Neuronal growth as diffusion in an effective potential

Nonlinear Growth Dynamics of Neuronal Cells Cultured on Directional Surfaces

During the development of the nervous system, neuronal cells extend axons and dendrites that form complex neuronal networks, which are essential for transmitting and processing information. Understanding the physical processes that underlie the formation of neuronal networks is essential for gaining a deeper insight into higher-order brain functions such as sensory processing, learning, and memory. In the process of creating networks, axons travel towards other recipient neurons, directed by a combination of internal and external cues that include genetic instructions, biochemical signals, as well as external mechanical and geometrical stimuli. Although there have been significant recent advances, the basic principles governing axonal growth, collective dynamics, and the development of neuronal networks remain poorly understood. In this paper, we present a detailed analysis of nonlinear dynamics for axonal growth on surfaces with periodic geometrical patterns. We show that axonal growth on these surfaces is described by nonlinear Langevin equations with speed-dependent deterministic terms and gaussian stochastic noise. This theoretical model yields a comprehensive description of axonal growth at both intermediate and long time scales (tens of hours after cell plating), and predicts key dynamical parameters, such as speed and angular correlation functions, axonal mean squared lengths, and diffusion (cell motility) coefficients. We use this model to perform simulations of axonal trajectories on the growth surfaces, in turn demonstrating very good agreement between simulated growth and the experimental results. These results provide important insights into the current understanding of the dynamical behavior of neurons, the self-wiring of the nervous system, as well as for designing innovative biomimetic neural network models.

Biased Random Walk Model of Neuronal Dynamics on Substrates with Periodic Geometrical Patterns

Neuronal networks are complex systems of interconnected neurons responsible for transmitting and processing information throughout the nervous system. The building blocks of neuronal networks consist of individual neurons, specialized cells that receive, process, and transmit electrical and chemical signals throughout the body. The formation of neuronal networks in the developing nervous system is a process of fundamental importance for understanding brain activity, including perception, memory, and cognition. To form networks, neuronal cells extend long processes called axons, which navigate toward other target neurons guided by both intrinsic and extrinsic factors, including genetic programming, chemical signaling, intercellular interactions, and mechanical and geometrical cues. Despite important recent advances, the basic mechanisms underlying collective neuron behavior and the formation of functional neuronal networks are not entirely understood. In this paper, we present a combined experimental and theoretical analysis of neuronal growth on surfaces with micropatterned periodic geometrical features. We demonstrate that the extension of axons on these surfaces is described by a biased random walk model, in which the surface geometry imparts a constant drift term to the axon, and the stochastic cues produce a random walk around the average growth direction. We show that the model predicts key parameters that describe axonal dynamics: diffusion (cell motility) coefficient, average growth velocity, and axonal mean squared length, and we compare these parameters with the results of experimental measurements. Our findings indicate that neuronal growth is governed by a contact-guidance mechanism, in which the axons respond to external geometrical cues by aligning their motion along the surface micropatterns. These results have a significant impact on developing novel neural network models, as well as biomimetic substrates, to stimulate nerve regeneration and repair after injury.

Feedback-controlled dynamics of neuronal cells on directional surfaces

The formation of neuronal networks is a complex phenomenon of fundamental importance for understanding the development of the nervous system. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body and axonal navigation toward target neurons. Axonal growth is guided by the interactions between the tip of the axon (growth cone) and its extracellular environmental cues, which include intercellular interactions, the biochemical landscape around the neuron, and the mechanical and geometrical features of the growth substrate. Here, we present a comprehensive experimental and theoretical analysis of axonal growth for neurons cultured on micropatterned polydimethylsiloxane (PDMS) surfaces. We demonstrate that closed-loop feedback is an essential component of axonal dynamics on these surfaces: the growth cone continuously measures environmental cues and adjusts its motion in response to external geometrical features. We show that this model captures all the characteristics of axonal dynamics on PDMS surfaces for both untreated and chemically modified neurons. We combine experimental data with theoretical analysis to measure key parameters that describe axonal dynamics: diffusion (cell motility) coefficients, speed and angular distributions, and cell-substrate interactions. The experiments performed on neurons treated with Taxol (inhibitor of microtubule dynamics) and Y-27632 (disruptor of actin filaments) indicate that the internal dynamics of microtubules and actin filaments plays a critical role for the proper function of the feedback mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which high-curvature geometrical features impart high traction forces to the growth cone. These results have important implications for our fundamental understanding of axonal growth as well as for bioengineering novel substrate to guide neuronal growth and promote nerve repair.

