Generation of field potentials and modulation of their dynamics through volume integration of cortical activity

Abstract

Field potentials (FPs) recorded within the brain, often called “local field potentials” (LFPs), are useful measures of net synaptic activity in a neuronal ensemble. However, due to volume conduction, FPs spread beyond regions of underlying synaptic activity, and thus an “LFP” signal may not accurately reflect the temporal patterns of synaptic activity in the immediately surrounding neuron population. To better understand the physiological processes reflected in FPs, we explored the relationship between the FP and its membrane current generators using current source density (CSD) analysis in conjunction with a volume conductor model. The model provides a quantitative description of the spatiotemporal summation of immediate local and more distant membrane currents to produce the FP. By applying the model to FPs in the macaque auditory cortex, we have investigated a critical issue that has broad implications for FP research. We have shown that FP responses in particular cortical layers are differentially susceptible to activity in other layers. Activity in the supragranular layers has the strongest contribution to FPs in other cortical layers, and infragranular FPs are most susceptible to contributions from other layers. To define the physiological processes generating FPs recorded in loci of relatively weak synaptic activity, strong effects produced by synaptic events in the vicinity have to be taken into account. While outlining limitations and caveats inherent to FP measurements, our results also suggest specific peak and frequency band components of FPs can be related to activity in specific cortical layers. These results may help improving the interpretability of FPs.

the field potential (FP) recorded within active neural tissue is an information-rich measure that arises mainly from synaptically driven transmembrane currents related to excitability fluctuations in ensembles of neurons (Buzsáki et al. 2012; Mitzdorf 1985). Decoding of FPs can extract information about motor commands (Bansal et al. 2012; Ince et al. 2010), sensory stimuli (Belitski et al. 2010; Kayser et al. 2007), and current behavioral and cognitive state (Fries et al. 2001; Lakatos et al. 2009, 2013; Scherberger et al. 2005; Steriade et al. 1993). This is true whether the FP is tightly or loosely phase-locked to an identified stimulus [so-called “evoked” and “induced” FPs (Makeig et al. 2004; Shah et al. 2004)], and it also holds for “spontaneous” activity, whose immediate antecedents are obscure (Fukushima et al. 2012; Lakatos et al. 2005).

Despite its strengths, the interpretation of FP is complicated by volume conduction, the spread of electric field generated by current sources in a conductive medium. Because FPs are typically recorded with an active recording electrode amplified against a reference signal from a distant electrode, the exact point of origin of the signal is unknown due to volume conduction (Einevoll et al. 2013; Nunez and Srinivasan 2006). The signal could in theory arise anywhere within the conductive medium housing the electrodes. Understanding of the principles of volume conduction allows one to track down the brain sources of electroencephalographic (EEG) and event-related potential (ERP) components (Godlove et al. 2011; Schroeder et al. 1995). This same understanding also prompts the use of techniques that isolate relatively local activity such as current source density (CSD) (Einevoll et al. 2013; Kajikawa and Schroeder 2011; Mitzdorf 1985; Nicholson and Freeman 1975, Pettersen et al. 2006) or the surface Laplacian, a two-dimensional approximation of the second derivative of potential, used to derive scalp current density (Nunez and Srinivasan 2006; Tenke and Kayser 2012). CSD analysis estimates the laminar profile and magnitude of current sources (net local outward currents) and sinks (net local inward currents) that produce fluctuation of voltage in the conductive medium of the extracellular space (Nicholson and Freeman 1975) and is in most experimental cases explored by analyzing the one-dimensional spatial pattern of FPs (but see Riera et al. 2012). Previous ERP/EEG studies have shown that ERP peak components and EEG frequency components (bands) have distinct scalp topographies and source localizations. The same conditions apply to the laminar-spatial patterns of FP and CSD signals within a cortical area. This is a critical problem for LFP studies, because it limits their ability to attribute experimental effects to specific cell populations (e.g., supragranular pyramidal cells) or input types (e.g., top-down vs. bottom-up).

Effects of volume conduction at a macro (>1 mm) scale are recognized as essential to the formation of the spatiotemporal profile of electromagnetic signals that can be measured in ERP and EEG recordings at the scalp (Nunez 1998; Nunez and Srinivasan 2006; Pascual-Marqui 1999), but these effects at a micro (< 1 mm) scale are poorly understood (Bédard and Destexhe 2011, Buzsáki et al. 2012, Einevoll et al. 2013, Riera et al. 2012). Recent reports on the related topic of the “spatial spread of the LFP” have outlined several variables that likely affect volume conduction beyond the boundaries of active (generator) tissue. Some have emphasized spatial variables, such as the size and shape of the generator substrate that is the population of neurons whose transmembrane currents are reflected in an FP, and the magnitude of the activation (Kajikawa and Schroeder 2011), whereas others have emphasized temporal variables, such as the synchrony of local cellular activity (Lindén et al. 2011; Reimann et al. 2013). Critically, in most circumstances, synchrony and strength of activation are confounded. On one hand, a given ensemble of neurons is expected to produce a stronger net response, as reflected by a larger FP signal magnitude, when they are activated simultaneously than when they are activated asynchronously (e.g., Reimann et al. 2013). On the other hand, we cannot directly relate the magnitude of FP signal to the degree of synchronous firing in a given neuron population because, as discussed above, synchrony is not a sole factor that determines spatial spread of the FP. In the present study, we deal with the spread of FP activity within hot spots of highly synchronized activation; i.e., we examine local neuronal responses to preferred stimuli that excite massive and synchronous activation of local neuron ensembles. In this circumstance, high synchrony can be assumed. Interestingly, such strong, synchronous, responses, likely encompassing multiple cortical layers, are reported in many papers. However, investigation of how local (i.e., submillimeter scale), well-correlated (i.e., sensory-evoked response) activity patterns of neuron populations in different layers of cerebral cortex contribute to the construction of a local FP began only recently.

A simulation study investigated the contributions of electrical activity of neurons and their compartments to LFP at such scales and suggested the potential role of the active membrane conductance that generates action potentials in shaping the spatiotemporal patterns of FP (Reimann et al. 2013). Regardless of types of membrane conductance and difference in patterns of constituent neurons, transmembrane currents of neuronal population can be translated to CSD features. In sensory systems, sensory stimuli effective in driving strong neuronal firing responses also evoked larger CSD responses (Fishman and Steinschneider 2006; Kajikawa and Schroeder 2011) and larger postsynaptic currents (Tan et al. 2004) that corroborate the close correspondence between CSD components and transmembrane currents.

Detailed understanding of the contribution of local activity to local FP is particularly important at this time, since studies are beginning to investigate laminar and sublaminar distributions of FPs in different frequency bands to help to sort out influences of feedforward and feedback projections (Buffalo et al. 2011; Spaak et al. 2012; Sundberg et al. 2012). Importantly in this regard, distinct frequency bands can be dissociated by their differential behavior in both event-related responses (Fries et al. 2011; Khayat et al. 2010; Ray and Maunsell 2011; Wilke et al. 2006) and ongoing activity (Lakatos et al. 2005; Magri et al. 2012). However, the spatiotemporal summation by which these microprocesses combine to form a local FP has not been directly investigated in physiological data. In an effort to advance the understanding and physiological interpretation of FP measures, this study examined the contribution of volume conduction to the generation of FPs at a sub-millimeter scale. We used a volume conductor model constructed on physical principles to spatially integrate “observed” CSD activity distributed across cortical depths to calculate a “predicted” spatiotemporal pattern of intracortical FPs and evaluated the similarity between the originally observed and model-derived FP patterns. Our results indicate that volume conduction integrates distributed activity across layers to shape the temporal pattern of an FP in any given layer. Our results also suggest that particularly in the infragranular layers where local FP generation is relatively weak, the form and frequency content of the locally recorded FP may largely reflect activity in other cortical layers.

