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Non-Markovian processes on heteroclinic networks.

Gray Manicom¹, Vivien Kirk¹, Claire Postlethwaite¹

¹Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand.

Chaos (Woodbury, N.Y.)

|March 12, 2024

Summary

This summary is machine-generated.

This study explores how noise-induced lift-off in heteroclinic networks affects Markov chain dynamics. We analyze the persistence of lift-off effects and propose methods to determine Markov chain order and transition probabilities.

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Area of Science:

Dynamical Systems and Chaos Theory
Stochastic Processes
Computational Neuroscience

Background:

Heteroclinic networks feature saddle equilibria connected by trajectories.
Noise can induce phenomena like 'lift-off' affecting state transitions.
The sequence of visited saddles can be modeled as a stochastic process.

Purpose of the Study:

Investigate the impact of lift-off at one saddle on subsequent saddle dynamics.
Determine the order of the associated Markov chain of states.
Calculate transition probabilities for the Markov chain.

Main Methods:

Review and extend Bakhtin's methods for saddle dynamics with noise.
Analyze the persistence of lift-off effects across saddles.
Develop a method to find a lower bound for Markov chain order.
Numerically simulate results for various noise amplitudes.

Main Results:

Lift-off at one saddle can influence dynamics at the next saddle.
Conditions for the persistence of lift-off effects were identified.
A method for bounding the Markov chain order was proposed.
Numerical simulations validated theoretical predictions across noise levels.

Conclusions:

Lift-off is a critical phenomenon influencing the Markovian or non-Markovian nature of state sequences in heteroclinic networks.
The study provides tools to analyze and predict the behavior of these systems under noise.
Understanding these dynamics is crucial for applications in fields like computational neuroscience.