RECTIFICATION AND INDUCTANCE IN THE SQUID GIANT AXON

Hysteresis in memristors produces conduction inductance and conduction capacitance effects

Memristors are devices in which the conductance state can be alternately switched between a high and a low value by means of a voltage scan. In general, systems involving a chemical inductor mechanism as solar cells, asymmetric nanopores in electrochemical cells, transistors, and solid state memristive devices, exhibit a current increase and decrease over time that generates hysteresis. By performing small signal ac impedance spectroscopy, we show that memristors, or any other system with hysteresis relying on the conductance modulation effect, display intrinsic dynamic inductor-like and capacitance-like behaviours in specific input voltage ranges. Both the conduction inductance and the conduction capacitance originate in the same delayed conduction process linked to the memristor dynamics and not in electromagnetic or polarization effects. A simple memristor model reproduces the main features of the transition from capacitive to inductive impedance spectroscopy spectra, which causes a nonzero crossing of current-voltage curves.

Advances in using ultrasound to regulate the nervous system

Ultrasound is a mechanical vibration with a frequency greater than 20 kHz. Due to its high spatial resolution, good directionality, and convenient operation in neural regulation, it has recently received increasing attention from scientists. However, the mechanism by which ultrasound regulates the nervous system is still unclear. This article mainly explores the possible mechanisms of ultrasound's mechanical effects, cavitation effects, thermal effects, and the rise of sonogenetics. In addition, the essence of action potential and its relationship with ultrasound were also discussed. Traditional theory treats nerve impulses as pure electrical signals, similar to cable theory. However, this theory cannot explain the phenomenon of inductance and cell membrane bulging out during the propagation of action potential. Therefore, the flexoelectric effect of cell membrane and soliton model reveal that action potential may also be a mechanical wave. Finally, we also elaborated the therapeutic effect of ultrasound on nervous system disease such as epilepsy, Parkinson's disease, and Alzheimer's disease.

Electrical Impedance Spectroscopy with Bacterial Biofilms: Neuronal-like Behavior

Negative capacitance at low frequencies for spiking neurons was first demonstrated in 1941 (K. S. Cole) by using extracellular electrodes. The phenomenon subsequently was explained by using the Hodgkin-Huxley model and is due to the activity of voltage-gated potassium ion channels. We show that Escherichia coli (E. coli) biofilms exhibit significant stable negative capacitances at low frequencies when they experience a small DC bias voltage in electrical impedance spectroscopy experiments. Using a frequency domain Hodgkin-Huxley model, we characterize the conditions for the emergence of this feature and demonstrate that the negative capacitance exists only in biofilms containing living cells. Furthermore, we establish the importance of the voltage-gated potassium ion channel, Kch, using knock-down mutants. The experiments provide further evidence for voltage-gated ion channels in E. coli and a new, low-cost method to probe biofilm electrophysiology, e.g., to understand the efficacy of antibiotics. We expect that the majority of bacterial biofilms will demonstrate negative capacitances.

The enigmatic HCN channels: A cellular neurophysiology perspective

What physiological role does a slow hyperpolarization-activated ion channel with mixed cation selectivity play in the fast world of neuronal action potentials that are driven by depolarization? That puzzling question has piqued the curiosity of physiology enthusiasts about the hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which are widely expressed across the body and especially in neurons. In this review, we emphasize the need to assess HCN channels from the perspective of how they respond to time-varying signals, while also accounting for their interactions with other co-expressing channels and receptors. First, we illustrate how the unique structural and functional characteristics of HCN channels allow them to mediate a slow negative feedback loop in the neurons that they express in. We present the several physiological implications of this negative feedback loop to neuronal response characteristics including neuronal gain, voltage sag and rebound, temporal summation, membrane potential resonance, inductive phase lead, spike triggered average, and coincidence detection. Next, we argue that the overall impact of HCN channels on neuronal physiology critically relies on their interactions with other co-expressing channels and receptors. Interactions with other channels allow HCN channels to mediate intrinsic oscillations, earning them the "pacemaker channel" moniker, and to regulate spike frequency adaptation, plateau potentials, neurotransmitter release from presynaptic terminals, and spike initiation at the axonal initial segment. We also explore the impact of spatially non-homogeneous subcellular distributions of HCN channels in different neuronal subtypes and their interactions with other channels and receptors. Finally, we discuss how plasticity in HCN channels is widely prevalent and can mediate different encoding, homeostatic, and neuroprotective functions in a neuron. In summary, we argue that HCN channels form an important class of channels that mediate a diversity of neuronal functions owing to their unique gating kinetics that made them a puzzle in the first place.

Simulations predict differing phase responses to excitation vs. inhibition in theta-resonant pyramidal neurons

Rhythmic activity is ubiquitous in neural systems, with theta-resonant pyramidal neurons integrating rhythmic inputs in many cortical structures. Impedance analysis has been widely used to examine frequency-dependent responses of neuronal membranes to rhythmic inputs, but it assumes that the neuronal membrane is a linear system, requiring the use of small signals to stay in a near-linear regime. However, postsynaptic potentials are often large and trigger nonlinear mechanisms (voltage-gated ion channels). The goals of this work were to 1) develop an analysis method to evaluate membrane responses in this nonlinear domain and 2) explore phase relationships between rhythmic stimuli and subthreshold and spiking membrane potential (Vmemb) responses in models of theta-resonant pyramidal neurons. Responses in these output regimes were asymmetrical, with different phase shifts during hyperpolarizing and depolarizing half-cycles. Suprathreshold theta-rhythmic stimuli produced nonstationary Vmemb responses. Sinusoidal inputs produced "phase retreat": action potentials occurred progressively later in cycles of the input stimulus, resulting from adaptation. Sinusoidal current with increasing amplitude over cycles produced "phase advance": action potentials occurred progressively earlier. Phase retreat, phase advance, and subthreshold phase shifts were modulated by multiple ion channel conductances. Our results suggest differential responses of cortical neurons depending on the frequency of oscillatory input, which will play a role in neuronal responses to shifts in network state. We hypothesize that intrinsic cellular properties complement network properties and contribute to in vivo phase-shift phenomena such as phase precession, seen in place and grid cells, and phase roll, also observed in hippocampal CA1 neurons.NEW & NOTEWORTHY We augmented electrical impedance analysis to characterize phase shifts between large-amplitude current stimuli and nonlinear, asymmetric membrane potential responses. We predict different frequency-dependent phase shifts in response excitation vs. inhibition, as well as shifts in spike timing over multiple input cycles, in theta-resonant pyramidal neurons. We hypothesize that these effects contribute to navigation-related phenomena such as phase precession and phase roll. Our neuron-level hypothesis complements, rather than falsifies, prior network-level explanations of these phenomena.

