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Brain activity generates extracellular voltage fluctuations recorded as local field potentials (LFPs). While known that the relevant micro-variables, the ionic currents across membranes, jointly generate the macro-variables, the extracellular voltage, neither the detailed biophysical knowledge nor the required computational power has been available to model these processes. We simulated the LFP in a model of the rodent neocortical column composed of >12,000 reconstructed, multi-compartmental and spiking cortical layer 4 and 5 pyramidal neurons and basket cells, including five million dendritic and somatic compartments with voltage- and ion-dependent currents, realistic connectivity and probabilistic AMPA, NMDA and GABA synapses. We found that, depending on a number of factors, the LFP reflects local and cross-layer processing and active currents dominate the generation of LFPs rather than synaptic ones. Spike-related currents impact the LFP not only at higher frequencies but lower than 50 Hz. This work calls for re-evaluating the genesis of LFPs.
Extracellular voltage recordings (Ve), the voltage difference between a point in the extracellular space and a reference electrode, are the primary method of monitoring brain processing in vivo. Such recordings are high-pass filtered to isolate spiking. Slower Ve-fluctuations (typically<300 Hz) referred to as local field potentials (LFPs), reflect the summed electric activity of neurons and associated glia and provide experimental access to the spatiotemporal activity of afferent, associational and local operations (Buzsáki, 2004). The relationship between electric activity of nerve and (presumably) glia cells and the LFP has remained mysterious (for a review, see (Buzsáki et al., 2012)). LFPs have traditionally been viewed as a reflection of cooperative postsynaptic activity (Lindén et al., 2011; Mitzdorf, 1985). Yet, even when synaptic activity is blocked, neural populations can show emergent activity associated with large LFP deflections (Buzsaki and Traub, 1996; Buzsaki et al., 1988; Jefferys and Haas, 1982). What is clear is that nonsynaptic events such as the spike afterpotential and intrinsic oscillatory membrane currents can contribute to the recorded LFP (Anastassiou et al., 2010, 2011; Belluscio et al., 2012; Buzsáki et al., 2012; Buzsaki et al., 1988; Ray and Maunsell, 2011; Schomburg et al., 2012).
A major advantage of extracellular recording techniques is that, in contrast to other methods used to study network activity, the biophysics related to these measurements are well understood (Buzsáki et al., 2012). This has enabled the development of reliable and quantitative mathematical models to elucidate how transmembrane currents give rise to the recorded electric potential (Gold et al., 2006; Lindén et al., 2011; Pettersen et al., 2007; Schomburg et al., 2012). In particular, models emulating realistic morphology, physiology and electric behavior as well as connectivity can provide insights into the origin of different kinds of extracellular signals since they allow precise control and access of all variables of interest. Here we use a very large-scale model consisting of more than 12 thousand morphologically and functionally realistic neurons, simulated using more than 5 million spatial compartments and 35 million discrete synaptic and membrane currents, connected with each other based on rules that capture many aspects of measured connectivity (Hill et al., 2012; Perin et al., 2011). In particular, we account for the presence of neocortical (S1, hindlimb area) excitatory (layer 4, L4, and layer 5, L5, pyramidal neurons) and inhibitory (L4 and L5 basket cells) neurons. We investigate the impact of slow (approximately 1 Hz) external activity impinging on neurons and its effect on the resulting LFP-signature.
Such rhythmic activity is relevant, for example, in the case of the most prominent of cortical processing, slow wave activity (SWA, 0.1-1 Hz). Found in humans (Achermann and Borbély, 1997) and animals (Steriade et al., 1993a, 1993b, 1993c), SWA involves large areas of neocortex, along with various subcortical structures, that are synchronized into cyclical periods of global excitation followed by widespread silence. SWA is a defining characteristic of slow wave, deep or non-REM sleep but also occurs under anesthesia and in isolated cortical preparations. Neocortical cells discharge during the trough of the LFP and remain silent during the peak of the LFP recorded from deep layers of cortex. Active and silent periods of this slow oscillation are referred to as UP (high conductance) and DOWN (low conductance) states. This robust neocortical oscillation coordinates various other rhythms, including spindles and delta waves (Steriade et al., 1993a, 1993b, 1993c) and faster activity (Mukovski et al., 2007).
While we do not attempt to emulate the biophysical details of SWA involving a multitude of internal and external inputs, our large-scale, bottom-up biophysical model provides insights into the origin of the LFP-signal, in the presence of active membrane conductances, realistic neural morphologies and network connectivity patterns.
Based on hundreds of morphologically and functionally reconstructed neurons (Druckmann et al., 2007; Hay et al., 2011) (Fig. S1), the network model was built to capture many aspects of connectivity (Fig. 1, (Hill et al., 2012; Oberlaender et al., 2011; Perin et al., 2011)). Neural membrane processing of every compartment of every neuron is reflected in Ve by superposing membrane current contributions from each neural compartment using the line-source approximation (Holt and Koch, 1999). That is, Ve at every location in extracellular space results from the linear summation of all membrane currents throughout the volume, scaled (to a first order inversely) by the distance to the current source (see Methods). In the present study, we focus on how the microscopic currents across each membrane sum to give rise to the macroscopic LFP signal and neglect any contributions that the LFP, in turn, might have on the voltage across each membrane (Anastassiou et al., 2010, 2011; Jefferys, 1995).
Comparison between the extracellular action potential (EAP) traces elicited during simulated administration of brief (10 ms) intracellular somatic current injections (Fig. 1, bottom row) and simulation as well as experimental observations (Gold et al., 2006) demonstrates that our single-neuron representations accurately reproduce the EAP-waveform even though their reconstruction was optimized to reproduce intracellular rather than extracellular events (Hay et al., 2011). In fact, accurate simulation of the EAP-waveform can be used as an additional (and often stricter) measure for the quality of the reconstruction of a neuron, especially for perisomatic compartments (Gold et al., 2007).
