PMC

Distribution of signal (open bars) and noise (dark bars) in extracellular action potential (EAP) data recorded from single units with Thomas tetrodes in visual cortex (N=61). Noise level is defined by per-channel RMS amplitude; the mean (range) was 21 (12–35) µV. Signal level is defined by the spatial minimum of EAP peak amplitude on the tetrode channel that registered the largest peak; the mean (range) was 91 (25–313) µV

(a) The distribution of element size of the finite element mesh, as a function of the distance from the tetrode tip. Colors indicate different domains in the finite element model: wire domain (red), gray matter in neighborhood of tetrode (cyan), and more distant elements (blue). The horizontal dotted line indicates the radius of the region of interest; the vertical dotted lines indicate the characteristic size of tetrode features (see details in text). (b) The marginal histogram of the data in (a). (c)–(d) Distribution of the element quality analyzed similarly to element size. The vertical dotted line indicates minimum element quality recommended for 3-d problems

The geometry of a Thomas tetrode and its finite element model. This tetrode is the first one listed in . (a) Scanning electron microscopy images of the cleaned, untreated tip of the tetrode with axial view (top) and lateral view (bottom). The lighter contact surfaces of the PtW wire leads contrast well with the quartz coat. (b) The critical geometry parameters used in the tetrode models (defined and listed for each reconstructed tetrode in ). (c) The tetrahedral mesh of the finite element model of the probe-brain system, viewed in three cross-sections; top, vertical on-axis cut showing overall dimensions; middle, same cut but zoomed-in on the tetrode tip area where elements representing the lead wires are visible; and bottom, cut in the x-y plane at the z-level where the core wires are exposed as contacts. Note that element size varies extensively; it is smallest near the tetrode tip

The lead fields of a Thomas tetrode, calculated by the finite element method for one of the tetrodes. The parameters of the tip geometry for this tetrode (#03–0591) are listed in . The images show the lead field strength in color scale for two planar cross-sections for the center lead on the left (x-y plane through the tip point; y-z plane through the vertical axis of the probe), and for one eccentric lead on the right (x-y plane through the center point of the exposed lead surface). The color bar indicates the lead field strength on a log scale after conversion to equivalent probe potentials (µV) that the probe would register for a dipole source whose dipole moment equals 5pA*m (see text for explanation) and whose vector is iso-oriented with the lead field vector at each position in space. The lead field vectors at each point in space are oriented approximately in the direction of the gradients

The lead fields of sharp electrodes modeled with a wide variety of tip shapes. Spatial scale and field strength are as in . Lead field geometry depends only weakly on the details of the tip geometry. The radius of the lead fields defined at a criterion field strength (or equivalent signal strength) is weakly anti-correlated with exposed tip area (models are vertically ordered with increasing tip area) and with the cone angle of the shaft, and does not depend significantly on the cone angle of the exposed tip. Most notably, the typical lead fields of our tetrodes (e.g., M5) are very similar in size to those of typical sharp electrodes (all other panels). Specifically, M1 and M2 are similar to tips fabricated by Hubel (), and M2 and M3 are similar to tips fabricated by Ainsworth (). The varied geometry parameters and the resulting field radii are summarized in . Other details of the lead field modeling were the same as used for the tetrodes (see ) and text

Determination of errors in dipole characterization attributable to errors in modeling probe geometry. To analyze the effects of geometry error in a controlled fashion, we compared localization calculations based on the tetrode actually used to record each dataset (“true data”), with parallel calculations carried out as if the recording tetrode was one of the other two measured tetrodes (“pseudo-data”). We then summarized this comparison for each pseudo-data vs. true-data pair by regressing values obtained for three parameters ((a): fractional MSE, (b): estimated cell-probe distance, (c): estimated dipole moment) from pseudo-data, against their values obtained from the true data. The three panels show how these regression slopes depend on the geometry error, as quantified by the ratio of contact separation for the pseudo-data tetrode vs. true-data tetrode. Unity ratio on the horizontal axis means no error in probe geometry; unity ratio on the vertical axis means no distortion in a dipole measure. The three tetrodes are the first three listed in , and their respective mean contact separation was 45 µm; 38 µm; and 29 µm. The dotted curves are the prediction of an idealized error analysis (see Section 4)

Schematics of dipole optimization procedure using translated lead fields of a moving tetrode. (a) During data collection, the tetrode moves down (down arrows) the path of penetration in steps of size Δz_step, yielding measurements of the same source at 4n_step spatial points (black dots). From the probe’s point of view used in the analysis, real movement of the tetrode is equivalent to virtual movement of the source in the opposite direction (up arrow). x_s and x_i, i={0,1,2,3}, denotes the Cartesian coordinates of the position of the source and the ith lead in the first step, respectively, relative to the (moving) tetrode tip (x₀). (b) The tetrode lead registers the extracellular action potential of a single unit that is characterized by a single dipole current source with moment p. The dipole moment vector is translated to a new position relative to the probe at each step (x_s + kΔz_step). At each step, the model prediction of the probe potential is the scalar (dot) product of the dipole moment vector of the source, p, and the lead field vector of the probe at that relative position, L_i(x). Thus for a fixed physical source position, the dipole interacts with the lead field in a set of translated virtual positions. See text for the details of dipole optimization