Interictal localization of the epileptogenic zone: Utilizing the observed resonance behavior in the spectral band of surrounding inhibition

Introduction

The gold standard for identification of the epileptogenic zone (EZ) continues to be the visual inspection of electrographic changes around seizures' onset by experienced electroencephalography (EEG) readers. Development of an epileptogenic focus localization tool that can delineate the EZ from analysis of interictal (seizure-free) periods is still an open question of great significance for improved diagnosis (e.g., presurgical evaluation) and treatment of epilepsy (e.g., surgical outcome).

Methods

We developed an EZ interictal localization algorithm (EZILA) based on novel analysis of intracranial EEG (iEEG) using a univariate periodogram-type power measure, a straight-forward ranking approach, a robust dimensional reduction method and a clustering technique. Ten patients with temporal and extra temporal lobe epilepsies, and matching the inclusion criteria of having iEEG recordings at the epilepsy monitoring unit (EMU) and being Engel Class I ≥12 months post-surgery, were recruited in this study.

Results

In a nested k-fold cross validation statistical framework, EZILA assigned the highest score to iEEG channels within the EZ in all patients (10/10) during the first hour of the iEEG recordings and up to their first typical clinical seizure in the EMU (i.e., early interictal period). To further validate EZILA's performance, data from two new (Engel Class I) patients were analyzed in a double-blinded fashion; the EZILA successfully localized iEEG channels within the EZ from interictal iEEG in both patients.

Discussion

Out of the sampled brain regions, iEEG channels in the EZ were most frequently and maximally active in seizure-free (interictal) periods across patients in specific narrow gamma frequency band (∼60-80 Hz), which we have termed focal frequency band (FFB). These findings are consistent with the hypothesis that the EZ may interictally be regulated (controlled) by surrounding inhibitory neurons with resonance characteristics within this narrow gamma band.

Chemical Inductor

A multitude of chemical, biological, and material systems present an inductive behavior that is not electromagnetic in origin. Here, it is termed a chemical inductor. We show that the structure of the chemical inductor consists of a two-dimensional system that couples a fast conduction mode and a slowing down element. Therefore, it is generally defined in dynamical terms rather than by a specific physicochemical mechanism. The chemical inductor produces many familiar features in electrochemical reactions, including catalytic, electrodeposition, and corrosion reactions in batteries and fuel cells, and in solid-state semiconductor devices such as solar cells, organic light-emitting diodes, and memristors. It generates the widespread phenomenon of negative capacitance, it causes negative spikes in voltage transient measurements, and it creates inverted hysteresis effects in current–voltage curves and cyclic voltammetry. Furthermore, it determines stability, bifurcations, and chaotic properties associated to self-sustained oscillations in biological neurons and electrochemical systems. As these properties emerge in different types of measurement techniques such as impedance spectroscopy and time-transient decays, the chemical inductor becomes a useful framework for the interpretation of the electrical, optoelectronic, and electrochemical responses in a wide variety of systems. In the paper, we describe the general dynamical structure of the chemical inductor and we comment on a broad range of examples from different research areas.

The thermodynamic theory of action potential propagation: a sound basis for unification of the physics of nerve impulses

The thermodynamic theory of action potential propagation challenges the conventional understanding of the nerve signal as an exclusively electrical phenomenon. Often misunderstood as to its basic tenets and predictions, the thermodynamic theory is virtually ignored in mainstream neuroscience. Addressing a broad audience of neuroscientists, we here attempt to stimulate interest in the theory. We do this by providing a concise overview of its background, discussion of its intimate connection to Albert Einstein's treatment of the thermodynamics of interfaces and outlining its potential contribution to the building of a physical brain theory firmly grounded in first principles and the biophysical reality of individual nerve cells. As such, the paper does not attempt to advocate the superiority of the thermodynamic theory over any other approach to model the nerve impulse, but is meant as an open invitation to the neuroscience community to experimentally test the assumptions and predictions of the theory on their validity.

A Frequency Domain Analysis of the Excitability and Bifurcations of the FitzHugh-Nagumo Neuron Model

The dynamics of neurons consist of oscillating patterns of a membrane potential that underpin the operation of biological intelligence. The FitzHugh-Nagumo (FHN) model for neuron excitability generates rich dynamical regimes with a simpler mathematical structure than the Hodgkin-Huxley model. Because neurons can be understood in terms of electrical and electrochemical methods, here we apply the analysis of the impedance response to obtain the characteristic spectra and their evolution as a function of applied voltage. We convert the two nonlinear differential equations of FHN into an equivalent circuit model, classify the different impedance spectra, and calculate the corresponding trajectories in the phase plane of the variables. In analogy to the field of electrochemical oscillators, impedance spectroscopy detects the Hopf bifurcations and the spiking regimes. We show that a neuron element needs three essential internal components: capacitor, inductor, and negative differential resistance. The method supports the fabrication of memristor-based artificial neural networks.

