Edited by: Andrea Nistri, Scuola Internazionale Superiore di Studi Avanzati, Italy
Reviewed by: Andrew Constanti, University College London, UK; Giorgio Rispoli, University of Ferrara, Italy
*Correspondence: Shiwei Huang
†Present Address: Shiwei Huang, Australian National University Medical School, The Australian National University, Canberra, ACT, Australia
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In their seminal works on squid giant axons, Hodgkin, and Huxley approximated the membrane leak current as Ohmic, i.e., linear, since in their preparation, sub-threshold current rectification due to the influence of ionic concentration is negligible. Most studies on mammalian neurons have made the same, largely untested, assumption. Here we show that the membrane time constant and input resistance of mammalian neurons (when other major voltage-sensitive and ligand-gated ionic currents are discounted) varies non-linearly with membrane voltage, following the prediction of a Goldman-Hodgkin-Katz-based passive membrane model. The model predicts that under such conditions, the time constant/input resistance-voltage relationship will linearize if the concentration differences across the cell membrane are reduced. These properties were observed in patch-clamp recordings of cerebellar Purkinje neurons (in the presence of pharmacological blockers of other background ionic currents) and were more prominent in the sub-threshold region of the membrane potential. Model simulations showed that the non-linear leak affects voltage-clamp recordings and reduces temporal summation of excitatory synaptic input. Together, our results demonstrate the importance of trans-membrane ionic concentration in defining the functional properties of the passive membrane in mammalian neurons as well as other excitable cells.
The non-voltage-gated component of excitable cell membranes, usually called the passive membrane, plays an important role in defining electrical properties of neurons. Passive membrane properties are controlled by the behavior of leak currents. Most leak channels in neurons are voltage-independent, 2-pore channels: the K+ permeable TASK (González et al.,
Leak channels have, by definition, a voltage-independent conductance, but leak currents do show a dependence on membrane potential, as they are driven by both electrical potentials of permeating ions and ionic concentration gradients. When a concentration gradient is taken into account, the associated electrical current rectifies with the membrane potential. The degree of rectification is dependent on the concentration gradient: the greater the concentration gradient, the greater the voltage-dependent current rectification. This phenomenon is well described by the Goldman-Hodgkin-Katz (GHK) current equation (Goldman,
But is Ohm's law truly a reasonable assumption of passive membrane currents? In support of a linear leak-current model, Roth and Häusser (
The GHK current equation has been shown to better represent voltage-gated currents than an Ohmic model. For example, voltage-gated Ca2+ current is much better modeled using the GHK current equation because of a 10,000-fold Ca2+ concentration gradient and a divalent charge (Hille,
In this study, we assess the importance of a GHK-based leak current in the passive membrane properties of single- and multi-compartmental neuronal neuron models. We show that passive membrane properties [membrane time constant (tau) and input resistance (Rn)] modeled using Ohm's Law do not change with membrane potential; however, in a GHK-based leak model with identical membrane parameters, tau and Rn vary nonlinearly with both membrane potential and ionic concentration. To validate the GHK-based leak model, we investigated passive membrane properties of cerebellar Purkinje neurons in current clamp mode. We found that the relationship between tau/Rn, membrane potential and ionic concentration, were consistent with our model predictions. Through modeling, we further show that nonlinear leak current can define the kinetics of voltage-gated ion channels and the amplitude synaptic summation. Combined, our computational and experimental evidence demonstrate the importance of non-linear leak currents in neuronal excitability.
Concentration-dependent passive membrane models are infrequently used, in part, because absolute permeability values of permeating ions (Figure
All models were constructed and implemented using Python (version 2.7.5) and NEURON (version 7.4) (Carnevale and Hines,
All ion channel models were adopted from published studies. Specifically, kinetics of Kv4 and hyperpolarization-activated, non-specific cationic current were adapted from Akemann and Knöpfel (
An Ohmic, non-specific leak current was used to describe the linear voltage-current relationship, while the combined K+ and Cl− GHK currents were used to describe the non-linear voltage-current relationship. K+ and Cl− absolute permeability values were calculated from the membrane conductance, the K+/Cl− permeability ratio, and the Erest using the GHK voltage equation (Equations 1, 3, and 4).
Mice (strain C57BL/6J 6w, Charles River) of either gender, age P17-22 and 2–4 months, were anesthetized and decapitated in accordance with the Science Council of Japan Guidelines for Proper Conduct of Animal Experiments, and with approval from the OIST Animal Resources Section.
Cerebellar sagittal slices (300 μm) were obtained at 34°C (Huang and Uusisaari,
In the recording chamber, slices were perfused with oxygenated ACSF at 2 ml/min, 34°C. Neurons were visualized using infrared differential interference contrast video microscopy (Olympus BX51WI microscope) with a 40x water-immersion objective lens.
