Frequency entrainment of squid axon membrane

Summary Sinusoidally varying stimulating currents were applied to space-clamped squid giant axon membranes in a double sucrose gap apparatus. Stimulus parameters varied were peak-to-peak current amplitude, frequency, and DC offset bias. In response to these stimuli, the membranes produced action potentials in varying patterns, according to variation of input stimulus parameters. For some stimulus parameters the output patterns were stable and obviously periodic with the periods being simple multiples of the input period; for other stimulus parameters no obvious periodicity was manifest in the output. The experimental results were compared with simulations using a computer model which was modified in several ways from the Hodgkin-Huxley model to make it more representative of our preparation. The model takes into account K⁺ accumulation in the periaxonal space, features of Na⁺ inactivation which are anomalous to the Hodgkin-Huxley model, sucrose gap hyperpolarization current, and membrane current noise. Many aspects of the experiments are successfully simulated but some are not, possibly because some very slow process present in the preparation is not included in the model.     

The influence of the chloride currents on action potential firing and volume regulation of excitable cells studied by a kinetic model

In excitable cells, the generation of an action potential (AP) is associated with transient changes of the intra- and extracellular concentrations of small ions such as Na⁺, K⁺ and Cl⁻. If these changes cannot be fully reversed between successive APs cumulative changes of trans-membrane ion gradients will occur, impinging on the cell volume and the duration, amplitude and frequency of APs. Previous computational studies focused on effects associated with excitation-induced changes of potassium and sodium. Here we present a model based study on the influence of chloride on the fidelity of AP firing and cellular volume regulation during excitation. Our simulations show that depending on the magnitude of the basal chloride permeability two complementary types of responsiveness and volume variability exist: (i) At high chloride permeability (typical for muscle cells), large excitatory stimuli are required to elicit APs; repetitive stimuli of equal strength result in almost identical spike train patterns (Markovian behavior), however, long excitation may lead to after discharges due to an outward directed current of intracellular chloride ions which accumulate during excitation; cell volume changes are large. (ii) At low chloride permeability (e.g., neurons), small excitatory stimuli are sufficient to elicit APs, repetitive stimuli of equal strength produce spike trains with progressively changing amplitude, frequency and duration (short-term memory effects or non-Markovian behavior); cell volume changes are small. We hypothesize that variation of the basal chloride permeability could be an important mechanism of neuronal cells to adapt their responsiveness to external stimuli during learning and memory processes.     

Subthreshold Voltage Noise Due to Channel Fluctuations in Active Neuronal Membranes

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na⁺ and K⁺ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.     

Dynamic excitation states and firing patterns are controlled by sodium channel kinetics in myenteric neurons: a simulation study

Enteric neurons located in the gastro-intestinal tract are of particular importance to control digestive functions such as motility and secretion. In our recent publication, we showed that mouse myenteric neurons exhibit 2 types of tetrodotoxin-resistant Na⁺ currents: a fast inactivating Na⁺ current produced by Nav1.5 channels, present in nearly all myenteric neurons, and a persistent Na⁺ current attributed to Nav1.9 channels, restricted to the intrinsic primary afferent neurons (IPANs). By combination of experimental recording and computer simulation we found that Nav1.5 contributed to the upstroke velocity of action potentials (APs), whereas Nav1.9 opposed AP repolarization. Here, we detailed the Na⁺, Ca²⁺ and K⁺ currents used in our computational model of IPAN. We refined the prototype cell to reproduce the sustained firing pattern recorded in situ. As shown in experimental conditions we demonstrated that Nav1.9 channels critically determine the up-state life-time and thus, are essential to sustain tonic firing.     

Dynamic excitation states and firing patterns are controlled by sodium channel kinetics in myenteric neurons: a simulation study

Enteric neurons located in the gastro-intestinal tract are of particular importance to control digestive functions such as motility and secretion. In our recent publication, we showed that mouse myenteric neurons exhibit 2 types of tetrodotoxin-resistant Na⁺ currents: a fast inactivating Na⁺ current produced by Nav1.5 channels, present in nearly all myenteric neurons, and a persistent Na⁺ current attributed to Nav1.9 channels, restricted to the intrinsic primary afferent neurons (IPANs). By combination of experimental recording and computer simulation we found that Nav1.5 contributed to the upstroke velocity of action potentials (APs), whereas Nav1.9 opposed AP repolarization. Here, we detailed the Na⁺, Ca²⁺ and K⁺ currents used in our computational model of IPAN. We refined the prototype cell to reproduce the sustained firing pattern recorded in situ. As shown in experimental conditions we demonstrated that Nav1.9 channels critically determine the up-state life-time and thus, are essential to sustain tonic firing.     

