- Poster presentation
- Open Access
FitzHugh-Nagumo to model a large number of diffusive coupled neurons
© Cattani and Canuto; licensee BioMed Central Ltd. 2013
- Published: 8 July 2013
- Mathematical Structure
- Diffusive Term
- Qualitative Behavior
- Length Interval
- Laplacian Matrix
The aim of our work is to investigate the dynamics of a neural network, in which neurons, individually described by the FitzHugh-Nagumo model , are coupled by a generalized diffusive term. The formulation we exploit is based on the general framework of graph theory, where neurons are represented by vertices and links by edges.
With the aim of defining the connection structure among the excitable elements, the discrete Laplacian matrix plays a fundamental role. Indeed, it allows us to model the instantaneous propagation of (electric) signals between neurons, which need not be physically close to each other. This approach enables us to address three fundamental issues. Firstly, each neuron is described using the well-known FitzHugh-Nagumo model which might allow to differentiate their individual behavior. Furthermore, exploiting the Laplacian matrix, a well defined connection structure is formalized. A thoroughly explained mathematical structure allows us to formally describe several fundamental features of interactions in neural populations. Indeed, the description of not only nearest neighbor interactions and the presence of inhibitory synapses has been achieved. Several simulations are performed to graphically present how the action potential within a network evolves. In general, placing the neuron in line and stimulating one of them, two waves which carry out the connection rule arise.
A continuum of neurons is the results of the limit process of letting N to infinity and a system of convection/reaction/diffusion partial differential equations describes the action potential and the recovery variable in the whole set of neurons.
Finally, our two approaches immediately extend to higher dimensions.
This work is the result of the initial stages of the first author's Ph.D. thesis project. I would like to thank Luigi Preziosi for his advice and guidance.
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.