The effect of ionic diffusion on extracellular potentials in neural tissue
Geir Halnes, Tuomo Mäki-Marttunen, Daniel Keller, Klas H. Pettersen, Gaute T. Einevoll
11 The effect of ionic diffusion on extracellular potentials in neuraltissue
Geir Halnes , ∗ , Tuomo M¨aki-Marttunen Daniel Keller , Klas H. Pettersen Gaute T. Einevoll ∗ E-mail: [email protected]
Abstract
In computational neuroscience, it is common to use the simplifying assumption that diffusive currents arenegligible compared to Ohmic currents. However, endured periods of intense neural signaling may causelocal ion concentration changes in the millimolar range. Theoretical studies have identified scenarioswhere steep concentration gradients give rise to diffusive currents that are of comparable magnitude withOhmic currents, and where the simplifying assumption that diffusion can be neglected does not hold. Wehere propose a novel formalism for computing (1) the ion concentration dynamics and (2) the electricalpotential in the extracellular space surrounding multi-compartmental neuron models or networks of such(e.g., the Blue-Brain simulator). We use this formalism to explore the effects that diffusive currents canhave on the extracellular (ECS) potential surrounding a small population of active cortical neurons. Ourkey findings are: (i) Sustained periods of neuronal output (simulations were run for 84 s) could changelocal ECS ion concentrations by several mM, as observed experimentally. (ii) For large, but realistic,concentration gradients, diffusive currents in the ECS were of the same magnitude as Ohmic currents.(iii) Neuronal current sources could induce local changes in the ECS potential by a few mV, whereasdiffusive currents could could induce local changes in the ECS potential by a few tens of a mV. Diffusivecurrents could thus have a quite significant impact on ECS potentials. (v) Potential variations causedby diffusive currents were quite slow, but could influence the comparable to those induced by Ohmiccurrents up to frequencies as high as 7Hz.
Introduction
During periods of intense neural signaling, ion concentrations in the extracellular space (ECS) can changelocally by several mM [1–5]. For example, the the extracellular K + -concentration can increase from atypical basal level of around 3mM and up to levels between 8 and 12 mM during non-pathological con-ditions [4, 6, 7]. Ion concentration shifts in the ECS will change neuronal reversal potentials and firingpatterns (see e.g., [8,9]). Too large deviances from basal levels can lead to pathological conditions such ashypoxia, anoxia, ischemia and spreading depression [10]. The extracellular ion concentration dynamicsdepends not only on transmembrane ionic sources, but also on ionic diffusion along extracellular concen-tration gradients. Diffusion plays an important role in maintaining local ion concentrations at healthylevels [11], but may also be involved in the propagation of e.g. epileptic seizures during pathologicalconditions [12].Diffusion of ions along concentration gradients carry electrical currents [13, 14], and may in princi-ple have measurable effects on electrical potentials (c.f. liquid junction potentials [15–17]). In manytheoretical approaches, it is assumed that diffusive currents are negligible compared to the Ohmic cur- a r X i v : . [ phy s i c s . b i o - ph ] M a y rents propelled by the electrical field (hereby termed field currents). This is, for example, an underlyingassumption when intracellular currents are computed with the cable equation, which is used in mostmulti-compartmental neural models (see e.g., [18]). It is also an underlying assumption in volume con-ductor theory, which is used to estimate extracellular potentials [19–22], and in standard current sourcedensity (CSD) theory, which predicts neural current sources from recordings of extracellular potentials(see e.g., [21, 23]).Ion concentration changes in the ECS often are accompanied by a slow negative potential shift, whichcan be in the order of a few millivolts [1, 2, 10, 24, 25]. Although Dietzel et al. argued that glial potas-sium buffering currents were the main source of these slow potential shifts, they also discussed possiblecontributions from diffusive currents [2]. Theoretical studies have also identified scenarios where large,but biologically realistic ion concentration gradients may induce diffusive currents that are comparable inmagnitude to Ohmic currents, both in the intracellular [26] and extracellular [14] space. As ion concen-trations in the ECS typically change at the time scale of seconds [2, 4, 14], it is unclear whether diffusivecurrents that originate from such changes can influence recorded extracellular potentials at frequencieshigher than the typical cut-off frequencies of 0.1-0.2Hz applied in most electrode systems (see e.g., [23]).However, if diffusive currents can have an impact on recorded extracellular potentials, it calls for a re-interpretation of extracellular voltage recordings, which have typically been assumed to predominantlyreflect transmembrane current sources (and i.e. not diffusion in the ECS). This was the topic of a recentdebate, where the omittance of diffusive currents in CSD theory was suggested as a possible explanationto an observed discrepancy between theoretical predictions and empirical measurements [27–31].In principle, one could combine experimental measurements of ion concentration gradients with priorknowledge of diffusion constants and typical tortuosities (hindrances) [32,33] to determine the magnitudeof the diffusive currents in the ECS. However, net diffusive currents generally depend on the joint move-ment of multiple ion species along their respective concentration gradients [14]. For example, in regionswith intense neural action potential (AP) firing, we would expect the extracellular [K + ] and [Na + ] toincrease and decrease, respectively. Accordingly, we would expect K + to diffuse out from such a regionand a similar amount of Na + to diffuse into the same region. The net transport of charge (diffusivecurrent) into/out from such a region would thus be much smaller than the charge transported by K + and Na + separately. A reliable experimental estimate of the net diffusive current would therefore requirean extreme accuracy in the measurement of the different concentration gradients, and is probably notfeasible. Therefore, questions about the relative importance of extracellular diffusive currents may bebetter addressed by a theoretical electrodiffusive framework. To give good predictions, such a frameworkshould ensure both biophysical realism and physical consistency, and should ensure (i) that extracellularion concentration gradients have values that are realistic in neural tissue, (ii) that ion concentrationgradients of different species are consistent in the sense that they ensure local electroneutrality in thebulk solution, (iii) that realistic extracellular electrical potentials consistently follows from the interplaybetween neural, transmembrane current sources, and extracellular diffusive sources.Several modelling frameworks based on the Poisson-Nernst-Planck equations are available for forhandling the time development of the ion concentration dynamics and electrical potential [34–38]. Theseframeworks typically require a very fine spatial and temporal resolution, and are not well suited forpredictions at a population level. Simplified models, on the other hand, have so far either been restrictedto only considering the ICS [26], or to be limited in the handling of neuronal features, such as ion channeldistributions and morphology [14]. The standard tool for simulating morphologically complex neuronsis the NEURON simulator, which is an effective tool for computing the dynamics of neuronal membranepotentials. There are methods toolboxes for computing extracellular fields surrounding neural modelsbased on the NEURON simulator (e.g., [39]), but no formalism has yet been developed to computethe extracellular ion concentration dynamics and diffusion in a way that can readily be combined withNEURON.In the current work, we have explored the possible effects that diffusive currents may have on the A B
Neurons ECS j nkM j kdn-1,n j kfn-1,n n+1n n-1 V n c k n c kn-1 V n-1 V n+1 c kn+1 j kf n,n+1 j kdn,n+1 C Neurons ECS i n M i ncap i dn-1,n i f n-1,n n+1n n-1 V n c k n c kn-1 V n-1 V n+1 c kn+1 i fn,n+1 i dn,n+1 Figure 1. The model system. (A)
A cortical column was subdivided into 15 depth intervals. Theedges n = 1 and n = 15 were assumed to be a constant background. The interior 13 depth intervalscontained a population of 10 neurons (only one shown in Figure) simulated with the NEURONsimulator. The output of specific ions into each depth interval n was computed for 10 neurons andsummed, yielding the total input of an ion species k to each depth interval. (B) Ion concentrationdynamics in a ECS subvolume n . J kMn denotes the total transmembrane efflux of ion species k into thedepth interval n of the whole population of neurons. J kf and J kd denote ECS fluxes betweenneighboring subvolume driven by electrical potential differences and diffusion, respectively. (C) Theextracellular potential could be calculated by demanding that the sum of currents into each ECSsubvolume should be zero. Currents were determined by summing the contributions from all ionicfluxes, and adding the capacitive current (black arrow).extracellular potential around a small population of cortical neurons. To do this, we developed a hybridformalism which is briefly summarized in Fig. 1. To simulate the neurodynamics, we used a previouslydeveloped multi-compartmental neural model of layer 5 pyramidal neurons with realistic morphologywhich was implemented in the NEURON simulator [40]), and driven by realistic synaptic input (Fig. 1A).When we had computed the neuronal output in this way, another, electrodiffusive framework was usedfor computing the dynamics of the ion concentrations and electrical potential in the ECS surrounding theneurons (Fig. 1B). The scheme was based on the Nernst-Planck equations for electrodiffusion. However,instead of deriving the extracellular potential from Poisson’s equation, we derived it from Kirchoff’scurrent law (demanding that all currents into a ECS subvolume should sum to zero), a physical constraintthat is valid at a larger spatiotemporal resolution (Fig. 1C). This framework represents a novel, efficientand generally applicable method for simulating extracellular dynamics surrounding multi-compartmentalneural models or networks of such (e.g., the Blue Brain simulator).This work is organized in the following way: In the result-section, the main focus is put on the in-vestigation of the possible role of diffusive currents on electrical potentials in the ECS. In the Discussionsection, we discuss possible implications that our findings will have for the interpretation of data fromextracellular recordings. The Discussion also includes an overview of the assumptions made in the pre-sented model, and on how the framework can be expanded to allow for more thorough investigations ofextracellular diffusive currents in neural tissue. A detailed derivation of the electrodiffusive formalism isfound in the Methods-section (which is found at the end of in this article).
