Isochrons and phaseless sets.

Winfree has developed mathematical models for his phase resetting experiments on biological clocks. These models lead him to ask a number of mathematical questions concerning dynamical systems. This paper deals with these mathematical questions. In Winfree's terminology we show the existence of isochrons and establish some of their properties.

The dynamics of density dependent population models.

The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bofurcations away from the equilibrium point is investigated and the existence of a "strange attractor" is demonstrated--i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).

Mapping the dynamics of a bursting neuron.

The anterior burster (AB) neuron of the lobster stomatogastric ganglion displays varied rhythmic behavior when treated with neuromodulators and channel-blocking toxins. We introduce a channel-based model for this neuron and show how bifurcation analysis can be used to investigate the response of this model to changes of its parameters. Two dimensional maps of the parameter space of the model were constructed using computational tools based on the theory of nonlinear dynamical systems. Changes in the intrinsic firing and oscillatory properties of the model AB neuron were correlated with the boundaries of Hopf and saddle-node bifurcations on these maps. Complex rhythmic patterns were observed, with a bounded region of the parameter plane producing bursting behavior of the model neuron. Experiments were performed by treating an isolated AB cell with 4-aminopyridine which selectively reduces gA, the conductance of the transient potassium channel. The model accurately predicts the qualitative changes in the neuronal voltage oscillations that are observed over a range of reduction of gA in the neuron. These results demonstrate the efficacy of dynamical systems theory as a means of determining the varied oscillatory behaviors inherent in a channel-based neural model. Further, the maps of bifurcations provide a useful tool for determining how these behaviors depend upon model parameters and comparing the model to a real neuron.

Bifurcation, bursting, and spike frequency adaptation.

Many neural systems display adaptive properties that occur on time scales that are slower than the time scales associated with repetitive firing of action potentials or bursting oscillations. Spike frequency adaptation is the name given to processes that reduce the frequency of rhythmic tonic firing of action potentials, sometimes leading to the termination of spiking and the cell becoming quiescent. This article examines these processes mathematically, within the context of singularly perturbed dynamical systems. We place emphasis on the lengths of successive interspike intervals during adaptation. Two different bifurcation mechanisms in singularly perturbed systems that correspond to the termination of firing are distinguished by the rate at which interspike intervals slow near the termination of firing. We compare theoretical predictions to measurement of spike frequency adaptation in a model of the LP cell of the lobster stomatogastric ganglion.

Dopamine modulation of two subthreshold currents produces phase shifts in activity of an identified motoneuron.

