Direction reversal of fluctuation-induced biased Brownian motion on distorted ratchets.

The biased movement of Brownian particles on a fluctuating two-state periodic potential made of identical distorted ratchets is studied. The purpose is to investigate how the direction of the particle movement is related to the asymmetry of the potential. In general, distorting one of the two linear arms of a regular symmetric ratchet (with equal arm lengths) can create a driving force for the Brownian particle to execute biased movement. The direction of the induced biased movement depends on the type of the distortion. It has been found that if one linear arm is kinked into two linear sub-arms, the direction of the movement can be either positive or negative depending on the frequency of the fluctuation and the location and the degree of the kink. In contrast, if one arm of the symmetric ratchet is replaced by a continuous nonlinear sinusoidal function, the movement is always unidirectional. Thus, for the latter case to generate the direction reversal phenomenon, the ratchets have to have an additional asymmetry. We also have found that two potentials with different distorted ratchets can generate identical fluxes if the distortions are polar symmetric about the mid-point of the arm(s) of the basic linear two-arm ratchet. The results are useful for designing experimental apparatuses for the separation of protein particles based on their sizes and charges and the viscosity of the medium.

A model of beta-cell mass, insulin, and glucose kinetics: pathways to diabetes.

Diabetes is a disease of the glucose regulatory system that is associated with increased morbidity and early mortality. The primary variables of this system are beta-cell mass, plasma insulin concentrations, and plasma glucose concentrations. Existing mathematical models of glucose regulation incorporate only glucose and/or insulin dynamics. Here we develop a novel model of beta -cell mass, insulin, and glucose dynamics, which consists of a system of three nonlinear ordinary differential equations, where glucose and insulin dynamics are fast relative to beta-cell mass dynamics. For normal parameter values, the model has two stable fixed points (representing physiological and pathological steady states), separated on a slow manifold by a saddle point. Mild hyperglycemia leads to the growth of the beta -cell mass (negative feedback) while extreme hyperglycemia leads to the reduction of the beta-cell mass (positive feedback). The model predicts that there are three pathways in prolonged hyperglycemia: (1) the physiological fixed point can be shifted to a hyperglycemic level (regulated hyperglycemia), (2) the physiological and saddle points can be eliminated (bifurcation), and (3) progressive defects in glucose and/or insulin dynamics can drive glucose levels up at a rate faster than the adaptation of the beta -cell mass which can drive glucose levels down (dynamical hyperglycemia).

Spontaneous secondary spiking in excitable cells.

Kepler & Marder (1993, Biol. Cybern.68, 209-214) proposed a model describing the electrical activity of a crab neuron in which a train of directly induced action potentials is sometimes followed by one or more spontaneous action potentials, referred to as spontaneous secondary spikes. We reduce their five-dimensional model to three dimensions in two different ways in order to gain insight into the mechanism underlying the spontaneous spikes. We then treat a slowly varying current as a parameter in order to give a qualitative explanation of the phenomenon using phase-plane and bifurcation analysis. We demonstrate that a three-dimensional model, consisting of a two-dimensional excitable system plus a slow inward current, is sufficient to produce the behaviour observed in the original model. The exact dynamics of the excitable system are not important, but the relative time constant and amplitude of the slow inward current are crucial. Using the numerical bifurcation analysis package AUTO (Doedel & Kernevez, 1986, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology), we compute bifurcation diagrams using the maximum amplitude of the slow inward current as the bifurcation parameter. The full and reduced models have a stable resting potential for all values of the bifurcation parameter. At a critical value of the bifurcation parameter, a stable tonic firing mode arises via a saddle-node of periodics bifurcation. Whether or not the models can exhibit transient or continuous spontaneous spiking depends on their position in parameter space relative to this saddle-node of periodics.

Subthreshold membrane resonance in neocortical neurons.