Results
We investigated the role that diffusion can have on extracellular potentials by comparing simulationswhere the ECS dynamics was predicted using the electrodiffusive formalism with simulations wherediffusive currents were assumed to be negligible, so that extracellular ion transports were solely due tofield currents. The model simulation upset is briefly introduced in the following section and in Fig. 1,while the details are postponed to the methods-section. A list of useful symbols and definitions can befound in Table 1.
Table 1. Useful symbols and definitionsSymbol Explanation Units J kMn Net membrane flux of species k into subvolume n mol/s I Mn Net ionic membrane current into subvolume n A I capn Capacitive current into subvolume n A V n Extracellular potential in subvolume n V J kfn − ,n Flux of k from subvolume n − n due to electrical migration mol/s I fn − ,n Electrical field current from subvolume n − n A J kdn − ,n Flux of k from subvolume n − n due to diffusion mol/s I dn − ,n Diffusive current from subvolume n − n A l c Length of ECS subvolume edge 100 µmA c Cross section area of ECS subvolume 600 µm Dynamics of a small neuronal population
Since diffusion takes place at a long time scale compared to the millisecond time scale of neuronal firing,we simulated our neural population for 84 s. The neurons were driven by Poissonian input trains through1000 synapses distributed uniformly over their membrane. The 10 neurons were simulated independently,receiving different input trains (but with the same statistics). Synaptic weights were tuned so that theinput evoked an average single-neuron AP firing rate of about 5 APs per second, which is a typical firingrate for cortical neurons [41].The time-development of the output of the neural population to three selected extracellular volumesis shown in Fig. 2 for the first 7s of the simulation. Fig. 2A shows the currents into the the ECSvolume containing the somatas. Here, we see clear signatures of neuronal APs: Neuronal depolarizationby a brief Na + current pulses (of about -50nA, since this current is leaving the ECS), and repolarizationby a brief K + current (of about +50nA, since this current enters the ECS). Generally, the subvolumecontaining the somatas received a significantly higher influx/efflux of ions (Fig. 2A) compared to thevolumes containing the apical trunk (Fig. 2B) and branches (Fig. 2C). These differences were mainlybecause the soma region contained a higher neuronal membrane area than other subvolumes. Similarly,the region where the apical dendrite branches, contained a larger membrane area than a part of the apicaldendritic trunk, so that the net transmembrane currents were bigger in Fig. 2C compared to Fig. 2B.Regional differences were also due to the distribution of specific ion channels in the neuronal model [40].For example, in the region containing the somata and initial parts of axons/dendrites, the density of Na + and K + -channels was much larger than in regions containing only dendrites. Accordingly, the Na + andK + currents leaving/entering the ECS in the soma-region (Fig. 2A1-2) were almost identical to the totalNa + and K + currents that left/entered the neuron as a whole (Fig. 2D1-2).The neural output in Fig. 2 (but for 84 s, and not only the 7 s shown in the figure) was used in allsimulations shown below. −100−500 A Soma i N a ( n A ) i K ( n A ) −1−0.50 i C a ( n A ) −101 i X ( n A ) −1000100 i c ap ( n A ) t(ms) i t o t ( n A ) −0.1−0.050 B Trunk t(ms) −0.6−0.4−0.2
C Apical t(ms) −100−500
D Total t(ms)
Figure 2. Output from the neuronal population.
Transmembrane currents into selectedextracellular volumes, including (A) the region containing the neuronal somatas ( n = 3), ( (B) theregion containing the trunk of the apical dendrite ( n = 7), and (C) the region where the apicaldendrites branched out ( n = 13). Currents were subdivided into ion specific currents (row 1-4) and thecapacitive current (row 5). The sum of all currents into a subvolume n is indicated in row 6. Thetransmembrane currents were defined as positive when crossing the membrane in the outward direction.Currents were summed over all neural segments (of all neurons) that occupied a given ECS-volume(depth level n ). The total transmembrane currents of the neuron as a whole (summed over all N − (D) . The effect of diffusion on slow extracellular potentials
Extracellular potentials may have many origins, including synaptic currents, passive neuronal membranecurrents, slow or fast active neuronal membrane currents [21,22], or glial buffering currents [2]. To explorewhether also diffusive currents in the ECS can be sources that influence the ECS potential, we first studiedthe long time-scale dynamics of the system. As we saw in Fig.2, the local neuronal current sources variedon a fast temporal time scale. However, the statistics of the input remained the same throughout the 84s simulation, and on a time resolution of seconds, the transmembrane current source (at a given location, n ) remained roughly constant with time. Fig.3A2 and B2 show the profiles of the transmembrane currentsources over the depth of the cortex when averaged over a time interval of 7 s. More or less identicalprofiles were observed if the average was taken over another time interval, regardless of where in the 84s simulation this interval was placed, or the length of the interval as long as it exceeded a few seconds(results not shown). In the soma-region ( n = 3) a net current entered the ECS (current source), whereasa net current left the ECS in all other subvolumes (black lines in Fig. 3A2 and B2). This observationindicates that there should be a net extracellular current in the positive z -direction (in order to completethe current loop). A1 −2 0 2051015 A2
100 150 200 A3 A4 A5
150 160 170 A6 −2 −1 0 A7 i Msyn i Mtot
0 0s16.8s33.6s50.4s67.2s 84s51015 B1 −2 0 2051015 B2 i M (nA)
140 150 160 B3 [Na + ] (mM) B4 [K + ] (mM) B5 [Ca ] (mM)
154 156 158 B6 [X − ] (mM) −2 −1 0 B7 V (mV)
Figure 3. Extracellular profiles in ion concentrations and potential.