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.

Effects of Ih and TASK-like shunting current on dendritic impedance in layer 5 pyramidal-tract neurons

Pyramidal neurons in neocortex have complex input-output relationships that depend on their morphologies, ion channel distributions, and the nature of their inputs, but which cannot be replicated by simple integrate-and-fire models. The impedance properties of their dendritic arbors, such as resonance and phase shift, shape neuronal responses to synaptic inputs and provide intraneuronal functional maps reflecting their intrinsic dynamics and excitability. Experimental studies of dendritic impedance have shown that neocortical pyramidal tract neurons exhibit distance-dependent changes in resonance and impedance phase with respect to the soma. We, therefore, investigated how well several biophysically detailed multicompartment models of neocortical layer 5 pyramidal tract neurons reproduce the location-dependent impedance profiles observed experimentally. Each model tested here exhibited location-dependent impedance profiles, but most captured either the observed impedance amplitude or phase, not both. The only model that captured features from both incorporates hyperpolarization-activated cyclic nucleotide-gated (HCN) channels and a shunting current, such as that produced by Twik-related acid-sensitive K⁺ (TASK) channels. TASK-like channel density in this model was proportional to local HCN channel density. We found that although this shunting current alone is insufficient to produce resonance or realistic phase response, it modulates all features of dendritic impedance, including resonance frequencies, resonance strength, synchronous frequencies, and total inductive phase. We also explored how the interaction of HCN channel current (I_h) and a TASK-like shunting current shape synaptic potentials and produce degeneracy in dendritic impedance profiles, wherein different combinations of I_h and shunting current can produce the same impedance profile.NEW & NOTEWORTHY We simulated chirp current stimulation in the apical dendrites of 5 biophysically detailed multicompartment models of neocortical pyramidal tract neurons and found that a combination of HCN channels and TASK-like channels produced the best fit to experimental measurements of dendritic impedance. We then explored how HCN and TASK-like channels can shape the dendritic impedance as well as the voltage response to synaptic currents.

Shock and detonation waves at an interface and the collision of action potentials

Action potentials in neurons are known to annihilate each other upon collision, while there are cases where they might penetrate each other. The fate of two waves upon collision is critically dependent on the underlying mechanism of propagation and therefore an understanding of possible outcomes of collision under different conditions is important. Previously, compression waves that travel within the plasma membrane of a neuron have been proposed as a thermodynamic basis for the propagation of action potentials. In this context, it was recently shown that two-dimensional compressive shock waves in the model system of lipid monolayers behave strikingly similar to action potentials in neurons and can even annihilate each other upon head-on collision. However, even a qualitative mechanism remained unclear. To this end, we summarise the fundamentals of shock physics as applied to an interface and recap how it explained the observation of threshold and saturation of shockwaves in the lipid monolayer (all - or - none). We then compare the theory with the soliton model that has the same fundamental premise, i.e. the conservation laws and thermodynamics, and was previously proposed as a model for the nerve pulse propagation. We elaborate on how the two approaches make different predictions with regards to collisions and the detailed structure of the wave-front. As a case study and a new qualitative result, we finally show that previously unexplained annihilation of shock waves in the lipid monolayer is a direct consequence of the nature of state changes, i.e. jump conditions, within these shockwaves.

A Physical Perspective to the Inductive Function of Myelin-A Missing Piece of Neuroscience

Starting from the inductance in neurons, two physical origins are discussed, which are the coil inductance of myelin and the piezoelectric effect of the cell membrane. The direct evidence of the coil inductance of myelin is the opposite spiraling phenomenon between adjacent myelin sheaths confirmed by previous studies. As for the piezoelectric effect of the cell membrane, which has been well-known in physics, the direct evidence is the mechanical wave accompany with action potential. Therefore, a more complete physical nature of neural signals is provided. In conventional neuroscience, the neural signal is a pure electrical signal. In our new theory, the neural signal is an energy pulse containing electrical, magnetic, and mechanical components. Such a physical understanding of the neural signal and neural systems significantly improve the knowledge of the neurons. On the one hand, we achieve a corrected neural circuit of an inductor-capacitor-capacitor (LCC) form, whose frequency response and electrical characteristics have been validated by previous studies and the modeling fitting of artifacts in our experiments. On the other hand, a number of phenomena observed in neural experiments are explained. In particular, they are the mechanism of magnetic nerve stimulations and ultrasound nerve stimulations, the MRI image contrast issue and Anode Break Excitation. At last, the biological function of myelin is summarized. It is to provide inductance in the process of neural signal, which can enhance the signal speed in peripheral nervous systems and provide frequency modulation function in central nervous systems.

A Model and Simulation With Therapeutic Device-Protocol Design Implications for Acute and Chronic Wounds

An understanding of healing processes for different tissues and organs, along with the development of appropriate therapeutic devices and treatment protocols, requires an appreciation for the mechanisms-of-action and sequencing of many interconnected chemical, electrical, mechanical, and optical activities. Unfortunately, the substantial contributions that endogenous electrical mechanisms-of-action provide in healing and regulation are often overlooked, resulting in a poor transfer of knowledge from science, to engineering, and finally, to therapy. The wide variety of healing processes, their therapeutic implications, and the devices and protocol designs that are most effective cannot be understood or addressed adequately without an understanding of the endogenous electrical mechanisms-of-action associated with wound healing. Achieving this level of understanding can be enhanced by the use of appropriate models and simulations that are based on physiological/biochemical system response characteristics.

