Mathematical modelling and numerical simulation of the morphological development of neurons

BACKGROUND: The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other. METHODS: A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described. RESULTS: Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development. CONCLUSION: A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity

Neurite, a finite difference large scale parallel program for the simulation of the electrical signal propagation in neurites under mechanical loading

With the growing body of research on traumatic brain injury and spinal cord injury, computational neuroscience has recently focused its modeling efforts on neuronal functional deficits following mechanical loading. However, in most of these efforts, cell damage is generally only characterized by purely mechanistic criteria, function of quantities such as stress, strain or their corresponding rates. The modeling of functional deficits in neurites as a consequence of macroscopic mechanical insults has been rarely explored. In particular, a quantitative mechanically based model of electrophysiological impairment in neuronal cells has only very recently been proposed (Jerusalem et al., 2013). In this paper, we present the implementation details of Neurite: the finite difference parallel program used in this reference. Following the application of a macroscopic strain at a given strain rate produced by a mechanical insult, Neurite is able to simulate the resulting neuronal electrical signal propagation, and thus the corresponding functional deficits. The simulation of the coupled mechanical and electrophysiological behaviors requires computational expensive calculations that increase in complexity as the network of the simulated cells grows. The solvers implemented in Neurite-explicit and implicit-were therefore parallelized using graphics processing units in order to reduce the burden of the simulation costs of large scale scenarios. Cable Theory and Hodgkin-Huxley models were implemented to account for the electrophysiological passive and active regions of a neurite, respectively, whereas a coupled mechanical model accounting for the neurite mechanical behavior within its surrounding medium was adopted as a link between lectrophysiology and mechanics (Jerusalem et al., 2013). This paper provides the details of the parallel implementation of Neurite, along with three different application examples: a long myelinated axon, a segmented dendritic tree, and a damaged axon. The capabilities of the program to deal with large scale scenarios, segmented neuronal structures, and functional deficits under mechanical loading are specifically highlighted

Biologically Plausible Models of Neurite Outgrowth

The growth of a neuronal dendritic tree depends on the neuron’s internal state and the environment within which it is situated. Different types of neuron develop dendritic trees with specific characteristics, such as the average number of terminal branches and the average length of terminal and intermediate segments. A key aspect of the growth process is the construction of the microtubule cytoskeleton within the dendritic tree. Neurite elongation requires assembly of microtubules from free tubulin at the growth cone. The stability of microtubule bundles is an important factor in determining how likely it is for a growth cone to split to form new daughter branches. Microtubule assembly rates and bundle stability are controlled by microtubule-associated proteins, principally MAP2 in dendrites. Extending previous work (Hely et al, J. Theor. Biol. 210:375-384, 2001) I have developed a mathematical model of neurite outgrowth in which elongation and branching rates are determined by the phosphorylation state of MAP2 at the tips of each terminal branch. Tubulin and MAP2 are produced in the cell body and transported along the neurite by a combination of diffusion and active transport. Microtubule (dis)assembly at neurite tips is a function of tubulin concentration. The rate of assembly depends on the amount of unphosphorylated MAP2 bound to the microtubules and linking them together. Phosphorylation of MAP2 destroys its linking capability and destabilises the microtubule bundles. Each terminal has a probability of branching that depends on the phosphorylation of MAP2 which, in turn, is a function of calcium concentration. Results from this model show that changes in the (de)phosphorylation rates of MAP2 affect the topology of the final dendritic tree. Higher phosphorylation promotes branching and results in trees with many short terminal branches and relatively long intermediate segments. Reducing phosphorylation promotes elongation and inhibits branching

The function of NaV1.8 clusters in lipid rafts

NaV1.8 is a voltage gated sodium channel mainly expressed on the membrane of thin diameter c-fibre neurons involved in the transmission of pain signals. In these neurons NaV1.8 is essential for the propagation of action potentials. NaV1.8 is located in lipid rafts along the axons of sensory neurons and disruption of these lipid rafts leads to NaV1.8 dependant conduction failure. Using computational modelling, I show that the clustering of NaV1.8 channels in lipid rafts along the axon of thin diameter neurons is energetically advantageous and requires fewer channels to conduct action potentials. During an action potential NaV1.8 currents across the membrane in these thin axons are large enough to dramatically change the sodium ion concentration gradient and thereby void the assumptions upon which the cable equation is based. Using scanning electron microscopy NaV1.8 is seen to be clustered, as are lipid raft marker proteins, on neurites at scales below 200nm. FRET signals show that the lipid raft marker protein Flotillin is densely packed on the membrane however disruption of rafts does not reduce the FRET signal from dense protein packing. Using mass spectrometry I investigated the population of proteins found in the lipid rafts of sensory neurons. I found that the membrane pump NaK-ATPase, which restores the ion concentrations across the membrane, is also contained in lipid rafts. NaK-ATPase may help to offset concentration changes due to NaV1.8 currents enabling the repeated firing of c-fibres, which is associated with spontaneous pain in chronic pain disorders.Open Acces

