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Snaking behavior of homoclinic solutions in a neural field model
BMC Neuroscience
Poster presentation Snaking behavior of homoclinic solutions in a neural field model Helmut Schmidt* and Stephen Coombes
0 References 1. Elvin AJ: Pattern Formation in a Neural Field Model, PhD thesis Massey University , Auckland , New Zealand. 2. 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
1 Address: School of Mathematical Sciences, University of Nottingham, University Park , Nottingham, NG7 2RD , UK
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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. (...truncated)