Squid Giant Axon Research Articles

An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to analyze because of the presence of the nonlinearity term, which is always very difficult to approximate. This paper aims to design a reliable scheme that consists of a combination of the nonstandard finite difference in time method, the Galerkin method and the compactness methods in space variables. This method is used to show that the solution of the problem exists uniquely. The a priori estimate from the existence process is applied to the scheme to show that the numerical solution from the scheme converges optimally in the L2 as well as the H1 norms. We proceed to show that the scheme preserves the decaying properties of the exact solution. Numerical experiments are introduced with a chosen example to validate the proposed theory.

From squid giant axon to automated patch-clamp: electrophysiology in venom and antivenom research.

Ion channels play a crucial role in diverse physiological processes, including neurotransmission and muscle contraction. Venomous creatures exploit the vital function of ion channels by producing toxins in their venoms that specifically target these ion channels to facilitate prey capture upon a bite or a sting. Envenoming can therefore lead to ion channel dysregulation, which for humans can result in severe medical complications that often necessitate interventions such as antivenom administration. Conversely, the discovery of highly potent and selective venom toxins with the capability of distinguishing between different isoforms and subtypes of ion channels has led to the development of beneficial therapeutics that are now in the clinic. This review encompasses the historical evolution of electrophysiology methodologies, highlighting their contributions to venom and antivenom research, including venom-based drug discovery and evaluation of antivenom efficacy. By discussing the applications and advancements in patch-clamp techniques, this review underscores the profound impact of electrophysiology in unravelling the intricate interplay between ion channels and venom toxins, ultimately leading to the development of drugs for envenoming and ion channel-related pathologies.

Fifty years of gating currents and channel gating.

We celebrate this year the 50th anniversary of the first electrophysiological recordings of the gating currents from voltage-dependent ion channels done in 1973. This retrospective tries to illustrate the context knowledge on channel gating and the impact gating-current recording had then, and how it continued to clarify concepts, elaborate new ideas, and steer the scientific debate in these 50 years. The notion of gating particles and gating currents was first put forward by Hodgkin and Huxley in 1952 as a necessary assumption for interpreting the voltage dependence of the Na and K conductances of the action potential. 20 years later, gating currents were actually recorded, and over the following decades have represented the most direct means of tracing the movement of the gating charges and gaining insights into the mechanisms of channel gating. Most work in the early years was focused on the gating currents from the Na and K channels as found in the squid giant axon. With channel cloning and expression on heterologous systems, other channels as well as voltage-dependent enzymes were investigated. Other approaches were also introduced (cysteine mutagenesis and labeling, site-directed fluorometry, cryo-EM crystallography, and molecular dynamics [MD] modeling) to provide an integrated and coherent view of voltage-dependent gating in biological macromolecules. The layout of this retrospective reflects the past 50 years of investigations on gating currents, first addressing studies done on Na and K channels and then on other voltage-gated channels and non-channel structures. The review closes with a brief overview of how the gating-charge/voltage-sensor movements are translated into pore opening and the pathologies associated with mutations targeting the structures involved with the gating currents.

In Memoriam: Lawrence B. Cohen (1939–2023)

In Memoriam: Lawrence B. Cohen (1939–2023)

Fine-structure constant and hysteresis regimes in the giant squid propagating action potential.

We present a phase space based phenomenological theory for the steady propagation of the action potential using the Rosenthal-Bezanilla experimental data. Time derivatives of giant squid action potential, its velocity of propagation, the radius of the axon and its capacitance, and the resistivity of axoplasm, yield the total ionic, capacitive, and membrane currents of the charge conserving cable equation. Evidence is presented that the sodium channels lattice has a polarization flip at the inception of the action potential and the corresponding flip reversal at the peak of the action potential while traversing a ferroelectric hysteresis loop. The polarization flip at the inception of the action potential is followed by capacitive polarization current known as gating current. The polarization flip at the peak of the action potential is preceded by a small sodium polarization current associated with sodium current inactivation. Ionic currents and sodium polarization ionic currentsare taken to have the familiar structure, as the product of maximum conductance, driving force, and fraction of open channels. Fractions of open channels are fitted in the lab frame by modified Avrami equations seeded with the value of the fine-structure constant α = 0.0072973... . The existence of sodium channels domains with two different symmetries, albeit only one being stable in squid axon, suggests the possibility of neurons with at least 2 stable states, the necessary condition for storage and retrieval of memories. It is expected that presented results will provide the framework for further analysis of thermodynamic phase changing behavior, the role of quantum mechanics in the flow of ions trough ionic channels and in particular the role of ferroelectric sodium channels lattice behavior in storage and retrieval of memories and nerve excitability.