MATERIALS AND METHODS

All experimental procedures were approved by the Institutional Animal Care and Use Committee of the Nathan S. Kline Institute.

Subjects.

Six macaque monkeys were implanted with a headpost and one or two recording chambers during aseptic surgeries. The chambers were oriented to make penetrations perpendicular to the lateral sulcus.

Stimuli.

Calibrated binaural tone stimuli were delivered through two free field speakers directed to ears (Tucker Davis Technologies). Tones ranging from 353.55 Hz to 32 kHz with 0.5-octave intervals (14 frequencies) were presented at 60 dB SPL (duration: 100 ms, stimulus onset asynchrony: 625 ms).

Recordings.

All recordings were done while monkeys were awake. We used electrodes of linear arrays of 23 electrical contacts spaced either 100 or 200 μm apart (0.3–0.5 MΩ at 1.0 kHz) to record signals at different cortical depths. FPs (0.1–500 Hz) were recorded from all contacts simultaneously and sampled at 2 kHz. As a reference (or indifferent) electrode, a metal pin immersed in saline filling the recording chamber was used. The best frequency (BF) of multiunit activity responses to tones was identified in each recording site (Kajikawa and Schroeder 2011). In the present study, responses to BF tones were analyzed.

Current source density.

CSD was calculated from FPs recorded from three adjacent electrode contacts by the second-order finite differences:

$$CSD\left(\mathit{z}\right)=\left[FP\left(\mathit{z}+d\mathit{z}\right)-2FP\left(\mathit{z}\right)+FP\left(\mathit{z}-d\mathit{z}\right)\right]/d{\mathit{z}}^2,$$ (1)

as the one-dimensional approximation of the second-order spatial derivative of the FP (Mitzdorf 1985), with an assumption of homogeneous conductivity (see discussion):

$$\nabla \left[\sigma \nabla \text{Φ}\left(\overrightarrow{\text{r}}\right)\right]=\sigma {\nabla }^2\Phi \left(\overrightarrow{r}\right)=\sigma \left(\frac{{\partial }^2\text{Φ}}{\partial {\text{x}}^2}+\frac{{\partial }^2\Phi }{\partial {\text{y}}^2}+\frac{{\partial }^2\Phi }{\partial {\text{z}}^2}\right)\cong \sigma \frac{{\partial }^2\Phi }{\partial {\text{z}}^2},$$ (2)

The generality of this method of CSD approximation (Eq. 1) has been questioned recently because of generator “edge” effects in rodent barrel cortex (Einevoll et al. 2013), with regard to traditional assumption of “transpositional invariance” to validate Eq. 1 performed only in vertical dimension across cortex (Mitzdorf 1985). However, the solution proposed by Einevoll and colleagues, “iCSD” (discussed below), was devised for the unique structure of rodent barrel cortex and does not generalize to the neocortex as a whole. Here, to support the last equality in Eq. 2 as an approximation, transpositional invariance is only a special case of generally required conditions that satisfy the inequality |∂V/∂z|≫|∂V/δx|,|∂V/∂y| along the vertical path of array electrodes (see discussion). As Fig. 2D will show, the condition is usually met when suprathreshold stimulation excites extensive regions of cortex (see also Reimann et al. 2013). Thereby, it is appropriate for a wide range of neocortical applications, including the present case. In the present study we used this standard approximation for compatibility with the vast majority of prior studies.

Fig. 2.

Volume conduction effects of 2 model local field potential (LFP) generators, differing in amplitude and orientation, acting either in isolation (A and B) or in combination (C and D). A: sink (cyan) and source (magenta) have equal magnitudes and opposite signs. Both have spatial distributions (i) delimited to a common circular area around x = 0 and y = 0 in x-y planes that differ in z positions: z = 0.125 and 0.375 for sink and source, respectively (arrows in ii). The absolute amplitude profile of sink or source within the x-y plane is shown in inset at bottom of Bi: the amplitude is constant within the inner half of the radius, circumscribed by cosine attenuation over the second half of the radius. The spatial FP profile generated by the source/sink pair shown in Ai was calculated using Eq. 5 and is shown in Aii. The color scale is clipped at the quarter of the peak amplitude. The unit of distance, although arbitrary, is common to horizontal and vertical dimensions. Note that the horizontal axis is compressed 4 times relative to the vertical axis. Due to symmetry, the same pattern appears for all planes along the vertical line at x = y = 0. Amplitudes of FP (red) and CSD (green) signals at the center of x-y planes are shown in Aiii. B: same format as A, except that the magnitudes of sink and source were one-fifth of those in A (inset at bottom of Bi) and z-positions of sink and source are z = −0.125 and −0.375, respectively. C: same format as Aii and Aiii, showing FP generated by superimposing both pairs of sinks and sources in A and B. D: spatial profile of the ratio of absolute values of horizontal derivative to vertical derivative of FP shown in Aii, Bii, and Ci. FP amplitudes are normalized to the peak amplitude of FP in A.

Spectral analyses of FP and CSD.

To derive the spectrotemporal patterns and frequency bands of those FP and CSD [FP(t) and CSD(t), respectively], we used the complex Morlet wavelet

$$\psi \left(t\prime \right)=\frac{1}{\sqrt[4]{\pi }}{e}^{i{\omega }_{0}t\prime }{e}^{\frac{t\prime }{2}},\left({\omega }_0=6\right),$$

to calculate the wavelet transforms of those signals (Torrence and Compo 1998):

$$\begin{array}{cc}W{T}_{FP}\left(t,f\right)=& \sqrt{\frac{\delta t}{s}}{\displaystyle \int FP\left(t\prime -t\right)\cdot {\psi }^{*}\left(\frac{t\prime -t}{s}\right)\mathrm{d}t\prime ,}\hfill \\ W{T}_{CSD}\left(t,f\right)=& \sqrt{\frac{\delta t}{s}}{\displaystyle \int CSD\left(t\prime -t\right)\cdot {\psi }^{*}\left(\frac{t\prime -t}{s}\right)\mathrm{d}t\prime ,}\hfill \\ \hfill f=& \frac{{\omega }_0+\sqrt{2+{\omega }_0^2}}{4\pi s},\hfill \end{array}$$ (3)

in which δt is the sampling interval. The complex Morlet wavelets were scaled to conserve the energy relationship between frequency components. The translation step was 1 ms. Center frequencies of the wavelet stepped from 1 to 256 Hz with 0.1-octave intervals. This dense frequency sampling leaves high redundancy between neighboring frequency bands of the wavelet transform and allows reconstruction of the original signal as (Farge 1992)

$$FP\left(t\right)=\mathrm{Re}\left[{\displaystyle {\int }_1^{256}W{T}_{FP}\left(f,t\right)\mathrm{d}f}\right]·$$

Power and phase spectrotemporal distributions were derived as the absolute values and phase angles of WT_FP(t,f) and WT_CSD(t,f).