A closer look at neuron interaction with track-etched microporous membranes

Microporous membranes support the growth of neurites into and through micro-channels, providing a different type of neural growth platform to conventional dish cultures. Microporous membranes are used to support various types of culture, however, the role of pore diameter in relation to neurite growth through the membrane has not been well characterised. In this study, the human cell line (SH-SY5Y) was differentiated into neuron-like cells and cultured on track-etched microporous membranes with pore and channel diameters selected to accommodate neurite width (0.8 µm to 5 µm). Whilst neurites extended through all pore diameters, the extent of neurite coverage on the non-seeded side of the membranes after 5 days in culture was found to be directly proportional to channel diameter. Neurite growth through membrane pores reduced significantly when neural cultures were non-confluent. Scanning electron microscopy revealed that neurites bridged pores and circumnavigated pore edges – such that the overall likelihood of a neurite entering a pore channel was decreased. These findings highlight the role of pore diameter, cell sheet confluence and contact guidance in directing neurite growth through pores and may be useful in applications that seek to use physical substrates to maintain separate neural populations whilst permitting neurite contact between cultures

Synapto-protective drugs evaluation in reconstructed neuronal network

Chronic neurodegenerative syndromes such as Alzheimer’s and Parkinson’s diseases, or acute syndromes such as ischemic stroke or traumatic brain injuries are characterized by early synaptic collapse which precedes axonal and neuronal cell body degeneration and promotes early cognitive impairment in patients. Until now, neuroprotective strategies have failed to impede the progression of neurodegenerative syndromes. Drugs preventing the loss of cell body do not prevent the cognitive decline, probably because they lack synapto-protective effects. The absence of physiologically realistic neuronal network models which can be easily handled has hindered the development of synapto-protective drugs suitable for therapies. Here we describe a new microfluidic platform which makes it possible to study the consequences of axonal trauma of reconstructed oriented mouse neuronal networks. Each neuronal population and sub-compartment can be chemically addressed individually. The somatic, mid axon, presynaptic and postsynaptic effects of local pathological stresses or putative protective molecules can thus be evaluated with the help of this versatile “brain on chip” platform. We show that presynaptic loss is the earliest event observed following axotomy of cortical fibers, before any sign of axonal fragmentation or post-synaptic spine alteration. This platform can be used to screen and evaluate the synapto-protective potential of several drugs. For instance, NAD+ and the Rho-kinase inhibitor Y27632 can efficiently prevent synaptic disconnection, whereas the broad-spectrum caspase inhibitor zVAD-fmk and the stilbenoid resveratrol do not prevent presynaptic degeneration. Hence, this platform is a promising tool for fundamental research in the field of developmental and neurodegenerative neurosciences, and also offers the opportunity to set up pharmacological screening of axon-protective and synapto-protective drugs

Inverting brain grey matter models with likelihood-free inference: a tool for trustable cytoarchitecture measurements

Effective characterisation of the brain grey matter cytoarchitecture with quantitative sensitivity to soma density and volume remains an unsolved challenge in diffusion MRI (dMRI). Solving the problem of relating the dMRI signal with cytoarchitectural characteristics calls for the definition of a mathematical model that describes brain tissue via a handful of physiologically-relevant parameters and an algorithm for inverting the model. To address this issue, we propose a new forward model, specifically a new system of equations, requiring a few relatively sparse b-shells. We then apply modern tools from Bayesian analysis known as likelihood-free inference (LFI) to invert our proposed model. As opposed to other approaches from the literature, our algorithm yields not only an estimation of the parameter vector $\theta$ that best describes a given observed data point $x_0$, but also a full posterior distribution $p(\theta|x_0)$ over the parameter space. This enables a richer description of the model inversion, providing indicators such as credible intervals for the estimated parameters and a complete characterization of the parameter regions where the model may present indeterminacies. We approximate the posterior distribution using deep neural density estimators, known as normalizing flows, and fit them using a set of repeated simulations from the forward model. We validate our approach on simulations using dmipy and then apply the whole pipeline on two publicly available datasets