Heterogeneities in intrinsic excitability and frequency-dependent response properties of granule cells across the blades of the rat dentate gyrus

The dentate gyrus (DG), the input gate to the hippocampus proper, is anatomically segregated into three different sectors, namely, the suprapyramidal blade, the crest region, and the infrapyramidal blade. Although there are well-established differences between these sectors in terms of neuronal morphology, connectivity patterns, and activity levels, differences in electrophysiological properties of granule cells within these sectors have remained unexplored. Here, employing somatic whole cell patch-clamp recordings from the rat DG, we demonstrate that granule cells in these sectors manifest considerable heterogeneities in their intrinsic excitability, temporal summation, action potential characteristics, and frequency-dependent response properties. Across sectors, these neurons showed positive temporal summation of their responses to inputs mimicking excitatory postsynaptic currents and showed little to no sag in their voltage responses to pulse currents. Consistently, the impedance amplitude profile manifested low-pass characteristics and the impedance phase profile lacked positive phase values at all measured frequencies and voltages and for all sectors. Granule cells in all sectors exhibited class I excitability, with broadly linear firing rate profiles, and granule cells in the crest region fired significantly fewer action potentials compared with those in the infrapyramidal blade. Finally, we found weak pairwise correlations across the 18 different measurements obtained individually from each of the three sectors, providing evidence that these measurements are indeed reporting distinct aspects of neuronal physiology. Together, our analyses show that granule cells act as integrators of afferent information and emphasize the need to account for the considerable physiological heterogeneities in assessing their roles in information encoding and processing.NEW & NOTEWORTHY We employed whole cell patch-clamp recordings from granule cells in the three subregions of the rat dentate gyrus to demonstrate considerable heterogeneities in their intrinsic excitability, temporal summation, action potential characteristics, and frequency-dependent response properties. Across sectors, granule cells did not express membrane potential resonance, and their impedance profiles lacked inductive phase leads at all measured frequencies. Our analyses also show that granule cells manifest class I excitability characteristics, categorizing them as integrators of afferent information.

Theoretical Modeling and Experimental Detection of the Extracellular Phasic Impedance Modulation in Rabbit Hearts

Theoretical cardiac electrophysiology focuses on the dynamics of the membrane and sarcoplasmic reticulum ion currents; however, passive (e.g., membrane capacitance) and quasi-active (response to small signals) properties of the cardiac sarcolemma, which are quantified by impedance, are also important in determining the behavior of cardiac tissue. Theoretically, impedance varies in the different phases of a cardiac cycle. Our goal in this study was to numerically predict and experimentally validate these phasic changes. We calculated the expected impedance signal using analytic methods (for generic ionic models) and numerical computation (for a rabbit ventricular ionic model). Cardiac impedance is dependent on the phase of the action potential, with multiple deflections caused by a sequential activation and inactivation of various membrane channels. The two main channels shaping the impedance signal are the sodium channel causing a sharp and transient drop at the onset of action potential and the inward rectifying potassium channel causing an increase in impedance during the plateau phase. This dip and dome pattern was confirmed in an ex-vivo rabbit heart model using high-frequency sampling through a monophasic action potential electrode. The hearts were immobilized using a myosin-inhibitor to minimize motion artifacts. We observed phasic impedance changes in three out of four hearts with a dome amplitude of 2 − 4Ω. Measurement of phasic impedance modulation using an extracellular electrode is feasible and provides a non-invasive way to gain insight into the state of cardiac cells and membrane ionic channels. The observed impedance recordings are consistent with the dip and dome pattern predicted analytically.

Active membrane conductances and morphology of a collision detection neuron broaden its impedance profile and improve discrimination of input synchrony

How neurons filter and integrate their complex patterns of synaptic inputs is central to their role in neural information processing. Synaptic filtering and integration are shaped by the frequency-dependent neuronal membrane impedance. Using single and dual dendritic recordings in vivo, pharmacology, and computational modeling, we characterized the membrane impedance of a collision detection neuron in the grasshopper Schistocerca americana. This neuron, the lobula giant movement detector (LGMD), exhibits consistent impedance properties across frequencies and membrane potentials. Two common active conductances g_H and g_M, mediated respectively by hyperpolarization-activated cyclic nucleotide-gated (HCN) channels and by muscarine-sensitive M-type K⁺ channels, promote broadband integration with high temporal precision over the LGMD's natural range of membrane potentials and synaptic input frequencies. Additionally, we found that a model based on the LGMD's branching morphology increased the gain and decreased the delay associated with the mapping of synaptic input currents to membrane potential. More generally, this was true for a wide range of model neuron morphologies, including those of neocortical pyramidal neurons and cerebellar Purkinje cells. These findings show the unexpected role played by two widespread active conductances and by dendritic morphology in shaping synaptic integration.NEW & NOTEWORTHY Neuronal filtering and integration of synaptic input patterns depend on the electrochemical properties of dendrites. We used an identified collision detection neuron in grasshoppers to examine how its morphology and two conductances affect its membrane impedance in relation to the computations it performs. The neuronal properties examined are ubiquitous and therefore promote a general understanding of neuronal computations, including those in the human brain.

Shunt peaking in neural membranes

Capacitance limits the bandwidth of engineered and biological electrical circuits because it determines the gain-bandwidth product (GBWP). With a fixed GBWP, bandwidth can only be improved by decreasing gain. In engineered circuits, an inductance reduces this limitation through shunt peaking but no equivalent mechanism has been reported for biological circuits. We show that in blowfly photoreceptors a voltage-dependent K⁺ conductance, the fast delayed rectifier (FDR), produces shunt peaking thereby increasing bandwidth without reducing gain. Furthermore, the FDR's time constant is close to the value that maximizes the photoreceptor GBWP while reducing distortion associated with the creation of a wide-band filter. Using a model of the honeybee drone photoreceptor, we also show that a voltage-dependent Na⁺ conductance can produce shunt peaking. We argue that shunt peaking may be widespread in graded neurons and dendrites.

