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Computational models of typical and atypical brain network development

Open AccessPublished:November 22, 2022DOI:https://doi.org/10.1016/j.biopsych.2022.11.012

      Abstract

      Over the last decade, the organization of brain networks at both micro and macro scales has become a key focus of neuroscientific inquiry. This has revealed fundamental features of brain network organization - small-worldness, modularity, heavy-tailed degree distributions – and has highlighted how these structural features support brain function. It is only recently however that the driving forces that shape brain networks over the course of development have begun to be explored. Here, we review recent efforts to gain insights into the mechanisms of brain development through generative modelling of both macro-scale human brain networks and micro-scale cellular connectomes in C. elegans and other organisms. We show how these mathematical models can begin to shed light on the biological processes that drive and constrain the development of brain networks. Finally, we show how generative network models can translate genetic and environmental differences into variability in developmental trajectories, leading to diverse cognitive and mental health outcomes in children and young people.

      Key words

      Introduction

      Over the course of a typical pregnancy, the fetal brain grows at an average rate of 250,000 neurons per minute, eventually forming 100 billion neurons, interconnected by hundreds of trillions of synapses (

      Ackerman, S. (1992) Discovering the brain. National Academies Press (US).

      ). Many of these connections are local, but neuronal populations also organize to form white-matter fiber bundles which allow them to connect across large anatomical distances. This white-matter growth and maturation can be detected non-invasively using diffusion-weighted MRI (
      • Conturo T.E.
      • Lori N.F.
      • Cull T.S.
      • Akbudak E.
      • Snyder A.Z.
      • Shimony J.S.
      • et al.
      Tracking neuronal fiber pathways in the living human brain.
      ), and while major white matter tracts are apparent already in utero, the large-scale connectivity of the human brain is further refined during the first two decades of life (
      • Cascio C.J.
      • Gerig G.
      • Piven J.
      Diffusion tensor imaging: application to the study of the developing brain.
      ,
      • Collin G.
      • van den Heuvel M.P.
      The ontogeny of the human connectome: Development and dynamic changes of brain connectivity across the life span.
      ,
      • Vértes P.E.
      • Bullmore E.T.
      Annual research review: Growth connectomics - the organization and reorganization of brain networks during normal and abnormal development.
      ).
      This protracted developmental period, and the sheer number and complexity of processes involved raises two fundamental questions for developmental neuroscience: (i) how can the resulting brains be so similar - reliably wired to perform the myriad task supporting human cognition? and (ii) how can the resulting brains be so different – allowing genetic and environmental forces to influence brain architecture and lead to neurodiverse outcomes?
      One strategy that may help us begin to address these questions is to consider the brain as a complex network whose nodes represent large-scale anatomical brain regions and whose links represent white-matter connections derived from neuroimaging data. Such a network perspective allows us to characterize the organization or architecture of connections between various regions of the brain. Crucially, it also allows us to ask about the mechanisms or driving forces which may shape brain networks into the architectures we observe. In particular we can seek to identify basic wiring rules based on biological and physical constraints, which determine the probability for any pair of brain regions to become connected at every point in developmental time (
      • Vértes P.E.
      • Bullmore E.T.
      Annual research review: Growth connectomics - the organization and reorganization of brain networks during normal and abnormal development.
      ). Such a network-based, connection-centric view of brain development is likely to be informative for two convergent reasons.
      First, we already know that many aspects of cognition in both health and disease depend critically on the large-scale network of interactions between spatially separated populations of neurons (
      • Roine U.
      • Roine T.
      • Salmi J.
      • Nieminen-von Wendt T.
      • Tani P.
      • Leppämäki S.
      • et al.
      Abnormal wiring of the connectome in adults with high-functioning autism spectrum disorder.
      ,
      • Vydrova R.
      • Komarek V.
      • Sanda J.
      • Sterbova K.
      • Jahodova A.
      • Maulisova A.
      • et al.
      Structural alterations of the language connectome in children with specific language impairment.
      ,
      • Griffiths K.R.
      • Grieve S.M.
      • Kohn M.R.
      • Clarke S.
      • Williams L.M.
      • Korgaonkar M.S.
      Altered gray matter organization in children and adolescents with ADHD: a structural covariance connectome study.
      ,
      • Hong S.J.
      • Vos de Wael R.
      • Bethlehem R.A.I.
      • Lariviere S.
      • Paquola C.
      • Valk
      • et al.
      Atypical functional connectome hierarchy in autism.
      ,
      • Siugzdaite R.
      • Bathelt J.
      • Holmes J.
      • Astle D.E.
      Transdiagnostic brain mapping in developmental disorders.
      ). It is reasonable therefore to ask whether the emergence of cognitive functions during childhood and adolescence is driven by network-level changes which can be captured by simple wiring rules.
      Second, networks are ubiquitous across many other domains, from social and transportation networks to protein interactions, and the internet. In neuroscience, networks have been used to understand large-scale brain organization (how macroscopic brain regions connect to one another), as well as the microcircuitry of neuronal connections. Strikingly, the application of network science to this broad spectrum of real-world systems has revealed that many of them share key organizational features – from the existence of modules and hub nodes to so-called small-world properties (
      • Bullmore E.T.
      • Sporns O.
      Complex brain networks: graph theoretical analysis of structural and functional systems.
      ). This implies that many insights gleaned from one specific organism will be generalizable, and that there is much to be gained by drawing on insights from other disciplines to accelerate progress in developmental neuroscience (
      • van den Heuvel M.P.
      • Bullmore E.T.
      • Sporns O.
      Comparative Connectomics.
      ).
      In what follows, we will review the growing literature on computational modelling of driving forces shaping brain networks into their characteristic topologies, at microscopic and especially macroscopic scales. In particular, we will present a rich body of research on so-called ‘generative network models’ and how they may be built on to understand typical and atypical trajectories of brain development in children and young people.