Bistability at the onset of neuronal oscillations.

The Hodgkin-Huxley (HH) model and squid axon (bathed in reduced Ca2+) fire repetitively for steady current injection. Moreover, for a current-range just suprathreshold, repetitive firing coexists with a stable steady state. Neuronal excitability, as such, shows bistability and hysteresis providing the opportunity for the system to perform as switchable between firing and non-firing states with transient input and providing the backbone as a dynamical mechanism for bursting oscillations. Some conditions for bistability can be derived by intricate analysis (bifurcation theory) and characterized by simulation, but conditions for emergence and robustness of such bistability do not typically follow from intuition. Here, we demonstrate with a semi-quantitative two-variable, V-w, reduction of the HH model features that promote/reduce bistability. Visualization of flow and trajectories in the V-w phase plane provides an intuitive grasp for bistability. The geometry of action potential recovery involves a late phase during which the dynamic negative feedback of [Formula: see text] inactivation and [Formula: see text] activation over/undershoot, respectively, their resting values, thereby leading to hyperexcitabilty and an intrinsically generated opportunity to by-pass the spiral-like stable rest state and initiate the next spike upstroke. We illustrate control of bistability and dependence of the degree of hysteresis on recovery timescales and gating properties. Our dynamical dissection reveals the strongly attracting depolarized phase of the spike, enabling approximations like the resetting feature of adapting integrate-and-fire models. We extend our insights and show that the Morris-Lecar model can also exhibit robust bistability.

Joule heating involving ion currents through channel proteins.

Ion currents associated with channel proteins in the presence of membrane potential are ubiquitous in cellular and organelle membranes. When an ion current occurs through a channel protein, Joule heating should occur. However, this Joule heating seems to have been largely overlooked in biology. Here we show theoretical investigation of Joule heating involving channel proteins in biological processes. We used electrochemical potential to derive the Joule's law for an ion current through an ion transport protein in the presence of membrane potential, and we suggest that heat production and absorption can occur. Simulation of temperature distribution around a single channel protein with the Joule heating revealed that the temperature increase was as small as <10-3 K, although an ensemble of channel proteins was suggested to exhibit a noticeable temperature increase. Thereby, we theoretically investigated the Joule heating of systems containing ensembles of channel proteins. Nerve is known to undergo rapid heat production followed by heat absorption during the action potential, and our simulation of Joule heating for a squid giant axon combined with the Hodgkin-Huxley model successfully reproduced the feature of the heat. Furthermore, we extended the theory of Joule heating to uncoupling protein 1 (UCP1), a solute carrier family transporter, which is important to the non-shivering thermogenesis in brown adipose tissue mitochondria (BATM). Our calculations showed that the Joule heat involving UCP1 was comparable to the literature calorimetry data of BATM. Joule heating of ion transport proteins is likely to be one of important mechanisms of cellular thermogenesis.

Recurrence quantification and bifurcation analysis of electrical activity in resistive/memristive synapse coupled Fitzhugh–Nagumo type neurons

Temporal response of excitable individual and coupled different neuronal circuits i.e., Cell-I and Cell-II, of Fitzhugh–Nagumo type have been numerically simulated with a view to understand and characterize the complexities of involved dynamical processes. FHN neuronal circuit Cell-I and Cell-II have been shown to exhibit supercritical Hopf bifurcation, when excited by an external steady stimulus. For time varying sinusoidal excitation, the temporal action potential behavior of these neuronal circuits, when analyzed using measures of recurrence quantification analysis (RQA) and bifurcation diagram reveals period-1, period-2, multi-periodic, chaotic and reverse period doubling characteristics as observed earlier experimentally in giant squid axon and other biological cell assemblies. Phase portrait, bifurcation diagrams and recurrence plot (RPs), of (i) unidirectional and (ii) bidirectional coupled FHN Cell-I and Cell-II, through a gap junction resistor, suggest the possibility of transition to intermittent state besides the transition of respective action potential from periodic to multi-periodic to chaotic state in an ordered or disordered fashion, depending on values of system parameters that control the complexities. Further, for both the configurations (i) and (ii), synchronization of FHN neuron Cell-I with Cell-II is also shown to occur for large values of coupling parameter, ξ. The dynamical effect of electromagnetic radiation on individual FHN neuron circuits has also been investigated using a flux controlled memristor that couples the magnetic flux with the individual FHN neurons. Phase portrait, bifurcation and RQA analysis have also been used to characterize the electrical activities of excitable cell assemblies involving memristive synapse coupled FHN neuronal Cell-I and Cell-II in settings (i) and (ii) respectively.

