Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents

Many neuron types exhibit preferred frequency responses in their voltage amplitude (resonance) or phase shift to subthreshold oscillatory currents, but the effect of biophysical parameters on these properties is not well understood. We propose a general framework to analyze the role of different ionic currents and their interactions in shaping the properties of impedance amplitude and phase in linearized biophysical models and demonstrate this approach in a two-dimensional linear model with two effective conductances g _L and g ₁. We compute the key attributes of impedance and phase (resonance frequency and amplitude, zero-phase frequency, selectivity, etc.) in the g _L ???g ₁ parameter space. Using these attribute diagrams we identify two basic mechanisms for the generation of resonance: an increase in the resonance amplitude as g ₁ increases while the overall impedance is decreased, and an increase in the maximal impedance, without any change in the input resistance, as the ionic current time constant increases. We use the attribute diagrams to analyze resonance and phase of the linearization of two biophysical models that include resonant (I _h or slow potassium) and amplifying currents (persistent sodium). In the absence of amplifying currents, the two models behave similarly as the conductances of the resonant currents is increased whereas, with the amplifying current present, the two models have qualitatively opposite responses. This work provides a general method for decoding the effect of biophysical parameters on linear membrane resonance and phase by tracking trajectories, parametrized by the relevant biophysical parameter, in pre-constructed attribute diagrams.     

White noise measurement of squid axon membrane impedance.

The use of white noise techniques for system identification is illustrated by the following characterization of the subthreshold membrane impedance of the squid giant axon, space-clamped in a double sucrose gap. Power spectra were also computed. Depolarization increases the resonance, shifts the resonant frequently upward and decreases the membrane's inductive reactance. Reduced external Ca⁺⁺ increases the resonance, shifts the resonant frequency downward and increases the inductive reactance.     

Resonance modulation,annihilation and generation of anti-resonance and anti-phasonance in 3D neuronal systems: interplay of resonant and amplifying currents with slow dynamics

Subthreshold (membrane potential) resonance and phasonance (preferred amplitude and zero-phase responses to oscillatory inputs) in single neurons arise from the interaction between positive and negative feedback effects provided by relatively fast amplifying currents and slower resonant currents. In 2D neuronal systems, amplifying currents are required to be slave to voltage (instantaneously fast) for these phenomena to occur. In higher dimensional systems, additional currents operating at various effective time scales may modulate and annihilate existing resonances and generate antiresonance (minimum amplitude response) and antiphasonance (zero-phase response with phase monotonic properties opposite to phasonance). We use mathematical modeling, numerical simulations and dynamical systems tools to investigate the mechanisms underlying these phenomena in 3D linear models, which are obtained as the linearization of biophysical (conductance-based) models. We characterize the parameter regimes for which the system exhibits the various types of behavior mentioned above in the rather general case in which the underlying 2D system exhibits resonance. We consider two cases: (i) the interplay of two resonant gating variables, and (ii) the interplay of one resonant and one amplifying gating variables. Increasing levels of an amplifying current cause (i) a response amplification if the amplifying current is faster than the resonant current, (ii) resonance and phasonance attenuation and annihilation if the amplifying and resonant currents have identical dynamics, and (iii) antiresonance and antiphasonance if the amplifying current is slower than the resonant current. We investigate the underlying mechanisms by extending the envelope-plane diagram approach developed in previous work (for 2D systems) to three dimensions to include the additional gating variable, and constructing the corresponding envelope curves in these envelope-space diagrams. We find that antiresonance and antiphasonance emerge as the result of an asymptotic boundary layer problem in the frequency domain created by the different balances between the intrinsic time constants of the cell and the input frequency f as it changes. For large enough values of f the envelope curves are quasi-2D and the impedance profile decreases with the input frequency. In contrast, for f ? 1 the dynamics are quasi-1D and the impedance profile increases above the limiting value in the other regime. Antiresonance is created because the continuity of the solution requires the impedance profile to connect the portions belonging to the two regimes. If in doing so the phase profile crosses the zero value, then antiphasonance is also generated.     

