Animals

All animal procedures were approved and performed in accordance with the ethical guidelines set by Virginia Tech Institutional Animal Care and Use Committee (IACUC). Mice were maintained in groups of five in a specific pathogen-free barrier facility in 12 h light/dark cycles. Male and female C.B.17 scid mice aged 6–9 weeks were used for intracranial tumor implantation. Female athymic nude mice aged 6–8 weeks were used for flank injections, maintenance and propagation of the GBM xenografts. Astrocyte-specific β1-integrin knockout FVB-N mice were generated as described previously31.

Patient-derived xenograft tumors

Patient-derived primary glioma tissue was maintained by serial passage in nude mice flanks as described previously4. Briefly, tumors were harvested after 14–18 days postinjection and glioma cells were mechanically and enzymatically dissociated. Cells were passed through a 40 µm filter and maintained as “gliospheres” in Dulbeccoʼs modified Eagle medium/nutrient mixture F-12 (DMEM/F-12), supplemented with 10 mg/ml fibroblast growth factor (FGF), 10 mg/ml epidermal growth factor (EGF), 260 mM l-glutamine, 2% B-27 Supplement without vitamin A (Invitrogen), 250 µM/ml amphotericin, and 50 mg/ml gentamycin (Fisher), and incubated in 10% CO 2 at 37 °C. Medium was changed daily for 2 days, then weekly. Gliospheres were maintained in vitro for 5–7 days before intracranial injection. For intracranial injections, cells were dissociated with Accutase (Sigma-Aldrich), counted, and then diluted in sterile phosphate-buffered saline (PBS) to get desired cells/unit volume.

Intracranial glioma injections

Human glioblastoma GBM14 and GBM22 tumors, previously established from human biopsies, were implanted into 6–9 weeks old immunodeficient C.B.17 scid mice as previously described4. Briefly, mice were anesthetized with 2–5% isoflurane and fixed to a stereotaxic apparatus (Leica Angleone stereotaxic model 39464710) followed by a midline scalp incision and a 0.5 mm burr hole 1.0–2.0 mm lateral and 0.5–1.0 mm anterior to bregma. Patient-derived xenograft tumor cells (2.0×105 cells in 2 μl of PBS, GBM22 and GBM14) were injected at a depth of 2.0–2.5 mm. Control mice were injected with sterile PBS. Body weight of animals was measured on alternate days and experiments were conducted between 12 and 20 days post-glioma implantation. To specifically target glioma cells to grow near dorsal hippocampus, we injected glioma intracranially at 1.25 mm lateral, 1.46 mm posterior and 1.4 mm ventral from bregma. A 10 µl syringe (World Precision Instruments #SGE010RNS) was used to infuse glioma cells at 11 nl/s rate.

GM6001 injections, cryofixation and in situ zymography

GM6001 or illomastat (ApexBio Cat# A4050), a broad-spectrum MMP inhibitor (Ki: 0.4 nM for MMP-1, 27 nM for MMP-3, 0.5 nM for MMP-2, 0.1 nM for MMP-8 and 0.2 nM for MMP-9), was administered intraperitoneally as a suspension of 100 mg/kg/day in 4% carboxymethylcellulose (CMC) in saline to a total volume of 200 µl57. Mice in the control group received 4% CMC in saline. Injection of GM6001 or CMC (control) was initiated 2 days after glioma implantation to let mice recover and allow glioma to settle in brain, followed by additional injections every 24 h for total 13 days. GM6001/CMC-treated mice were sacrificed to harvest brains for in situ zymography on 14 days after glioma implantation. Mice were deeply anesthetized with intraperitoneal injection with a cocktail of ketamine (100 mg/kg of body weight) and xylazine (10 mg/kg of body weight) followed by transcardial perfusion with ice-cold PBS for 2 min and a quick dissection of brain in ice-cold N-methyl-D-glucamine (NMDG)-based cutting solution. To achieve rapid freezing of tissue without forming ice crystals, we trimmed out one-fourth brain from caudal side to decrease the tissue size. Brain tissues were immersed in tissue freezing medium (TFM5, Electron microscopy Sciences) in a mold (4566 Tissue tek Cryomold) followed by placing the mold on 2-propanol-filled beaker surrounded by dry ice. The total tissue freezing time from brain mounting in TFM to its solidification did not exceed one and half minute, which ensured the preservation of enzymatic activity in the tissue. Sections were stored at −80 °C and moved to −20 °C for overnight before cryo-sectioning. Coronal cryo-sections (15 µm) were cut using cryostat (Leica CM 1850 UV) and mounted on glass slides (Superfrost plus microslides, Cat# 48311-703). For electrophysiology experiments, GM6001-treated mice were used (one mouse per day) from 14th day post-glioma implantation and the remaining mice were routinely injected with GM6001 until used.