A pilot investigation of the efficacy and safety of magnesium chloride and ethanol as anesthetics in Loligo vulgaris embryos.

The inclusion of cephalopods in the legislation related to the use of animals for experimental purposes has been based on the precautionary principle that these animals have the capacity to experience pain, suffering, distress, and lasting harm. Recent studies have expanded this view and supported it. Handling cephalopod mollusks in research is challenging and whenever more invasive procedures are required, sedation and/or anesthesia becomes necessary. Therefore, finding adequate, safe, and effective anesthetics appears mandatory. Several substances have been considered in sedating cephalopods, in some instances applying those utilized for fish. However, species-specific variability requires more detailed studies. Despite long-lasting experience being linked to classic studies on squid giant axons, evidence of action on putative anesthetic substances is scarce for Loligo vulgaris and particularly for their embryos. The aim of the current study was to evaluate effects elicited by immersion of squid embryos in anesthetic solutions and examine whether these forms display a similar reaction to anesthetics as adults do. Different concentrations of ethanol (EtOH; 2, 2.5, and 3%) and magnesium chloride (MgCl2; 1, 1.5, and 1.8%) were tested by adopting a set of indicators aimed at exploring the physiological responses of squid embryos. Forty-two embryos of the common squid Loligo vulgaris (stages 27-28) were assigned to three conditions (EtOH, MgCl2, and controls) and video recorded for 15min (5min before, 5min during, and 5min after immersion in the anesthetic solutions). In each group, the heart rate, respiratory rate, buoyancy, chromatophore activity, and tentacles/arms responses were assessed to evaluate the embryos' vitality and responsiveness to stimulation. Both substances provoked a decrease in heart and respiratory rates and inhibited buoyancy, chromatophores, and tentacles/arms responses; no adverse effects were observed. EtOH had a faster onset of action and faster recovery than MgCl2, being potentially more adequate as an anesthetic for shorter procedures. Even though MgCl2 caused a longer muscle relaxation, the reversibility was not confirmed for the 1.8% concentration; however, lower concentrations triggered similar results as the ones obtained with the highest EtOH concentrations. We have shown that the late developmental stages of Loligo vulgaris embryos could represent a good model to evaluate anesthetics for cephalopods since they can display similar reactions to anesthetics as adults animals do.

"You've Got to Work on This Axon": J. Z. Young and Squid Giant Axon Preparations in 20th -Century Neurobiology.

Employing and extending Hans-Jörg Rheinberger's analytical concept of epistemic things, this essay proposes one reason why squid giant axons, unusually large invertebrate nerve fibers, had such great impacts on twentieth-century neurobiology. The 1930s characterizations of these axons by John Zachary Young reshaped prevailing assumptions about nerve cells as epistemic things, I argue. Specifically, Young's preparations of these axons, which consisted of fibers attached to laboratory technologies, highlighted similarities between giant axons and more familiar ones via lines of comparative study common to aquatic biology. Young's work convinced other biologists that the squid giant fibers were, in fact, axons, despite their unusual fused (syncytial) structures, thereby promoting further studies, such as intracellular measurements, made possible by the fiber's size. Tracing direct relations between preparations of squid axons and broader interpretations of neurons as epistemic things, this paper renders an actors' category, "preparations," into an analytical one. In turn, it offers glimpses into how aquatic organisms shaped twentieth-century neurobiology and how local experiments can drive broader, disciplinary changes.

Quantum mechanics, universal scaling, and ferroelectric hysteresis regimes in the giant squid axon propagating action potential: a phase space approach

Experimental data for giant squid axon propagating action potential is examined in phase space and it is parsed into three distinct parts: properties of individual sodium and potassium channels, morphing of sodium channels and two narrow regions pertaining to ferroelectric polarization flips of sodium channels lattice. Plots of ionic currents vs. potential exhibit quasilinear segments yielding temperature dependent maximum conductance constants and the related time rates. Plotting ionic time rates as Boltzmann kinetic rates yields activation energies similar to activation energies of 0.6 eV for rate-limiting bio-chemical metabolic reactions.

The Giant Axon of the Squid: A Simple System for Axonal Transport Studies.