Electrical Resonance in the θ Frequency Range in Olfactory Amygdala Neurons

The cortical amygdala receives direct olfactory inputs and is thought to participate in processing and learning of biologically relevant olfactory cues. As for other brain structures implicated in learning, the principal neurons of the anterior cortical nucleus (ACo) exhibit intrinsic subthreshold membrane potential oscillations in the θ-frequency range. Here we show that nearly 50% of ACo layer II neurons also display electrical resonance, consisting of selective responsiveness to stimuli of a preferential frequency (2–6 Hz). Their impedance profile resembles an electrical band-pass filter with a peak at the preferred frequency, in contrast to the low-pass filter properties of other neurons. Most ACo resonant neurons displayed frequency preference along the whole subthreshold voltage range. We used pharmacological tools to identify the voltage-dependent conductances implicated in resonance. A hyperpolarization-activated cationic current depending on HCN channels underlies resonance at resting and hyperpolarized potentials; notably, this current also participates in resonance at depolarized subthreshold voltages. KV7/KCNQ K⁺ channels also contribute to resonant behavior at depolarized potentials, but not in all resonant cells. Moreover, resonance was strongly attenuated after blockade of voltage-dependent persistent Na⁺ channels, suggesting an amplifying role. Remarkably, resonant neurons presented a higher firing probability for stimuli of the preferred frequency. To fully understand the mechanisms underlying resonance in these neurons, we developed a comprehensive conductance-based model including the aforementioned and leak conductances, as well as Hodgkin and Huxley-type channels. The model reproduces the resonant impedance profile and our pharmacological results, allowing a quantitative evaluation of the contribution of each conductance to resonance. It also replicates selective spiking at the resonant frequency and allows a prediction of the temperature-dependent shift in resonance frequency. Our results provide a complete characterization of the resonant behavior of olfactory amygdala neurons and shed light on a putative mechanism for network activity coordination in the intact 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 f _max. 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 (-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. This might be the underlying biophysical mechanism of receptive fields whose dimensions are large for rapid signals but contract to a smaller area for slow signals as suggested by Detwiler et al. (1978).     

Specific binding of peanut agglutinin to GM1-doped solid supported lipid bilayers investigated by shear wave resonator measurements

This study deals with the specific interaction between the lectin peanut agglutinin (PNA) from Arachis hypogaea and the ganglioside G_M1 which was incorporated in a solid supported lipid bilayer immobilized on a gold electrode placed on top of an AT-cut quartz crystal. Bilayer formation was reached by self-assembly processes. The first monolayer consists of octanethiol attached to the gold surface via chemisorption and the second monolayer was immobilized by vesicle fusion on the preformed hydrophobic surface. We managed to keep unspecific binding to a minimum by using a phospholipid matrix consisting of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC). Lectin binding to ganglioside G_M1 containing membranes was determined by a decrease of the resonant frequency of the quartz crystal. The minimum amount of receptor within the membrane which is necessary to obtain a complete protein monolayer was found to be less than 2 mol%. The adsorption isotherm of PNA to G_M1 was recorded and analyzed to be of Langmuir type, exhibiting a binding constant of PNA to the ganglioside of 8.3 ⋅ 10⁵ M^–1. The good agreement of the calculated Langmuir adsorption isotherm with the obtained experimental data implies that protein multilayers are not formed and that interactions between the adsorbents can be neglected. Furthermore, the association constants of two different saccharides, β-Galp-(1 → 3)-GalNAc exhibiting a strong binding to PNA in solution, and β-D-galactose with a much lower affinity were estimated by determining the equilibrium concentration of PNA attached to the surface. Moreover we were able to remove the attached lectin monolayer by digestion of the protein with pronase causing an increase in the resonant frequency which almost reversed the frequency shift to lower frequencies during adsorption. An even more complex system was built up by the use of digoxigenin-labeled PNA which also binds to the solid supported membrane containing the receptor G_M1. The immobilized lectin was recognized by anti-digoxigenin-F_ab-fragments, which is measurable by a further decrease of the resonant frequency. For all binding processes we found larger frequency shifts for a complete protein monolayer than predicted by Sauerbrey's equation, clearly showing that in addition to mass loading viscoelastic changes occur at the lipid-protein interface. Received: 22 July 1996 / Accepted: 12 September 1996     

A Long-Term Blockade of L-Type Calcium Currents Upregulates the Number of Ca2+ Channels in Skeletal Muscle