In situ gelatinolytic activity of MMP-2/9 was performed on above-mentioned frozen sections using a commercial kit (EnzChek Gelatinase Assay kit, Molecular Probes, Cat# E-12055). In brief, sections were thawed on ice for 30 min and followed by 3–5 washes with PBS to remove TFM. Sections were incubated with DQ gelatin fluorescein conjugate (20 µg/ml), at 37 °C for 1 h in a dark and humid chamber to optimize gelatinolytic activity. Cleavage of gelatin by active tissue MMPs exposes the fluorescein molecules (495/515 nm). Substrate concentration was optimized for the assay and 20 µg/ml was used for all the experiments. 1,10-phenanthroline (1 mM in dimethyl sulfoxide (DMSO), Sigma-Aldrich) was used as a nonspecific inhibitor of MMP (Supplementary Fig. 2c). To remove excess fluorogenic substrate, sections were washed 3–5 times and fixed in 4% paraformaldehyde (PFA) for 5 min. Further, sections were incubated with primary antibody specific for astrocytes (Chicken GFAP, Cat# ab4674, Abcam, 1:1000) and neurons (Rabbit NeuN, Cat# ABN78, Millipore, 1:500; Mouse PV, Cat# PV 235, Swant, 1:1000), and biotinylated WFA (Vector laboratories, Cat# B-1355, 1:500) for 1 h at room temperature. Appropriate fluorophore-conjugated secondary antibodies and Alexa Fluor® 555-conjugated streptavidin (Vector laboratories, Cat# S32355, 1:500) were used to detect the primary antibodies and WFA, respectively. Sections were examined and images were taken using Nikon A1 confocal microscope within 48 h of staining procedure. Images were acquired at various magnifications and data were quantified using Nikon Elements analysis program associated with Nikon A1 confocal microscope.