Axonal growth on surfaces with periodic geometrical patterns

The formation of neuron networks is a complex phenomenon of fundamental importance for understanding the development of the nervous system, and for creating novel bioinspired materials for tissue engineering and neuronal repair. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body towards target neurons. Axonal growth is guided by environmental stimuli that include intercellular interactions, biochemical cues, and the mechanical and geometrical features of the growth substrate. The dynamics of the growing axon and its biomechanical interactions with the growing substrate remains poorly understood. In this paper, we develop a model of axonal motility which incorporates mechanical interactions between the axon and the growth substrate. We combine experimental data with theoretical analysis to measure the parameters that describe axonal growth on surfaces with micropatterned periodic geometrical features: diffusion (cell motility) coefficients, speed and angular distributions, and axon bending rigidities. Experiments performed on neurons treated Taxol (inhibitor of microtubule dynamics) and Blebbistatin (disruptor of actin filaments) show that the dynamics of the cytoskeleton plays a critical role in the axon steering mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which high-curvature geometrical features impart high traction forces to the growth cone. These results have important implications for our fundamental understanding of axonal growth as well as for bioengineering novel substrates that promote neuronal growth and nerve repair.

Neuronal Growth and Formation of Neuron Networks on Directional Surfaces

The formation of neuron networks is a process of fundamental importance for understanding the development of the nervous system and for creating biomimetic devices for tissue engineering and neural repair. The basic process that controls the network formation is the growth of an axon from the cell body and its extension towards target neurons. Axonal growth is directed by environmental stimuli that include intercellular interactions, biochemical cues, and the mechanical and geometrical properties of the growth substrate. Despite significant recent progress, the steering of the growing axon remains poorly understood. In this paper, we develop a model of axonal motility, which incorporates substrate-geometry sensing. We combine experimental data with theoretical analysis to measure the parameters that describe axonal growth on micropatterned surfaces: diffusion (cell motility) coefficients, speed and angular distributions, and cell-substrate interactions. Experiments performed on neurons treated with inhibitors for microtubules (Taxol) and actin filaments (Y-27632) indicate that cytoskeletal dynamics play a critical role in the steering mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which geometrical patterns impart high traction forces to the growth cone. These results have important implications for bioengineering novel substrates to guide neuronal growth and promote nerve repair.

Neuron dynamics on directional surfaces

Geometrical features play a very important role in neuronal growth and the formation of functional connections between neuronal cells. Here, we analyze the dynamics of axonal growth for neuronal cells cultured on micro-patterned polydimethylsiloxane surfaces. We utilize fluorescence microscopy to image axons, quantify their dynamics, and demonstrate that periodic geometrical patterns impart strong directional bias to neuronal growth. We quantify axonal alignment and present a general stochastic approach that quantitatively describes the dynamics of the growth cones. Neuronal growth is described by a general phenomenological model, based on a simple automatic controller with a closed-loop feedback system. We demonstrate that axonal alignment on these substrates is determined by the surface geometry, and it is quantified by the deterministic part of the stochastic (Langevin and Fokker-Planck) equations. We also show that the axonal alignment with the surface patterns is greatly suppressed by the neuron treatment with Blebbistatin, a chemical compound that inhibits the activity of myosin II. These results give new insight into the role played by the molecular motors and external geometrical cues in guiding axonal growth, and could lead to novel approaches for bioengineering neuronal regeneration platforms.

Variations of Elastic Modulus and Cell Volume with Temperature for Cortical Neurons

Neurons change their growth dynamics and mechanical properties in response to external stimuli such as stiffness of the local microenvironment, ambient temperature, and biochemical or geometrical guidance cues. Here we use combined atomic force microscopy (AFM) and fluorescence microscopy experiments to investigate the relationship between external temperature, soma volume, and elastic modulus for cortical neurons. We measure how changes in ambient temperature affect the volume and the mechanical properties of neuronal cells at both the bulk (elastic modulus) and local (elasticity maps) levels. The experimental data demonstrate that both the volume and the elastic modulus of the neuron soma vary with changes in temperature. Our results show a decrease by a factor of 2 in the soma elastic modulus as the ambient temperature increases from room (25 °C) to physiological (37 °C) temperature, while the volume of the soma increases by a factor of 1.3 during the same temperature sweep. Using high-resolution AFM force mapping, we measure the temperature-induced variations within different regions of the elasticity maps (low and high values of elastic modulus) and correlate these variations with the dynamics of cytoskeleton components and molecular motors. We quantify the change in soma volume with temperature and propose a simple theoretical model that relates this change with variations in soma elastic modulus. These results have significant implications for understanding neuronal development and functions, as ambient temperature, cytoskeletal dynamics, and cellular volume may change with variations in physiological conditions, for example, during tissue compression and infections in vivo as well as during cell manipulation and tissue regeneration ex vivo.

