The large number of variables involved in many biophysical models can obscure potentially simple dynamical mechanisms governing the properties of its solutions and the transitions between them. To address this issue, we extend a novel model reduction method, based on "scales of dominance," to multi-compartment models. We use this method to systematically reduce the dimension of a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation -- tonic spiking, intermediate bursting and strong bursting. A coarse analysis of the scales of dominance in a trajectory of this sixteen-variable model leads to a globally-reduced, nine-variable model. In a finer analysis we divide the trajectory into intervals dominated by a smaller number of variables, resulting in a locally-reduced hybrid model whose dimension varies between two and six for different temporal regimes. Both reduced models exhibit the same modes of oscillation as the sixteendimensional model over a comparable parameter range. The reduced models highlight low-dimensional organizing structure in the dynamics of the PD neuron, and the dependence of its oscillations on parameters such as the maximal conductances of calcium currents. In some intervals, the low-dimensional structure allows for a geometric analysis of the trajectory in the phase space, thus enabling predictions of the behavior of the full biophysical model. Our technique could be used to build hybrid low-dimensional models from any large multi-compartment conductance-based model in order to analyze transitions between different modes of activity.
A major challenge in contemporary computational neuroscience is the analysis of high-dimensional biophysically-realistic models. Mathematical approaches in computational neuroscience have progressed from analysis of abstract neural systems at steady state (Amit and Tsodyks 1991; Wilson and Cowan 1972) to more biologically realistic situations in which systems are in rhythmic or chaotic states (Kopell and LeMasson 1994; Terman 1991). A further increase in the biophysical sophistication of neural models has been driven by the availability of increasingly detailed electrophysiological and anatomical data about neuronal dynamics. Bifurcation theory (Guckenheimer and Holmes 1983; Strogatz 2001) is particularly helpful in understanding the qualitative change in the behavior of dynamical systems models as parameters are varied. However, the direct application of bifurcation theory becomes prohibitively difficult as the complexity of the model increases beyond a few dynamical variables, and heuristic arguments are commonly used to justify restricting analysis to smaller, approximate models (Fitzhugh 1961; Kopell et al. 2000; Meunier 1992). In this study, we use a new approach aimed at systematically determining appropriate reduced models from a detailed biophysical model without a priori simplifications. We begin by identifying temporally-localized sub-regimes of the detailed model's dynamics according to simple objective criteria. Within each of these sub-regimes we make a minimal local approximation to the full model--typically of much lower dimension. We then examine how the pattern of transitions between these sub-regimes depends on parameters and use these patterns to distinguish qualitatively different solutions to the model equations.
We use this novel reduction approach (Clewley et al. 2005) to study spiking and bursting in a conductance-based model of the pyloric dilator (PD) neuron, a member of the pacemaker ensemble of the pyloric network in the well-characterized crustacean stomatogastric nervous system (Marder and Bucher 2007; Nusbaum and Beenhakker 2002). The PD neuron typically spikes tonically when it is synaptically isolated from the network but, in some preparations, is also capable of producing rhythmic bursts of action potentials and thus it is considered a conditional burster (Marder 1984; Miller and Selverston 1982). A recent modeling study by Soto-Trevino et al. (2005) produced a biophysically-realistic model of the PD neuron and its electrically coupled counterpart, the anterior burster (AB) neuron. In this model, the proper activity of the PD model neuron, both in isolation and as part of the network, is crucially based on the fact that ionic currents responsible for spike production are spatially segregated from other voltage-gated ionic currents. This realistic model of the PD neuron suggests that several parameters can be responsible for the presence of these two types of output--tonic spiking or bursting--in different preparations. One such set of parameters is the maximal conductance of calcium currents, which have been shown to be modified by exogenous neuromodulators (Johnson et al. 2003). This focus on local parameter variation and bifurcation is complementary to the broader search for parameter ranges in which these behavioral regimes can be found (Prinz et al. 2004b).
We identify a transient critical time interval (referred to as a "critical sub-regime" of the model) that occurs during the inter-spike interval and involves only a subset of ionic currents that are essential for the transition. Using the mathematical technique of "dominant scales", developed by Clewley et al. (2005), we analyze the interactions among this select subset of ionic currents in order to characterize the differences in tonic and bursting activity. Our analysis elucidates the biological mechanisms underlying the change in qualitative behavior of the model by determining local and low-dimensional approximations to a high-dimensional biophysical model and serves as an example for the multitude of similar transitions in neuronal outputs that could be similarly investigated.