Acute slice electrophysiology

After cervical dislocation, mice were quickly decapitated and brains were dissected out and kept in an ice-cold ACSF (135 mM NMDG, 1.5 mM KCl, 1.5 mM KH 2 PO 4 , 23 mM choline bicarbonate, 25 mM d-glucose, 0.5 mM CaCl 2 , 3.5 mM MgSO 4 ; pH 7.35, 310 ± 5 mOsm) (Sigma-Aldrich) saturated with carbogen (95% O 2 + 5% CO 2 ). Coronal slices (300 μm) were prepared using Leica VT 1000P tissue slicer and slices were allowed to recover for 1 h in ACSF (125 mM NaCl, 3 mM KCl, 1.25 mM NaH 2 PO 4 , 25 mM NaHCO 3 , 2 mM CaCl 2 , 1.3 mM MgSO 4 , 25 mM glucose, pH 7.35, 310 ± 5 mOsm) at 32 °C. Afterwards slices were kept at room temperature until used for recordings. Individual slices were placed in a recording chamber continuously superfused with ACSF at a flow rate of 2 ml/min. Glioma cells grow rapidly and expand to primary motor and somatosensory cortical areas after 12–14 days postinjection. Tumor mass in these cortical areas was visually identified by their unique appearance as described previously6. The pyramidal cells and FSNs in cortical layers 3–5 were identified under an upright microscope (Leica DMLFSA) with ×40 water immersion lens and infrared illumination. Whole-cell voltage-clamp and current-clamp recordings were achieved using an Axopatch 200B amplifier (Molecular Devices). Patch pipettes of 3–5 MΩ open-tip resistance were created from standard borosilicate capillaries (WPI, 4IN THINWALL Gl 1.5OD/1.12ID) using Narishige PP-83 and HEKA PIP 6 vertical pipette pullers. Patch pipettes were filled with an intracellular solution of 134 mM potassium gluconate, 1 mM KCl, 10 mM 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES), 2 mM adenosine 5′-triphosphate magnesium salt (Mg-ATP), 0.2 mM guanosine 5′-triphosphate sodium salt (Na-GTP) and 0.5 mM ethylene glycol tetraacetic acid (EGTA) (pH 7.4, 290–295 mOsm). Unless otherwise stated, we added 20 µl Lucifer yellow (Sigma Cat# L0259, 20 mg/ml stock solution in deionized water) in intracellular buffer just before the recording for post-experiment identification of the cells. Patch pipettes were visually guided using MM-225 micromanipulator (Sutter Instrument, Navato, CA). A potential damage to the PNNs was minimized during the patching procedure by applying minimum positive pressure while approaching cells. Whole-cell recordings were made once >5–10 GΩ seal was achieved. For voltage-clamp recordings, the membrane potential was clamped at −70 mV. The membrane capacitance (Cm) and series resistance were not compensated unless otherwise stated. For few experiments, we superfused ACSF containing 50 µM D-2-amino-5-phosphonovalerate (D-AP-5) and 20 µM 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX) to block glutamatergic neurotransmission and confirmed the blockage by observing disappearance of spontaneous EPSCs. Data were acquired using Clampex 10.4 software and Axon Digidata 1550A interface (Molecular Devices), filtered at 5 kHz, digitized at 10–20 kHz and analyzed using Clampfit 10.6 software (Molecular Devices). Unless otherwise stated, during all the recordings carbogen-bubbled ACSF was continuously superfused (2 ml/min) and all recordings were made at 32–33 °C using an inline feedback heating system (Cat# TC 324B, Warner Instruments).

Measurement of intrinsic properties: The resting membrane potential (Vm) was measured by setting I = 0 mode immediately after achieving whole cell. Action potential (AP) threshold current was calculated by injecting 2−200 pA current pulses (10 ms duration) with 2 pA increment in each step. Minimum current required to generate first AP where onwards each subsequent higher current leads to AP generation was noted as threshold current. Most of the cells fired APs within this current range; however, cells that did not fire within this range were lumped in single group of 200 pA firing threshold. To calculate input resistance (R in ), we injected 15 hyperpolarization current steps (−100 pA each for 1000 ms) and recorded the steady-state membrane voltage deflection (ΔV). The R in was measured as a ratio (ΔV/I) of steady-state change in the membrane voltage (ΔV) and the corresponding injected current (I). Membrane capacitance (Cm) was also calculated from the same 15 voltage responses of hyperpolarization current steps. All 15 voltage traces were averaged and the discharging phase of voltage was fit with a simple exponential decay and the slowest time constant was used for calculating the Cm58. The excitability of neurons was assessed using the input−output curve obtained by applying increasing step currents of different magnitudes (−100 to 180 pA, 15 steps with 20 pA increment each step, step duration 1000 ms) and counting number of APs using Clampfit 10.6 program. We observed that the amplitudes of APs shortened as the firing frequency increased; therefore, we set a minimum 15 mV deflection from the steady-state response as a qualifying criterion for a spike to be identified as an AP. To check the saturating/maximum firing frequency (AP frequency) supported by FSNs in our experimental conditions, we kept increasing the magnitude of injected current until the firing frequency did not increase any further with the higher currents. The observed saturating/maximum firing frequency was found to be achieved by injecting 400−500 pA; therefore, we capped the highest injected current to 500 pA.

