Spatiotemporal Dynamics of Lateral Na⁺ Diffusion in Apical Dendrites of Mouse CA1 Pyramidal Neurons

Ethical approval

The study was conducted in accordance with the guidelines of the Heinrich Heine University Düsseldorf, as well as the European Community Council Directive (2010/63/EU). All experiments were communicated and approved by the Animal Welfare Office at the Animal Care and Use Facility of the Heinrich Heine University (Institutional Act No. O52/05). In accordance with the recommendations of the European Commission (published in: Euthanasia of experimental animals, Luxembourg: Office for Official Publications of the European Communities, 1997; ISBN 92-827-9694-9), animals, all younger than 10 d, were killed by decapitation.

Preparation of organotypic tissue slice cultures

Organotypic tissue slice cultures were prepared from Balb/C mice [postnatal days (P) 6–9; both sexes] following a modified protocol originally described by Stoppini and colleagues (Stoppini et al., 1991; Lerchundi et al., 2019). Tissue slice cultures for experiments illustrated in Figure 1E (5 and 17 mM internal Na⁺) were prepared from C57BJ/6J mice. Animals were decapitated, and their brains quickly removed and placed into ice-cold artificial cerebral spinal fluid (ACSF) containing the following (in mM): 130 NaCl, 2.5 KCl, 2 CaCl₂, 1 MgCl₂, 1.25 NaH₂PO₄, 26 NaHCO₃, and 10 glucose, bubbled with 95% O₂/5% CO₂, pH 7.4, and an osmolarity of 310 ± 5 mOsm/L. Subsequently, brains were separated into hemispheres and cut into 250-µm-thick slices in a parasagittal orientation using a vibrating blade microtome (HM 650V, Thermo Fisher Scientific).

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Figure 1.

FLIM-based analysis of [Na⁺]_i and line scan recording of [Na⁺]_i changes. A, Color-coded image of ING2 fast fluorescence lifetime (FL) of a CA1 neuron. Color code on the right illustrates [Na⁺]_i as determined by in situ calibration. The dotted box, region enlarged on the right; PP, patch pipette. Top right, Dotted lines indicate 10 µm long ROIs from which FL was analyzed. Note that pipette and soma (encircled by magenta) are depicted with the same color code as dendrites; calibration properties, however, differ in these compartments. B, Box plots illustrating baseline [Na⁺]_i along primary dendrites at different distances from the soma (n = 8, N = 6). C, Color-coded image of ING2 FL of a cell exposed to TTX. Again, pipette and soma are depicted with the same color code as dendrites, but calibration properties differ. Top right, Current-clamp recording in control and with TTX. D, Box plots showing [Na⁺]_i in control (16 recordings in 8 cells) and with TTX (24 recordings in 12 cells; p = 0.006, Mann–Whitney test). E, Box plots showing [Na⁺]_i in primary dendrites in control conditions from cells patched with a pipette saline containing either 5, 11 or 17 mM Na⁺. B, D, E, Shown are single data points (diamonds), mean (squares), median (horizontal line), quantiles (box), and SD (whiskers). D, Gray diamonds, primary dendrites; red diamonds, secondary dendrites; statistical significance in indicated by asterisks with 0.01 < **p < 0.001. F, Left, Projection of an SBFI-filled primary dendrite. Red line, position of the scan line; SOI 0, segment exhibiting the largest relative change in fluorescence. IP, iontophoresis pipette; arrowhead indicates flow of bath perfusion. Top Right, False color-coded (x,t) image of the baseline-corrected line scan, depicting [Na⁺]_i changes induced by glutamate iontophoresis (100 ms). Vertical box, position of SOI 0. Center, Glutamate-induced somatic inward current. Bottom right, Glutamate-induced change in [Na⁺]_i in SOI 0. Black, baseline corrected trace (subjected to 50 Hz low-pass FFT); red, filtered trace as given by the Python-based program. G, Somatic inward current, its calculated first deviation and the accompanying change in [Na⁺]_i in SOI 0. H, Glutamate-induced dendritic [Na⁺]_i transients and somatic currents taken under control conditions (left), after wash in of glutamate receptor blockers APV and NBQX (middle), and after washout of the blockers (right). I, Peak-normalized, averaged traces from experiments conducted at 22°C (n = 18, N = 18; red) and at 32°C (n = 15, N = 11; green). J, Box plot comparing the decay time constants (τ) of [Na⁺]_i recovery at 22 and 32°C. Shown are individual data points (diamonds), means (squares), medians (horizontal lines), quantiles (boxes), and standard deviations (whiskers). Data was statistically analyzed using a Mann–Whitney test (p = 0.704). K, Scatterplot showing correlation between Δ[Na⁺]_i and τ of the signal measured at SOI 0 in experiments taken from DIV 10–25 primary dendrites. The dotted line shows a linear fit (R² < 0.001; slope = 0.02 mM/s).

