The study and comparison of sequences of characters from a finite alphabet is relevant to various areas of science, notably molecular biology. The measurement of sequence similarity involves the consideration of the different possible sequence alignments in order to find an optimal one for which the "distance" between sequences is minimum. By associating a path in a lattice to each alignment, a geometric insight can be brought into the problem of finding an optimal alignment. This problem can then be solved by applying a dynamic programming algorithm. However, the computational effort grows rapidly with the number N of sequences to be compared (O(I N ), where is the mean length of the sequences to be compared).It is proved here that knowledge of the measure of an arbitrarily chosen alignment can be used in combination with information from the pairwise alignments to considerably restrict the size of the region of the lattice in consideration. This reduction implies fewer computations and less memory space needed to carry out the dynamic programming optimization process. The observations also suggest new variants of the multiple alignment problem.
An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tissue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane. The phase model is motivated by Rinzel and Lee's dissection of a model for neuronal parabolic bursting (J. Math. Biol. 25, 653-675 (1987)). Rapid spiking is represented canonically by a one-variable phase equation that is coupled bi-directionally to a two-variable slow system. The model is analyzed in the slow-variable phase plane, using quasi steadystate assumptions and formal averaging. We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spiking. The relative speed of activation and inactivation of the slow variables strongly influences the burst pattern as well as other dynamics. We find conditions of the bistability of solutions between continuous spiking and bursting. Although the phase model is simple, we demonstrate that it captures many dynamical features of more complex biophysical models.
Many physical and biological phenomena involve accumulation and discharge processes that can occur on significantly different time scales. Models of these processes have contributed to understand excitability self-sustained oscillations and synchronization in arrays of oscillators. Integrate-and-fire (I+F) models are popular minimal fill-and-flush mathematical models. They are used in neuroscience to study spiking and phase locking in single neuron membranes, large scale neural networks, and in a variety of applications in physics and electrical engineering. We show here how the classical first-order I+F model fits into the theory of nonlinear oscillators of van der Pol type by demonstrating that a particular second-order oscillator having small parameters converges in a singular perturbation limit to the I+F model. In this sense, our study provides a novel unfolding of such models and it identifies a constructible electronic circuit that is closely related to I+F.
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