      Generative Network Models

      In the context of network science, a generative model is a set of rules which can be used to generate a network with a required set of characteristics (

      Vértes, P. E. (2020), The Cognitive Neurosciences, Sixth Edition. MIT Press. Edited by Poeppel, D., Mangun, G. R. & Gazzaniga, M. S. ISBN: 9780262043250

      ). The simplest network model is the so-called Erdös-Rényi model, where the wiring rule is simply that each node connects to every other node with a fixed probability p (
      • Erdős P.
      • Rényi A.
      On Random Graphs. I.
      ). The resulting networks are called ‘random networks’ because their connections display no structure apart from a characteristic network density that is tuned by the parameter p.
      Real-world networks on the other hand tend to display some obvious structure, and the last two decades have seen a string of seminal papers that first highlight the ubiquity and functional importance of a particular network feature, and then propose simple wiring rules that may give rise to it.
      For example, Barabási and Albert observed that many real-world networks have a heavy-tailed degree distribution, where most nodes have just a few connections but certain so-called hub nodes have a very high degree of connectivity (
      • Barabasi A.-L.
      • Albert R.
      Emergence of scaling in random networks.
      ). This deviates sharply from the degree-distribution expected for an Erdös-Rényi network, so the authors proposed a new generative model based on the idea of preferential attachment: each new node added to the network will preferentially attach to pre-existing nodes with high degree, therefore increasing their connectivity even further and generating a number of highly connected hub nodes in the final network (
      • Barabasi A.-L.
      • Albert R.
      Emergence of scaling in random networks.
      ). Preferential attachment has intuitive interpretations in many real-world contexts. For example, in social networks, new individuals joining a community may preferentially link to individuals who are already well-connected, reflecting their popularity or various leadership roles they fulfil.
      Around the same time, Watts and Strogatz observed (
      • Watts D.J.
      • Strogatz S.H.
      Collective dynamics of 'small-world' networks.
      ) that many complex networks exhibit the small-world property of simultaneously having high clustering (where any given node’s immediate neighbours are likely to also be interconnected) and high efficiency (where it only takes a few steps, on average, to traverse the network, see 17). Small-world architectures emerge naturally, for example, in situations where nodes tend to form densely connected communities but occasionally form connections between communities also – as is the case in social networks. The authors highlighted that this type of organization is not only ubiquitous – from social networks to the C. elegans neural network – but is also of functional relevance, conferring “enhanced signal-propagation speed, computational power, and synchronizability” to the network. Interestingly, they also show that small-worldness emerges easily, when starting from a locally connected ‘lattice’ network and randomly rewiring a small fraction of connections (
      • Watts D.J.
      • Strogatz S.H.
      Collective dynamics of 'small-world' networks.
      ).
      Since these seminal papers, a number of other models have been proposed to capture individual network properties such as modularity (
      • Kaiser M.
      • Hilgetag C.C.
      Development of multi-cluster cortical networks by time windows for spatial growth.
      ), or the preponderance of short-range connections in spatially embedded networks (
      • Chen B.L.
      • Hall D.H.
      • Chklovskii D.B.
      Wiring optimization can relate neuronal structure and function.
      ,
      • Cherniak C.
      Component placement optimization in the brain.
      ,
      • Barthélemy M.
      Spatial networks.
      ,
      • Ercsey-Ravasz M.
      • Markov N.T.
      • Lamy C.
      • Van Essen D.C.
      • Knoblauch K.
      • Toroczkai Z.
      • Kennedy H.
      A predictive network model of cerebral cortical connectivity based on a distance rule.
      ,
      • Song H.F.
      • Kennedy H.
      • Wang X.-J.
      Spatial embedding of structural similarity in the cerebral cortex.
      ). Taken together, this body of work has prepared the ground for more applied uses of generative network models (GNM), which seek to quantitatively fit a range of network features observed in a real world network. In such applied settings, the process for designing generative models can be summarized in the following five-step recipe (

      Vértes, P. E. (2020), The Cognitive Neurosciences, Sixth Edition. MIT Press. Edited by Poeppel, D., Mangun, G. R. & Gazzaniga, M. S. ISBN: 9780262043250