Identification of FSNs with PNNs in peritumoral cortex

As evident by our immunohistochemical staining results, disintegration of PNNs was reliably found within ~0.5 mm of the tumor border. We randomly patched neurons within this range and later confirmed the disintegrated status of PNN by WFA staining and measuring the distance of the patched cell from the growing border of glioma. (Supplementary Fig. 3a, b). We added Lucifer yellow dye in the internal patch solution for postrecording identification to determine the structural integrity of PNNs around the recorded cell and the distance from the tumor border. On completion of recordings, patch pipette was carefully retracted to minimize the membrane damage and slices were fixed in 4% PFA overnight and later stained for WFA and DAPI to check the presence of PNNs around the patched neurons (Supplementary Fig. 3). The slices were incubated with biotinylated WFA (1:200) for 1.5 h followed by three rinses with PBS, and subsequently incubated for 1 h with Alexa Fluor® 555-conjugated streptavidin (Vector laboratories, Cat# S32355, 1:200). Then, the slices were rinsed thrice with PBS, stained with DAPI for 3 min (Cat# D1306, Life Technologies, 1:1000 diluted from 2 mg/ml stock) and mounted on the glass slides using mounting medium (SlowFade Gold antifade reagent, Cat# S36936, Invitrogen) and cover glasses (20 × 50-1, Cat# 12-548-5E, Fisher). The distance of patched cell from the tumor border was also measured in the images and only those cells falling within 500 µm of tumor border were used for analysis. Glioma mass was identified as densely packed amorphous mass of DAPI-positive cells (Supplementary Fig. 3).

Acute slice PNN degradation and identification of PNN+ cells

Chondroitinase ABC (ChABC) from Proteus vulgaris (Cat# C3667, Sigma-Aldrich) was reconstituted in a 0.01% bovine serum albumin aqueous solution according to the manufacturer’s instruction to make 1 U/40 µl stock solution. Aliquots of 2 U were prepared and stored at −20 °C until used. We performed ECM/PNN degradation by two methods depending on the experimental requirement.

For one set of experiments, we pretreated recovered slices with ChABC and thereafter performed recordings on the ChABC-treated slices. In brief, on completion of post-slicing recovery, 2–3 cortical half slices were incubated in 3 ml of 0.5 U ChABC per ml of ACSF in an incubation chamber continuously supplied with carbogen at 33 °C for 45 min. Then, the slices were rinsed with and incubated in ACSF until used for electrophysiological recordings. These parameters of PNNs digestion by ChABC (enzyme concentration—0.5 U/ml, incubation time—45 min, incubation temperature—33 °C) reliably degraded PNNs but spared traces around the FSNs thereby allowing the identification of PNN-positive cells (Supplementary Fig. 6a, b). Similarly, previously separated contralateral halves of the ChABC-treated slices were incubated in 3 ml of ACSF without ChABC in the incubation chamber. Then, both the ChABC-treated and nontreated slices were kept in ACSF together until used for the recordings.

For the second set of experiments, we performed real-time digestion of PNNs in acute slices simultaneously with the electrophysiological recordings. We randomly patched FSNs and recorded their intrinsic properties and subsequently monitored their baseline membrane voltage (I = 0) for 5–10 min. The cells with resting membrane potential unstable and greater than −60 mV were discontinued. After baseline membrane potential recording, we superfused 1 U/ml ChABC solution for 50 min while strictly maintaining the bath temperature at 32–33 °C as it appeared to be critical for PNN degradation. This protocol also reliably degraded PNNs but left identifiable traces around the FSNs to confirm the presence of PNNs around patched cells prior to the digestion (Supplementary Fig. 7). On completion of recordings, slices were fixed in 4% PFA overnight followed by WFA staining as described above. The FSNs expressing PNNs in the ChABC-treated slices were identified by a low WFA intensity and disintegrated PNN around the cell body of LY-filled cell (Supplementary Fig. 7a, ChABC) compared to their non-ChABC-treated counterpart exhibiting intense WFA staining around the cell body and proximal dendrites of LY-filled cell (Supplementary Fig. 7a, ACSF). The recorded excitatory neurons appeared as the LY-filled cells with intact PNNs and high WFA intensity background in nontreated slices (Supplementary Fig. 7b, ACSF) and as the LY-filled cells with nearby low WFA intensity and with only identifiable traces of PNNs in ChABC-treated slice (Supplementary Fig. 7b, ChABC).