The prevailing view is that the LFP primarily reflects postsynaptic currents for frequencies lower than approximately 100-150 Hz (Nunez and Srinivasan, 2006) which stems from the recognition that extracellular currents from many individual compartments must overlap in time to induce a measurable signal with such overlap primarily occurring for synaptic events (Elul, 1971; Logothetis and Wandell, 2004). This assumption, in turn, has motivated the study of LFPs using models that account for morphologically realistic but passive neurons with the statistics of postsynaptic currents and their spatial distribution emulating experimental observations. Yet, the presence of active conductances along the neural membrane is a highly nonlinear (either voltage- or ion-dependent) contributor of extracellular currents that cannot be accounted for via passive elements.
Fig. 2 shows the outcome of a large-scale simulation in which slow (1 Hz) external excitatory (AMPA and NMDA) and inhibitory (GABAA) synaptic activity impinged along both L4 and L5 pyramidal neurons (Fig. 2a). For the active membrane simulation, this elicits spiking (Fig. 2b) which, in turn, gives rise to local and global postsynaptic activity (Fig. 2c, d). We define the depolarizing (hyperpolarizing) part of the external 1 Hz stimulation as UP (DOWN) state. The spike frequency (Fig. 2d) of the different cell types considered in our simulations agrees with experimental observations in rodents during SWA (Fanselow and Connors, 2010; Haider et al., 2006; Luczak et al., 2007, 2009; Sanchez-Vives and McCormick, 2000).
To understand the different components contributing to the LFP, we considered three scenarios, each of which has identical spatiotemporal postsynaptic currents (PSC). We define the PSC to be the postsynaptic membrane current flowing at the synapse in response to the synaptic-associated conductance change, Isyn(t) = gsyn(t)(Vm-Vrev), with gsyn being the synaptic conductance, Vm the membrane potential and Vrev the reversal potential (Koch, 1999). In the first scenario, we only consider the LFP caused by these currents from the roughly 15 million synapses (Fig. 2e) by ignoring all non-synaptic currents in the calculation of the LFP. Thus, the simulation reflects purely PSC activity and only accounts for the way synapses are arranged in space in the absence of neurons. In Fig. 2f we replay the identical PSC input along every neuron as in the full simulation (Fig. 2g) but, in a more complex scenario than in Fig. 2e, compute the LFP contributed by synapses plus the morphologically accurate but passive cables. Finally, the last scenario includes synapses as well as the morphology supplemented by all active membrane conductances (Fig. 2g).
If we compute the LFP only from synaptic conductances (Fig. 2e), excitatory input (mainly along the basal dendrites (Hill et al., 2012)) on L4 and L5 pyramids gives rise to a negative LFP-deflection extending across L4 and L5 at the onset of UP. The LFP-negativity attenuates during the UP state due to synaptic depression (see Methods). During the DOWN state, synaptic activity is much reduced, resulting in an LFP close to zero.
How do morphological features of neurons impact the LFP? In Fig. 2f, we replayed the pattern of PSC activation of Fig. 2e but this time included morphologically detailed neurons (Fig. 1; Fig. S1) with passive membranes. In this setup, the LFP-contributors are by definition limited to PSC and related passive ‘return’ currents, i.e., currents induced along the neural membrane by impinging synaptic input due to charge conservation (Buzsáki et al., 2012). (Notably, the impact of return currents is absent in the simulation shown in Fig. Fig. 2e.) All sodium, potassium and calcium currents have been blocked. Oscillatory external inputs (Fig. 2a) give rise to oscillatory intracellular depolarization (similar to Fig. 2c). Yet LFP-features, such as the amplitude or the temporal width in the two layers, change drastically compared to Fig. 2e. The presence of passive membranes markedly attenuates the amplitude and the temporal width of the LFP-waveform (note the voltage scale bar in Fig. 2e is five-fold larger than in Fig. 2f and 2g). This reduction is due to the impact of return currents of opposite sign that cancel out the extracellular impact of locally impinging synaptic input and lowpass filtering of passive membranes. In particular, the LFP-waveform changes as a function of depth. This is especially true during the first 50-100 ms of UP.
How do voltage- and ion-specific membrane conductances found in all of these neurons shape the LFP? The short answer is a lot, in particular compared to the passive cable simulation (Fig. 2f). The LFP amplitude in the active case (Fig. 2g; mid L5 at approx. 1100 μm cortical depth; mean amplitude: 0.8 mV (active) vs. 1.3 mV (passive); mean half-wave width: 60 ms (active) vs. 130 ms (passive); see also upcoming sections and Fig. 4) is substantially attenuated. This is caused by the active conductances giving rise to a leakier membrane, especially at the onset and during UP, that, in turn, manifests itself in spatially extended extracellular multi-poles of smaller amplitude (Fig. S2). During DOWN, the difference between active and passive membrane leakiness is much attenuated because the membrane conductances in the state of near absence of synaptic input are almost identical (Fig. S2). In general, adding return currents (via the inclusion of passive morphologies) and, in a subsequent step, increasing membrane leakiness (via the inclusion of active membrane conductances) leads to attenuation of the LFP-amplitude and spatiotemporal width.
Given the linearity of the extracellular resistive milieu ((Anastassiou et al., 2011; Logothetis et al., 2007) but also see (Bedard et al., 2004)), the LFP plotted in Fig. 2e-g is the sum of extracellular contributions from synapses and neurons distributed across two layers. In Fig. 3, we segregate the LFP-contribution of each neural type (top to bottom: L4 pyramids, L5 pyramids, L4/5 basket cells) for the case shown in Fig. 2g. We observe that the LFP-contributors within both layers are currents associated with L4 and L5 pyramids. More specifically, in L4, L4 pyramids contribute 46±18% of the LFP (L5 pyramids contribution: 45±18%) while in L5, L5 pyramids contribute 52±20% (L4 pyramids contribution: 39±18%). These results support the view that, under the conditions studied here, the LFP does not reflect only local population processing but also outer-layer activity (Fig. 3a, b), especially in L4. The LFP in L5 is larger than in L4 due to the large size of L5 pyramidal neurons as well as the powerful synaptic drive they receive along their basal (mainly) and apical dendrites (Fig. 2g). This elicits membrane currents along the whole depth axis (Fig. 3b) so that, while perisomatic compartments still contribute mostly to the LFP, the apical dendrites of these neurons also contribute to the LFP in L4, especially during the transition from DOWN to UP, i.e., during the highly synchronous barrage of excitation impinging on L5 pyramidal neurons.
