^{1}

^{1}

^{2}

^{2}

^{2}

^{1}

^{2}

Edited by: Olaf Sporns, Indiana University, USA

Reviewed by: Brian Mulloney, University of California Davis, USA; Ronald L. Calabrese, Emory University, USA

*Correspondence: Roman Borisyuk, School of Computing and Mathematics, University of Plymouth, Plymouth PL4 8AA, UK. e-mail:

This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.

In this paper we develop a computational model of the anatomy of a spinal cord. We address a long-standing ambition of neuroscience to understand the structure–function problem by modeling the complete spinal cord connectome map in the 2-day old hatchling

A key to understanding the operation of any central nervous neuronal network is knowledge of the architecture of that network: where the neurons, dendrites, axons, and synapses are located in a three-dimensional structure. This detailed architecture of inter-neuronal connections provides a framework into which accumulated experimental information can be mapped. It may then be possible to model the activity of the network using these connections. In most cases, the morphology of nervous systems, or even regions of nervous systems, is highly complex, making it extremely challenging to define the real connection architecture of the component networks. Within the vertebrates, there is now extensive information on the brainstem and spinal cord neurons and networks that control locomotion (e.g., Ziskind-Conhaim et al.,

In the developing frog tadpole spinal cord, we have detailed information on the brainstem and spinal neurons active during swimming, their physiology, synaptic connections, and morphology (Roberts et al.,

We have used a “developmental” approach to modeling the connectome of the young

Studying and modeling the connectome of the tadpole spinal cord is interdisciplinary and involves diverse aspects from informatics, mathematics, and biology. In this paper we focus on methodological aspects of modeling the connectome which are relevant to neuroinformatics. Starting from our accumulated experimental data, we present a theoretical study in which we consider axon growth modeling, our reconstruction algorithm and analysis of the resulting connectome before providing a preliminary assessment of the possible significance of the connectome for understanding the functioning of the tadpole spinal cord. We are not, at this stage, describing a functional model of the spinal cord.

The anatomical details used to inform the connectome modeling were obtained using a variety of techniques. These details have been published previously but, since they are fundamental to construction of the connectome, they are briefly reviewed here. Most measurements were made in isolated nervous systems of chemically fixed tadpoles where individual neurons had been filled with neurobiotin. Filling was done using whole-cell recording electrodes. After the recording, tadpoles were fixed and processed to show the neurobiotin. The CNS was then removed and mounted between glass coverslips so that neurons could be viewed, traced, and photographed from either side at ×200 on a bright-field microscope (Li et al.,

_{L} and R_{R}), separated by the ventral floor plate (dark gray rectangle, R_{F}). Examples of the cell body positions (ellipses), dendrites (thick lines), and axon projections (thin lines) are illustrated. Neuron types are listed on the right: RB, Rohon Beard sensory neuron; dla, dlc, dorsolateral ascending and dorsolateral commissural sensory interneurons; dIN, cIN, aIN, descending, commissural, and ascending premotor interneurons; mn, motoneurons.

Longitudinal distributions of somata for the populations of the main spinal cord neuron types are given in Yoshida et al. (

As an initial measure of dendrite distributions, the dorsal and ventral-most extents of dendritic branching have been measured for a sample of neurons of each type. For each neuron, the dendritic field was assumed to extend evenly between these dorso-ventral extremes. Measurements of individual fields were then combined to provide an overall dorso-ventral distribution for each neuron type (Li et al.,

In this Section we consider our simple mathematical model of axon growth (Li et al.,

A simple mathematical model is described by a system of three difference equations (i.e., time is discrete) corresponding to the growth angle θ and two positions of the growth cone, the structure which forms the tip of the growing axon and which characterizes the direction of growth. The rostro-caudal (RC) horizontal position (coordinate) is denoted by the

The model equations are:

where, _{n}_{n}_{n} is the current growth angle. Parameters are: _{n} is independent and uniformly distributed in the interval (−α, α), where α is a parameter. Parameter Δ is the elongation of the axon in one iteration (Δ = 1 μm). The parameter γ characterizes the tendency of an axon to grow straight in a horizontal direction. The parameter

This simple model is used below to reconstruct a connectome of the spinal cord. Parameter values were calculated using an optimization procedure to fit the model to available experimental measurements of both ascending and descending axons of different cell-types. The axon growth procedure does not start from the cell body so the initial portion of the axon is ignored (for details see Li et al.,

The connectome we describe includes almost all the spinal cord cell-types. Since these spinal neurons form populations that extend uninterrupted into the caudal part of the hindbrain, and since this caudal part of the hindbrain is needed for generating sustained locomotion (Li et al.,

