A hybrid computational model to predict chemotactic guidance of growth cones

Some Mechanical Constraints to the Biomimicry with Peripheral Nerves

Novel high technology devices built to restore impaired peripheral nerves should be biomimetic in both their structure and in the biomolecular environment created around regenerating axons. Nevertheless, the structural biomimicry with peripheral nerves should follow some basic constraints due to their complex mechanical behaviour. However, it is not currently clear how these constraints could be defined. As a consequence, in this work, an explicit, deterministic, and physical-based framework was proposed to describe some mechanical constraints needed to mimic the peripheral nerve behaviour in extension. More specifically, a novel framework was proposed to investigate whether the similarity of the stress/strain curve was enough to replicate the natural nerve behaviour. An original series of computational optimizing procedures was then introduced to further investigate the role of the tangent modulus and of the rate of change of the tangent modulus with strain in better defining the structural biomimicry with peripheral nerves.

Toward computational neuroconstructivism: a framework for developmental systems neuroscience

Brain development is underpinned by complex interactions between neural assemblies, driving structural and functional change. This neuroconstructivism (the notion that neural functions are shaped by these interactions) is core to some developmental theories. However, due to their complexity, understanding underlying developmental mechanisms is challenging. Elsewhere in neurobiology, a computational revolution has shown that mathematical models of hidden biological mechanisms can bridge observations with theory building. Can we build a similar computational framework yielding mechanistic insights for brain development? Here, we outline the conceptual and technical challenges of addressing this theory gap, and demonstrate that there is great potential in specifying brain development as mathematically defined processes operating within physical constraints. We provide examples, alongside broader ingredients needed, as the field explores computational explanations of system-wide development.

Mathematical models of neuronal growth

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites-axons and dendrites-to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically.

Theory for Durotactic Axon Guidance

During the development of the nervous system, neurons extend bundles of axons that grow and meet other neurons to form the neuronal network. Robust guidance mechanisms are needed for these bundles to migrate and reach their functional target. Directional information depends on external cues such as chemical or mechanical gradients. Unlike chemotaxis that has been extensively studied, the role and mechanism of durotaxis, the directed response to variations in substrate rigidity, remain unclear. We model bundle migration and guidance by rigidity gradients by using the theory of morphoelastic rods. We show that, at a rigidity interface, the motion of axon bundles follows a simple behavior analogous to optic ray theory and obeys Snell's law for refraction and reflection. We use this powerful analogy to demonstrate that axons can be guided by the equivalent of optical lenses and fibers created by regions of different stiffnesses.

Strain stiffening of peripheral nerves subjected to longitudinal extensions in vitro

The mechanical response of peripheral nerves is crucial to understand their physiological and pathological conditions. However, their response to external mechanical solicitations is still partially unclear, since peripheral nerves could behave in a quite complex way. In particular, nerves react to longitudinal strains increasing their stiffness to keep axons integrity and to preserve endoneural structures from overstretch. In this work, the strain stiffening of peripheral nerves was investigated in vitro through a recently introduced computational framework, which is able to theoretically reproduce the experimental behaviour of excised tibial and sciatic nerves. The evolution and the variation of the tangent modulus of tibial and sciatic nerve specimens were quantitatively investigated and compared to explore how stretched peripheral nerves change their instantaneous stiffness.

A Quantitative Investigation on the Peripheral Nerve Response within the Small Strain Range

Peripheral nerves are very complex biological structures crucial to linking the central nervous system to the periphery of the body. However, their real behaviour is partially unknown because of the intrinsic difficulty of studying these structures in vivo. As a consequence, theoretical and computational tools together with in vitro experiments are widely used to approximate the mechanical response of the peripheral nervous tissue to different kind of solicitations. More specifically, particular conditions narrow the mechanical response of peripheral nerves within the small strain regime. Therefore, in this work, the mechanical response of nerves was investigated through the study of the relationships among strain, stress and displacements within the small strain range. Theoretical predictions were quantitatively compared to experimental evidences, while the displacement field was studied for different values of the tissue compressibility. This framework provided a straightforward computational assessment of the nerve response, which was needed to design suitable connections to biomaterials or neural interfaces within the small strain range.