Strings on a Violin: Location Dependence of Frequency Tuning in Active Dendrites

Strings on a violin are tuned to generate distinct sound frequencies in a manner that is firmly dependent on finger location along the fingerboard. Sound frequencies emerging from different violins could be very different based on their architecture, the nature of strings and their tuning. Analogously, active neuronal dendrites, dendrites endowed with active channel conductances, are tuned to distinct input frequencies in a manner that is dependent on the dendritic location of the synaptic inputs. Further, disparate channel expression profiles and differences in morphological characteristics could result in dendrites on different neurons of the same subtype tuned to distinct frequency ranges. Alternately, similar location-dependence along dendritic structures could be achieved through disparate combinations of channel profiles and morphological characteristics, leading to degeneracy in active dendritic spectral tuning. Akin to strings on a violin being tuned to different frequencies than those on a viola or a cello, different neuronal subtypes exhibit distinct channel profiles and disparate morphological characteristics endowing each neuronal subtype with unique location-dependent frequency selectivity. Finally, similar to the tunability of musical instruments to elicit distinct location-dependent sounds, neuronal frequency selectivity and its location-dependence are tunable through activity-dependent plasticity of ion channels and morphology. In this morceau, we explore the origins of neuronal frequency selectivity, and survey the literature on the mechanisms behind the emergence of location-dependence in distinct forms of frequency tuning. As a coda to this composition, we present some future directions for this exciting convergence of biophysical mechanisms that endow a neuron with frequency multiplexing capabilities.

Accommodation to hyperpolarization of human axons assessed in the frequency domain

Human axons in vivo were subjected to subthreshold currents with a threshold impedance amplitude profile to allow the use of frequency domain techniques to determine the propensity for resonant behavior and to clarify the relative contributions of different ion channels to their low-frequency responsiveness. Twenty-four studies were performed on the motor and sensory axons of the median nerve in six subjects. The response to oscillatory currents was tested between direct current (DC) and 16 Hz. A resonant peak at ∼2-2.5 Hz was found in the response of hyperpolarized axons, but there was only a small broad response in axons at resting membrane potential (RMP). A mathematical model of axonal excitability developed using DC pulses provided a good fit to the frequency response for human axons and indicated that the hyperpolarization-activated current Ih and the slow potassium current IKs are principally responsible for the resonance. However, the results indicate that if axons are hyperpolarized by more than -60% of resting threshold, the only conductances that are appreciably active are Ih and the leak conductance, i.e., that the activity of these conductances can be studied in vivo virtually in isolation at hyperpolarized membrane potentials. Given that the leak conductance dampens resonance, it is suggested that the -60% hyperpolarization used here is optimal for Ih As expected, differences between the frequency responses of motor and sensory axons were present and best explained by reduced slow potassium conductance GKs, up-modulation of Ih, and increased persistent Na(+) current INaP (due to depolarization of RMP) in sensory axons.

Analytical applications for pore-forming proteins

Proteinaceous nanometer-scale pores are ubiquitous in biology. The canonical ionic channels (e.g., those that transport Na(+), K(+), Ca(2+), and Cl(-) across cell membranes) play key roles in many cellular processes, including nerve and muscle activity. Another class of channels includes bacterial pore-forming toxins, which disrupt cell function, and can lead to cell death. We describe here the recent development of these toxins for a wide range of biological sensing applications. This article is part of a Special Issue entitled: Pore-Forming Toxins edited by Mauro Dalla Serra and Franco Gambale.

The Effect of the Nonlinearity of the Response of Lipid Membranes to Voltage Perturbations on the Interpretation of Their Electrical Properties. A New Theoretical Description

Our understanding of the electrical properties of cell membranes is derived from experiments where the membrane is exposed to a perturbation (in the form of a time-dependent voltage or current change) and information is extracted from the measured output. The interpretation of such electrical recordings consists in finding an electronic equivalent that would show the same or similar response as the biological system. In general, however, there is no unique circuit configuration, which can explain a single electrical recording and the choice of an electric model for a biological system is based on complementary information (most commonly structural information) of the system investigated. Most of the electrophysiological data on cell membranes address the functional role of protein channels while assuming that the lipid matrix is an insulator with constant capacitance. However, close to their melting transition the lipid bilayers are no inert insulators. Their conductivity and their capacitance are nonlinear functions of both voltage, area and volume density. This has to be considered when interpreting electrical data. Here we show how electric data commonly interpreted as gating currents of proteins and inductance can be explained by the nonlinear dynamics of the lipid matrix itself.

Interplay of intrinsic and synaptic conductances in the generation of high-frequency oscillations in interneuronal networks with irregular spiking

High-frequency oscillations (above 30 Hz) have been observed in sensory and higher-order brain areas, and are believed to constitute a general hallmark of functional neuronal activation. Fast inhibition in interneuronal networks has been suggested as a general mechanism for the generation of high-frequency oscillations. Certain classes of interneurons exhibit subthreshold oscillations, but the effect of this intrinsic neuronal property on the population rhythm is not completely understood. We study the influence of intrinsic damped subthreshold oscillations in the emergence of collective high-frequency oscillations, and elucidate the dynamical mechanisms that underlie this phenomenon. We simulate neuronal networks composed of either Integrate-and-Fire (IF) or Generalized Integrate-and-Fire (GIF) neurons. The IF model displays purely passive subthreshold dynamics, while the GIF model exhibits subthreshold damped oscillations. Individual neurons receive inhibitory synaptic currents mediated by spiking activity in their neighbors as well as noisy synaptic bombardment, and fire irregularly at a lower rate than population frequency. We identify three factors that affect the influence of single-neuron properties on synchronization mediated by inhibition: i) the firing rate response to the noisy background input, ii) the membrane potential distribution, and iii) the shape of Inhibitory Post-Synaptic Potentials (IPSPs). For hyperpolarizing inhibition, the GIF IPSP profile (factor iii)) exhibits post-inhibitory rebound, which induces a coherent spike-mediated depolarization across cells that greatly facilitates synchronous oscillations. This effect dominates the network dynamics, hence GIF networks display stronger oscillations than IF networks. However, the restorative current in the GIF neuron lowers firing rates and narrows the membrane potential distribution (factors i) and ii), respectively), which tend to decrease synchrony. If inhibition is shunting instead of hyperpolarizing, post-inhibitory rebound is not elicited and factors i) and ii) dominate, yielding lower synchrony in GIF networks than in IF networks.