The effects of a long-term blockade of L-type Ca²⁺ channels on membrane currents and on the number of dihydropyridine binding sites were investigated in skeletal muscle fibers. Ca²⁺ currents (I _Ca) and intramembrane charge movement were monitored using a voltage-clamp technique. The peak amplitude of I _Ca increased by more than 40% in fibers that were previously incubated for 24 hr in solutions containing the organic Ca²⁺ channel blocker nifedipine or in Ca²⁺-free conditions. A similar incubation period with Cd²⁺, an inorganic blocker, produced a moderate increase of 20% in peak I _Ca. The maximum mobilized charge (Q _max) increased by 50% in fibers preincubated in Ca²⁺-free solutions or in the presence of Cd²⁺. Microsomal preparations from frog skeletal muscle were isolated by differential centrifugation. Preincubation with Cd²⁺ prior to the isolation of the microsomal fraction doubled the number of ³H-PN200-110 binding sites and produced a similar increase in the values of the dissociation constant. The increase in the number of binding sites is consistent with the increase in the peak amplitude of I _Ca as well as with the increase in Q _max. Received: 31 August 1998/Revised: 7 December 1998     

Mechanisms of postinhibitory rebound and its modulation by serotonin in excitatory swim motor neurons of the medicinal leech

Postinhibitory rebound (PIR) is defined as membrane depolarization occurring at the offset of a hyperpolarizing stimulus and is one of several intrinsic properties that may promote rhythmic electrical activity. PIR can be produced by several mechanisms including hyperpolarization-activated cation current (I_h) or deinactivation of depolarization-activated inward currents. Excitatory swim motor neurons in the leech exhibit PIR in response to injected current pulses or inhibitory synaptic input. Serotonin, a potent modulator of leech swimming behavior, increases the peak amplitude of PIR and decreases its duration, effects consistent with supporting rhythmic activity. In this study, we performed current clamp experiments on dorsal excitatory cell 3 (DE-3) and ventral excitatory cell 4 (VE-4). We found a significant difference in the shape of PIR responses expressed by these two cell types in normal saline, with DE-3 exhibiting a larger prolonged component. Exposing motor neurons to serotonin eliminated this difference. Cs⁺ had no effect on PIR, suggesting that I_h plays no role. PIR was suppressed completely when low Na⁺ solution was combined with Ca²⁺ -channel blockers. Our data support the hypothesis that PIR in swim motor neurons is produced by a combination of low-threshold Na⁺ and Ca²⁺ currents that begin to activate near –60 mV.     

Two mechanisms of calcium oscillations in adipocytes

In non-excitable cells, several kinds of agonist-induced oscillations of cytosolic Ca²⁺ concentration ([Ca²⁺]_i) are known which differ in their form and generation mechanism. The oscillation source is, as a rule, the regulation of Ca²⁺ mobilization from intracellular stores through inositol 1,4,5-trisphosphate (IP₃) receptors (IP₃R) and in some cases through ryanodine receptors (RyR). In the present work, oscillations in single mature adipocytes of mice epididymal fat on the ninth day of cultivation are studied. Cells were stimulated by acetylcholine (ACh) or by fetal bovine serum (FBS). ACh at a concentration of 0.1–5 μM evoked a rise in [Ca²⁺]_i to a peak and subsequent oscillations whose peaks and troughs declined along with increasing amplitude while frequency decreased. In most cells oscillations lasted less than 5 min. The new constant or interspike level exceeded the initial one or was equal to it (at 1 μM ACh). The removal of ACh stopped oscillations immediately. An inhibitor of phospholipase C (U73122) or of IP₃R (Xestospongin C) did not affect the pattern of responses, which means that the generation of oscillations does not depend on IP₃. At the same time, suppression of responses by ryanodine, which blocks RyR, was observed. Besides, oscillatory responses were abolished by inhibitors of phosphatidylinositol 3-kinase, NO synthase, and cGMP-dependent protein kinase. FBS (1%) initiated oscillations characterized by return of [Ca²⁺]_i after each peak to the baseline level, occurring prior to stimulation, and by maintenance of roughly constant amplitude and frequency (of the order of 1 min⁻¹). Oscillations persisted longer (more than 15 min in 87% of cells) than with ACh. Repeated stimulation of cells by FBS revealed a strongly reduced sensitivity after 1 h of rest, whereas responses to ACh partially restored within 3 min. Investigation of the involvement of IP₃R and RyR in FBS-induced oscillations gave completely inverse results relative to ACh and demonstrated a leading role of IP₃R without a considerable contribution of RyR and of its activation pathways. With both stimuli, Ca²⁺ entry through the plasma membrane was necessary only as a support of oscillations. The results show that in adipocytes different agonists can engage distinct subsystems of Ca²⁺ signaling, each of them generating oscillations with a specific temporal pattern.     

