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Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons
FitzHugh-Nagumo 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 well-known FitzHugh-Nagumo 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 self-sustained 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), 99-111.
doi: 10.1016/j.laa.2011.06.035. |
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N. Burić and D. Todorović, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E (3), 436 (2012), 99-111.
doi: 10.1103/PhysRevE.67.066222. |
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A. Cattani, Generalized Diffusion to Model Biological Neural Networks, Ph.D thesis, Politecnico di Torino, ongoing. |
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R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
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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), 500-544. |
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J. D. Murray, Mathematical Biology I, An Introduction third edition, Springer-Verlag, New York, 2002. |
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Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHugh-Nagumo neurons on random networks, Progress of Theoretical Physics Supplements, 161 (2006), 389-392.
doi: 10.1143/PTPS.161.389. |
[8] |
A. C.Scott, The electrophysics of a nerve fiber, Review of Modern Physics, 47 (1975), 487-533. |
show all references
References:
[1] |
R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs, Linear Algebra Appl., 436 (2012), 99-111.
doi: 10.1016/j.laa.2011.06.035. |
[2] |
N. Burić and D. Todorović, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E (3), 436 (2012), 99-111.
doi: 10.1103/PhysRevE.67.066222. |
[3] |
A. Cattani, Generalized Diffusion to Model Biological Neural Networks, Ph.D thesis, Politecnico di Torino, ongoing. |
[4] |
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[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), 500-544. |
[6] |
J. D. Murray, Mathematical Biology I, An Introduction third edition, Springer-Verlag, New York, 2002. |
[7] |
Y. Oyama, T. Yanagita and T. Ichinomiya, Numerical analysis of FitzHugh-Nagumo neurons on random networks, Progress of Theoretical Physics Supplements, 161 (2006), 389-392.
doi: 10.1143/PTPS.161.389. |
[8] |
A. C.Scott, The electrophysics of a nerve fiber, Review of Modern Physics, 47 (1975), 487-533. |
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