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An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
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Path integrals and large deviations in stochastic hybrid systems.

Paul C Bressloff1, Jay M Newby2

  • 1Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 16, 2014
PubMed
Summary
This summary is machine-generated.

We developed a path-integral method for stochastic hybrid systems, enabling the calculation of most probable escape paths and quasipotentials for metastable states. This approach aids in understanding complex systems like neural networks.

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Area of Science:

  • Mathematical Physics
  • Computational Neuroscience
  • Biophysics

Background:

  • Stochastic hybrid systems combine continuous dynamics with discrete Markov processes.
  • These systems model phenomena in ion channels, gene networks, and neural networks.
  • Understanding escape dynamics from metastable states is crucial in these fields.

Purpose of the Study:

  • To develop a path-integral representation for stochastic hybrid systems.
  • To derive a large deviation action principle for these systems.
  • To determine most probable escape paths and calculate quasipotentials.

Main Methods:

  • Constructing a path-integral representation for solutions.
  • Deriving a large deviation action principle.
  • Minimizing an action functional over all trajectories from a metastable state.

Main Results:

  • The path-integral representation allows for the derivation of a large deviation action principle.
  • Minimizing the action functional identifies most probable escape paths.
  • Evaluating the action functional yields the quasipotential for mean first passage time calculations.

Conclusions:

  • The developed path-integral method provides a powerful tool for analyzing stochastic hybrid systems.
  • This framework is applicable to understanding escape dynamics in complex systems, such as bistable neural networks.
  • The method facilitates the calculation of key quantities like quasipotentials and most probable paths.