The squid giant axon has a long history of being a superb experimental system in which to investigate a wide range of questions concerning intracellular transport. In this protocol we describe the method used for dissecting the axon to preserve its viability in vitro, and the technique for injecting exogenous materials into the living axon. Now that the squid genome is emerging, and the CRISPR/cas9 system has been successfully applied to knock-out squid genes, the giant axon will resume its place in the scientific pantheon of powerful experimental systems in which to address biological questions pertaining to all eukaryotes.

Research on inductive neuron model and its dynamic characteristics

The size of neuron is on a mesoscopic scale. In this paper, considering the inductance characteristics of the neuron, the classical circuit model and mesoscopic circuit model of neuron including neuron membrane inductance, membrane capacitance, potassium ion memristor and chloride ion resistance are established. Based on the classical circuit theory and the quantum theory of mesoscopic circuit, the expression of neuron membrane voltage response under external impulse excitation is derived. Substituting the electrophysiological parameters of the squid neuron into the expression of membrane voltage, we find that the membrane voltages in both models first increase and then decrease, and finally reach their corresponding resting states of zero value, and their energy values are concentrated mainly in a range of 0–30 Hz in which the brainwave frequency is. Further comparisons show that the peak value of membrane voltage and the time required to reach the peak value (peak time) in the mesoscopic circuit model are lower than those in the classical circuit model, and are closer to the experimental results after the squid axon has been stimulated, indicating that the mesoscopic circuit model can better reflect the physiological characteristics of the stimulated neurons. Based on the mesoscopic circuit model, the peak value of membrane voltage increases and the peak time decreases with the increase of external excitation intensity. Parameters such as membrane voltage peak and peak time are more sensitive to the neuron membrane capacitance. The mesoscopic circuit model of the neuron is of great significance in understanding the excitability of the stimulated neuron and also in promoting the development of quantum neural networks inspired by brain function.

A Morphological Peak-Detector for Single-Unit Neural Recording Acquisition Systems

The increasing demand for measurement systems in neuroscience with the ability to acquire signals at neuron-level resolution has led to the development of techniques based on innovative organic biosensors. These single-unit extracellular neural recording systems provide useful information on neural behavior in terms of the action potential (AP) firing mechanism. In this work, we propose a processing technique specifically designed to accurately and reliably extract single-cell APs generated by free and attached membranes from signals acquired by extracellular recording devices. The simultaneous presence of overlapped APs, due to the nonideal coupling between cell and surface device, involves a distortion of the signal acquired by the device. Such distortion makes APs detection challenging. The proposed approach consists of a morphological peak detector characterized by high selectivity and based on three stages: a denoising phase for reducing wideband noise performed by the empirical mode decomposition (EMD) technique, and a threshold-based classifier for the identification of all the possible peaks that could correspond to an AP and a morphological classifier based on the support vector machine technique that improves the performances of the whole peak detection algorithm increasing its selectivity. The proposed method was designed on simulated data obtained by taking cell-to-cell variability into account in order to adapt the proposed method to the heterogeneity of the biological reality rather than the average behavior of the cells. This point is crucial to design an approach suitable for applications <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in vivo</i> . Finally, the morphological classifier was tested on simulated and semisimulated data obtained from experimental acquisitions measured from the axons of a giant squid in response to current stimulation, achieving F1-score > 89% for both scenarios.

Line Tensions Caused by Voltage-Induced Axon Swelling Propel Non-Myelinated Action Potentials

Voltage-induced swelling of action potential (AP) accompanies shrinkage along the tubular axon. Longitudinal tensions appear at both boundaries of AP of nerve axon, and the line tension at the front boundary has been shown to be greater than that at the rear boundary. The difference of tensions has been certified by previously observed diphasic heat behavior of AP. The propagation velocity of AP is determined by the balance between the difference of tensions and the hydrodynamic resistance of AP by ambient water. The bilayer lipid membrane is liberated from the strong electrostriction of resting potential to depolarized AP. An elastic model of both boundaries of AP has been proposed, which describes the relation between diphasic heat behavior of AP and generated line tensions. The propagation velocity of HEK-293 cells, squid giant axon (SGA), and garfish olfactory nerve (GON) has been analyzed and reliably explained by the proposed line tension model. The energy cost of propelling APs of SGA and GON has been shown to be greater than that of the electric energy.