Distribution of differentsystem variables over the depth of the cortex for the situation where diffusion was assumed to be zero (A) , and for the situation with diffusion included (B) . The distribution of transmembrane currentsources (temporal average) is shown in (A2) and (B2). Panels A3-A6 and B3-B6 show the distributionsof ECS ion concentrations at different time points. Panels A7 and B7 show the low pass filtered ECSpotential ( (cid:104) V (cid:105) ). For (cid:104) V (cid:105) (A7, B7), the temporal average was taken over the time interval 16.8s prior tothe value indicated in the legend. For the current sources (A2,B2), the temporal average was taken overthe first 16.8 ms of the simulation (the statistics of current sources did not change during thesimulation, and any time interval > s gave nearly identical results for the temporal average).As the neurodynamics resulted in influxes or effluxes of different ion species, ionic concentrationsin the ECS varied during the 84 s simulations. Fig.3 shows the extracellular concentration profiles atselected time points in the cases when (i) diffusion was not included (Fig.3A3-6), and (ii) when diffusion was included (Fig.3B3-6). In the case (i) with no diffusion, changes in ECS ion concentrations weresharp in the soma region ( n = 3), as the transmembrane sources were larger there. For example, the Na + concentration decreased significantly, while K + increased significantly in the soma region. This effectwas less pronounced in the case (ii) when diffusion was included, as diffusion acted to smoothen the ionconcentrations over the cortical depth.In the case (ii) when diffusion was included, the K + -concentration in the soma region increased from abasal level of 3 mM to slightly above 10 mM during the simulation. These final ECS concentrations werein line with the maximum local changes that have been observed experimentally during non-pathologicalconditions [4, 6, 7]. Hence, we expect that the extracellular diffusive transports obtained with such ionconcentration gradients are realistic. In the case (i) when diffusion was not included, the local ionconcentration changes in the soma-regions were unrealistically high. However, this was of no concern forour simulation of other variables, as no system properties depended on the ECS ion concentrations in thecase when diffusion was not included.To explore whether diffusion could have an impact on ECS potentials, we calculated the low passfiltered ECS potential ( (cid:104) V (cid:105) ) in the cases without (i) and with (ii) diffusion included (Fig. 3A7 and B7).Our measure for ( (cid:104) V (cid:105) ) at a time t was simply the average V taken over the time interval between two ofthe selected time points in Fig. 3 (i.e. (cid:104) V (cid:105) at time t was the temporal average of V between t − . s and t ). As the figure shows, (cid:104) V (cid:105) varied from around 0 mV in the basal dendrites and soma-region, to about-2 mV in the apical dendrites. Basically the profile of (cid:104) V (cid:105) reflected the profile of the transmembranecurrent (Fig. 3A2 and B2). As a net current entered the ECS in the soma region, and left the ECSalong the apical dendrites, an extracellular current in the positive z -direction was required in order to close the loop . A positive current in the z -direction, is consistent with a negative voltage gradient in thepositive z -direction. The (cid:104) V (cid:105) profile obtained in our simulations were qualitatively similar to what havebeen seen experimentally, where profiles of sustained voltages which vary by a few mV across the depthof the cortex have been observed (see e.g. [2]).When diffusion was not included, the (cid:104) V (cid:105) profiles did not vary with time (profiles for different t coincide in Fig. 3A7 shows). This was as expected, since (cid:104) V (cid:105) in this case was determined solely bytransmembrane current sources, which, as we saw in Fig. 3A2 were constant over time (for a timeresolution of seconds). The (cid:104) V (cid:105) profiles obtained when diffusion was included were different (Fig. 3B7).In the latter case, (cid:104) V (cid:105) did vary with time, especially in the soma region, where the local ECS potentialwas reduced by approximately 0.2 mV during the time course of the simulation. Thus, Fig. 3B7 showsthat diffusion can indeed induce visible changes in the ECS potential. Several previous, experimentalstudies have observed slow negative potential shift measured in the ECS that coincide with changes inion concentrations, and in particular increases in extracellular K + [1, 2, 24, 25]. Mechanisms behind the impact of diffusion on extracellular potentials
To explore the relationship between diffusion and < V > in further detail, we plotted the extracellularfluxes of all ion species (K + , Na + , Ca , and X - ) in the cases (i) without and (ii) with diffusion includedat the selected time points (Fig. 4). We also plotted the net electrical current associated with the ionicfluxes. To be able to compare the electrical current directly with ionic fluxes, we represented the currentsas a flux of positive unit charges e + . When diffusion was not included, all transports in the ECS weredriven by the electrical field (Fig. 4A). The main transports were those mediated by the most abundantion species in the ECS, which in our simulation were Na + and X - . Due to the negative potential gradientbetween the soma and apical dendrites (3A7), Na + was driven away from the soma, while X - (havingthe opposite valence) was driven towards the soma. Both these transport amounted to a net electricalcurrent away from the soma, i.e. the flux of positive unit charges was positive in depth intervals with n >
3, and negative in depth intervals with n < (cid:104) V (cid:105) profile varied with time (as we saw in Fig. 3B7). Unlike the field currents, thediffusive transports were not determined by which ion species that were most abundant in the ECS, buton the concentration gradients, which were largest for Na + and K + . Due to the high decrease/increaseof Na + /K + caused by AP firing in the soma region, extracellular diffusion drove Na + into the somaregion, and K + out from the soma region (Fig. 4B2). Due to the opposite directionality of these twomain diffusive transports, the net charge transport (as represented by a diffusive flux of unit charges, e + )was smaller than the transports of Na + and K + separately. However, diffusive transports still gave riseto a net electrical transport that was smaller, but of the same order of magnitude as the Ohmic current(compare fluxes of e + in Fig. 4B1 and B2).Fig. 4B3 shows the total ionic transports ( j f + j d ) in the case (ii) when diffusion was included. Asexpected, the extracellular ionic transports differed quite significantly from the case when diffusion wasnot included (Fig. 4A). However, the net transport of unit charges in the system were identical in thetwo cases (compare fluxes of e + in Fig. 4A with that in Fig. 4B3). There is a simple physical argumentto why this had to be the case: Since the formalism was based on Kirchoff’s law (the sum of electricalcurrents into any given ECS volume were zero), the transmembrane current sources into/out from anextracellular volume had to be balanced by extracellular currents out from/into the same volume. As oursimulations were set up so that the neurodynamics (and thus the transmembrane current sources) were j f μ mol/(m s) +− j f j d j tot Figure 4. Extracellular Ion fluxes.
Low pass filtered extracellular fluxes in the case when diffusionwas assumed to be zero (A) , and in the case when diffusion was included (B) . In the latter case, thetotal flux (B3) was subdivided into a field-driven (B1) and diffusive (B2) component. Curves to theright/left of the blue, vertical lines represent fluxes in the positive/negative z -direction respectively.Electrical currents were represented as fluxes of positive unit charges j e + = i/F (rightmost column).The scale bar and legend apply to all fluxes. The low pass filtered fluxes were computed as thetemporal mean of the real fluxes, taken over the 16.8s prior to the value indicated in the legend.identical in the cases with and without diffusion, also the net extracellular current had to be identical. Effect of diffusion on extracellular potentials at shorter time scales
