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Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis
Grégory Faye1, James Rankin, Pascal Chossat
1NeuroMathComp Laboratory, INRIA, ENS Paris, 2004 Route des Lucioles, BP 93, 06902, Sophia-Antipolis, France. gregory.faye@inria.fr
Journal of Mathematical Biology
|April 25, 2012
Summary
Localized solutions in neural networks, crucial for working memory, were proven to exist using normal form theory and numerical continuation. These findings map the parameter regions where these memory-encoding solutions persist.
Area of Science:
- Computational Neuroscience
- Mathematical Neuroscience
- Dynamical Systems Theory
Background:
- Spatially localized solutions in neural networks are theorized to represent working memory.
- Understanding the conditions for their existence and stability is key to understanding memory mechanisms.
- Previous models often relied on simplified network structures.
Purpose of the Study:
- To mathematically prove the existence of spatially localized solutions in an unbounded neural network model.
- To analyze the stability of these localized solutions.
- To determine the parameter space where these solutions are stable and persistent.
Main Methods:
- Utilizing a neural field equation with a wizard hat spatial connectivity.
- Applying normal form theory for a reversible Hopf bifurcation.
- Employing numerical continuation to compute solution branches and analyze snaking behavior.
Main Results:
- Existence of localized solutions proven via normal form theory, linked to homoclinic orbits.
- Stability analysis of these solutions presented.
- Identification of parameter regions defining the persistence of localized solutions, including snaking dynamics.
Conclusions:
- The study confirms the existence of localized solutions in a detailed neural network model, providing a mathematical basis for working memory.
- Normal form theory and numerical continuation are effective tools for analyzing complex neural dynamics.
- The identified parameter regions offer insights into the robustness and conditions for memory representation.
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