Determining k channel activation curves from k channel currents often requires the goldman-hodgkin-katz equation

Sex-Dependent Changes in Gonadotropin-Releasing Hormone Neuron Voltage-Gated Potassium Currents in a Mouse Model of Temporal Lobe Epilepsy

Temporal lobe epilepsy (TLE) is the most common focal epilepsy in adults, and people with TLE exhibit higher rates of reproductive endocrine dysfunction. Hypothalamic gonadotropin-releasing hormone (GnRH) neurons regulate reproductive function in mammals by regulating gonadotropin secretion from the anterior pituitary. Previous research demonstrated GnRH neuron hyperexcitability in both sexes in the intrahippocampal kainic acid (IHKA) mouse model of TLE. Fast-inactivating A-type (I A) and delayed rectifier K-type (I K) K+ currents play critical roles in modulating neuronal excitability, including in GnRH neurons. Here, we tested the hypothesis that GnRH neuron hyperexcitability is associated with reduced I A and I K conductances. At 2 months after IHKA or control saline injection, when IHKA mice exhibit chronic epilepsy, we recorded GnRH neuron excitability, I A, and I K using whole-cell patch-clamp electrophysiology. GnRH neurons from both IHKA male and diestrus female GnRH-GFP mice exhibited hyperexcitability compared with controls. In IHKA males, although maximum I A current density was increased, I K recovery from inactivation was significantly slower, consistent with a hyperexcitability phenotype. In IHKA females, however, both I A and I K were unchanged. Sex differences were not observed in I A or I K properties in controls, but IHKA mice exhibited sex effects in I A properties. These results indicate that although the emergent phenotype of increased GnRH neuron excitability is similar in IHKA males and diestrus females, the underlying mechanisms are distinct. This study thus highlights sex-specific changes in voltage-gated K+ currents in GnRH neurons in a mouse model of TLE and suggesting potential sex differences in GnRH neuron ion channel properties.

Development and prenatal exposure to androgens alter potassium currents in gonadotropin-releasing hormone neurons from female mice

Pulsatile gonadotropin-releasing hormone (GnRH) release is critical for reproduction. Disruptions to GnRH secretion patterns may contribute to polycystic ovary syndrome (PCOS). Prenatally androgenized (PNA) female mice recapitulate many neuroendocrine abnormalities observed in PCOS patients. PNA and development induce changes in spontaneous GnRH neuron firing rate, response to synaptic input, and the afterhyperpolarization potential of the action potential. We hypothesized potassium currents are altered by PNA treatment and/or development. Whole-cell patch-clamp recordings were made of transient and residual potassium currents of GnRH neurons in brain slices from 3-week-old and adult control and PNA females. At 3 weeks of age, PNA treatment increased transient current density versus controls. Development and PNA altered voltage-dependent activation and inactivation of the transient current. In controls, transient current activation and inactivation were depolarized at 3 weeks of age versus in adulthood. In GnRH neurons from 3-week-old mice, transient current activation and inactivation were more depolarized in control than PNA mice. Development and PNA treatment interacted to shift the time-dependence of inactivation and recovery from inactivation. Notably, in cells from adult PNA females, recovery was prolonged compared to all other groups. Activation of the residual current occurred at more depolarized membrane potentials in 3-week-old than adult controls. PNA depolarized activation of the residual current in adults. These findings demonstrate the properties of GnRH neuron potassium currents change during typical development, potentially contributing to puberty, and further suggest PNA treatment may both alter some typical developmental changes and induce additional modifications, which together may underlie aspects of the PNA phenotype. There was not any clinical trial involved in this work.

Unimolecular Helix-Based Transmembrane Nanochannel with a Smallest Luminal Cavity of 1 Å Expressing High Proton Selectivity and Transport Activity

Natural protein channels have evolved with exquisite structures to transport ions selectively and rapidly. Learning from nature to construct biomimetic artificial channels is always challenging. Herein we present a unimolecular transmembrane proton channel by quinoline-derived helix, which exhibited highly selective and ultrafast proton transport behaviors. This helix-based channel possesses a small luminal cavity of 1 Å in diameter, which could efficiently reject the permeation of cations, anions or water molecules but only permits the translocation of protons owing to the size effect. The proton flow rate exceeded 10⁷ H⁺ s^-1 channel^-1 and reached the same magnitude with gramicidin A. Mechanism investigation revealed that the directionally arrayed NH-chain inside the synthetic channel played a pivotal role during the proton flux. This work not only presented a helix-based channel with the smallest observable nanopore, but also unveiled an unexplored pathway for realizing efficient transport of protons via the consecutive NH-chain.

