Manual Delay-Coupled Complex Systems: and Applications to Lasers (Springer Theses)

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All authors reviewed the manuscript. Reprints and Permissions. By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Advanced search. Skip to main content. Article metrics Citations 3. Altmetric 2. Theoretical tools to relate emergent behaviour to structure and function are lacking, and by separating SC from FC we make it even more difficult to analyse emergent behaviour. A key challenge is how to best use concepts and operalizations of connectivity to develop quantitative descriptions of emergence that will enable us to understand emergent behaviour better, and overcome the aforementioned barriers to this goal.

To apply the concept of connectivity to understand the behaviour of real-world complex systems, it is necessary to quantify connectivity, yet this poses several challenges. The level of connectivity observed within a system depends on the lens through which one observes it. Due to the inability of existing tools to measure connectivity directly, it is often necessary to infer connectivity from an alternative set of indirect measurements.

Indirect measurements can lead to subjectivity and uncertainty in our understanding of connectivity and dynamical feedbacks operating within a system, and furthermore, can impede appropriate operational definitions of the fundamental unit. There is also a timescale issue: at finer timescales measurements of connectivity may tend towards descriptors of SC rather than FC.

FC is more than just inferring what is happening between snap-shots, but trying to determine the actual processes operating to produce fluxes Bracken et al. A variety of tools for studying connectivity exist: the challenge here is both to recognise the limitations that these tools place on our understanding of connectivity, and to develop new tools where those limitations significantly constrain our understanding. Within all the disciplines explored here, an abstraction of the system is required in order to study connectivity. Most commonly, this abstraction involves representing the system as a network.

In recent years networks have emerged as an invaluable tool for describing and quantifying complex systems, especially due to the ease with which networks can be used to represent hierarchical organization over multiple scales Clauset et al. The representation of a system as a network provides the opportunity — using network-based tools — to disentangle noise and stochasticity from non-random patterns and mechanisms, in order to gain a better understanding of how these systems function and evolve Menichetti et al. Here, the network-based approach underpins our conceptual framework for connectivity studies due to its ubiquity.

The simplest network-based abstraction of a system is a graph consisting of nodes and links. More detailed abstractions can be achieved through the use of weighted links and directional links. In an unweighted network, links are binary entities, where they are either present or not. In a weighted network links have associated weights that record their strengths relative to one another Newman, a. In a directional network, the flow of information or material is in one direction, from one node to another, whereas in an undirected network, flow is in either direction between two nodes.

A system can also be represented as a bipartite network, in which two different classes of nodes can be represented, and links between nodes are allowed across but not within two classes Daminelli et al. Furthermore, whilst some disciplines can be represented using non-spatial networks, space is critical in others.

Network-based representation of structural and functional connectivity. Systems Biology strives to understand biological entities as complex systems through analysis of the interdependencies of the components of these systems; i. Over the past two decades, new high-throughput technologies have enabled mathematical modelling, statistical analysis and numerical simulation of biological systems with unprecedented detail for each level of description.

At the core of systems thinking in Biology is the concept of networks see, e. Together with Statistical Physics e. An important prerequisite for the success of networks in Systems Biology is that the fundamental units are biologically well defined. Metabolic networks, for example, are a compilation of biochemical knowledge the set of chemical reactions catalyzed by enzymes and the inventory of genes encoding enzymes derived from a carefully annotated genome.

The most prominent category of gene regulatory networks, transcriptional regulatory networks, is obtained from genes in the genomes together with the binding sites for transcription factors. As is often the case, many little issues are hidden in the details underneath these general definitions. Similarly regarding gene regulatory networks, there is a multitude of other regulatory mechanisms in addition to transcription factors.

Thus, even in the comparatively simple case of bacterial gene regulation, we cannot assume the transcriptional regulatory network to be a complete representation of the regulatory apparatus see for example the debate in Marr et al. Even though the networks compiled at the gene-regulatory, metabolic or protein interaction level can be thought of as vast compilations of biological information, it should be noted that the underlying databases will drift with time: more biological knowledge is accumulated, but also the biological categories used to derive suitable network representations of the system will be refined and altered.

As a consequence, the network properties will drift as well Beber et al. Additionally, the diversity of databases, data formats, models and computational tools and the lack of standards on these levels is currently an enormous barrier to progress in the field Chavan et al. Thus, when adopting a network representation many details are necessarily omitted, and decisions about these details can have a severe effect on network structures. A key strategy in addressing this challenge is to separate the interdependencies of system components i.