Profound alterations in the intrinsic excitability of cerebellar Purkinje neurons following neurotoxin 3-acetylpyridine (3-AP)-induced ataxia in rat: new insights into the role of small conductance K+ channels

Alterations in the intrinsic properties of Purkinje cells (PCs) may contribute to the abnormal motor performance observed in ataxic rats. To investigate whether such changes in the intrinsic neuronal excitability could be attributed to the role of Ca(2+)-activated K(+) channels (K(Ca)), whole cell current clamp recordings were made from PCs in cerebellar slices of control and ataxic rats. 3-AP induced profound alterations in the intrinsic properties of PCs, as evidenced by a significant increase in both the membrane input resistance and the initial discharge frequency, along with the disruption of the firing regularity. In control PCs, the blockade of small conductance K(Ca) channels by UCL1684 resulted in a significant increase in the membrane input resistance, action potential (AP) half-width, time to peak of the AP and initial discharge frequency. SK channel blockade also significantly decreased the neuronal discharge regularity, the peak amplitude of the AP, the amplitude of the afterhyperpolarization and the spike frequency adaptation ratio. In contrast, in ataxic rats, both the firing regularity and the initial firing frequency were significantly increased by the blockade of SK channels. In conclusion, ataxia may arise from alterations in the functional contribution of SK channels, to the intrinsic properties of PCs.     

High frequency electrostimulation of excitable cells

Reactions of nerve fibers to high frequency electrical stimulation are examined with three nerve models. Switching on the signal produces a single AP at the threshold current. Stronger currents lead into a region of repetitive firing. The firing rate depends on the current and the fibers more distant from the electrode will have a lower rate. The AP's are not synchronized. In the "House-Urban" cochlear implant a 16 kHz carrier is used for stimulation. It is modulated by electrical signals derived from sound pressure. An analysis of the modulation shows which signals can produce APs synchronized with the source signal.     

The High Energy Demand of Neuronal Cells Caused by Passive Leak Currents is Not a Waste of Energy

It is estimated that maintenance of the resting potential of neurons consumes between 15 % (in gray matter) and 44 % (in fully myelinated white matter) of the brain’s total energy budget [1]. This poses the intriguing question why evolution has not strived to lower the permeability of passive ion channels to cut the high resting-state energy budget of the brain. Based on a conceptual mathematical model of neuronal ion currents and action potential (AP) firing we demonstrate that a neuron endowed with small leak currents and correspondingly low energy consumption by the Na⁺/K⁺-ATPase in the resting state may indeed recapitulate all features of normal AP firing. However, the activation and inactivation of such a “low-energy-cost neuron” turns out to be extremely sensitive to small fluctuation of Na⁺ currents associated with Na⁺-dependent secondary-active transport that is indispensable for the metabolic integrity of the cell and neurotransmitter recycling. We provide evidence that sufficiently large leak currents function as important stabilizers of the membrane potential and thus are required to allow robust AP firing. Our simulations suggest that the energy demand of the Na⁺/K⁺-ATPase needed to counterbalance passive leak currents cannot be significantly dropped below observed values.     

Systematic characterization of the ionic basis of rabbit cellular electrophysiology using two ventricular models

Several mathematical models of rabbit ventricular action potential (AP) have been proposed to investigate mechanisms of arrhythmias and excitation-contraction coupling. Our study aims at systematically characterizing how ionic current properties modulate the main cellular biomarkers of arrhythmic risk using two widely-used rabbit ventricular models, and comparing simulation results using the two models with experimental data available for rabbit. A sensitivity analysis of AP properties, Ca²⁺ and Na⁺ dynamics, and their rate dependence to variations (±15% and ±30%) in the main transmembrane current conductances and kinetics was performed using the Shannon et al. (2004) and the [Mahajan et?al., 2008a] and [Mahajan et?al., 2008b] AP rabbit models. The effects of severe transmembrane current blocks (up to 100%) on steady-state AP and calcium transients, and AP duration (APD) restitution curves were also simulated using both models. Our simulations show that, in both virtual rabbit cardiomyocytes, APD is significantly modified by most repolarization currents, AP triangulation is regulated mostly by the inward rectifier K⁺ current (I_K1) whereas APD rate adaptation as well as [Na⁺]_i rate dependence is influenced by the Na⁺/K⁺ pump current (I_NaK). In addition, steady-state [Ca²⁺]_i levels, APD restitution properties and [Ca²⁺]_i rate dependence are strongly dependent on I_NaK, the L-Type Ca²⁺ current (I_CaL) and the Na⁺/Ca²⁺ exchanger current (I_NaCa), although the relative role of these currents is markedly model dependent. Furthermore, our results show that simulations using both models agree with many experimentally-reported electrophysiological characteristics. However, our study shows that the Shannon et al. model mimics rabbit electrophysiology more accurately at normal pacing rates, whereas Mahajan et al. model behaves more appropriately at faster rates. Our results reinforce the usefulness of sensitivity analysis for further understanding of cellular electrophysiology and validation of cardiac AP models.     