Frequency band signals of FP and CSD were calculated as the sums of 16 center frequencies (δ: 1.0–2.9 Hz, θ: 3.0–8.8 Hz, γ₁: 27.7–81 Hz, γ₂: 83.9–256 Hz) or 8 center frequencies (α: 9.1–15.9 Hz, β: 17.0–25.8 Hz) of WT_FP(t,f) and WT_CSD(t,f) along the frequency axis (Kajikawa and Schroeder 2011). Derived signals were ∼1.6 or 0.8 octaves wide with small overlaps between frequency bands and were complex-valued, the same as WT(t,f). The real part of signals (e.g., ℜ[WT(t,f)]) was identical to the transform with real part of wavelet (e.g., replacing Ψ(t′) in Eq. 2 with ℜ[Ψ(t′)]). Because the original signals were real-valued, real parts of derived signals corresponded to the filtered signals and were used as frequency band-limited signals (Farge 1992). Absolute amplitudes of the complex transforms were used as the power of signals.

Volume conductor model.

The volume conductor model assumed that CSD at different depths contributed to the FP by volume conduction. Under the assumption of homogeneous tissue conductivity, volume-conducted signals decay in proportion to the inverse of distance with no assumption of frequency-dependent gain change; this predicted decay is matched by empirical findings (Kajikawa and Schroeder 2011; Schroeder et al. 1992, 1998). We calculated FP at a certain depth d_k that could be generated by a given spatial distribution of CSD at every moment as

$$vcFP\left(k,t\right)=A{\displaystyle \sum _j\frac{CSD\left(d_j,t\right)}{\sqrt{h^2+{\mid d_{\mathrm{j}}-d_{\mathrm{k}}\mid }^2}},h=r_{h2d}·d,}$$ (4)

in which d (= d_k+1 − d_k) is the spatial distance between neighboring contacts of array electrode. The parameter h represents a displacement distance of the center of mass of CSD from the array electrode assuming all CSD components were vertically aligned across cortical depths. The idea behind the displacement is as follows.

In general, there are many possible solutions of sources that could generate topographic or depth patterns of FP in solving Eq. 2. In the present case, the only boundary was the line of the array electrodes with depth-distributed signals with surrounding open space. It was not definitive in finding particular CSD solutions because there are many possible configurations of CSD. However, to be consistent with the columnar structure of cortex, one constraint we could presume was that CSD distributed vertically across cortical layers. That reduced the problem to one of determination of the finite horizontal spread. However, again, any horizontal spread of CSD could be replaced by a variety of other CSD to create the same spatial pattern of FP. Thus it would be an ideal approach to reduce the arbitrariness of horizontal spread and model it as simply as possible, since at this time we cannot realize more complex (parametric) models in detail or justify them.

Responses to suprathreshold pure tones in auditory cortex must distribute as medial-lateral stripes defined by isofrequency contours. The shapes of stripes were kept similar across cortical layers and horizontally finite, constrained by the anterior-posterior extent of an isofrequency domain. Any such spatial distributions of sources may be substituted by one or several charge densities located at a horizontal point somewhere inside of responsive zones to generate the same field potential patterns. That is equivalent to what is done at a different spatial scale in source estimation for EEG/magnetoencephalography (MEG). The anchoring point around which spread is estimated is a single point at the center of mass in a horizontal distribution of sources. Its exact position may remain unknown. Only its distance from electrodes is needed in deriving FP at the electrodes. Thus we included the horizontal displacement parameter h to represent the horizontally delimited distribution of activity in the simplest way. Although such a simplified model can introduce error (uncertainty), it is preferable to a model with more unknown parameters, unless they can be empirically validated.

The output of the resultant model becomes equivalent to that generated by a dipole-like substrate (i.e., having a vertical distribution), horizontally displaced at an average distance h from the recording electrode array across depths (Kajikawa and Schroeder 2011; Somogyvari et al. 2012). In Eq. 4, r_h2d was the sole free parameter that could influence the shape of vcFP(k,t). Its values were chosen to maximize the similarity (see below) of spatiotemporal patterns between the calculated and the observed FP. The parameter h = r_h2d·d may be considered to be the distance to the center of horizontally distributed sources generating FP from the electrodes. Note that because of the linearity of both wavelet decomposition and volume conduction, the order of calculating them is interchangeable. However, once the power or phase of wavelet-decomposed signals is derived, these values cannot be entered into the volume conductor model because of the nonlinearity of power/phase calculations.

Within a “column,” vertical profiles of CSD that contributed to the FP at a given depth were derived as follows. The spatiotemporal CSD profiles were weighted along the depth axis by their distances from the origins of CSD with an additional displacement h. Wavelet-derived frequency band signals were derived in a similar manner. The spatiotemporal profiles of CSD power were derived after distance-weighing.

Depth distributions of the power of CSD contributions to granular FP responses and phase differences of CSD from granular FP responses were derived as follows. The mean of the CSD power over a 150-ms period following onset of tone was calculated for each CSD and subtracted by mean power during a 150-ms period preceding the tone onset. For data sampled with 200-μm intervals, missing data points were linearly interpolated. For each frequency band, such distribution was normalized by the power at the depth of the granular sink. Since depths of cortical layers relative to electrode positions differed between penetration sites, depths of granular sinks were aligned to zero. Median and confidence intervals at each depth were derived after alignment.

Whereas the field generated by dipoles decays with the inverse of the square of distances, 1/|r|², the volume conductor model treated individual poles of multipoles as separate monopoles whose field decayed with the inverse of the distance, 1/|r|, not squared. However, it should be noted that superimposition of electric fields generated by two monopoles of opposite signs decaying with 1/|r| becomes the same as an electric field that decays with 1/|r|² at distances relatively larger than the dimension of the dipole. Therefore, although the model treats CSD features as spatially distributed monopoles, it can generate the field potentials generated by dipoles, as well.

This implementation was preferred because recordings in the present study were done within the zones of active generators or dipoles, where the dipole field approximation's relevance is questionable.

Similarity score.

To quantify the similarity between temporal patterns of FP and CSD, and the similarity between the spatiotemporal patterns of the observed FP and CSD or model-derived FP, we calculated the similarity score as described below. Temporal patterns of FP, CSD, and model-derived FP at individual recording channels were expressed as vectors ν⃗_FP, ν⃗_CSD, and ν⃗_vcFP of n_T independent samples, where n_T = sample duration × sampling rate. Spatiotemporal profiles were expressed as matrices M_FP, M_CSD, and M_vcFP, whose number of rows corresponded to the number of recording contacts of electrodes (23 and 21 for FP and CSD, respectively). Spectrotemporal patterns of signals at each channel were expressed similarly in matrices WT_FP, WT_CSD, and WT_vcFP, whose number of rows corresponded to the number of wavelet center frequencies. Before calculation of the similarity, these vectors and matrices were normalized by the root of mean squared values [e.g., ν⃗/mean(|ν⃗|²) or M/mean(|M|²)] to compensate for their difference in the signal power and to analyze shapes in isolation. For matrices, the mean was calculated across both rows and columns. Similarity between those patterns was calculated as the inner product between vectors or the Frobenius inner products of matrices after normalization. The similarity value ranged from −1 for completely mirrored shapes of opposite polarities to 1 for identical shapes of same polarities regardless of magnitudes. Although similarity could depend on the periods of FP, CSD, or their wavelet transforms, values were derived using signals between −30 and 170 ms from the onset of sound. However, the similarity score is model free in that it can be evaluated for any models or analysis that creates spatiotemporal patterns as long as it keeps dimensions (the number of depth channels and time bins) constant.

