Mathematical foundations of the dendritic growth models

Abstract

At present two growth models describe successfully the distribution of size and topological complexity in populations of dendritic trees with considerable accu- racy and simplicity, the BE model (Van Pelt et al. in J. Comp. Neurol. 387:325–340, 1997) and the S model (Van Pelt and Verwer in Bull. Math. Biol. 48:197–211, 1986). This paper discusses the mathematical basis of these models and analyzes quantita- tively the relationship between the BE model and the S model assumed in the literature by developing a new explicit equation describing the BES model (a dendritic growth model integrating the features of both preceding models; Van Pelt et al. in J. Comp. Neurol. 387:325–340, 1997). In numerous studies it is implicitly presupposed that the S model is conditionally linked to the BE model (Granato and Van Pelt in Brain Res. Dev. Brain Res. 142:223–227, 2003; Uylings and Van Pelt in Network 13:397–414, 2002; Van Pelt, Dityatev and Uylings in J. Comp. Neurol. 387:325–340, 1997; Van Pelt and Schierwagen in Math. Biosci. 188:147–155, 2004; Van Pelt and Uylings in Network. 13:261–281, 2002; Van Pelt, Van Ooyen and Uylings in Modeling Dendritic Geometry and the Development of Nerve Connections, pp 179, 2000). In this paper we prove the non-exactness of this assumption, quantify involved errors and determine the conditions under which the BE and S models can be separately used instead of the BES model, which is more exact but considerably more difficult to apply. This study leads to a novel expression describing the BE model in an analytical closed form, much more efficient than the traditional iterative equation (Van Pelt et al. in J. Comp. Neurol. 387:325–340, 1997) in many neuronal classes. Finally we propose a new algorithm in order to obtain the values of the parameters of the BE model when this growth model is matched to experimental data, and discuss its advantages and improvements over the more commonly used procedures.