Traveling waves in a piecewise-linear reaction-diffusion model of excitable medium

One-dimensional autowaves (traveling waves) in excitable medium described by a piecewise-linear reaction-diffusion system have been investigated. Two main types of wave have been considered: a single pulse and a periodic sequence of pulses (wave trains). In a two-component system, oscillations are due to the second component of the reaction-diffusion system, while in a one-component system, they are caused by external periodic excitation (forcing). Using semianalytical solutions for the wave profile, the shape and velocity of autowaves have been found. It is shown that the dispersion relation for oscillating sequences of pulses has an anomalous character.     

The free energy of biomembrane and nerve excitation and the role of anesthetics

In the electromechanical theory of nerve stimulation, the nerve impulse consists of a traveling region of solid membrane in a liquid environment. Therefore, the free energy necessary to stimulate a pulse is directly related to the free energy difference necessary to induce a phase transition in the nerve membrane. It is a function of temperature and pressure, and it is sensitively dependent on the presence of anesthetics which lower melting transitions. We investigate the free energy difference of solid and liquid membrane phases under the influence of anesthetics. We calculate stimulus-response curves of electromechanical pulses and compare them to measured stimulus-response profiles in lobster and earthworm axons. We also compare them to stimulus-response experiments on human median nerve and frog sciatic nerve published in the literature.     

The dependence of impulse propagation speed on firing frequency, dispersion, for the Hodgkin-Huxley model.

Propagation speed of an impulse is influenced by previous activity. A pulse following its predecessor too closely may travel more slowly than a solitary pulse. In contrast, for some range of interspike intervals, a pulse may travel faster than normal because of a possible superexcitable phase of its predecessor's wake. Thus, in general, pulse speeds and interspike intervals will not remain constant during propagation. We consider these issues for the Hodgkin-Huxley cable equations. First, the relation between speed and frequency or interspike interval, the dispersion relation, is computed for particular solutions, steadily propagating periodic wave trains. For each frequency, omega, below some maximum frequency, omega max, we find two such solutions, one fast and one slow. The latter are likely unstable as a computational example illustrates. The solitary pulse is obtained in the limit as omega tends to zero. At high frequency, speed drops significantly below the solitary pulse speed; for 6.3 degrees C, the drop at omega max is greater than 60%. For an intermediate range of frequencies, supernormal speeds are found and these are correlated with oscillatory swings in sub- and superexcitability in the return to rest of an impulse. Qualitative consequences of the dispersion relation are illustrated with several different computed pulse train responses of the full cable equations for repetitively applied current pulses. Moreover, changes in pulse speed and interspike interval during propagation are predicted quantitatively by a simple kinematic approximation which applies the dispersion relation, instantaneously, to individual pulses. One example shows how interspike time intervals can be distorted during propagation from a ratio of 2:1 at input to 6:5 at a distance of 6.5 cm.     

Traveling Wave Solutions of a Nerve Conduction Equation

We consider a pair of differential equations whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh-Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse-shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.     

A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance that allows calculation of both the speed and shape of the traveling pulses, in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses that is valid with good accuracy in a wide range of the parameters.     

Spatial stability of traveling wave solutions of a nerve conduction equation.

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.     

Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.     

Integration of ascending and descending inputs in the auditory midbrain of anurans.

In anurans, the midbrain torus semicircularis is involved in auditory processing and audio-motor integration. In this study, we examined the influence of descending forebrain projections on the auditory response properties and hence the audiomotor transmission of mesencephalic interface neurons. In order to investigate response integration, we performed intracellular recordings from torus neurons in an isolated brain preparation of Discoglossus pictus and Bombina orientalis and stimulated the auditory nerve, striatum, and the dorsal thalamus electrically with single pulses. Stimulation of all three sites could evoke responses in torus neurons that were either excitatory, inhibitory, or a mixture of both, with durations of up to several hundred milliseconds. Further, striatum and thalamus were activated by pulse trains (10-20 Hz, 50 pulses) immediately before stimulating the auditory nerve with single pulses. Thus, responses of torus neurons to "auditory" input were facilitated or suppressed for up to 2 min by striatum stimulation or only suppressed by thalamus stimulation. Intracellular labeling of recorded neurons revealed that response modulation by descending input mostly occurred in laminar nucleus neurons. These results suggest that descending forebrain projections to mesencephalic audiomotor interface neurons may play an important role in modifying acoustically guided behavior in anurans.     