Neurons in the brain engage in collective oscillations at different frequencies. Gamma and high-gamma oscillations (30–100 Hz and higher) have been associated with cognitive functions, and are altered in psychiatric disorders such as schizophrenia and autism. Our understanding of how high-frequency oscillations are orchestrated in the brain is still limited, but it is necessary for the development of effective clinical approaches to the treatment of these disorders. Some neuron types exhibit dynamical properties that can favour synchronization. The theory of weakly coupled oscillators showed how the phase response of individual neurons can predict the patterns of phase relationships that are observed at the network level. However, neurons in vivo do not behave like regular oscillators, but fire irregularly in a regime dominated by fluctuations. Hence, which intrinsic dynamical properties matter for synchronization, and in which regime, is still an open question. Here, we show how single-cell damped subthreshold oscillations enhance synchrony in interneuronal networks by introducing a depolarizing component, mediated by post-inhibitory rebound, that is correlated among neurons due to common inhibitory input.

Resonance in neocortical neurons and networks

Neocortical networks produce oscillations that often correspond to characteristic physiological or pathological patterns. However, the mechanisms underlying the generation of and the transitions between such oscillatory states remain poorly understood. In this study, we examined resonance in mouse layer V neocortical pyramidal neurons. To accomplish this, we employed standard electrophysiology to describe cellular resonance parameters. Bode plot analysis revealed a range of resonance magnitude values in layer V neurons and demonstrated that both magnitude and phase response characteristics of layer V neocortical pyramidal neurons are modulated by changes in the extracellular environment. Specifically, increased resonant frequencies and total inductive areas were observed at higher extracellular potassium concentrations and more hyperpolarised membrane potentials. Experiments using pharmacological agents suggested that current through hyperpolarization-activated cyclic nucleotide-gated channels (I(h) ) acts as the primary driver of resonance in these neurons, with other potassium currents, such as A-type potassium current and delayed-rectifier potassium current (Kv1.4 and Kv1.1, respectively), contributing auxiliary roles. The persistent sodium current was also shown to play a role in amplifying the magnitude of resonance without contributing significantly to the phase response. Although resonance effects in individual neurons are small, their properties embedded in large networks may significantly affect network behavior and may have potential implications for pathological processes.

History-dependent excitability as a single-cell substrate of transient memory for information discrimination

Neurons react differently to incoming stimuli depending upon their previous history of stimulation. This property can be considered as a single-cell substrate for transient memory, or context-dependent information processing: depending upon the current context that the neuron "sees" through the subset of the network impinging on it in the immediate past, the same synaptic event can evoke a postsynaptic spike or just a subthreshold depolarization. We propose a formal definition of History-Dependent Excitability (HDE) as a measure of the propensity to firing in any moment in time, linking the subthreshold history-dependent dynamics with spike generation. This definition allows the quantitative assessment of the intrinsic memory for different single-neuron dynamics and input statistics. We illustrate the concept of HDE by considering two general dynamical mechanisms: the passive behavior of an Integrate and Fire (IF) neuron, and the inductive behavior of a Generalized Integrate and Fire (GIF) neuron with subthreshold damped oscillations. This framework allows us to characterize the sensitivity of different model neurons to the detailed temporal structure of incoming stimuli. While a neuron with intrinsic oscillations discriminates equally well between input trains with the same or different frequency, a passive neuron discriminates better between inputs with different frequencies. This suggests that passive neurons are better suited to rate-based computation, while neurons with subthreshold oscillations are advantageous in a temporal coding scheme. We also address the influence of intrinsic properties in single-cell processing as a function of input statistics, and show that intrinsic oscillations enhance discrimination sensitivity at high input rates. Finally, we discuss how the recognition of these cell-specific discrimination properties might further our understanding of neuronal network computations and their relationships to the distribution and functional connectivity of different neuronal types.

Mode locking in a periodically forced resonate-and-fire neuron model

The resonate-and-fire (RF) model is a spiking neuron model which from a dynamical systems perspective is a piecewise smooth system (impact oscillator). We analyze the response of the RF neuron oscillator to periodic stimuli by expressing the firing events in terms of an implicit one-dimensional time map. Based on such a firing map, we describe mode-locked solutions and their stability, leading to the so-called Arnol'd tongues. The boundaries of these tongues correspond to either local bifurcations of the firing time map or grazing bifurcations of the discontinuity of the flow. Despite the fact that the periodically driven RF system shows periodic firing, its behavior may become chaotic when the forcing frequency is near the resonant frequency. We compare these results to numerical simulations of the model undergoing sinusoidal forcing. Furthermore, upon varying a system parameter, the RF system can be reduced to the integrate-and-fire system and in this case we show the consistency of the results on mode-locked solutions.