SC from correlations among dynamical observations such as gene activities or metabolite concentrations i. For a wide range of organisms in all domains of life, the genome i. The richness of biology in terms of molecular components and interactions has for a long time prevented the development of systems-level descriptions.

For a long time the detailed mathematical modelling of small subsystems e. This is the defining feature of Theoretical Biology. The abstraction of biological systems in terms of nodes and links has paved the way for more qualitative approaches, which, however, allow us to address the scale of a whole cell. These network approaches have become one of the cornerstones of Systems Biology. One of the reasons for the success of the network view in Systems Biology is that structural properties and snapshots of biological function are typically measured in independent ways: transcriptome profiles as activity states of gene regulatory networks which are compiled from information on transcription factors and binding sites , metabolic fluxes or metabolite concentrations as activity states of metabolic networks which are compiled from known biochemical reactions and the enzymes identified in the genome.

For a particular observation of biological activity — for example a transcriptome profile — the effective networks, representing the currently active part of a biological system, can be viewed as a representation of FC. Examples include the predictive power of elementary flux modes in metabolism Stelling et al. As mentioned above, the independence of data resources behind SC and FC are of high relevance to the success of network approaches in Systems Biology.

Nevertheless, the possibility of predicting a link i. Examples of such link prediction or network inference approaches include the inference of gene regulatory networks from gene expression patterns Marbach et al. In the literature on network inference, e. In Claussen et al. One should also emphasise that all these methods have been designed for data-rich situations and do not necessarily yield convincing results see, e. In Systems Biology, memory effects in the classical sense of learning and adaptive networks are highly reduced, because the relevant time scales of dynamical behaviour and network adaptation i.

Still, we can expect the networks to be shaped by evolution to optimize or enhance certain functional properties. Network architecture can facilitate the spread of time scales contributing to a certain biological function, which is seen on the scale of few nodes e. In Kashtan and Alon the relationship between modularity and time scales is investigated in more detail.

Emergent behaviour is a key concept from the theory of complex systems. With this remark, Kitano emphasizes the danger of building up Systems Biology directly from the toolbox of complex systems see also the more general remark by Keller, Nevertheless, many examples of self-organised patterns — and thus of emergent behaviour — come from biology, both on the intracellular level e.

Networks can play an important role on this level as well. A prominent example is the food foraging network formed by the slime mould Physarum polycephalum , which connects spatially distributed food sources in an efficient network layout Tero et al. This example points to an important difference between patterns in Physics and Chemistry on the one hand and patterns in Biology on the other. In Physics and Chemistry, patterns are often a by-product of the nonlinear interactions of system components.

In Biology patterns often have undergone a clear evolutionary tuning and this might serve a system-level function such as the network connecting food sources in the case of Physarum , the aggregation of cells as a step to a multicellular organism in the case of Dictyostelium or the spatial organization of cell division in E.

The regulatory components must therefore have evolved to yield stable functional patterns and we can thus expect a deep relationship between regulatory components and properties of spatiotemporal patterns. An example is strong and non-monotonous dependence of the density of spiral waves which regulates the size of the later-stage multicellular aggregates on the intracellular feedback loop regulating the production of the main signalling substance, cAMP Sawai et al. Generally speaking, in Systems Biology the structural networks come from two main sources: 1 they are obtained from repositories of accumulated biological information; 2 they are derived from high-throughput data.

As in most other disciplines discussed here, it is highly instructive to discriminate between the cases, where the system properties define the units i. Typically, for networks extracted from accumulated biological information the fundamental units are not dictated by the measurement process, but rather by the biological system itself. As outlined above, metabolic networks can be seen as an example of this category.

Protein interaction networks, on the other hand, are an example of the other type of SC: There, a link the physical binding of two proteins has originally become a fundamental unit due to its accessibility via high-throughput measurements. The biological relevance of such protein-protein interactions is not the defining criterion. In fact, a link between two proteins might be part of the network, even though the two proteins are located in different cellular compartments and therefore will never actually have a chance of interacting. Functional connectivity, which is a representation of the current state of, for example, a biological cell, is typically measured via the high-throughput technologies discussed above.

