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Modeling neurodevelopmental gradients and connectivity formation
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Spatiotemporal ontogeny of brain wiring

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Populating the synthetic brain with neurons. Pillars of the current computational modeling approach are the studies of (19, 36). The brain was modeled as a 2D square, reminiscent of previous modeling work (24, 36). The size of the synthetic brain was set to 50 × 50 units for all the different brain connectomes (Fig. 3A). Variations of the exact size of the synthetic brain did not substantially influence the results. Each unit surface in the synthetic brain represents a structure of the developing brain, for instance, the cortical plate of mammalian cortices. Each unit surface can host N neurons. The population of the synthetic brain with neurons is achieved as follows. In the scenarios realizing a heterochronous development, a distinct time window was assigned to each surface unit. This time window dictates the probability of n neurons migrating at the corresponding unit surface at time point t and was defined as $P(i,t,a)=(t2λ(tλ−1)2)1m(i)α$ with $m(i)=1k+1$, $λ=−log(2)log(m)$, i denotes a unique integer identifying a time window assigned to a unit surface, k denotes the total number of time windows considered in the simulation, and α constitutes a parameter controlling the width of the time window of each unit surface of the synthetic brain (α > 0). Values for the parameter α closer to 0 lead to more narrow time windows and, hence, less overlap between the set of time windows of the surface units. Larger values for α lead to broader time windows, hence leading to more overlap between the set of time windows of the surface units (19). Here, we explored a range of values for parameter α, ranging from 0.2 to 0.8 with an increment of 0.2.

For each time window, the function P(i,t,a) was scaled to max[P(i,t,a)] for t = 0, …, 1 so that the outcome of the function is bound to [0 1] and can, thus, serve as a probability. The temporal resolution of the simulations was set to 0.05, where one developmental “tick” marked a temporal increment of 0.05. Thus, the simulations unfolded across a time frame defined by t = 0, 0.05, 0.1, …, 1. For each unit surface, the distance from the neurodevelopmental root(s)/origin(s) was computed. The total number of time windows was defined on the basis of the unique number of minimum distances from the neurodevelopmental root(s). The unique minimum distances of each unit surface served as a criterion for assigning to them a time window i. Time windows were assigned in such a way so that surface units with closer distances to the neurodevelopmental root(s) were assigned to time windows with the lowest corresponding unique integer i (i = 1, 2, …, k). Thus, surface units close to the neurodevelopmental root(s) were more likely to be populated with neurons at early stages of the simulations, that is, at low t values (Fig. 3A). The total number of neurons to populate the synthetic brain was increased exponentially, as empirical evidence dictates (37). Thus, the number of neurons was increased on the basis of the following relation: nt = ninit(1 + r)t, where ninit is the initial number of neurons, here set to 100, r is the rate of growth, here set to 0.2, and t is the current time point of the simulation. We should note that the exact phenomena of neurogenesis, such as number of divisions and apoptosis, were not within the scope of the current investigation and were not modeled. Instead, neuronal populations were positioned in the synthetic brain as described above, corresponding to the release and accumulation of neuronal populations across the developing brain (22). Modeling phenomena such as cell proliferation and apoptosis is feasible (38, 39), and thus, this future extension of our modeling approach is plausible.

In total, three scenarios were simulated, that is, heterochronous and spatially ordered, heterochronous and spatially random, and tautochronous development. In the first scenario, the population of the brain with neurons proceeded in a spatially ordered fashion as co-centric rings from the root(s) (Fig. 3B). In the second scenario, the population of the brain took place in a heterochronous fashion but in a spatially random manner. This was achieved by randomizing the relations between the position of the unit surfaces from the root(s) and their assigned time window. This randomization destroyed the spatially ordered co-centric population of the brain (Fig. 3C). Last, a tautochronous scenario consisted of the simultaneous placement of the neurons in the synthetic brain and was followed by subsequent connectivity formation. Note that, in all scenarios, the total number of neurons of the final “adult” synthetic brain contained the exact same number of neurons.

Connectivity formation. Once a neuron was placed in the synthetic brain, it could form a connection. Neurons eligible for establishing connections, that is, neurons that have not already established a connection, were selected uniformly and at random. Connectivity formation was established in a stochastic manner. A direction across the 2D space was randomly sampled from a uniform distribution, and if a straight line from the neuron that was about to develop a connection toward that direction was intersecting a circle placed around each of the other neurons in the synthetic brain, then a connection was established. A radius equal to the surface unit was used to define the circle dictating if a connection would be established. Each neuron could form one connection, corresponding to the fact that projection neurons have one axon. Axon collaterals were not modeled. Each neuron had a maximum capacity of constituting the target of up to 100 connections. If a neuron did not form a connection during a developmental time point, then it could retry establishing a connection during the next time point.

Constructing the synthetic connectomes. The empirical connectomes are interregional connectomes. Therefore, for a proper comparison of the synthetic and empirical connectomes, the following procedure was followed. First, the synthetic brain was parcellated with a Voronoi tessellation to “brain regions.” The number of the regions was dictated by the number of regions of each empirical brain connectome (Fig. 3E). Ten different parcellations were applied to each synthetic brain to avoid parcellation-dependent results. Second, a synthetic, directed and weighted connectome was assembled by creating a region-to-region synthetic connectivity matrix. A connection from region A to region B was formed by summing the number of neurons contained in region A that form connections with those in region B. The connectivity density of the synthetic connectomes, that is, the number of connections present in a connectome out of all possible connections given a number of regions, was equated to the connectivity density of each empirical brain connectome. Hence, the synthetic connectomes had the same number of regions and connections as the empirical connectomes. For the synthetic human connectomes, since the human empirical connectome is undirected, connectomes were symmetrized by computing the average of the weight of the connection from regions A to B and the weight of the connection from regions B to A.

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