(a) The error in the optimal dipole fits. The histogram shows the sample distribution of the fractional mean-squared error, i.e., the summed squared difference between model fit and EAP data, as a fraction of the summed squared data. Histograms from the “exact-probe” cells (dark bars; N=43), to the “approximated-probe” cells (open bars; N=18) are stacked. For definition of these subsets, see Section 2. The quality of the fit and the size of the error are indicated by two examples in the insets, one with a slightly better than typical fit (cell 68; fMSE 0.009), and the other with an atypically poor fit (cell 28, fMSE 0.094). Further examples of the typical in the sample are illustrated by the 4 cells shown in ; the fMSE for those cells were 0.034 for cell 4; 0.015 for cell 45; 0.039 for cell 6; 0.018 for cell 20. (b) Definition of the scatter radius—a measure of the error in dipole localization (see text for details). (c) The joint distribution, in log-log coordinates, of the fMSE error in the dipole fit (horizontal axis) and the error in dipole localization (vertical axis). Arrows indicate cells 6 (third example in ) as well as cells 28 and 68, the same two as in (a). The horizontaland vertical dotted lines indicate the sample medians. Note that the localization error (abscissa) and the fitting error (ordinate) are not correlated.

Analysis of the spatial variation of the size and shape of the extracellular action potential (EAP) waveform. (a) Example data set recorded from a single unit (cell 4) with a tetrode (06–3200, see ) in 9 equally spaced positions. Rows, labeled by nominal depth along the penetration, show the mean spike waveforms registered by the 4 channels of the tetrode; data from different channels are organized in separate columns. The vertical line indicates the sampling spatial EAP amplitudes at a fixed moment in time (here at the peak). By convention, negative extracellular potentials are plotted above the zero line. (b) Definition of the positive and negative peaks and the width of an EAP. We define an index of the spatial variation of these features by their relative spatial modulation, i.e., by the ratio of the difference of the spatial maximum and minimum to the spatial mean. (c) The distribution of the index of spatial variation of EAP size (the negative EAP amplitude) in the entire sample (N=61) of visual cortical neurons. The sample median was 0.51. (d) The distribution of an index of the spatial variation of EAP shape (the relative size of negative peak over positive peak) in the entire sample. The sample median was 0.15. (e) The distribution of another index of the spatial variation of EAP shape (width). The sample median was 0.10. Note that the horizontal scales in panels (c), (d), and (e) differ

Characterization of the diversity of the spatial variation of the EAP amplitude recorded with a stepped tetrode. (a) EAP variation is shown for four cells (tetrode channels are labeled by color); each represents high or low values of either of two shape indices: the index of non-monotonicity and the index of peak distinctness (see text for their definition). The continuous lines are the predictions from the optimal dipole fit. (b) The joint distribution of two indices of spatial EAP variation in the entire sample (N=61). Each symbol represents one cell: up triangles, monotonic increasing EAPs; down triangles, monotonic decreasing EAPs; 'x' symbols, discernible EAP peaks. The vertical line separates the cells of monotonic EAPs from cells of peaking EAPs. Examples featured in the text are numbered. (c) The linear spatial span of EAP’s. Analysis of the distribution of the measured linear span of detectable signal levels in the entire sample (‘All’). The stick and ball model (inset) helps to explain how such distributions depend on the relative size of the full linear span of spatial samples (sticks with fixed length l) detectability radius of neurons (ball with radius r). The thickened portion of a stick indicates the range of detectable EAP’s recorded from an isolated single cell within the full linear span of spatial samples. Their relative position defines 4 subsets of neurons: neurons whose spikes were isolated either (t₁) at all positions sampled; or (t₂) beginning with the first sample and ending before the last; or (t₃) beginning after the first sample and continuing through to the last; or (t₄) beginning after the first sample, and ending before the last. The vertical axes in (t₁)-(t₄) are omitted for clarity but the same vertical scale applies as in ‘All’

L-curve regularization of the optimal dipole for a single visual neuron. (a) The joint distribution of the size (abscissa) of the locally optimal dipole source and the associated residual error (ordinate) between the predicted and measured EAP’s. Each of these model dipoles (fine dots) was calculated by at one of the node points of a regular cylindrical mesh covering a finite volume surrounding the recording tetrode. Large dots mark data in the lower bound subset, the red line indicates the tangent envelope, and the smooth cyan curve is a 4-parameter empirical L-curve fitted to the lower bound. The L-shaped pattern suggests two model regimes. On the steep left segment, models increasingly well capture genuine physical features of the data at the cost of small increases in the dipole size (and, implicitly, cell-probe separation). The shallow segment to the right corresponds to the noise limit that allows no further improvement in capturing physical features. The optimal equivalent dipole is defined as the data point nearest to the corner point of the L (open circle). The cyan circular halo indicates the neighborhood of the 30 data nearest to the corner point. The point with the largest curvature on the tangent envelope (red circle) corresponds to a similar dipole. (b) Dipole localization of neurons via L-curve regularization is robust. The mapped locations of the locally optimal dipoles that fell within the lower bound subset are plotted in small symbols; the open circle marks the globally optimal dipole. Confirming that the optimization principle was realized, the 30 nearest neighbors of the L-curve corner (indicated by cyan halo in (a)), are mapped in a compact volume (cyan symbols) that represents a choke-point in the distribution; dipoles from the noise-limited flat right limb of the L-curve are mapped in an explosively expanded volume. Cell 10. (c)-(d): another example, illustrating the unstable nature of the prediction by the tangent envelop in discrete sample. In contrast to the corner of the L-curve (cyan circle), the point with the largest curvature on the tangent envelope (red circle) corresponds to physiologically unrealistic dipole parameters). Cell 37