Revisiting the memristor concept within basic circuit theory

SummaryIn this paper, we revisit the memristor concept within circuit theory. We start from the definition of the basic circuit elements; then, we introduce the original formulation of the memristor concept and summarize some of the controversies on its nature. We also point out the ambiguities resulting from a nonrigorous usage of the flux linkage concept. After concluding that the memristor is not a fourth basic circuit element, prompted by claims in the memristor literature, we look into the application of the memristor concept to electrophysiology, realizing that an approach suitable to explain the observed inductive behavior of the giant squid axon had already been developed in the 1960s, with the introduction of “time‐variant resistors.' We also discuss a recent memristor implementation in which the magnetic flux plays a direct role, concluding that it cannot strictly qualify as a memristor, because its v‐i curve cannot exactly pinch at the origin. Finally, we present numerical simulations of a few memristors and memristive systems, focusing on the behavior in the φ‐q plane. We show that, contrary to what happens for the most basic memristor concept, for general memristive systems, the φ‐q curve is not single‐valued or not even closed.

Implementation of Hodgkin-Huxley Neuron Model With the Novel Memristive Oscillator

In this brief, in order to simplify the implementation of the classical Hodgkin-Huxley neuron (HH) model, a novel floating memristor emulator is introduced, which consists of four electronic components and can be considered the simplest implementation in the existing literature. Firstly, the schematic diagrams of the proposed novel HH neuron-based memristive oscillator is presented, and the working mechanism and the bursting phenomenon of which are analyzed. Then, the simulation model of the proposed memristive oscillator emulator is developed in MULTISIM using several off-the-shelf components, which is used to implement a HH neuron model. Moreover, the electrical characteristics of it are examined. In particular, the relationship between the ion channels concentration and the variable voltages of the neuron model is described in detail. Simulation results confirm that the proposed model demonstrates rich bursting oscillation behaviors which is more suitable for simulating the giant squid axon compared to the existing model in the literature, which can be utilized to transmit the neuron information and establish the neural network.

A simple parametric representation of the Hodgkin-Huxley model.

The Hodgkin-Huxley model, decades after its first presentation, is still a reference model in neuroscience as it has successfully reproduced the electrophysiological activity of many organisms. The primary signal in the model represents the membrane potential of a neuron. A simple representation of this signal is presented in this paper. The new proposal is an adapted Frequency Modulated Möbius multicomponent model defined as a signal plus error model in which the signal is decomposed as a sum of waves. The main strengths of the method are the simple parametric formulation, the interpretability and flexibility of the parameters that describe and discriminate the waveforms, the estimators’ identifiability and accuracy, and the robustness against noise. The approach is validated with a broad simulation experiment of Hodgkin-Huxley signals and real data from squid giant axons. Interesting differences between simulated and real data emerge from the comparison of the parameter configurations. Furthermore, the potential of the FMM parameters to predict Hodgkin-Huxley model parameters is shown using different Machine Learning methods. Finally, promising contributions of the approach in Spike Sorting and cell-type classification are detailed.

Improved Hodgkin-Huxley type model for neural action potentials.

The simple Goldman-Hodgkin-Katz model for resting-state membrane potentials has been generalized to provide a new nonlinear theoretical model for action potentials in perfused axons. Our minimalistic model appeals naturally to physically based electrodiffusion principles to describe electric-current densities inside sodium and potassium-ion channels whereas the 1952 Hodgkin-Huxley model describes such current densities in an ad hoc way. Although the two models share similar schemes for the kinetics of ion-channel gating, our relaxation times for channel gating are simpler, being independent of membrane potential. Like the theoretical model of Hodgkin and Huxley, based primarily on experimental data at [Formula: see text], our dynamical system behaves as a 4-dimensional resonator exhibiting subthreshold oscillations. Although our present analysis refers to experiments at [Formula: see text], re-parameterizations of this model should permit consideration of action potentials at alternative temperatures. The predicted speed of propagating action potentials in giant axons of squid at [Formula: see text] is in excellent agreement with the Hodgkin-Huxley experimental value at [Formula: see text]. In cases where our model predictions differ from those of the Hodgkin-Huxley model, new experiments will be required to determine which model is more accurate.

Nanopore sensing: A physical-chemical approach

Protein nanopores have emerged as an important class of sensors for the understanding of biophysical processes, such as molecular transport across membranes, and for the detection and characterization of biopolymers. Here, we trace the development of these sensors from the Coulter counter and squid axon studies to the modern applications including exquisite detection of small volume changes and molecular reactions at the single molecule (or reactant) scale. This review focuses on the chemistry of biological pores, and how that influences the physical chemistry of molecular detection.