A fully coupled transient excited state model for the sodium channel

Summary The axon membrane is simulated by standard Hodgkin-Huxley leakage and potassium channels plus a coupled transient excited state kinetic scheme for the sodium channel. This scheme for the sodium channel is as proposed previously by the author. Simultations are presented showing the form of the action potential, threshold behavior, accommodation, and repetitive firing. It is seen that the form of the individual action potential, its all-or-none nature, and its refractory period are well simulated by this model, as they are by the standard Hodgkin-Huxley model. However, the model differs markedly from the Hodgkin-Huxley model with respect to repetitive firing and accommodation to stimulating currents of slowly rising intensity, in ways that are anomn to be related to those features of the sodium inactivation which are anomalous to the H-H model. The tendency for repetitive firing is highly dependent on that parameter which primarily determintes the existence of the inactivation shift in voltage clamp experiments, in such a way that the more pronounced the inactivation shift, the less the tendency for repetitive firing,. The tendency for accommodation is highly dependent on that parameter which primarily determines the “τ_c − τ_h” separation, in such a way that the greater the separation the greater the tendency for the membrane to accommodate without firing action potentials to a slowly rising current.     

A transient excited state model for sodium permeability changes in excitable membranes.

In this paper we explore the properties of a mathematical model for the passive sodium permeability system of excitable membranes. This model is distinguished by the explicit inclusion of a rate constant which depends not on instantaneous voltage, but on rate of voltage change. Actually, the model is a rather modest modification of the Hodgkin-Huxley model, but displays some behaviors which the H-H model does not. Among these behaviors are a pronounced inactivation shift (for certain parameter values), a difference between inactivation time constant as measured by turning off a sodium current under sustained depolarization and as measured by double pulse experiments, skip runs under sustained current stimulation, and accommodation to slowing rising currents.     

Temporal Modulation of Sodium Current Kinetics in Neuron Cells by Near-Infrared Laser

Near-infrared laser provides a novel nerve stimulation modality to regulate the cell functions. Understanding its physiological effect is a prerequisite for clinic laser therapy applications. Here, the whole-cell sodium (Na) channel kinetics of neuron cell was employed to determine the temporal roles of infrared laser. The Na currents were elicited by electrical pulses that were synchronized at the rising and falling edges of the 980 nm laser pulses, respectively, to investigate the different infrared effect on cell functions. The time constants of activation (τ _m) and inactivation (τ _h) kinetics were extracted from fitting of the Na current (m³h) according to the Hodgkin–Huxley (HH) model. By comparing the time constants without and with the laser irradiation, we obtained that laser pulses changed the Na current kinetics by accelerating τ _h-phase and slowing down τ _m-phase at the beginning of the laser pulse, whereas both phases were accelerated at the end of the pulse. After relating the ratios of the time constants to the temperature characteristics of Na channel by Q ₁₀, we found that the accelerating in Na current kinetics could be related to the average temperature of extracellular solution in the corresponding time span by choosing Q ₁₀ = 2.6. The results of this study demonstrated that there was a positive correlation between the acceleration of the Na current kinetics and increases in temperature of the extracellular solution.     

The Effect of Time Delay in a Two-Patch Model with Random Dispersal

We consider a two-patch model for a single species with dispersal and time delay. For some explicit range of dispersal rates, we show that there exists a critical value τ _c for the time delay τ such that the unique positive equilibrium of the system is locally asymptotically stable for τ∈[0,τ _c ) and unstable for τ>τ _c .     

A comparative analysis of models of Na+ channel gating for mammalian and invertebrate nonmyelinated axons: Relationship to energy efficient action potentials