The hippocampus and part of the adjacent cortex were isolated, washed in Hanks’ balanced salt solution (Sigma-Aldrich), and transferred onto Biopore membranes (Millicell standing insert, Merck Millipore). Membranes were then placed into a 6-well plate and maintained in an incubator at the interface between humidified air containing 5% CO₂ and culture medium at 36°C. Culture medium contained the following: minimum essential medium (MEM; Sigma-Aldrich), 20% heat-inactivated horse serum (Thermo Fisher Scientific), 1 mM ʟ-glutamine, 0.01 mg/ml insulin, 14.5 mM NaCl, 2 mM MgSO₄, 1.44 mM CaCl₂, 0.00125% ascorbic acid, and 13 mM d-glucose. Slices were kept in the incubator until use between 3 and 50 d in vitro and the medium was changed every 3 d. Unless otherwise mentioned, experiments were carried out at room temperature (22 ± 1°C), and slices were constantly perfused with standard ACSF at a rate of 2.5 ml/min. In experiments conducted at 32 ± 1°C, slices were perfused with ACSF containing the following (in mM): 138 NaCl, 2.5 KCl, 2 CaCl₂, 1 MgCl₂, 1.25 NaH₂PO₄, 18 NaHCO₃, and 10 glucose, bubbled with 95% O₂/5% CO₂, pH 7.4.

Electrophysiology, dye loading, and glutamate iontophoresis

Whole-cell patch-clamp measurements were performed using a pipette solution containing the following (in mM): 114 KMeSO₃, 32 KCl, 10 HEPES [4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid], 10 NaCl, 4 Mg-ATP, 0.4 Na₃-GTP, pH adjusted to pH 7.3. To study the influence of pipette [Na⁺] on dendritic [Na⁺]_i, pipette solutions contained either 4 or 16 mM NaCl with KCl concentrations adapted to maintain isoosmolarity. Cells were held in the current-clamp mode in “zero current” conditions or in the voltage-clamp mode at −70 mV (liquid junction potential not corrected) using an EPC10 amplifier (HEKA Elektronik) and PatchMaster-Software (HEKA Elektronik). Electrophysiological data was analyzed using OriginPro software (OriginPro 2021).

For multiphoton imaging of Na⁺, the membrane-impermeable forms of the fluorescent chemical Na⁺ indicators ION-NaTRIUM-Green-2 (ING2 TMA⁺ salt; ION Biosciences/MoBiTec) or Na⁺-binding benzofuran-isophthalate (SBFI K⁺ salt; TEFLabs) were added to the intracellular saline at a final concentration of 50 µM (ING2) or 1 mM (SBFI). The dyes were loaded into individual neurons using whole-cell patch clamp for at least 30 min before starting imaging experiments as reported previously (Gerkau et al., 2019). Standard visualization of filled neurons was performed using maximum intensity projections from z-stacks (1 µm steps).

