Inter-modular Connections Research Articles

Modular and Hierarchically Modular Organization of Brain Networks

Brain networks are increasingly understood as one of a large class of information processing systems that share important organizational principles in common, including the property of a modular community structure. A module is topologically defined as a subset of highly inter-connected nodes which are relatively sparsely connected to nodes in other modules. In brain networks, topological modules are often made up of anatomically neighboring and/or functionally related cortical regions, and inter-modular connections tend to be relatively long distance. Moreover, brain networks and many other complex systems demonstrate the property of hierarchical modularity, or modularity on several topological scales: within each module there will be a set of sub-modules, and within each sub-module a set of sub-sub-modules, etc. There are several general advantages to modular and hierarchically modular network organization, including greater robustness, adaptivity, and evolvability of network function. In this context, we review some of the mathematical concepts available for quantitative analysis of (hierarchical) modularity in brain networks and we summarize some of the recent work investigating modularity of structural and functional brain networks derived from analysis of human neuroimaging data.

Age-related changes in modular organization of human brain functional networks

Graph theory allows us to quantify any complex system, e.g., in social sciences, biology or technology, that can be abstractly described as a set of nodes and links. Here we derived human brain functional networks from fMRI measurements of endogenous, low frequency, correlated oscillations in 90 cortical and subcortical regions for two groups of healthy (young and older) participants. We investigated the modular structure of these networks and tested the hypothesis that normal brain aging might be associated with changes in modularity of sparse networks. Newman's modularity metric was maximised and topological roles were assigned to brain regions depending on their specific contributions to intra- and inter-modular connectivity. Both young and older brain networks demonstrated significantly non-random modularity. The young brain network was decomposed into 3 major modules: central and posterior modules, which comprised mainly nodes with few inter-modular connections, and a dorsal fronto-cingulo-parietal module, which comprised mainly nodes with extensive inter-modular connections. The mean network in the older group also included posterior, superior central and dorsal fronto-striato-thalamic modules but the number of intermodular connections to frontal modular regions was significantly reduced, whereas the number of connector nodes in posterior and central modules was increased.

Modular synchronization in complex networks

We study the synchronization transition (ST) of a modified Kuramoto model on two different types of modular complex networks. It is found that the ST depends on the type of intermodular connections. For the network with decentralized (centralized) intermodular connections, the ST occurs at finite coupling constant (behaves abnormally). Such distinct features are found in the yeast protein interaction network and the Internet, respectively. Moreover, by applying the finite-size scaling analysis to an artificial network with decentralized intermodular connections, we obtain the exponent associated with the order parameter of the ST to be beta approximately 1 different from beta(MF) approximately 1/2 obtained from the scale-free network with the same degree distribution but the absence of modular structure, corresponding to the mean field value.

Modularity and the units of evolution

While many developmental processes (e. g., gene networks or signaling pathways) are astonishingly conserved during evolution, they may be employed differently in different metazoan taxa or may be used multiply in different contexts of development. This suggests that these processes belong to building blocks or modules, viz., highly integrated parts of the organism, which develop and/or function relatively independent from other parts. Such modules may be relatively easy to dissociate from other modules and, therefore, could also serve as units of evolution. However, in order to further explore the implications of modularity for evolution, the vague notion of “modularity” as well as its relation to concepts like “unit of evolution” need to be more precisely specified. Here, a module is characterized as a certain type of dynamic pattern of couplings among the constituents of a process. It may or may not form a spatially contiguous unit. A unit of selection is defined as a unit of those constituents of a reproducing process/system, which exists in different variants and acts as a non-decomposable unit of fitness and variant reproduction during a particular selection process. The more general notion of a unit of evolution is characterized as a nondecomposable unit of constituents with reciprocal fitness dependence, be it due to fitness epistasis or due to the lack of independent variability. Because such fitness dependence may only be observed for some combinations of variants, several constituents may act as a unit of evolution only with a certain probability (coevolution probability). It is argued, that under certain conditions modules are likely to act as units of evolution with high coevolution probabilities, because there is likely to be a close tie between the pattern of couplings of the constituents of a reproducing system and their interdependent fitness contributions. Moreover and contrary to the traditional dichotomy of genes versus organisms as units of selection, modules tend to be more important in delimiting actual units of selection than either organisms or genes, because they are less easily disrupted by recombination than organisms, while having less contextsensitive fitness values than genes. Finally, it is suggested that the evolution of modularity is self-reinforcing, because the flexibility of intermodular connections facilitates the recombination among modules and their multiple employment in new contexts.

Structure and flexibility of the multiple domain proteins that regulate complement activation.

