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Snaking behavior of homoclinic solutions in a neural field model
BMC Neuroscience volume 10, Article number: P297 (2009)
Introduction
In our work, we investigate stationary homoclinic solutions of a neural field model with Mexican hat connectivity. Homoclinic solutions, often called "bumps," represent local activity of neural tissue in a state of global quiescence and are related to short-term memory. The solutions have a snake-like shape in the bifurcation diagram. Therefore the evolution of multiple bump solutions are often called "snaking." The scaled model is reduced to parameters of the firing rate function that represents the averaged spike rate of neurons. It can be presented in either an integro-differential equation or an ordinary differential equation (ODE). We investigate the range of parameters in which single bump and multiple bump solutions exist.
Method
To cope with our model we choose an ansatz developed in [1]. It makes use of the integrability of the ODE and of physiologically reasonable boundary conditions. Further, the symmetry of the system is exploited. This approach allows us to reduce the free parameters of the solutions to one. The remaining free parameter is determined by continuation of the boundary conditions and checking the resulting solutions for symmetry. As to general firing rate functions, this method has proven to be advantageous in comparison to shooting methods. In addition to this we investigate the stability of the homoclinic solution by using an ansatz presented in [2]. It approximates the firing rate function to a step function and delivers 2N eigenvalues for N-bump solutions.
Conclusion
The neural field model with Mexican hat connectivity produces stable single and multiple bump solutions. The existence of these solutions depends on parameters shaping the firing rate function. It turned out that homoclinic solutions exist only for low firing thresholds. Regarding the fact that just one neuron type is involved, it is still arguable that our results foster insights to the physical basis of short-term memory.
References
Elvin AJ: Pattern Formation in a Neural Field Model, PhD thesis. Massey University, Auckland, New Zealand
Coombes S, Owen MR: Evans functions for integral neural field equations with Heaviside firing rate function. SIAM Journal on Applied Dynamical Systems. 34: 574-600.
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Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Schmidt, H., Coombes, S. Snaking behavior of homoclinic solutions in a neural field model. BMC Neurosci 10 (Suppl 1), P297 (2009). https://doi.org/10.1186/1471-2202-10-S1-P297
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DOI: https://doi.org/10.1186/1471-2202-10-S1-P297