Notes on: Universal resilience patterns in complex networks, Gao et al. 2016

Resiliency Overview

• Resiliency is a system’s ability to maintain its overall structure and function when disturbances act upon its network.
• Some examples include environmental systems, technological systems, and biological systems that are subject to environmental changes, errors, or failures. More resilient systems will be stable despite such perturbations.
• Loss of resiliency occurs when some bifurcation converts a stable fixed point into an unstable fixed point. Bifurcations occur when a changing parameter results in one or more fixed points undergoing stability transitions.
• Below is the equation of a 1D dynamical system (a flow on the line) with an adjustable parameter β.

Improved Model of Resiliency

• Gao et al. developed a higher-dimensional mathematical model to allow for better prediction of network resilience than past models.
• The main equation of this model is given below. Aij represents a weighted adjacency matrix, N is the number of nodes in the network, and x = (x1,…,xN)T is a vector representing the activities at each node. The function F(xi) describes the activity of each given node xi independent of the influences of other nodes. The function G(xi,xj) is a rule describing the influence of node xj on node xi.

• Disturbances to networks described by this equation might include gaining nodes or edges, losing nodes or edges, changes in edge weights, or changes in the values of parameters that are part of the functions F and G.

Resiliency Model Applied to Ecological Networks

• Below, this resiliency model describes abundance of species. The function xi(t) is the abundance of a species i. The parameter Bi represents the incoming migration rate of each species. The second term considers logistic growth with carrying capacities Ki as well as the Allee effect. According to the Allee effect, small populations, where xi<Ci will decline (negative growth). Finally, the third term describes mutualistic interactions between species xi and xj (i.e. plant-pollinator relationships). At high abundances of xi and xj, this term tends to plateau. The weighted adjacency matrix Aij represents the strengths of these mutualistic interactions. Di, Ei, and Hj are further adjustable parameters.

• The authors tested the resiliency of experimentally mapped ecological networks using this model. First, a fraction fn of the plant nodes was removed, simulating plant extinctions. Then, a fraction fI of the pollinator nodes was removed, simulating pollinator extinctions. Finally, the weights of the matrix Aij were randomly rescaled such that the average weight decreased to a fraction fw of the original average weight.
• Initially, the ecological networks each only possessed a single stable fixed point, xH. These fixed points described high average abundances among the species. Upon applying perturbations, the networks maintained stability until thresholds were reached. Each threshold was dependent on its network and the type of perturbation applied (plant node removal, pollinator node removal, or changes to edge weights). After passing the thresholds, each network gained an additional stable fixed point, xL. These fixed points described low average abundances among the species (an undesirable state).
• For example, network1 lost resilience after 35% of its pollinators were removed, while Network5 lost resilience only after 80% of its pollinators were removed.

Resiliency Model Applied to Gene Regulatory Networks

• Below, the resiliency model describes the activity of genes in coli and S. cerevisiae using Michaelis-Menten kinetics. The function xi(t) is the activity of a gene i. The first term represents the degradation rate (f=1) of a protein (from gene i). Alternatively, if the xi in the first term is squared (f=2), this term can represent dimerization. The second term describes gene expression levels, where the Hill coefficient h represents the level of cooperativity in the regulation of gene i by protein j. The weighted adjacency matrix Aij represents regulatory connections among the genes.

• By computing the average activity of all genes in a cell, the resiliency model provided a method of predicting cell viability. If a cell’s , then the model predicts that the cell will die. If a cell’s , then the model predicts that the cell will be viable.
• Once again, depending on the gene regulatory network and the types of perturbations, resilience loss occurred after thresholds (each specific to the network and the perturbation types) were crossed. The types of perturbations for this application included gene deletions, changes to the weights of regulatory interactions, and global environmental changes.

My Digital Art Sampling

Self-Replicating Cursors

Transcendent Entomologies

Hodgkin-Huxley Neuron

Year 1,000,000

Jupiter Brain

Don’t Spook the Snuffler

Endodermal

Kaleidaelectrica

Notes on: A Topological Representation of Branching Neuronal Morphologies, Kanari et al. 2017

Challenge

• Neural morphologies involve complicated branching structures.
• Traditionally, classification of such morphologies has involved qualitative assessment of microscopic images.
• Topological data analysis (TDA) can classify geometric structures that are built from well-understood pieces such as spheres, cylinders, and tori. However, this sort of analysis computationally intensive when applied to neurons since their morphologies are so complex.
• Feature extraction has been applied to quantitatively classify neural morphologies. This is a technique that allows a large, but potentially redundant, dataset to be converted into a reduced set of features that can be more easily analyzed.
• Neurons are reconstructed as a set of points in connected by edges and then experts identify the relevant morphological characteristics to subject to the feature extraction algorithms. Unfortunately, the initial feature identification is quite subjective and can lead to different results. In addition, a great deal of information loss occurs in the processing of the data.
• Neither TDA nor feature extraction are ideal for studying neural morphologies.