The rapidly activating, voltage gated Na⁺ current, I_Na, has recently been measured in mammalian nonmyelinated axons. Those results have been incorporated in simulations of the action potential, results that demonstrate a significant separation in time during the spike between I_Na and the repolarizing K⁺ current, I_K. The original Hodgkin and Huxley (1952) model of Na⁺ channel gating, m³h, where m and h are channel activation and inactivation, respectively, has been used in this analysis. This model was originally developed for invertebrate nonmyelinated axons, squid giant axons in particular. The model has not survived challenges based on results from invertebrate preparations using a double-step voltage clamp protocol and measurements of gating currents, results that demonstrate a kinetic link between activation and inactivation leading to a delayed onset of inactivation following a voltage step. These processes are independent of each other in the Hodgkin and Huxley (1952) model. Application of the double-step protocol to the m³h model for mammalian I_Na results reveals a surprising prediction, an apparent delay in onset of inactivation even though activation and inactivation are uncoupled in the model. Other results, most notably gating currents, will be required to demonstrate such a link, if indeed it exists for mammalian Na⁺ channels. The information obtained will be significant in determining the way in which the Na⁺ channel is sequestered away from its open state during repolarization, thereby allowing for a separation in time between I_Na and I_K during a spike, an energetically efficient mechanism of neuronal signaling in the mammalian brain.     

Protons and calcium alter gating of the hyperpolarization-activated cation current (Ih) in rod photoreceptors

We investigated the effects of protons and calcium ions on the voltage-dependent gating of the hyperpolarization-activated, nonselective cation channel current, I_h, in rod photoreceptors. I_h is a cesium-sensitive current responsible for the peak-plateau sag during the rod response to bright light. The voltage dependence of I_h activation shifted about 5 mV per pH unit, with external acidification producing positive shifts and alkalinization producing negative shifts. Increasing external [Ca²⁺] from 3 to 20 mM resulted in a large (∼17 mV) positive shift in I_h activation. External [Ca²⁺] (20 mM) blocked pH-induced shifts in activation. Cytoplasmic acidification produced by 25 mM sodium acetate led to a negative shift in inactivation (−9 mV) and internal alkalinization produced with 20 mM ammonium chloride resulted in a positive shift (+6 mV). Surface charge binding and screening theory (Gouy-Chapman-Stern) accounted for the observed shifts in I_h activation, with the best fit achieved when protons and calcium ions were assumed to bind to distinct sites on the membrane. Since light induces changes in the retinal ionic environment, these results permit us to gauge the degree to which rod light responses could be modified via alterations in I_h activation.     

Internal noise enhanced oscillation in a delayed circadian pacemaker

The effect of internal noise in a delayed circadian oscillator is studied by using both chemical Langevin equations and stochastic normal form theory. It is found that internal noise can induce circadian oscillation even if the delay time τ is below the deterministic Hopf bifurcation τ_h. We use signal-to-noise ratio (SNR) to quantitatively characterize the performance of such noise induced oscillations and a threshold value of SNR is introduced to define the so-called effective oscillation. Interestingly, the τ-range for effective stochastic oscillation, denoted as Δτ_EO, shows a bell-shaped dependence on the intensity of internal noise which is inversely proportional to the system size. We have also investigated how the rates of synthesis and degradation of the clock protein influence the SNR and thus Δτ_EO. The decay rate K_d could significantly affect Δτ_EO, while varying the gene expression rate K_e has no obvious effect if K_e is not too small. Stochastic normal form analysis and numerical simulations are in good consistency with each other. This work provides us comprehensive understandings of how internal noise and time delay work cooperatively to influence the dynamics of circadian oscillations.     

Multifractal analysis of K+ channel activity

The Multifractal version of the Detrended Fluctuation Analysis was used for the study of non-stationary dwell time series of Ca²⁺-activated K⁺ channels (K_Ca channels) in cultured kidney Vero cells and of voltage-dependent K⁺ channels (K_v channels) in mollusc (Lymnaea stagnalis) neurons. The data obtained can briefly be summarized as follows: (i) The generalized fluctuation function F _q (l) strongly depends on the index (order) q; for monofractal time series, such dependence is nonexistent; (ii) The relationship between the scaling exponent τ commonly employed in standard multifractal analysis and q is characterized by two slopes and a transitory region, whereas monofractal processes are characterized by the linear dependence; (iii) The relationship between the singularity spectrum f(h) and the Hurst exponent h is bell-shaped, while in the case of monofractal processes it is represented by a single point f(h) = 1. Random mixing of the time series resulted in the narrowing of the spectrum f(h) and a shift of f(h) towards the value more characteristic of stochastic (monofractal) processes (h ～ 0.5). It is concluded that the activities of both K_Ca channels in kidney Vero cells and of K_V channels in mollusc (Lymnaea stagnalis) neurons can be characterized as multifractal processes.     