Modeling squid axon K+ channel by a nucleation and growth kinetic mechanism

A kinetic model accounting for all salient features of the K⁺ channel of the squid giant axon, including the rising phase of the ON gating charge and the Cole-Moore effect, is provided. Upon accounting for a significant feature distinguishing K⁺, Na⁺ and Ca^2 + channels from channel-forming peptides modeled in our previous 2016 BBA paper, the nucleation-and-growth kinetic model developed therein is extended to simulate ON ionic and gating currents of the K⁺ channel of the squid giant axon at different depolarization potentials by the use of only two free parameters. K⁺ channel opening is considered to proceed by progressive aggregation of single subunits, while they are moving their gating charge outward under depolarizing conditions within their tetrameric structure; K⁺ channel closing proceeds in the opposite direction, by repolarization-induced disaggregation of subunits, accompanied by inward movement of their gating charge.     

Action Potential Energy Efficiency Varies Among Neuron Types in Vertebrates and Invertebrates

The initiation and propagation of action potentials (APs) places high demands on the energetic resources of neural tissue. Each AP forces ATP-driven ion pumps to work harder to restore the ionic concentration gradients, thus consuming more energy. Here, we ask whether the ionic currents underlying the AP can be predicted theoretically from the principle of minimum energy consumption. A long-held supposition that APs are energetically wasteful, based on theoretical analysis of the squid giant axon AP, has recently been overturned by studies that measured the currents contributing to the AP in several mammalian neurons. In the single compartment models studied here, AP energy consumption varies greatly among vertebrate and invertebrate neurons, with several mammalian neuron models using close to the capacitive minimum of energy needed. Strikingly, energy consumption can increase by more than ten-fold simply by changing the overlap of the Na⁺ and K⁺ currents during the AP without changing the APs shape. As a consequence, the height and width of the AP are poor predictors of energy consumption. In the Hodgkin–Huxley model of the squid axon, optimizing the kinetics or number of Na⁺ and K⁺ channels can whittle down the number of ATP molecules needed for each AP by a factor of four. In contrast to the squid AP, the temporal profile of the currents underlying APs of some mammalian neurons are nearly perfectly matched to the optimized properties of ionic conductances so as to minimize the ATP cost.     

Spike-Frequency Adaptation of a Generalized Leaky Integrate-and-Fire Model Neuron

Although spike-frequency adaptation is a commonly observed property of neurons, its functional implications are still poorly understood. In this work, using a leaky integrate-and-fire neural model that includes a Ca²⁺-activated K⁺ current (I _AHP), we develop a quantitative theory of adaptation temporal dynamics and compare our results with recent in vivo intracellular recordings from pyramidal cells in the cat visual cortex. Experimentally testable relations between the degree and the time constant of spike-frequency adaptation are predicted. We also contrast the I _AHP model with an alternative adaptation model based on a dynamical firing threshold. Possible roles of adaptation in temporal computation are explored, as a a time-delayed neuronal self-inhibition mechanism. Our results include the following: (1) given the same firing rate, the variability of interspike intervals (ISIs) is either reduced or enhanced by adaptation, depending on whether the I _AHP dynamics is fast or slow compared with the mean ISI in the output spike train; (2) when the inputs are Poisson-distributed (uncorrelated), adaptation generates temporal anticorrelation between ISIs, we suggest that measurement of this negative correlation provides a probe to assess the strength of I _AHP in vivo; (3) the forward masking effect produced by the slow dynamics of I _AHP is nonlinear and effective at selecting the strongest input among competing sources of input signals.     