Theoretical Models of Neural Development

Constructing a functioning nervous system requires the precise orchestration of a vast array of mechanical, molecular, and neural-activity-dependent cues. Theoretical models can play a vital role in helping to frame quantitative issues, reveal mathematical commonalities between apparently diverse systems, identify what is and what is not possible in principle, and test the abilities of specific mechanisms to explain the data. This review focuses on the progress that has been made over the last decade in our theoretical understanding of neural development.

A unified approach to model peripheral nerves across different animal species

Peripheral nerves are extremely complex biological structures. The knowledge of their response to stretch is crucial to better understand physiological and pathological states (e.g., due to overstretch). Since their mechanical response is deterministically related to the nature of the external stimuli, theoretical and computational tools were used to investigate their behaviour. In this work, a Yeoh-like polynomial strain energy function was used to reproduce the response of in vitro porcine nerve. Moreover, this approach was applied to different nervous structures coming from different animal species (rabbit, lobster, Aplysia) and tested for different amount of stretch (up to extreme ones). Starting from this theoretical background, in silico models of both porcine nerves and cerebro-abdominal connective of Aplysia were built to reproduce experimental data (R² > 0.9). Finally, bi-dimensional in silico models were provided to reduce computational time of more than 90% with respect to the performances of fully three-dimensional models.

Biomaterials and computation: a strategic alliance to investigate emergent responses of neural cells

Synergistic use of biomaterials and computation allows to identify and unravel neural cell responses.

Multi-phasic bi-directional chemotactic responses of the growth cone

The nerve growth cone is bi-directionally attracted and repelled by the same cue molecules depending on the situations, while other non-neural chemotactic cells usually show uni-directional attraction or repulsion toward their specific cue molecules. However, how the growth cone differs from other non-neural cells remains unclear. Toward this question, we developed a theory for describing chemotactic response based on a mathematical model of intracellular signaling of activator and inhibitor. Our theory was first able to clarify the conditions of attraction and repulsion, which are determined by balance between activator and inhibitor, and the conditions of uni- and bi-directional responses, which are determined by dose-response profiles of activator and inhibitor to the guidance cue. With biologically realistic sigmoidal dose-responses, our model predicted tri-phasic turning response depending on intracellular Ca²⁺ level, which was then experimentally confirmed by growth cone turning assays and Ca²⁺ imaging. Furthermore, we took a reverse-engineering analysis to identify balanced regulation between CaMKII (activator) and PP1 (inhibitor) and then the model performance was validated by reproducing turning assays with inhibitions of CaMKII and PP1. Thus, our study implies that the balance between activator and inhibitor underlies the multi-phasic bi-directional turning response of the growth cone.

Topographical strategies to control neural outgrowth

In this work a synergistic approach is used to investigate how directional anisotropic surfaces (i.e., nanogratings) control the alignment of PC12 neurites. Finite Element models were used to assess the distribution of stresses in non-spread growth cones and filopodia. The stress field was assumed to be the main triggering cause fostering the increase and stabilization of filopodia, so the local stress maxima were directly related to the neuritic orientation. Moreover, a computational framework was implemented within an open source Java environment (CX3D), and in silico simulations were carried out to reproduce and predict biological experiments. No significant differences were found between biological experiments and in silico simulations (alignment angle, p = 0.4685; tortuosity, p = 0.9075) with a standard level of confidence (95%).

Deterministic control of mean alignment and elongation of neuron-like cells by grating geometry: a computational approach

Neuron-like cells are driven by their surrounding environment through local topography. A causal mechanotransductive web of topography-force relationships influences and controls complex cellular phenomena such as growth and alignment. This work aimed to provide a computational framework able to model the behaviour of neuron-like (PC12) cells on gratings, accounting for the twofold ability of topographical cues to simultaneously align and enhance the growth of cells. In particular, starting from the mechanical behaviour of the growth cone and filopodia, the effect of grating geometry (e.g., the periodicity and the size of grooves and ridges) on the neuritic mean alignment angle and on the outgrowth rate of cells was explored through theoretical tools and combinatorial simulations, which were able to predict (R(2) > 0.9) experimental data in a time range of 72-120 hours.