The influence of external potassium on the inactivation of sodium currents in the giant axon of the squid, Loligo pealei

Isolated giant axons were voltage-clamped in seawater solutions having constant sodium concentrations of 230 mM and variable potassium concentrations of from zero to 210 mM. The inactivation of the initial transient membrane current normally carried by Na⁺ was studied by measuring the Hodgkin-Huxley h parameter as a function of time. It was found that h reaches a steady-state value within 30 msec in all solutions. The values of h _∞, τ_h, α_h,and β_h as functions of membrane potential were determined for various [K _o]. The steady-state values of the h parameter were found to be inversely related, while the time constant, τ_h, was directly related to external K⁺ concentration. While the absolute magnitude as well as the slopes of the h _∞ vs. membrane potential curves were altered by varying external K⁺, only the magnitude and not the shape of the corresponding τ_h curves was altered. Values of the two rate constants, α_h and β_h, were calculated from h_∞ and τ_h values. α_h is inversely related to [K_o] while β_h is directly related to [K_o] for hyperpolarizing membrane potentials and is independent of [K_o] for depolarizing membrane potentials. Hodgkin-Huxley equations relating α_h and β_h to E_m were rewritten so as to account for the observed effects of [K_o]. It is concluded that external potassium ions have an inactivating effect on the initial transient membrane conductance which cannot be explained solely on the basis of potassium membrane depolarization.     

Reexcitation mechanisms in epicardial tissue: Role of Ito density heterogeneities and INa inactivation kinetics

Dispersion of action potential repolarization is known to be an important arrhythmogenic factor in cardiopathies such as Brugada syndrome. In this work, we analyze the effect of a variation in sodium current (I_Na) inactivation and a heterogeneous rise of transient outward current (I_to) in the probability of reentry in epicardial tissue. We use the Luo-Rudy model of epicardial ventricular action potential to study wave propagation in a one-dimensional fiber. Spatial dispersion in repolarization is introduced by splitting the fiber into zones with different strength of I_to. We then analyze the pro-arrhythmic effect of a variation in the relaxation time and steady-state of the sodium channel fast inactivating gate h. We quantify the probability of reentry measuring the percentage of reexcitations that occurs in 200 beats. We find that, for high stimulation rates, this percentage is negligible, but increases notably for pacing periods above 700 ms. Surprisingly, with decreasing I_Na inactivation time, the percentage of reexcitations does not grow monotonically, but presents vulnerable windows, separated by values of the I_Na inactivation speed-up where reexcitation does not occur. By increasing the strength of L-type calcium current I_CaL above a certain threshold, reexcitation disappears. Finally, we show the formation of reentry in stimulated two-dimensional epicardial tissue with modified I_Na kinetics and I_to heterogeneity. Thus, we confirm that while I_to dispersion is necessary for phase-2 reentry, altered sodium inactivation kinetics influences the probability of reexcitation in a highly nonlinear fashion.     

Kinetics of the sodium current through EDTA-modified calcium channels of the molluscan neuron somatic membrane

The kinetics of the slow current carried by sodium ions through potential-dependent calcium channels after addition of EDTA to calcium-free external solution was investigated in experiments by the intracellular dialysis method on isolatedHelix pomatia neurons. The activation kinetics of this current was similar to that of the calcium current and could be described by the use of the square of the activation variable m in Hodgkin-Huxley equations. The decay (inactivation) kinetics of the induced sodium current during prolonged depolarization is biexponential in character. It is suggested that decay of the sodium currents takes place as a result of two independent processes: potential-dependent inactivation with a time constant τ_h～1 sec, taking place as far as a certain steady-state level h_∞, and a decrease in current connected with Na⁺ accumulation inside the cell during passage of the current and a consequent change in the sodium electrochemical potential (τ_c～10 sec). It is concluded that modification of the calcium channels, so that they acquire the ability to conduct sodium, has no significant effect on the gating mechanisms responsible for opening and closing of the channels.     