Statistics.

All statistical tests were done nonparametrically with a criterion level of P = 0.01, except for ad hoc comparisons at P = 0.05. Ninety-five percent confidence intervals were derived by bootstrap using the Matlab function “bootstrp” with 1,000 resampling (see Figs. 4 and 6, A and B).

Fig. 4.

Effects of spatiotemporal summation on the similarity between the observed and the volume conductor model-derived FP. A–C: median similarity score of temporal patterns between the observed FP responses and the FP responses derived by the volume conductor model are plotted as a function of the number of proximal CSD signals included in the model for Sg (A), Gr (B), and Ig signals (C). Dotted lines are 95% confidence intervals (n = 130), derived by bootstrap. D: box plots showing the channel increment size (in medians and quartiles) necessary for the similarity score to reach 90% of its maximum net change.

Fig. 6.

Spread of CSD over cortical depths and layers in each frequency band. Columns represent plots of 6 frequency bands (δ, θ, α, β, γ₁, and γ₂) as indicated at top. A: spread of CSD power that contributed to the Gr layer FP in the volume conductor model. CSD power was estimated as an increase in the mean power during 0–150 ms relative to the mean during −150–0 ms. For each laminar profile, CSD signals across depth were weighted by the inverse of the distance from the Gr layer (schematic illustration at right). Their powers were then normalized by the magnitude of the weighted Gr CSD power. Median value of normalized CSD power is plotted against depth relative to the Gr layer sink; positive values extend above the Gr layer. B: increase in the similarity between the observed and the volume conductor model-derived Gr layer FP as a function of the number of proximal CSD signals included in the model. Channel increment (abscissa) equal to 1 corresponds to the similarity between the observed FP and colocated CSD calculated from the FP. Data plotted in A and B are median values and their 95% confidence intervals (n = 130, bootstrap). C–E: laminar distributions of increments of distance-weighted CSD power (by frequency band) in Sg, Gr, and Ig layers (green channels in the schematic illustrations at right) at the peak timing of FP power (by frequency band) in Sg (C), Gr (D), and Ig (E) layers (red channels in the schemes at right), respectively. Increments were from the mean CSD power during a baseline period of −150 to −10 ms from the sound onset and were normalized by the total power of all frequency bands in Gr layer at each peak timing of a layer, showing the power of 3 layers' contributions to that layer. Box plots show the median and quartiles (n = 130). Note that ranges on the x-axes differ between plots. Where normalized CSD power differed significantly between layers (Friedman's nonparametric repeated-measures ANOVA, P < 0.05), asterisks indicate pairs of layers that had significant (Tukey's honestly significant difference test; *P < 0.05) or borderline significant (**0.05 < P < 0.1) differences indicated by post hoc analyses.

RESULTS

Sensory-evoked FP and CSD signals at a given location in the brain often differ in temporal pattern even though both signals are considered to reflect local synaptic activity. We first elaborate the difference between these signals in both temporal and spectral domains. We then describe how the volume conductor model bridges the gap between the two signals. We further show that the FP integrates local and distant activity in individual frequency bands independently of the spatial distributions of the CSD activity of other frequency bands.

Differences between colocated FP and CSD signals.

First, we show the difference in temporal patterns between FP and CSD. Figure 1 shows examples of FP and colocated CSD responses recorded simultaneously from 1) supragranular (Sg), 2) granular (Gr), and 3) infragranular (Ig) layers in an A1 penetration, along with spectrotemporal patterns of signal power and phase difference. FPs in the different layers share similar patterns starting with onset negativity (Fig. 1A), but after this point waveforms diverge between Sg and other layers. This is not surprising because FP patterns are expected to change systematically as recordings traverse an active FP generator region (Mitzdorf 1985; Schroeder et al. 1995), and the strongest CSD activity was found in the Sg layer. The point of interest here, however, is the contrast between colocated FP and CSD signals, because it reveals effects of volume conduction. The FP-CSD contrast is described systematically by comparison of their temporal patterns (B), low-frequency power plots (C vs. D), and phase differences (E). In Sg layers, both FP and CSD signals maintained negative deflections after the onset until 100 ms. Low-frequency power had similar peaks. In Gr layers, although temporal patterns looked similar, the onset negativity occurred earlier in CSD than in FP and had a nearly 90° phase difference in high-frequency bands. In Ig layers, the larger positive/negative fluctuations in FP were more rapid than the source/sink fluctuations in the CSD signal. Thus the peak FP power was higher in a higher frequency band than peak CSD power.

Fig. 1.

Differences in spectrotemporal patterns between field potential (FP) and current source density (CSD). A: spatiotemporal profiles of FP (left) and CSD (right) responses to the best frequency tone. Horizontal lines delineate borders between cortical layers. Three arrows indicate supragranular (Sg), granular (Gr), and Infragranular (Ig) channels whose signals correspond to those shown in B–E (i: Sg, ii: Gr, iii: Ig). B: temporal patterns of FP (red) and CSD (green). Vertical lines indicate the onset and offset of tones. Superimposed traces were rescaled to match the onset negative amplitudes. C and D: spectrotemporal power distributions of the FP (C) and CSD (D) shown in B. Also in each, red and green contours circumscribe regions of spectrotemporal pattern where the power was above 10% of the peak in each profile. E: spectrotemporal distribution of phase difference (|Δθ|) between FP and CSD. Red and green contours in C and D are superimposed. F: histograms of the similarity score (n = 130), calculated by the Frobenius inner product of temporal patterns of the FP and CSD signals. Scores of 1 correspond to identical FP and CSD signal shapes, and scores of −1 correspond to identical shapes of opposite polarity. Arrows point to median scores. Freq., frequency.

To assess the dissimilarity between FP and CSD across penetration sites, we quantified the similarity score as an inner product of vectors representing temporal patterns of two signals in three cortical layers (Fig. 1E). The score changes from 1 when FP and CSD had identical temporal patterns to −1 when two patterns were of opposite polarity and mirror images of one another. The scores for 3 pairs in Fig. 1B were 0.85 (i), 0.44 (ii), and −0.08 (iii), respectively. The distributions of the score were skewed toward 1 in all cortical layers. Median scores were 0.81, 0.75, and 0.33 for Sg, Gr, and Ig layers, respectively (n = 130), with significant difference [Friedman's nonparametric repeated-measures ANOVA, χ_γ²(df = 2, n = 116) = 30.4, P < 0.01, excluding penetration sites missing Ig layer]. This difference was attributable to the lower scores of Ig layer compared with other layers (Tukey's honestly significant difference test, P < 0.05). Thus temporal patterns of FP and CSD were more dissimilar in Ig layers than in Gr and Sg layers. However, even in Gr and Sg layers, the scores were distributed over wide ranges. These results suggested the generality of the differences in spectrotemporal patterns between colocated FP and CSD signals. Below, we show how the discrepancies between the two signals are explained by volume conduction and how their effect is evaluated using the similarity score.