      ):
      • 1.
        Make a list of stylized facts observed in the data
      • 2.
        Propose a plausible but parsimonious mechanism that could lead to the emergence of the stylized facts identified in step 1
      • 3.
        Fit the parameters of the model and compare to other models
      • 4.
        Validate the model on independent data
      • 5.
        Think of what the model doesn’t capture, then return to step 1, with this new observation acting as an additional stylized fact
      While these steps are easy to state, in practice both the validity and the utility of a network model will depend on the investigator’s ingenuity in selecting appropriate stylized facts and in proposing a mechanism that strikes an appropriate balance between parsimony and plausibility. In particular, when choosing stylized facts, investigators look for empirical observations (‘facts’) that are ‘stylized’ in that they capture broad features (but not necessarily the fine details) of the network. These stylized facts ought to be statistically robust (
      • Clauset A.
      • Shalizi C.R.
      • Newman M.E.J.
      Power-law distributions in empirical data.
      ), functionally relevant (
      • Watts D.J.
      • Strogatz S.H.
      Collective dynamics of 'small-world' networks.
      ) and surprising when compared to appropriate null models (
      • Rubinov M.
      Constraints and spandrels of interareal connectomes.
      ).
      Like many other real-world networks, brain networks at both macro and micro scales have hub nodes and a heavy-tail degree distribution, exhibit high modularity, and are small-world networks with high clustering and high efficiency (
      • Bullmore E.T.
      • Sporns O.
      Complex brain networks: graph theoretical analysis of structural and functional systems.
      ). In addition, the brain is also a spatial network, and as such the distribution of connection lengths is heavily skewed towards short connections (
      • Barthélemy M.
      Spatial networks.
      ,
      • Bullmore E.T.
      • Sporns O.
      The economy of brain networks.
      ). The challenge then is to capture all these properties simultaneously with relatively simple, parsimonious, and biologically plausible models containing just one or two factors – and therefore just one or two free parameters.

      Economic wiring principles in human brain networks

      Ever since Cajal’s work in the 19th century (

      S. Ramón y Cajal (1995). Histology of the nervous system of man and vertebrates. Oxford Univ. Press,

      ) it has been acknowledged that nervous systems are wired in an economical way, that does not waste material or energy on unnecessarily long connections. It therefore seems natural to start with a model implementing this simple spatial constraint first. Indeed a number of such models were put forward (
      • Chen B.L.
      • Hall D.H.
      • Chklovskii D.B.
      Wiring optimization can relate neuronal structure and function.
      ,
      • Cherniak C.
      Component placement optimization in the brain.
      ,
      • Barthélemy M.
      Spatial networks.
      ,
      • Ercsey-Ravasz M.
      • Markov N.T.
      • Lamy C.
      • Van Essen D.C.
      • Knoblauch K.
      • Toroczkai Z.
      • Kennedy H.
      A predictive network model of cerebral cortical connectivity based on a distance rule.
      ,
      • Song H.F.
      • Kennedy H.
      • Wang X.-J.
      Spatial embedding of structural similarity in the cerebral cortex.
      ,
      • Kaiser M.
      • Hilgetag C.C.
      Spatial growth of real-world networks.
      ). For example, in 2004 Kaiser and Hilgetag proposed a generative model (
      • Kaiser M.
      • Hilgetag C.C.
      Spatial growth of real-world networks.
      ) where the probability of connecting a new node i to a pre-existing node j decreases exponentially with the distance dij between the two nodes:
      Pij=βeαdij


      As the authors pointed out, the distance penalty in this model is not only motivated from an evolutionary perspective but could also easily be implemented biologically, for example through gradients of guidance molecules, which decay exponentially with distance (
      • Kaiser M.
      • Hilgetag C.C.
      Spatial growth of real-world networks.
      ). Despite its simplicity, the model is able to capture not just the predominance of short-range connections but also the high clustering observed empirically in brain networks.
      While the model generates networks that are significantly more brain-like than random Erdös-Rényi networks, it is also quite clear that the drive to minimize wiring cost is not the only constraint relevant in shaping brain networks. Kaiser and Hilgetag also published an elegant study a couple of years later explicitly showing that both the known wiring diagram of the nematode C. elegans and the Macaque connectome obtained from tract-tracing studies featured more long-range connections than expected if the networks had been optimized purely to preserve wiring (
      • Kaiser M.
      • Hilgetag C.C.
      Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems.
      ). Subsequent papers (
      • Chen Y.
      • Wang S.
      • Hilgetag C.C.
      • Zhou C.
      Trade-off between multiple constraints enables simultaneous formation of modules and hubs in neural systems.
      ,
      • Chen Y.
      • Wang S.
      • Hilgetag C.C.
      • Zhou C.
      Features of spatial and functional segregation and integration of the primate connectome revealed by trade-off between wiring cost and efficiency.
      ) describe that both these networks could be rewired to minimize connection cost, but that this would come at the expense of reducing network efficiency - increasing the number of steps it would take to traverse the network. Building on Watts and Strogatz’ insights regarding the key role of a network’s efficiency in its function, several studies showed that the efficiency of human brain networks also correlates with IQ across the population, such that higher-IQ individuals have, on average, brain with higher network efficiency (32, 33 – but see also 34 which fails to find such a link between functional network efficiency and IQ). Taken together, these and other studies suggest that brain development may be operating under competing constraints of reducing energy expenditure (by limiting long-range connections) but increasing information processing (for which some long-range connections are beneficial) – see (
      • Bullmore E.T.
      • Sporns O.
      The economy of brain networks.
      ) and references therein.
      Building on this idea of economic trade-offs in brain wiring, Vértes and colleagues proposed a family of generative models explicitly embodying an economic trade-off between the cost Di,j and value Ki,j of forming each connection (
      • Vértes P.E.
      • Alexander-Bloch A.F.
      • Gogtay N.
      • Giedd J.
      • Rapoport J.L.
      • Bullmore E.T.
      Simple models of human brain functional networks.
      ). As the network grows, the probability of connecting two nodes I and j at each time-step takes the form:
      Pi,jDi,jKi,j