Comparatively, L4/5 basket cells, making up only 13% of all cells with their temporally narrow EAPs (Fig. 1, bottom; (Schomburg et al., 2012)) and fairly symmetric and localized dendritic arbors, contribute very little to the LFP in either layers (basket cell contribution is 9±2% in L4 and 9±6% in L5; Fig. 3c). The negligible contribution of L4/5 basket cells to the LFP is in stark contrast to their particularly high level of activity (their spiking rate reaches up to 75 Hz during UP, Fig. 2d) compared to L4 and L5 pyramidal neurons in our simulations.
The simulated LFP-contributions of L4 and L5 pyramids capture a common experimental observation: the positive LFP-deflections in the perisomatic region and the negative LFP-deflection along basal and apical dendrites during somatic depolarization periods (especially at UP-onset; see also Fig. 4) can be attributed to the location of excitatory and inhibitory conductances along the elongated morphology of neurons in combination with the presence of return currents (Buzsáki et al., 2012).
To quantify differences in the spatial extent of the LFP between the passive (Fig. 2f) and active membrane (Fig. 2g) simulations, we fit the sum of two, spatially displaced, Gaussian functions (independent variable: location along the depth axis) of opposite sign to the mean LFP depth profile during UP (Fig. 4a-c) and determined the amplitude, peak location and the LFP length scale (described by the half width of each of the Gaussians). We found that the amplitude changes by approx. 50-300%, the location by 100-300 μm, and the spatial width by 30-40% (values determined 50 ms after onset of UP; Fig. 4d). Differences between active and passive are even greater during the first 50 ms of UP-states (Fig. 4a) but we chose to compare LFP depth profiles after synaptic activity had propagated throughout the network. Thus, in both layers, the presence of spiking and spike-related currents drastically alters LFP depth-characteristics (amplitude, spatial and temporal constellation) with differences being more pronounced in L5 especially during the first 100 ms of UP (Fig. 4a). On the other hand, in L4, the LFP-traces for the active and passive simulation are more similar, suggesting the LFP there reflects not only active membrane processing but also synaptic and passive processes.
Current source density (CSD) analysis estimates the negative second order spatial derivative of the LFP along the depth-axis of the recordings. Per definition, the CSD represents the volume density of the net current entering or leaving the extracellular space (Nicholson and Freeman, 1975) and is used as a measure of synaptic input eliciting so-called current sinks (for excitatory inputs) and sources (for inhibitory inputs). In contrast to the LFP that is a distance-weighted superposition of currents within a small volume, the CSD crucially depends on local events along the depth-axis. Thus, it is a better measure for processes occurring along the extent of L4 and L5 pyramids.
We calculated the one-dimensional CSD along the 1 mm depth-axis covering L4 and L5 (Figs. 2e-g and and3;3; sinks are in blue and sources in red). In the presence of active membrane conductances, sodium influx and potassium efflux associated with spiking gives rise to sinks and sources, respectively, in the vicinity of cell bodies. The oscillatory pattern of impinging synaptic inputs gives rise to a temporally oscillatory CSD of the same frequency as well as an intricate spatial structure of the waxing and waning of two sources (one in each layer) and one sink (in L5) with a length scale of approximately 250 μm. The aforementioned LFP-differences (amplitude, spatial and temporal variance) are also reflected in the CSD-characteristics with passive membranes resulting in temporally wider CSD and differential sink-source constellation along the depth axis (Fig. 2f, g). Notably, a current source is present in deep L5 for active membranes at UP-onset (Fig. 2g, red areas) that vanishes in the passive case. More generally, passive membranes exaggerated the strength and spatial reach of the induced multi-poles along pyramidal neurons (Fig. 2f, g and Fig. 4a-d). Examination of the CSD-contribution of the individual neural types (Figs. 3 and S3) revealed that the presence of active vs. passive membranes altered the overall sink-source constellation and individual neural type contributions. Yet, for the stimulation scenarios examined in this paper, the contribution of L5 pyramids continues to dominate also in terms of CSD (Figs. 3b and S3b).
Which CSD, passive (Fig. 2f) or active (Fig. 2g), is closer to CSDs obtained in vivo? Answering this question involves comparing CSDs during various brain states that can differ greatly. Riera and co-workers (Riera et al., 2012) recently conducted detailed experiments in rat somatosensory barrel cortex and measured the CSD along the depth axis of barrel cortex during single whisker deflections. In Fig. 4e, we plot the CSD for (left to right) the passive simulation (mean of the data shown in Fig. 2f aligned at UP-onset; Fig. 4a), the active membrane simulation (mean of the data shown in Fig. 2f aligned at UP-onset; Fig. 4a) and the grand average measured by Riera and colleagues (their Fig. 3). We observe how at UP-onset and during the first 10-20 ms, sink-source constellation in L4 and L5 is similar to in vivo experiments. Subsequently, following synaptic depression in L5 attributed to particularly synchronous spiking, the two scenarios markedly differ for the next 10-20 ms with the sink-source constellation inverting. Finally, after equilibration of synaptic weights in L4, the active membrane simulation becomes almost identical to experiments. Notably, the resemblance between simulated and measured CSDs is greatly diminished when assuming identical synaptic input but passive membranes (Fig. 4e, left), with the sink in L5 being exaggerated and the source almost absent from L4. (The resemblance becomes even poorer when comparing the experimental CSD to the PSC case shown in Fig. 2e.) While this comparison needs to be extended across multiple brain states, it suggests that active membrane conductances have a powerful influence on the CSD.
