A model of dendritic growth for a cortical pyramidal neuron driven by activity-independent and -dependent mechanisms.

(a) Schematic of the soma of a pyramidal neuron (orange triangle) with 12 randomly distributed potential synapses from presynaptic axons (circles) with correlated activity patterns indicated by color. (b) Schematic of activity-independent and -dependent mechanisms. Soma and synapses correspond to box in a. Signaling molecules diffusing from potential synapses (1) attract dendrite growth and promote synapse formation (2) independent of firing pattern (3). Over time, poorly synchronized synapses depress and are pruned from the dendrite (4), while well-synchronized synapses remain stable (5). After a branch retracts, the dendrite is less sensitive to the growth field at that location (5). (c) Change in weight of one synapse (green) following the stimulation of itself (green bolt) and of two nearby synapses (purple bolts). Left: Schematic of the developing dendrite from b with bolts indicating synaptic activation. Right: Presynaptic accumulator (top), postsynaptic accumulator (middle), and change in synaptic weight (bottom) as a function of time (see Methods Kirchner and Gjorgjieva (2021) for details of the plasticity rule). Dashed line (bottom) indicates zero change.

Figure 1–Figure supplement 1. The growth field is similar to two-dimensional heat diffusion.

Figure 1–Figure supplement 2. Detailed illustration of the dendritic growth mechanism.

Parameters of the minimal plasticity model (Kirchner and Gjorgjieva, 2021) and the synaptotrophic growth model.

Balanced growth and retraction generate morphologically stable dendrites.

(a) Three example dendrites at five time points from our simulations. For clarity of presentation, connected synapses are not displayed. (b) Total length of dendritic tree as a function of time. (c) Length of dendrite added (green) and removed (red) as a function of time. (d) Morphological stability (correlation between the dendrite shape at time t and t − 4.5 hours) as a function of time. (e) Average number of dendrite intersections as a function of distance from the soma (the Sholl diagram). Data from basal dendrites in the developing mouse medial prefrontal cortex superimposed, normalized to the maximum (blue; ref. (Kroon et al., 2019)). All lines represent averages across 32 simulations with nine dendrites each. Shaded area indicates two standard deviations.

Figure 2–video 1. Example of a simulation in which several dendrites develop in parallel.

Synapse formation and removal predominate in distinct phases of dendrite development.

(a) Three examples of dendrites at the beginning (t = 9 hours) and end (t =72 hours) of the simulation. Green circles indicate formed synapses. (b) Total number of connected synapses as a function of time. Orange arrow highlights overshoot and subsequent pruning. (c) Added (green) and pruned synapses (red) as a function of time. (d) Average synaptic weights of synapses that ultimately stabilize (solid black; final weight more than 0.5) or are removed (dashed black; final weight less than 0.5) as a function of time. All lines represent averages across 32 simulations with nine dendrites each. Shaded area indicates two standard deviations.

Stable morphology is obtained through selective removal of synapses and dendritic input selectivity.

(a,b) Dendritic trees before (a, 9 hours) and after (b, 72 hours) removal of synapses (Figure 3). Connected synapses colored corresponding to activity group, which represents activity correlations (Figure 1b). (c) Left: Schematic illustrating the difference between Euclidean and tree distance. Note that we compute the Euclidean distance between synapses from different trees. Right: Correlation between pairs of synapses as a function of the Euclidean distance (blue) and tree distance (red). (d) Input selectivity of dendrites (defined as the fraction of the activity group with the highest representation) as a function of time. Dashed line indicates chance level. All lines represent averages across 32 simulations with nine dendrites each. Shaded area indicates two standard deviations. (e) Fraction of connected synapses per activity group early (t = 9 hours) and late (t = 72 hours) in the simulation. Each dot represents one of the five activity groups on one of the nine dendrites from the 32 simulations, resulting in 5 × 9 × 32 = 1440 data points. (f) Left: Schematic of different levels of overlap (rows) between the convex hulls of two dendrites, referring to the smallest convex sets that contain the dendrite. Right: Signal correlation (correlation between fractions of synapses from the same activity groups) for different levels of dendritic overlap. Error bars indicate the standard error of the mean, computed from 1152 pairs of dendrites from 32 simulations.

Dendritic arborization is controlled by the ratio of neurotrophic factors.