For three-dimensional reconstruction of dye-filled dendrites, z-stacks were taken at 0.2 µm spacing between focal planes. Z-stacks were reconstructed and visualized using the ImageJ software (National Institutes of Health). Stacks were imported into the Huygens Software (Huygens Professional, SVI imaging) for deconvolution using an experimentally determined point spread function (PSF). Deconvoluted stacks were reconstructed and projected as a three-dimensional structure with the Imaris software (Center for Bio-Image Informatics, University of California). Dendrites were traced and reconstructed using the Imaris filament tool to determine their diameter and to identify presumptive spines.

For local iontophoresis of glutamate, a fine-tipped, high resistance (90–120 MΩ) borosilicate glass pipette was pulled out (PP-830, Narishige) and filled with 150 mM glutamate. Pipettes were coupled to an iontophoresis module (MVCS-M-45, NPI Electronic), and a retain current was applied to prevent leakage of glutamate. The pipette tip was lowered into the tissue slice under visual control and placed close to a chosen dendrite. The tip of the iontophoresis pipette was positioned upstream relative to the bath perfusion and directed against the perfusion flow to minimize the extracellular diffusion of glutamate along the chosen dendritic section.

Fluorescence lifetime imaging of Na⁺ with ING2

Fluorescence lifetime imaging (FLIM) of ING2 was performed as described earlier (Meyer et al., 2022), employing a laser-scanning microscope based on a Fluoview300 system (Olympus Deutschland), or a Nikon A1-R MP system (Nikon Europe), equipped with a water immersion objective (NIR Apo 60×, NA 1.0, Nikon Instruments Europe). Laser pulses (100 fs) were generated at 80 MHz by a mode-locked Titan Sapphire laser (Mai Tai or Mai Tai DeepSee, Newport, Spectra Physics); excitation wavelength was 840 nm. Amplitude weighted, average fluorescence lifetimes (FL) were measured using time-correlated single photon counting (TCSPC) with a spatial resolution of 0.57 × 0.57 µm per pixel and analyzed employing MultiHarp 150 electronics and SymPhoTime 64 acquisition software version 2.8 (both PicoQuant). Fluorescence emission was bandpass-filtered (534/30 or 540/25 nm, AHF Analysentechnik) and directed to a phosphomolybdic acid (PMA) hybrid photodetector (PicoQuant). Images of dye-filled neurons represent fast lifetime (fLT) images as generated by the SymPhoTime64 software (Meyer et al., 2022). Data analysis was performed using average fluorescence lifetimes (τ_AVG) from defined regions of interest (ROIs) drawn along dendrites. Decay curves were analyzed by reconvolution of the instrument response function (IRF) and biexponential fitting resulting in the amplitude weighted τ_AVG (Meyer et al., 2022).

For determination of dendritic Na⁺ concentrations ([Na⁺]_i), z-stacks (1 µm steps, each 15 or 30 frames) were taken. Photons were collected until >1,000,000 photons were reached for each plane, and maximal projections of fLT images were calculated. Further analysis of the extracted τ_AVG, including conversion of FL into [Na⁺]_i (see below) and statistical analysis of the data, was commenced using Excel (Excel 2019, Microsoft Corporation) and Origin software (OriginPro 2021).

To convert FL into [Na⁺]_i, ING2 was calibrated in situ as described earlier (Meier et al., 2006; Langer and Rose, 2009; Meyer et al., 2022). The properties of Na⁺ indicator dyes (including apparent binding affinity to Na⁺) are different under in vitro conditions as compared with those inside cells (Despa et al., 2000; Langer and Rose, 2009; Lamy and Chatton, 2011; Naumann et al., 2018; Meyer et al., 2019). Indeed, when employing in situ calibration parameters determined for ING2, the FL obtained from ROIs positioned over patch pipettes were too low for a conversion into meaningful [Na⁺]. In contrast, using in vitro calibrations parameters established earlier (Meyer et al., 2022) resulted in a value of 11.0 mM that was in good agreement with the known pipette [Na⁺] (11.2 mM). In somata, in vitro calibration parameters resulted in rather high apparent [Na⁺]_i (>40 mM), while in situ calibration indicated an apparent [Na⁺]_i significantly below that of the pipette solution (0–5 mM). These results suggest that in cell bodies, held in whole-cell patch-clamp mode and thus in direct contact with the pipette saline, neither of the two calibrations is suited to reliably convert FL into [Na⁺]_i. Our FL analysis, therefore, did not consider somata.