In whole-cell current clamp recordings, borosilicate glass electrodes of 5–7 MΩ were filled with an internal solution containing (in mM): 140 potassium gluconate, 10 KCl, 10 HEPES, 10 EGTA, 4 MgATP, 0.4 NaGTP, 10 phosphocreatine, 8 biocytin (pH adjusted to 7.3 with KOH). Purkinje neurons were identified by their distinctive morphology and position within the cerebellar cortex.
A resistance seal of ≥4 GΩ was required before entering whole-cell, patch-clamp configuration. Signals were amplified and low-pass filtered at 5 kHz using a Cornerstone BVC-700 A amplifier (Dagan), and were recorded at 40 kHz using a custom interface written in Labview acquisition software (National Instruments). After obtaining whole-cell configurations, Purkinje neurons were hyperpolarized to −65 mV.
Cerebellar Purkinje neurons were perfused with TTX (1 μM), ZD7288 (30–50 μM), and synaptic blockers (DNQX, 10 μM and GABAzine, 10 μM) to block voltage-gated Na+, hyperpolarization-activated cationic current (Ih), and AMPA and GABAa receptor currents, respectively. High intracellular EGTA (10 mM) was used to minimize Ca2+-dependent K+ currents, which contribute to spontaneous firing of Purkinje neurons (Edgerton and Reinhart,
We observed that the Erest of cerebellar Purkinje neurons after pharmacological treatment could be permanently shifted by repeated current injections. Therefore, after each current pulse step, data were discarded if a deviation of more than 2 mV from the Erest occurred. The voltage trace of each current pulse step is the average of at least four replicates.
Electrophysiological data were analyzed using Python 2.7, Pylab 2.7, and R 3.01. Data are given as means ± standard error and/or 95% confidence interval, and statistical significance was tested using Student's
Passive membrane properties of an isopotential cell are conventionally measured by applying a small current pulse to produce a membrane voltage deflection from Erest. From the transient proportion of the voltage deflection, tau can be estimated using an exponential decay (see Methods); from the steady-state proportion, Rn can be calculated. We created two leak-current models, one described by a linear circuit and the other by a K+ and Cl− GHK current model, to compare the difference in passive membrane properties. These models are used to demonstrate non-linear properties of the GHK current model, they are not intended to fit the experimental data exactly.
In the Ohmic model, membrane potential changes proportionally with injected current. As a result, injecting a positive and a negative current pulse of the same magnitude (± pulses) produces two identical voltage curves, one of which is inverted. (Figure
Why does voltage deflection decrease with depolarization and increase with hyperpolarization from Erest? In the GHK leak model, Erest lies between the reversal potentials of IK and ICl. Therefore, IK causes hyperpolarization while ICl causes depolarization. The relative proportion of the two leak currents is determined by the ratio of the degree of current rectification, determined by the concentration gradient, and their absolute permeability ratio. While in the GHK model the permeability ratio of K+ and Cl− is close to 1, the K+ gradient is significantly higher than that of Cl− (60-fold vs. 13-fold, respectively). As a result, K+ rectification is stronger than that of Cl− (Figure
The crucial question addressed by this study is whether the non-linear leak model is a better fit for the passive membrane responses to electrophysiological stimulation. Using patch clamp electrophysiology in the whole-cell configuration and the current clamp protocol as described for simulations, the passive properties of cerebellar Purkinje neurons were measured. The magnitude of voltage deflections from the current injection protocol was consistent with the GHK model simulation (Figure
Central to the GHK model prediction is that tau and Rn should be influenced by ionic concentration change. In the GHK model, changing a single ionic concentration can result in two different tau/Rn-voltage relationships depending on which property of the model is conserved. Using intracellular Cl− concentration as an example, at fixed Pratio, the tau/Rn-voltage curve shifts right, toward the Cl− reversal potential (Figure
In Purkinje neurons patched with 30 mM Cl− -filled pipettes, smaller voltage deflections at hyperpolarizing current-clamp were observed (Figure
The effect of concentration on the tau-voltage relationship was less conclusive. The slope of the higher Cl− data was steeper, contrary to our prediction (β10mM = −0.5, β30mM = −1.2,
The resting membrane potentials measured under the 10 and 30 mM Cl− settings were −63 ± 3 and −67 ± 2 mV, respectively, while we expected that the latter would have a more depolarized potential. The underlying mechanism of this unclear, however we suspect that it is a homeostatic response trying to maintain a set point resting potential.
