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J Rinzel

Publications and source records attributed to J Rinzel.

66 records · Page 4Linked to original sources
PubMed1984

Beyond a pacemaker's entrainment limit: phase walk-through.

Desynchronization occurs when a pacemaker is externally stimulated at too high a rate, beyond the entrainment limit. One may then observe phase walk-through: the pacemaker and stimulus phase difference, rather than lock to a fixed pattern, may cycle repeatedly through all values. By means of a simple but nonlinear one-variable model we describe qualitatively the properties of entrainment, loss of entrainment, and phase walk-through for rhythmically stimulated pacemakers. We obtain an expression for the cycle length (beat period) of the repetitive phase walk-through. This shows, in contrast to the simplest treatment, that in general beat frequency is not a simple linear function of the difference between the pacemaker and stimulus frequencies. Our results are illustrated in the context of a model experimental pacemaker, the rhythmically flashing firefly. We discuss how these modeling results apply in a much more general setting.

Animals↗
PubMed1983

A theoretical basis for large coefficient of variation and bimodality in neuronal interspike interval distributions.

We consider the classic Stein (1965) model for stochastic neuronal firing under random synaptic input. Our treatment includes the additional effect of synaptic reversal potentials. We develop and employ two numerical methods (in addition to Monte Carlo simulations) to study the relation of the various parameters of the model to the shape of the theoretical interspike interval distribution. Contrary to the results of Tuckwell (1979) we are unable to account, on the basis of substantial synaptic inhibition and with parameter settings in the known physiologic range, for experimental interspike interval distributions which exhibit large coefficients of variation or bimodality. We therefore introduce a time varying threshold into the model, which readily allows for such distributions and which has physiological justification.

Action Potentials↗
PubMed1981

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.

Animals↗
PubMed1980

Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator.

1. Repetitive firing in space-clamped squid axons bathed in low Ca and stimulated by a just suprathreshold step of current can be annihilated by a brief depolarizing or hyperpolarizing pulse of the proper magnitude applied at the proper phase. 2. In response to such perturbations, membrane potential and ionic currents show damped oscillations toward a steady state. 3. For other, non-annihilating, perturbations repetitive firing resumes with unaltered frequency but with phase resetting. 4. Experimental findings are compared with calculations for the space-and current-clamped Hodgkin-Huxley equations. Annihilation of repetitive firing to a steady state corresponds to a solution trajectory perturbed off a stable limit cycle and into the domain of attraction of a coexistent stable singular point. 5. Experimentally and theoretically the nerve exhibits hysteresis with two different stable modes of operation for a just suprathreshold range of bias current: the oscillatory repetitive firing state and the time-independent steady state. 6. Analogy is made to a brief synaptic input (excitatory or inhibitory) which may start or stop a biological pace-maker.

Action Potentials↗
PubMed1978

Repetitive activity and Hopf bifurcation under point-stimulation for a simple FitzHugh-Naguma nerve conduction model.

In response to point-stimulation with a constant current, a neuron may propagate a repetitive train of action potentials along its axon. For maintained repetitive activity, the current strength I must be, typically, neither too small nor too large. For I outside some range, time independent steady behavior is observed following a transient phase just after the current is applied. We present analytical results for a piecewise linear FitzHugh-Nagumo model for a point-stimulated (non-space-clamped) nerve which are consistent with this qualitative experimental picture. For each value of I there is a unique, spatially nonuniform, steady state solution. We show that this solution is stable except for an interval (I*, I(*)) of I values. Stability for I too small or too large corresponds to experiments with sub-threshold I or the excessive I which leads to 'nerve block'. For I = I*, I(*) we find Hopf bifurcation of spatially nonuniform, time periodic solutions. We conclude that (I*, I(*)) lies interior to the range of I values for repetitive activity. The values of I* and I(*) and their dependence on the model parameters are determined. Qualitative differences between results for the point-stimulated configuration and the space-clamped case are discussed.

Action Potentials↗
PubMed1975

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.

Mathematics↗
PubMed1975

Voltage transients in neuronal dendritic trees.

An analytical method is outlined for calculating the passive voltage transient at each point in an extensively branched neuron model for arbitrary current injection at a single branch. The method is based on a convolution formula that employs the transient response function, the voltage response to an instantaneous pulse of current. For branching that satisfies Rall's equivalent cylinder constraint, the response function is determined explicitly. Voltage transients, for a brief current injected at a branch terminal, are evaluated at several locations to illustrate the attenuation and delay characteristics of passive spread. A comparison with the same transient input terminal input, the fraction of input charge dissipated by various branches in the neuron model is illustrated. These fractions are independent of the input time course. For transient synaptic conductance change at a single branch terminal, a numerical example demonstrates the nonlinear effect of reduced synaptic driving potential. The branch terminal synaptic input is compared with the same synaptic conductance input applied to the soma on the basis of excitatory postsynaptic potential amplitude at the soma and charge delivered to the soma.

Action Potentials↗
PubMed1974

Transient response in a dendritic neuron model for current injected at one branch.

Mathematical expressions are obtained for the response function corresponding to an instantaneous pulse of current injected to a single dendritic branch in a branched dendritic neuron model. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. The response function when used in a convolution formula enables one to compute the voltage transient at any specified point in the dendritic tree for an arbitrary current injection at a given input location. A particular numerical example, for a brief current injection at a branch terminal, illustrates the attenuation and delay characteristics of the depolarization peak as it spreads throughout the neuron model. In contrast to the severe attenuation of voltage transients from branch input sites to the soma, the fraction of total input charge actually delivered to the soma and other trees is calculated to be about one-half. This fraction is independent of the input time course. Other numerical examples, which compare a branch terminal input site with a soma input site, demonstrate that, for a given transient current injection, the peak depolarization is not proportional to the input resistance at the injection site and, for a given synaptic conductance transient, the effective synaptic driving potential can be significantly reduced, resulting in less synaptic current flow and charge, for a branch input site. Also, for the synaptic case, the two inputs are compared on the basis of the excitatory post-synaptic potential (EPSP) seen at the soma and the total charge delivered to the soma.

Animals↗
PubMed1973

Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model.

Mathematical solutions and numerical illustrations are presented for the steady-state distribution of membrane potential in an extensively branched neuron model, when steady electric current is injected into only one dendritic branch. Explicit expressions are obtained for input resistance at the branch input site and for voltage attenuation from the input site to the soma; expressions for AC steady-state input impedance and attenuation are also presented. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. Numerical examples illustrate how branch input resistance and steady attenuation depend upon the following: the number of dendritic trees, the orders of dendritic branching, the electrotonic length of the dendritic trees, the location of the dendritic input site, and the input resistance at the soma. The application to cat spinal motoneurons, and to other neuron types, is discussed. The effect of a large dendritic input resistance upon the amount of local membrane depolarization at the synaptic site, and upon the amount of depolarization reaching the soma, is illustrated and discussed; simple proportionality with input resistance does not hold, in general. Also, branch input resistance is shown to exceed the input resistance at the soma by an amount that is always less than the sum of core resistances along the path from the input site to the soma.

Dendrites↗
PubMed1973

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.

Electric Conductivity↗