1. Using whole cell recording techniques, we studied subthreshold and suprathreshold voltage responses to oscillatory current inputs in neurons from the sensorimotor cortex of juvenile rats. 2. Based on firing patterns, neurons were classified as regular spiking (RS), intrinsic bursting (IB), and fast spiking (FS). The subthreshold voltage-current relationships of RS and IB neurons were rectifying whereas FS neurons were almost ohmic near rest. 3. Frequency response curves (FRCs) for neurons were determined by analyzing the frequency content of inputs and outputs. The FRCs of most neurons were voltage dependent at frequencies below, but not above, 20 Hz. Approximately 60% of RS and IB neurons had a membrane resonance at their resting potential. Resonant frequencies were between 0.7 and 2.5 Hz (24-26 degrees C) near -70 mV and usually increased with hyperpolarization and decreased with depolarization. The remaining RS and IB neurons and all FS neurons were nonresonant. 4. Resonant neurons near rest had a selective coupling between oscillatory inputs and firing. These neurons selectively fired action potentials when the frequency of the swept-sine-wave (ZAP) current input was near the resonant frequency. However, when these neurons were depolarized to -60 mV, spike firing was associated with many input frequencies rather than selectively near the resonant frequency. 5. We examined three subthreshold currents that could cause low-frequency resonance: IH, a slow, hyperpolarization-activated cation current that was blocked by external Cs+ but not Ba2+; IIR, an instantaneously activating, inwardly rectifying K+ current that was blocked by both Cs+ and Ba2+; and INaP, an quickly activating, inwardly rectifying persistent Na+ current that was blocked by tetrodotoxin (TTX). Voltage-clamp experiments defined the relative steady state activation ranges of these currents. IIR (activates below -80 mV) and INaP (activates above -65 mV) are unlikely to interact with each other because their activation ranges never overlap. However, both currents may interact with IH, which activated variably at potentials between -50 and -90 mV in different neurons. 6. We found that IH produces subthreshold response. Consistent with this, subthreshold resonance was blocked by external Cs+ but not Ba2+ or TTX. Application of Ba2+ enlarged FRCs and resonance at potentials below -80 mV, indicating that IK,ir normally attenuates resonance. Application of TTX greatly diminished resonance at potentials more depolarized than -65 mV, indicating that INaP normally amplifies resonance at these potentials. 7. The ZAP current input may be viewed as a model of oscillatory currents that arise in neocortical neurons during synchronized activity in the brain. We propose that the frequency selectivity endowed on neurons by IH may contribute to their participation in synchronized firing. The voltage dependence of the frequency-selective coupling between oscillatory inputs and spikes may indicate a novel mechanism for controlling the extent of low-frequency synchronized activity in the neocortex.

Models of subthreshold membrane resonance in neocortical neurons.

1. We obtained whole cell data from sensorimotor cortical neurons of in vitro slices (juvenile rats) and observed a low-frequency resonance (1-2 Hz) in their voltage responses. We constructed models of subthreshold membrane currents to determine whether a hyperpolarization-activated cation current (IH) is sufficient to account for this resonance. 2. Parameter values for a basic IH (BH) model were estimated from voltage-clamp experiments at room temperature. The BH model formed a component of a reduced membrane (RM) model. On numerical integration, the RM model exhibited voltage sags and rebounds to injected current pulses; maximal voltage responses to injected oscillatory currents occurred near 2 Hz. 3. We compared the experimentally measured frequency-response curves (FRCs) at room temperature with the theoretical FRCs derived from the RM model. The theoretical FRCs exhibited resonant humps with peaks between 1 and 2 Hz. At 36 degrees C, the theoretical FRCs peaked near 10 Hz. The characteristics of theoretical and observed FRCs were in close agreement, demonstrating that IH is sufficient to cause resonance. Thus we classified IH as a resonator current. 4. We developed a technique, the reactive current clamp (RCC), to inject a computer-calculated current corresponding to a membrane ionic current in response to the membrane potential of the neuron. This enabled us to study the interaction of an artificial ionic current with living neurons (electronic pharmacology or EP-method). Using the RCC, a simplified version of the BH model was used to cancel an endogenous IH (electronic antagonism) and to introduce an artificial IH (electronic expression) when an endogenous IH was absent. Antagonism of IH eliminated sags and rebounds, whereas expression of IH endowed neurons with resonance and the frequency-selective firing that accompanies resonance in neurons with an endogenous IH. Previous investigations have relied on the specificity of pharmacological agents to block ionic channels, e.g., Cs+ to block IH. However, Cs+ additionally affects other currents. This represents the first time an in vitro modeling technique (RCC) has been used to antagonize a specific endogenous current, IH. 5. A simplified RM model yielded values of the resonant frequency and Q (an index of the sharpness of resonance), which rose almost linearly between -55 and -80 mV. Resonant frequencies could be much higher than fH = (2 pi tau H) - 1 where tau H is the activation time constant for IH. 6. In the FRCs, resonance appeared as a hump at intermediate frequencies because of low- and high-frequency attenuations due to IH and membrane capacitance, respectively. Changing the parameters of IH altered the low-frequency attenuation and, hence, the resonance. Changes in the leak conductance affected both the low- and high-frequency attenuations. 7. We modeled an inwardly rectifying K+ current (IIR) and a persistent Na+ current (INaP) to study their effects on resonance. Neither current produced resonance in the absence of IH. We found that IIR attenuated, whereas INaP amplified resonance. Thus IIR and INaP are classified as attenuator and amplifier currents, respectively. 8. Resonators and attenuators differ in that they have longer and shorter time constants, respectively, compared with the membrane time constant. Therefore, an increase in the leak conductance decreases the membrane time constant, which can transform an attenuator into a resonator, altering the frequency response. This suggests a novel mechanism for modulating the frequency responses of neurons to inputs. 9. These investigations have provided a theoretical framework for detailed understanding of mechanisms that produce resonance in cortical neurons. Resonance is one aspect of the intrinsic rhythmicity of neurons. The rhythmicity due to IH resonance is latent until it is revealed by oscillatory inputs. (ABSTRACT TRUNCATED)