(a) Interactions between molecular factors underlie a local activity-dependent plasticity rule for synaptic change (Equation 1, (Kirchner and Gjorgjieva, 2021)). Neurotrophins (BDNF and proBDNF) bind to different neurotrophin receptors, and a cleaving protease (MMP9) converts proBDNF to BDNF in an activity-dependent manner. (b) Schematic illustrating the impact of different concentrations of BDNF on synaptic change. Upon stimulation of a synapse (top), proBDNF and BDNF is released into extracellular space (middle), where proBDNF can be cleaved into BDNF by MMP9. Depending on the neurotrophin ratio, computed as BDNF/(BDNF + proBDNF), the synapse is stabilized (left) or depressed and hence eventually removed (right). (c) Maximally possible stable density of synapses as a function of the initial concentration of BDNF. Stable (no pruning; green) and unstable (pruning occurs; red) areas are indicated. (d) Three examples of dendrites with superimposed synapses (green) with high initial BDNF concentration (49%), the baseline concentration (45%, same as Figure 1-Figure 3) and low initial BDNF (40%). Symbols correspond to locations marked in panel c. (e-g) Averages for density of synapses on the dendrite (e), number of connected synapses (f) and total length of dendrite (g) as a function of time for dendrites from the three conditions shown in d. (h-i) Average number of dendrite intersections (h) and synapses (i) as a function of distance from the soma for dendrites from the three conditions shown in d. (j) Global selectivity as a function of time for dendrites from the three conditions shown in d. All lines represent averages across 32 simulations with nine dendrites each.

Morphological variability emerges from the interaction of activity-dependent and -independent factors.

(a) Example of three dendrites with identical initial conditions but different random seeds. The colors illustrate that initial growth is governed by activity-independent factors, while later growth is governed by activity-dependent factors. (b) Schematic illustrating how variability is introduced into model: activity-dependent via the patterns of spontaneous activity (orange), and activity-independent via fluctuations in both the extrinsic growth stimulating field (purple 1) and the intrinsic mechanisms underlying dendrite growth (purple 2; see Methods). (c,d) Total length (c) and number of synapses (d) as a function of time for dendrites with identical initial conditions but different random seeds. Each gray line corresponds to one dendrite from one of 32 simulations. Bold line represents average. (e,f) Percentage in change of morphological similarity (e) and similarity of connected synapses (f) as a function of time for simulations where activity-dependent (orange) or -independent (purple) factors vary. Lines represent averages across 32 simulations with nine dendrites each. Shaded area indicates two standard deviations. (g,h) Final length as a function of number of major branches (g) and maximal length in the first 18 hours of the simulation (h). Lines indicate linear regression.

Figure 6–Figure supplement 1. Total tree length increases with the number of stems.

Dendritic morphology approximately minimizes cable length.

(a) Schematic illustrating optimal (top) and random (bottom) wiring to connect a given set of synapses. (b) The convex hull of all synapses connected to a dendrite with the proportionality of optimal length (bottom). (c) Total tree length as a function of convex hull area in the optimal, simulated and random scenario. Each dot corresponds to one of 288 dendrites from 32 simulations. Lines correspond to analytic predictions for the average density across all simulations. (d) Total tree length against the number of branch points in log scale, both for data and theoretical optimum. Data extracted from (Cuntz et al., 2012). (e) Total tree length in the data (black, average of n=13,112), our simulations (blue, average of 288 dendrites from 32 simulations), and the random baseline (green, analytically computed) relative to theoretical optimum (pink, analytically computed).

The growth field is similar to two-dimensional heat diffusion.

By iteratively convolving the potential synapses in the growth field with a Gaussian filter, the growth field reaches a steady-state that resembles a two-dimensional heat diffusion with point sources. Over time, individual point sources disappear and reappear to mimic when the corresponding synapses are connected or pruned from the dendrite.

Four scenarios of asynchronous dendritic growth can be modeled as a scout agent representing the tip of a dendrite exploring a two-dimensional grid.

A scout agent (yellow dot) has reached the location of a potential synapse (blue dot). Scenario 1: The scout agent will extend the dendrite and form a new synapse if nothing else happens. Scenario 2: To prevent overlap, if a second branch from the same dendrite blocks the path to the potential synapse, the original branch retracts. Scenario 3: If a branch from another dendrite reaches the potential synapse first, the original branch retracts. Scenario 4: If a new potential synapse becomes available adjacent to the dendrite (1) so that growth is not possible (since the branch cannot form immediately adjacent to two other parts of the dendrite), the corner flips (2), and the synapse forms.

Total tree length increases with the number of stems.

Average (solid black) and individual (gray circles) total tree lengths as a function of the number of stems. N = 701. Data from the Allen Cell Types Database (2015).