Gonadotropin-Releasing Hormone (GnRH) Neuron Potassium Currents and Excitability in Both Sexes Exhibit Minimal Changes upon Removal of Negative Feedback

Gonadotropin-releasing hormone (GnRH) drives pituitary secretion of luteinizing hormone and follicle-stimulating hormone, which in turn regulate gonadal functions including steroidogenesis. The pattern of GnRH release and thus fertility depend on gonadal steroid feedback. Under homeostatic (negative) feedback conditions, removal of the gonads from either females or males increases the amplitude and frequency of GnRH release and alters the long-term firing pattern of these neurons in brain slices. The neurobiological mechanisms intrinsic to GnRH neurons that are altered by homeostatic feedback are not well studied and have not been compared between sexes. During estradiol-positive feedback, which is unique to females, there are correlated changes in voltage-gated potassium currents and neuronal excitability. We thus hypothesized that these same mechanisms would be engaged in homeostatic negative feedback. Voltage-gated potassium channels play a direct role in setting excitability and action potential properties. Whole-cell patch-clamp recordings of GFP-identified GnRH neurons in brain slices from sham-operated and castrated adult female and male mice were made to assess fast and slow inactivating potassium currents as well as action potential properties. Surprisingly, no changes were observed among groups in most potassium current properties, input resistance, or capacitance, and this was reflected in a lack of differences in excitability and specific action potential properties. These results support the concept that, in contrast to positive feedback, steroid-negative feedback regulation of GnRH neurons in both sexes is likely conveyed to GnRH neurons via mechanisms that do not induce major changes in the biophysical properties of these cells.

Protonic conductor: better understanding neural resting and action potential

With the employment of the transmembrane electrostatic proton localization theory with a new membrane potential equation, neural resting and action potential is now much better understood as the voltage contributed by the localized protons/cations at a neural liquid- membrane interface. Accordingly, the neural resting/action potential is essentially a protonic/cationic membrane capacitor behavior. It is now understood with a newly formulated action potential equation: when action potential is <0 (negative number), the localized protons/cations charge density at the liquid-membrane interface along the periplasmic side is >0 (positive number); when the action potential is >0, the concentration of the localized protons and localized nonproton cations is <0, indicating a "depolarization" state. The nonlinear curve of the localized protons/cations charge density in the real-time domain of an action potential spike appears as an inverse mirror image to the action potential. The newly formulated action potential equation provides biophysical insights for neuron electrophysiology, which may represent a complementary development to the classic Goldman-Hodgkin-Katz equation. With the use of the action potential equation, the biological significance of axon myelination is now also elucidated as to provide protonic insulation and prevent any ions both inside and outside of the neuron from interfering with the action potential signal, so that the action potential can quickly propagate along the axon with minimal (e.g., 40 times less) energy requirement.NEW & NOTEWORTHY The newly formulated action potential equation provides biophysical insights for neuron electrophysiology, which may represent a complementary development to the classic Goldman-Hodgkin-Katz equation. The nonlinear curve of the localized protons/cations charge density in the real-time domain of an action potential spike appears as an inverse mirror image to the action potential. The biological significance of axon myelination is now elucidated as to provide protonic insulation and prevent any ions from interfering with action potential signal.

Estradiol Enhances the Depolarizing Response to GABA and AMPA Synaptic Conductances in Arcuate Kisspeptin Neurons by Diminishing Voltage-Gated Potassium Currents