      where Di,j represents the ‘cost’ of the putative connection and is some decreasing function of the distance between nodes i and j, and where Ki,j represents the ‘value’ of this connection in terms of network function. A straightforward extension of the preferential attachment model by Barabási and Albert suggests for example that the value term Ki,j might correspond to the product of the two nodes’ degree. However by fitting these models to large-scale brain networks, Vértes et al showed that the best fitting model for the probability of nodes i and j connecting to each other at each timestep is:
      Pi,jdi,jηki,jγ


      where di,j is the distance between nodes i and j, and ki,j is the number of common neighbours between node i and j (see Figure 1A). Intuitively, the first term in the model is a distance-penalty based on the cost of maintaining long-range connections, and the second term is a bias towards clustering or linking similar regions with a large overlap in their existing connections (
      • Vértes P.E.
      • Alexander-Bloch A.F.
      • Gogtay N.
      • Giedd J.
      • Rapoport J.L.
      • Bullmore E.T.
      Simple models of human brain functional networks.
      ). This second factor ki,j can also be called a ‘homophily’ factor by analogy to social networks, where two people with many friends in common are likely to also become friends, even if they live relatively further apart. This factor therefore tends to promote the formation of hubs and to connect up hubs even if they are far apart.
      Figure thumbnail gr1
      Figure 1(A) Schematic of the generative network model proposed in Vértes et al (
      • Vértes P.E.
      • Alexander-Bloch A.F.
      • Gogtay N.
      • Giedd J.
      • Rapoport J.L.
      • Bullmore E.T.
      Simple models of human brain functional networks.
      ). Edges are added at successive timesteps t (see illustration at the top of the panel). The probability of adding a given edge connecting nodes i and j is given by Pi,j which is a function of the distance di,j between the nodes and a function of the number ki,j of neighbours i and j already share in common (see illustration at the bottom of the panel). (B) Phase diagram of the model shown in (A). The orange and purple arrows show sections through phase space, varying only η or γ, respectively, while the other parameter is held at its optimal value estimated in healthy volunteers (HV). Schematics of the networks obtained at various points along these sections are also shown. Increasing parameter η results in fewer long-range connections, while increasing parameter γ results in the emergence of hub nodes. The optimal parameters are slightly detuned when fitting the model to brain networks for people with childhood onset schizophrenia (COS). (C) Schematic representation of three families of stylized facts. Top: the model seeks to capture aspects of the topology of the empirical adult network, such as the degree distribution. Middle: the model seeks to capture aspects of the topography of the empirical adult network, such as the location of hubs. This can be quantified by correlating the modelled and empirical degree across nodes. Bottom: the model seeks to capture aspects of network growth, such as how a given network metric varies over developmental time. (D) Generative models can offer a mechanistic link from genetic and environmental factors, to altered wiring rules, to varying brain network organization, to behavioural differences.
      Figure thumbnail gr2
      Figure 2Association between wiring parameters (η and γ) and cognitive outcomes in the Centre for Attention Learning and Memory (CALM) cohort of children (N=270). (A) Using a partial least squares regression, Akarca et al. (
      • Akarca D.
      • Vértes P.E.
      • Bullmore E.T.
      • Astle D.E.
      the CALM team
      A generative network model of neurodevelopmental diversity in structural brain organization.
      ) identify a linear combination of various markers of cognitive performance in childhood (cognition score) which correlates significantly with a linear combination of the wiring parameters (parameter score), r = 0.191, p = 1.63x10-3. (B) This panel shows the loadings or contributions of the nine individual cognitive tests onto the final cognitive score. (C) This panel shows the loading of each wiring parameter onto the final parameter score. We see that large-magnitude wiring parameters (e.g., large negative η values and large positive γ values) are associated with increased cognitive scores.
      The model has two parameters η and γ which calibrate the relative importance of the ‘cost’ and ‘value’ factors. Fitting paramters η and γ to brain networks derived from neuroimaging data allows the model to simultaneously reproduce the empirically observed degree distribution and distance distribution of connections in human brain networks, as well as their level of clustering, modularity and efficiency (Figure 1B). Interestingly the same model also proved to have wide applicability across different modalities, scales and species. While the model was initially developed in the context of functional brain networks derived from resting-state fMRI data, it is arguably more natural to develop GNMs for anatomical brain networks based on DTI data (
      • Betzel R.F.
      • Bassett D.S.
      Generative models for network neuroscience: prospects and promise.
      ). In 2016, Betzel and colleagues showed that the same model also provides an excellent fit to anatomical brain networks, and is only outperformed by a closely related model which includes a normalized version of the homophily term, called ‘Matching Index’ (
      • Betzel R.F.
      • Avena-Koenigsberger A.
      • Goni J.
      • He Y.
      • de Reus M.A.
      • Griffa A.
      • et al.
      Generative models of the human connectome.
      ). Recent work also found that the same models, trading off connectivity cost versus homophily, provide a good fit to anatomical brain networks in the mouse (

