Knapper, Drake

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ScholarWorks @ Georgia State University

GSURC Poster Presentation ProposalDrake E. Knapper1 TitleQuantitative and Qualitative Stability Anaylsis of Polyrhythmic Circuits2 AuthorsDrake E. Knapper, Justus T. C. Schwabedal, and Andrey L. Shilnikov3 Faculty SponserAndrey Shilnikov, Professor, Neuroscience Institute, Department of Mathematics and Statistics4 Abstract4.1 IntroductionIn many applications, circuits of relaxation oscillators are used to model behavior in an oscillitory system.The main motivation for this research is the application of simplified networks of relaxation oscillatorsto neurons. Many neuronal models are complex, and thus computationally expensive. We wish to showthe simplified model we use exhibits similar dynamics as the more complex networks, to simplify study ofneuronal behavior.4.2 MethodIn this study we use a novel qualitative method of plotting basins of attraction of stable fixed points, in orderto find polyrhythmic stability as well as network bifurcations with parameter variations. We also highlighta method used previously in a different model, and show its advantages and disadvantages.4.3 ResultsThe multiple approaches taken in the study all yield the same results, validating our qualitative anaylsesof these circuits. Whe have various forms of bifurcations in the network that all correspond to differnetparameter changes, that are detected in the qualitative analysis, as well as the quantitative and using acombonation of the methods, we can now predict when a bifurcation will occur in the network.4.4 ConclusionUsing the knowledge of the polyrhythmic behavior these network exhibit, we can focus on figuring outalgorithms for learning, and enforcing specific behavior. Also, future analysis of larger networks, and differentmodels is now simplified with our novel qualitative methods.1

Quantitative and Qualitative Stability Analysis of Polyrhythmic Circuits

Georgia State University

GSURC Poster Presentation Proposal
Drake E. Knapper
1 Title
Quantitative and Qualitative Stability Anaylsis of Polyrhythmic Circuits
2 Authors
Drake E. Knapper, Justus T. C. Schwabedal, and Andrey L. Shilnikov
3 Faculty Sponser
Andrey Shilnikov, Professor, Neuroscience Institute, Department of Mathematics and Statistics
4 Abstract
4.1 Introduction
In many applications, circuits of relaxation oscillators are used to model behavior in an oscillitory system.
The main motivation for this research is the application of simplified networks of relaxation oscillators
to neurons. Many neuronal models are complex, and thus computationally expensive. We wish to show
the simplified model we use exhibits similar dynamics as the more complex networks, to simplify study of
neuronal behavior.
4.2 Method
In this study we use a novel qualitative method of plotting basins of attraction of stable fixed points, in order
to find polyrhythmic stability as well as network bifurcations with parameter variations. We also highlight
a method used previously in a different model, and show its advantages and disadvantages.
4.3 Results
The multiple approaches taken in the study all yield the same results, validating our qualitative anaylses
of these circuits. Whe have various forms of bifurcations in the network that all correspond to differnet
parameter changes, that are detected in the qualitative analysis, as well as the quantitative and using a
combonation of the methods, we can now predict when a bifurcation will occur in the network.
4.4 Conclusion
Using the knowledge of the polyrhythmic behavior these network exhibit, we can focus on figuring out
algorithms for learning, and enforcing specific behavior. Also, future analysis of larger networks, and different
models is now simplified with our novel qualitative methods.
1


https://scholarworks.gsu.edu/cgi/viewcontent.cgi?article=1360&amp;context=gsurc

Quantitative and Qualitative Stability Analysis of Polyrhythmic Circuits

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