Fluorescence intensity imaging of SBFI and automated analysis of line scans

Imaging was performed at the multiphoton system described above for FLIM at an excitation wavelength of 790 nm. Fluorescence emission <700 nm was directed to the internal FluoView photomultiplier tubes (Olympus). Line scans were performed on dye-filled dendrites with a scanning frequency of 233–747 Hz. ImageJ was used to generate line scan projections and projections of z-stacks which were later used for three-dimensional morphological analysis.

Unless stated otherwise, line scans were analyzed using a custom Python-based script for unbiased detection of peak time point and peak amplitude of fluorescence changes (code see Repositorium). The program was operated through the Spyder software (version 5.2.2) over anaconda navigator (version 2.3.2). For this, line scan images were converted into text files using ImageJ and Excel software (Microsoft Corporation), resulting in a pixel-specific listing of fluorescent intensity values over time. The files were then analyzed by the script.

First, the program established segments of interest (SOIs) by averaging intensity values of n neighboring pixels along the line scan, e.g., for an n of 50, starting from pixel 1–50, 51–100, and so forth. Next, the average baseline fluorescence intensity (F₀) of each SOI was determined from the first 500 ms of each recording (during which no stimulation was performed), after which traces were normalized to F₀ (ΔF/F₀). Subsequently, an automated correction for photobleaching was performed by fitting a biexponential decay function ((y=(x0A1*e−xt1)+(A2*e−xt2)+y0) with t₁ > 0.2 and t₁ + t₂ < 10) through the data points between 0 and 500 ms (“baseline period”) and the last quarter of each trace (i.e., when changes in fluorescence induced by glutamate had fully recovered) and subtracting this fit from the respective data trace. Afterward, traces were filtered using a Butterworth low-pass filter (cutoff, 1.5; sampling frequency, 40 Hz; order, 3) and a rolling average (window of 25 data points).

Finally, the peak time point and peak amplitude of glutamate-induced changes in fluorescence intensity were determined by automatically extracting the lowest point of each trace (note that SBFI intensity decreases with increasing [Na⁺]). This point had to obey the following criteria: (1) signal-to-noise ratio (calculated by the division of the peak by the 1% percentile of the trace) larger than 20%; (2) occurrence between 500 ms (time point at which iontophoresis was triggered) and 3,000 ms; (3) baseline drift less than 10%; (4) peak amplitude larger than two times the standard deviation of the noise during baseline conditions. If these criteria were not met, the data trace was discarded. The remaining traces (unfiltered baseline-corrected traces, Butterworth-filtered traces and traces subjected to the rolling average filter) were then plotted using Origin software (OriginPro 2021). Moreover, peak amplitudes and time points determined by the program were imported into the Origin software for further analysis.

Analysis of Na⁺ diffusion within dendrites

Lateral Na⁺ diffusion in dendrites was analyzed following procedures similar to those described by Santamaria et al. (2006). First, we established SOIs of 1.02 µm length (5 pixels) along dendritic line scans using the Python-based automated analysis and performed a temporal binning to obtain frequencies of 50 Hz. Furthermore, conversion of ΔF/F₀ values into changes in [Na⁺]_i was performed via the software, using calibration parameters established as described above and assuming the baseline [Na⁺]_i as determined before by FLIM (see above and results). As a last step, measurements were normalized using Equation 1:fN(x,t)=f(x,t)∑f(x,t)dx,(1) where ƒ_N(x,t) is the normalized concentration distribution over the length of the imaged dendrite. Concentration distributions (ƒ(x,t)) were thereby taken every 0.02 s (dt = 0.02 s) and each SOI had a size of dx = 1.02 µm (5 pixels). Measurements from a given dendrite order (primary or secondary dendrite) and time in culture were then pooled for further analysis.