Synaptic and intrinsic properties interact to sculpt neuronal output. Kisspeptin neurons in the hypothalamic arcuate nucleus help convey homeostatic estradiol feedback to central systems controlling fertility. Estradiol increases membrane depolarization induced by GABA_A receptor activation in these neurons. We hypothesized that the mechanisms underlying estradiol-induced alterations in postsynaptic response to GABA, and also AMPA, receptor activation include regulation of voltage-gated potassium currents. Whole-cell recordings of arcuate kisspeptin neurons in brain slices from ovariectomized (OVX) and OVX+estradiol (OVX+E) female mice during estradiol negative feedback revealed that estradiol reduced capacitance, reduced transient and sustained potassium currents, and altered voltage dependence and kinetics of transient currents. Consistent with these observations, estradiol reduced rheobase and action potential latency. To study more directly interactions between synaptic and active intrinsic estradiol feedback targets, dynamic clamp was used to simulate GABA and AMPA conductances. Both GABA and AMPA dynamic clamp-induced postsynaptic potentials (PSPs) were smaller in neurons from OVX than OVX+E mice; blocking transient potassium currents eliminated this difference. To interrogate the role of the estradiol-induced changes in passive intrinsic properties, different Markov model structures based on the properties of the transient potassium current in cells from OVX or OVX+E mice were combined in silico with passive properties reflecting these two endocrine conditions. Some of tested models reproduced the effect on PSPs in silico, revealing that AMPA PSPs were more sensitive to changes in capacitance. These observations support the hypothesis that PSPs in arcuate kisspeptin neurons are regulated by estradiol-sensitive mechanisms including potassium conductances and membrane properties.SIGNIFICANCE STATEMENT Kisspeptin neurons relay estradiol feedback to gonadotropin-releasing hormone neurons, which regulate the reproductive system. The fast synaptic neurotransmitters GABA and glutamate rapidly depolarize arcuate kisspeptin neurons and estradiol increases this depolarization. Estradiol reduced both potassium current in the membrane potential range typically achieved during response to fast synaptic inputs and membrane capacitance. Using simulated GABA and glutamate synaptic inputs, we showed changes in both the passive and active intrinsic properties induced by in vivo estradiol treatment affect the response to synaptic inputs, with capacitance having a greater effect on response to glutamate. The suppression of both passive and active intrinsic properties by estradiol feedback thus renders arcuate kisspeptin neurons more sensitive to fast synaptic inputs.

Another interpretation of the Goldman-Hodgkin-Katz equation based on Ling's adsorption theory

According to standard membrane theory, the generation of membrane potential is attributed to transmembrane ion transport. However, there have been a number of reports of membrane behavior in conflict with the membrane theory of cellular potential. Putting aside the membrane theory, we scrutinized the generation mechanism of membrane potential from the view of the long-dismissed adsorption theory of Ling. Ling's adsorption theory attributes the membrane potential generation to mobile ion adsorption. Although Ling's adsorption theory conflicts with the broadly accepted membrane theory, we found that it well reproduces experimentally observed membrane potential behavior. Our theoretical analysis finds that the potential formula based on the GHK eq., which is a fundamental concept of membrane theory, coincides with the potential formula based on Ling's adsorption theory. Reinterpreting the permeability coefficient in the GHK eq. as the association constant between the mobile ion and adsorption site, the GHK eq. turns into the potential formula from Ling's adsorption theory. We conclude that the membrane potential is generated by ion adsorption as Ling's adsorption theory states and that the membrane theory of cellular potential should be amended even if not discarded.

A Novel Method for the Description of Voltage-Gated Ionic Currents Based on Action Potential Clamp Results-Application to Hippocampal Mossy Fiber Boutons

Action potential clamp (AP-clamp) recordings of the delayed rectifier K(+) current I K and the fast-activated Na(+) current I Na in rat hippocampal mossy fiber boutons (MFBs) are analyzed using a computational technique recently reported. The method is implemented using a digitized AP from an MFB and computationally applying that data set to published models of I K and I Na. These numerical results are compared with experimental AP-clamp recordings. The I Na result is consistent with experiment; the I K result is not. The difficulty with the I K model concerns the fully activated current-voltage relation, which is described here by the Goldman-Hodgkin-Katz dependence on the driving force (V-E K) rather than (V-E K) itself, the standard model for this aspect of ion permeation. That revision leads to the second-a much steeper voltage dependent activation curve for I K than the one obtained from normalization of a family of I K records by (V-E K). The revised model provides an improved description of the AP-clamp measurement of I K in MFBs compared with the standard approach. The method described here is general. It can be used to test models of ionic currents in any excitable cell. In this way it provides a novel approach to the relationship between ionic current and membrane excitability in neurons.

Non-linear leak currents affect mammalian neuron physiology

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.