Above, we predicted that realistic concentration variations and the diffusive currents that they evokedcould cause observable variations in sustained extracellular voltage profiles, as earlier discussed in [2].As we saw in Fig. 4, diffusive charge transport out of the soma region was of the same magnitude asfield driven charge transport (compare fluxes of e + in Fig. 4B1 and B2). The assumption that diffusivecurrents is negligible compared to field currents is this not warranted in the scenarios studied here, whichhad large, but realistic extracellular ion concentration gradients.The key differences between diffusive and field driven currents are clear if we explore their effects at ahigher temporal resolution. In Fig. 5A1 we have plotted the extracellular voltage at two selected depthintervals (soma n = 3 and apical dendrite n = 13), in the cases (i) without and (ii) with diffusion included.The figure shows how V varies over the entire simulation of 84 s. It was mainly included to illustratethe general time course of V during the simulation. In Fig. 5B1 we see that the fluctuations in the fieldcurrent (in the positive z -direction) out from the soma and apical dendrites had similar time course as V .As Fig. 5C1 shows, this was definitely not the case for the diffusive current. While field current densitiescould vary by ∼ A/m over the time course of a millisecond, it took about 30s of neural activity tobuild up an extracellular diffusive current density of ∼ . A/m (Fig. 5C1). Early in the simulation, thediffusive current out from the soma-region increased with time in an approximately linear fashion. Withtime, diffusion between neighbouring subvolumes acted to smoothen the ECS ion concentration gradients.Due to this flattening of extracellular concentration gradients, the diffusive current out from the somapeaked after around 30 s, and decreased slowly after that. The concentration build-up was slower in thedendritic region, and the diffusive current out of the apical dendrite still increased in a linear fashion atthe end of the 84 s simulation 5C1.Fig. 5A2-4 shows the time course of V at selected, shorter (40 ms) time intervals, which includeinclude a few neuronal APs. In the soma-region, APs caused an initial decrease in the ECS voltage(current into the neurons discharge the ECS), followed by an increase in the ECS voltage when theneuron repolarized. Since currents always form closed loops, an inward current in the soma region evokedoutward currents along the dendritic branches. Therefore, AP-signatures in the apical ECS region hadthe opposite temporal profiles (increase followed by decrease) compared to what we observed in thesoma-region (decrease followed by increase). Although the fluctuation in V during APs were of the sameorder of magnitude in the soma- and apical region, field currents were generally largest in the somaregion (or out from the soma-region). The explanation to this has to do with the opposite extracellularAP-profiles in the subvolumes containing the soma versus those containing dendrites. In neighboringdendritic subvolumes, the AP profiles were very similar. Accordingly, the difference in V between thesubvolumes were small, and gave rise to small electrical currents through the ECS. Conversely, the soma-region had AP profiles that were opposite from its neighbours. During APs, there were thus large voltagedifferences between the soma ( n = 3) and its neighboring subvolumes, leading to to strong field currents.These observations are in line with previous investigations of extracellular AP-signatures (see e.g., [20]).At an early stage in the simulation, when ion concentrations did not diverge very much from thebasal concentrations, the time development of V was almost identical in the cases without (i) and with(ii) diffusion (Fig. 5A2). However, as ion concentration built up in the system, V was gradually shiftedto more negative values in the case when diffusion was included (Fig. 5A3-A4). After 82 s, V wasshifted with about -0.2 mV in the case when diffusion was included compared to the case with only fieldcurrents, consistent with what we also saw in Fig.3. These shifts took place at a slow time scale, sothat V was close to parallel in the diffusive and non-diffusive cases over the short 40 ms time intervalsplotted in Fig. 5A2-A4. Related to the shifts in V , also the field currents showed closely parallel timecourses in the diffusive and non-diffusive cases (Fig. 5B2-B4). At this short time courses, the diffusivecurrents remained roughly constant, and single APs evoked no visible fluctuations in diffusive currents(Fig. 5C2-C4).As we have seen, the maximum magnitude of the field currents were much larger than the diffusivecurrents during dramatic events such as APs (compare Fig. 5B2-B4 with Fig. 5C2-C4). In addition,we saw that diffusive currents had no visible impact on the temporal development of brief extracellularvoltage signals, and would e.g. not have any impact on AP shapes detected in extracellular recordings(such as multi-unit array recordings). However, the high amplitude field currents were brief in duration,and for most of the time-course (i.e. between APs), the magnitude of field currents and diffusive currentswere similar. We therefore hypothesized that diffusive currents could give a significant contribution to0 (cid:19) (cid:21)(cid:19) (cid:23)(cid:19) (cid:25)(cid:19) (cid:27)(cid:19)(cid:237)(cid:22)(cid:237)(cid:21)(cid:237)(cid:20)(cid:19)(cid:20)(cid:21) (cid:36)(cid:20) (cid:57) (cid:3) (cid:11) (cid:80) (cid:57) (cid:12) (cid:21) (cid:21)(cid:17)(cid:19)(cid:21) (cid:21)(cid:17)(cid:19)(cid:23)(cid:237)(cid:21)(cid:237)(cid:20)(cid:17)(cid:24)(cid:237)(cid:20)(cid:237)(cid:19)(cid:17)(cid:24)(cid:19)(cid:19)(cid:17)(cid:24) (cid:36)(cid:21) (cid:22)(cid:19) (cid:22)(cid:19)(cid:17)(cid:19)(cid:21) (cid:22)(cid:19)(cid:17)(cid:19)(cid:23)(cid:237)(cid:21)(cid:237)(cid:20)(cid:17)(cid:24)(cid:237)(cid:20)(cid:237)(cid:19)(cid:17)(cid:24)(cid:19)(cid:19)(cid:17)(cid:24) (cid:36)(cid:22) (cid:27)(cid:21) (cid:27)(cid:21)(cid:17)(cid:19)(cid:21) (cid:27)(cid:21)(cid:17)(cid:19)(cid:23)(cid:237)(cid:21)(cid:237)(cid:20)(cid:17)(cid:24)(cid:237)(cid:20)(cid:237)(cid:19)(cid:17)(cid:24)(cid:19)(cid:19)(cid:17)(cid:24) (cid:36)(cid:23) (cid:19) (cid:21)(cid:19) (cid:23)(cid:19) (cid:25)(cid:19) (cid:27)(cid:19)(cid:237)(cid:21)(cid:19)(cid:237)(cid:20)(cid:19)(cid:19)(cid:20)(cid:19) (cid:76) (cid:73) (cid:3) (cid:11) (cid:36) (cid:18) (cid:80) (cid:21) (cid:12) (cid:37)(cid:20) (cid:21) (cid:21)(cid:17)(cid:19)(cid:21) (cid:21)(cid:17)(cid:19)(cid:23)(cid:237)(cid:20)(cid:19)(cid:237)(cid:24)(cid:19)(cid:24) (cid:37)(cid:21) (cid:22)(cid:19) (cid:22)(cid:19)(cid:17)(cid:19)(cid:21) (cid:22)(cid:19)(cid:17)(cid:19)(cid:23)(cid:237)(cid:20)(cid:19)(cid:237)(cid:24)(cid:19)(cid:24) (cid:37)(cid:22) (cid:27)(cid:21) (cid:27)(cid:21)(cid:17)(cid:19)(cid:21) (cid:27)(cid:21)(cid:17)(cid:19)(cid:23)(cid:237)(cid:20)(cid:19)(cid:237)(cid:24)(cid:19)(cid:24) (cid:37)(cid:23) (cid:19) (cid:21)(cid:19) (cid:23)(cid:19) (cid:25)(cid:19) (cid:27)(cid:19)(cid:237)(cid:19)(cid:17)(cid:21)(cid:19)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:25) (cid:76) (cid:71) (cid:3) (cid:11) (cid:36) (cid:18) (cid:80) (cid:21) (cid:12) (cid:38)(cid:20) (cid:21) (cid:21)(cid:17)(cid:19)(cid:21) (cid:21)(cid:17)(cid:19)(cid:23)(cid:19)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:25) (cid:38)(cid:21) (cid:22)(cid:19) (cid:22)(cid:19)(cid:17)(cid:19)(cid:21) (cid:22)(cid:19)(cid:17)(cid:19)(cid:23)(cid:19)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:25) (cid:38)(cid:22) (cid:27)(cid:21) (cid:27)(cid:21)(cid:17)(cid:19)(cid:21) (cid:27)(cid:21)(cid:17)(cid:19)(cid:23)(cid:19)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:25) (cid:38)(cid:23) (cid:86)(cid:82)(cid:80)(cid:68)(cid:3)(cid:81)(cid:82)(cid:71)(cid:76)(cid:73)(cid:73)(cid:68)(cid:83)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:81)(cid:82)(cid:71)(cid:76)(cid:73)(cid:73)(cid:86)(cid:82)(cid:80)(cid:68)(cid:3)(cid:71)(cid:76)(cid:73)(cid:73)(cid:68)(cid:83)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:71)(cid:76)(cid:73)(cid:73) t(s) t(s) t(s) t(s) Figure 5. Dynamics on a shorter time-scale . The time development of (A) the extracellularpotential, (B) the field current, and (C) the diffusive current in ECS volumes surrounding the soma( n = 3, full lines) and apical dendrite ( n = 13, dotted lines). The first column (A1-C1) shows the signalfor the entire 84s simulation, while the remaining panels show the signal in selected, brief intervalsduring the simulations. Red lines show the signal obtained when diffusion was assumed to be zero,while blue lines show the signal obtained with the full electrodiffusive formalism. Field currents variedat the same time-scale as V (ms), while diffusive currents varied very slowly (s).the low frequency part of extracellular voltage recordings, which are often sampled down to frequenciesof 0.1-0.2Hz (see e.g., [23]). Below, we explore which frequency components of the extracellular potentialthat may be influenced by diffusive sources. Effect of diffusive currents on power spectra
Fig. 6 shows the power spectrum of V at the depth level of the soma n = 3 (i.e. of the signal seen in Fig.5A1). To see if the power spectrum varied over the time course of the simulation, we split the signal intofour intervals of 21 s, which are shown in Fig. 6A, B, C and D.The effect of diffusion on the power spectrum was largest in the first 21 s of the simulation, i.e. inthe period when the diffusive current out from the soma increased steeply with time (Fig. 5C1). Fig.6A1 shows the full frequency spectrum for these initial 21 s, while Fig. 6A2-A4 are close-ups of selected1frequency intervals of the signal in Fig. 6A1. When comparing the cases (i) without and (ii) with diffusionincluded, we found that diffusion had a strong impact on the power spectrum for frequencies around 1Hz6A2, and even clearly influenced the power of frequencies almost up to 10 Hz (Fig. 6A3). For higherfrequencies, the power spectra obtained without and with diffusion included were more or less identical(Fig. 6A4). In the final stages of the simulation (Fig. 6D), the range of frequencies that were influencedby diffusion terminated at lower frequencies. The power spectra obtained without and with diffusionclosely coincided at 7Hz (Fig. 6D3), but diffusion still had a strong impact on frequencies up to 1 Hz(Fig. 6D2). −1 0 1 2 3−10 −8−6 l og ( po w e r) −0.2 0 0.2−8 −7−6 −7−6 −7−6 −1 0 1 2 3−10−8−6 l og ( po w e r) −0.2 0 0.2−8−7−6 0.8 0.85 0.9−8−7−6 1.8 1.805 1.81−8−7−6−1 0 1 2 3−10−8−6 l og ( po w e r) −0.2 0 0.2−8−7−6 0.8 0.85 0.9−8−7−6 1.8 1.805 1.81−8−7−6−1 0 1 2 3−10−8−6 log(freq(Hz)) l og ( po w e r) −0.2 0 0.2−8−7−6 log(freq(Hz)) 0.8 0.85 0.9−8−7−6 log(freq(Hz)) 1.8 1.805 1.81−8−7−6 log(freq(Hz)) Diff offDiff on Figure 6. Power spectrum for the ECS potential in the soma region
