Collective neuronal growth and self organization of axons

Abstract

I describe an assembly of neurons that synthesizes a chemorepellant molecule and transfers this molecule to the growth cones by axonal transport. From the growth cone, the repellant diffuses in nearby regions, achieving a certain concentration profile, δ, with a gradient. The growth cone migrates preferentially toward the regions of low δ. I show that this process self-consistently generates a profile of finite width, with a relatively sharp front of neuronal growth.

In the developing neural tissue, axons grow under the influence of various chemical agents operating at very low concentration (1): chemoattractants/repellants. The growth cone (terminal portion of the axon) progresses via lamellipodium motion. For one cone moving on a solid substrate, the details of this motion (implying both an advancing mode and a receding mode) have been studied in detail in Leipzig, Germany (2).

General Aims

In the present paper, I am interested in collective features (many neurons growing simultaneously) and larger spatial distances (x). The (ultimate) hope is to understand how random processes achieve the correct wiring of certain brain sectors. Sharp gradients are observed from their final consequences: For instance, through the borders of visual areas (see, e.g., ref. 3) or in the primate cortex (4). The sharp gradients are classically interpreted by assuming nonlinear chemical kinetics (5–7). Many types of signaling molecules participate, but I wish to point out here that a single repellant secretion from the growth cones may, by itself, lead to collective behavior and sharp gradients.

Our starting point is a set of neurons, with their main body localized in a thin region: here a plane (x = 0) with ν neurons per squared meter. Each neuron emits an axon of length l(t) (after a time t) terminating in a growth cone at a certain distance x of the source (Fig. 1). Our aim is to understand the spatial distribution G_t(x) of the cones.

Fig. 1.

A layer of neurons (N) emitting axons with a growth cone (M). A repellant is synthesized in the layer of neurons, transported to the growth cone, and secreted in the region around the layer of neurons. The growth cones grow toward the region of low repellant level: They are pushed to the right.

We postulate that

1. The neuron nuclei synthesize a certain chemorepellant (of concentration δ_N near the cell soma).

2. The chemorepellant is transported along the axon on microtubules to the growth cone with a fixed motor velocity V.

3. At the growth cone, the repellant acts as a “guidance cue” (6, 7), diffuses freely in the glial matrix, and is ultimately scavenged. This generates an average concentration δ(x, t) that (in the simplest limit) is proportional to the local growth cone density G_t(x).

4. Each growing cone has a random sequence of jumps (of size a and correlation time θ) but, superimposed on this, we expect a drift velocity: The cone drifts toward the region of low chemorepellant content.

In Transport Model, we construct a self-consistent picture of this cascade of processes. We will demonstrate in Self-Consistent Profiles how this leads to sharp gradients.

Transport Model

Convection from Nucleus to Growth Cone.

We postulate that the length l of the axon increases linearly with time,

and that the repellant is transported (along microtubules) at a fixed velocity V. This leads us to write the following relationship between the concentration at the tip δ_M(t) and the concentration at the start δ_N(t):

where f is the “useful fraction” of synthesized chemorepellant.

We expect a growth velocity W much smaller than the transport velocity V. Thus, we can replace Eq. 2 with

i.e., convection is fast.

Secretion and Free Diffusion of the Inhibitor from the Growth Cone.

Let us call r the distance between the secreting growth cone M and an observation point P. In a “local steady state,” the concentration due to M at P is

where a is the size of the growth cone and κ⁻¹ is an extinction length related to the lifetime τ of the repellant in the matrix

(D being the diffusion constant of the chemorepellant). We assume that κ⁻¹ is smaller than the width of our profiles. Neuronal cues can have a long shelf lifetime (in vitro). But we do assume that, in vivo, the lifetime is on the order of minutes.

Adding the contributions of all of the cones to Eq. 4, we arrive at a concentration profile δ(x) proportional to the local cone profile G_t(x):

Growth Cone Distribution G_t(x).

The transport equation for the cone is of the form

where the current J is the sum of two terms: a standard cone diffusion term (with a diffusion constant D_c = a²/θ) and a drift term of velocity

with k = D_c 1/C₀, where C₀ is a threshold concentration for repellant action. Ultimately, from Eqs. 7, 8, and 6, the cone distribution is ruled by

The effective diffusion constant D_eff is dependent on G:

where ϕ is defined in Eq. 6.