Novel description of ionic currents recorded with the action potential clamp technique: application to excitatory currents in suprachiasmatic nucleus neurons

The traditional method of recording ionic currents in neurons has been with voltage-clamp steps. Other waveforms such as action potentials (APs) can be used. The AP clamp method reveals contributions of ionic currents that underlie excitability during an AP (Bean BP. Nat Rev Neurosci 8: 451-465, 2007). A novel usage of the method is described in this report. An experimental recording of an AP from the literature is digitized and applied computationally to models of ionic currents. These results are compared with experimental AP-clamp recordings for model verification or, if need be, alterations to the model. The method is applied to the tetrodotoxin-sensitive sodium ion current, INa, and the calcium ion current, ICa, from suprachiasmatic nucleus (SCN) neurons (Jackson AC, Yao GL, Bean BP. J Neurosci 24: 7985-7998, 2004). The latter group reported voltage-step and AP-clamp results for both components. A model of INa is constructed from their voltage-step results. The AP clamp computational methodology applied to that model compares favorably with experiment, other than a modest discrepancy close to the peak of the AP that has not yet been resolved. A model of ICa was constructed from both voltage-step and AP-clamp results of this component. The model employs the Goldman-Hodgkin-Katz equation for the current-voltage relation rather than the traditional linear dependence of this aspect of the model on the Ca(2+) driving force. The long-term goal of this work is a mathematical model of the SCN AP. The method is general. It can be applied to any excitable cell.

Efficient calculation of the quasi-static electrical potential on a tetrahedral mesh and its implementation in STEPS

We describe a novel method for calculating the quasi-static electrical potential on tetrahedral meshes, which we call E-Field. The E-Field method is implemented in STEPS, which performs stochastic spatial reaction-diffusion computations in tetrahedral-based cellular geometry reconstructions. This provides a level of integration between electrical excitability and spatial molecular dynamics in realistic cellular morphology not previously achievable. Deterministic solutions are also possible. By performing the Rallpack tests we demonstrate the accuracy of the E-Field method. Efficient node ordering is an important practical consideration, and we find that a breadth-first search provides the best solutions, although principal axis ordering suffices for some geometries. We discuss potential applications and possible future directions, and predict that the E-Field implementation in STEPS will play an important role in the future of multiscale neural simulations.

A novel analysis of excitatory currents during an action potential from suprachiasmatic nucleus neurons

A new application of the action potential (AP) voltage-clamp technique is described based on computational analysis. An experimentally recorded AP is digitized. The resulting Vi vs. ti data set is applied to mathematical models of the ionic conductances underlying excitability for the cell from which the AP was recorded to test model validity. The method is illustrated for APs from suprachiasmatic nucleus (SCN) neurons and the underlying tetrodotoxin-sensitive Na(+) current, INa, and the Ca(2+) current, ICa. Voltage-step recordings have been made for both components from SCN neurons (Jackson et al. 2004). The combination of voltage-step and AP clamp results provides richer constraints for mathematical models of voltage-gated ionic conductances than either set of results alone, in particular the voltage-step results. For SCN neurons the long-term goal of this work is a realistic mathematical model of the SCN AP in which the equations for I(Na) and I(Ca) obtained from this analysis will be a part. Moreover, the method described in this report is general. It can be applied to any excitable cell.

Tuning of EAG K(+) channel inactivation: molecular determinants of amplification by mutations and a small molecule

Ether-à-go-go (EAG) and EAG-related gene (ERG) K⁺ channels are close homologues but differ markedly in their gating properties. ERG1 channels are characterized by rapid and extensive C-type inactivation, whereas mammalian EAG1 channels were previously considered noninactivating. Here, we show that human EAG1 channels exhibit an intrinsic voltage-dependent slow inactivation that is markedly enhanced in rate and extent by 1–10 µM 3-nitro-N-(4-phenoxyphenyl) benzamide, or ICA105574 (ICA). This compound was previously reported to have the opposite effect on ERG1 channels, causing an increase in current magnitude by inhibition of C-type inactivation. The voltage dependence of 2 µM ICA-induced inhibition of EAG1 current was half-maximal at −73 mV, 62 mV negative to the half-point for channel activation. This finding suggests that current inhibition by the drug is mediated by enhanced inactivation and not open-channel block, where the voltage half-points for current inhibition and channel activation are predicted to overlap, as we demonstrate for clofilium and astemizole. The mutation Y464A in the S6 segment also induced inactivation of EAG1, with a time course and voltage dependence similar to that caused by 2 µM ICA. Several Markov models were investigated to describe gating effects induced by multiple concentrations of the drug and the Y464A mutation. Models with the smallest fit error required both closed- and open-state inactivation. Unlike typical C-type inactivation, the rate of Y464A- and ICA-induced inactivation was not decreased by external tetraethylammonium or elevated [K⁺]_e. EAG1 channel inactivation introduced by Y464A was prevented by additional mutation of a nearby residue located in the S5 segment (F359A) or pore helix (L434A), suggesting a tripartite molecular model where interactions between single residues in S5, S6, and the pore helix modulate inactivation of EAG1 channels.