Volume conductor model.

We explain the volume conductor model first and then fill the gap between FP and CSD by applying the model to experimental data. CSD signals reflect transmembrane current flow patterns that are localized to the spatial extent of neuronal processes (1). The local one-dimensional CSD profile is customarily derived from a laminar FP profile using Eq. 1 by approximating σ∇²Φ($\vec{r}$) = −q($\vec{r}$), in which σ is the tissue conductivity (assumed to be spatially uniform), Φ($\vec{r}$) is the FP, and q($\vec{r}$) is CSD. FP reflects spatially distributed events q($\vec{r}\prime$) through volume conduction. When the boundary condition of Φ(∞) = 0 is met, σ∇²Φ(r⃗) = −q(r⃗) can be solved as

$$\Phi \left(\vec{\mathrm{r}}\right)=\frac{1}{4\pi \sigma }{\displaystyle \int \frac{q\left(\vec{r}-\vec{r}\prime \right)}{\mid \vec{r}-\vec{r}\prime \mid }\mathrm{d}}\vec{r}\prime ·$$ (5)

Since auditory cortical FP responses to BF tones are several hundreds of microvolts and scalp ERPs centimeters away from the auditory cortex are generally on the order of 1 μV, we used this solution as an approximate model of a local volume conductor. There is no constraint on the spatial distribution of q($\vec{r}$) within a body. The volume conductor model summates spatially distributed CSD signals after weighing them with the inverse of their distances from the point of measurement as in Eq. 5, which weakens distant CSD to some degree.

Figure 2 shows how Eq. 5 governs volume conduction when there are two different FP generators, each consisting of sink and source, operating in isolation (one strong source/sink, Fig. 2A; one weak sink/source, Fig. 2B) or together (Fig. 2C). Sinks and sources modeled here have round shapes delimited to two parallel flat planes spread horizontally (xy; Fig. 2, Ai and Bi). FP profiles were calculated based on Eq. 5. Because of symmetry of sink and source shapes around their centers, FP profiles formed in planes that contain x-y centers of sink and source are identical. When there is one FP generator, inversion of FP polarity occurs at the mid z-position between sink and source (Fig. 2, Aii and Bii). Above and below the inversion, the FP spreads with constant polarity with amplitude peak positions aligned vertically to sink and source (Fig. 2, Aiii and Biii). However, when two FP generators coexist, FP distributions from single generators superimpose, and an FP distribution reflecting an interaction of the two generators emerges. Ventral spread of the negative FP below the inversion of the single strong generator (Fig. 2A) is reduced in the combined case (Fig. 2C) and is interrupted by small positivity due to the lower generator's source. Compared with the positive peak below the inversion in Fig. 2B, the positive peak at the corresponding z-position is smaller in Fig. 2C. Thus, within the cortex where activity spreads over cortical layers through volume conduction, the FP in any one layer is “contaminated” by signals of activity generated in other layers. We utilized the volume conductor model to specify this effect quantitatively.

Figure 3 shows how the volume conductor model links sensory-evoked FP and CSD responses. At all penetration sites, we simultaneously recorded FP at 23 depths spanning the superficial to deep layers of auditory cortex, as illustrated by the representative FP profile in Fig. 3C, from which the spatiotemporal CSD profile in Fig. 3A was derived. The volume conductor model-derived FP profile for these signals (described below) is shown in Fig. 3B. Figure 3D illustrates the gradual change in the FP produced by the systematically increasing spatial range of the CSD profile that is incorporated into the volume conductor model to derive Gr layer FP waveform. This is shown in increasing order from top to bottom, starting from a summation of one channel that corresponds to Gr layer CSD itself. Whereas temporal patterns of the observed FP and CSD (bottom red and top green traces in Fig. 3D) differed in their temporal patterns, the FP generated by the model (traces between the top and bottom in Fig. 3D) became successively closer approximations of the observed FP as the spatial range of CSD estimates (incorporated by the model) was increased. In this example, the model broadened the onset negativity and delayed the timing of the following positive peak (asterisks in Fig. 3D) from 38 to 62 ms. This effect indicates that the Gr layer FP was influenced by current generators (indexed by CSD) in other layers. By performing the same model calculation over all depths of FP recording, we calculated a model-derived spatiotemporal FP profile (Fig. 3B), which resembles the profile of the observed FP (Fig. 3C) more closely than it does the CSD profile. Accordingly, the spectrotemporal pattern of the model-derived FP for the Gr layer also bears greater similarity to that of the observed FP than that of the CSD signal (Fig. 3, E–G).

Fig. 3.

Implementation of the volume conductor model illustrated for a site in A1. A: a spatiotemporal CSD profile (current sinks, red; current sources, blue). B: the volume conductor model-derived FP profile (positive, red; negative, blue). C: a spatiotemporal FP profile (positive, red; negative, blue). CSD profile in A was derived from FP profile in C. FP profile in B was derived from CSD profile in A. Equations at left indicate how the CSD the profile was derived from the observed FP profile and how the model-derived FP was reconstructed from the CSD profile. Dashed lines delineate borders between Gr, Sg, and Ig layers. D: CSD (top, green) and FP waveforms (bottom, red) in Gr layer. Waveforms displayed in between (middle) are of FP derived by volume conductor model with an increasing number of neighboring CSD channels taken into the model, as indicated by the brackets at right of the CSD plot. Asterisks label positive peaks in CSD and FP (see text). E–G: spectrotemporal patterns of CSD power (E), observed FP power (F), and derived FP power (G) in Gr layer.

For the example shown in Fig. 3, the similarity score of the observed FP with the colocated CSD signal was 0.38 and increased to 0.95 when computed using the model-derived FP. Also, across penetration sites (n = 130), the similarity over the 150-ms-long temporal patterns between the observed FP in Gr layer and the model-derived FP increased as a function of the number of proximal CSD signals (Fig. 4B). Improvement of the score through use of the model waveform was also observed for FP in Sg and Ig layers (Fig. 4, A and C).

In all layers, the score reached asymptote by integrating signals within <1 mm. This suggested that FP responses to the BF tones within the auditory cortex were shaped by activity of variable temporal patterns occurring in the gray matter over the range of a millimeter where strong activation is spread. It also may be noted that the model-derived FP in Sg layer, where local CSD activity was stronger than in other layers (see Fig. 6C), still increased its similarity to the observed FP by integrating activity over several hundreds of micrometers. These results suggested that even at loci where the CSD signals were strong, the effect of volume conduction would not be negligible due to activity in their vicinity.

Notably, the FPs in Ig layers (which are farthest from Sg layers) had the lowest similarity scores. FPs in these layers required the widening of spatial coverage in the model to include CSD in the Sg layers in order for the similarity score to asymptote. We estimated the channel increments needed for the scores to reach 90% of the net changes, for all 3 layers in each penetration site (Fig. 4D). Those channel increments differed significantly between layers [Friedman's nonparametric repeated-measures ANOVA, χ_γ²(df = 2, n = 116) = 38.2, P < 0.01, excluding sites missing Ig layer data]. This difference was attributable to the channel increments needed significantly more in Ig layer than in other layers (Tukey's honestly significant difference test, P < 0.05). Thus, to define the physiological processes generating FPs recorded in loci of relatively weak synaptic activity (e.g., the Ig layers), stronger synaptic events in the vicinity (e.g., those in the Sg layers) have to be taken into account.