How do LFP-characteristics change with input statistics? Synaptic input correlation crucially impacts the spatial extent of the LFP (Lindén et al., 2011; Pettersen et al., 2007; Schomburg et al., 2012). We performed simulations where we either eliminated (‘uncorrelated’ case; Fig. 5a-c) or further enhanced (‘super-synchronized’ case; Fig. 5d-f) the temporal correlation of impinging synaptic input compared to the simulations shown in Fig. 2g (termed the ‘control’ case). Importantly, the ‘uncorrelated’ and ‘super-synchronized’ simulations have identical number of PSCs impinging at the same locations as the ‘control’ simulation. Only their timing is shifted, reflecting a decrease or an increase in input correlation (see Methods and Fig. S4). As synaptic drive becomes more correlated, the LFP-amplitude increases (Fig. 5c vs. 5f). To quantify such differences, we use the same method as introduced in Fig. 4 and report amplitude, location and spatial width of the two spatially displaced Gaussian functions 50 ms after UP-onset (Fig. 5g-h; see also Table S1).
For example, the amplitude of the LFP-negativity (fit by a Gaussian) Aneg, increases with input correlation: 0.12 mV (uncorrelated) vs. 0.36 mV (control) vs. 0.50 mV (super-synchronized) (Table S1). We see that the extent of the amplitude-decrease for passive vs. active membranes depends on cell type, with the greatest effect observed for L5 pyramids due to their size and strong synaptic drive.
As witnessed by Figs. 2, ,44 and and5,5, identical synaptic input causes larger LFP-amplitudes for passive than for active membranes for almost all input correlation scenarios considered. For example, for the ‘control’ simulation, identical synaptic activity gave rise to Aneg=0.99 mV and Apos=0.68 mV for passive membranes vs. Aneg=0.50 mV and Apos=0.46 mV for active membranes (Table S1). Increased input correlation generally resulted in an increase in the length scale of the LFP, both for active and passive membranes, with L5 pyramids most strongly affected (compare spatial width w in Fig. 5g vs. 5h; Table S1). Again, passive membrane simulations have a larger spatial extent than active ones (manifested in the negative slope in almost all w-related panels in Fig. 4d and 5g, h).
So far our analyses have focused on the LFP- and CSD-features along the cortical depth axis. Assuming extracellular recording sites are situated along the center of the cortical disk, how do LFP-characteristics change along the radial axis, that is, tangential to the cortical sheet? In Fig. 6, we segmented the population into concentric cylinders of radii R and calculated the LFP-amplitude contributed in the center of L4 (left column) and L5 (right column) as a function of R. Accounting only for the Ve-contribution of pyramidal neurons within a certain layer, we adopted the approach introduced in (Lindén et al., 2011) (their Fig. 5) to calculate the LFP-contribution for the uncorrelated (stars) and control (circles) case for active (red) and passive (black) membrane conductances. Briefly, we defined the LFP-amplitude σ as the std of the LFP-signal (Fig. 6a and 6b) and the LFP-saturation distance R* (Fig. 6c and 6d; blue triangles) as the radius where the LFP-amplitude reaches 95 % of its maximum value with neurons located father from R* contributing minimally to the LFP. (Importantly, LFP-amplitude σ is not the same as A reported in Figs. 4d and 5g-h). Similar to (Lindén et al., 2011), we found that increasing input correlation increased R*. Yet, as for the analyses along the cortical depth axis, the presence of active membranes reduced R* (active vs. passive in L4: uncorrelated, 89 vs. 184 μm; control, 187 vs. 278 μm; L5: uncorrelated, 212 vs. 249 μm; control, 315 vs. 319 μm) especially in L4. Interestingly, for uncorrelated input in L5 and passive membranes, R* from our simulations (249 μm) is in agreement with the value reported by Linden et al. (approximately 200 μm; their Fig. 5c).
So far, we focused on the LFP-contribution of different cell types. Given the critical role of active membranes, which channels impact the LFP the most and under which conditions? To address this question we calculated the LFP-contribution of synaptic input as well as specific ions sodium (Na), potassium (K) and calcium (Ca) of the different cell types separately and show them for two cases, ‘uncorrelated’ and ‘control’ (Fig. 7). (Performing the same analyses for the ‘super-synchronized’ case yields very similar results to ‘control’.) Specifically, we define as LFP-contribution the normalized portion of the LFP-signal attributed to the current passing from a particular conductance integrated over the time bin (resulting in charge). We calculated the LFP-contribution of specific conductances in two locations, the center of L4 and L5. For the ‘uncorrelated’ case (Fig. 7a), synaptic excitatory and inhibitory currents contribute under 15-20% to the LFP. Fast sodium currents, especially from local pyramidal neurons, contribute about 30 %, with the rest of the contribution stemming from slower potassium currents. Interestingly, while L5 pyramids expectedly (due to the presence of thick apical dendrites) contribute to the LFP recorded in L4, L4 pyramids also contribute to the LFP recorded in L5, mainly via K-related currents. The main contribution of L4/5 basket cells is in L5 where sodium and potassium currents constitute about 30 % of the total current, yet it needs to be pointed out that the LFP-amplitude for uncorrelated input is small (see Fig. 5g and traces in Fig. 7).
How do these contributions change with input correlation? For the ‘control’ case (Fig. 7b), we observe how spiking Na- and K-currents from L5 pyramids dominate the LFP 20-40 ms from UP-onset both in L4 and L5. In fact, in L4, the LFP-contribution from postsynaptic input impinging on L5 pyramids is larger than the LFP-contribution of postsynaptic input impinging along L4 pyramids. Concurrently, there is a strong activation of Na- and K-related currents through spiking of L5 pyramids that prominently contribute to the LFP in L4. It is after the initial transient of 40 ms that synapses of L5 pyramids depress at which point Na- and K-related currents of L4 pyramids begin dominating (approx. 60-80 %) the LFP-signal in L4. In L5, within-layer pyramids dominate the LFP throughout the UP-DOWN cycle with two main differences to L4 activity: firstly, synaptic currents contribute more (approx. 15-20 %) than in any other case during UP and, secondly, L4/5 basket cells have a significant (even if short-lived) impact on the LFP 50 to 70 ms from UP-onset (approx. 30-40 %) where dense local connectivity (Fig. 1) and the massive bolus of postsynaptic activity induces high spiking rates (Fig. 2). Finally, we found IPSCs to contribute approximately 10 % of the total (excitatory and inhibitory) synaptic contribution, i.e., under the conditions studied here excitatory input dominates the synaptic contribution.
