In hippocampal slice models of epilepsy, two behaviors are seen: short bursts of electrical activity lasting 100 msec and seizure-like electrical activity lasting seconds. The bursts originate from the CA3 region, where there is a high degree of recurrent excitatory connections. Seizures originate from the CA1, where there are fewer recurrent connections. In attempting to explain this behavior, we simulated model networks of excitatory neurons using several types of model neurons. The model neurons were connected in a ring containing predominantly local connections and some long-distance random connections, resulting in a small-world network connectivity pattern. By changing parameters such as the synaptic strengths, number of synapses per neuron, proportion of local versus long-distance connections, we induced "normal," "seizing," and "bursting" behaviors. Based on these simulations, we made a simple mathematical description of these networks under well-defined assumptions. This mathematical description explains how specific changes in the topology or synaptic strength in the model cause transitions from normal to seizing and then to bursting. These behaviors appear to be general properties of excitatory networks.
The PyDSTool software environment is designed to develop, simulate, and analyze dynamical systems models, particularly for biological applications. Unlike the engineering application focus and graphical specification environments of most general purpose simulation tools, PyDSTool provides a programmatic environment well suited to exploratory data- and hypothesis-driven biological modeling problems. In this work, we show how the environment facilitates the application of hybrid dynamical modeling to the reverse engineering of complex biophysical dynamics; in this case, of an excitable membrane. The example demonstrates how the software provides novel tools that support the inference and validation of mechanistic hypotheses and the inclusion of data constraints in both quantitative and qualitative ways. The biophysical application is broadly relevant to models in the biosciences. The open source and platform-independent PyDSTool package is freely available under the BSD license from http://sourceforge.net/projects/pydstool/. The hosting service provides links to documentation and online forums for user support.
We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the bifurcation analysis of fixed points and invariant curves of a Poincaré return maps for phase lags between cells. We explore different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps. Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our analysis does not require knowledge of the equations modeling the system and provides a powerful qualitative approach to studying detailed models of rhythmic behavior. Thus, our approach is applicable to a wide range of biological phenomena beyond motor control.
Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that aids the identification of structure in a class of systems of nonlinear ordinary differential equations (ODEs) that are commonly applied in dynamical models of physical processes. This "dominant scale" technique prioritizes consideration of the influence that distinguished "inputs" to an ODE have on its dynamics. On this basis a sequence of reduced models is derived, where each model is valid for a duration that is determined self-consistently as the system's state variables evolve. The characteristic time scales of all sufficiently dominant variables are also taken into account to further reduce the model. The result is a hybrid dynamical system of reduced differential-algebraic models that are switched at discrete event times.The technique does not rely on explicit small parameters in the ODEs and automatically detects changing scale separation both in time and in "dominance strength" (a quantity we derive to measure an input's influence). Reduced regimes describing the full system have quantified domains of validity in time and with respect to variation in state variables. This enables the qualitative analysis of the system near known orbits (e.g., to study bifurcations) without sole reliance on numerical shooting methods.These methods have been incorporated into a new software tool named Dssrt, which we demonstrate on a limit cycle of a synaptically driven Hodgkin-Huxley neuron model. Key words. multiple scale methods, computational methods, bifurcation analysis, methods for differential-algebraic equations, analytic approximation of solutions, biophysical neural networks AMS subject classifications. 33F05, 34E13, 37M20, 65L80, 74H10, 92-08, 92C20, 93B35 1. Introduction. Systems of ordinary differential equations (ODEs) arise commonly as models in the natural sciences. Many of these models involve nonlinear dynamics and exhibit complex behavior [26,52]. To understand this behavior, it is useful to have tools that allow us to focus on salient features of the equations and to dissect the qualitative structure of their dynamics.There is an assumption of one or more explicit small parameters in many popular models that exhibit multiple scales, for instance the oscillators of van der Pol [64], FitzHugh [18], Morris and LeCar [51], and Wilson and Cowan [71] and other weakly coupled oscillators [32,41,45,49]. Typically these systems are amenable to a standard use of geometrical singular perturbation theory [14,34] and invariant manifold theory [70]. The systems are first understood at "fast" and "slow" singular limits as the small parameter vanishes. Subsequently, technical results are invoked that prove qualitatively similar orbits lie close to the singular orbits when these subsystems are matched [50] and the small parameter is...
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