tilt for β = 90° and 45° cases, but β = 0° results in rapid increase in the distance estimation error with the increasing tilt. Even in this worst case, distance estimation error remained under 20 μm up to α = 20° for all examined distances (d = 20, 40, 60 μm), thus we concluded, that the distance estimation gives reasonable results up to α = 20° deviation for any case.
Application of sCSD method for in vivo data
Spatio-temporal potential patterns of action potentials from 13 putative neurons were analysed (see Supporting Information).
Cell-to-electrode distance estimations
During the application of the sCSD method, the distance interval to be examined should be specified a priori. Based on the results of Henze et al. (2000), 200 μm distance was the maximum, from where the potential was measurable. Thus, the interval to examine was set to 1−200 μm. However, Henze et al. (2000) also showed that EC potential of spikes reaches the 60 μV amplitude from the 60 μm vicinity of the electrode, and this amplitude is required for successful spike clustering. In our data, the minimum of the clustered spike amplitudes was 45 μV, so it was likely that the EC measured neurons lay slightly farther than 60 μm from the electrode. This assumption should be verified, to support the accuracy of the distance estimation.
Figure 7 shows the results of the distance estimations. The maximal estimated distance was 72 μm, which is in accordance with the results of Henze et al. (2000). In one case, the estimated distance was unrealistically low, 1 μm. Because in this case there was a sharp phase reversal in the potential near the soma, it was likely that a dendritic process was too close to a contact point. Thus, one of the basic assumptions of CCM, that the line source is parallel to the electrode, was violated in this case. This putative spike was excluded from the further analysis. The 2D positions of the remaining 13 putative neurons are shown in Fig. 7A. The linear electrode did not permit the determination of the angle of the cell, so the distance and the cortical depth is shown.
It is known that the potential is decreasing function of the distance. In case of a monopole source, the amplitude should be proportional to the inverse of the distance, but the actual course of the decay strongly depends on the spatial source distribution. However, the observability of this attenuation could be decreased by the possible heterogeneity of the neural sources and the amplitude threshold applied during the spike detection. Thus, the attenuation of the amplitude is only partially recognizable in Fig. 7B.
Spatio-temporal dynamics of spikes
The estimated distances made it possible to calculate the inverse solutions in every time moment during the spike. So, it allowed the determination of CSD distributions not only in the moment of highest amplitude negative potential peak, but in a longer time window during the course of the whole action potential. Four-millisecond-long time windows were cut around the negative peak of the EC action potential in each cluster and averaged pointwise. The generated spatio-temporal EC potential maps characteristic to the dynamics of action potentials on individual neurons with high signal to noise ratio, where fine details can also be reliably recognized. Considering the large number of spikes averaged in these clusters, a few percentage of difference could be highly significant, but hardly recognizable either on the traditional amplitude vs time graph or on color-coded maps with linear color-scale. In order to make visible these fine details of the spikes, a special colormap was created using the arctangent function. This colormap highlights the small changes in the potential and compresses the scale on larger amplitudes. Figure 8 shows the spatio-temporal potential patterns and the spatio- temporal CSD maps resulted from the sCSD method for six neurons. On the X-axis the time (4 ms), and on the Y-axis the cortical depth (500–2000 μm) is shown. Although each map, so each putative neuron has many individual properties, some common features are also recognizable.
In seven cells, instantaneous and spatially attenuating negative potentials appeared in the time of the largest negative peak, mostly in the basal direction. Due to their instantaneous appearance, these are simply the EC fields of the large sink at the soma. This field in the EC medium should be present in all cases, but become visible only due to the lack of significant cellular sources at that position. Significant positive currents, mainly from the apical dendrites, compensate locally the effect of a larger but farther sink, thus this effect remained observable only in the basal direction. The T matrix in Eqn (5) describes the smearing out of this EC field with the distance, thus the application of T−1 removes its effect while generating the CSD map. In the examples shown in Fig. 8, the passive EC propagation is visible on spike A and B in the basal direction, on spike D in the apical direction, and to a lesser extent on spikes C and F also. In all cases, this effect vanished from the CSD map.
The apical action potential BP was observable in all sCSD maps. Similar to the apical BP, BP sink-waves were also observable in the basal direction in some cases and sink-waves preceding the action potential, propagating towards the soma as well. To quantify the observability of the BP action potentials and the observability of the forward-propagating (FP) sink-waves, first the significance of the small-amplitude sCSD patterns of the cluster averages were tested. Each point of the spatio-temporal sCSD patterns was tested against the zero-mean null hypothesis, by one sample Student's t-test. Supporting Information Figs. (subplot h-j) show the space-time points of cluster averages, where the amplitude of the mean differed from 0 on the significance level P = 0.99. Considering the 1264 space-time points of each cluster mean, this significance level would yield only 13 sparsely distributed false-significant points on average, if the record did not contain real data. After the significance test, the space-time sCSD maps were divided into four quadrants according to the temporal and spatial relationship to the maximal amplitude point: the apical after-spike; the basal after-spike; the apical before-spike and the basal before-spike; quadrants. The measurements and the fitted simulations of Gold et al. (2006) showed that the position of the maximal amplitude was not farther to the soma than 20 μm. Because this is negligible distance compared with our 100 μm, resolution, we assumed, that the maximal amplitude marks the position of the soma. The wave propagation was considered to be observable in a given quadrant, if significant (P = 0.99) amplitude maxima of sinks appear by monotonously increasing or decreasing time lag at least three neighboring channels including the soma. This latest assumption corresponds to minimal 200 μm propagation length from the soma, taking into account that the basal dendrites of layer V pyramidal cells typically extend to 250 μm in length (Antic, 2003).