Power spectra for V inthe case when diffusion was set to zero (red lines) and in the case when diffusion was included (bluelines). Rows (A)-(D) represent different 21s time intervals of the simulations, so that e.g. (A) showsthe power spectrum for V between t = 0 and t = 21 s , and (C) is the power spectrum for V between t = 42 s and t = 63 s . Column 1 shows the full power spectrum, while columns 2-4 show close ups ofselected frequency intervals. Diffusive currents had strongest impact in the first simulation interval (A),where it had a notable impact on the power spectrum up to frequencies as high as 7Hz (A3). The somaregion had n = 3.2Fig. 7 shows the frequency spectra for the extracellular potential in the region of the apical dendrites n = 13. In this region, extracellular ion concentrations built up very slowly, and diffusion had a smallerimpact. Here, only the very low frequency part (0.1 Hz) of the power spectrum was significantly influencedby including diffusion in the ECS dynamics. −1 0 1 2 3−10−8−6 l og ( po w e r) −0.2 0 0.2−8−7−6−5 −1 0 1 2 3−10−8−6 l og ( po w e r) −0.2 0 0.2−8−7−6−5 0.8 0.85 0.9−8−7−6−5 1.8 1.805 1.81−7.5−7−6.5−6−5.5−1 0 1 2 3−10−8−6 l og ( po w e r) −0.2 0 0.2−8−7−6−5 0.8 0.85 0.9−8−7−6−5 1.8 1.805 1.81−7.5−7−6.5−6−5.5−1 0 1 2 3−10−8−6 log(freq(Hz)) l og ( po w e r) −0.2 0 0.2−8−7−6−5 log(freq(Hz)) 0.8 0.85 0.9−8−7−6−5 log(freq(Hz)) 1.8 1.805 1.81−7.5−7−6.5−6−5.5 log(freq(Hz)) Diff offDiff on Figure 7. Power spectrum for the ECS potential in the apical region
Power spectra for V inthe case when diffusion was set to zero (red lines) and in the case when diffusion was included (bluelines). Rows (A)-(D) represent different 21s time intervals of the simulations, so that e.g. (A) showsthe power spectrum for V between t = 0 and t = 21 s , and (C) is the power spectrum for V between t = 42 s and t = 63 s . Column 1 shows the full power spectrum, while columns 2-4 show close ups ofselected frequency intervals. Diffusive currents had relatively small effects on the frequency spectrum inthe apical region, since concentration changes were quite small here. The apical region considered here,had n = 13.3 Diffusive currents without neuronal sources
A major component of the ECS diffusive currents did not depend directly on transmembrane neuronalsources. To show this, we ran an additional simulation, where we removed the neuronal sources midways inthe simulation, i.e. after 42s. After this, extracellular transports were solely evoked by the concentrationgradients that had built up in the ECS during the 42s of neuronal activity. In the case when diffusionwas not included in the system, the ECS voltage gradient instantly turned to zero when the neuronalsources were removed, and there were no extracellular transports (results not shown).In the case when diffusion was included in the system, the remaining concentration gradients evokeddiffusive currents through the cortical column (Fig. 8). The diffusive currents evoked a non-zero voltagegradient through the column, so that the ECS potential in the soma region was about 0.1 mV morenegative than the potential in the apical region. This diffusion-generated potential difference is knownas the liquid junction potential (Fig. 8A5). To explain this effect, consider an example where two poolsof salt solutions, so that pool 1 contains NaCl and pool 2 contains KCl. If these are set in contact witheach other, Na + will will diffuse from pool 1 to pool 2, while K + will diffuse from pool 2 to pool 1. SinceK + and Na + do not have identical diffusion constants ( D K > D Na ), diffusion will lead to net transportof charge from pool 2 to pool 1. It is known that the local charge separation associated with this processis extremely small [42], as charge accumulation is rapidly counteracted by an induced electrical potentialdifference between the two pools, which drives charge in the opposite direction from diffusion (c.f., thepotential profile seen in Fig. 8). After the neuronal output was removed, the ionic composition was sothat it evoked a net diffusive current out from the soma region. Accordingly, the electrical potential inthe soma region decreased, which induced a net field current into the soma region. In this case, therewas no neuronal current source, and the sum of the diffusive and field driven currents was always zero.In the absence of neuronal sources, diffusive transports gradually reduced the ECS concentrationgradients. This was a quite slow process, which happened over a time course of tens of seconds (Fig.8A1-A4). The extracellular (liquid junction-) potential gradient decreased accordingly (Fig. 8A5). Thisgives us valuable insight in the diffusive processes, as the simulations in this case are solely dependent onthe ECS concentration gradients, and does not depend on momentary concentration fluctuation generatedby neuronal output, or by the neural processes that originally generated these concentration gradients.In Fig. 8B-C we have explored the power spectrum of the diffusion evoked ECS potential (in thesoma region). The black, straight lines indicate that the local V decays exponentially with time. Fig.8B shows the power spectrum for the time period between 42s and 63s, i.e. for the first 21s after theneuronal sources had been removed. When we compared the power-spectrum of the exponential decaywith the previous power spectra obtained for this time period (i.e. when neuronal sources were present),we see that the removal of neuronal sources increased the power for frequencies up to about 7Hz. Wealso plotted the power spectrum for the following 21s of activity, i.e. for the time period between 63sand 84s (Fig. 8C). In this time interval, the ECS concentration gradients were smaller. The absence ofneuronal sources then only increased the power for frequencies up to about 1Hz.An increase in power for low frequencies was an expected outcome of removing the neuronal sources.In the presence of neuronal sources, the slow ECS potential profile was more or less preserved over time.The observed increase in power when neuronal sources were removed, reflected the gradual decay of theliquid junction potential V . However, we did not expect the range over which this decay process coulddominate to stretch to as high frequencies as 1Hz or above.Based on the analysis of Figures 6-8 we conclude that, as a generality, the contribution of diffusivesources to extracellular potentials are not negligible. However, a comparison between Fig. 8B andFig. 8C, also shows that the powers of the extracellular V that are affected by diffusive processes dependstrongly on the ion concentration gradients in the system. Whether diffusive effects needs to be accountedfor when interpreting extracellular recordings, thus depend on the extracellular ion concentration changesthat are expected under the relevant experimental conditions (see Discussion for more on this).4
140 145 150 A1 A2 A3
154 156 158 A4 −0.1−0.05 0 A5 B2 −1 0 1 2 3−10−9−8−7−6−5 l og ( po w e r) B3 B4 −1 0 1 2 3−10−9−8−7−6−5 log(freq(Hz)) l og ( po w e r) −0.2 0 0.2−8−7.5−7−6.5−6−5.5 log(freq(Hz)) 0.8 0.85 0.9−8−7.5−7−6.5−6 log(freq(Hz)) 1.8 1.805 1.81−8−7.5−7−6.5−6 log(freq(Hz)) Diff offDiff onNo nrnsDiff on n mM mVmMmM B1C1 C3 C4 mM142
C2Na+ K+ X− VCa2+
Figure 8. Extracellular dynamics in the case of no neuronal sources.
The neuronaltransmembrane sources were removed midways (after 42s) in the (84s) simulation. (A) shows thedistribution of different system variables over the depth of the cortex for different time points after theneurons were removed. Extracellular ion concentration gradients ( (A1)-(A4) evoked diffusive currentsthat gave rise to liquid junction potentials in the ECS ( (A5) ). (cid:104) V (cid:105) was the temporal mean taken overthe 8.4s prior to the value indicated in the legend. (B-C) Power spectra for V in the time window t = 42 s to t = 63 s (B) and for t = 63 s to t = 84 s (C) . (B1) and (C1) show the full power spectrum,while B2-B4) and (C2)-(C4) show close-ups of selected frequency intervals. Black, straight lines showthe power spectra for the simulation when neuronal transmembrane sources were removed, and indicatean exponential decay of (cid:104) V (cid:105) . Cases when neuronal sources were included (same as previous figures: bluelines = diffusion, red line = no diffusion) were included for comparison. Power spectra were for thesoma region ( n = 3). Discussion
We tested the hypothesis that neuronal activity could generate extracellular ion concentration gradientssufficiently large to induce diffusive currents of the same order of magnitude as field driven currents in theECS. To explore this, we simulated the extracellular transport of ions in a cortical column, resulting fromthe activity of a small population of layer 5 pyramidal cells. We compared simulations when diffusive5currents were included with simulations where diffusive currents were set to zero. Our findings were sur-prising. Not only could the slow component of diffusive currents be of roughly similar magnitude as fielddriven currents, but diffusive currents could influence the power spectrum of extracellular potentials upto frequencies as high as almost 10 Hz. We note that the simulated shifts in ECS ion concentrations werein the upper range of concentration shifts that have been observed under non-pathological, experimentalconditions. Presumably, the role of diffusion was therefore in the upper range of what could be expectedunder physiological conditions, and it is possible that there are many cases when it is warranted to neglectdiffusive currents (however, this should be verified in each specific case). We still conclude that, as agenerality, diffusive currents can not be assumed to have a negligible impact on extracellular potentials,unlike what has been assumed in many previous theoretical analysis [19, 20, 22, 23, 27].