Immunohistochemistry

Animals of different experimental groups were injected with a mixture of ketamine and xylazine (100 mg/kg and 10 mg/kg, respectively) and subsequently trancardialy perfused with PBS followed by 4% PFA. The brains were dissected out and stored overnight in 4% PFA at 4 °C. Next, the brains were transferred to PBS for 48 h and 50-µm-thick floating sections were cut using vibratome (5100MZ, Campden instruments). The sections were either used for IHC immediately or stored at −20 °C in a custom-made cryoprservative medium (10% (v/v) 0.2 mM phosphate buffer, 30% (v/v) glycerol, 30% (v/v) ethylenglycol in deionized water, pH 7.2–7.4) for later uses. To minimize procedure-related variability, we preformed staining in a large batch of duplicate sections from 5 to 7 mice of each treatment group. The sections were retrieved from −20 °C storage, rinsed with PBS, and permeabilized and blocked by incubating them in blocking buffer (0.5% Triton X-100 and 10% goat serum in PBS) for 2 h at RT. Then the sections were incubated overnight at 4 °C with appropriate primary antibodies and biotinylated WFA (Cat# B-1355, Vector Laboratories) in diluted blocking buffer (1:3 of blocking buffer and PBS). On the next day, the sections were incubated with appropriate secondary antibodies and Alexa Fluor® 555-conjugated streptavidin (Cat# S32355, ThermoFisher Scientific, 1:500) in diluted blocking buffer for 2 h at RT in dark. Further, the sections were rinsed with PBS and occasionally stained with DAPI (10 µg/ml in PBS for 3 min) followed by two rinses with PBS. The sections were mounted on the glass slides (Fisherfinest 25 × 25 × 1, Cat# 12-544-2) covered with cover glass and the edges of slides were sealed with nail polish. The antibodies used were rabbit NeuN (Cat# ABN78, Millipore, 1:500), mouse PV (Cat# PV 235, Swant, 1:1000), and chicken GFAP (Cat# ab4674, Abcam, 1:1000). Images were acquired at different magnifications using Nikon A1 confocal microscope and quantification was performed by inbuilt NIS-Elements AR analysis program. High-magnification images in all three panels of Figs. 2a and 4e are snapshots of 3D volume images of 30–40-µm-thick z-stacks. Figure 5f and Supplementary Figs. 1b, 3b (panel images), 3c, and 2a (top right) are snapshots of 3D volume image of ~10-µm-thick z-stacks.

Cell counting

NeuN+, PV+ and PNN+ cells were counted from 100 µm × 100 µm square region in ×10 magnification single plane images. In brief, a grid of 100 µm × 100 µm boxes was drawn and the boxes containing both glioma mass and brain parenchyma were defined as the tumor border. The number of cells in each subsequent square away from the tumor border was counted and arranged with respect to their distance from the tumor border and their cortical layer location. Each cortical layer has characteristic density of neurons; therefore, spatial localization of glioma in a specific cortical layer(s) may affect the cell density differently. To overcome these variations we analyzed cortical sections in which glioma was present in all the layers and counted cells in 100 µm × 100 µm boxes in each layer in all the comparative test groups such as peritumoral, contralateral, and sham. We restricted the analytical comparisons to layers 3–5 due to highest density of PV/WFA-expressing neurons. Average number of cells per 0.01 mm2 area (corresponding to 100 µm × 100 µm box) for each spatial bin in a brain slice was calculated by averaging the numbers of cells in 3–5 boxes. Similarly, average numbers of cells per 0.01 mm2 area for each spatial bin were calculated from 10 to 15 brain slices from minimum five mice (minimum two brain slices from each mouse) and averaged to obtain mean ± SEM and the numbers of analyzed brain slices were considered as n for statistical analysis. To compare the relative cell density of excitatory and inhibitory neurons in the PTC and contralateral cortex, we normalized them with their respective cell densities in the cortex of sham-treated mice.