Applying these requirements to the sCSD maps (Table 2), the dendritic BP into the apical direction is observable in all neurons (n = 13). Surprisingly, the BP into the basal direction is recognizable in 11 neurons (85%) as well. While the signs of apical FP regenerative potential were observable only in three cases, the FP from the basal direction was more significant, it was observed in eight neurons (62%).
Table 2. Summary of the observed BPs and FPs in the neurons. The number of neuron are shown, in which the BPs and FPs were observed in the apical and the basal dendrites. The maximal observed length of the propagation, the average length and the average speed of the propagation with the standard deviation (SD) are shown as well. The minimal 200 μm (3-channels) propagation length of the sink peak with 0.99 significance were the requirements to count the wave propagation as observable. The high standard deviations for the speed are the result of the large differences between the individual cells. BP, back-propagation; FP, forward-propagation
Number of neurons out of 13
Max distance (μm)
Mean distance (μm)
0.57 ± 0.4
0.39 ± 0.23
0.3 ± 0.13
0.48 ± 0.19
The speed of BP was typically different into apical and basal directions – on average, the apical wave was faster, it propagates with 0.57±0.4 m/s (mean and standard deviation), while the basal speed was 0.39±0.23 m/s. The high standard deviation values show the observed diversity of cells. The difference between the apical and the basal speed does not necessarily mean real difference in the speed of electrical wave propagation along the dendritic processes. Because we measure only the projection of the wave propagation speed to the apical-basal axis, the difference could be originated, at least partially, from the different average angle between the electrode and the apical and basal dendrites, respectively. Furthermore, in three cases no asymmetry was detectable between the apical and the basal propagation speed. The spatial attenuation of apical BP was much slower than the basal BP and the FP both in apical and basal directions (Fig. 9).
The sCSD analysis reveals some fine details in spatio-temporal dynamics of CSD during action potential initialization. In one case (spike C) small-amplitude spike-like marks, spikelets, appear at 400 and 800 μm basal distances from the soma. The amplitudes of these spikelets reach only 2.5% of the main peak in the potential map, and slightly more pronounced, above 3% on the CSD map. The sCSD method reveals similar spikelets in three other cases (altogether seven spikelets from four clusters), from which spike A and B are shown in Fig. 8. The amplitude of these spikelets reaches 6% of the main peak. The occurrence of these spikelets on a potential map showed that they could not be artifacts of the sCSD calculation. They appear on different channels in different spikes, for example in spike A at least two spikelets are recognizable on channels 11 and 15, while in spike B the spikelet is observed on channel 16. The appearance of the spikelets on different channels showed, that they are not the results of the cross-talking between recording channels. Because both the potential map and the CSD are calculated from cluster averages of spikes, we wanted to exclude the possibility, that the spikelets are results of high-amplitude spikes from other cells, co-occurring to the clustered one with lower frequency. The amplitude histogram in the case of spike A, 15th channel is shown in Fig. 10E. Very few outlier are recognizable on this histogram, but it is clear that the whole, Gaussian-shaped distribution is shifted right from zero, thus the average is not determined by few high- amplitude outliers. We have carefully checked the histogram in case of all spikelets and concluded the same. Next, we have examined the possibility whether the spikelets are results of tightly correlated but farther located cells. The originally high-amplitude spikes of these assumptive far cells would produce small spikes at the location of the electrode with high frequency. On the one hand, the simulations of Miller-Larsson (1985) showed that the temporal length of EC measured spikes is increasing with the radial distance from the cell. On the other hand, Somogyvári et al. (2005) showed, that the spatial width of the spikes is also increasing with the distance from the cell. The amplitudes of these spikelets reach only a few percentages of the main spikes, which would indicate high distances if their origin were normal action potentials with normal amplitudes. In contrast to this situation, the small-amplitude spikelet was faster and narrower than the main spike in each cases. Thus, they cannot be originated from high distances. After excluding the close, high-amplitude, low-frequency and the far, high-amplitude, high-frequency sources, we concluded that the spikelets are the results of close, small-amplitude and high-frequency sources. In most cases, these spikelets tightly follow the assumptive initiation of the spikes near the axon initial segment and appear earlier than the dendritic BP or even the somatic spike (Fig. 10D). Considering their discontinuous spatial distribution, the most probable morphological source of these spikelets are the Ranvier-nodes of the axons or axon terminals.
As a summary of our results, both the spatial distribution and the temporal sequence of the events can be well demonstrated on spike A (Figs 8A, and 10A and B). The first significant amplitude appears one channel below the soma, at the presumptive position of axon initial segment (Fig. 10B, AxI). This phenomenon was more general, the first significant amplitude sink typically appears earlier on the channel 100 μm basal to soma, than on the soma (Fig. 10C). Next, the spiklets on far basal channels appear if observable. There was only one case (spike C), in which one of the observed spikelets appears before all the other significant amplitudes at the proximal part of the cell. The fitted propagation speed from the AxI to the spikelets was much higher than to the soma. This finding is consistent with the assumption that the spikelets belong to the Ranvier-nodes, as the propagation speed on a myelinated axon should be much higher that in the dendrite (Fig. 10D). As a next step, the action potential sCSD reaches its peak amplitude at the presumptive position of the soma. The BP into the dendrites starts from here, and proceeds faster into the apical and a bit slower into the basal direction (Fig. 10B).
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 IR should be determined separately. In perspective, this new method raises the possibility of identifying synaptic inputs, which causes a cell fire.
This research was supported by the Hungarian Science Foundation (OTKA K 81354) and the French-Hungarian grants ANR-TÉT Neurogen and ANR-TÉT Multisca. We thank György Buzsáki for his helpful comments on the manuscript, and Tamás Kiss and Nicholas F. Vogel for their help in manuscript preparation.