1. The lateral pyloric (LP) neuron is a component of the 14-neuron pyloric central pattern generator in the stomatogastric ganglion of the spiny lobster, Panulirus interruptus. In the pyloric rhythm, this neuron fires rhythmic bursts of action potentials whose phasing depends on the pattern of synaptic inhibition from other network neurons and on the intrinsic postinhibitory rebound properties of the LP cell itself. Bath-applied dopamine excites the LP cell and causes its activity to be phase advanced in the pyloric motor pattern. At least part of this modulatory effect is due to dopaminergic modulation of the intrinsic rate of postinhibitory rebound in the LP cell. 2. The LP neuron was isolated from all detectable synaptic input. We measured the rate of recovery after 1-s hyperpolarizing current injections of varying amplitudes, quantifying the latency to the first spike following the hyperpolarizing prepulse and the interval between the first and second action potentials. Dopamine reduced both the first spike latency and the first interspike interval (ISI) in the isolated LP neuron. During the hyperpolarizating pre-steps, the LP cell showed a slow depolarizing sag voltage that was enhanced by dopamine. 3. We used voltage clamp to analyze dopamine modulation of subthreshold ionic currents whose activity is affected by hyperpolarizing prepulses. Dopamine modulated the transient potassium current IA by reducing its maximal conductance and shifting its voltage dependence for activation and inactivation to more depolarized voltages. This outward current is normally transiently activated after hyperpolarization of the LP cell, and delays the rate of postinhibitory rebound; by reducing IA, dopamine thus accelerates the rate of rebound of the LP neuron. 4. Dopamine also modulated the hyperpolarization-activated inward current Ih by shifting its voltage dependence for activation 20 mV in the depolarizing direction and accelerating its rate of activation. This enhanced inward current helps accelerate the rate of rebound in the LP cell after inhibition. 5. The relative roles of Ih and IA in determining the first spike latency and first ISI were explored using pharmacological blockers of Ih (Cs+) and IA [4-aminopyridine (4-AP)]. Blockade of Ih prolonged the first spike latency and first ISI, but only slightly reduced the net effect of dopamine. In the continued presence of Cs+, blockade of IA with 4-AP greatly shortened the first spike latency and first ISI. Under conditions where both Ih and IA were blocked, dopamine had no additional effect on the LP cell. 6. We used the dynamic clamp technique to further study the relative roles of IA and Ih modulation in dopamine's phase advance of the LP cell. We blocked the endogenous Ih with Cs+ and replaced it with a simulated current generated by a computer model of Ih. The neuron with simulated Ih gave curves relating the hyperpolarizing prepulse amplitude to first spike latency that were the same as in the untreated cell. Changing the computer parameters of the simulated Ih to those induced by dopamine without changing IA caused only a slight reduction in first spike latency, which was approximately 20% of the total reduction caused by dopamine in an untreated cell. Bath application of dopamine in the presence of Cs+ and simulated Ih (with control parameters) allowed us to determine the effect of altering IA but not Ih: this caused a significant reduction in first spike latency, but it was still only approximately 70% of the effect of dopamine in the untreated cell. Finally, in the continued presence of dopamine, changing the parameters of the simulated Ih to those observed with dopamine reduced the first spike latency to that seen with dopamine in the untreated cell. 7. We generated a mathematical model of the lobster LP neuron, based on the model of Buchholtz et al. for the crab LP neuron.

Dopamine modulation of transient potassium current evokes phase shifts in a central pattern generator network.

Bath application of dopamine modifies the rhythmic motor pattern generated by the 14 neuron pyloric network in the stomatogastric ganglion of the spiny lobster, Panulirus interruptus. Among other effects, dopamine excites many of the pyloric constrictor (PY) neurons to fire at high frequency and phase-advances the timing of their activity in the motor pattern. These responses arise in part from direct actions of dopamine to modulate the intrinsic electrophysiological properties of the PY cells, and can be studied in synaptically isolated neurons. The rate of rebound following a hyperpolarizing prestep and the spike frequency during a subsequent depolarization are both accelerated by dopamine. Based on theoretical simulations, Hartline (1979) suggested that the rate of postinhibitory rebound in stomatogastric neurons could vary with the amount of voltage-sensitive transient potassium current (IA). Consistent with this prediction, we found that dopamine evokes a net conductance decrease in synaptically isolated PY neurons. In voltage clamp, dopamine reduces IA, specifically by reducing the amplitude of the slowly inactivating component of the current and shifting its voltage activation curve in the depolarized direction. 4-Aminopyridine, a selective blocker of IA in stomatogastric neurons, mimics and occludes the effects of dopamine on isolated PY neurons. A conductance-based mathematical model of the PY neuron shows appropriate changes in activity upon quantitative modification of the IA parameters affected by dopamine. These results demonstrate that dopamine excites and phase-advances the PY neurons in the rhythmic pyloric motor pattern at least in part by reducing the transient K+ current, IA.

An improved parameter estimation method for Hodgkin-Huxley models.

We consider whole-cell voltage-clamp data of isolated currents characterized by the Hodgkin-Huxley paradigm. We examine the errors associated with the typical parameter estimation method for these data and show them to be unsatisfactorally large especially if the time constants of activation and inactivation are not sufficiently separated. The size of these errors is due to the fact that the steady-state and kinetic properties of the current are estimated disjointly. We present an improved parameter estimation method that utilizes all of the information in the voltage-clamp conductance data to estimate steady-state and kinetic properties simultaneously and illustrate its success compared to the standard method using simulated data and data from P. interruptus shal channels expressed in oocytes.