Such as state can be an activity pattern of all genes or a list of concentrations of metabolites available in the cell at a certain moment in time. Neuroscience is the study of the nervous system and is centred on the brain. Connectivity in neuroscience is mainly studied using networks, with a range of networks being studied, depending on what is adopted as the network components.

The techniques used by neuroscientists range from molecular and cellular studies of individual nerve cells to imaging of sensory and motor tasks in the brain. These techniques have enabled researchers to investigate the nervous system more fully, including how it is structured, how it works, how it develops, how it malfunctions, and how it can be changed. For the sake of brevity, here we focus on an intermediate level description where the nodes of the network are well circumscribed areas that are bounded by borders that are sharply defined by changes in both structure anatomically definable changes in the local architecture and function e.

Within Neuroscience, it is usual to focus on a level of description that is characteristic of the structural organization of the brain and the functions it performs, and for didactic purposes and for practical reasons, it is usual to focus on a network description that has neither too few nor too many elements. There are roughly billion neurons in a brain, so a network description based on individual neurons or sub-cellular entities like synapses will lead to a network of too many components to be of practical use. When one studies quantitatively the way the constituent neurons are arranged in space with respect to each other the cytoarchitecture and what neurotransmitters they express receptor density distribution a clear pattern emerges: the brain is divided into a few hundred areas about to areas for the cortical mantle 1.

Within each one of these areas the cytoarchitecture and receptor density distributions are fairly uniform with only relatively slow and gradual variation. In contrast, at the borders between these areas there is a rapid transition, so identifying either the rapid changes in cytoarchitecture or receptor density properties allows an accurate and objective identification of the borders and hence an accurate delineation of these areas Zilles et al. For the purposes of the following discussion, the cytoarchitectonic areas are the fundamental unit for neuroscience, at least at the level of nodes for the networks focused on here.

Functional connectivity per se is defined in terms of quantitative measures of linked activity, computed from time series of regional brain activations. Both methods provide indirect correlates of brain activity with time constants of many minutes for PET and a few seconds for fMRI.

These changes are slow — orders of magnitude slower than the few millisecond transit time for the activity between areas. With improvements in the accuracy of these methods, it has become clearer that the foci of brain activity coincides with the cytoarchitectonic areas, with initial demonstrations emphasizing responses to well defined stimuli as these excite the early cytoarchitectonic areas in each sensory hierarchy.

Within Neuroscience, memory comes under different forms, each characterised by a different temporal scale. Modality specific sensory memory allows continuity in perception and it typically decays within a second. Short-term memory allows us, possibly through rehearsal, to remember recent events and has a characteristic decay time around a minute. Short-term memory is very likely based on FC reverberations of patterns of activity that maintain resemblance to the original pattern for only a short time before they become indistinguishable from the background neural activity leaving no permanent trace.

In general, references to memory that are applicable to connectivity are about long-term memory, which allows us to recall events over longer time periods. The process of establishing long term memories is facilitated by the transfer of memory related activity from a temporal store centred around the hippocampus to a more permanent storage in the cortex Bontempi et al. The consolidated memory must be associated with a change in SC of the network, but its long-lasting anatomical imprint is diffusely stamped in many nodes of the anatomical network. These diffused changes are however organised so that spatiotemporal patterns of electrical activity in the network re- construct the experience as and when needed in normal life or as a persistent and inescapable replay of dramatic events in pathology.

Understanding memory is a challenge for neuroscience because of the diffuse nature of its anatomical imprint and the labile nature of the electrical activity associated with its recollection; currently, it is difficult to capture in its totality by any one or a combination of the different neuroimaging modalities.

Detailed analysis of connectivity patterns of the network can be used to identify sub-networks that correspond to individual sensory and other networks. These sub-networks can be derived from the anatomical structure and the SC e. Large-scale organization also emerges from a decomposition of the full network derived from the FC of resting state fMRI van den Heuvel et al. Analyses of these data reveal a natural decomposition of the full network into distinct sub-networks. The role of each sub-network is evident from the known specialization of its component nodes. The decomposition based on the resting activity reveals sub-networks for each sensory modality; networks that are known to be critical for the implementation of supramodal cognitive functions including attention, working memory and the default mode network a network that becomes more active when the subject is not occupied with a specific task or monitoring the external environment.

It therefore appears that properties of the mind are correlated with emergent behaviour of the functional networks, and are consistent with the properties of the physical brain, as these were determined by wet brain anatomy and electrophysiology.

