Growth models

DeNSE provides several models to describe neuronal growth; its purpose is to provide a versatile ensemble of algorithms to reproduce the development of cultured neurons in complex environments and on various substrates. Thus, a lot of care went into the conception of a broad range of models, from simple models reproducing well-known properties of random walks, to more elaborate models accounting for most of the observed behaviors.

Moreover, DeNSE aims at providing a comprehensive access to state-of-the-art models in neuronal growth. For that reason, all models implemented in previous simulators such as NETMORPH or NeuroMac have been adapted and upgraded to be used out-of-the-box.

Models and parameters are detailed below and a brief description of relevant biological elements is presented at the end of this document.

For the full list of implemented models and the way to enable them see the models table

For more details on the precise implementation of the models, see the Models structure page in the Developer space.

Branching models

During the neuronal development, new growth cones can emerge from a neurite through two different mechanisms:

  • bifurcation or splitting, where a single growth cone divides into two separate growth cones which start elongating in different directions;

  • lateral or interstitial branching, where a growth cone emerges de novo from the side of an existing branch.

By default, all branching mechanisms are turned off so they need to be set explicitely by the user.

Bifurcation or splitting mechanism

In that situation, a growth cone divides into two child branches of comparable diameter, both elongating in a new direction which differs significantly from that of the initial parent branch.

Such mechanism for the apparition of additional growth cone thus differs significantly from the interstitial branching, which will be seen later on, in the sense that none of the child branches is the exact continuity of the parent, and as such, the situation is almost symmetrical compared to the interstitial branching.

Diameters of child branches

The regulation of the diameter of the child growth cones in growth cone splitting event is determined by 3 parameters by the empirical relation established in [Chklovskii2003] and [Shefi2004] according to experimental measurements. The diameters \(d_1\) and \(d_1\) of the two child branches are thus related to the diameter \(d_0\) of the parent branch through the relation:

(1)\[d^{\eta}_0 = d^{\eta}_1 + d^{\eta}_2,\]

with \(\eta = \nu + 2\) in the original paper.

In DeNSE, the stochasticity in the splitting process is introduced by making the ratio between the new diameters Gaussian-distributed, with an average diameter ratio \(\frac {d_1}{d_2} = r_{avg}\) and a standard deviation \(\sigma_d\), both defined by the user. Therefore branching diameters are determined by the 3 following paramters:

  • diameter_ratio_avg: mean ratio \(r_{avg}\) between the diameters of the two sibling branches,

  • diameter_ratio_std: deviation \(\sigma_d\) of the ratio between the sibling branches

  • diameter_eta_exp: \(\eta\) exponent of the relation above (rename to diameter_split_exponent)

Van Pelt model

This branching mechanism was modelled by Van Pelt’s research group and formalized in a set of equations with phenomenological parameters. The model implemented in DeNSE is equivalent to the “BEST” model from [VanPelt2002], which is condensed in the equations:

\[\begin{split}\begin{gathered} \frac{dn}{dt} = n(t)p(n, t) \\ p(n, t) = \frac{B}{T} e^{-t/T} n^{-E} \end{gathered}\end{split}\]

where n is the average number of growth cones in the neurite and \(p(n, t)\) is the branching rate.

The parameters have the following meaning:

  • \(B\) the number of expected branching events at a given segment (number, no unit)

  • \(E\) the competition among growth cones in the neurite (number, no unit)

  • \(T\) the characteristic time of branching events (time unit)

  • \(S\) the competition among growth cones by centrifugal order (number, no unit)

The previous equations define the probability of branching for a neurite without selection a branching cone. The each cone \(i\) can be selected based on parameter \(S\) (the competition factor) and its centrifugal order \(\gamma_i\), according to the probability:

\[p_i = \frac{2^{-\gamma_i S}}{\sum_j 2^{-\gamma_j S}}\]

After a duration \(\Delta t \gg T\), the expected number of growth cones in the neurite, if none have been destroyed by other processes, is:

\[n_{final} = (1 + BE)^{1/E}\]

This model can be activated by setting "use_van_pelt" to True in the neurite parameters.

Interstitial or lateral branching

This latter mechanism is only present in the DeNSE simulator, where it is implemented through the uniform and FLPL branching models.

Contrary to the bifurcation (or splitting) mechanism, this situation presents a fully asymmetric case where a new branch emerges de novo from a location on an existing dendritic or axonal tree. Because of this, instead of having two similar branches linked through the scaling relation in Equation (1), the child branch emerges with a diameter of a fraction of the parent branch.

All interstitial branching models share 3 parameters regarding the geometrical properties of the new branches that emerge from the main branch:

  • lateral_branching_angle_mean is the mean angle between the child branch and the parent (rename to interstitial_angle_avg?),

  • lateral_branching_angle_std is the standard deviation of this angle (rename to interstitial_angle_std?),

  • interstitial_diameter_ratio (or another name) gives the diameter of the child branch as a fraction of the local parent diameter. @todo

Lateral branching models can be turned on or off using the "use_uniform_branching" or "use_flpl_branching" entries in the neurite parameters. @TODO : Why two parameters and what is the difference ?

Growth cones

Generic properties

The parameters of the growth cones can also be set through set_object_properties(), either through the params argument, to set the properties of all growth cones in the neuron, or separately through the axon_params or dendrite_params arguments.

The main properties are:

  • filopodia_finger_length, the length of the filopodia (determines how far the growth cone will sense its surroundings.

  • filopodia_min_number (@todo change name, number of filopodia does not change), the number of filopodia.