Axonal Noise as a Source of Synaptic Variability

Post-synaptic potential (PSP) variability is typically attributed to mechanisms inside synapses, yet recent advances in experimental methods and biophysical understanding have led us to reconsider the role of axons as highly reliable transmission channels. We show that in many thin axons of our brain, the action potential (AP) waveform and thus the Ca⁺⁺ signal controlling vesicle release at synapses will be significantly affected by the inherent variability of ion channel gating. We investigate how and to what extent fluctuations in the AP waveform explain observed PSP variability. Using both biophysical theory and stochastic simulations of central and peripheral nervous system axons from vertebrates and invertebrates, we show that channel noise in thin axons (<1 µm diameter) causes random fluctuations in AP waveforms. AP height and width, both experimentally characterised parameters of post-synaptic response amplitude, vary e.g. by up to 20 mV and 0.5 ms while a single AP propagates in C-fibre axons. We show how AP height and width variabilities increase with a ¾ power-law as diameter decreases and translate these fluctuations into post-synaptic response variability using biophysical data and models of synaptic transmission. We find for example that for mammalian unmyelinated axons with 0.2 µm diameter (matching cerebellar parallel fibres) axonal noise alone can explain half of the PSP variability in cerebellar synapses. We conclude that axonal variability may have considerable impact on synaptic response variability. Thus, in many experimental frameworks investigating synaptic transmission through paired-cell recordings or extracellular stimulation of presynaptic neurons, causes of variability may have been confounded. We thereby show how bottom-up aggregation of molecular noise sources contributes to our understanding of variability observed at higher levels of biological organisation.     

Bifurcation of periodic solutions of Hodgkin-Huxley model for the squid giant axon

The Hodgkin-Huxley model of the space-clamped squid giant axon is shown to admit unstable periodic solutions for current stimuli less than the stimulus at which the rest state becomes linearly unstable. The periodic solutions are demonstrated both by bifurcation theory and by numerical integration. The presence of subcritical unstable oscillations explains the discontinuous behaviour of the amplitude of the repetitive response as a function of current stimulus     

Axoplasmic RNA species synthesized in the isolated squid giant axon

Isolated squid stellate nerves and giant fiber lobes were incubated for 8 hr in Millipore filtered sea water containing [³H]uridine. The electrophoretic patterns of radioactive RNA purified from the axoplasm of the giant axon and from the giant fiber lobe (cell bodies of the giant axon) demonstrated the presence of RNA species with mobilities corresponding to tRNA and rRNA. The presence of labeled rRNAs was confirmed by the behavior of the large rRNA component (31S) which, in the squid, readily dissociates into its two constituent moyeties (17S and 20S). Comparable results were obtained with the axonal sheath and the stellate nerve. In all the electrophoretic patterns, additional species of radioactive RNA migrated between the 4S and the 20S markers, i.e. with mobilities corresponding to presumptive mRNAs. Chromatographic analysis of the purified RNAs on oligo(dT)cellulose indicated the presence of labeled poly(A)⁺ RNA in all tissue samples. Radioactive poly(A)⁺ RNA represented approximately 1% of the total labeled RNA in the axoplasm, axonal sheath and stellate nerve, but more than 2% in the giant fiber lobe. The labeled poly(A)⁺ RNAs of the giant fibre lobe showed a prevalence of larger species in comparison to the axonal sheath and stellate nerve. In conclusion, the axoplasmic RNAs synthesized by the isolated squid giant axon appear to include all the major classes of axoplasmic RNAs, that is rRNA, tRNA and mRNA.Special Issue dedicated to Prof. Holger Hydén.     

Effects of yohimbine on squid axons.

Yohimbine, an indolealkylamine alkaloid, reduces the amplitude of the sodium current in the squid giant axon. For doses that reduce sodium current amplitude by up to 50%, there is no significant change in the kinetics or in any of the voltage-dependent parameters associated with sodium channels. The effective equilibrium constant for yohimbine binding to the sodium channel is 3 x 10(-4) M. Repetitive depolarizing pulses increase the inhibition of squid axon sodium current by yohimbine. This use-dependent inhibition is enhanced by increasing the concentration of yohimbine, by increasing the frequency of pulsing, and by increasing the magnitude or the duration of depolarization. It is reduced by hyperpolarizing prepulses. This behavior can be explained by a model wherein yohimbine binds more readily to open sodium channels than to closed sodium channels and wherein the Hodgkin-Huxley kinetic parameters are modified by the binding of the drug. This type of model may also explain the tonic and use-dependent inhibition previously described by others for local anesthetics.     