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At this point in time neuroscience has achieved a fair understanding of how elements necessary for complex purposeful behaviour such as attention and memory are implemented, yet it remains a mystery how these are combined to give rise to consciousness which can be considered as the most significant emergent property of all. Understanding how consciousness emerges, from the activity of neurons and the organization and function of the networks they form, constitutes the Holy Grail of modern neuroscience and perhaps of science of the 21 st century. A recent synthesis of results from many neuroimaging studies provides a tentative step in this direction within a unified framework that explains how memory and attention are managed in awake state and sleep states and how they help maintain what appears to be a neural representation of self Ioannides Anatomical connectivity relates to the number of connections linking two anatomical nodes, i.

Quantifying FC requires methods that can separate the contributions from individual cytoarchitectonic areas at temporal resolutions that are typical of the processing time within and the transfer between areas. The distance between neighbouring cytoarchitectonic areas is typically a few millimetres and the typical transfer time between two areas is of the order of 10 milliseconds. The resulting networks are purely structural, and when they are analyzed, e. Direct measures of brain activity rely on electrophysiology.

The EEG records the electrical potential difference on the scalp and the MEG records the magnetic field just outside the head. They are both generated by electrical activity in a very large number of neurons that are activated synchronously and they are arranged in a similar way in space, so that the resulting effect summates constructively. While in principle the EEG and MEG records carry similar information, the analysis of MEG signals requires less detailed modelling of the conductivity profile of the head to identify the generators accurately compared with the EEG analysis.

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Coincidence in spatial location and timing of activations in the brain. The demonstration of the coincidence in spatial location and timing of the earliest visually evoked top and spatial attention bottom related activations responses to images presented in the left visual field.

No matter how regional time series are derived, mathematical methods must then be employed to extract from each pair of time series a quantitative measure of the functional link between two brain areas, usually in two stages. First, one must define an appropriate measure of linked activity.

Laser chimeras as a paradigm for multistable patterns in complex systems

For example using time-delayed mutual information as a non-linear measure enables identification and quantification of linkages between areas in real time for example in relation to an external stimulus or event and enables assessment of reactive delays. The second stage of addressing the connectivity problem is the technical problem of using graph theory tools to put together the pair-wise links into a more global network.

Specific problems can be tackled using a subset of the entire network through judicious choice of what cytoarchitectonic area to include and careful design of experiments. While empirical Neuroscience section Neuroscience deals with measuring and functionally interpreting connectivity on many scales, the aspects of Computational Neuroscience, which we address here, deal with structure-function relationships on a more abstract, aggregated level. Generic models of network topology, as well as simple abstract models of the dynamical units, play an important role.

In Computational Neuroscience the idea of relating network architecture with dynamics and, consequently, function has long been explored e. On the level of network architecture, a particularly fruitful approach has been to compare empirically observed networks with random graphs. The field was revolutionized in the late s by the publication of two further models of random graphs: a model of small-world graphs Watts and Strogatz, uniting high local clustering with short average distances between nodes and a model of random graphs with a broad power-law shaped degree distribution Barabasi and Albert, Within Computational Neuroscience, the fundamental unit is typically defined as being individual neurons Vladimirov et al.

The discussion below focusses on the latter case. In contrast to the discussion in section Neuroscience the fundamental unit is not necessarily identified with the cortical areas, but is more flexible, allowing aggregates of cortical areas or even abstract ones derived from the raw data from fMRI to be the fundamental units that constitute the nodes of a network. Such cortical areas can also be defined by anatomical means and neurobiological knowledge as for example in the cortical areas of the cortical areas network of the cat or the macaque; see Hilgetag et al. In Computational Neuroscience, SC refers to brain network connectivity derived from anatomical and other data, at the level of the fundamental unit.

FC refers to relationships among nodes inferred from the dynamics. Typical observables for FC are co-activations or sequential activations of nodes. A node can be excited active , refractory resting and susceptible, waiting for an excitation in the neighbourhood. Upon the presence of such a neighbouring excitation, a susceptible node changes to the active state for a single time step, then goes into the refractory state, from which it moves to the susceptible state with a probability p at each time step.