      Carozza, S., Holmes, J., Vértes, P. E., Bullmore, T. E., Arefin, T. M., Pugliese, A., et al. (2022). Early adversity changes the economic conditions of structural brain network organization. bioRxiv 2022.06.08.495303; doi: https://doi.org/10.1101/2022.06.08.495303

      ), and even to the formation of functional networks in in-vitro cortical cultures (

      Akarca, D., Dunn, A. W. E., Hornauer, P. J., Ronchi, S., Fiscella, M. Wang, C., et al. (2022) Homophilic wiring principles underpin neuronal network topology in vitro. bioRxiv 2022.03.09.483605; doi: https://doi.org/10.1101/2022.03.09.483605

      ).
      As described in step 5 of our recipe, a key part of generative network modelling is to keep designing increasingly accurate models. One approach is to apply the recipe iteratively, identifying features of the empirical network that are not well captured by the best current models, and considering these as additional stylized facts for the next iteration. A second, more hypothesis-driven approach is to include additional domain-specific factors, which we expect might improve model-fit or explain more complex features of the network. In the next section we will explore both of these approaches in the context of cellular scale connectomes in C. elegans, before translating the resulting insights back to human brain network modelling.

      Lessons from modelling cellular-scale brain networks

      A striking result in network neuroscience is that the same organizational principles (e.g. modules, hubs, small-world networks) seem to apply across scales and species, from large-scale human brain networks all the way down to the cellular-scale of the nematode C. elegans (
      • van den Heuvel M.P.
      • Bullmore E.T.
      • Sporns O.
      Comparative Connectomics.
      ). It is natural therefore to ask whether the same wiring rules also underpin these diverse networks. C. elegans offers a unique opportunity to explore this question, as it is to date the only organism for which a full wiring diagram exists (
      • White J.G.
      • Southgate E.
      • Thomson J.N.
      • Brenner S.
      The structure of the nervous system of the nematode Caenorhabditis elegans.
      ,
      • Varshney L.R.
      • Chen B.L.
      • Paniagua E.
      • Hall D.H.
      • Chklovskii D.B.
      Structural Properties of the Caenorhabditis elegans Neuronal Network.
      ). In addition, we have access to the birth-time of each individual neuron (node) in the network so that we can move from generative network models (which seek to replicate stylized facts about the final, adult network) to growth models (which seek to capture growth curves for specific properties). For example in 2013, Nicosia*, Vértes* et al showed that the growth of the C. elegans neural network undergoes a transition from an accelerated to a constant increase in the number of links (synaptic connections) as a function of the number of nodes (neurons), and that this transition happens at the time of hatching (
      • Nicosia V.
      • Vértes P.E.
      • Schafer W.R.
      • Latora V.
      • Bullmore E.T.
      Phase transition in the economically modeled growth of a cellular nervous system.
      ). They also found that a trade-off model of the form Pi,jDi,jKi,j was again able to provide a close fit to a range of stylized facts (Figure 1C), including: (i) aspects of the final network topology: distribution of nodal degree, local efficiency, and connection distance. (ii) aspects of the final network topography: anatomical location of high-degree nodes, and connection densities within and between anatomically defined ganglia. (iii) the characteristic shape of the growth curve, switching from accelerated to constant increase in connections as new neurons are born after hatching.
      Unsurprisingly, accurately modelling these increasingly sophisticated stylized facts was only made possible by including additional domain-knowledge into the model.
      A first crucial step was to account for the fact that the connection distance between any pair of neurons will be shorter at earlier stages of development, as the worm grows in length. Therefore di,j was estimated as the Euclidean distance between neurons i and j at the time of birth of the neurons, as extrapolated based on the worm’s elongation during development. This dynamic assessment of connection cost was crucial especially to matching the shape of the growth curve.
      A second novel aspect of this model is that the best-fitting value term Ki.j is not a homophily term in this case, but a simpler ‘preferential attachment’ term, such that a newborn neuron j is more likely to connect to an existing node i if it has higher degree ki, as defined by the eventual degree of each node in the adult network. From a biological perspective, new neurons might ‘know’ about the final degree of other neurons if this was genetically predetermined and signaled in some way. This idea is supported by the finding, across scales and species, that high-degree hub nodes display a shared transcriptomic signature (43 and references therein). Such an implicit genomic patterning is likely what allows the model to match the topography of the neural network (i.e. the anatomical location of its eventual hubs and other features).
      Recently, a string of papers have asked how these considerations of anatomical growth and genetic patterning translate to human brain networks. While early work on generative models only attempted to match topological characteristics (global clustering, global efficiency, modularity and degree distribution) more recent work has asked whether the same models also provide good fits for topography (
      • Akarca D.
      • Vértes P.E.
      • Bullmore E.T.
      • Astle D.E.
      the CALM team
      A generative network model of neurodevelopmental diversity in structural brain organization.
      ,