Quantification of IHC and ISZ images

To evaluate the glioma-induced PNN/interstitial matrix degradation, we quantified the IHC data of WFA intensity in stratum pyramidale and stratum radiatum of hippocampal CA2 region and PTC. In brief, we averaged the mean fluorescence intensities of randomly drawn 4–6 regions of interest (ROIs) of 15 µm × 15 µm area in each slice. In the PTC areas, we placed these ROIs onto individual PNNs to overcome the potential discrepancy due to different density of PNNs and neurons in similar dimension areas. Subsequently, the mean intensities from 8 to 12 brain slices from minimum five mice were averaged to obtain mean ± SEM and the numbers of analyzed brain slices were considered as n for statistical analysis. To quantify gelatinase activity (fluorescein intensity) and WFA intensity in the ISZ experiments, we drew concentric circles in the PTC at every 200 µm starting from the glioma border to spatially bin the distance-dependent effect of glioma. Next, we drew random 3–5 ROIs of 20 µm × 20 µm dimension in each 200 µm bin area and averaged it to obtain mean intensity which was considered as n. Mean fluorescence intensities of each area from 2 to 3 slices per animal (5–7 mice in each treatment group) were used to calculate mean ± SEM for graphical representation and statistical analysis.

Line profile intensity and PNN integrity analysis

To quantify the structural integrity of PNNs, we acquired high-magnification (40 × 5) single plane images of individual PNNs in different experimental groups. A polyline was drawn along the entire periphery of cell (stained with NeuN/PV/gelatinase) with PNNs (Fig. 2c). WFA fluorescence intensity of this line shows high intensity peaks and low intensity drops on regular intervals representing areas covered by PNN-CSPGs and holes in the PNNs, respectively. We set a threshold of ~50% of highest fluorescence intensity (red two-headed arrow in Fig. 2d) and counted the number of peaks above the threshold for 10–20 PNNs in each experimental group using Clampfit 10.6 program. A line profile intensity graph showing almost flat line with only few low intensity peaks indicates disintegrated PNN by glioma as exemplified in Fig. 2d. This scheme was adopted to generate data in Figs. 2e and 5h. For representing different stages of PNN and cell disintegration in the PTC in Supplementary Fig. 1d, we drew a straight line across the cell in different areas of the PTC, and the fluorescence intensities of WFA, PV, NeuN and DAPI along this line were plotted. Different patterns of PV and NeuN intensity peaks flanked by WFA peaks indicate whether a cell is present inside the PNN and the status of PNN.

HH modeling of PNN

We developed a first-order HH differential equation model using Matlab. This model was first described in the seminal work by Hodgkin and Huxley, and modified by Abbot and Kepler59,60. We used the same parameter values as Abbot and Kepler59; resting membrane potential, V R (−65 mV); sodium reversal potential, E Na (50 mV); potassium reversal potential, E k (−77 mV); leak potential, E l (−54.4 mV); specific membrane capacitance, Cm (1 µF/cm2); maximum potassium conductance, G k (36e−3/ohm/cm2); maximum sodium conductance, G Na (120e−3/ohm/cm2); leak conductance, G l (0.3e−3/ohm/cm2); external applied current density, J ext (0.2e−4 A/cm2); and temperature (18.5 °C). This similar methodology, both model interactions and parameter values, has been used in recent studies61,62,63. A step size of 1 µs was used for the numerical method.

Briefly, Hodgkin and Huxley modeled the single-cell action potential using nonlinear voltage-dependent channel resistors in series with the respective ionic Nernst potential to charge a parallel membrane capacitor. This circuit model results in the following differential equation:

$${\mathrm{d}} V_{\mathrm{m}}\left( t \right) / {\mathrm{d}}t = 1 / {\mathrm{Cm}} \left( J_{\mathrm{ext}} - J_{\mathrm{Na}} - J_{\mathrm{k}} - J_{\mathrm{leak}} \right),$$

$${\mathrm d} V_{\mathrm m} \left( t \right) / {\mathrm d} t = 1/{\mathrm {Cm}} \left( {J_{{\mathrm{ext}}} - g_{{\mathrm {Na}}}\left( {V_{\mathrm m} - E_{{\mathrm {Na}}}} \right) - g_{\mathrm k}\left( {V_{\mathrm m} - E_{\mathrm k}} \right) - J_{{\mathrm {leak}}}} \right).$$

The sodium and potassium current through their respective variable resistor are governed by three gating variables: n for potassium, m for sodium deactivation, and h for sodium inactivation.

$$g_{\mathrm{k}} \left( t \right) = G_{\mathrm{k}} m^{4} ( V_{\mathrm{m}} )\, {\mathrm{and}}$$

$$g_{{\mathrm {Na}}}\left( t \right) = G_{{\mathrm {Na}}} n ^{3}\left( {V_{\mathrm{m}}} \right)h\left( {V_{\mathrm{m}}} \right),$$

where g k and g Na denote the time-varying potassium and sodium conductance, respectively.