Furthermore, spontaneous excitations are possible with a small probability f. A more global perspective includes learning, i. In its simplest form, such a co-evolution of structural and FC is given by Hebbian learning rules Hebb, , where — qualitatively speaking — frequently used network links persist, while rarely used links are degraded. The co-evolution of SC and FC offers an interesting possibility for the overarching perspective of self-organization and emergent behaviours, as the system now can, in principle, tune itself towards phase transition points, maximizing its flexibility and its pattern formation capacities.

This concept is called self-organised criticality and goes back to the pioneering work by Bak et al. Phase transition points of a dynamical system are choices of parameters, which position the system precisely at the boundary between two dynamical regimes e. At such points a small change of the parameter value can induce drastic changes in system behaviour.

As already suggested with the example of waves around hubs, the concepts of self- organization and pattern formation may provide a useful theoretical framework for describing the interplay of SC and FC. Returning to network topology, a wide range of descriptors of connectivity is used in Computational Neuroscience and other disciplines. Another common quantifier of connectivity is the average degree i. Beyond these simple quantifiers, the connection pattern in a graph can be characterized in a multitude of ways, for instance via clustering coefficients Watts and Strogatz, , centrality measures Newman, and the matching index or topological overlap Ravasz et al.

Geomorphologists study the origin and evolution of landforms. Geomorphic surface processes comprise the action of different geomorphic agents or transporting media, such as water, wind and ice which move sediment from one part of the landscape to another thereby changing the shape of the Earth. Therefore, looking at potential sediment pathways connections and transport processes has always been one of the core tasks in Geomorphology.

Chorley and Kennedy, ; Brunsden and Thornes, However, since the beginning of the 21 st century connectivity research experienced a huge boom as geomorphologists started to develop new concepts on connectivity to understand better the complexity of geomorphic systems and system response to change. It is widely recognised that investigating connectivity in geomorphic systems provides an important opportunity to improve our understanding of how physical linkages govern geomorphic processes Van Oost et al.

Connectivity further reflects the feedbacks and interactions within the different system components under changing conditions Beuselinck et al. However, to date most - if not all - of the existing connectivity concepts in geomorphology represent a palimpsest of traditional system thinking based on general systems theory e.

Landforms are the product of a myriad of processes operating at different spatial and temporal scales: defining a fundamental unit for the study of connectivity is therefore particularly difficult. Geomorphologists have traditionally drawn structural boundaries between the units of study which are often obvious by visible sharp gradients in the landscape, for example channel-hillslope or field boundaries.

This imposition of structural boundaries has led to the separate consideration of these landscape compartments, rather than looking at the interlinkages between them, which results in an incomplete picture when it comes to explain large-scale geomorphic landscape evolution. Bracken et al. However, this framework provides no insight into how the fundamental unit may be defined. Its size and demarcation is highly dependent on i the processes involved and ii the spatial and temporal scale of study i. If, for example, the temporal scale of analysis is considerably greater than the frequency of key processes i.

Alternatively, if the temporal scale over which sediment connectivity is evaluated is less than the frequency at which key sediment-transport-related processes within the study domain operate, then sediment connectivity will be perceived to be lower Bracken et al. The size of a fundamental unit in Geomorphology is thus dependent on the underlying research question and may range from plot- e. However, geomorphic processes tend to vary between spatial scales, which leads to one of the key problems in geomorphology, i.

Consideration of how fundamental units make up landscapes has a long history in geomorphology. Vertically, the upper boundary of a geomorphic cell is defined by the atmosphere, while the lower boundary is generally formed by the bedrock layer of the lithosphere. Laterally, geomorphic cells are delimited from neighbouring cells with a change in environmental characteristics that determine hydro-geomorphic boundary conditions e.

Geomorphic feedbacks between structural and functional connectivity.


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Schematic diagram of feedbacks between structural and functional connectivity source: Wainwright et al. Landscapes can be perceived as systems exhibiting a distinct type of memory, i. Thus, a critical issue when separating SC and FC is determining the timescale at which a change in SC becomes dynamic i. Past geologic, anthropogenic and climatic controls upon sediment availability, for example, influence contemporary process-form relationships in many environments Brierley, such as embayments e.

Hine et al. Poeppl et al. In most geomorphic systems the imprint of memory and the timescales over which feedbacks affect connectivity are too strong for a separation of SC and FC. However, this philosophical position has not yet made its way into approaches to measuring connectivity. A challenge when developing quantitative descriptions of the structural-functional evolution of connectivity in geomorphic systems is thus how to incorporate memory effects.