Temporal frequency (‘1/f’) and distance (‘1/r’) scaling of LFP-signals can reveal aspects of neural processing (Bédard et al., 2006; Katzner et al., 2009; Miller et al., 2009; Milstein et al., 2009; Pritchard, 1992; Rasch et al., 2009). What sort of scaling do our simulations exhibit? Using the Ve-traces recorded in depths ranging from 500 to 1700 μm (representative Ve-traces shown in Fig. 8a; blue: PSC only, black: passive membranes, red: active membranes) we initially calculated the power spectral density (PSD) P (‘control’ simulations in Fig. 8b; line: mean, shaded area: std). We calculate the best fit (see Table S2) to P(f) ![[proportional, variant]](/corehtml/pmc/pmcents/x221D.gif)
1/ fa with f being the frequency and α the scaling exponent for two bandwidths: < 40 Hz (Fig. 8c, bottom) and 40-1000 Hz (Fig. 8c, top). α is consistently smaller across all cases of input correlation for low frequencies compared to high ones (circles: mean; error bars: s.e.m.), with the differences in α between all cases being small for <40 Hz (Table S3). For 40-1000 Hz, α is similar between PSC and passive membrane simulations, while substantially reduced for active membranes (Table S3). For example, for the ‘control’ simulation with active membranes, α=2.0±0.4, while for passive membranes, α=3.7±0.1. (For <40 Hz, for the ‘control’ simulation, α=1.0±0.2 and 0.9±0.1, respectively.) Notably, experimental recordings exhibit α close to 2 (Miller et al., 2009; Milstein et al., 2009), with α smaller at lower frequencies (Miller et al., 2009). We conclude that α is crucially shaped not only by postsynaptic currents but also by membrane characteristics in the 40-1000 Hz range.
How do individual neurons and the associated micro-variables give rise to such frequency-scaling evident in the macro-variables, i.e., the LFP? To address this question, we defined a single-cell frequency scaling exponent for all L5 pyramidal neurons (the population with the strongest LFP-contribution) where P(f) 1/ fβ and calculated the mean Ve of all 5,364 L5 pyramidal neurons at three different locations relative to the soma (Fig. 8d-e shows the ‘control’ simulation). The PSD as well as its frequency scaling differs substantially depending on whether only PSC, passive cable structures or active membranes contribute to the LFP. PSC and passive membranes consistently give rise to steeper scaling and larger β (approx. 2.5-3; Fig. 8e,f; Table S4) for all simulations while for active membranes β is smaller (approx. 1-2; Table S4). The PSD decreases drastically as a function of frequency for passive membranes while much less so for active (Fig. 8e). More surprisingly, differences in PSD as well as frequency-scaling for active vs. passive membranes persist for frequencies <100 Hz (Fig. 8e). This suggests that spiking and spike-related currents contribute to low LFP-bandwidths traditionally considered to reflect purely synaptic activity, an observation that agrees with experiments demonstrating LFPs generated via nonsynaptic events (Anastassiou et al., 2010; Buzsáki et al., 2012; Chrobak et al., 2000).
The spatial extent of LFPs changes substantially between cases (Figs. 4 and and5).5). We analyzed the LFP-contribution of L5 pyramids to three bandwidths (<50 Hz, 50-100 Hz and 800-1000 Hz; Fig. 8g), as a function of distance r between the soma and the electrode, i.e., P(r) 1/ rγ with P(r) the distance-dependent PSD in a particular bandwidth, and γ the distance-dependent exponent (Figs. 8g-i). In agreement with (Lindén et al., 2011; Pettersen et al., 2007; Schomburg et al., 2012), we found that for passive membranes, γ<2 for r<100 μm, increasing to γ≈3 for larger distances (Fig. 8i). This observation was robust for all bandwidths and input correlations we examined. In the presence of active membrane conductances, PSD distance-scaling changes substantially closer than 100 μm (Fig. 8h, i) with γ≈3 for all distances and input correlation scenarios. This suggests that active membrane conductances in L5 pyramids consistently generates extracellular multi-poles (Pettersen et al., 2007; Riera et al., 2012). Notably, PSC simulations, consistent with the point-like nature of synaptic input, give rise to monopoles close to the recording electrode and dipoles when measured farther away. As illustrated in Fig. 8h (and already suggested by Figs. 8b and 8e), PSD not only differs in the higher bandwidths where spiking currents dominate but, surprisingly, also below 50 Hz. Given the identical synaptic activity between PSC, passive and active membrane simulations, these differences are attributed to the active membrane properties that not only give rise to a leakier membrane but fundamentally alter the sink-source constellation.
We use a novel, large-scale computational model with more than 5 million compartments to study the extracellular signature of active brain tissue, the LFP. The model accounts for biophysically characterized and morphologically reconstructed neurons interconnected based on rules supported by experimental data. Traditionally, the LFP has been assumed to reflect postsynaptic currents and associated passive return currents, with the final extracellular field mainly shaped by neural morphology and synaptic input. Our simulations challenge this picture. With identical synaptic input waxing and waning at 1 Hz, active membrane conductances cause markedly different LFP-signatures than passive cable structures or only postsynaptic activity without any passive or active membranes. These differences are not merely due to the amount of current flowing through the membrane but also by the radically altered spatial constellation of extracellular sinks and sources. In agreement with recent work (Lindén et al., 2011; Pettersen et al., 2007; Schomburg et al., 2012), we find that the LFP length-scale depends on the temporal coordination of the oscillatory inputs. Importantly, spiking and spike-related currents impact the LFP not only in the higher bandwidths but also in lower ones (<50 Hz) traditionally thought to reflect purely postsynaptic activity.