For calculation of diffusion coefficients, we assumed a one-dimensional diffusion process along the axis of the dendrite (Santamaria et al., 2006), which is described by Equation 2:⟨xexp2⟩=2D0t,(2) where 〈x²_exp〉 is the variance at each time point, D₀ the diffusion coefficient, and t is time. The variance was calculated for each time point using Equation 3:⟨xexp2⟩=∑xx2fN(x,t)dx,(3) We implemented this numerical integration in Matlab using a trapezoidal integration method. Finally, the apparent diffusion coefficient D_app was calculated as follows:Dapp=⟨xexp2⟩−⟨xexp2⟩02t,(4) where 〈x²_exp〉₀ is the variance at the initial condition.

In our experiments, the increase in dendritic [Na⁺]_i was induced by iontophoretic glutamate application which did not represent a strictly localized point source for Na⁺. Moreover, glutamate application resulted in a Na⁺ influx for several hundreds of millisecond. Resulting from that, the rising phase of the signal consists of both an influx and lateral diffusion of Na⁺. We assume that after the peak is reached, the spread of Na⁺ is due to diffusion. Therefore, the concentration distribution which was measured at the peak of the overall signal was defined as initial distribution.

To further verify our results and to assess the effect of boundary conditions set by the experiments such as the duration of Na⁺ influx and the restricted length of dendritic segments imaged, we simulated a normal 2D diffusion process using the following:C(x,tD)=14πDNa+tDe−x2/4DNa+tD,(5) where C(x,t_D) is the concentration spread over the length of the dendrite at a certain time (t_D), x is the distance, and D_Na+ is either 600 µm²/s (intraD_Na+), a value reported earlier for unhindered Na⁺ diffusion in cellular structures (Kushmerick and Podolsky, 1969) or was simulated for the most effective fit of the experimental data (effD_Na+). The variance 〈x²〉 is as follows:⟨x2⟩=2DNa+tD,(6) To take the initial distribution into account, we calculated the time which the simulation needed to reach that distribution (t_shift) given 〈x²_exp〉₀:tshift=⟨xexp2⟩02DNa+.(7) We then simulated a normal diffusion process for 3 s after t_shift using Equation 5. Thereby, t_shift was associated to the peak time point (t = 0) of the experiment. For calculation of the variance, performed using Equation 6, we took the length of the measured dendrite into account. Results of the simulation thus show the variance which is expected within the boundary conditions of the experiment during a normal diffusive process with an effD_Na+ = 200–800 µm²/s (Table 1).

Simulation of lateral spread of Na⁺

To simulate the lateral spread of Na⁺ in dendrites, we used a reconstructed morphology of a pyramidal neuron, obtained from NeuroMorpho.Org (RRID:SCR_002145; Buchin et al., 2022; morphology #NMO_276156) with a total of 2,895 compartments (Tecuatl et al., 2024). The morphology file contains an integer as compartment identifier, type of compartment (soma, axon, basal dendrite, apical dendrite, etc.), coordinates (x, y, z), radius, and the parent compartment of each compartment. The coordinates of two adjacent compartments were used to calculate the length of each compartment. Thus, the length and radius vary from compartment to compartment according to the morphological reconstruction. The compartment's identifier and parent compartment's identifier were used to couple different compartments according to their positioning in the morphology. Overall, this information is enough to spatiotemporally model the spread of [Na⁺]_i using the actual morphology where the currents are distributed following the reported electrophysiology of pyramidal cells (Spruston, 2008). As explained below, although the effect of the radius of each compartment is incorporated in the electrical coupling between neighboring compartments, the diffusion of Na⁺ flux between neighboring compartments only depend on intraD_Na+, separation between two compartments, and the concentration gradient between the neighboring compartments. The treatment of a compartment as a point follows from the assumption that Na⁺ stabilizes very quickly across the width of the compartment due to the large value of intraD_Na+.