Detecting chaotic structures in noisy pulse trains based on interspike interval reconstruction

The nonlinear prediction method based on the interspike interval (ISI) reconstruction is applied to the ISI sequence of noisy pulse trains and the detection of the deterministic structure is performed. It is found that this method cannot discriminate between the noisy periodic pulse train and the noisy chaotic one when noise-induced pulses exist. When the noise-induced pulses are eliminated by the grouping of ISI sequence with the genetic algorithm, the chaotic structure of the chaotic firings becomes clear, and the noisy chaotic pulse train could be discriminated from the periodic one.     

On the role of subthreshold dynamics in neuronal signaling

The role of subthreshold dynamics in neuronal signaling is examined using periodic pulse train stimulation of the Fitzhugh-Nagumo (FN) model of nerve membrane excitability and results from the squid giant axon as an experimental data base. For a broad range of stimulus conditions the first pulse in a pulse train elicited an action potential, whereas all subsequent pulses elicited subthreshold responses, both in the axon and in the FN model. These results are not well described by the Hodgkin and Huxley 1952 model. Various different patterns of subthreshold responses, including chaotic dynamics, can be observed in both systems-the FN model and the axon-depending upon stimulus conditions. For some conditions action potentials are occasionally interspersed among the subthreshold events with randomly occurring interspike intervals. The randomness is directly attributable to the underlying subthreshold chaos-deterministic chaos-rather than to a stochastic noise source. We conclude that this mechanism may contribute to multimodal interspike interval histograms which have been observed from individual neurons throughout the nervous system.     

Do circulating plasma AVT and/or cortisol levels control pulsatile urea excretion in the gulf toadfish (Opsanus beta)?

Previous work has shown that pulsatile urea excretion at the gills of the gulf toadfish is due to periodic activation of a facilitated diffusion transport system with molecular and pharmacological similarity to the UT-A transport system of the mammalian kidney. In mammals, AVP and glucocorticoids are two important endocrine regulators of this system. The present study focused on the potential role of circulating AVT (the teleost homologue of AVP) and cortisol levels as possible triggers for urea pulses. Long-term (34-84 h) monitoring of plasma levels by repetitive sampling at 2-h intervals from chronic cannulae in individual toadfish demonstrated that circulating AVT concentrations are low (10(-12)-10(-11) M), and show no relationship to the occurrence of natural urea pulses. In contrast, plasma cortisol levels decline greatly prior to natural pulses and rise rapidly thereafter. AVT injections into the caudal artery or ventral aorta elicited pulse events, but these were extremely small (1-10%) relative to natural pulses, and occurred only at unphysiological dose levels (10(-9) M in the plasma). AVP was a partial agonist, but isotocin, insulin-like growth factor-1, and atrial natriuretic peptide were without effect at the same concentration. Artificially raising plasma cortisol levels by cortisol injection tended to reduce responsiveness to AVT. Pharmacological reduction of plasma cortisol levels by metyrapone injection elicited small pulses similar to those caused by AVT. Following such pulse events, AVT was ineffective in inducing pulses. We conclude that decreases in circulating cortisol play an important permissive role in urea pulsing, but that circulating AVT levels are not involved.     

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.     

Delayed initiation of SS1 pulses in the sea anemone Calliactis parasitica: evidence for a fourth conducting system.

1. Single electrical shocks to the column sometimes elicit a series of 1-6 pulses in the SS1 (ectodermal slow system) but the first pulse does not appear until 5-28 s after stimulation. These pulses occur in addition to the early SS1 pulse which follows every shock and which has a conduction delay of less than 1 s. 2. The threshold of the delayed SS1 response is different from the thresholds of the three known conducting systems (through-conducting nerve net, SS1, and SS2). 3. In the case of stimulation of the column, the delayed SS1 pulses do not arise at the point of stimulation but probably originate in the tentacles or upper column. The pulse origin can shift during a single burst. 4. The pathway from the point of stimulation to the site of origin of delayed SS1 pulses is endodermal. We propose that this pathway represents a fourth conducting system (Delayed Initiation System--DIS). The DIS must connect, across the mesogloea, with the ectodermal SS1. The long pulse delay and repetitive firing may derive from pacemaker activity in the DIS. The DIS pacemakers closely resemble the pacemakers connected to the through-conducting nerve net. The DIS may be neuronal. 5. Delayed SS1 pulse bursts from unattached anemones showed an earlier onset, and more pulses/burst, than those from attached anemones. 6. Delayed SS1 pulses can also be evoked by electrical, and in some cases mechanical, stimulation of the pedal disc, tentacles, and pharynx, but there are regional differences in the number of pulses evoked, in their delay, and in their site of origin.     

