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Gaussdiffusion processes for modeling the dynamics of a couple of interacting neurons
FitzHughNagumo equations with generalized diffusive coupling
1.  Department of Mathematical Sciences, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 
This approach enables us to address three fundamental issues. Firstly, each neuron is described using the wellknown FitzHughNagumo model which might allow to differentiate their individual behaviour. Furthermore, exploiting the Laplacian matrix, a well defined connection structure is formalized. Finally, random networks and an ensemble of excitatory and inhibitory synapses are considered.
Several simulations are performed to graphically present how dynamics within a network evolve. Thanks to an appropriate initial stimulus a wave is created: it propagates in a selfsustained way through the whole set of neurons. A novel graphical representation of the dynamics is shown.
References:
[1] 
R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs,, Linear Algebra Appl., 436 (2012), 99. doi: 10.1016/j.laa.2011.06.035. Google Scholar 
[2] 
N. Burić and D. Todorović, Dynamics of FitzHughNagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 436 (2012), 99. doi: 10.1103/PhysRevE.67.066222. Google Scholar 
[3] 
A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). Google Scholar 
[4] 
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S00063495(61)869026. Google Scholar 
[5] 
A. L. Hodgkin and A. F.Huxley, A quantitative description of membrane current and its application in conduction and excitation in nerve,, J. Physiol., 117 (1952), 500. Google Scholar 
[6] 
J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002). Google Scholar 
[7] 
Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHughNagumo neurons on random networks,, Progress of Theoretical Physics Supplements, 161 (2006), 389. doi: 10.1143/PTPS.161.389. Google Scholar 
[8] 
A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487. Google Scholar 
show all references
References:
[1] 
R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs,, Linear Algebra Appl., 436 (2012), 99. doi: 10.1016/j.laa.2011.06.035. Google Scholar 
[2] 
N. Burić and D. Todorović, Dynamics of FitzHughNagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 436 (2012), 99. doi: 10.1103/PhysRevE.67.066222. Google Scholar 
[3] 
A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). Google Scholar 
[4] 
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S00063495(61)869026. Google Scholar 
[5] 
A. L. Hodgkin and A. F.Huxley, A quantitative description of membrane current and its application in conduction and excitation in nerve,, J. Physiol., 117 (1952), 500. Google Scholar 
[6] 
J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002). Google Scholar 
[7] 
Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHughNagumo neurons on random networks,, Progress of Theoretical Physics Supplements, 161 (2006), 389. doi: 10.1143/PTPS.161.389. Google Scholar 
[8] 
A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487. Google Scholar 
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