The h channel mediates location dependence and plasticity of intrinsic phase response in rat hippocampal neurons

The presence of phenomenological inductances in neuronal membrane has been known for more than one-half a century. Despite this, the dramatic contributions of such inductive elements to the amplitude and, especially, phase of neuronal impedance, and their roles in modulating temporal dynamics of neuronal responses have surprisingly remained unexplored. In this study, we demonstrate that the h channel contributes a location-dependent and plastic phenomenological inductive component to the input impedance of CA1 pyramidal neurons. Specifically, we show that the h channels introduce an apparent negative delay in the local voltage response of these neurons with respect to the injected current within the theta frequency range. The frequency range and the extent of this lead expand with increases in h current either through hyperpolarization, or with increasing distance of dendritic location from the soma. We also demonstrate that a spatially widespread increase in this inductive phase component accompanies long-term potentiation. Finally, using impedance analysis, we show that both location and activity dependence of intrinsic phase response are attributable not to changes in a capacitive or a leak component, but to changes in h-channel properties. Our results suggest that certain voltage-gated ion channels can differentially regulate internal time delays within neurons, thus providing them with an independent control mechanism in temporal coding of neuronal information. Our analyses and results also establish impedance as a powerful measure of intrinsic dynamics and excitability, given that it quantifies temporal relationships among signals and excitability as functions of input frequency.

Electricity and mechanics of biomembrane systems: Flexoelectricity in living membranes

Flexoelectricity provides a reciprocal relationship between electricity and mechanics in membranes, i.e., between membrane curvature and polarization. Experimental evidence of biomembrane flexoelectricity (including direct and converse flexoelectric effect) is reviewed. Biological implications of flexoelectricity in membrane transport, membrane contact, mechanosensitivity, electromotility and hearing are underlined. Flexoelectricity enables membrane structures to function like soft micro- and nano-machines, sensors and actuators, thus providing important input to molecular electronics applications.

From overshoot to voltage clamp

In 1939, A.L. Hodgkin and I found that the nerve action potential shows an "overshoot"--that is, the interior of the fibre becomes electrically positive during an action potential. In 1948, we did our first experiments with a voltage clamp to investigate the current-voltage relations of the nerve membrane. Between those dates, we spent much time speculating about the mechanism by which ions cross the membrane and how the action potential is generated. This article summarizes these speculations, none of which has been previously published.

Effect of lithium and sodium valproate ions on resting membrane potentials in neurons: an hypothesis

In an attempt to understand the therapeutic effects of lithium and sodium valproate in stabilizing the moods in manic depressive illness, the well-known Goldman-Hodgkin-Katz (G-H-K) equation is modified to include a fourth ion, such as a lithium ion or a sodium ion. The modified G-H-K equation is used to calculate the resting membrane potential in neurons. These calculations show that the resting membrane potential is depolarized depending upon the relative concentration of the lithium ion and upon its relative permeability. These calculations suggest that the resting membrane potential may be hyperpolarized in bipolar patients before treatment, and that the lithium ion perhaps depolarizes the resting membrane potential back to the normal level. They further support the prevailing hypothesis that manic-depressive illness may be caused by the hyperpolarization of the resting membrane potential, which, in turn, may be caused by the changes in ionic conductance (permeability) of the membranes. Sodium ions in sodium valproate do not significantly affect the resting membrane potential since they do not significantly change in the serum.

Resonance, oscillation and the intrinsic frequency preferences of neurons

The realization that different behavioural and perceptual states of the brain are associated with different brain rhythms has sparked growing interest in the oscillatory behaviours of neurons. Recent research has uncovered a close association between electrical oscillations and resonance in neurons. Resonance is an easily measurable property that describes the ability of neurons to respond selectively to inputs at preferred frequencies. A variety of ionic mechanisms support resonance and oscillation in neurons. Understanding the basic principles involved in the production of resonance allows for a simplified classification of these mechanisms. The characterization of resonance and frequency preference captures those essential properties of neurons that can serve as a substrate for coordinating network activity around a particular frequency in the brain.

Cable theory in neurons with active, linearized membranes

This investigation aims at exploring some of the functional consequences of single neurons containing active, voltage dependent channels for information processing. Assuming that the voltage change in the dendritic tree of these neurons does not exceed a few millivolts, it is possible to linearize the non-linear channel conductance. The membrane can then be described in terms of resistances, capacitances and inductances, as for instance in the small-signal analysis of the squid giant axon. Depending on the channel kinetics and the associated ionic battery the linearization yields two basic types of membrane: a membrane modeled by a collection of resistances and capacitances and membranes containing in addition to these components inductances. Under certain specified conditions the latter type of membrane gives rise to a membrane impedance that displays a prominent maximum at some nonzero resonant frequency fmax. We call this type of membrane quasi-active, setting it apart from the usual passive membrane. We study the linearized behaviour of active channels giving rise to quasi-active membranes in extended neuronal structures and consider several instances where such membranes may subserve neuronal function: 1. The resonant frequency of a quasi-active membrane increases with increasing density of active channels. This might be one of the biophysical mechanisms generating the large range over which hair cells in the vertebrate cochlea display frequency tuning. 2. The voltage recorded from a cable with a quasi-active membrane can be proportional to the temporal derivative of the injected current. 3. We modeled a highly branched dendritic tree (delta-ganglion cell of the cat retina) using a quasi-active membrane. The voltage attenuation from a given synaptic site to the soma decreases with increasing frequency up to the resonant frequency, in sharp contrast to the behaviour of passive membranes.(ABSTRACT TRUNCATED AT 400 WORDS)

Ion conductances of the surface and transverse tubular membranes of skeletal muscle

A combination voltage clamp and admittance analysis of single skeletal muscle fibers showed that moderate depolarizations activated a steady-state negative sodium conductance in both the surface and transverse tubular membranes. The density of the voltage-dependent channels was similar for the surface and tubular conductances. The relaxation times associated with the negative conductance were in the millisecond range and markedly potential dependent. The negative tubular conductance has the consequence of increasing the apparent steady-state radial space constant to large values. This occurs because the positive conductance is counterbalanced by the maintained inward-going sodium current. The enhancement of the space constant by a negative conductance provides a means for the nearly simultaneous activation of excitation-contraction coupling.