In this review we summarise more than 10 years of biophysical exploration into the structural biology of the regulators of complement activation (RCA). The five human proteins responsible for regulation of the early events of complement are homologous and are composed largely from building blocks called "complement control protein (CCP) modules". Unlike most multiple domain proteins they do not contain any of the other widely occurring module types. This apparent simplicity of RCA structure, however, is belied by their sophistication of function. In fact, the structures of the individual CCP modules exhibit wide variations on a common theme while the extent and nature of intermodular connections is diverse. Some neighbouring modules within a protein stabilise each other and some co-operate to form specific binding surfaces. The degree of true "modularity" of CCPs is open to debate. The study of RCA proteins clearly illustrates the value of combining complementary structural biology techniques. The results could have implications for folding, evolution, flexibility and structure-function relationships of other molecules in the large, diverse and little understood category of multiple domain proteins.

Random fields and probability distributions with given marginals on randomly correlated systems: a general method and a problem from theoretical neuroscience

A class of families of marginal probabilities on sets of discrete random variables is studied and a necessary and sufficient condition for the consistency of the given marginals is provided. This result allows one to verify the consistency of the marginals through a Boltzmann statistical analysis. The procedure is then applied in order to verify the hypotheses assumed in a recent model of neocortical associative areas, according to which connected modules of neurons are simultaneously active with probability higher than chance, and inter-modular connections are very diluted. The verification becomes a typical problem of extremely diluted spin systems in Boltzmann-Gibbs ensemble. The results presented here justify the assumptions made in the neuroscientific theory, and an upper bound to the inter-modular activity correlation is found.

Associative memory in a multi-modular network.

Recent imaging studies suggest that object knowledge is stored in the brain as a distributed network of many cortical areas. Motivated by these observations, we study a multimodular associative memory network, whose functional goal is to store patterns with different coding levels--patterns that vary in the number of modules in which they are encoded. We show that in order to accomplish this task, synaptic inputs should be segregated into intramodular projections and intermodular projections, with the latter undergoing additional nonlinear dendritic processing. This segregation makes sense anatomically if the intermodular projections represent distal synaptic connections on apical dendrites. It is then straightforward to show that memories encoded in more modules are more resilient to focal afferent damage. Further hierarchical segregation of intermodular connections on the dendritic tree improves this resilience, allowing memory retrieval from input to just one of the modules in which it is encoded.

The importance of nonlinear dendritic processing in multimodular memory networks

Recent imaging studies testify that objects are stored in the brain in a distributed network of discrete cortical areas. Motivated by these observations, we study a multi-modular associative memory network, whose functional goal is to store patterns of different coding levels, i.e., patterns that vary in the number of modules in which they are encoded. We show that in order to accomplish this task, the synaptic inputs to each neuron should be segregated to intramodular projections whose activations are summed up linearly, and inter-modular projections whose activations are summed up in a nonlinear fashion. This may reflect dendritic processing of distal synaptic connections. Moreover, we find that further hierarchical segregation of inter-modular connections on the dendritic tree improves memory retrieval from partial input cues, and improves resilience of the network to afferent damage.

Associative memory properties of multiple cortical modules

The existence of recurrent collateral connections between pyramidal cells within a cortical area and, in addition, reciprocal connections between connected cortical areas, is well established. In this work we analyse the properties of a tri-modular architecture of this type in which two input modules have convergent connections to a third module (which in the brain might be the next module in cortical processing or a bi-modal area receiving connections from two different processing pathways). Memory retrieval is analysed in this system which has Hebb-like synaptic modifiability in the connections and attractor states. Local activity features are stored in the intra-modular connections while the associations between corresponding features in different modules present during training are stored in the inter-modular connections. The response of the network when tested with corresponding and contradictory stimuli to the two input pathways is studied in detail. The model is solved quantitatively using techniques of statistical physics. In one type of test, a sequence of stimuli is applied, with a delay between them. It is found that if the coupling between the modules is low a regime exists in which they retain the capability to retrieve any of their stored features independently of the features being retrieved by the other modules. Although independent in this sense, the modules still influence each other in this regime through persistent modulatory currents which are strong enough to initiate recall in the whole network when only a single module is stimulated, and to raise the mean firing rates of the neurons in the attractors if the features in the different modules are corresponding. Some of these mechanisms might be useful for the description of many phenomena observed in single neuron activity recorded during short term memory tasks such as delayed match-to-sample. It is also shown that with contradictory stimulation of the two input modules the model accounts for many of the phenomena observed in the McGurk effect, in which contradictory auditory and visual inputs can lead to misperception.