Correlations of rates of insulin release from islets and plateau fractions for beta-cells.

Pancreatic beta-cells in intact islets of Langerhans perfused with various glucose concentrations exhibit periodic bursting electrical activity (BEA) consisting of active and silent phases. The fraction of the time spent in the active phase is called the plateau fraction and appears to be strongly correlated with the rate of release of insulin from islets as glucose concentration is varied. Here this correlation is quantified and a theoretical development is presented in detail. Experimental rates of insulin release are correlated with "effective" plateau fractions over a range of glucose concentrations. There are a number of different models for BEA in pancreatic beta-cells and a method is developed here to quantify the dependence of a glucose dependent parameter on glucose concentration. As an example, the plateau fractions computed from the Sherman-Rinzel-Keizer model are matched with experimental plateau fractions to obtain a relationship between the model's glucose-dependent parameter, beta, and glucose concentration. Knowledge of the relationships between beta and glucose concentration and between experimental measurements of rates of insulin release and plateau fractions permits the determination of theoretical rates of insulin release from the model.

Low-threshold calcium current and resonance in thalamic neurons: a model of frequency preference.

1. We constructed a mathematical model of the subthreshold electrical behavior of neurons in the nucleus mediodorsalis thalami (MDT) to elucidate the basis of a Ni(2+)-sensitive low-frequency (2-4 Hz) resonance found previously in these neurons. 2. A model that included the low- and high-threshold Ca2+ currents (IT and IL), a Ca(2+)-activated K+ current (IC), a rapidly inactivating K+ current (IA), a voltage-dependent K+ current which we call IKx, and a voltage-independent leak current (Il), successfully simulated the low-threshold spike observed in MDT neurons. This model (the MDT model) and a minimal version of the model containing only IT and I1 (the minimal MDT model) were used in the analysis. 3. An impedance function was derived for a linearized version of the MDT model. This showed that the model predicts a low-frequency (2-4 Hz) resonance in the voltage response to "small" oscillatory current inputs (producing voltage changes of < 10 mV) when the membrane potential is between -60 and -85 mV. 4. Further examination of the impedances for the MDT and minimal MDT models shows that IT underlies the frequency- and voltage-dependent resonance. The slow inactivation of IT results in an attenuation of voltage responses to low frequencies, resulting in a band-pass behavior. The fast activation of IT amplifies the resonance and modulates the peak frequency but does not, in itself, cause resonance. 5. When voltage responses are small (< 10 mV), the strength and voltage-dependence of resonance of the minimal MDT model are determined by the steady-state window conductance, gw, due to IT. This steady-state conductance arises where the steady-state activation, m(infinity2)(V), and inactivation, h(infinity) (V), curves overlap. Parallel shifts in the inactivation curve can eliminate or enhance resonance with little effect on the IT-dependent low-threshold spike evoked after hyperpolarizing current pulses. When the peak magnitude of gw was large, the minimal MDT model showed spontaneous oscillations at 3 Hz with amplitudes > 30 mV. 6. Large oscillatory current inputs evoked significantly nonlinear voltage responses in the minimal MDT model, but the 2- to 4-Hz frequency selectivity (predicted from the linearized impedance) remained. 7. We conclude that the properties of the low-threshold Ca2+ current, IT, are sufficient to explain the Ni(2+)-sensitive 2- to 4-Hz resonance seen in MDT neurons.(ABSTRACT TRUNCATED AT 400 WORDS)

Consequences of 4-aminopyridine applications to trigeminal root ganglion neurons.