Anomalous diffusion for neuronal growth on surfaces with controlled geometries

Geometrical cues are known to play a very important role in neuronal growth and the formation of neuronal networks. Here, we present a detailed analysis of axonal growth and dynamics for neuronal cells cultured on patterned polydimethylsiloxane surfaces. We use fluorescence microscopy to image neurons, quantify their dynamics, and demonstrate that the substrate geometrical patterns cause strong directional alignment of axons. We quantify axonal growth and report a general stochastic approach that quantitatively describes the motion of growth cones. The growth cone dynamics is described by Langevin and Fokker-Planck equations with both deterministic and stochastic contributions. We show that the deterministic terms contain both the angular and speed dependence of axonal growth, and that these two contributions can be separated. Growth alignment is determined by surface geometry, and it is quantified by the deterministic part of the Langevin equation. We combine experimental data with theoretical analysis to measure the key parameters of the growth cone motion: speed and angular distributions, correlation functions, diffusion coefficients, characteristics speeds and damping coefficients. We demonstrate that axonal dynamics displays a cross-over from Brownian motion (Ornstein-Uhlenbeck process) at earlier times to anomalous dynamics (superdiffusion) at later times. The superdiffusive regime is characterized by non-Gaussian speed distributions and power law dependence of the axonal mean square length and the velocity correlation functions. These results demonstrate the importance of geometrical cues in guiding axonal growth, and could lead to new methods for bioengineering novel substrates for controlling neuronal growth and regeneration.

Role of geometrical cues in neuronal growth

Geometrical cues play an essential role in neuronal growth. Here, we quantify axonal growth on surfaces with controlled geometries and report a general stochastic approach that quantitatively describes the motion of growth cones. We show that axons display a strong directional alignment on micropatterned surfaces when the periodicity of the patterns matches the dimension of the growth cone. The growth cone dynamics on surfaces with uniform geometry is described by a linear Langevin equation with both deterministic and stochastic contributions. In contrast, axonal growth on surfaces with periodic patterns is characterized by a system of two generalized Langevin equations with both linear and quadratic velocity dependence and stochastic noise. We combine experimental data with theoretical analysis to measure the key parameters of the growth cone motion: angular distributions, correlation functions, diffusion coefficients, characteristics speeds, and damping coefficients. We demonstrate that axonal dynamics displays a crossover from an Ornstein-Uhlenbeck process to a nonlinear stochastic regime when the geometrical periodicity of the pattern approaches the linear dimension of the growth cone. Growth alignment is determined by surface geometry, which is fully quantified by the deterministic part of the Langevin equation. These results provide insight into the role of curvature sensing proteins and their interactions with geometrical cues.

Dendrite and Axon Specific Geometrical Transformation in Neurite Development

We propose a model of neurite growth to explain the differences in dendrite and axon specific neurite development. The model implements basic molecular kinetics, e.g., building protein synthesis and transport to the growth cone, and includes explicit dependence of the building kinetics on the geometry of the neurite. The basic assumption was that the radius of the neurite decreases with length. We found that the neurite dynamics crucially depended on the relationship between the rate of active transport and the rate of morphological changes. If these rates were in the balance, then the neurite displayed axon specific development with a constant elongation speed. For dendrite specific growth, the maximal length was rapidly saturated by degradation of building protein structures or limited by proximal part expansion reaching the characteristic cell size.

Effects of surface asymmetry on neuronal growth

Detailed knowledge of how the surface physical properties, such as mechanics, topography and texture influence axonal outgrowth and guidance is essential for understanding the processes that control neuron development, the formation of functional neuronal connections and nerve regeneration. Here we synthesize asymmetric surfaces with well-controlled topography and texture and perform a systematic experimental and theoretical investigation of axonal outgrowth on these substrates. We demonstrate unidirectional axonal bias imparted by the surface ratchet-based topography and quantify the topographical guidance cues that control neuronal growth. We describe the growth cone dynamics using a general stochastic model (Fokker-Planck formalism) and use this model to extract two key dynamical parameters: diffusion (cell motility) coefficient and asymmetric drift coefficient. The drift coefficient is identified with the torque caused by the asymmetric ratchet topography. We relate the observed directional bias in axonal outgrowth to cellular contact guidance behavior, which results in an increase in the cell-surface coupling with increased surface anisotropy. We also demonstrate that the disruption of cytoskeletal dynamics through application of Taxol (stabilizer of microtubules) and Blebbistatin (inhibitor of myosin II activity) greatly reduces the directional bias imparted by these asymmetric surfaces. These results provide new insight into the role played by topographical cues in neuronal growth and could lead to new methods for stimulating neuronal regeneration and the engineering of artificial neuronal tissue.