Model limitations
The simplified model setup used in the current study has several limitations. Firstly, cortical columnscontain several neuron species, whose somata are located in different cortical layers. The small populationof 10 layer 5 pyramidal cells used in the current study, will likely create a bias towards strong concentrationgradients in layer 5 (or in the soma region n = 3 in Fig. 1). Secondly, synaptic connections betweenneurons were not considered in the current study. Such connections could induce a level of synchronyin the neuronal firing, which likely would have an impact on the power spectrum of the ECS potential[41]. Thirdly, the multi-compartmental neuronal model used in this study [40] (and most other multi-compartmental models) does not include ionic uptake mechanisms (such as Na + /K + -pumps). Suchmechanisms would generally act to maintain the ECS ion concentrations closer to the basal levels, andalso astrocytes are known to play a major role in the maintenance of the ECS [14,43]. As such mechanismswere not included in the current study, the simulated shifts in ion concentrations were most likely largerthan those that would naturally occur from the neuronal activity. A natural way to improve the modelwould be to incorporate the effect of neuronal and glial ionic uptake-mechanisms, as well as more of thecortical complexity. If appropriately expanded in this way, the model could ideally replicate the exactrelationship between neuronal activity and extracellular ion concentration dynamics, and could then beused to identify the exact experimental conditions under which diffusion is likely to have an impact onECS potentials, and the conditions for which diffusive currents can rightfully be neglected.In the following sub-sections we discuss some of these model limitations in further detail. We also arguethat, despite its limitations, the modelling study presented here gives strong support to the conclusionthat, as a generality, diffusive currents can not be assumed to have a negligible impact on extracellularpotentials. Interpretation of the extracellular potential
In the current work, V was computed in the following way: First, we determined what the current density i f needed to be between two ECS subvolumes in order for Kirchoff’s current law to be fulfilled (Eq. 11),assuming that i f was uniformly distributed over the intersection ( A c ) between the subvolumes. Next,we computed what V needed to be in each subvolume in order to give the correct values for i f betweensubvolumes (c.f., Eq. 8). The obtained values for V thus represented average potentials over the entireECS subvolumes (with volumes l c × A c ).Experimentally, ECS potentials depend on the distances between the recording electrode and theneuronal current source [44]. For example, ECS signatures of action potentials are much higher closeto the neural membrane, while slower signals can have a longer spatial reach [20, 41]. Accordingly, it islikely that currents are not uniformly distributed over ECS cross sections A c , but are larger in regionsthat are in the vicinity of neuronal membranes. A direct comparison between V as determined by theformalism presented here (averaged over a ECS subvolume), and V measured by point electrodes is thusnot feasible. However, in both cases V was determined by neuronal current sources, and followed the6same time course as these signals (Fig. 5). In addition, the estimates of V in the current work showedsustained ECS profiles (Fig. 3) that were qualitatively similar to those observed experimentally [1,2]. Wethus believe that the large scale (volume averaged) V in the current study represents a useful variablefor assessing relative contribution of field currents and diffusive currents at a tissue level.In our model, we assumed that the average cortex surface area per neuron was about 300 µm [41].This number is species- and region specific (e.g. a smaller surface area of 125 µm per neuron was usedin [20]). Essentially, changing A c (while keeping the population size constant) would amount to changingthe average distance between an arbitrary point in the ECS and the neuronal sources. The computedamplitudes in V thus depended on the ECS cross section area A c . However, the results regarding therelative contributions from diffusive versus field driven currents did not depend qualitatively on A c . Tosee why, we can start by noting that the neuronal current sources at each depth layer n were unalteredwhen the ECS volume was changed. Accordingly, the total ECS current I tot = i tot A c = ( i d + i f ) A c in the z -direction would also be unaltered. An increase in A c by a factor α , would reduce ECS ion concentrationgradients by the same factor α (as the same number of ions would enter an enlarged volume, concentrationchanges would be smaller). This would lead to a reduction i d → i d /α in the the diffusive current density,but at the same time an increase in the ECS cross section area over which diffusion occurs A c → αA c .The net diffusive current I d = i d A c would thus be invariant to changes in A c . The same must thereforehold for the field current I f . In analog to the situation with i d , this would imply a reduction in thecurrent density i f by a factor α , caused by a decrease in the voltage gradients by the same factor α .Thus, the amplitudes of ECS potentials would scale approximately linearly with A c , but but the relativeroles of diffusive versus field currents in generating these potentials would remain the same (the scalingis not strictly linear due to possible variations in the ECS conductivity, see further below). This wasverified in additional simulations, where we varied A c (results not shown).We also ran test simulations to explore how the results depended on size of the neuronal population(results not shown, but described here). If we used a population of 100 neurons instead of 10 neurons, andscaled up the ECS volume accordingly by a factor 10, the power of the V signal was generally reduced(results not shown). This was because the neurons were not firing in synchrony, and thus producedpartially uncorrelated membrane currents within a given subvolume. The summation rules for correlatedversus uncorrelated signals says that N correlated signals, all with standard deviations (amplitudes) σ ,will give a summed signal with standard deviation σN , whereas N uncorrelated signals with standarddeviations σ will give a summed signal with standard deviation σ (cid:112) ( N ). Thus, only fully correlatedsources would give a linear increase in the net signal. A similar effect was shown in a previous study,where the high frequency part of the ECS potential was found to scale sublinearly with the number of APselicited in a volume [20]. Although changes in population size resulted in negative shifts in the powerspectra, the qualitative findings regarding the relative contribution of diffusion to such power spectrawere not significantly changed. Extracellular conductivity
Physically, the ECS conductivity ( σ ) is determined by the number of free charge carriers, weighted bytheir mobility and valence. Thus, in our simulations σ was a varying function of the ECS concentrations(c.f., Eq. 9). However, as absolute variations in ion concentrations were relatively small compared to theinitial ion concentrations (at least of the most abundant species), and as these variations typically wereasymmetric (decreases in K + were accompanied by increases in Na + ), variations in σ were only by a fewpercent compared to the initial value. By running test simulations where we kept σ fixed at the initialvalue, we could confirm that these variations had no significant effect on our simulation results.With the initial ion concentrations that we used, we obtained an ECS conductivity of σ . S/m . Inthe literature, there are quite some variations in values that are given for the ECS conductivity, and alsovariations in how this quantity is defined. In analysis of neural tissue, Chen and Nicholson [4] operatedwith an apparent conductivity, defined as σ (cid:48) = α/λ σ , which was used to describe extracellular electrical7currents through neural tissue (as a whole) at a relatively large spatial scale. In that study, σ repre-sented the conventional conductivity associated with the extracellular bath solution. The extracellulartortuosity ( λ = 1 .
6) represented the hindrances imposed on moving ions by e.g. neuronal processes,while α represented the fraction of the neural tissue that was actually ECS [32, 33]. The apparent ECSconductivity was in that work estimated to be 0.1 S/m [4]. In our simulations, we considered the ECSas a separate domain, and thus explicitly accounted for the fact that ECS currents only moved througha fraction α = 0 . σ = 0 . S/m in our ECS domain. For other comparisons,previous computational studies of local field potentials and current source density estimates, have usedthe value σ (cid:48) = 0 . S/m [20, 21, 41]. However, also in those studies, σ (cid:48) represented the conductivity forelectrical currents through neuronal tissue as a whole. This conductivity would therefore correspond toa conductivity of σ = 1 . S/m in our ECS domain. Our estimate for σ thus lies between the previouslyestimated values for σ , and is relatively close to the value used in Chen and Nicholson’s work [4]. Ion exchange between neurons and the ECS