Contributions of different frequency bands to volume-conducted signals.

Below we expand the consideration of volume conduction to the relationships between components of FP and CSD. Sensory cortical responses are not only distributed across layers but also are transient and of broad spectral content. A CSD signal q(r⃗,t) can be considered as a linear summation of the CSD frequency bands q_f(r⃗,t) as q(r⃗,t) = $\sum _f{q}_f(\vec{r},t)$. By incorporating this into Eq. 5, FP signals decomposed into frequency bands in the volume conductor model are

$$\Phi \left(\vec{r},t\right)=\frac{1}{4\pi \sigma }{\displaystyle \int \frac{{\displaystyle \sum _f{q}_f\left(\vec{r}-\vec{r}\prime ,t\right)}}{\mid \vec{r}-\vec{r}\prime \mid }}\mathrm{d}\vec{r}\prime ·$$

Due to linearity of both volume conduction for spatially distributed CSD signals and spectral transforms for frequency bands in each CSD signal, those operations are interchangeable. Thus the FP due to volume conduction can be derived separately for different frequency bands as

$${\Phi }_f\left(\vec{r},t\right)=\frac{1}{4\pi \sigma }{\displaystyle \int \frac{q_f\left(\vec{r}-\vec{r}\prime ,t\right)}{\mid \vec{r}-\vec{r}\prime \mid }}\mathrm{d}\vec{\mathrm{r}}\prime ·$$

Figure 5 illustrates how spatiotemporal patterns of CSD activity in the different frequency bands contribute to the infragranular FP response. Figure 5, A and B, shows the spatiotemporal profiles of CSD responses and their power over a wide frequency range (δ∼γ, 1–256 Hz) and in five 1.6-octave (δ, θ, γ₁, and γ₂)- and 0.8-octave-wide (α and β) frequency bands. Due to division by the distance from the point of interest, the volume conductor model weakens distant CSD before summation. Therefore, actual signals in the Sg layer CSD components were stronger than what is plotted in Fig. 5, A and B, regardless of frequency band. In each frequency band, summation of CSD responses weighted inversely by the distance from the Ig layer along depth reflects the Ig layer FP response derived by the volume conductor model (orange traces in Fig. 5C).

Fig. 5.

Decomposition of volume conductor model into constituent frequency bands. A: spatiotemporal CSD profile in a wide band (left; δ∼γ) and in its constituent frequency bands (δ: 1.0–2.9 Hz; θ: 3.0–8.8 Hz; α: 9.1–15.9 Hz; β: 17.0–26.7 Hz; γ₁: 27.7–81 Hz; γ₂: 83.9–256 Hz), as indicated at top (sink, red; source, blue). Horizontal dashed lines delineate the borders between Gr, Sg, and Ig layers. CSD signals at different depths were divided by individual distances from the Ig layer for derivation of the Ig layer FP signal by the volume conductor model. B: spatiotemporal profile of broadband CSD response power (left) and its composition in the frequency bands after dividing by distance from the Ig layer as in A. C: temporal patterns of CSD signals (green), volume conduction model-derived FP signals (vcFP; orange), and the observed FP signals (red) in the Ig layer. Within each row of A–C, gains of δ∼γ and θ were the same, gains of δ, α, and β were twice that of δ∼γ, and gains of γ₁ and γ₂ were 3 times that of δ∼γ. Scales of individual rows were adjusted to compare different signals within each frequency band and to reveal detailed features visually. Vertical dashed lines with arrows highlight the peaks that were of opposite polarity (δ, θ, and α) or the same polarity (β) between CSD and FP responses.

In the example shown in Fig. 5, the Ig layer CSD response (green in Fig. 5C, δ∼γ) was negative from 10 to 100 ms after the onset of sound and had a negative peak at 50 ms followed by a positive peak at 130 ms. The FP response (red in Fig. 5C, δ∼γ) had a positive peak at 50 ms followed by a negative peak around 130 ms. Temporal pattern differences between CSD and FP signals were reflected in opposite polarities essentially throughout the response time course across frequency bands (δ, θ, α, and β). β-Band CSD and FP signals also differed in frequency.

These temporal pattern discrepancies between CSD and FP signals across frequency bands were due to volume conduction. In most frequency bands, powers of signals in Gr and Sg layers were comparable to or even stronger than the power in the Ig layer (Fig. 5B, δ, θ, α, β, and γ₁). In each band, by spatially summating the CSD signals across depths, the volume conductor model reproduced an FP response that greatly resembled the observed FP, but with a polarity opposite to that of the colocated Ig layer CSD signal (orange in Fig. 5C). Thus the temporal pattern of the Ig layer FP response reflected not the local generator currents, but rather those in other cortical layers via volume conduction.

We quantified the relative contributions of CSD signals across depths to the granular FP for all experiments (penetration sites). In each penetration site, the depth distributions of CSD response power, weighted by the inverse of distances from the Gr layer sink, were calculated separately for the same six frequency bands as was done for Fig. 5. In general, sensory-evoked CSD signals are distributed across cortical layers in sensory cortices and have broad frequency content (Kajikawa and Schroeder 2011; Lakatos et al. 2005; Maier et al. 2011).

Figure 6A illustrates the laminar distribution of CSD power by frequency band that contributed to the Gr layer FP response. Because the power of the Gr layer CSD responses differed significantly between frequency bands [median values from low- to high-frequency bands: 0.49, 0.41, 0.076, 0.032, 0.012 and 0.86 × 10⁻³ mV/mm², Friedman's test, χ_γ²(df = 5, n = 130) = 621, P < 0.01], distributions were normalized (within band) by the Gr layer CSD power. The depth distributions of power tended to peak near the depth of the Gr layer sink, consistent with the weighting by the inverse of distance. Power of CSD responses did not diminish to zero within the estimated boundaries of gray matter (∼1 mm above and below the Gr layer), presumably due to non-uniformities in tissue conductance. Notably, confidence intervals in four frequency bands approached and exceeded 1 at locations within 1 mm above (Fig. 6A, δ, θ, α, and β), and medians of three frequency bands exceeded 1 at depths below (Fig. 6A, δ, θ, and α). These locations above and below the Gr layer correspond to Sg and Ig layer activity, suggesting large contributions of CSD responses in the Sg and Ig layers to Gr layer FP responses in those frequency bands. Even for high-frequency bands (γ₁ and γ₂), clear CSD contributions arose from areas several hundred microns above and below the Gr layer. Thus, across the constituent frequency bands, intracortical FPs were susceptible to contamination by generator currents distributed above and below the recording site.

