Stability of Traveling Fronts in a Neural Field Model
Abstract
:1. Introduction
2. Materials and Methods
2.1. Coupling Function, Gain Function, and Front Solutions Considered
- on an interval , and ,
- is decreasing on ,
- on ,
- is continuous on , and is integrable on ,
- has a unique minimum on such that , and is strictly increasing on .
2.2. Formation of the 5th Order ODE
2.3. Matching Conditions for the ODE
2.4. Forms for Eigenfunction
2.4.1. Eigenfunction Forms When ()
- If , then is the transition between either cases 1 and 2, cases 2 and 4, or cases 3 and 4.
- For the first two transitions mentioned above, the repeated roots are . The solution form is
- If the transition is between case 3 and case 4, the solution form is the same as case 3.
- If , then is the transition between either cases 4 and 5, cases 6 and 8, or cases 7 and 8.
- For the transition from 4 to 5 or 6 to 8, the repeated roots are . The solution form is the same as case 6.
- For the transition from 7 to 8, the solution form is the same as case 7.
2.4.2. Eigenfunction Forms When ()
2.5. Evans Function for Stability Analysis
3. Results
3.1. Case L1 Analysis
3.2. Case L2 Analysis
3.3. Case L3 Analysis
3.4. Case L4 Analysis
3.5. Case L5 Analysis
3.6. Case L6 Analysis
4. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Details of (16c), (16d) in Lemma (1)
Appendix B. Examples of M(γ) Used in the Evans Function in Equation (38)
Appendix C. Explicit Matrix: L1 Case, α = 0, γ ∈
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Macaluso, D.; Guo, Y. Stability of Traveling Fronts in a Neural Field Model. Mathematics 2023, 11, 2202. https://doi.org/10.3390/math11092202
Macaluso D, Guo Y. Stability of Traveling Fronts in a Neural Field Model. Mathematics. 2023; 11(9):2202. https://doi.org/10.3390/math11092202
Chicago/Turabian StyleMacaluso, Dominick, and Yixin Guo. 2023. "Stability of Traveling Fronts in a Neural Field Model" Mathematics 11, no. 9: 2202. https://doi.org/10.3390/math11092202
APA StyleMacaluso, D., & Guo, Y. (2023). Stability of Traveling Fronts in a Neural Field Model. Mathematics, 11(9), 2202. https://doi.org/10.3390/math11092202