1. The effects of 4-aminopyridine (4-AP) on the electrical properties of 30 trigeminal root ganglion (TRG) neurons were determined from the membrane voltage responses to step and sinusoidal current injections using intracellular microelectrode techniques in in vitro slice preparations (guinea pigs). 2. Comparisons of results from 4-AP applications (0.05-5 mM) with those from tetraethylammonium (TEA) applications (0.1-10 mM) revealed very different actions of these agents. Both agents produced an increase in input resistance and a decrease in threshold for spike generation. Applications of 4-AP increased subthreshold oscillations of the membrane potential and enhanced the repetitive spike firing evoked by intracellular injections of current pulses. However, TEA applications blocked the potential oscillations and did not exaggerate repetitive spike discharges. Spontaneous spike activity or bursts were observed in four neurons that received 4-AP applications. 3. Membrane properties were determined in 20 of the 30 neurons by fitting impedance data in the frequency domain with a four-parameter membrane model by the use of computer-intensive techniques. In the majority of neurons, the time-invariant and time-dependent membrane conductances decreased during 4-AP application. The time constant for the time-dependent conductance also decreased, suggesting that the closing of K+-channels was facilitated in the membrane. 4. Applications of 4-AP in a dose range of 50 microM-5 mM produced rapid (approximately tens of seconds) responses of the neurons, resulting in a dose-dependent increase of the impedance magnitude functions and in a leftward shift of the resonant "humps" to lower frequencies. This shift indicates that the TRG neuronal membrane is capable of producing large voltage responses to current inputs at low frequencies. Recovery from the effects of 4-AP was slow (usually greater than 30 min). 5. Applications of 4-AP at high doses (greater than or equal to 1 mM) and at various imposed membrane potentials in four neurons resulted in poorly reversible unspecific changes in certain membrane parameters (increased input capacitance and conductance) and an insensitivity of the input conductance to the imposed membrane potential. These effects could be interpreted as membrane breakdown. 6. The tendencies of TRG neurons to fire repetitively and in bursts of spikes during 4-AP application result from the increased oscillatory behavior of their membrane potentials and changes in membrane resonance induced by presumed blockade of K+ channels.(ABSTRACT TRUNCATED AT 400 WORDS)

Voltage dependence of membrane properties of trigeminal root ganglion neurons.

1. Membrane potentials of trigeminal root ganglion neurons were varied systematically by intracellular injections of long-lasting step currents to determine the voltage dependence of their membrane electrical properties. The complex impedance and impedance magnitude functions were first determined using oscillatory input currents superimposed on these step currents. 2. Systematic step variations in the membrane potential led to qualitative changes in the impedance magnitude functions. Depolarization of neurons exhibiting resonance at their initial resting membrane potentials resulted in a reduction in the resonance behavior. Hyperpolarization of these neurons to membrane potentials of about -80 to -90 mV led to a disappearance of the resonant peak but increased the maximum of the impedance magnitude. 3. The complex impedance data were fitted with a neuronal model derived from linearized Hodgkin-Huxley-like equations, yielding estimates for the membrane properties. The four parameters of the model were 1) a time invariant, resting membrane conductance, Gr, 2) a voltage- and time-dependent conductance, GL, 3) a time constant, tau u, for the unknown ionic channels that are activated by the 2- to 5-mV oscillatory perturbation of the stepped membrane potential, and 4) Ci, the input capacitance. 4. The results of the curve-fitting procedures suggested that all parameters depended on membrane voltage. The most voltage-dependent parameters were GL and tau u throughout a 25- to 30-mV range that was subthreshold to the production of action potentials. Both Gr and GL increased with subthreshold depolarization. 5. These impedance data suggest the very important role of the membrane potential of the trigeminal root ganglion neurons on their abilities to synthesize and filter inputted electrical signals.