The procedure for assigning cell body distribution is applied independently to left and right side of the CNS. The region of CNS being considered here is defined by RC coordinates from 600 to 4000 μm (measured from the tadpole midbrain; the region from 600 to ∼850 μm is caudal hindbrain, see above). The length of this region we divide to small spatial steps

Suppose that cell-types are labeled by numbers: 1 – means RB cell-type, 2 – dlc cell-type, 3 – aIN cell-type, 4 – cIN cell-type, 5 – dIN cell-type, 6 – mn cell-type. First of all the number of cells of each type (1–6) inside the subinterval is calculated according to the longitudinal distribution of neuron cell body numbers (per 100 μm) which are denoted by _{1},…,_{6}. Thus, neurons of cell type 1 (i.e., RB) are allocated at the first _{1} positions with a gap _{2} positions with a gap _{i}_{i + 1} or is empty. Swap means that position _{i + 1} or is empty and the position (_{i}

After filling all subintervals with appropriate numbers of neurons, the total number of neurons of each cell type can be calculated. These numbers, shown for one side of the body in Table

Cell type | # |
---|---|

1 (RB) | 107 |

2 (dlc) | 86 |

3 (aIN) | 97 |

4 (cIN) | 271 |

5 (dIN) | 135 |

6 (mn) | 248 |

The dendrite of each neuron is allocated the same RC position as its cell body. The dendrite is represented by a vertical bar, the coordinates of whose ventral and dorsal extremes are randomly selected within the overall DV interval of 100 μm. Values are based on experimental measurements that provide distributions of pairs of low (_{k}_{k}

Cell body and dendrite distributions are summarized in a text file. In this file, numbers of neurons on the left side range from 1 to 944 and the numbers of neurons on the right side range from 945 to 1,888. For each neuron the cell type is given as well as the rostro-caudal coordinate of the cell body (in a range from 600 to 4,000 μm). It is assumed that the RC coordinate of the cell body coincides with the RC coordinate of the dendrite. The DV position of the dendrite is represented by the most dorsal coordinate of the dendrite in μm and the most ventral coordinate of the dendrite in μm (in a range from 0 to 100). It should be noted that neurons of RB cell-type have no dendrites and, therefore, dorsal and ventral positions are set to zero.

Once the neuron cell bodies and dendrites are positioned, the process of axon generation starts and axons are grown for each neuron. All axons are generated sequentially and independently from each other using our simple model of axon growth (see above; Li et al.,

To start growth of an axon the dorso-ventral coordinates of the initial point (_{0}, _{0}) and the initial angle of the axon growth (θ_{0}) are specified. After that, by applying the iterative formulas (1), the first point of axon (_{1}, _{1}) will be generated and continuing these iterations the whole axon will be generated: (_{i}_{i}

In the current version of the connectome reconstruction model, axon growth starts from a point whose RC coordinate is the same as the RC coordinate of the cell body. For commissural neurons (dlc and cIN) the initial stage of axon growth from the cell body to the ventral floor plate, crossing the floor plate and growing up the opposite side of the spinal cord to a branching point is not considered. Instead, the initial point of axon growth has the same RC coordinate as the cell body but on the opposite side. The DV coordinate of the start point, the initial angle of axon growth, and the length of generated axon are randomly selected based on distributions from experimental data (for details see Li et al.,

Once the axons are grown for all the neurons, the connectome is completed by assigning the locations of synapses. Every time that a growing axon crosses the dendrite of another neuron, a synapse can form with a probability 0.46 (this value is based on experimental observations: Li et al.,

The connectome model shows a complex structure of connections. In this section we first present some approaches to visualization of the connectome. We then present some analysis to allow us to illustrate the kinds of information that the connectome model yields. In considering the spinal cord, some particular features of the connectome should be noted. For example, the fact that neurons are distributed on two sides of the spinal cord is very important; the cell bodies of dlc and cIN cell-types are located on one side but their axons project to the opposite side. The neurons of these cell-types therefore make their connections onto neurons on the opposite side. Figure

Schematically the connectome can be described in the following way. There are two networks, one on each side of the CNS: left side and right side networks. Each network contains 944 neurons of six cell-types. The left side network contains non-commissural neurons with mutual connections and commissural neurons which also receive connections from non-commissural neurons but send their connections to the right side network. The right side network has the same organization. Thus, the connectome model contains left side and right side networks of mutually coupled excitatory (RB, dIN, mn) and inhibitory (aIN) neurons and these two networks are connected by inhibitory commissural neurons (dlc and cIN cell-types).