Furthermore, when distinguishing between SC and FC, the challenge is to achieve the balance between scientific gains and losses, further depending on the spatio-temporal scale of interest and the applied methodology. The conceptualization of landforms as the outcome of the interactions of structure, function and memory implies that landscapes are organised in a hierarchical manner as they are seen as complex macroscopic features that emerge from myriad microscopic factors processes which form them at different spatio-temporal scales Harrison, For example, river meander development e.

Church, or dune formation e. Baas, can be seen as emergent properties of geomorphic systems that are governed by manifold microscale processes e.

Laser chimeras as a paradigm for multistable patterns in complex systems

In Geomorphology, emergence thus becomes the basis on which qualitative structures landforms arise from the self- organisation of quantitative phenomena processes Harrison, operating at a range of different spatial and temporal scales. In order to get a grasp on emergent behaviour of geomorphic systems recent advances in Geomorphology are based on chaos theory and quantitative tools of complex systems research e.

Coco and Murray, ; combined approaches: e. Combining numerical models with new data-collection strategies and other techniques as also discussed in 3. Murray et al. However, to date this hypothesis remains untested and is being subject to further inquiry. Yet, the potential appears to exist that connectivity may help to understand how geospatial processes produce a range of fluxes that come together to produce landscape form.

In Geomorphology, it is only possible to measure i the morphology of the landscape itself from which SC is quantified or ii fluxes of material that are a result of FC and event magnitude. Few standard methods exist to quantify FC directly Bracken et al. One of the key challenges to measure connectivity is to define the spatial and temporal scales over which connectivity should be assessed, which may depend on how the fundamental unit is defined. Furthermore, data comparability is often constrained by the measurement design including the types of technical equipment involved.

Changes in SC can be quantified at high spatial and temporal resolutions using several novel methods that have been developed or improved over the past years. Structure-from-Motion SfM photogrammetry and laser scanning are techniques that create high-resolution, three-dimensional digital representations of the landscape.

Sediment transport processes FC are traditionally measured using erosion plots for small-scale measurements to water sampling for suspended sediment and bedload traps in streams and rivers for large-scale measurements e. Recently, new techniques have been developed to trace and track sediment with higher spatial and temporal resolution. Sediment tracers, which can either occur naturally in the soil or be applied to the soil, have been increasingly used to quantify erosion and deposition of sediments.

Furthermore, laboratory experiments allow sediment tracking in high detail by using a combination of multiple high-speed cameras, trajectories and velocities of individual sand particles under varying conditions Long et al. However, it is highly questionable if measuring water and sediment fluxes provides sufficient information to infer adequately FC, since these data solely represent snapshots of fluxes instead of reflecting system dynamics incl.

Besides measuring landscape structure and sediment fluxes to infer connectivity, different types of indices and models are used.


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Connectivity indices mainly use a combination of topography and vegetation characteristics to determine connectivity Borselli et al. These indices are static representations of SC, which are useful for determining areas of high and low SC within the study areas. Because indices are static, they do not provide information about fluxes.

Suchfunktion

Different types of models e. Landscapes are composed of interconnected ecosystems that mediate ecological processes and functions — such as material fluxes and food web dynamics, and control species composition, diversity and evolution. The importance of connectivity within ecology has been recognised for decades e. Connectivity is now recognised to be an important determinant of many ecological processes Kadoya, including population movement Hanski, , changes in species diversity Cadotte, , metacommunity dynamics Koelle and Vandermeer, and nutrient and organic matter cycling Laudon et al.

For example, in marine ecology, identifying and quantifying the scale of connectivity of larval dispersal among local populations i. Regardless of the scale at which connectivity is defined within Ecology, there is nonetheless consensus that connectivity affects most population, community, and ecosystem processes Wiens ; Moilanen and Hanski Hierarchy theory provides a clear tool for dealing with spatial scale, and suggests that all scales are equally deserving of study Cadenasso et al. It is therefore critical that the fundamental unit be defined clearly as well as relationships that cross scales Ascher, The fundamental unit is typically defined as being the ecosystem — a complex of living organisms, their physical environment, and their interrelationships in a particular unit of space Weathers et al.

In this respect, an ecosystem can be a single gravel bar, a whole river section, or the entire catchment, or an ecosystem can be a plant, a vegetation patch, or a mosaic of patches, depending on the spatiotemporal context and the specific questions.