Effects of environmental features on flocking behavior

Social animals or insects in nature often exhibit a form of emergent collective behavior known as flocking. Animal flocks are strongly affected by the spatial heterogeneity representing static environmental factors, such as landscape. To investigate how this heterogeneity affects flocking behavior, I constructed a two-dimensional, lattice-based environment, in which the spatial heterogeneity was generated using a neutral landscape model. A value ranging from 0.0 to 1.0 was assigned to each lattice site. High values represent favorable environmental conditions, and thus, individuals tend to be attracted to the area with the highest values. The value H controlled the degree of heterogeneity and the value h represented the amplitude of the heterogeneity. To characterize the effect of the spatial heterogeneity, I introduced the order parameter, ?, defined as the absolute value of the average unit velocity. When all individuals move in the same direction, the value of ? is 1, while the value of ? becomes 0 in the case where all individuals move randomly. In this simulation, the flock size, N, has the values of 20, 40, 60, 80 and 100. I investigated the average value of ? and ?_s, for 1500 < t < 2000. The simulation results showed that the value of ?_s increased with an increase in H, regardless of N, until becoming saturated. When the value of h was higher, the value of ?_s became lower.     

Translocation of non-interacting heteropolymer protein chains in terms of single helical propensity and size

In this work we present an analytical framework to calculate the average translocation time τ required for an ideal proteinogenic polypeptide chain to cross over a small pore on a membrane. Translocation is considered to proceed as a chain of non-interacting amino acid residues of sequence {X_j} diffuses through the pore against an energy barrier Δℱ, set by chain entropy and unfolding-folding energetics. We analyze the effect of sequence heterogeneity on the dynamics of translocation by means of helical propensity of amino acid residues. In our calculations we use sequences of fifteen well-known proteins that are translocated which span two orders of magnitude in size according to the number of residues N. Results show non-symmetric free energy barriers as a consequence of sequence heterogeneity, such asymmetry in energy may be useful in differentiated directions of translocation. For the fifteen polypeptide chains considered we found conditions when sequence heterogeneity has not a significant effect on the time scale of translocation leading to a scaling law τ ∝ N^ν, where ν ∼ 1.6 is an exponent that holds for most ground state energies. We also identify conditions when sequence heterogeneity has a great impact on the time scale of translocation, in consequence, no more scaling laws for τ there exist.     

A New Preparation Method of Coenzyme A with Microorganisms

The structure of aggregates formed by heating dilute BSA solution was analyzed with the fractal concept using light scattering methods. BSA was dissolved in HEPES buffer of pH 7.0 and acetate buffer of pH 5.1 to 0.1% and 0.001% solutions, respectively, and heated at 95°C, varying the heating time t_a. The fractal dimension D_f of the aggregate in the solution was evaluated from static light scattering experiments. The polydispersity exponent τ and the average hydrodynamic radius <R_h> of the aggregates were calculated from dynamic light scattering experiments using master curves obtained by Klein et al. The values of D_f and τ of heat-induced aggregates of BSA at pH 7.0 were about 2.1 and 1.5, respectively, the values of which agreed with those predicted by the reaction-limited cluster–cluster aggregation (RLCCA) model. On the other hand, D_f of heat-induced aggregates at pH 5.1 was about 1.8, which agreed with that predicted by the diffusion-limited cluster–cluster aggregation (DLCCA) model. The dependence of <R_h> for the sample of pH 7.0 on t_a was similar to that of the polystyrene colloids reported previously.     

Statistical inference on markov process of neuronal impulse sequences

To clarify the stochastic properties of the neuronal impulse sequences, we have proposed a measure of statistical dependency d _i (T=τ) and an equation ? _m of the matrices of the serial correlation coefficients. Markov properties of the interval sequences could be provided with d _i (T=τ) and ? _m , which represent the necessary and sufficient condition for the statistical dependence. A method to estimate the order of Markov process with the use of d _i (T=τ) and ? _m was found to be useful in practice. This was proved by the interval sequences of the 0-th, 1-st, and 2nd order semi-Markov process generated by computer. It was also found that the order of Markov process of neuronal impulse sequence is an important parameter representing the pattern of the sequence. This was proved with computer simulation by semi-Markov model of impulse sequence.     

