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Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain
Xiaofeng Liao1, Shaowen Li, Guanrong Chen
1Department of Computer Science and Engineering, Chongqing University, China. xfliao@cqu.edu.cn
Summary
This study reveals that Hopf bifurcation occurs in a two-neuron model with distributed delays when the mean delay is the bifurcation parameter. This leads to new periodic solutions in neural network dynamics.
Area of Science:
- Computational Neuroscience
- Mathematical Biology
- Dynamical Systems
Background:
- Neural networks exhibit complex dynamics influenced by delays.
- Understanding bifurcations is crucial for analyzing neural oscillations and information processing.
Purpose of the Study:
- Investigate the dynamic behavior of a general two-neuron model with distributed delays.
- Determine the conditions for Hopf bifurcation in this model.
Main Methods:
- Frequency domain approach applied to the characteristic equation.
- Analysis of bifurcation parameter (mean delay).
- Nyquist criterion and graphical Hopf bifurcation theorem for stability analysis.
Main Results:
- Existence of a bifurcation parameter confirmed.
- Hopf bifurcation occurs for a strong kernel when mean delay is the bifurcation parameter.
- Periodic solutions bifurcate from equilibrium beyond a critical delay value.
Conclusions:
- The two-neuron model demonstrates Hopf bifurcation, leading to periodic neural activity.
- The findings provide insights into the emergence of oscillations in neural systems.
- Theoretical results are validated through numerical simulations.