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On a spike train probability model with interacting neural units
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.
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A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). 
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R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S00063495(61)869026. 
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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. 
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J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002). 
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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. 
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A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487. 
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. 
[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. 
[3] 
A. Cattani, Generalized Diffusion to Model Biological Neural Networks,, Ph.D thesis, (). 
[4] 
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophysical Journal, 1 (1961), 445. doi: 10.1016/S00063495(61)869026. 
[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. 
[6] 
J. D. Murray, Mathematical Biology I, An Introduction, third edition, (2002). 
[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. 
[8] 
A. C.Scott, The electrophysics of a nerve fiber,, Review of Modern Physics, 47 (1975), 487. 
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