Dale’s Principle Is Necessary for an Optimal Neuronal Network’s Dynamics

Author(s)
Eleonora Catsigeras

Affiliation(s)

Instituto de Matemática y Estadstica “Prof. Ing. Rafael Laguardia”, Universidad de la República, Montevideo, Uruguay.

Instituto de Matemática y Estadstica “Prof. Ing. Rafael Laguardia”, Universidad de la República, Montevideo, Uruguay.

ABSTRACT

We study a mathematical model of biological neuronal
networks composed by any finite number *N* ≥ 2 of non-necessarily
identical cells. The model is a deterministic dynamical system governed by
finite-dimensional impulsive differential equations. The statical structure of
the network is described by a directed and weighted graph whose nodes are certain
subsets of neurons, and whose edges are the groups of synaptical connections among
those subsets. First, we prove that among all the possible networks such as their
respective graphs are mutually isomorphic, there exists a dynamical optimum.
This optimal network exhibits the richest dynamics: namely, it is capable to
show the most diverse set of responses (*i.e.* orbits in the future) under external stimulus or signals. Second, we prove that
all the neurons of a dynamically optimal neuronal network necessarily satisfy
Dale’s Principle, *i.e.* each neuron
must be either excitatory or inhibitory, but not mixed. So, Dale’s Principle is
a mathematical necessary consequence of a theoretic optimization process of the
dynamics of the network. Finally, we prove that Dale’s Principle is not
sufficient for the dynamical optimization of the network.

Cite this paper

E. Catsigeras, "Dale’s Principle Is Necessary for an Optimal Neuronal Network’s Dynamics,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 15-29. doi: 10.4236/am.2013.410A2002.

E. Catsigeras, "Dale’s Principle Is Necessary for an Optimal Neuronal Network’s Dynamics,"

References

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[13] E. Catsigeras and P. Guiraud, “Integrate and Fire Neural Networks, Piecewise Contractive Maps and Limit Cycles,” Journal of Mathematical Biology, 2013, in Press. http://dx.doi.org/10.1007/s00285-012-0560-7

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j.apm.2006.04.008

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[17] M. Megas, Z. S. Emri, T. F. Freund and A. I. Gulyas, “Total Number and Distribution of Inhibitory and excitatory Synapses on Hippocampal CA1 Pyramidal Cells,” Neuroscience, Vol. 102, No. 3, 2001, pp. 527-540. http://dx.doi.org/10.1016/S0306-4522(00)00496-6

[18] J. Van Mill, “Domain Invariance,” In: M. Hazewinkel, Ed., Encyclopedia of Mathematics, Springer, Berlin, 20012003. http://www.springer.com/mathematics/book/978-1-4020-0609-8

[1] P. Strata and R. Harvey, “Dale’s Principle,” Brain Research Bulletin, Vol. 50, No. 5-6, 1999, pp. 349-350. http://dx.doi.org/10.1016/S0361-9230(99)00100-8

[2] M. F. Bear, B. W. Connors and M. A. Paradiso, “Neuroscience—Exploring the Brain,” 3rd Edition, Lippincott Williams & Wilkins, Philadelphia, 2007

[3] G. Burnstock, “Cotransmission,” Current Opinion in Pharmacology, Vol. 4, No. 1, 2004, pp. 47-52. http://dx.doi.org/10.1016/j.coph.2003.08.001

[4] L. E. Trudeau and R. Gutiérrez, “On Cotransmission & Neurotransmitter Phenotype Plasticity,” Molecular Interventions, Vol. 7, No. 3, 2007, pp. 138-146. http://dx.doi.org/10.1124/mi.7.3.5

[5] R. E. Mirollo and S. H. Strogatz, “Synchronization of Pulse Coupled Biological Oscillators,” SIAM Journal on Applied Mathematics, Vol. 50, No. 6, 1990, pp. 16451662. http://dx.doi.org/10.1137/0150098

[6] L. Gómez and R. Budelli, “Two-Neuron Networks II: Leaky Integrator Pacemaker Models,” Biological Cybernetics, Vol. 74, No. 2, 1996, pp. 131-137. http://dx.doi.org/10.1007/BF00204201

[7] W. Mass and C. M. Bishop, “Pulsed Neural Networks,” MIT Press, Cambridge, 2001.

[8] G. B. Ermentrout and D. H. Terman, “Mathematical Foundations of Neuroscience,” In: Interdisciplinary Applied Mathematics, Springer, New York, 2010. http://link.springer.com/book/10.1007/978-0-387-87708-2/page/1

[9] W. Gerstner and W. Kistler, “Spiking Neuron Models,” Cambridge University Press, Cambridge, 2002. http://dx.doi.org/10.1017/CBO9780511815706

[10] K. K. Lin, K. C. A. Wedgwood, S. Coombes and L.-S. Young, “Limitations of Perturbative Techniques in the Analysis of Rhythms and Oscillations,” Journal of Mathematical Biology, Vol. 66, No. 1-2, 2013, pp. 139-161. http://dx.doi.org/10.1007/s00285-012-0506-0

[11] E. M. Izhikevich, “Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting,” MIT Press, Cambridge, 2007.

[12] A. J. Catllá, D. G. Schaeffer, T. P. Witelski, E. E. Monson and A. L. Lin, “On Spiking Models for Synaptic Activity and Impulsive Differential Equations,” SIAM Review, Vol. 50, No. 3, 2008, pp. 553-569. http://dx.doi.org/10.1137/060667980

[13] E. Catsigeras and P. Guiraud, “Integrate and Fire Neural Networks, Piecewise Contractive Maps and Limit Cycles,” Journal of Mathematical Biology, 2013, in Press. http://dx.doi.org/10.1007/s00285-012-0560-7

[14] V. D. Milman and A. D. Myshkis, “On the Stability of Motion in the Presence of Impulses (in Russian),” Siberian Mathematical Journal, Vol. 1, No. 2, 1960, pp. 233237.

[15] G. T. Stamov and I. Stamova, “Almost Periodic Solutions for Impulsive Neural Networks with Delay,” Applied Mathematical Modelling, Vol. 31, No. 7, 2007, pp. 12631270. http://dx.doi.org/10.1016/

j.apm.2006.04.008

[16] R. F. Schmidt and G. Thews, “Human Physiology,” SpringerVerlag, Berlin, 1983.

[17] M. Megas, Z. S. Emri, T. F. Freund and A. I. Gulyas, “Total Number and Distribution of Inhibitory and excitatory Synapses on Hippocampal CA1 Pyramidal Cells,” Neuroscience, Vol. 102, No. 3, 2001, pp. 527-540. http://dx.doi.org/10.1016/S0306-4522(00)00496-6

[18] J. Van Mill, “Domain Invariance,” In: M. Hazewinkel, Ed., Encyclopedia of Mathematics, Springer, Berlin, 20012003. http://www.springer.com/mathematics/book/978-1-4020-0609-8