As expected, when the volume conductor model integrated individual frequency bands of CSD responses separately, the similarity score between the Gr layer FP responses and the volume conductor-modeled FP increased gradually with the number of channels proximal to the Gr layer for all frequency bands (Fig. 6B). This further suggested that the presence of a frequency band in the FP signal reflected CSD activity in that frequency band but might not necessarily be local. It can be noted in Fig. 6B that as the frequency band increased, the asymptotic similarity values decreased. This was presumably due to lower signal power for higher frequency bands in the proximity of the recording and increasing susceptibility of FP to the surrounding activity beyond the spatial range reflected in CSD. It could also more simply relate to signal-to-noise ratios. It also may be noted in Fig. 6B that the similarity score of γ₂ reached asymptote with fewer channel increments than in other frequency bands (θ–γ₁). This was presumably due to the fact that the power of γ₂ was smaller than that of other bands in all cortical layers and that the contribution of CSD γ₂ power in Sg layer to FP power in all layers was small (Fig. 6, C–E).

When the FP (or its power) peaks, CSD activity is expected to peak at the same time. Due to volume conduction, such CSD activity could be either colocated with FP or located somewhere else. Because the frequency bands of FP power have different peak timing [medians are 128, 94, 41, 28.5, 22, and 14 ms for δ, θ, α, β, γ₁, and γ₂ of the Gr layer FP responses, Friedman's test, χ_γ²(df = 5, n = 129) = 385, P < 0.01], we addressed the layer pattern of distance-weighted CSD power that contributed to FP peaks for individual frequency bands separately. Figure 6, C–E, shows the normalized CSD power distributed across Sg, Gr, and Ig layers for six frequency bands at the timing of peak FP power in Sg (C), Gr (D), and Ig layers (E). For Sg layer FP peaks, the largest CSD contributors were in Sg layer regardless of frequency band (Fig. 6C). For Gr layer FP peaks, the largest CSD contributors were in Sg layers for δ and β, Gr layer for γ₂, Sg and Gr layers for α and γ₁, and all layers for θ (Fig. 6D). For Ig layer FP peaks, Ig layer CSD contributions appeared largest for δ and θ. Contributions of Sg, Gr, and Ig layers were not discernibly different for α, β, and γ₁. These results suggest that depending on the position of recording relative to the distributed pattern of CSD, FP reflects gray matter activity differentially. Furthermore, even within a given location, distributed CSD activity contributing to an FP differs between frequency bands.

DISCUSSION

Prior findings (reviewed by Kajikawa and Schroeder 2011) indicate that a FP signal recorded at any point within the brain can be influenced by signals generated elsewhere in the brain. Volume conduction clearly impacts FP analysis at such a macro (>1 mm) scale; however, the effects of volume conduction at the micro (<1 mm) scale are poorly understood. To define the extent of this problem, we first made a direct comparison of the spectrotemporal patterns of FP signals across cortical layers to those of underlying local synaptic currents as indexed by CSD analysis. We then explored the physiological origins of the FP on the micro scale using a volume conductor model that systematically dissects the contributions of local synaptic currents (estimated by CSD) over adjacent cortical layers distributed over distances above and below the FP recording site.

Volume conduction is literally an integral operation of CSD over a tissue volume that shapes the temporal pattern of the locally recorded FP. At every moment, electrical activities occurring within a conductive medium contribute to every other locus within that medium. Although such contributions attenuate with distance, FP signals are clearly not a simple reflection of local activity. An extreme case is illustrated by the scalp-recorded auditory brain stem responses, or ABR (Jewett and Williston 1971). The ABR recorded from a vertex scalp electrode, outside of the brain and centimeters away from the auditory brain stem nuclei, contains multiple peaks generated by different nuclei. The cycles of those peaks are much faster than the uppermost frequency considered in this study (256 Hz). However, the same principle applies to detecting intracranially recorded FP and scalp-recorded ABR signals, that is, to register the difference in electrical potential between active and reference sites.

Volume conduction effects are advantageous for the ABR, because they permit assessment of an otherwise inaccessible part of the early auditory pathways. The same is true in general for scalp EEG, ERP, and intracranial FP signals. However, these effects potentially limit the specificity of those signals. Even for FPs recorded within a cortical area, the neural origins of response components are unclear in that different peaks may reflect activity generated in different cortical layers or even in different brain areas. Whether a FP is analyzed in terms of the amplitude and timing of peaks and/or by the power/phase of signals in its constituent frequency bands, the analyses are inevitably susceptible to volume conduction to some degree. Our results demonstrate explicitly how linear superposition of generator contributions over a few millimeters of active tissue, irrespective of active or passive (e.g., return) currents, can be expected to impact event-related FP signals.

Sensory-evoked responses in sensory areas generally contain power over a wide spectral range (Lakatos et al. 2005, 2007; Maier et al. 2011). However, volume-conducted activity can differ between frequency bands. Band-limited FP response signals were best modeled by combining CSD responses in the same bands over all laminar locations above and below the FP recording site, regardless of frequency range. In the cortex, the laminar distribution of CSD activity may differ between frequency bands (Maier et al. 2011). Also, CSD temporal patterns may differ between layers. Both these sources of variability affect the frequency content of an FP.

It has been suggested that the conductivity in the brain may vary with frequency of signals, which would impact gain differentially across frequencies. Gabriel et al. (1996) showed gradual increase in brain tissue conductivity over the frequency range including 1–256 Hz. This would suggest that magnitudes of low-frequency CSD components may be overestimated relative to the magnitudes of higher frequency components. However, more recent studies report that tissue conductivity or signal gain modulations over low-frequency ranges are largely flat (Logothetis et al. 2007), and this would argue that differential effects between frequency bands are modest to undetectable over the frequency range addressed in the present study.

An important point of controversy merits emphasis here. On the basis of both modeling (Bedard et al. 2004) and dual intracellular/extracellular recordings in rodents (Bedard et al. 2010), Bedard and colleagues argue that significant inhomogeneity of impedance in brain tissue gives rise to filtering of FP's over space in a way that distorts the frequency content of the original signal. One prediction that can be made for tissues with significant inhomogeneity in conductivity (Bedard et al. 2004) is a phase/time shift of field potential components over space. Tissue permittivity alone, which is frequency dependent as shown by Gabriel et al. (1996), also predicts such an effect irrespective of whether the permittivity itself is frequency dependent or not. However, these predictions are not supported by direct studies of field potential gradients in several laboratories. Neither Logothetis et al. (2007) in whole brain recordings nor Nelson et al. (2013) in brain slice recordings detected phase shift for artificial signals over a frequency range that included that analyzed in the present study. Kajikawa and Schroeder (2011) detected no change in either the peak timing or frequency content of sensory-evoked FP responses, over tens of millimeter distances along recording tracks passing through both gray and white matter. Stangl and Fromherz (2008) found no detectable change in peak timing of FPs across distance in slices. It may be noted that the latter two studies avoid technical complications, such as cross talk, that can complicate data interpretation in studies with artificial signals, and that these studies also observed volume conduction of spatially declining FPs consistent with the idea of a conductive medium. Although it is fair to state that this issue remains controversial, it seems to us that despite the strengths of the theoretical/modeling approaches to date, the more simple direct measurements of FP gradients in neural tissue produce findings consistent with the classical assumption that brain is primarily a resistive tissue with very small extracellular capacitance in the low-frequency ranges (Nunez and Srinivasan 2006).

A recent article (Einevoll et al. 2013) questions the validity of two key assumptions required for deriving CSD based only on a one-dimensional derivative (Eq. 1): 1) the homogeneity of tissue conductance and 2) the transpositional invariance of activity. Because of the modular structure of the neocortex, those assumptions clearly have limitations. However, these assumptions are robust and can tolerate certain violations, whereas others can be compensated for, as described below.