Axonal excitability revisited

The original papers of Hodgkin and Huxley (J. Physiol. 116 (1952a) 449, J. Physiol. 116 (1952b) 473, J. Physiol. 116 (1952c) 497, J. Physiol. 117 (1952d) 500) have provided a benchmark in our understanding of cellular excitability. Not surprisingly, their model of the membrane action potential (AP) requires revisions even for the squid giant axon, the preparation for which it was originally formulated. The mechanisms they proposed for the voltage-gated potassium and sodium ion currents, IK, and INa, respectively, have been superceded by more recent formulations that more accurately describe voltage-clamp measurements of these components. Moreover, the current-voltage relation for IK has a non-linear dependence upon driving force that is well described by the Goldman-Hodgkin-Katz (GHK) relation, rather than the linear dependence on driving force found by Hodgkin and Huxley. Furthermore, accumulation of potassium ions in the extracellular space adjacent to the axolemma appears to be significant even during a single AP. This paper describes the influence of these various modifications in their model on the mathematically reconstructed AP. The GHK and K+ accumulation results alter the shape of the AP, whereas the modifications in IK and INa gating have surprisingly little effect. Perhaps the most significant change in their model concerns the amplitude of INa, which they appear to have overestimated by a factor of two. This modification together with the GHK and the K+ accumulation results largely remove the discrepancies between membrane excitability of the squid giant axon and the Hodgkin and Huxley (J. Physiol. 117 (1952d) 500) model previously described (Clay, J. Neurophysiol. 80 (1998) 903).     

The early phase of sodium channel gating current in the squid giant axon

A fast component of displacement current which accompanies the sodium channel gating current has been recorded from the membrane of the giant axon of the squid Loligo forbesii. This component is characterized by relaxation time constants typically shorter than 25 µs. The charge displaced accounts for about 10% (or 2 nC/cm²) of the total displacement charge attributed to voltage-dependent sodium channels. Using a low noise, wide-band voltage clamp system and specially designed voltage step protocols we could demonstrate that this component: (i) is not a recording artifact; (ii) is kinetically independent from the sodium channel activation and inactivation processes; (iii) can account for a significant fraction of the initial amplitude of recorded displacement current and (iv) has a steady state charge transfer which saturates for membrane potentials above + 20 mV and below – 100 mV This component can be modelled as a single step transition using the Eyring-Boltzmann formalism with a quantal charge of 1 e^– and an asymmetrical energy barrier. Furthermore, if it were associated with the squid sodium channel, our data would suggest one fast transition per channel. A possible role as a sodium channel activation trigger, which would still be consistent with kinetic independence, is discussed. Despite uncertainties about its origin, the property of kinetic independence allows subtraction of this component from the total displacement current to reveal a rising phase in the early time course of the remaining current. This will have to be taken into account when modelling the voltage-dependent sodium channel.     

Characterization of the ATP-promoted aspects of Na+-Ca2+ exchange present in squid retinal nerve axolemma

Using an in vitro system which consists of an axolemma-rich vesicle fraction prepared from squid retinal nerve fibers, an Na⁺-Ca²⁺ exchange process has been characterized and appears identical with that reported in squid giant axon. This exchange is absolutely dependent on the establishement of an Na⁺ gradient, shows monovalent and divalent cation specificity and is highly sensitive to monensin, A23187 and valinomycin but not to ouabain, digitoxigenin, vanadate, pentylenetetrazole, tetrodotoxin or tetraethylammonium. Furthermore, it was found that the exchange process is enhanced by the addition of ATP. This ATP-promoted aspects of Na⁺-Ca²⁺ exchange shares many similar characteristics with Na⁺-Ca²⁺ ATP hydrolysis and may indicate a common mechanism for both activities via a protein phosphorylation-dephosphorylation event.     

Modeling temporal behavior of postnatal cat retinal ganglion cells

During development, mammalian retinal ganglion cells (RGCs) go through marked ontogenetic changes with respect to their excitable membrane properties. Voltage-clamp studies conducted in our laboratory have shown that the amplitude, voltage-dependence and kinetics of activation and inactivation (where present) of Na(+), K(+) and Ca(2+) conductances all exhibit developmental changes during a time when the firing patterns of mammalian ganglion cells shift from being transient to being predominantly sustained in nature. In order to better understand the contribution of each conductance to the generation of spikes and spiking patterns, we have developed a model based on our experimental data. For simplicity, we have initially used experimental data obtained from postnatal ganglion cells. At this age the ontogenetic changes observed in the characteristics of the various ionic currents are complete. Utilizing the methods adopted by Hodgkin and Huxley for the giant squid axon, we have determined rate equations for the activation and inactivation properties of the I(A), I(K dr), I(Na), I(Ca L), I(Ca N), and I(leak) currents in postnatal cat RGCs. Combining these with a simplified model of the calcium-activated potassium current (I(KCa)), we have solved and analysed the resulting differential equations. While spikes and spiking patterns resembling experimental data could be obtained from a model in which [Ca(2+)i] was averaged across the whole cell, more accurate simulations were obtained when the diffusion of intracellular Ca(2+) was modeled spatially. The resulting spatial calcium gradients were more effective in gating I(KCa), and our simulations more accurately matched the recorded amplitude and shape of individual spikes as well as the frequency of maintained discharges observed in mammalian postnatal RGCs.