The neuron was modeled using Hodgkin–Huxley type conductance based current equations together with dynamic intra- and extracellular ion concentrations. The reversal potentials for various currents are made variable such that they depend on the instantaneous values of concentrations of relevant ions inside and outside the cell. The membrane potential of ith compartment, V⁽ⁱ⁾, is modeled with the following rate equation.Cm(i)dV(i)dt=IK+DR(i)+IK+A(prox)(i)+IK+A(dist)(i)+Ih(i)+IL(i)+ICoup(i)+(INa+Stim(i)−INKA(i))/f,(8) where the equations for the delayed rectified K⁺ (I_K+DR), proximal A (I_K+A(prox)), distal A (I_K+A(dist)), and h (I_h) currents are taken from (Migliore et al., 1999, 2004) and are given as follows. We remark that ignoring I_K+A(prox), I_K+A(dist), and I_h currents does not change results from the simulation significantly and that these currents are included only for completion, considering the previously reported currents in different compartments (Spruston, 2008). The leak (I_L) and coupling (I_Coup) currents between neighboring compartments are also given below. I_Na+Stim and I_NKA represent localized Na⁺ stimulus (Na⁺ is injected to raise [Na⁺]_i locally) and NKA, respectively. f = 0.0445 (Barreto and Cressman, 2011) converts current from μA/cm² to mM/s. Note that our experiments are performed in the presence of TTX; therefore, fast voltage-gated Na⁺ current is not included in the model. Note that in the equations for different currents, we have dropped the superscript (i) for clarity; however, these currents are computed using voltage and intra- and extracellular ion concentrations of that compartment:IK+DR=gK+DR⋅n⋅(V−VK+),(9) where g_K+DR= 40 mS/cm².

The gating variable x (e.g., n, l, etc.) for various currents are given by the following functional form:dxdt=−1τx(V)(x−x∞(V)),(10) with n_∞= 1 / (1 + α_n); τ_n= 50β_n / (1 + α_n); α_n= exp(−11(V − 13)); β_n= exp(−08(V − 13)); τ_n= 2 ms if τ_n < 2 ms.IK+A(prox)=gK+Ap⋅n⋅l⋅(V−VK+),(11) where g_K+Ap = 48 mS/cm², 48 mS/cm², and 4.8 mS/cm² for soma, basal dendrites, and axon, respectively (note that the morphology file contains information about the type of compartment):Forapicaldendrites,gK+Ap={48.(1+d/100)ford≤100μmfromsoma0ford>100μmfromsoma, n_∞= 1 / (1 + α_n); τ_n=4β_n / (1 + α_n); α_n= exp(−0.038(1.5 + 1 / (1 + exp(V + 40) / 5)) · (V − 11)); β_n= exp(−0.038(0.825 + 1 / (1 + exp(V + 40) / 5)) · (V − 11)); l_∞= 1 / (1 + α_l); τ_l= 0.26 · (V + 50); α_l= exp(0.11(V + 56)); τ_n= 1 ms if τ_m < 1 ms; τ_l= 2 ms if τ_l< 2 ms. The distance (d) of a compartment from the soma is calculated using the (x, y, z) coordinates of both compartments:IK+A(dist)=gK+Ad⋅n⋅l⋅(V−VK+),(12) I_K+A(dist)·is only considered in apical dendrites with the following:gK+Ad={0ford≤100μmfromsoma48.(1+d/100)ford>100μmfromsoma, n_∞= 1 / (1 + α_n); τ_n= 2β_n / (1 + α_n); α_n= exp(−0.038(1.8 + 1 / (1 + exp(V + 40) / 5)) · (V + 1)); β_n= exp(−0.038(0.7 + 1 / (1 + exp(V + 40) / 5)) · (V + 1)); l_∞= 1 / (1 + α_l); τ_l= 0.26 · (V + 50); α_l= exp(0.11(V + 56)); τ_n= 1 ms if τ_m < 1 ms; τ_l= 2 ms if τ_l < 2 ms.