      Arnatkevičiūtė, A., Fulcher, B.D., Oldham, S., Tiego, J., Paquola, C., Gerring, Z.M, et al. (2021) Genetic influences on hub connectivity of the human connectome. Natuure Communications, 12, 4237. https://doi.org/10.1038/s41467-021-24306-2

      ,

      Oldham, S., Fulcher, B. D., Aquino, K., Arnatkevičiūtė, A., Paquola, C., Shishegar, R. & Fornito, A. (2021) Modeling spatial, developmental, physiological, and topological constraints on human brain connectivity. Science Advances, 8:22, doi: 10.1126/sciadv.abm6127

      ). This can be evaluated by correlating nodal properties (such as degree) between the empirical and synthetic networks, to assess for example whether high-degree hubs of the synthetic network are located in the correct anatomical regions. Interestingly, these papers reach opposite conclusions with Arnatkevičiūtė and colleagues reporting no correlation whatsoever (

      Arnatkevičiūtė, A., Fulcher, B.D., Oldham, S., Tiego, J., Paquola, C., Gerring, Z.M, et al. (2021) Genetic influences on hub connectivity of the human connectome. Natuure Communications, 12, 4237. https://doi.org/10.1038/s41467-021-24306-2

      ) while Akarca and colleagues report moderate correlations both for node degree and for other network properties (
      • Akarca D.
      • Vértes P.E.
      • Bullmore E.T.
      • Astle D.E.
      the CALM team
      A generative network model of neurodevelopmental diversity in structural brain organization.
      ). Although this is an area of active research, we speculate that the discrepancy may relate to technical differences in parcellation or tractography methods as well as differences in how best-fitting model parameters are estimated and/or in how the network growth is initialized, based on a ‘seed’ of initial connections (
      • Akarca D.
      • Vértes P.E.
      • Bullmore E.T.
      • Astle D.E.
      the CALM team
      A generative network model of neurodevelopmental diversity in structural brain organization.
      ).
      Interestingly, in keeping with the insights from C. elegans, Arnatkevičiūtė and colleagues show that moderate fits to network topography can also be recovered by incorporating genetic constraints, effectively replacing the spatial distance penalty in the model by a penalty for connecting regions with dissimilar transcriptomic profiles. Similar results are also obtained by Oldham and colleagues (

      Oldham, S., Fulcher, B. D., Aquino, K., Arnatkevičiūtė, A., Paquola, C., Shishegar, R. & Fornito, A. (2021) Modeling spatial, developmental, physiological, and topological constraints on human brain connectivity. Science Advances, 8:22, doi: 10.1126/sciadv.abm6127

      ), who show that this genetic constraint alone can match or even outperform trade-off models (although based on a different formulation from the trade-off models previously studied). One intriguing possibility is that the genetic patterning of brain regions already includes both a spatial constraint and topographic information, effectively providing a biological implementation of the proposed trade-off between cost and value. This genomic hypothesis is also consistent with recent work showing that the process of cell division sets up a spatial patterning of gene expression profiles across the brain, and that this map can be used to efficiently encode axonal routes (
      • Kerstjens S.
      • Michel G.
      • Douglas R.j.
      Constructive connectomics: How neuronal axons get from here to there using gene-expression maps derived from their family trees.
      ). It is worth noting that preferential connection of regions with similar transcriptomic profiles also aligns with the observation that regions with similar cytoarchitecture (potentially indexed by transcriptomic similarity) tend to preferentially connect (
      • Beul S.F.
      • Grant S.
      • Hilgetag C.C.
      A predictive model of the cat cortical connectome based on cytoarchitecture and distance.
      ,
      • Beul S.F.
      • Barbas H.
      • Hilgetag C.C.
      A predictive structural model of the primate connectome.
      ,
      • Goulas A.
      • Uylings H.B.
      • Hilgetag C.C.
      Principles of ipsilateral and contralateral cortico-cortica connectivity in the mouse.
      ).
      Separately, Oldham and colleagues also evaluated the importance of anatomical growth in human brain networks by modelling how the pairwise distance between brain regions increases over developmental time (