Quantification of membrane properties of trigeminal root ganglion neurons in guinea pigs.

Passive and active (voltage- and time-dependent) membrane properties of trigeminal root ganglion neurons of decerebrate guinea pigs have been determined using frequency-domain analyses of small-amplitude perturbations of membrane voltage. The complex impedance functions of trigeminal ganglion neurons were computed from the ratios of the fast Fourier transforms of the intracellularly recorded voltage response from the neuron and of the input current, which had a defined oscillatory waveform. The impedance magnitude functions and corresponding impedance locus diagrams were fitted with various membrane models such that the passive and active properties were quantified. The complex impedances of less than one-quarter of the 105 neurons which were investigated extensively could be described by the complex impedance function for a simple RC-electrical circuit. In such neurons, the voltage responses to constant-current pulses, using conventional bridge-balance techniques, could be fitted with single exponential curves, also suggesting passive membrane behavior. A nonlinear least-squares fit of the complex impedance function for the simple model to the experimentally observed complex impedance yielded estimates of the resistance of the electrode, and of input capacitance (range, 56 to 490 pF) and input resistance (range, 0.8 to 30 M omega) of the neurons. The majority of trigeminal ganglion neurons were characterized by a resonance in the 50- to 250-Hz bandwidth of their impedance magnitude functions. Such neurons when injected with "large" hyperpolarizing current pulses using bridge-balance techniques showed membrane voltage responses that "sagged" (time-dependent rectification). Also, repetitive firing commonly occurred with depolarizing current pulses; this characteristic of neurons with resonance in their impedance magnitude functions was not observed in neurons with "purely" passive membrane behavior. A nonlinear least-squares fit of a five-parameter impedance fitting function based on a membrane model to the impedance locus diagram of a neuron with resonance yielded estimates of its membrane properties: input capacitance, the time-invariant part of the conductance, the conductance activated by the small oscillatory input current, and the relaxation time constant for this conductance. The ranges of the estimates for input capacitance and input resistance were comparable to the ranges of corresponding properties derived for neurons exhibiting "purely" passive behavior.(ABSTRACT TRUNCATED AT 400 WORDS)

Impedance profiles of peripheral and central neurons.

The electrical impedance of trigeminal ganglion cells (in vivo) and hippocampal CA1 neurons (in vitro) of guinea pigs was measured in the frequency range of 5-1250 Hz using intracellular recording techniques with single microelectrodes and computerized methodology. The transfer functions of the electrode and the electrode-neuron system were computed from the ratio of fast Fourier transforms of the output voltage response from the neuron and input current composed of sine waves with rapidly increasing frequency which displaced membrane potential by 2-5 mV. We believe these to be the first measurements of complex impedance and transfer functions in peripheral and central neurons of vertebrates and the first use of such input current functions. The majority of trigeminal ganglion cells did not exhibit electrical behaviour ascribable to a simple resistance-capacitance (RC) circuit but showed a hump at low frequencies (5-250 Hz) in the computed transfer function, probably attributable to resonance. The transfer function in less than 20% of the trigeminal neurons could be fitted approximately to a theoretical transfer function (resistance in series with a parallel RC circuit model) providing values for electrode resistance, effective input resistance, and effective input capacitance. The transfer functions measured in hippocampal CA1 neurons were characterized by a rapid fall-off in the low frequency range (less than 200 Hz). Impedance locus plots approximate the locus corresponding to a series RC circuit in parallel with a parallel RC circuit.

A mathematical model for spreading cortical depression.

A mathematical model is derived from physiological considerations for slow potential waves (called spreading depression) in cortical neuronal structures. The variables taken into account are the intra- and extracellular concentrations of Na+, Cl-, K+, and Ca++, together with excitatory and inhibitor transmitter substances. The general model includes conductance changes for these various ions, which may occur at nonsynaptic and synaptic membrane together with active transport mechanisms (pumps). A detailed consideration of only the conductance changes due to transmitter release leads to a system of nonlinear diffusion equations coupled with a system or ordinary differential equations. We obtain numerical solutions of a set of simplified model equations involving only K+ and Ca++ concentrations. The solutions agree qualitatively with experimentally obtained time-courses of these two ionic concentrations during spreading depression. The numerical solutions exhibit the observed phenomena of solitary waves and annihilation of colliding waves.