I_h is included in the soma and apical dendrites and is given by the following:Ih=gh⋅l⋅(V−Vh),(13) where V_h = −30 mV and g_h = 0.05 · (1 + 3d / 100). Here d is the distance from the soma.

τ_l= β_l / (0.013(1 + α_l)); l_∞= 1 / (1 + exp((V − V_half) / 8)); α_l= exp(0.08316(V − V_half)); β_l= exp(0.033(V − V_half)).

The leak current I_L is composed of three components; Na⁺ leak (I_L,Na+), K⁺ leak (I_L,K+), and Cl⁻ leak (I_L,Cl−):IL=IL,K++IL,Na++IL,Cl−.(14) And I_L,K+= g_L (V − V_K+); I_L,Na+ = 0.35g_L (V – V_Na+); I_L,Cl− = g_L (V − V_Cl−), where g_L = 10³ / R_m is the leak conductance and R_m = 50,000 Ωcm² is the resistivity of the membrane.

Neighboring compartments are electrically coupled with each other through a coupling current (I_Coup):ICoup(i)=∑jγj(i)(V(j)−V(i)),(15) where the summation is over the nearest neighbors. The identifiers of a compartment and its parent compartment are used to calculate the number of nearest neighbors. If compartment (i) has length L⁽ⁱ⁾ and radius r⁽ⁱ⁾ and compartment (j) has length L^(j) and radius r^(j), then the intercompartmental resistance is the sum of the two resistances from the middle of each compartment with respect to the junction between them, i.e., R=RaL(i)2π(r(i))2+RaL(j)2π(r(j))2 , where Ra = 400 Ωcm² is the intracellular resistivity. To compute intercompartmental conductance γ, we take the inverse of the above expression and divide the result by the total area of compartment (i) (Dayan and Abbott, 2001).

The dynamics of intracellular Na⁺ in the ith compartment and K⁺ in its corresponding extracellular compartment (K_o) are formulated by the following equations (Cressman et al., 2009):d[K+]o(i)dt=fIK+,T(i)−2βINKA(i)−Iastro(i)−Idiff(i)+DK+Δx2∑j([K+]o(j)−N[K+]o(i)),(16) d[Na+]i(i)dt=fINa+,T(i)−3βINKA(i)+INa+Stim(i)+intraDNa+Δx2∑j([Na+]i(j)−N[Na+]i(i)).(17) As mentioned above, f converts current density into flux. I_Na+,T and I_K+,T, respectively, are the total Na⁺ and K⁺ currents in the compartment or its corresponding extracellular compartment, due to different channels described above. That is:IK+,T(i)=IK+,DR(i)+IK+,A(prox)(i)+IK+,A(dist)(i)+IL,K+(i),(18) INa+,T(i)=IL,Na+(i),(19) We consider the ratio of intra- to the extracellular volume for each compartment (β) to be 6. I_NKA, I_astro, and I_diff represent NKA, buffering of K⁺ by an astrocyte from the extracellular space of the compartment, and diffusion of K⁺ between the extracellular space of the compartment and bath solution. In other words, each intracellular compartment has its corresponding extracellular compartment, which has its associated astrocytic compartment and diffusion flux with the bath solution. As before, although we have dropped the superscript (i) for clarity, ion fluxes for a given compartment are calculated using the ion concentrations in that compartment and its corresponding extracellular space. I_Na+Stim represents intracellular injection of Na⁺ in a ∼5 μm segment of the dendrite for 500 ms to raise local [Na⁺]_i to the desired value.

The equations for NKA and K⁺ diffusion between the extracellular space and bath are given by Cressman et al. (2009); Ullah et al. (2009):INKA=(ρ1+exp((25−[Na+]i)/3)(11+exp((8−[K+]o)),(20) Idiff=ε([K+]o−[K+]o,∞),(21) The K⁺ uptake by astrocytes is modeled by the following equation:Iastro=Gastro1+exp((18−[K+]o)/2.5),(22) where ρ = 1.25 mM/s and G_astro = 67 mM/s is the maximum strength of NKA and astrocytic buffering, respectively. ε = 1.33/s is the diffusion constant for K⁺ from the extracellular space to the bath solution and [K+]o,∞ is the steady state K⁺ concentration in the bath and is set to 3 mM under physiological conditions.