The layer-oriented approach to declarative languages for biological modeling

We present a new approach to modeling languages for computational biology, which we call the layer-oriented approach. The approach stems from the observation that many diverse biological phenomena are described using a small set of mathematical formalisms (e.g. differential equations), while at the same time different domains and subdomains of computational biology require that models are structured according to the accepted terminology and classification of that domain. Our approach uses distinct semantic layers to represent the domain-specific biological concepts and the underlying mathematical formalisms. Additional functionality can be transparently added to the language by adding more layers. This approach is specifically concerned with declarative languages, and throughout the paper we note some of the limitations inherent to declarative approaches. The layer-oriented approach is a way to specify explicitly how high-level biological modeling concepts are mapped to a computational representation, while abstracting away details of particular programming languages and simulation environments. To illustrate this process, we define an example language for describing models of ionic currents, and use a general mathematical notation for semantic transformations to show how to generate model simulation code for various simulation environments. We use the example language to describe a Purkinje neuron model and demonstrate how the layer-oriented approach can be used for solving several practical issues of computational neuroscience model development. We discuss the advantages and limitations of the approach in comparison with other modeling language efforts in the domain of computational biology and outline some principles for extensible, flexible modeling language design. We conclude by describing in detail the semantic transformations defined for our language.

Membranes with the same ion channel populations but different excitabilities

Electrical signaling allows communication within and between different tissues and is necessary for the survival of multicellular organisms. The ionic transport that underlies transmembrane currents in cells is mediated by transporters and channels. Fast ionic transport through channels is typically modeled with a conductance-based formulation that describes current in terms of electrical drift without diffusion. In contrast, currents written in terms of drift and diffusion are not as widely used in the literature in spite of being more realistic and capable of displaying experimentally observable phenomena that conductance-based models cannot reproduce (e.g. rectification). The two formulations are mathematically related: conductance-based currents are linear approximations of drift-diffusion currents. However, conductance-based models of membrane potential are not first-order approximations of drift-diffusion models. Bifurcation analysis and numerical simulations show that the two approaches predict qualitatively and quantitatively different behaviors in the dynamics of membrane potential. For instance, two neuronal membrane models with identical populations of ion channels, one written with conductance-based currents, the other with drift-diffusion currents, undergo transitions into and out of repetitive oscillations through different mechanisms and for different levels of stimulation. These differences in excitability are observed in response to excitatory synaptic input, and across different levels of ion channel expression. In general, the electrophysiological profiles of membranes modeled with drift-diffusion and conductance-based models having identical ion channel populations are different, potentially causing the input-output and computational properties of networks constructed with these models to be different as well. The drift-diffusion formulation is thus proposed as a theoretical improvement over conductance-based models that may lead to more accurate predictions and interpretations of experimental data at the single cell and network levels.

Going native: voltage-gated potassium channels controlling neuronal excitability

In this review we take a physiological perspective on the role of voltage-gated potassium channels in an identified neuron in the auditory brainstem. The large number of KCN genes for potassium channel subunits and the heterogeneity of the subunit combination into K(+) channels make identification of native conductances especially difficult. We provide a general pharmacological and biophysical profile to help identify the common voltage-gated K(+) channel families in a neuron. Then we consider the physiological role of each of these conductances from the perspective of the principal neuron in the medial nucleus of the trapezoid body (MNTB). The MNTB is an inverting relay, converting excitation generated by sound from one cochlea into inhibition of brainstem nuclei on the opposite side of the brain; this information is crucial for binaural comparisons and sound localization. The important features of MNTB action potential (AP) firing are inferred from its inhibitory projections to four key target nuclei involved in sound localization (which is the foundation of auditory scene analysis in higher brain centres). These are: the medial superior olive (MSO), the lateral superior olive (LSO), the superior paraolivary nucleus (SPN) and the nuclei of the lateral lemniscus (NLL). The Kv families represented in the MNTB each have a distinct role: Kv1 raises AP firing threshold; Kv2 influences AP repolarization and hyperpolarizes the inter-AP membrane potential during high frequency firing; and Kv3 accelerates AP repolarization. These actions are considered in terms of fidelity of transmission, AP duration, firing rates and temporal jitter. An emerging theme is activity-dependent phosphorylation of Kv channel activity and suggests that intracellular signalling has a dynamic role in refining neuronal excitability and homeostasis.