We found that L4 pyramids impacted the LFP and CSD within both layers, with their extracellular contribution greatly affected by the presence or absence of active membranes. Conversely, L5 pyramids with their large somata, thick apical dendrites and strong synaptic input contribute not only to the LFP within L5 but also to the LFP in L4, especially at the onset of coordinated synaptic input. Given their large size and powerful synaptic input, it is conceivable that L5 pyramids could also contribute to the LFP in other layers such as L2/3 or L6 not simulated here. Thus, while the LFP reflects processing of neurons whose cell bodies are situated within that layer, the extended nature of pyramidal neurons gives rise to multi-poles that reach into nearby layers. Importantly, we found this to be broadly true in simulations exhibiting varying degrees of input correlation.
In agreement with others (Pettersen et al., 2007; Schomburg et al., 2012), we find that L4/5 basket cells with their fairly low density (compared to excitatory neurons), localized and symmetric dendritic arbor, spatially uniform synaptic input, the small temporal width of their somatic spikes and lack of strong afterpotentials have only a small impact on the LFP and CSD even though their spike frequency is substantially higher than that of their excitatory neighbors (Fig. 3c). Of course, this does not suggest that extracellular action potentials from individual basket cells are small.
When considering LFP characteristics such as amplitude and spatiotemporal width, we observed that these are markedly shaped by the impinging pattern of postsynaptic currents and membrane characteristics. Increasing model complexity from only postsynaptic to using fully reconstructed active neurons attenuates the LFP-amplitude, alters its spatiotemporal width and changes the sink-source location. Additionally, our findings regarding the LFP-length scale (depending on input correlation, approximately 200-600 μm along the cortical depth and 100-300 μm tangentially) points to the necessity of large-scale models to study the origin and functionality of the LFP.
How do these observations compare with LFPs recorded during whisker stimulation (Riera et al., 2012)? Such stimulation triggers prominent thalamocortical input into L4 in somatosensory cortex (Brecht and Sakmann, 2002). At UP-onset, and during the first 10-20 ms, the sink-source constellation in L4 and L5 was similar to experiments. Following the onset of synaptic depression in L5, the CSD became markedly different for the next 10-20 ms with sink-source constellation inverting. Finally, after equilibration of synaptic weights in L4, the simulated CSD became almost identical to experiments. Given synaptic activation in our network was not designed to emulate whisker stimulation we are lead to the conclusion that while network computation requires inclusion of synaptic, morphological and membrane characteristics, connectivity patterns and features of synaptic dynamics such as plasticity rules are crucial not only for network processing but also to fully account for extracellular sinks and sources.
Sodium and potassium currents prominently contribute to the LFP in both layers with K currents dominating (approx. 40-60 %) the LFP during the UP-DOWN cycle. While fast Na currents of local neurons contribute less than K ones, their contribution to the LFP is greater (approx. 10-20 %) than that of postsynaptic currents (< 10 % in most cases). Thus, while it is true that synaptic input is reflected in the LFP in that it initiates and sustains the intracellular and membrane currents along neurons, our simulations show that the LFP-signal does not directly reflect postsynaptic activity. Instead, it predominantly reflects active membrane conductances activated by impinging postsynaptic input.
This observation challenges the classic view that LFPs are primarily a reflection of synaptic currents based on the number of activated synapses within a volume of brain tissue being typically much larger than the number of spikes (per unit time) within the same volume. Why do our simulations show such strong contribution of active membrane currents? The main reason is that during an individual spike, charge fluxes across the neural membrane at the perisomatic region (axon initial segment, soma, etc.) are much stronger than individual PSCs (Koch, 1999). While the strongest charge fluxes occur within 1-2 ms of every spike (according to the standard Hodgkin-Huxley model) a cascade of slower spiking currents (mainly K- but also Ca-dependent) with much longer time scales is co-activated. These slower active membrane conductances crucially contribute to the LFP as observed in Fig. 7. On the other hand, fast synaptic currents (AMPA- and GABAA-type) die out rapidly while the slower ones (NMDA-type) have a fairly small contribution (the AMPA vs. NMDA component of every excitatory synapse is about 1-to-0.7; (Ramaswamy et al., 2012)). (Notably, not all presynaptic inputs give rise to PSCs, (Markram, 1997; Ramaswamy et al., 2012)). Finally, active conductances contribute much more to the LFP than passive ones because they are mainly located in the perisomatic region along large compartments (i.e., low axial resistance) such as the soma and near dendrites (especially for L5 pyramidal neurons) so that the associated return currents are spread along the whole morphology of the neuron. As a consequence, EAP-amplitude is approximately proportional to the sum of the dendritic cross-sectional areas of all dendritic branches connected to the soma. Therefore, neurons with thick dendrites connected to the soma produce large EAPs and have the largest ‘radius of visibility’ (Pettersen and Einevoll, 2008). At the same time, PSCs are mainly located along thin dendrites (i.e., much higher axial resistance) preventing return currents from spreading along the whole neural morphology.
Another important observation stemming from our simulations is the input specificity of the LFP composition. While the LFP during the first 50-80 ms from UP-onset is dominated by K currents originating from L5 pyramids for temporally coordinated input (Fig. 7b), this switches to K currents from L4 pyramids for uncorrelated input (Fig. 7a). Moreover, while basket cells generally do not contribute markedly to the LFP, this changes briefly 50 to 70 ms after UP-onset. Thus, the LFP-composition is not static but time- and state-dependent and is crucially impacted by the impinging input and the sort of subthreshold and spiking activity it induces (especially proximally to the recording site).
What are the functional (computational) ramifications of these observations? Coherence between spiking and specific LFP bands has been used to infer the relationship between synaptic input (hitherto considered to be reflected in the LFP) and neural output (spiking) and thereby specific mechanisms of information processing within and across brain regions (Fries et al., 1997; Lee et al., 2005; Montgomery et al., 2008; O’Keefe and Recce, 1993; Rutishauser et al., 2010; Womelsdorf et al., 2006). This raises the question of the extent to which the locally generated LFP (or particular bandwidths of it) represent actual synaptic input impinging on local neurons rather than spiking output (Buzsáki et al., 2012). For example, it was recently shown that spiking coherence to ripples during sharp waves in CA1 is partly attributed to spiking-currents shaping the ripple-signal (Belluscio et al., 2012; Schomburg et al., 2012).