Genetic and physiological evidence concerning the development of chemically sensitive voltage-dependent ionophores in L6 cells

The electrophysiological properties of a tissue culture muscle line, L₆, and a K⁺ resistant mutant (MK₁) derived from L₆ were determined to elucidate certain aspects of membrane differentiation and function. MK₁ was selected as a clone of myoblasts resistant to the toxic effects of 55 mM K⁺. The resting potentials of L₆ and MK₁ myoblasts and myotubes were K⁺ dependent and equal. The amplitudes of the action potentials were equal in normal medium, but 27.7 mM K⁺ interfered with or eliminated the ability of L₆ myotubes to produce action potentials. MK₁ myotubes produced nearly normal action potentials under these conditions. Thus, the K⁺ resistant myoblasts differentiate into myotubes which have an action potential generating mechanism much less sensitive to K⁺ than the normal mechanism. Also, both d-tubocurarine and α-bungarotoxin enhance the amplitude of the action potentials produced by L₆ myotubes in the presence of 27.7 mM K⁺; these compounds do not enhance the amplitude of the action potentials produced by MK₁ myotubes under the same conditions. It is proposed that as a consequence of differentiation a type of ionophore present in myoblasts becomes a voltage-dependent ionophore in myotubes. Furthermore, these voltage-dependent ionophores can be chemically sensitive.     

Frequency dependence of the frog skin impedance

At frequencies between 20 Hz and 1 kHz the impedance locus of the isolated frog skin is circular; below 20 Hz the resistive component of the impedance is frequently greater than would be expected from extrapolation of the high-frequency locus. At frequencies greater than twice the highest frequency at which there are deviations from the circular locus the variation of impedance Z with angular frequency ω is closely described by the equation Z = r₁ + r₀/[1+(jωτ)^1-α], where j is √?1, r¹ and r⁰ are resistance, τ is a time constant and α a constant in the range 0.02–0.14.     

Nerve membrane electrical characteristics near the resting state

Summary An experimental analysis of the squid axon membrane impedance in the vicinity of the resting state and as a function of frequency is presented. Particular attention was devoted to the measurement of theresonance frequency, for which the absolute magnitude of the impedance attains its maximum value, in different, extracellular solutions, at various temperatures and in the presence of constant depolarizations or hyperpolarizations.The variations in the concentration of sodium, potassium and divalent ions and the addition of tetrodotoxin, changed markedly the maximum impedance but had little effect, at a fixed temperature, on the resonance frequency, whose temperature dependance is described by aQ ₁₀ variable from 3.7 (around 4 °C) to 1.9 (around 15 °C). Substitution of heavy water decreased the resonance frequency by a factor 1.25, fairly independent of temperature. Steady depolarizations or hyperpolarizations produced large variations of the resonance frequency, with strong temperature dependance.The results indicate that the resonance frequency is directly related to the membrane permeability changes which take place quite independently of the composition of the extra cellular solution and are governed by the electric field existing within the membrane structure rather than by the total membrane potential, to which membrane-solution boundary potentials can give a large contribution.     

Auditory Sensitivity of the Crickets <Emphasis Type="Italic">Teleogryllus commodus</Emphasis> and <Emphasis Type="Italic">T. oceanicus</Emphasis>

WHEN Johnstone, Saunders and Johnstone¹ measured the vibrational amplitude of the auditory tympana of the crickets Teleogryllus commodus and T. oceanicus in response to sound stimuli, they obtained curves with a sharp resonant peak at 5.0 kHz for both species (although in their text they described this resonant peak as occurring at 4.0 kHz¹). They suggested that tympanic resonance is wholly responsible for the sharp tuning characteristics of primary auditory fibres in orthopterans^2,3 and that “the functional significance of this tuning may be associated with the fact that the maximum energy in the (orthopteran) mating call lies in the 4.5–5.5 kHz range”¹.     