Commonly, only a subset of the transmembrane currents in a multi-compartmental neuron model areion specific. In the model that we used [40], non-specific currents included the passive (leakage) current,synaptic currents, and the currents through a non-specific ion channel ( I h ). For simplicity, we assumedthat all non-specified currents were carried by an unspecified ion species X - , which to a large degreehad the role that Cl - has in the biological system. We are aware that this is an inaccurate assumption.However, we do not believe it to be of any significance for our main results. The arguments for this are(i) that ECS ion concentrations did not have any significant effects on the ECS conductivity (as discussedabove), and (ii) that diffusive currents in the ECS were mainly mediated by K + and Na + gradients. Theinfluence of the X - -dynamics was therefore rather minor, and subdividing the currents into different ionicspecies would likely not have any significant impact on the results. The ECS also contains other ionspecies than the ones included in our simulation, such as phosphorous and magnesium. However, theconcentrations of these species are typically low compared to those of the main charge carriers, so thatthe omittance of these is unlikely to be a concern with our model.A more critical issue is that real neurons contain Na + /K + -exchangers. As these typically work ata slower pace than the mechanisms involved in fast time scale neurodynamics, they are typically notincluded in neuronal models, and were not included in the model that we used [40]. In addition toneural ion pumps, also glial cells, and particularly astrocytes, are involved in the maintenance of theextracellular space [3, 4, 7, 14, 43]. Significant changes in ECS ion concentrations are therefore likely tooccur only in cases when the neuronal activity level is too intense for such clearance mechanisms to keepup. We are therefore aware that the ionic concentration gradients like those we predicted in Fig. 3 arelikely to be an overestimation of the ion concentration gradients that would realistically build up duringthe relatively moderate AP-firing activity of the neuronal population applied in our model. However, thekey conclusions regarding the contribution of diffusive currents depended on ion concentration gradientsin the ECS, but not directly on the neuronal activity responsible for building up such gradients. The ionconcentration changes that occurred during our simulations were in the range of experimentally observedvalues during non-pathological conditions [4, 6, 7, 45]. We therefore believe that our key conclusions arequalitatively sound.The exact shape of the (cid:104) V (cid:105) -profiles (Fig. 3) depended strongly on how the synaptic input wasdistributed over the cortical depth. In all simulations (Figs. 2-8) synapses were distributed uniformlyover the neuronal membrane. We did run some test simulations to explore if the main conclusionsdepended critically on this modelling choice (results not shown, but described here). Profiles similarto those in Fig. 3 were obtained when only the apical dendrites contained synapses. In the case whensynapses were found exclusively in basal dendrites and soma, (cid:104) V (cid:105) -profiles increased gradually from 0mV8in the soma and basal dendrites to +2mV in the most superficial layers, and were almost a mirror imagefrom what we observed with the uniform synapse distribution. Accordingly, the direction of electricaltransports were reversed. However, all cases gave rise to qualitatively similar conclusion regarding therelative importance of extracellular diffusive currents. We therefore only included the scenario with auniform synapse distribution in our main results.The main objective of this work was to explore the effect of diffusion on the dynamics of the ECSpotential. We therefore compared simulations obtained when we set j dk = 0 (i.e. the case with purelyOhmic extracellular currents), with simulations where the extracellular ion concentration- and voltagedynamics were derived using the full electrodiffusive scheme. In the modelling setup, we assumed thatthere was no feedback between the ECS to the neurons. That is, we did not account for changes in neuralreversal potentials due to changes in ECS ion concentrations [8, 9], or ephaptic effects of ECS potentialson neuronal membrane potentials [46–49]. This simplification gave us the advantage that we could haveexactly the same neurodynamics when comparing the ECS dynamics in the cases without or with diffusionincluded in the ECS. Hence, we could be sure that the observed differences between the two scenarioswere due to extracellular diffusion, and not indirect effects stemming from altered neurodynamics. Ina more realistic scenario, the shifts in ECS ion concentrations that occurred during the time course ofthe simulation would induce changes in the neuronal firing patterns. Most likely, the increases in K + in the ECS would make neurons more excitable, and increase the AP firing rate. However, when itcomes to the conclusions regarding the relative contribution of diffusive versus field currents for a givenunderlying neurodynamics, we do not believe that the inclusion of feedback from the ECS dynamics tothe neurodynamics would induce any qualitative changes in our findings.Theoretically, the sum of currents through a closed surface, such as a neuron, should be zero. In Fig.2D6, we saw that the total sum of transmembrane neural currents was small but not zero. This wasa numerical inaccuracy that could be improved by using smaller time steps in the Neuron-simulation.However, a non-zero input current is, however, not inconsistent with the ECS-formalism presented inthis work. In principle, there is no demand that the selected ECS-volume should not contain partialmembranes, such as e.g., a neural axon entering from a neuron located outside the column selected here.The formalism was based on Kirchoff’s current law, but had a leaky boundary at n = 1. The small netcurrent entering/leaving the ECS from the neuronal sources (Fig. 2D6) gave rise to a corresponding smallcurrent leaving/entering the system at n = 1 (results not shown). The boundary condition at n = N ensured that no net current could leave the system there. A novel mathematical framework for simulating electrodiffusion in neural tis-sue
The mathematical framework presented here represents a novel, general framework that can be usedto compute the dynamics of ion concentrations and the electrical potential in the ECS surroundingmulti-compartmental neuronal models or networks of such (such as e.g. the Blue-Brain-Simulator [50]).The framework was particularly adapted to study dynamics at a large spatiotemporal scale (for activepopulations on neurons at the level of neural tissue). For this purpose, the framework is significantlymore computationally efficient than other electrodiffusive frameworks based on the Poisson-Nernst-Planckequations [34–38]. A future ambition is to expand this framework so that it (i) can account for neuronaland glial ionic uptake mechanisms (ion pumps), and (ii) can be used to simulate general, three dimensionaltransport processes in neural tissue. Such a generalized framework would be of undisputable value for thefield of neuroscience, as it can be applied to explore pathological conditions related to ion concentrationshifts in neural tissue [10, 12, 51, 52], and for further exporations of possible effects that diffusive currentscan have on recorded extracellular fields.9
Methods
Extracellular dynamics
From a methods-development point of view, the main contribution of the current work was the formulationof a novel formalism for computing the ion concentration dynamics and the electrical potential in theECS surrounding a neural population. For simplicity, we assumed that spatial variation only occurredin one spatial direction ( z -direction), and that we had radial homogeneity. This simplification may bewarranted in several brain regions, such as within a cortical column. Continuity equation
The formalism represents a way of solving the continuity equation for the ionic concentrations ( c kn (mol/m )) in a system as sketched in Fig. 1B. The ECS is subdivided into a number of N subvol-umes of length l c and cross section area A c (in the application used in the current work, we had N = 15).In each subvolume n , the concentrations of all present ion species k are assumed to be known. Ionsmay enter the subvolume either via (i) transmembrane fluxes from neurons that exchange ions with thesubvolume ( j kM ), (ii) diffusive fluxes between neighboring subvolumes ( j kd ), or (iii) field fluxes betweenneighboring subvolumes ( j kf ). The formalism computes the ECS fluxes, and is general to the choice ofneuronal sources. For now, we therefore assume that the transmembrane fluxes j kM for all ion species aswell as the transmembrane capacitive current (which will be relevant below) are known (e.g., determinedfrom a separate simulation using the e.g., the NEURON simulator). The continuity equation is: A c l c ∂c kn ∂t = J kMn + J kdn − ,n − J kdn,n +1 + J kfn − ,n − J kfn,n +1 (1)Here, A c l c is the volume of a subvolume, so that the left hand side of Eq. 1 represents the time dependentchange of the number of particles (in mol/s) of species k in subvolume n . The extracellular fluxes aredescribed by the Nernst-Planck equation: J kdn − ,n = − A c D k l c ( c kn ) − c kn − ) (2)and J kfn − ,n = − A c D k z k ψl c c kn − + c kn V n − V n − ) , (3)We have used the notation that J n − ,n denotes the flux from subvolume n − n . The factor ψ = RT /F is defined in terms of the gas constant ( R = 8 . / (mol K)), the absolute temperature( T ), and Faraday’s constant. Furthermore, D k = ˜ D k /λ is the effective diffusion constant for ion species k , where ˜ D k is the diffusion constant for ion species k in dilute solvents, and λ is the extracellulartortuosity, which represents miscellaneous hindrances to motion through neuronal tissue [4, 33]. In thecurrent work, we used the standard values [53]: ˜ D K = 1 . × − m /s , ˜ D Na = 1 . × − m /s , ˜ D Ca = 0 . × − m /s and ˜ D X = 2 . × − m /s (for the unspecified ion species, we used the diffusionconstant for Cl - ). These values were modified with a tortuosity of λ = 1 . n = 1 and n = N ) represent a background where ion concen-trations remain constant. The continuity equation then governs the ion concentration dynamics in allthe N − K of different ion species, the continuity equation(Eq. 1) for n = 2 , , ..., N − k = 1 , , ..., K gives us K ( N −
2) conditions for the K ( N −
2) ionconcentrations c kn in the N − N state variables for V n in all subvolumes (including the edges). We thus need N additional constraintsto fully specify the system.0 Derivation of the extracellular potential