Effects of changing external potassium and chloride concentrations on the photoresponses of Bufo bufo rods

1. Intracellular responses to light were recorded from Bufo bufo rods in different ionic media. 2. The exposure of the retina to high external [K+] depolarized the rod and modified the time course of the photoresponse. The prominent initial transient of rod responses to bright flashes was drastically reduced in 5 mM-external [K+] and completely disappeared in 26 mM. In high external [K+] the kinetics of responses to dim flashes were considerably slower than in control conditions. 3. When external [Cl-] was changed from 120.6 to 10.6 mM the resting membrane potential decreased and the size of photoresponses increased. Changes in the kinetics similar to those described in high external [K+] were also observed. 4. In many cases exposure of the retina to low external [Cl-] induced oscillations of the resting membrane potential that sometimes became sustained. This instability of the membrane completely disappeared upon restoring to normal conditions. 5. The present results may be explained by assuming the existence of a voltage- and time-dependent conductance active near the dark level of membrane potential. This hypothesis can be represented by an equivalent electrical circuit that includes an inductance (Detwiler, Hodgkin & McNaughton, 1980).

Non-linear dielectric properties of nerve membranes

The unique feature of the measurements of membrane capacitance and conductance of nerve axons is the facts that these measurements must be carried out in the presence of a bias potential (-60 mV) across the membrane and also that measurements are performed in the presence of time dependent and non-linear currents. The bias potential creates an electrical field of 80 kV/cm in the membrane which produces a highly ordered structure at the resting state. Because of this, the AC field which is used for capacitance and conductance measurements should be considered a perturbation of the ordered structure rather than the force causing a transition from a random distribution to a polarized state. If the bias potential is removed by depolarization of the membrane, the ionic permeabilities of the membrane increases and Na and K ions will flow across it. Therefore, measurements of capacitance and conductance of depolarized membranes will be carried out in the presence of these ionic currents. First of all, K+ current is known to create an inductive reactance. Also Na current is believed, although still uncertain, to produce a capacitance component. Therefore, these reactances must be differentiated from the real membrane capacitance due to the dielectric properties of the components of the membrane. In order to do so, it is desirable to conduct dielectric measurements in the absence of ionic currents after application of appropriate toxins.

Impedance changes elicited by electrocutaneous stimulation

Electrical stimulation of the skin in the cavum conchae is followed by bilateral changes in the acoustic impedance of the ear due to reflex contraction of the stapedius muscle. A stimulator and a special surface electrode for electrical stimulation of the skin have been developed. The apparatus produces square-wave constant-current pulses independent of the electrical impedance of the electrode and the skin, and its hazard-proof. The effect of five independently variable stimulus parameters upon the elicited impedance changes has been examined in order to find a stimulus producing a large and reproducible impedance change without causing discomfort to the subject. Although a pronounced habituation phenomenon tends to mask the cause to effect relation of the various parameters, the results indicate that an appropriate stimulus is a pulse train of 0.5 s duration, consisting of individual pulses of a frequency of 200 Hz and 1 ms duration. The stimulus is applied at 10-second intervals.

Electrical properties of squid axon membrane. II. Effect of partial degradation by phospholipase A and pronase on electrical characteristics

Passive electrical characteristics of perfused squid axon membrane are investigated. In a previous publication, we reported that the capacitance of intact squid axon membrane is partly frequency dependent. We extended the same measurement to perfused axons. We found that the electrical characteristics of perfused axon membrane are essentially the same as those of intact axons. In this work, we investigated the effects of phospholipase A and pronase on the membrane capacitance. Phospholipase A is known to block the sodium activation and pronase to eliminate the sodium inactivation. Phospholipase A is found to increase the frequency dependent as well as the frequency independent capacitances. Our tentative conclusion is that this enzyme perturbs the lipid structure and decreases its thickness. Pronase is found to increase the frequency dependent capacitance slightly while the capacitance of the lipid layer remains unaltered. Although voltage clamp data indicate that the pronase disrupts the excitatory mechanism extensively, this enzyme has relatively little effect on the overall membrane capacitance.

Anomalous reactances in electrodiffusion systems

A frequency response analysis of a constrained diffusion boundary has been made by linearizing the Nernst-Planck equations for a small applied AC current. The number of time constants and their dependence on ionic concentrations and electric field as well as membrane parameters such as dielectric constant, thickness, etc. have been evaluated by this method. Numerical solutions have been carried out for cases when the Planck charging time can be neglected and the results are presented in the form of impedance loci. These impedance loci show that if the membrane separates two univalent electrolytes with a common anion it will exhibit a combined capacitative inductive response with a 90 degrees phase angle. The dependence of these anomalous reactances on ionic concentrations and the electric field is consistent with the behavior of the Hodgkin-Huxley axon suggesting that a homogeneous electrodiffusion regime could be adequate as a basic model for the kinetic behavior of biological membranes.

Anode break excitation in space-clamped squid axons

Strength-duration curves for space-clamped squid axons, using square wave anode breaks as stimuli, established the existence of four distinct regions. For the average experimental axon the intersection of the first two regions, tau(1), occurs at about 7 msec. This agrees with computations based on the Hodgkin-Huxley (HH) equations and corresponds to the accommodation time constant found previously for a linearly rising ramp, as given by the HH equations and as found experimentally. The second break in the curve, tau(2), at about 200 msec, and the third break, tau(3), at 1 sec, are far beyond the range of the HH equations and may be the counterpart in the excitability of the long time constants, which have been apparent from a number of other types of experiments. The regions of the curve before 1 msec and beyond 2 or 3 sec are quite variable and may represent breakdown. Rheobase increases in both experimental and computed axons when temperature is raised. In both experimental and computed axons tau(1) descreases slightly when the temperature is raised from 10 to 15 degrees C. At 20 and 25 degrees C, tau(1) of the experimental axon increases markedly.