Table 1 Summary of connectivity challenges across different disciplines. Extent to which connectivity challenges are an issue: * do not present a challenge; ** presents a challenge but progress has been made; *** presents a major challenge.

Systems Biology

**

• FUs are biologically well defined, based on the biological system rather than measurement process.

• A full inventory of functional elements is still missing; also interdependences between different levels of cellular organizations (i.e. among fundamental units from different networks) are often neglected.

**

• Generally, a clear time scale separation ensures a distinction between SC and FC.

• While FC is a network representation of cellular states (e.g. correlations between metabolite concentrations or gene expression levels), SC is shaped by evolution on a much slower time scale.

**

• Concepts such as modularity and hierarchy may provide a starting point for addressing emergent behaviour within the current knowledge of SC and FC, but a true incorporation of the many (spatial and temporal) scales will require novel multiscale methods.

• Key challenge: understanding the relation of regulatory mechanisms and emergent collective behaviour.

**

• Explosive growth of high-throughput methods providing access to many facets of structural and functional connectivities, but has led to a dramatic diversification of databases, methods and nomenclatures, resulting in a strong need of data and method integration.

Neuroscience

*

• Defining the fundamental unit of the brain is clear-cut and depends on the level (scale) at which one is working.

• The fundamental unit is commonly defined as being the cortical area, although is alternatively defined as the neuron.

**

• Different techniques are used to distinguish between SC and FC, but FC is defined (using various techniques) using indirect correlates of brain activity.

• Memory affects the relation between SC and FC through its long-lasting anatomical import on the structure of the network.

• Understanding the effects of memory remains a key challenge because of the diffuse nature of the anatomical imprint.

**

• Great progress has been made in understanding concepts such as attention and emotion as emergent properties of neural activity.

• The big challenge today is to push this understanding to its limit and offer a convincing account of how neural activity and its organization in the networks of the brain eventually leads to consciousness.

**

• A range of techniques exist to measure connectivity

• Due to differences in spatial and temporal resolution between different measurement techniques, hybrid approaches are becoming popular to overcome the challenge of measuring SC and FC at appropriate resolutions.

• The main challenge with the [premature] use of hybrid methods is that the final result may be limited to what the least sensitive method can provide.

• Measurements of FC are inferred from high-resolution snapshots, rather than being measured directly.

Computational Neuroscience

*

• The fundamental unit is typically defined as being individual neurons or cortical areas and form the nodes of a network. Identifying these fundamental units is done using anatomical means and neurobiological knowledge.

**

• It is increasingly common to directly and quantitatively compare SC and FC (such separations are readily undertaken and pose no major challenges).

• A limitation is that SC/FC correlations only tend to look in one direction only – the effect of structure on function.

*

• Network-based approaches are used, allowing for the interplay of SC and FC allow for the exploration of self-organization and pattern formation – both important characteristics of emergent behaviour.

*

• Measurements of connectivity are undertaken using a wide range of network-based descriptors of connectivity.

Geomorphology

***

• No clear or consistent definition of the fundamental unit (it is dependent on the research question).

***

• Separating SC and FC in a meaningful way is challenging because of the myriad of processes operating over a multitude of spatial and temporal scales.

• Separating SC and FC is compounded by the imprint of memory and timescales over which it affects connectivity for a meaningful separation of SC and FC.

***

• Tools to explore how SC and FC lead to emergent behaviour are lacking.

• Multi-method approaches focussing on the interactions of SC and FC over relevant spatio-temporal scales may aide in understanding emergent behaviour.

**

• Approaches to measure SC are well developed, and make use of high-resolution techniques where appropriate.

• No direct measurement techniques for FC are available and rely on inferring connectivity from snapshots of information.

• Modelling is often used as a surrogate for direct measurements of FC.

Ecology

**

• Conceptually the fundamental unit is defined as the ecosystem.

• In operational terms, defining the FU is more difficult.

**

• Linking and separating SC and FC is common, but made challenging where there are time lags in the response of ecological function to changes in ecological structure and vice versa.

**

• Many attempts to explain emergent behavior (e.g. using advection-diffusion models) have produced realistic patterns, but at the expense of realistic processes.

• A challenge is to study emergent behaviour using model structures that are not inherently designed to produce patterns.

**

• A range of techniques exist for measuring SC based on simple indices of patch connectivity, through to network-based approaches.

• Measuring FC poses much more of a challenge and requires dealing with complex phenomena that are difficult to quantify and tend to reply upon inferring FC connectivity based on a series of empirical measurements through time.

• Advances being made in measuring SC and FC through the use of weighted monopartite and bipartite networks.

Social Network Science

*

• Traditionally the fundamental unit is defined as the person, although more recently the definition has become less certain with some researchers now using the interaction as the unit of study.

***

• The focus tends to be on the connectivity of structural networks.

• Approaches to look at dynamics (FC) are limited in terms of analytical power.

• In social networks culture is a type of memory effect that affects function (i.e. the response of an individual to social interactions), and the complexity that an evolving mix of cultures brings to social networks is a significant challenge.

***

• The emergence of the network property is conditional on entire network interactions, and the challenge of adopting models of social behaviour that recognise the diversity of social interactions across a population complicates matters further.

***

• Measuring connectivity is a major challenge due to ethical, practical and philosophical constraints.