Qualitative features of the dynamics of excitable/oscillatory processes are shared by broad classes of neuronal models. These features are expressed in models for single cell behavior as well as for ensemble activity, and they include excitability and threshold behavior, beating and bursting oscillations and phase-locking, bistability and hysteresis, etc. Our goal here is to illustrate, by exploiting a specific model of excitable membrane, some of the concepts and techniques which can be used to understand, predict, and interpret biophysically these dynamic phenomena. The mathematical methods include numerical integration of the model equations, graphical/geometric representation of the dynamics (phase plane analysis), and analytic formulae for characterizing thresholds and stability conditions. The concepts are from the qualitative theory of nonlinear differential equations and nonlinear oscillations, and from perturbation and bifurcation theory. In this brief chapter, we will not consider the spatio-temporal aspects of distributed systems so the methods apply directly only to a membrane patch, to a spatially uniform, equipotential cell, or to a network with each cell type perfectly synchronized.
One may be familiar with models which exhibit one or two of the different dynamic behaviors, e.g. generation of individual or repetitive action potentials. However, one should realize that a given model, even a seemingly simple one, may display a great variety of response characteristics when a broad range of parameters is considered. This means that a given cell or ensemble may behave in many different modes, e.g. as a generator of single pulses, as a bursting pacemaker, as a bistable ``plateauing'' cell, or as a beating oscillator, depending upon the physiological conditions (neuromodulator or ionic concentrations) or stimulus presentations (applied currents or synaptic inputs). The nonlinear nature of the models provides the substrate for this broad repertoire; in contrast, linear models may be characterized by exponential and/or oscillatory time courses over their entire parameter ranges. It is important when studying a nonlinear model that stimulus-response properties be considered over ranges of the biophysical parameters.
Here, we show that a simple, but biophysically reasonable, two-current excitable membrane model is sufficiently robust to exhibit such behavioral richness, as parameters are systematically varied. By adjusting channel densities, activation dynamics, and stimulus intensities, we find that the cell can exhibit quite different threshold characteristics for spike generation (finite or infinite latency, with or without intermediate amplitude responses) and for onset of repetitive firing (finite or zero minimum frequency). The cell shows various types of bistable behavior: two different rest states in one case, and, in another case, a rest state with a coexistent oscillatory response around a depolarized level. The latter situation can provide a mechanism for rhythmic bursting when additional slower processes (e.g., slow channel kinetics, or a channel affected by slow ion accumulation) respond differently at the two potential levels. The spike-generating dynamics significantly influence the burst's waveform so there can be several different types of bursting depending on the nature of the fast dynamics; e.g., parabolic bursting does not depend on bistability in the spike-generating processes. Finally, by considering the phase-resetting behavior for a self-oscillatory cell, we show that the response to a single brief, arbitrarily-timed, perturbing stimulus can often be used to predict phase-locking responses to periodic stimulation, and to predict the synchronization properties of weakly coupled cells.
The underlying qualitative structure for these behaviors will be revealed here with graphical phase plane analysis, complemented by a few analytic formulae. The concepts we will cover include steady states, trajectories, limit cycles, stability, domains of attraction, and bifurcation of solutions. Phase plane characteristics and system dynamics will be interpreted biophysically in terms of activation curves, current-voltage relations, etc. A user-friendly program XPP (developed by G.B.Ermentrout) for X-windows computers allows one to interactively generate, explore, and visualize most of the behaviors described here in the same spirit as an experimental ``setup". Its numerical procedures are summarized in Appendix B. The concepts apply to higher order systems, and in many cases appropriate projections of phase space, motivated by differences in time scales for certain variables, can lead to similar insights.