Patch clamp recordings of passive properties of mammalian neurons are routinely conducted for neuronal excitability investigations. Why hasn't passive membrane non-linearity been observed? First, as demonstrated in our simulation and whole-cell recordings, passive non-linearity is only apparent when voltage deflections from two different holding potentials are compared. At the same holding potential, the difference between positive and negative current-induced voltage traces is minimal. Second, dendritic branching increases local input resistance (Rinzel and Rall,
To illustrate the scale of errors caused by assuming an Ohmic leak instead of the GHK leak, we compared voltage-clamp simulations of the mammalian voltage-gated potassium current Kv4 under both settings (Figure
The results thus far have dealt with non-linear voltage responses due to DC current injection; however,
As a guide to the significance of the difference in temporal summation of EPSPs, we re-simulated this in the presence of a hyperpolarization-dependent, non-specific cationic current (Ih) that had the same maximal conductance as that of the leak currents (see Methods). Ih was chosen because it is known to also dampen temporal summation (Magee,
Computational and experimental evidence presented here demonstrates that pharmacologically isolated passive membrane current is better modeled using the GHK than the Ohmic equation. The results suggest that non-linear leak currents are an important parameter for defining membrane electrical properties and that ionic concentration should be considered in neuronal modeling in general.
Our GHK model assumes that leak currents are generated solely from ion permeation through membrane pores, but physiological passive membranes are far from simplistic. As described earlier, leak currents are generated by specific channels, some of which may have more complex properties than those captured by the GHK model (Jentsch et al.,
The primary difficulty with the GHK model is to determine the absolute permeability value for permeating ions. In our model, two out of the three most abundant physiological ions were used in the GHK model. This is so that an exact pair of K+ and Cl− absolute permeability values could be obtained for subsequent model simulations. Ignoring the Na+ leak current may have caused an over-estimation of the effects of K+ and Cl− currents, and have rendered the Cl− current excitatory because the resting potential lies between the Nernst potential of the two ions. But the ability to simulate the effect of ionic concentration on Erest outweighed the need to have a more complete, non-linear leak-current model. Furthermore, the consistency between our experimental and simulated data suggests that a non-linear sodium leak current would not have altered the conclusion of this study.
We used the chord conductance to calculate permeability values; however, slope conductance is equally valid for the same calculation (Thompson,
It is important to note that a cocktail of drugs was perfused to isolate the passive membrane in the experimental investigation. How these drugs affect leak channel expression during the recordings, which lasted over 40 min per cell, is unknown. Therefore, values of passive membrane properties measured experimentally may not be inherently meaningful, but relative changes in them are. This also extends to Erest, which slightly hyperpolarized with increasing intracellular Cl− concentration, although we predicted the opposite. We suspect that this is due to cellular compensatory mechanisms to prevent depolarization from Erest.
The main findings of this study contradict the observation of membrane linearity in cerebellar Purkinje neurons reported by Roth and Häusser (
Clay (
In somato-gastric neurons, White and Hooper (
The non-linearity of passive membrane differs substantially between types of preparations, for example, it is smaller in squid axons relative to mammalian neurons. This non-linearity becomes more significant as the holding potential deviates from the resting potential of the patch-clamped cell or when the K+/Cl− permeability ratio deviates from 1. Because electrophysiology recordings of neurons are routinely conducted under a range of voltages and the physiological K+/Cl− permeability ratio is commonly greater than 1, the use of Ohmic leak in passive membrane modeling is almost always inaccurate.
The non-linear relationship between Rn/tau and membrane potential, affect many routine electrophysiological measurements. If observed, they may be interpreted as evidence for the activation of voltage-gated channels by depolarization as the non-linearity has a similar potential-dependent effect on Rn and tau (Figure
We have demonstrated that under conditions where major resting ionic conductances are blocked, passive membrane properties can change with membrane potential due to their non-linear dependence on ionic concentration. This effect is usually ignored, which leads to incorrect characterization of passive and active properties of the cell. Our results suggest that, prior to investigating active properties, it is necessary to determine passive membrane properties of the cell type of interest by fully examining the effect of ion concentration via a GHK-based current model of leak currents. Our model takes into account ionic concentration at each integration time step, which can be used to model concentration-dependent neuronal computation, as well as ion homeostasis in other excitable cells.
SWH constructed models, acquired data, and analyzed both simulation and experimental data. SHH, constructed models and analyzed simulation data. SWH, SHH and ED drafted and revised the article.
This work was supported by funding from the Okinawa Institute of Science and Technology Graduate University.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
We are grateful to Bernd Kuhn and Alexander Mikheyev for supporting this work. We also thank Marylka Y. Uusisaari, Iain Hepburn, Mario Negrello, and Minh-Son To for helpful discussions. Lastly we thank Alexander Mikheyev and Steven D. Aird for editing the manuscript.