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

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.

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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

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