In the following, we derive expressions for the ECS potential ( V n ) based on the principle of Kirchoff’scurrent law, and the assumption that the bulk solution is electroneutral [14]. To do this, we multiplythe continuity equation (Eq. 1) by F z k , take the sum over all ion species k , and obtain the continuityequation for electrical charge: ∂q n ∂t = I Mn + I dn − , − I dn,n +1 + I fn − ,n − I fn,n +1 (4)Here, we have transformed fluxes/concentrations into electrical currents/charge densities by use of therelations: I Mn = F (cid:88) k (cid:0) z k J kMn (cid:1) , (5) q n / ( A c l c ) = ρ n = F Σ k (cid:0) z k c kn (cid:1) , (6) I dn − ,n = F Σ k (cid:0) z k J kdn − ,n (cid:1) = − A c F Σ k (cid:18) z k D k l c ( c kn − c kn − )) (cid:19) (7)and I fn − ,n = F (cid:88) k (cid:16) z k J kfn − ,n (cid:17) = − A c l c σ n − ,n ( V n − V n − ) , (8)where z k is the valence of ion species k and F = 96 , . / mol is Faraday’s constant. In Eq.8, wealso defined the conductivity (units (Ω m ) − )for currents between two subvolumes n − n as: σ ( n − , n ) = F (cid:88) k (cid:18) D k ( z k ) ψ c kn − + c kn (cid:19) (9)At time scales of > ns , bulk solutions can be assumed to be electroneutral [54]. For our purpose,bulk electroneutrality implies that any net ionic charge entering an ECS-subvolume, must be identicalto the charge that enters a capacitive neural membrane within this subvolume. This is also an implicitassumption in the cable equation, upon which the Neuron-simulator is based (see e.g., [14, 18, 26, 55]).With this assumption at hand, the continuity equation for charge (Eq. 4) becomes useful for us, as it isgoverned by a constraint that we did not have at the level of ion concentrations (Eq. 1). Electroneutralityin the bulk solution implies that the net charge entering an ECS subvolume (the time derivative of q n inEq. 4) must be identical to the charge which accumulates at the neuronal membrane and gives rise tothe neuronynamics. This means that the time derivative of q n must be equal to the capacitive currentthat we know from the NEURON-simulator: ∂q n ∂t = − I capn (10)Thus, q n (in Eq. 4) is not a state variable, but an entity known from the Neuron-simulation (i.e. aninput condition to the ECS). With this at hand, we can rewrite Eq. 4)on the form: − I capn − I Mn = I dn − , − I dn,n +1 + I fn − ,n − I fn,n +1 (11)We now see that Eq. 11 is simply Kirchoff’s current law, and states that the net current into an ECSvolume n is zero, cf. Fig. 1C. If we insert Eq. 8 for I f , Eq. 11 becomes: σ n − ,n V n − − ( σ n − ,n + σ n,n +1 ) V n + σ n,n +1 V n +1 = l c A c (cid:0) − I capn − I Mn − I dn − , + I dn,n +1 (cid:1) (12)1We note that I Mn was defined as the net ionic transmembrane current (Eq. 5), and that it does notinclude the capacitive current. We further note that Eq. 12 for a subvolume ( n ) depends on the voltagelevels in the two neighbouring subvolumes ( n − n + 1), and thus only gives us N − N − n = 1 and n = N ). As we may chose an arbitrary reference point for the voltage, we may take the first criterion tobe: V = 0 (13)As the second criterion, we impose a boundary condition stating that no net electrical current isallowed to pass between the subvolumes n = N − n = N (i.e. no net electrical current enters/leavesthe system from/to the constant background). Since there may be a diffusive current between these twosubvolumes ( c kN − is not constant), this criterion implies that we must define V N so that the field currentis opposite from the diffusive current ( I dN − ,N + I fN − ,N = 0). If we insert for I f (c.f., Eq. 8), thiscondition becomes: σ N − ,N V N − − σ N − ,N V N = l c A c I dN − , (14)The conductivities ( σ ) and the diffusive currents ( I d ) are defined by ionic concentrations in the ECS,whereas we assumed that the neuronal output ( I cap and I M ) was known. Equations 12-14 thus give us N equations for the N voltage variables V n . In matrix form, we can write the system of equations (Eq.12-14) as: A m,n V n = M n , (15)where the vector M n has elements: M n = n = 1 l c A c (cid:0) − I capn − I Mn − I dn − , + I dn,n +1 (cid:1) for n = 1 , , ..., N − l c A c (cid:0) I dN − , (cid:1) for n = N (16)and where A m,n = A , A , · · · A , A , A , · · · A , A , A , · · · · · · A N − ,N − A N − ,N − A N − ,N · · · A N,N − A N,N (17)is a tridiagonal matrix. The diagonal above the main diagonal is given by: A n,n +1 = (cid:40) n = 1 σ n,n +1 for n = 2 , , ..., N − A n,n − = (cid:110) σ n,n +1 for n = 2 , , ..., N (19)The main diagonal is given by: A n,n = n = 1 − ( σ ( n − , n ) + σ ( n, n + 1)) for n = 2 , , ..., N − − σ ( N − , N ) for n = N (20)2For each time step in the simulation, we can determine V n by solving the algebraic equation set: V n = A − m,n M n , (21)When we ran simulations where diffusion was not included, j d was simply set to zero in the continuityequation (Eq. 1), and in the equation where the ECS potential is derived (Eq.16). Initial Conditions
As initial conditions, we assumed that all ECS volumes were at potential V n = 0. The initial ionconcentrations were also identical in all ECS subvolumes. We used c K = 3 mM , c Na = 150 mM , c Ca = 1 . mM . These ion concentrations are quite typical for the ECS solutions [56]. To obtain an initialcharge density of zero in the bulk solution ( (cid:80) z k c k = 0), we computed that the initial concentration forthe unspecified anion was c X = 155 . mM . This value for c X is close to typical ECS concentrations forCl - , and the unspecified ion X - can be seen as essentially taking the role that Cl - has in real systems. Power spectrum analysis
The power spectra (Figs. 6-8) were computed with the fast Fourier-transform in MATLAB (http://se.mathworks.com/).
Neuronal population dynamics
In the current work, we applied the electrodiffusive formalism presented above to predict the extracellularion concentration dynamics and electrical potential surrounding a small population of 10 layer 5 pyramidalcells. The neural simulation used in this study was briefly introduced in the Results section, but ispresented in more detail here.
Pyramidal cell model
As neural model, we used the thick-tufted layer 5 pyramidal cell model by Hay et al. [40], which wasimplemented in the NEURON simulation environment [57]. The model was morphologically detailed (ithad 196 sections, each of which was divided to 20 segments), and had an extension of slightly less than1300 µm from the tip of the basal dendrite to the tip of the apical dendrites. It contained ten active ionchannels with different distributions over the somatodendritic membrane, including two Ca -channels( i CaT and i CaL ), five K + -channels ( i KT , i KP , i SK , i Kv . , and i m ) and two Na -channels ( i NaT and i NaS ). In addition, in included a non-specific ion channel ( I h ) and the non specific leakage current i leak .We refer to the original publication for further model details [40]. Synapse model
The neurons were driven by Poissonian input trains through 1000 synapses. The synapses were uniformlydistributed across the membrane so that the expected number of synapses in a segment was proportionalto its membrane area. A population of 10 neurons was simulated by running 10 independent simulationswith the same neural model. In the independent simulations, we applied different input trains (but withthe same Poissonian statistics).The AMPA synapses were modelled as α -shaped synaptic conductances: I ( t ) = (cid:26) g max ( t − t ) /τ × exp(( t − t ) /τ ) , when t ≥ t , when t < t , (22)where t represents the time of onset. The time constant was set to τ = 1 . g max = 0 . µ S was tuned so that the input evoked an average single-neuron AP firing rate of about5 APs per second, which is a typical firing rate for cortical neurons [41].3
Population output to the ECS
The expansion of the cell morphology in the applied computational model [40] was such that the maximalspatial distance between two segments (from tip of basal dendrite to tip of apical dendrite) was < µm .We therefore assumed a cortical depth of 1500 µm , and subdivided it into N = 15 depth intervals of length l c = 100 µm , so that the neurons populated the interior 13 subvolumes. Each neural segment was assignedas belonging to a particular subvolume n , determined by the spatial location of the segment midpoint. Inthe setup, the soma was placed in subvolume n = 3, the basal dendrites were in subvolumes n = 2 , n = 3 , ...,
14. The multi-compartmental model also includeda short axon, which was, however, not based on the reconstruction and hence had no fixed coordinates.We assigned the axonal segments into the same subvolume as the soma, n = 3. The edge-subvolumes 1and 15 contained no neural segments (see Fig. 1A).The transmembrane current density ( i kMseg ) of ion species k is available in the Neuron simulationenvironment. It was multiplied by the surface area of the segment ( A seg ) to get the net current, anddivided by Faraday’s constant ( F ) to get a net ion flux with units mol/s. During the neural simulation, wegrouped all currents that were carried by a specific ion species into the net transmembrane influx/effluxof this ion species. We assumed that all non-specific currents, including the synaptic currents ( i leak , i h and i syn ) were carried by a non specific anion that we denoted X - . In this way we could compute the netefflux of each ion species into a subvolume n : J CaMn = 12 F (cid:88) seg (cid:0) ( i CaTseg + i CaLseg ) A seg (cid:1) (23) J NaMn = 1 F (cid:88) seg (cid:0) ( i NaTseg + i NaSseg ) A seg (cid:1) J KMn = 1 F (cid:88) seg (cid:0) ( i KTseg + i KPseg + i SKseg + i Kv . seg + i mseg ) A seg (cid:1) J XMn = − F (cid:88) seg (cid:0) ( i leakseg + i hseg + i synseg ) A seg (cid:1) . Here, the sum was taken over all neural segments ( seg ) of all 10 neurons contained in n . The factor2 in the denominator in the extression for J CaMn was due to Ca having valence 2, and the negativesign in the expression for J XMn was due to X − having valence -1. We also kept track of the (non-ionic)capacitive currents, as required by the electrodiffusive formalism (Eq. 12). I capn = (cid:88) seg (cid:0) i capseg A seg (cid:1) (24)Following [41], we assumed that the average cortex surface area per neuron was about 300 µm . As wehad 10 neurons, and as only about 20 % of cortical tissue is extracellular volume, the ECS subvolumesused in our simulations had surface area A c = 600 µm and length l c = 100 µm (Fig. 1B).We wanted to simulate the extracellular ion concentration dynamics for steady-state neuronal activityover 84 s. Due to the immense amount of data associated with recording all ionic currents in 13 depthlayers over such a long time period, we only simulated the neurons for 7 s, and looped this data 12 timesto obtain 84 s of neuronal output. References
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