1) Cytoarchitecture of cerebral cortex is inhomogeneous, consisting of intertwined neuronal and glial processes and extracellular spaces in between. Myelin density is inhomogeneous across cortical layers (Hackett et al. 2001). Such structural features create non-uniform profiles of conductivity in neocortex (Goto et al. 2010) and in striatum (Nelson et al. 2013). Goto et al. (2010) explored conductivity profiles of mouse barrel cortex in detail. On one hand, they showed that conductivity differed between cortical layers and between tangential and radial orientations, and taking this into account altered the derived CSD. On the other hand, because the variability of conductivity was small, alteration of the spatial CSD patterns by inhomogeneous spatial conductivity profile (the key variable in the present case) was also small.

Where conductivity changes, false-positive sinks/sources may contaminate signals derived by conventional CSD analysis. In that case, whenever otherwise similar field potentials, irrespective of whether they are stimulus evoked or spontaneous, occur at conductivity boundaries, the same patterns must also appear in CSD at the conductivity gap, whose amplitudes are proportional to the amplitudes of the field potential responses (Geselowitz 1967). However, such CSD components proportional to the field potential do not typically appear within the bounds of a column of cortex (i.e., between the pial surface and white matter boundaries). Therefore, fluctuation of tissue conductance over ∼100–200 μm, the spacing of electrodes used in the present study, can be considered negligible.

If we assume deviation of homogeneous conductivity σ_norm from true inhomogeneous conductivity σ_true, errors to CSD and the volume conductor model could be considered. Multiplication of σ_norm would overestimate (or underestimate) CSD by amount of σ_norm/σ_true at loci where σ_true were smaller (or larger) than σ_norm. Similarly, after division by σ_norm, contribution of given CSD would be underestimated (or overestimated) by σ_true/σ_norm at loci where σ_true were smaller (or larger) than σ_norm. As a consequence, at loci where σ_norm > σ_true, the contribution of CSD overestimated by σ_norm/σ_true to FP was downestimated by σ_true/σ_norm before integration. In other words, errors in CSD derivation due to the assumption of homogeneous conductivity should be canceled in the volume conductor model.

2) Because field potential is distributed in three-dimensional volume, derivation of a CSD approximation only in one dimension is generally inaccurate to some extent. As we described earlier (see materials and methods), our one-dimensional CSD analyses require an inequality |∂V/∂z| ≫ |∂V/∂x|,|∂V/∂y|, being satisfied locally along all depths of array electrodes to be a good approximation. Where the inequality holds true, it is expected for the ratio of absolute values of electric fields, or gradients of field potentials between horizontal and vertical orientations (field ratio, |∂V/∂r|/|∂V/∂z|) to be near zero. For the model of pairs of CSD sinks/sources used in Fig. 2, we plotted the spatial pattern of field ratio in Fig. 2D. The orientation for the numerator's differentiation, ∂V/∂r, can be along any direction from the center of sinks/sources within horizontal planes due to the symmetry of sinks/sources around the center. The plot shows that the field ratio around the center of sinks/sources and above and below the depths of the FP inversion is close to zero, thus satisfying |∂V/∂z| ≫ |∂V/∂x|,|∂V/∂y|. Thereby, within the area of strong activity, the inequality condition to validate one-dimensional CSD as good approximation is met, even though the condition degrades near the periphery of sinks/sources in horizontal dimensions.

In general, inequality is guaranteed near the maxima and minima of topographic patterns of FP, where |∂V/∂r| ≈ 0 holds. When sink/source components are aligned vertically like cortical columns, the resultant FP topography has maxima/minima in horizontal planes at vertical positions above or below where sinks/sources are strongest. In the case of the auditory cortex, such loci would be found within stripes of isofrequency contours for responses to tones (Phillips et al. 1994), as our recordings were done in the present study.

However, even at where the inequality is guaranteed, one-dimensional CSD still is an approximation. Using a rodent whisker barrel as a model system, Pettersen et al. (2006) attempted a more accurate derivation of CSD for cases in which modeling the volume of cortical activation is possible, called the inverse CSD (iCSD). This innovation is clearly useful in the whisker barrel; however, the highly specialized barrel system represents more of an extreme case than a model of the neocortex in general. In particular, the loud tones as used in the present study activate a much larger horizontal extent of auditory cortex than that excited by a pure tone at threshold intensity (Phillips et al. 1994), resulting in activation of a cortical extent wider than several square millimeters (Bakin et al. 1996; Tanji et al. 2010). When the cortical area of activation is relatively large compared with the cortical thickness, as for cases in the present study, the model-based CSD becomes equivalent to one-dimensional CSD (Pettersen et al. 2006; Reimann et al. 2013).

On the other hand, misuse of the iCSD method by applying incorrect models could result in erroneous CSD estimates. With its high flexibility in modeling, the method can deduce CSD for any complex structure. However, even when the model structure is not optimal, the method derives a CSD estimate as if activity is generated in the modeled mass. Extreme case could be observed when iCSD is applied to far field potentials that are monopolar over wide distances above or below cortical dipole activity (Kajikawa and Schroeder 2011; Schroeder et al. 1992, 1995). Application of any model with iCSD to such field potential results in monopole CSD (e.g., Hunt et al. 2011), even though field potentials of opposite polarity is usually present for every generator (Joyce and Rossion 2005). In contrast, CSD derived by conventional methods vanish for far field potentials, as they should (Kajikawa and Schroeder 2011).

With the iCSD method, the linearity between frequency bands holds true and the spatial density and the temporal patterns that are different between frequency bands as revealed in the present study can be incorporable in the derivation of CSD. Thus it would be possible to estimate more accurate CSD for different frequency bands. However, without the ability to validate such models, it would be difficult to guarantee better estimations. In addition, if accurate modeling of such activity patterns is possible, modeling for iCSD itself may substitute the need to derive CSD. For real complex data, the best estimate of CSD would still be an approximation, and approximated CSD still captures aspects of real data, unless it was ill-modeled, e.g., to violate the linearity of frequency bands or volume conduction.

In conclusion, our results illustrate at a submillimeter scale how local synaptic currents combined with local volume conduction generate FP responses to sensory input. Even though both the CSD and the volume conduction model are approximate physiological and physical descriptions (Einvevoll et al. 2013; Tenke and Kayser 2012), our findings confirm that even with a strong local synaptic response, intracortical FP responses in a typical recording montage have potential contributions from all cortical layers. As suggested by earlier studies (Kajikawa and Schroeder 2011; Schroeder et al. 1995), when the local FP signal generator is weak, susceptibility to volume conduction effects over distance increases. Additional measurements such as spatial topography of signal components or CSD analysis are needed to localize the precise cellular elements and physiological processes that generate specific FP components.

GRANTS

This work was supported by National Institutes of Health Grants R21-DC012918, MH060358, and DC011490.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

Y.K. and C.E.S. conception and design of research; Y.K. performed experiments; Y.K. analyzed data; Y.K. and C.E.S. interpreted results of experiments; Y.K. prepared figures; Y.K. drafted manuscript; Y.K. and C.E.S. edited and revised manuscript; Y.K. and C.E.S. approved final version of manuscript.