The last terms in Equations 16 and 17 represent the diffusion of extracellular K⁺ and intracellular Na⁺ between the neighboring compartments with diffusion coefficient of D_K+ = 250 μm²/s and intraD_Na+ = 600 μm²/s, respectively. Δx is the separation between two neighboring compartments, calculated using their (x, y, z) coordinates. No-flux boundary conditions are applied to the ending compartments. For example, a compartment at the end of a branch can exchange Na⁺ with its parent compartment (since the compartments at branch ends do not have daughter compartments). The same approach is also applied to the extracellular compartments. Furthermore, the movement of Na⁺ due to electric drift (electromigration) is ignored as it is significantly smaller than the Fickian diffusion and does not affect our results qualitatively (data not shown).

Following Cressman et al. (2009), we assume that the net flow of Na⁺ into the cell is matched by the flow of K⁺ out of the cell, thus giving the intracellular K⁺ concentration [K⁺]_i:[K+]i=140+(10−[Na+]i),(23) where the constants 140 and 10 mM represent the values for the intracellular K⁺ and Na⁺ concentrations at rest. We also assume that the total amount of Na⁺ is conserved as follows:[Na+]o=144−β([Na+]i−10),(24) where [Na⁺]_o represents the instantaneous value of extracellular Na⁺ and 144 mM is the extracellular Na⁺ concentration at rest. We remark that our model is a very simple representation of a complex reality. In addition to the simplification employed in Equations 23 and 24 and ignoring Cl⁻ dynamics and the effect of electric drift on the diffusion, we lumped K⁺ exchange between the ECS and astrocytes into a simple sigmoidal function, ignoring the complex behavior of various channels and other ion species. However, adding such complexity to the model would not change our main conclusions about the diffusion of intracellular Na⁺ in the neuron. Furthermore, we have shown previously that the intra- and extracellular ion dynamics from this simpler formalism closely matches that from the more extended model where [Na⁺]o, [K⁺]_i, and Cl⁻ are modeled explicitly (Wei et al., 2014).

The reversal potentials for different ion currents are determined by the instantaneous values of their intra- and the extracellular concentrations according to the Nernst equation, with VK+=26.64ln([K+]o[K+]i) ; VNa+=26.64ln([Na+]o[Na+]i) ; and VCl−=26.64ln([Cl−]i[Cl−]o) .

The above equations were solved in Matlab version 2022b using Euler method with a timestep of 0.002 ms and were also confirmed using Python. Both versions of the code are available from the authors upon request.

Experimental design and statistical analysis

Experiments were performed on tissue derived from neonatal animals of both sexes. Each set of experiments was performed on at least four different slices obtained from at least three different animals. Unless specified differently, n represents the number of analyzed cells; N represents the number of slices. Power analysis was conducted using G*Power 3.1.9.7. The effect size was calculated as the difference of means divided by the standard deviation of the control group. The α error probability was set to <0.05 with a resulting minimum power of 0.8.

Statistical analysis was performed by the OriginPro software (OriginPro 2021). Datasets were first tested for normal distribution using the Shapiro–Wilk test. Normally distributed datasets were statistically analyzed by one-way ANOVA followed by a post hoc Bonferroni for unpaired datasets and a paired Student’s t test for paired datasets. Unpaired data which was not normally distributed was analyzed with a Mann–Whitney (MWU) test. Data are illustrated in box-and-whisker plots or scatter (x, y) plots unless otherwise specified. Box-and-whisker plots indicate the median (line), mean (square), quantile range (25/75, box), and SD (whiskers); p values are indicated as follows: *p < 0.05; **0.05 > p > 0.01; and ***p < 0.001.