Stochastic Models of Neuronal Growth

Neurons are the basic cells of the nervous system. During their growth neurons extend two types of processes: axons and dendrites, which navigate to other neurons and form complex neuronal networks that transmit electrical signals throughout the body. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body towards target neurons. During growth axons sense their environment, process information, and respond to external stimuli by adapting their growth dynamics. Environmental stimuli include intercellular interactions, biochemical cues, and the mechanical and geometrical features of the growth substrate. Axonal dynamics is controlled both by deterministic components (tendency to grow in certain preferred directions imparted by surface mechanical and geometrical properties), and a random deviation from these growth directions due to stochastic processes. Several types of stochastic processes are involved in the growth cone dynamics including: intercellular signaling, detection of biomolecules at very small concentrations, biochemical reactions taking place in the growth cone, signal transduction, formation of lamellipodia and filopodia, and the polymerization of actin filaments and microtubules that steer the growth cone. Despite recent impressive advances in our understanding of how neurons grow and form functional connections, a fully quantitative description of axonal dynamics is still missing. In this paper, we present a review of our work on a certain class of stochastic models. We show that Langevin and Fokker-Planck stochastic differential equations provide a powerful theoretical framework for describing how collective axonal dynamics and the formation of neuronal network emerge from biophysics of single neurons and their interactions. We combine this theoretical analysis with experimental data to extract key parameters of axonal growth: diffusion (cell motility) coefficients, speed and angular distributions, mean square displacements, as well as mechanical parameters describing the cell-substrate coupling. These results have important implications for our understanding of fundamental mechanisms combined fluorescence microscopy with AFM force mapping to produce systematic, high-resolution elasticity and fluorescence maps for several types of neuronal cells, including cortical, embryonic cDRGs, and P-19 (mouse embryonic carcinoma stem cells). We have also performed combined AFM and fluorescence experiments to investigate the relationship between external temperature, soma volume and elastic modulus for cortical neurons 59]. Our results show an increase by a factor of 2 in the soma elastic modulus as the ambient temperature decreases from physiological (37ºC) to room (25ºC) temperature We have also found that the soma volume is increasing with temperature by a factor of 1.3 during the temperature sweep from 25 0 C to 37 0 C Our results show that both the variations in elastic modulus and cell volume are highly reduced if the activity of molecular motors inside the neurons is inhibited, through treatment with chemical compounds (Taxol and Blebbistatin) that are known to have an inhibitory effect on the cytoskeleton dynamics 59].