The channel gating variables approach their steady-state values according to fitted equations using channel population on (α) and off (β) probabilities.

$${\mathrm{d}}n\left( t \right)/{\mathrm{d}}t = \alpha _{n}\left( {1 - n} \right) - {\mathrm{\beta }}_{n}\left( n \right),$$

$${\mathrm{d}}m\left( t \right)/{\mathrm d}t = \alpha _{m}\left( {1 - m} \right) - {\mathrm{\beta }}_{m}\left( m \right),$$

$${\mathrm d}h\left( t \right) / {\mathrm d}t = \alpha _{h}(1 - h) - \beta _{h}\left( h \right),$$

where,

$${{\alpha }}_{n} = \phi \left( 0.10 - 0.01\,{\mathrm d}V \right) {/} \left( {\mathrm{exp}}\left( 1 - 0.1\,{\mathrm d}V \right) - 1 \right),$$

$${\beta }_{n} = \phi \left( {0.125} \right)( {\mathrm{exp}} ( - {\mathrm{d}} V {/} 80 )) ,$$

$${\alpha }_{m} = \phi ( 2.5 - 0.1\,{\mathrm{d}} V ){/} ( {\mathrm exp} ( 2.5 - 0.1\,{\mathrm{d}} V) - 1 ),$$

$${\beta }_{m \,=} \,\phi \left( 4 \right){\exp}\left( - {\mathrm{d}} V{/}18 \right),$$

$${\alpha }_{h} = \phi ( {0.07} ){\mathrm{exp}} \left(-{\mathrm d}V{/}20 \right),$$

$${\beta }_{h} = \phi ( 1 ){/} ( {\mathrm{exp}} ( 3.0 - 0.1\,{\mathrm d}V ) + 1 ),$$

$$\phi = 3^{( {( {T - 6.3} ){{/}}10} )} ; {\mathrm{d}}V = ( {v} - {Vr} ) \times {1000}.$$

The initial condition for all simulations was: (Vr, n(0), m(0), h(0)) = (−65 mV, 0.3177, 0.0529, 0.5961).

Each simulation sampled from a (capacitance, external current) space. The capacitance ranged from 0.8 to 1.5 µF/cm2 with a step size of 0.01 µF/cm2; the external current ranged from 0.2−4 to 1.4−2. The firing frequency as a function of capacitance and external current was calculated by using Matlab’s findpeaks function on the voltage time series output for each simulation. Firing frequency was calculated as the number of peaks >20 mV in the 1000 ms simulation.

Statistical analysis

Data are represented as box and whisker plots unless otherwise stated in the specific figures. The central lines and plus signs within box and whisker plot represent the medians and means, respectively; the two ends of the rectangles represent first and third quartiles. The upper and lower whiskers extend to the highest and lowest values in the data set, respectively. Individual data points are represented by dots. Statistical details of experiments are mentioned in the figure legends and all the details of statistical analysis including test statistics, P values, post-hoc comparisons, and 95% confidence intervals of mean differences are summarized in Supplementary Table 1. Data are sufficiently normal distributed and variance within groups is sufficiently similar to be used for parametric tests. Experimental designs with two treatment groups were analyzed by two-tailed unpaired or paired t test. Welch’s correction was applied where variances of both the groups were statistically different. Experimental designs with more than two groups were analyzed using one-way ANOVA or two-way ANOVA followed by Tukey’s or Sidak’s post-hoc multiple comparison tests. Statistically significant difference between groups were notified in graphs using asterisk(s) (*P < 0.05, **P < 0.01, ***P < 0.001, ****P < 0.0001) and occasionally number sign (#P < 0.05, ##P < 0.01, ###P < 0.001, ####P < 0.0001). Data analysis was performed using GraphPad Prism 7.0, Microsoft Excel, and Origin 2016 (OriginLab).

Code availability

Matlab codes will be provided on reasonable request to the corresponding author.