Each side of the connectome contains about 30,000 internal connections (both excitatory and inhibitory synapses) and there are about 30,000 inhibitory connections from left to right and the same from right to left. One way to visualize the pattern of outgoing synapses from the axons of neurons in the connectome is shown in Figure

A second way to visualize the synapses in the connectome is from the perspective of the post-synaptic neurons, by mapping the incoming synapses onto their dendrites (Figure

The total number of connections in the connectome is 122,221 of which 51,530 are ascending and 70,691 are descending. The distribution of these connections between neuron types is illustrated in Figure

Presynaptic neurons | Post-synaptic neurons | Total from | |||||
---|---|---|---|---|---|---|---|

RB | dlc | aIN | cIN | dIN | mn | ||

RB | 0 | 1,842 | 744 | 1,373 | 1,408 | 246 | 5,613 |

dlc | 0 | 187 | 2,780 | 6,982 | 3,266 | 6,057 | 19,272 |

aIN | 0 | 757 | 3,044 | 7,336 | 3,879 | 5,769 | 20,785 |

cIN | 0 | 1,585 | 6,275 | 15,798 | 8,507 | 12,056 | 44,221 |

dIN | 0 | 247 | 3,576 | 8,955 | 4,447 | 7,706 | 24,931 |

mn | 0 | 0 | 1,169 | 2,573 | 718 | 2,939 | 7,399 |

Total to | 0 | 4,618 | 17,588 | 43,017 | 22,225 | 34,773 | 122,221 |

To analyze the pattern of incoming synaptic connections, we examined the distances from which particular neuron types receive incoming synapses from other particular types. These distances were calculated by the following procedure. A neuron of cell type

As well as examining the patterns of incoming and outgoing synapses, we have used the connectome to explore the probabilities of connection between neurons of different types (Table

Presynaptic neurons | Post-synaptic neurons | |||||
---|---|---|---|---|---|---|

RB | dlc | aIN | cIN | dIN | mn | |

RB | 0.00 | 0.33 | 0.13 | 0.24 | 0.25 | 0.04 |

dlc | 0.00 | 0.01 | 0.14 | 0.36 | 0.17 | 0.31 |

aIN | 0.00 | 0.04 | 0.15 | 0.35 | 0.19 | 0.28 |

cIN | 0.00 | 0.04 | 0.14 | 0.36 | 0.19 | 0.27 |

dIN | 0.00 | 0.01 | 0.14 | 0.36 | 0.18 | 0.31 |

mn | 0.00 | 0.00 | 0.16 | 0.35 | 0.10 | 0.40 |

total | 0.00 | 0.04 | 0.14 | 0.35 | 0.18 | 0.28 |

In addition to details of the distributions of neurons, their axons and particularly their synaptic connections, the connectome model can also be analyzed to allow comparison with connections found in other systems.

Each part of the connectome on either side of the CNS can be characterized as a scale-free network. This property was shown by calculating the degree distribution of the connectome in which the nodes are individual neurons. Each node’s degree is defined as the total number of connections (incoming plus outgoing) for each node. The degree distribution of a scale-free network follows a power law. Figure

We have also analyzed the distribution of overall connection lengths, combining lengths from all neuron types and ignoring the direction of connection. It has recently been shown that this distribution is similar for different neural networks of different animals (Kaiser et al.,

Lastly, the edge density (or connectivity) characterizes how many potential connections between network nodes are really present. For a directed network, each of

A key feature of the connectome modeling that we describe here is that it is based on detailed (though of course still not complete) knowledge of the neuronal components of the comparatively simple tadpole nervous system (Roberts et al.,

With the exception of

One simple prediction from the connectome is the number of synapses an individual neuron is likely to receive. In the case of motoneurons, this can be compared to an independent experimental estimate. Assuming an equal distribution of synapses, the connectome would predict an average number of synapses per mn as: total synapses onto mn (34,773) per total mn (496) = 70 synapses. An experimental estimate based on combined light and electron microscopy data estimated an overall average of 118–236 per mn depending on synapse spacing on each dendrite (Roberts et al.,

The tadpole connectome model describes neuronal networks connecting between the two sides of the CNS. Biologically, this interconnection underpins control of motor responses like rhythmic swimming which alternate on left and right sides (Roberts et al.,

Modeling the connectome of the tadpole is clearly an ongoing process and very much linked to accumulating the biological data that is needed to guide its construction. We are in the process of introducing what is the next stage in axon growth modeling, a gradient model of axon guidance. The importance of this new model will be its greater basis in biological reality, replacing rather artificial parameters of growth with more realistic responses to morphogen gradients. A further addition to be introduced when suitable biological data are available will be a more realistic description of the dendritic branches of the various spinal cord neuron types. It is likely that this will alter the DV pattern of connections, but it is too soon to predict whether the change will be rather subtle or more significant.

The construction and analysis of models of nervous system networks continues to be an area of huge interest (e.g., see: Bullmore and Sporns,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work is supported by BBSRC grants (BB/G006369/1 and BB/G006652/1).