Another question arises regarding how perturbing rhythmic LFP-activity such as theta with tetanic stimulation at particular phases of theta induces potentiation or depression of synaptic strength (Hölscher et al., 1997; Hyman et al., 2003; Pavlides et al., 1988). Other studies relate cognitive alteration to perturbation of neocortical UP-DOWN states (Marshall et al., 2006) or hippocampal sharp waves (Girardeau et al., 2009). While our population model does not attempt to reproduce any particular LFP-rhythm, it does link the LFP to biophysical processing. Thus, it can become a useful tool towards addressing the involvement of particular mechanisms during particular LFP-bandwidths and -phases and how perturbing them crucially alters other processing and, ultimately, cognitive function.
When modeling the impact of active membranes on LFP power scaling, we found an inverse power law (Miller et al., 2009; Milstein et al., 2009) with scaling exponent α depending on input correlation and bandwidth of interest. Passive membrane consistently resulted in larger exponents for higher bandwidths (40-1000 Hz). When zooming in to the level of individual L5 pyramids by calculating the scaling exponent β, active membrane contributions differ substantially from passive membrane ones not just for higher bandwidths but, importantly, down to low frequencies (<50 Hz). Interestingly, β compares much better to α in the 40-1000 Hz range than below 40 Hz for synaptic only and passive membranes. Yet, in the presence of active membrane conductances, β becomes comparable to α both in the lower and higher bandwidth (especially so for the control and super-synchronized scenarios) suggesting very similar scaling between the entire population and L5 pyramidal neurons irrespective of their exact location within L5.
We also looked at PSD distance-scaling (exponent γ) – within a 100 μm radius, PSD scales with γ≈2, characteristic of a dipole. For larger distances, γ≈3. A recent study elegantly illustrated that as long as γ>2, the contribution of successive more distant populations of neurons to the LFP saturates, that is, the LFP has a finite spatial reach (Lindén et al., 2011). In our simulations, for active membranes, PSD consistently scales with distance as γ≈3. To generalize, for smaller distances, postsynaptic currents contribute as monopoles (γ≈1), the presence of passive membranes gives rise to return currents and an additional pole (γ≈2) while active conductances give rise to leakier membranes resulting in a third pole (γ≈3). For larger distances, power scaling of active and passive membranes is similar (γ≈3). Concurrently, an increase in input correlation results in an increase in LFP-amplitude and, importantly, length scale. Thus, while the LFP is a good estimator of local neural processing, the volume it is representative for (within the same layer) can change substantially.
The present biophysical model does not include glial and astrocytic processes likely to be important for slowly fluctuating components of the LFP nor do we include nonmyelinated presynaptic axonal compartments (though (Gold et al., 2006; Schomburg et al., 2012) and our own modeling indicate they contribute minimally to the LFP). Likewise, we neglected contributions of presynaptic terminals; given their small size, it is likely that the associated local return currents will render their contribution nugatory. Diffusion was also excluded in our simulations, which can lead to 1/f-scaling (Bédard and Destexhe, 2009). Finally, in our simulations we assumed a purely resistive and homogeneous extracellular medium. While there is evidence in favor of a purely ohmic extracellular medium for frequencies <500 Hz, at least one study has emphasized a capacitive component (Bedard et al., 2004) which, if true, may alter some of the findings in terms of the LFP-contributions of all processes involved. Moreover, even for the purely resistive case, conductivity experiments have shown that the extracellular medium is inhomogeneous, i.e., resistivity gradients exist (Goto et al., 2010). While the model can be extended to account for such observations, our primary goal was to account for the conventional biophysical processes related to LFP generation and the impact of active membrane conductances in particular.
Despite these limitations, our model reproduces a number of observations. First, external synaptic input gives rise to spike frequencies compatible with in vivo observations during slow-wave activity. The simulated EAP-waveforms from our pyramids and basket cells agree with experimental observations (Gold et al., 2006). Our simulations suggest the LFP-contribution of fast spiking basket cells is small, as also shown in (Lindén et al., 2011; Schomburg et al., 2012). Furthermore, our active simulations generate LFPs and CSDs that agree both in terms of spatial constellation (Riera et al., 2012) as well as spectral content (Miller et al., 2009; Milstein et al., 2009) with in vivo observations especially after UP-onset. Using passive morphologies, we were able to reproduce the observation that LFP-power scales differently within versus outside a 100 μm radius from the recording electrode (Lindén et al., 2011). This changed substantially in the presence of active membranes. Finally, increasing input correlation resulted in larger LFP-amplitudes and length scales, both for active and passive membranes. Feynman once famously wrote: ‘what I cannot create, I do not understand’. It is our belief that the present approach is a necessary step towards unraveling the biophysics of LFPs and the workings of brain circuitry, in general.
The model and simulations were developed using the software and hardware infrastructure of the Blue Brain Facility including data, models and workflows for modeling rat (P12-16) cortical S1 microcircuitry. Network simulations were performed using NEURON software (Hines and Carnevale, 1997) running on a Blue Gene P supercomputer on 1024 nodes and 4096 CPUs. 4 seconds of simulated time took approx.. 3 hours to compute. A collection of tools and templates written in HOC and NMODL were employed to handle the setup and configuration on the parallel machine architecture (Hines et al., 2008).
Electrophysiology and reconstruction protocols are described in (Hay et al., 2011). Briefly, the firing response was obtained from slice whole-cell patch clamp recordings in rat S1. For L4 and L5 pyramidal neurons, protocols were identical to (Hay et al., 2011). For the basket cells, we used some additional stimulation protocols (Toledo-Rodriguez et al., 2004). After the experiment, brain slices were fixed and incubated overnight. Morphological reconstruction was performed from well-stained neurons exhibiting only few cut neurite branches.
