Borisyuk R et al. (2014), A developmental approach to predicting neuronal...

XB-ART-48639

PLoS One 2014 Feb 03;92:e89461. doi: 10.1371/journal.pone.0089461.

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A developmental approach to predicting neuronal connectivity from small biological datasets: a gradient-based neuron growth model.

Borisyuk R , Al Azad AK , Conte D , Roberts A , Soffe SR .

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Relating structure and function of neuronal circuits is a challenging problem. It requires demonstrating how dynamical patterns of spiking activity lead to functions like cognitive behaviour and identifying the neurons and connections that lead to appropriate activity of a circuit. We apply a "developmental approach" to define the connectome of a simple nervous system, where connections between neurons are not prescribed but appear as a result of neuron growth. A gradient based mathematical model of two-dimensional axon growth from rows of undifferentiated neurons is derived for the different types of neurons in the brainstem and spinal cord of young tadpoles of the frog Xenopus. Model parameters define a two-dimensional CNS growth environment with three gradient cues and the specific responsiveness of the axons of each neuron type to these cues. The model is described by a nonlinear system of three difference equations; it includes a random variable, and takes specific neuron characteristics into account. Anatomical measurements are first used to position cell bodies in rows and define axon origins. Then a generalization procedure allows information on the axons of individual neurons from small anatomical datasets to be used to generate larger artificial datasets. To specify parameters in the axon growth model we use a stochastic optimization procedure, derive a cost function and find the optimal parameters for each type of neuron. Our biologically realistic model of axon growth starts from axon outgrowth from the cell body and generates multiple axons for each different neuron type with statistical properties matching those of real axons. We illustrate how the axon growth model works for neurons with axons which grow to the same and the opposite side of the CNS. We then show how, by adding a simple specification for dendrite morphology, our model "developmental approach" allows us to generate biologically-realistic connectomes.

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	Figure 2. Generalization of values from limited biological datasets.(A) Piece-wise linear approximation of the cumulative distribution function for cIN axon length. Red stars are points on the cumulative distribution function whose horizontal co-ordinates relate to biological measurements (nâ€Š=â€Š46). The blue line shows the piece-wise-linear approximation of the cumulative distribution. The yellow dot shows the generated axon length corresponding to random value w. (B) Two-dimensional generalization of dorso-ventral axon start point and axon initial outgrowth angle for two examples of tadpole spinal neurons (cIN upper, aIN lower). Coloured symbols are measured values; grey symbols are generated values. (Algorithm parameter values: and ; see SI for details.).
	Figure 3. Cost function components and optimization of growth parameters for tadpole aIN neurons.(A) Ten axon trajectories. Histogram (left) showing the dorso-ventral distribution of interpolated points along the length of a set of real axons (right: viewed laterally as in Fig. 4C; red symbols indicated intermittently measured points with all axons starting at the right). The proportion of points accumulated at each dorso-ventral level (e.g. cyan bar) is shown in the appropriate 10 Âµm bin. (B) Like A, but for a set of model axons. (C) Tortuosity in single axons. Red lines indicate the direct (chord) length; red symbols indicate measured points on the path of a real axon (left); paths of model axons (right, blue) were re-sampled at 10 Âµm intervals. (D) The random component in the cost function needed for optimization produces an uneven surface (illustrated for two dimensions: dorsal and ventral sensitivity). White arrows indicate multiple slopes from the start point of a search for a minimum cost function value (Global minimum). (E) Histogram of a sample containing 1000 repetitive calculations of the cost function for a single set of axon growth parameter values (All examples in Aâ€“E are for tadpole aINs).
	Figure 4. The two-dimensional environment for axon growth.(A) Side view of the head end of the tadpole showing hindbrain and spinal cord (buff). (B) Diagram of a section of the CNS to show the main parts including the central canal surrounded by a glial cell layer and the ventral floor plate, surrounded in turn by the layer of neuronal somata. Lying outside the soma layer are the marginal zones, in which most axons grow, and the dorsal tracts containing sensory neuron axons, separated from the marginal zone by a barrier formed by the dli column, a column of dorsolaterally-situated sensory pathway somata (red line), and bounded dorsally by the column of RB sensory neuron soma (yellow line). (C) The CNS opened like a book along dorsal midline (dotted line in B). Graphs on the left show the gradients originating at the dorsal edge (GD, green) and near the midline of the ventral floor-plate (GV, blue). There is also a longitudinal polarity (GR, not illustrated). Lines on the right (purple) indicate the dorso-ventral positions of a series of barriers to axon growth (see text for further details).
	Figure 5. Measured axon projections of two of the tadpole neuron types: aINs (dark blue) with uncrossed, primary ascending axons and secondary descending axons; and cINs (light blue) with crossing, primary ascending axons and secondary descending axons.In each case, examples are shown in situ, within the growth environment and also individually to illustrate their basic morphology.
	Figure 6. Stages of axon growth and model axon projections.(A) Flowchart summarising the sequence of stages in modelling axon growth for neurons with crossed or uncrossed axons. Rectangles denote axon growth stages; ovals denote values obtained using generalization procedures. Note that a secondary axon can branch from the âorientationâ or âmainâ region of a primary axon. (B) Illustration of the main stages of axon growth described in A. In these examples, both primary axons are ascending. Asterisks indicate branch points. (C) Axon projections generated by the growth model for uncrossed aINs (dark blue) and crossing cINs (light blue). Ten examples of each type are shown in situ with some of each type separated to show their individual morphology. Compare to real examples in Figure 5. Bar charts compare the proportions of the main growth for the primary axon projections in real and model axons in 10 Âµm dorso-ventral bins (projections sampled every 1 Âµm).
	Figure 7. The influence of the random variable and initial angle on ascending axon growth.Groups of 25 ascending axons, grown using aIN parameters. Note: All axons start at 2000 Âµm rostro-caudally and from a range of dorso-ventral positions. The fixed point of stability for aINs is indicated (red line shows ). (AâC) Axon starts are randomly distributed dorso-ventrally. As the random variable is increased (values of Î± indicated), trajectories become more variable: for , axon trajectories approach ; for and , trajectories increasingly deviate from . (D)(E) Axons have lengths, dorso-ventral start positions and initial angles distributed according to generalized aIN values. With , no axons reach within the length of the axon. With , the value optimized for aINs, the tendency of growth towards is much less obvious, but the axon trajectories are much more realistic. (F) Ascending axons of real aINs, aligned rostro-caudally to match the model axons.
	Figure 8. Connectome generation.A. Diagram with two longitudinally-running axons passing the dendrites (vertical bars) of three aIN neurons. Synapses (circles) can form (with probabilityâ=â0.46) where an axon meets a dendrite. B. Part of the growth field showing a region of a partial connectome formed by populations of just two neuron types (aIN, dark blue and cIN, light blue). Asterisks indicate dendrites of cINs.
	Figure 1. Features of the axon growth process (A) Three consecutive points of a growing axon are shown.Direction of growth during each step (Î”) is defined by the growth angle (Î¸). The dashed line shows the trajectory if based only on axon â€œstiffnessâ€ (keeping the same direction) where there is no influence causing it to deviate. (B) Rostro-caudal (GRC) and dorso-ventral (GDV) gradients and their projections to the direction of growth between two consecutive points of a growing axon.

References [+] :

Borisyuk, Stochasticity and functionality of neural systems: mathematical modelling of axon growth in the spinal cord of tadpole. 2008, Pubmed, Xenbase