      Oldham, S., Fulcher, B. D., Aquino, K., Arnatkevičiūtė, A., Paquola, C., Shishegar, R. & Fornito, A. (2021) Modeling spatial, developmental, physiological, and topological constraints on human brain connectivity. Science Advances, 8:22, doi: 10.1126/sciadv.abm6127

      ). As with C. elegans, they showed that this dynamic treatment of spatial constraints substantially improved model fit across a range of trade-off models taking distance into account. As and when relevant data become available it will be interesting to implement a dynamic version of genetic constraints, taking into account transcriptomic differentiation of brain regions over developmental time. Indeed, it is likely that time-related constraints, much like spatial ones, might also be genetically encoded.
      It is worth noting that, separate from generative modelling, there is a large body of work on computational modelling of brain development (
      • Beul S.F.
      • Goulas A.
      • Hilgetag C.C.
      Comprehensive computational modelling of the development of mammalian cortical connectivity underlying an architectonic type principle.
      ,
      • Beul S.F.
      • Hilgetag C.C.
      Systematic modelling of the development of laminar projection origins in the cerebral cortex: Interactions of spatio-temporal patterns of neurogenesis and cellular heterogeneity.
      ,
      • Van Ooyen A.
      • Van Pelt J.
      • Corner M.A.
      Implications of activity dependent neurite outgrowth for neuronal morphology and network development.
      ,

      Van Ooyen, A. (2003) Modeling neural development. MIT press. doi: 10.7551/mitpress/4703.001.0001

      ,
      • Koene R.A.
      • Tijms B.
      • van Hees P.
      • Postma F.
      • de Ridder A.
      • Ramakers G.J.A.
      • et al.
      NETMORPH: A framework for the stochastic generation of large scale neuronal networks with realistic neuron morphologies.
      ,
      • Zubler F.
      • Douglas R.
      A framework for modeling the growth and development of neurons and networks.
      ,
      • Kassraian-Fard P.
      • Pfeiffer M.
      • Bauer R.
      A generative growth model for thalamocortical axonal branching in primary visual cortex.
      ,
      • Goulas A.
      • Betzel R.F.
      • Hilgetag C.C.
      Spatiotemporal ontogeny of brain wiring.
      ) which explicitly considers biophysical properties and processes such as neurogenesis (
      • Beul S.F.
      • Goulas A.
      • Hilgetag C.C.
      Comprehensive computational modelling of the development of mammalian cortical connectivity underlying an architectonic type principle.
      ,
      • Beul S.F.
      • Hilgetag C.C.
      Systematic modelling of the development of laminar projection origins in the cerebral cortex: Interactions of spatio-temporal patterns of neurogenesis and cellular heterogeneity.
      ), neurite growth (
      • Van Ooyen A.
      • Van Pelt J.
      • Corner M.A.
      Implications of activity dependent neurite outgrowth for neuronal morphology and network development.
      ) or cell division (
      • Kerstjens S.
      • Michel G.
      • Douglas R.j.
      Constructive connectomics: How neuronal axons get from here to there using gene-expression maps derived from their family trees.
      ). These are not the focus of the current review, but we refer the reader, for example, to recent work comparing detailed mechanistic simulations to generative modelling (
      • Kassraian-Fard P.
      • Pfeiffer M.
      • Bauer R.
      A generative growth model for thalamocortical axonal branching in primary visual cortex.
      ). As with the genomic encoding of spatial constraints described above, it is likely that some of the features used in generative models (such as homophily) can be derived from more biologically detailed ontogenic simulations (
      • Goulas A.
      • Betzel R.F.
      • Hilgetag C.C.
      Spatiotemporal ontogeny of brain wiring.
      ).

      Modelling individual variability in human brain networks

      So far we have seen that computational models with just two parameters have surprising explanatory power for how the complexity of brain wiring may reliably emerge. However for these models to become useful accounts of developmental processes in the brain, they will need also to allow for individual variability and, crucially, they will need to link this variability to genetic and environmental factors.
      Vértes et al provided a first indication that this may be possible by showing that many of the abnormal properties of brain functional networks in people with schizophrenia could be captured by the original model Pi,jdi,jηki,jγ with only a small detuning of model parameters η and γ (Figure 1B; 31). Subsequently Zhang and colleagues showed, in the context of anatomical brain networks, that best-fitting parameters η and γ varied between individuals in a way that correlates with polygenic risk for schizophrenia (
      • Zhang X.
      • Braun U.
      • Harneit A.
      • Zang Z.
      • Geiger L.S.
      • Betzel R.F.
      • et al.
      Generative network models of altered structural brain connectivity in schizophrenia.
      ). Most recently, working with ex-vivo neuroimaging of mouse connectomes, Carozza and colleagues also showed that early life adversity (in the form of a maternal separation paradigm) can influence best-fitting η and γ parameters, and thereby shift the constraints that govern the formation of the structural connectome at the level of the individual (

      Carozza, S., Holmes, J., Vértes, P. E., Bullmore, T. E., Arefin, T. M., Pugliese, A., et al. (2022). Early adversity changes the economic conditions of structural brain network organization. bioRxiv 2022.06.08.495303; doi: https://doi.org/10.1101/2022.06.08.495303