Single-neuron computational modeling is described in (Hay et al., 2011). Briefly, neurons were represented as a compartmental, conductance-based model using reconstructed morphologies from rat S1. The compartments were separated in 4 zones: axon initial segment (AIS), soma, basal dendrites and apical dendrites (Fig. 1). The full axon was not simulated, only the AIS (Fig. 1, bottom row). Synapses at the postsynaptic cells were activated after spike detection in AIS in the control case and prerecorded spike trains otherwise. A conduction delay based on axonal path distance to the soma (assuming spike conduction velocity 300 μm/ms (Stuart et al., 1997)) was accounted for. Passive membrane capacitance was 1 μF/cm2 for the soma, AIS and dendrites, while for pyramids 2 μF/cm2 for basal and apical dendrites to correct for dendritic spine area. Axial resistance was 100 Ω cm for all compartments. Input resistance Rin was 225±41 MΩ for L4 pyramids and 74±35 MΩ for L5 pyramids. For basket cells, Rin=379±210 MΩ. The resting potential was −74.1±0.1 mV for L4 pyramids, −73.8±0.1 mV for L5 pyramids and −71.6±1.4 mV for basket cells.
Up to 10 active membrane conductance types were accounted with kinetics taken from the published ion channel models or from published experimental data (Hay et al., 2011). The reversal potentials for sodium and potassium were 50 mV and −85 mV, respectively, and −45 mV was used for the Ih current. Ion currents were modeled using the Hodgkin-Huxley formalism.
Connectivity patterns were implemented as presented in (Hill et al., 2012). Briefly, reconstructed cells from L4 and L5 were placed in a hexagonal volume with a radius of 320 μm, matching biological densities of approx. 240,000 per mm3 in L4 and 90,000 per mm3 in L5 (J. Gonzalez-Soriano, J. DeFelipe, L. Alonso-Nanclares; personal communication). Every axonal part closer than 3 μm to a dendrite is detected and synapses are placed at a 5% subset of these appositions. The subset is chosen such that the number of synapses per connection and synaptic bouton densities match biological values. Spatial distributions of synapses placed in such manner are known to match biological distribution for a number of intracortical pathways with a mean error<8%.
All 15,137,757 synapses were modeled using conductance changes. AMPA- and NMDA-type synapses accounted for excitation. For AMPA receptor (AMPAR) kinetics, the synaptic conductance was 0.3±0.2 nS. The rise and decay time constants were 0.2±0.05 ms and 1.7±0.18 ms, respectively. For NMDAR kinetics, conductance was 0.2±0.1 nS with rise and decay times, 0.29±0.23 ms and 43±1.2 ms, respectively. The reversal potential of AMPAR and NMDAR was 0 mV. For inhibitory GABAA synapses, the mean conductance was 0.66±0.2 nS with the rise and decay time constants, 0.2±0.05 ms and 8.3±2.2 ms. Time constant for recovery from depression and time constant for recovery from facilitation were adopted (Angulo et al., 1999; Gupta et al., 2000) and assigned to each putative inhibitory synaptic location identified by the collision-detection algorithm. The GABAA reversal was −80 mV. External input is mediated by distributing additional excitatory and inhibitory synapses randomly (uniform distribution) across all cells and activating them independently with a temporally modulated frequency. External synapses accounted for approximately 5% of the total number of synapses.
To measure spiking synchrony, we calculated the mean of the normalized joint peristimulus time (PST) histogram at a lag of 0 ms, i.e., the mean cross-covariance of PST histograms of cell pairs, normalized by the product of their std. To generate the histograms, we used a bin width of 1 ms. As the covariance would be affected by the change in firing rates between simulated UP and DOWN, we limited the analysis to spikes elicited during UP. To remove synchrony from the simulation (uncorrelated case), we first generated artificial spike trains by moving all spikes of the control case to times randomly chosen between 0 and 4000 ms. This generated independent stationary Poisson spike trains with the same number of spikes as in the control case. This spike train was then used to drive synapses in a simulation. The external input was also present, but with a constant rate equal to the mean of the rate in the control case. To increase synchrony (super-synchronized case), we moved all spike times of the control case to the nearest multiple of 5 ms. External input in this case was identical to the control case.
The extracellular contribution of transmembrane currents of all neural compartments (approx. 410 compartments per cell, >5,000,000 in total) was calculated via the line-source approximation, LSA (Holt and Koch, 1999). Briefly, assuming a purely homogeneous and resistive (3.5 Ω m) extracellular medium, Laplace’s equation applies ![[big down triangle, open]](/corehtml/pmc/pmcents/x25BD.gif)
2Ve=0. At the boundaries, (1/ρ)Ve = Jm with ρ being the resistivity and Jm the transmembrane current density. LSA assumes each cylindrical compartment of the spatially discretized neuron as a line (a cylinder of infinitesimally small diameter) with a constant current density along the line. The Ve contributed by current Ij of each neural compartment j evenly distributed over the line segment of length Δsj and the overall extracellular voltage becomes:
with rj being the radial distance from line-segment, hj the longitudinal distance from the end of the line-segment, and lj=Δsj+hj the distance from the start of the line-segment. The LSA was found to be accurate except at very small distances (a few μm) from the cable.
Calculation of Ve using the LSA took place on a separate computer cluster (SGI, Fremont, CA) and took approx. 1 hour. The CSD was estimated as the negative second spatial derivative along the depth axis. We also calculated the CSD via iCSD (Łéski et al., 2011) and the outcome remained very similar. We thus used the conventional CSD definition.
The work was supported by NINDS, HFSP, SNSF, the Mathers Charitable Foundation and funding to the Blue Brain Project by the ETH Board and EPFL. Financial support for the CADMOS and Blue Gene/P system provided by the Canton of Geneva, Canton of Vaud, Hans Wilsdorf Foundation, Louis-Jeantet Foundation, University of Geneva, University of Lausanne, and EPFL. Special thanks go to G. Buzsáki, E. Schomburg, A. Shai, Y. Billeh, J. Taxidis, members of the Blue Brain Consortium, in particular Michael Hines, James King, Eilif Muller, Srikant Ramaswamy, Felix Schürmann and Werner van Geit.
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