Topological Morphology Descriptor (TMD)

• The TMD is an algorithm that maps the original images of dendritic trees to a topological representation with less information loss than other methods, though it does simplify the trees somewhat.
• The TMD takes a partially ordered set of branch points (“parent” nodes with multiple further connections or “children”) and leaves (parent nodes with no further connections) as input.
• Branch points and leaves are ordered by the “parent-child” relations.

The Persistence Barcode and Persistence Diagram

• This output of the TMD is a “persistence” barcode and an equivalent persistence diagram.
• In the persistence barcode below (left), the distance or “lifetime” along a component (in this case a component is an individual dendritic branch) is given on the horizontal axis. This distance is measured from the component’s “birth” or most distant point from the soma to its “death” or closest point to the soma.
• In the equivalent example of a persistence diagram below (right), each component’s death distance is plotted against corresponding birth distances.
• Note that many possible functions f can be used to generate a persistence barcode and diagram, but here, they are ordered by lifetime with units of microns.
• In the figure below, the highlighted red component is used to exemplify what an individual component looks like when analyzed in this way.

• The TMD method allows analysis of any treelike structure, unlike previous techniques which were not generalizable for the various types of neural morphology. TMD is much more biologically useful and computationally inexpensive.
• TMD allows a reliability measure to be assigned to proposed groupings of neuronal trees. This helps develop a diversity profile that reflects the morphological variety of neurons.

Details of the TMD Algorithm

• T denotes a rooted tree embedded in (a tree with a central node or root R).
• N is the set of nodes in a tree T. Furthermore, N is defined as the union of branch points B and leaves L. Individual nodes are referred to in lowercase as n
• Nodes with the same parent are called siblings.
• Sets of nodes are denoted as A.
• Subtrees are designated Tn and centered at node n. Each subtree possesses leaves Ln. Individual leaves are .
• The function f could be defined by the radial distance, the branch length, the path distance, or other characteristics. In this case, it is defined as the radial distance from the soma (or root R).
• The function v (below) is computed by the TMD algorithm. It finds a maximum value for f as a function of x where x belongs to the set of leaves.

• Recall that (here) f is the radial distance from the soma. As such, v(n) will find the most distant leaves.
• The algorithm starts with the leaves and follows each leaf’s path down towards the soma.
• Leaves are associated with their respective nodes in the set of nodes A. When a node (with one or more siblings) is encountered, all but the “oldest” sibling is removed from that node. The oldest sibling is the sibling that extends farthest from the soma. After passing through all the nodes on a particular path, the algorithm reaches the soma and designates that path as a component. The component is added to the persistence barcode.
• Below are some examples of what various paths along the dendritic tree might look like. Once again, this is achieved by a process of removing the younger siblings emerging from each node along a given path.

• Once all the paths have been traversed, a set of components is outputted (recall that a component is an individual branch, not the entire path). The terminus (birth) and parent node (death) of each component is known. The lifetime of each component is the distance between its birth and death. This information allows a persistence barcode and an equivalent persistence diagram to be plotted.
• To reiterate, the persistence barcode and diagram describe all the individual components in a given neural tree.
• The TMD algorithm has linear complexity with respect to the number of nodes.
• If a tree is rotated or translated, the TMD algorithm will still give the same result.

Comparing Persistence Diagrams

• There are standard topological distance measures between persistence diagrams, but this are not suited to dendritic trees. The authors defined an alternative distance measure called dbar in the space of persistence barcodes. However, there were still issues with this metric because it converted the barcode into a one-dimensional distribution and so lost some important information for classifying neural morphologies.
• The better method for comparing persistence barcodes is to convert each barcode into a matrix of pixel values (an image) by using a discretization algorithm (in this case a sum of Gaussian Kernels) and then to employ machine learning for classifying the images.