Use-dependent block of sodium channels in frog myelinated nerve by tetrodotoxin and saxitoxin at negative holding potentials

Na+ currents were measured in myelinated frog nerve fibres in the presence of nanomolar concentrations of tetrodotoxin (TTX) or saxitoxin (STX) in the extracellular solution. The Na+ currents declined during a train of depolarizing pulses if the fibre was held at hyperpolarizing potentials between the pulses. At a pulse frequency of 0.8 Hz, the peak Na+ currents were reduced to 70 or 60% of the initial value in 9.3 nM TTX and 3.5 nM STX solutions, respectively. A decline of Na+ currents was also observed in two-pulse experiments. The peak Na+ current during a second test pulse did not depend on the duration (0.2 to 12 ms) of the first pulse. It decreased with increasing interval between the pulses, reached a minimum and increased again. The results are interpreted with a use-dependent blockage of Na+ channels by TTX or STX at negative holding potentials. The effects were described quantitatively, assuming a fast affinity increase of toxin receptors at Na+ channels triggered by Na+ activation followed by slow toxin binding to channels and relaxation of the receptor affinity.     

Stability of periodic travelling wave solutions of a nerve conduction equation

Summary Nagumo's nerve conduction equation has travelling wave solutions of pulse type and periodic wave type. We consider the stability of the latter ones. We denote byL(c) the minimum spatial period of a periodic travelling wave solution whose propagation speed isc. It is shown that this travelling wave solution is unstable ifL′(c)<0.     

A presettable multichannel digital timer.

A digital timer is described which generates a number of pulses whose delays with respect to a periodic reference pulse can be independently preset by means of thumbwheel switches. The timing intervals are crystal-controlled and can be varied over a wide range--typically 0.1 ms to 99.99 s. It is shown how a pulse train of up to 46 pulses may be obtained, the delay of each pulse in the train being individually presettable. Digital integrated circuits are used throughout, except for the output stage of enhanced drive capability.     

Decay of the slow calcium current in twitch muscle fibers of the frog is influenced by intracellular EGTA

The mechanism(s) of the decay of slow calcium current (ICa) in cut twitch skeletal muscle fibers of the frog were studied in voltage-clamp experiments using the double vaseline-gap technique. ICa decay followed a single exponential in 10 mM external Ca2+ and 20 mM internal EGTA solutions in all pulse protocols tested: single depolarizing pulses (activation protocol), two pulses (inactivation protocol), and during a long pulse preceded by a short prepulse (400 ms) to 80 mV (tail protocol). In single pulses the rate constant of ICa decay was approximately 0.75 s-1 at 0 mV and became faster with larger depolarizations. ICa had different amplitudes during the second pulses of the inactivation protocol (0 mV) and of the tail protocol (-20 to 40 mV) and had similar time constants of decay. The time constant of decay did not change significantly at each potential after replacing 10 mM Ca2+ with a Ca2+-buffered solution with malate. With 70 mM intracellular EGTA and 10 mM external Ca2+ solutions, ICa also decayed with a single-exponential curve, but it was about four times faster (approximately 3.5 s-1 at 0 mV pulse). In these solutions the rate constant showed a direct relationship with ICa amplitude at different potentials. With 70 mM EGTA, replacing the external 10 mM Ca2+ solution with the Ca2+-buffered solution caused the decay of ICa to become slower and to have the same relationship with membrane potential and ICa amplitude as in fibers with 20 mM EGTA internal solution. The mechanism of ICa decay depends on the intracellular EGTA concentration: (a) internal EGTA (both 20 and 70 mM) significantly reduces the voltage dependence of the inactivation process and (b) 70 mM EGTA dramatically increases the rate of tubular calcium depletion during the flow of ICa.