Neuronal Growth: A Bistable Stochastic Process

Abstract

The fundamentally stochastic nature of neuronal growth has hardly been addressed in neuroscience. We report on the stochastic fluctuations of a neuronal growth cone's leading edge movement, the basic step in neuronal growth. Describing the edge movement as a stochastic bistable process leads to an isotropic noise parameter that is successfully used to test the model. An analysis of growth cone motility confirms the model, and predicts that linear changes of the bistable potential, as known from stochastic filtering, result in directed growth cone translocation.

Active gels as a description of the actin‐myosin cytoskeleton

“…The only other type of cells that we are aware of where a similar analysis can be done are neural growth cones. Both the velocity field and the friction force have been mapped by the group of Kaes (Betz et al, 2006), and the active stress can be estimated using the procedure applied to the keratocyte cell. One finds an active stress ⌬µ ϳ 50 Pa smaller than for the keratocyte, but in this case the measured elastic modulus of the actin cytoskeleton is smaller and the ratio ⌬µ / E is of the order 0.5.…”

Neuronal alignment on asymmetric textured surfaces

“…The motion of the growth cone is described by the following Langevin equation, m _ v ¼ Àav þ F þ nðtÞ, where m is an effective mass, t is the time, a is a Stokes drag coefficient, F is a constant force, and nðtÞ is a random force which has zero mean hni ¼ 0 and is Markovian hnðtÞ Á nðt 0 Þi ¼ m 2 Cdðt À t 0 Þ, where C represents the strength of the noise and d is the Dirac delta function. 18 The random force in the model causes the stochastic motion of the growth cone observed experimentally as it explores the local environment. In the absence of such a force, nðtÞ ¼ 0, the model is deterministic and the equilibrium solution is motion with a constant velocity, v 0 ¼ F=c; where the reduced drag c ¼ a=m is introduced.…”

“…Similar models have been adopted for axon growth in the literature for symmetric surfaces. 18,19 To interpret the observed alignment of the axonal growth along the asymmetric surfaces, we derived the expected distribution of the tangent angles from a biased random walk model. The motion of the growth cone is described by the following Langevin equation, m _ v ¼ Àav þ F þ nðtÞ, where m is an effective mass, t is the time, a is a Stokes drag coefficient, F is a constant force, and nðtÞ is a random force which has zero mean hni ¼ 0 and is Markovian hnðtÞ Á nðt 0 Þi ¼ m 2 Cdðt À t 0 Þ, where C represents the strength of the noise and d is the Dirac delta function.…”

Mathematical Modeling of Axonal Formation Part I: Geometry

“…This type of model also allows for the simulation of axons in a network assigning synapses under strict probabilistic rules, ultimately leading to the reconstruction of various anatomical architectures such as the spinal cord. Other mathematical tools that have been used to model specific features of axonogenesis at both the extracellular and intracellular level are described in Maskery and Shinbrot (2005), Maskery et al (2004), Odde et al (1996), Betz et al (2006), Goodhill et al (2004).…”

“…More recently, mechanisms such as growth cone velocity (forward and backward motion) have been incorporated into models (Betz et al 2006). The growth cone velocity is believed to be governed by the polymerization at the leading edge of the lamellipodium, where forward motion of the lamellipodium depends largely on the polymerization of actin filaments (Maskery and Shinbrot 2005).…”