Self-Consistent Profiles

“Strong” Regimes.

The spatial distribution of growth cones is ultimately controlled by the equation for transport, Eq. 9.

In Eq. 10 for the diffusion coefficient, we find a classical term D₀ and a nonlinear term, which can lead to sharp fronts. From now on, we concentrate on the “strong” regime, where the nonlinear term dominates: (ϕ/C₀)Ga ≫ 1.

If the profile at time t has a certain characteristic thickness x̄, then G ∼ x̄⁻¹, and the condition for strong regimes reads

From now on, we shall use the dimensionless parameter:

In the strong regime, the effective diffusion constant becomes small when G→0. (We call this a “hypodiffusive” case.) This implies that the profile G_t(x) terminates abruptly at a certain distance.

Self-Consistent Width x̄(t).

Eq. 9 can be solved exactly in the strong regime, giving a parabolic profile for G_t(x):

The structure of x̄ can be understood by a simple scaling argument. Because G ∼ 1/x̄, we expect

Remembering that D₀ ∼ a²/θ, this gives

Thus, x̄ increases only slowly with time. Using Eq. 12, we can now return to the criterion (Eq. 11) for the strong regime, which becomes

We are interested in times much longer than the jump time θ; thus, we need ψ ≫ 1. Let us take an example with a distance between neuronal bodies in the source ν^−1/2 ∼ 10 μm and a repellant lifetime in the matrix τ = 10² sec. With an inhibitor diffusion coefficient D_i ∼ 10⁻¹⁰ m²/sec, this leads to κ⁻¹ = 100 μm. Take f = 0.1 (1/10 of the effector is convected to the cone) and δ_N ∼ C₀ (the production is tuned to produce a significant effect on the jumps). Then we find ψ ≅ 100.

The Extreme Front.

Our predicted form of the growth cone density G_t(x) in the strong regime is qualitatively shown in Fig. 2: The cone density vanishes linearly at a certain distance x = x̄(t). This can again be seen in Eq. 12, but it can also be explained simply by a scaling argument. Consider the vicinity of the front x = x̄(t) − y, with y small. In this region we may roughly say that the profile is transported with a local velocity v = dx̄/dt. The current J of Eq. 9 must vanish in a reference frame moving at velocity v. This implies

Because D_eff(G) is linear in G (Eq. 10, in the strong regime), this gives a finite slope ∂G/∂y.

Fig. 2.

Qualitative shape of the growth cone density G. In the “strong regime” (a), G_t(x) vanishes linearly at the front position x = x̄. The area under the curve is equal to 1. In realistic regimes (b), the front is smoothed out over a distance λ.

Of course, there is always a small region near y = 0 (0 < y < λ), where the condition for the strong regime fails because G is small. In this region, linear diffusion holds, and the singularity is smoothed out. Starting from the discussion in “Strong” Regimes, we find

Thus, indeed λ ≪ x̄ in the strong regime.

Discussion

The main conclusions of this (highly tentative) model are (i) a growth band of width increasing only slowly with time: x̄ ∼ t^1/3 (the exponent is smaller than standard diffusion) and (ii) a relatively sharp front in the strong regime.

Of course, if we go to very thick layers (violating the condition λ < x̄ in Eq. 17), the front will have a small classical tail at x > x̄.

Let us return now to the starting assumptions.

1. We looked only at the effect of a secreted repellant. Of course, in the real world, the growth cone is also sensitive to many types of signaling molecules, which can originate from other territories in the developing brain. But our point is to show that a single, autogenerated repellant can generate sharp fronts. Coordinated growth need not be always due to attractive cues.

2. We ignored any degradation of the repellant during transport on the microtubules.

3. We assumed that the repellant spreads around each growth cone in a region (of size κ⁻¹), small compared with the profile width. This is crucially dependent on the lifetime τ. If the reverse were true (κx̄ < 1), most of the inhibitor would be spread in a useless region.

4. Our description of the cone “jumps” is primitive when compared with the detailed studies of ref. 2, but this is hopefully not critical at the long time scales involved here.