Change in Ca<Superscript>2+</Superscript>-responses of cultivated brown adipocytes at adrenergic activation

The thermogenic capability of brown adipose tissue is controlled by noradrenaline. By interacting with α1- and β-adrenoreceptors of adipocytes, noradrenaline (NA) increases the intracellular concentration of Ca²⁺ ([Ca²⁺]_i) and cAMP. The changes in [Ca²⁺]_i under the action of NA and selective agonists of α1- and β-adrenoreceptors, i.e., cirazoline and isoproterenol (IP), are recorded on individual cells of the primary culture of adipocytes during the day in vitro (DIV) 1, DIV 3, and DIV 6. The change in [Ca²⁺]_i under the effect of IP as compared to the response to cirazoline in cells of DIV 1 is characterized by a higher amplitude and shorter duration of impulses in the entire diapason of the used physiological concentrations. After DIV 3, these differences are insignificant and, after DIV 6, the differences in kinetics are nearly absent. For all three agonists, the kinetics of the [Ca²⁺]_i change in the proliferating and differentiated cells is significantly different; i.e., the response amplitude increases with the age of the culture and the duration of transitory response decreases, while sensitivity to agonists of adrenoreceptors increases. It can be seen from the rise in [Ca²⁺]_i with an inhibitor of Ca²⁺-ATPase of the endoplasmic reticulum thapsigargin in calcium-free medium that the source of calcium ions in the endoplasmic reticulum rises with the growth and development of cells in culture, while the rate at which Ca²⁺ is pumped out of cells, which characterizes the activity of Ca²⁺-ATPase of the plasma membrane, increases.     

The influence of extracellular calcium on the response of fly photoreceptors

Summary This paper presents a systematic investigation of the influence of the extracellular concentration of calcium ([Ca²⁺]₀) on the electrophysiological response of the fly's photoreceptors (R1–R6) to light. The hemisected heads of flies were perfused with a standard medium containing 10^–4 mol/1 CaCl₂ and in this medium the intracellularly recorded response of the cell was virtually identical to the normal response obtained in vivo. All the effects of changing the [Ca²⁺]₀ could be reversed within 5 min by perfusing the eye with the standard medium.Changing the [Ca²⁺]₀ did not influence the frequency with which quantum bumps occurred or the resting membrane potential, but did lead to changes in the latency and amplitude of the response and, most significantly, in the repolarization time (t _r). The plot oft _r versus the [Ca²⁺]₀ revealed that the value oft _r changes significantly in two distinct regions representing a [Ca²⁺]₀ of between 2×10^–8 and 10^–7 mol/l and 10^–4 and 10^–2 mol/l, respectively. Lowering the [Ca²⁺]₀ did not affect the amplitude of the response but did lead to a drastic increase int _r which was accompanied by an increase in latency and peak time. Raising the [Ca²⁺]₀ led to a reduction in the duration and amplitude of the response. The latter effect is evidence of reduction in the sensitivity of the photoreceptor cell which is dependent on the [Ca²⁺]₀.It is postulated that two types of binding site for calcium exist, high affinity binding sites (HABS) and low affinity binding sites (LABS), which modulate the functioning of ion channels in the cell membrane that are activated as a consequence of light absorption. The results indicate that the sensitivity of the photoreceptor cell is determined by the degree of saturation of the LABS.     

The electrical changes accompanying fertilization and cortical vesicle secretion in the medaka egg

The electrical properties of the egg of the medaka, Oryzias latipes, were studied before, during, and after fertilization. The resting potential of the unfertilized egg averaged ?39 ± 9 mV in Yamamoto's Ringers (Y. Ringers), but 20% of the values were between ?50 and ?60 mV. Fertilization triggers a small depolarization of 4 ± 3 mV in 10% Y. Ringers with an average duration of 20 ± 10 sec. The amplitude of this depolarization is independent of [Na⁺]_o, [Ca²⁺]_o, and [Cl^?]_o, so it appears to be due to a nonspecific leak triggered by sperm-egg fusion. The depolarization is followed by a longer hyperpolarizing phase with an average amplitude of 31 ± 12 mV. Recovery from this hyperpolarization has a fast phase lasting 155 ± 18 sec, followed by a slower phase which reaches a steady average membrane potential of ?19 ± 1 mV by 9 min after fertilization. The membrane resistance falls 10-fold during the first 2 min after fertilization, from 40 (1520 kΩ-cm²) to 3 MΩ. This is largely due to an increase in the K⁺ conductance. At the peak of the hyperpolarization, the membrane potential exhibits a 28 mV/decade [K⁺]_o dependence and a 6 mV/decade [Na⁺]_o dependence. The membrane resistance slowly recovers over the next 8 min to a value about 30% larger than before fertilization. The relation of current vs voltage was linear before, during, and after fertilization and indicated a reversal potential of ?98 ± 20 mV for the hyperpolarization peak. The egg's capacitance averaged 0.04 ± 0.01 μF (0.9 μF/cm²) before fertilization and approximately doubles within 90 sec after fertilization. It then decreases over a 9-min period, reaching a value 25% smaller than before fertilization.