  • sensing_angle the typical aperture angle of the growth cone; in normal conditions, the filopodia will be distributed evenly in this angular range to sense the environment; typical values are around 70 to 90°.

  • max_sensing_angle, the maximum aperture angle of the filopodia, even when it is stuck, the growth cone cannot widen more than this value to send filopodia “further back”.

  • speed_growth_cone, the average extension speed of the growth cone (this value can be modified by specific properties of extension models, see below).

Influence of the neurite

Growth cone are not isolated units but interact through the common architecture of their neurite. This can be seen for instance through changes in the speed of individual growth cones as their number increases.

A typical way of accounting for this fact is through an ad hoc change of the speed of the growth cones according to the following equation:

\[v(t) = v_0 n(t)^{-s_d}\]

where:

  • \(v(t)\) is the speed of a growth cone at time \(t\)

  • \(v_0\) is the original speed when a single growth cone is present

  • \(n(t)\) is the number of growth cone supported by the neurite at time \(t\)

  • \(s_d\) is the “speed decay” factor (available through the speed_decay)

This feature is available for all extension components except the resource-based (see Extension component).

The resource-based model, on the other hand, does not require this ad hoc feature since the speed of the growth cones directly depend on the amount of resource available to each cone, and that they are intrinsically competing for this resource. Thus, the resource-based model provides a mechanistic approach to the same phenomenon.

Elongation models

Beyond the standard properties shared by neurons or neurites, the precise models underlying how the tips of the neurites (the growth cones) move around can be selected independently for dendrites and the axons through the "growth_cone_model" parameter of eac set of parameters (dendrite and axon).

These models determine how the growth cones (of respectively dendrites and axon) extend, interact with their surroundings, and select a new direction during elongation. Thus, a single model is composed of three combined subcomponents:

  • An Extension component which determines the length of the progression step that the growth cone will make. Depending on the model, this step can stay constant or change over time.

  • A Steering component: which uses information about the growth cone surroundings to determine the probability of going in each of the directions where it is projecting filopodia.

  • A Direction selection component which determines how, from the probabilities of going in each direction, the growth cone will choose a specific angle.

The list of components implemented is shown in the table below:

Extension

Steering

Direction selection

constant

pull-only

noisy-maximum

gaussian-fluctuations

memory-based

noisy-weighted-average

resource-based

self-referential-forces

run-and-tumble

Note that each of these components can be combined with any of the other components. The list of all possible combinations can be obtained directly through the get_models() function, which lists them using their abbreviated names for convenience.

As an example a standard random-walk can be implemented by combining together the constant extension component with the pull-only steering method and the noisy-weighted-average direction selection. The complete model is thus named constant_pull-only_noisy-weighted-average and abbreviated cst_po_nwa for convenience.

In order to make things easier to remember, standard models such as the random-walk are directly available through their given names. These models include:

  • netmorph-like, which reproduces the behavior implemented in the NETMORPH simulator and consists in constant_memory-based_noisy-maximum, or cst_mem_nm in short.

  • run-and-tumble, which consists in constant_pull-only-run-and-tumble or cst_po_rt for short.

  • self-referential-forces, which reproduces the behavior detailed in Torben-Nielsen’s paper and simulator (NEUROMAC) and consists in constant_self-referential-forces_noisy-weighted-average or cst_srf_nwa for short.

  • simple-random-walk, constant_pull-only_noisy-weighted-average for full name and cst_po_nwa for short.

The full list of abbreviations is shown in the table below:

Extension (full name)

Abbreviated version

constant

cst

gaussian-fluctuations

gfluct

resource-based

res

Steering (full name)

Abbreviated version

pull-only

po

memory-based

mem

self-referential-forces

srf

Direction (full name)

Abbreviated version

noisy-maximum

nm

noisy-weighted-average

nwa

run-and-tumble

rt

Biological parameters

Neurite mechanical properties

These are mostly associated to the properties of mictotubules, characterized by:

  • a flexural rigidity \(\kappa\) associated to a bending energy \(dU = \frac{\kappa}{2} \left(\frac{d\theta}{ds}\right) ds\) [Rauch2013]

  • a persistence length \(l_P > 700 \mu m\) [Rauch2013]

Growth Cone Guidance

But can also be associated to actin properties, such as:

  • its maximum treadmilling speed \(v_t \approx 5 nm/s\) [Etienne2015]

References

Chklovskii2003

Chklovskii & Stepanyants (2003). Power-law for axon diameters at branch point. BMC Neuroscience 4(1), 18. https://doi.org/10.1186/1471-2202-4-18, http://arxiv.org/abs/physics/0302039

Rauch2013(1,2)

Rauch, Heine, Goettgens & Käs (2013). Forces from the rear: Deformed microtubules in neuronal growth cones influence retrograde flow and advancement. New Journal of Physics http://doi.org/10.1088/1367-2630/15/1/015007

Etienne2015

Étienne, Fouchard, Mitrossilis, Bufi, Durand-Smet & Asnacios (2015). Cells as liquid motors: mechanosensitivity emerges from collective dynamics of actomyosin cortex, PNAS, 112(9), 2740-2745. http://doi.org/10.1073/pnas.1417113112

Shefi2004

Shefi, Harel, Chklovskii, Ben-Jacob, Ayali (2004). Biophysical constraints on neuronal branching, Neurocomputing, 58(60), 487-495

VanPelt2002

Van Pelt & Uylings (2002). Branching rates and growth functions in the outgrowth of dendritic branching patterns, Network, 13, 261-281