Modeling the electrical behavior of anatomically complex neurons using a network analysis program: Excitable membrane

We present methods for using the generalpurpose network analysis program, SPICE, to construct computer models of excitable membrane displaying Hodgkin-Huxley-like kinetics. The four non-linear partial differential equations of Hodgkin and Huxley (H-H; 1952) are implemented using electrical circuit elements. The H-H rate constants, and , are approximated by polynomial functions rather than exponential functions, since the former are handled more efficiently by SPICE. The process of developing code to implement the H-H sodium conductance is described in detail. The Appendix contains a complete listing of the code required to simulate an H-H action potential. The behavior of models so constructed is validated by comparison with the space-clamped and propagating action potentials of Hodgkin and Huxley. SPICE models of multiply branched axons were tested and found to behave as predicted by previous numerical solutions for propagation in inhomogeneous axons. New results are presented for two cases. First, a detailed, anatomically based model is constructed of group Ia input to an -motoneuron with an excitable soma, a myelinated axon and passive dendrites. Second, we simulate interactions among clusters of mixed excitable and passive dendritic spines on an idealized neuron. The methods presented in this paper and its companion (Segev et al. 1985) should permit neurobiologists to construct and explore models which simulate much more closely the real morphological and physiological characteristics of nerve cells.     

Local anesthetic action of carboxylic esters: Evidence for the significance of molecular volume and for the number of sites involved

Summary The effects of the homologous series of carboxylic esters, methyl propionate to methyl decanoate, on the steadystate inactivation of the sodium current in squid axons have been studied. The esters moved the relationship between the inactivation parameter,h , and the membrane potential in the hyperpolarizing direction, thus reducing the number of sodium channels available at the resting potential. The concentration dependence of the shift at the mid-point of the curve ofh against potential has been measured for all esters except decanoate, which was almost inactive. Two aspects of these concentration dependences suggest that molecular volume is an important determinant of the effectiveness of each ester. Firstly, there is a sharp decline in activity above methyl hexanoate. This cut-off in activity resembles that for hydrocarbons where it has been suggested [e.g., Haydon, D.A., Urban, B.W. 1983)J. Physiol. (London) 341:411–427] to a result from a decrease in uptake with increasing molecular volume. (Further data for the hydrocarbonsn-butane ton-heptane are reported here.) Secondly, the smallest compounds, methyl propionate and methyl butyrate, are less effective than would be predicted if equal membrane concentrations of each ester produced the same shift. The aqueous concentration dependences for these esters indicate that below methyl hexanoate, as the series is descended, progressively higher membrane concentrations are required to produce a given shift. This would be expected if the volume of ester in the membrane, rather than the number of molecules, is important.Differences between the effects of the ester series on steady-state inactivation and on the reduction of the peak sodium current suggest that, in the unclamped squid axon, excitability is influenced by at least two distinct mechanisms in which at least two sites of action are involved.     

Vibrational resonance in groundwater-dependent plant ecosystems

We report the phenomenon of vibrational resonance in a single species and a two species models of groundwater-dependent plant ecosystems with a biharmonic oscillation (with two widely different frequencies ω and Ω, Ω ≫ ω) of the water table depth. In these two systems, the response amplitude of the species biomass shows multiple resonances with different mechanisms. The resonance occurs at both low- and high-frequencies of the biharmonic force. In the single species bistable system, the resonance occurs at discrete values of the amplitude g of the high-frequency component of the water table. Furthermore, the best synchronization of biomass and its carrying capacity with the biharmonic force occurs at the resonance. In the two species excitable and time-delay model, the response amplitude (Q) profile shows several plateau regions of resonance, where the period of evolution of the species biomass remains the same and the value of Q is inversely proportional to it. The response amplitude is highly sensitive to the time-delay parameter τ and shows two distinct sequences of resonance intervals with a decreasing amplitude with τ.