      ). Separately, Akarca and colleagues showed that subtle individual variations in best-fitting η and γ parameters were also associated with a range of cognitive scores in a large cohort of children with increased neurodevelopmental risk of poor cognitive outcomes (
      • Akarca D.
      • Vértes P.E.
      • Bullmore E.T.
      • Astle D.E.
      the CALM team
      A generative network model of neurodevelopmental diversity in structural brain organization.
      ).
      Taken together, these studies suggest that both genetic and environmental risk factors can influence the brain’s wiring rules by calibrating the trade-off between connectivity cost and homophily. For example, a slightly lower eta value in some individuals will lead to reduced distance-penalty during network growth, such that relatively more long-distance connections are allowed to develop. In turn, the resulting topological differences in the developing brain networks can lead to a diversity of cognitive and behavioural outcomes (Figure 1D).

      Outlook and future directions

      Generative network modelling is uniquely placed to offer a mechanistic explanation for how genetic and environmental factors shape developing brain networks and predispose certain children and young people to mental health and cognitive disorders.
      Future work in this area will likely focus on exploring in more detail the impact of genetic and other factors on wiring rules. At the same time, models will seek to capture increasingly complex features of brain networks, including features from algebraic topology (
      • Courtney O.T.
      • Bianconi G.
      Weighted growing simplicial complexes.
      ,

      Helm, A., Blevins, A.S., Bassett, D. S. (2021) The growing topology of the C. elegans connectome. arXiv:2101.00065v2

      ) and multiplex representations of brain networks (

      Bentley, B., Branicky, R., Barnes, C. L., Chew, Y. L., Yemini, E., Bullmore, E. T., et al. (2016). The multilayer connectome of Caenorhabditis elegans. PLoS Computational Biology, 12, e1005283. doi:10.1371/ journal.pcbi.1005283

      ). As described above, this work will depend crucially on understanding the methodological factors - from parcellation to model-fitting - that influence model performance.
      It will also be important to move towards a growth modelling framework and seek to capture the developmental trajectory of brain network organization, as opposed to just the final state. This will likely require adapting our models in at least three ways. First, we will need to include time-resolved aspects of brain anatomy, as suggested by (
      • Nicosia V.
      • Vértes P.E.
      • Schafer W.R.
      • Latora V.
      • Bullmore E.T.
      Phase transition in the economically modeled growth of a cellular nervous system.
      ,

      Oldham, S., Fulcher, B. D., Aquino, K., Arnatkevičiūtė, A., Paquola, C., Shishegar, R. & Fornito, A. (2021) Modeling spatial, developmental, physiological, and topological constraints on human brain connectivity. Science Advances, 8:22, doi: 10.1126/sciadv.abm6127

      ). Second, we will need to adapt models to account not just for the creation of binary edges between brain regions but for the tuning of connection weights that occurs through synaptic pruning and plasticity. And third we will likely need to allow for some connections to get weaker rather than stronger over time, since pruning is well-known to be a key component of early brain development (
      • Collin G.
      • van den Heuvel M.P.
      The ontogeny of the human connectome: Development and dynamic changes of brain connectivity across the life span.
      ). Finally, it is also possible that the parameters or even wiring rules themselves cannot be considered constant over developmental time (
      • Betzel R.F.
      • Avena-Koenigsberger A.
      • Goni J.
      • He Y.
      • de Reus M.A.
      • Griffa A.
      • et al.
      Generative models of the human connectome.
      ), which would add yet another layer of complexity.
      Ultimately, as we develop these increasingly sophisticated models of both typical and atypical brain development, these may suggest early interventions to steer network growth away from undesirable states.

      Uncited reference

      • Latora V.
      • Marchiori M.
      Efficient behavior of small-world networks.
      ,
      • van den Heuvel M.P.
      • Stam C.J.
      • Kahn R.S.
      • Hulshoff Pol H.E.
      Efficiency of Functional Brain Networks and Intellectual Performance.
      ,
      • Li Y.
      • Liu Y.
      • Li J.
      • Qin W.
      • Li K.
      • Yu C.
      • Jiang T.
      Brain anatomical network and intelligence.
      ,
      • Kruschwitz J.D.
      • Waller L.
      • Daedelow L.S.
      • Walter H.
      • Veer I.M.
      General, crystallized and fluid intelligence are not associated with functional global network efficiency: A replication study with the human connectome project 1200 data set.
      ,

      Fornito, A., Arnatkevičiūtė, A. & Fulcher, B. (2019) Bridging the gap between connectome and transcriptome. Trends in Cognitive Sciences, 23(1), 34-50.

      .

      Acknowledgement:

      P.E.V. is a fellow of MQ: Transforming Mental Health (grant no. MQF17_24). All research from the Department of Psychiatry at the University of Cambridge is made possible by the NIHR Cambridge Biomedical Research Centre and the NIHR East of England Applied Research Centre. The views expressed are those of the author and not necessarily those of the NHS, the NIHR or the Department of Health. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.

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