The adaptive stochasticity hypothesis: Modeling equifinality, multifinality, and adaptation to adversity

Abstract

Neural phenotypes are the result of probabilistic developmental processes. This means that stochasticity is an intrinsic aspect of the brain as it self-organizes over a protracted period. In other words, while both genomic and environmental factors shape the developing nervous system, another significant-though often neglected-contributor is the randomness introduced by probability distributions. Using generative modeling of brain networks, we provide a framework for probing the contribution of stochasticity to neurodevelopmental diversity. To mimic the prenatal scaffold of brain structure set by activity-independent mechanisms, we start our simulations from the medio-posterior neonatal rich club (Developing Human Connectome Project, n = 630). From this initial starting point, models implementing Hebbian-like wiring processes generate variable yet consistently plausible brain network topologies. By analyzing repeated runs of the generative process (>10⁷ simulations), we identify critical determinants and effects of stochasticity. Namely, we find that stochastic variation has a greater impact on brain organization when networks develop under weaker constraints. This heightened stochasticity makes brain networks more robust to random and targeted attacks, but more often results in non-normative phenotypic outcomes. To test our framework empirically, we evaluated whether stochasticity varies according to the experience of early-life deprivation using a cohort of neurodiverse children (Centre for Attention, Learning and Memory; n = 357). We show that low-socioeconomic status predicts more stochastic brain wiring. We conclude that stochasticity may be an unappreciated contributor to relevant developmental outcomes and make specific predictions for future research.

Keywords: brain development; early adversity; generative modeling; stochasticity; structural connectome.

Fig. 1.

Schematic of dissimilarity procedure. (A) A schematic illustration of how stochasticity in the developing simulation leads to variable outcomes. For example, the same wiring constraints can lead to quite dissimilar outcomes (i.e., multifinality, Top Right) and different wiring constraints can lead to similar outcomes (i.e., equifinality, Top Left). (B) For each parameter combination, we ran 625 repeated simulations. To calculate the dissimilarity between these network outcomes, two measures were produced. The first was a topological dissimilarity measure, which captures the dissimilarity in global measures of network topology between each pairwise combination of network outcomes at each parameter combination (Top). The second was an embedding dissimilarity measure, which captures the dissimilarity in edge existence between each pairwise combination of network outcomes at each parameter combination (Bottom).

Fig. 2.

Weaker brain wiring constraints increase multifinality in network outcomes. (A) The topological dissimilarity landscape is given across the wiring parameter space. Purple corresponds to highly topologically dissimilar networks while blue corresponds to low topological dissimilarity. (B) A scatter plot of the topological dissimilarity as a function of the wiring parameters shows that $\gamma$ most drives topological dissimilarity. (C) Variable outcomes are more likely with weaker topological constraints on the network development (highlighted by the light purple wider funnel) and vice versa for highly constrained networks (highlighted by the light blue narrower funnel). (D) The embedding dissimilarity landscape is given across the wiring parameter space. Green corresponds to highly dissimilar networks in terms of embeddings while yellow corresponds to low embedding dissimilarity (E) A scatter plot of the topological dissimilarity as a function of the wiring parameters shows that $\eta$ most drives embedding dissimilarity. (F) Variable outcomes are more likely with less embedding constraints on the network development (highlighted by the light green wider funnel) and vice versa for highly constrained networks (highlighted by the light yellow narrower funnel).

Fig. 3.

Up-regulating developmental noise increases variability in network outcomes. Injecting stochasticity (A) early, (B) middle, or (C) late in the generative process increased the topological dissimilarity across repeated runs of each parameter combination, with middle and late noise exerting a greater impact (ANOVA F_2,1872, = 134.44, P = 2.774 × 10⁻⁵⁵), especially at higher values of $\gamma$ . Injecting stochasticity (D) early, (E) middle, or (F) late in the generative process also increased the embedding dissimilarity, with early noise exerting a greater impact (ANOVA F_2,1872, = 140.92, P = 9.792 × 10⁻⁵⁸), especially at lower values of $\eta$.

Fig. 4.

Weaker wiring constraints confer relative resilience to simulated attacks. (A) A schematic demonstration of the robustness testing protocol. Robustness is estimated by quantifying how much network communicability decreases upon the removal of network nodes. (B) The $\beta$ coefficient computed from a targeted attack regime, which preferentially removes network hubs, across the parameter space. More constrained networks (Top Left of the landscape) are less robust, as the gradient of change is greater. Weakly constrained networks (Bottom Right of the landscape) exhibit less change. To the right of the landscape, the communicability trajectories of networks with the least (Left) and most (Right) robustness to change are presented. (C) A schematic showing that weakly constrained networks (which achieve more multifinality, as indicated by the funnel width), are relatively more robust to attack.

Fig. 5.

Similar wiring constraints and deterministic development lead to equifinality in network topology. A SVM was trained to distinguish simulations run with different parameters. The SVM sought to correctly classify the runs of the simulations using their topological properties. The mean misclassification rate refers to the mean proportion of the sample that was incorrectly classified across all pairwise comparisons, and was obtained using cross-validation. (A) A density plot of equifinality, measured using the misclassification rate of the SVM, by the distance between the $\gamma$ parameters of the two simulations. (B) The equifinality exhibited by a simulation, indexed using the mean misclassification rate of the SVM, by the topological dissimilarity of that simulation. The mean was taken over all pairwise comparisons of that simulation.

Fig. 6.

Wiring parameters and model fits vary with SES. In n = 357 children, we split groups into high and low SES groupings. We find that the wiring parameter $\eta$ is greater in negative magnitude in high SES children, suggesting more constrained connectivity (Left). We find no difference in the $\gamma$ parameter between groups (Middle). Model fits are better in low SES networks, due to the connectome being more randomly organized and therefore more easily simulated by the generative model (Right).

Fig. 7.

Contribution of developmental stochasticity to multifinality, equifinality, and robustness of brain networks. Each cone represents a stochastic manifold, or the range of possible paths a developing network may take. The width of the cone indicates the range of phenotypic outcomes that may be achieved (multifinality). At weak wiring constraints, the cones are wide (indicating more stochastic development), while at strong wiring constraints, they are narrow (indicating more deterministic development). The relative angle of the cones indicates the likelihood of ending up in a similar range of phenotypic values (equifinality). Stochasticity is inversely proportional to the equifinality achieved; weak constraints more often produce diverse and unique phenotypic outcomes. Finally, stochastically developing networks tend to be more robust to perturbation and change than their deterministically developing counterparts.