Dimension reduction of dynamics on modular and heterogeneous directed networks.

Dimension reduction is a common strategy to study nonlinear dynamical systems composed by a large number of variables. The goal is to find a smaller version of the system whose time evolution is easier to predict while preserving some of the key dynamical features of the original system. Finding such a reduced representation for complex systems is, however, a difficult task. We address this problem for dynamics on weighted directed networks, with special emphasis on modular and heterogeneous networks. We propose a two-step dimension-reduction method that takes into account the properties of the adjacency matrix. First, units are partitioned into groups of similar connectivity profiles. Each group is associated to an observable that is a weighted average of the nodes' activities within the group. Second, we derive a set of equations that must be fulfilled for these observables to properly represent the original system's behavior, together with a method for approximately solving them. The result is a reduced adjacency matrix and an approximate system of ODEs for the observables' evolution. We show that the reduced system can be used to predict some characteristic features of the complete dynamics for different types of connectivity structures, both synthetic and derived from real data, including neuronal, ecological, and social networks. Our formalism opens a way to a systematic comparison of the effect of various structural properties on the overall network dynamics. It can thus help to identify the main structural driving forces guiding the evolution of dynamical processes on networks.

Firing rate distributions in spiking networks with heterogeneous connectivity.

Mean-field theory for networks of spiking neurons based on the so-called diffusion approximation has been used to calculate certain measures of neuronal activity which can be compared with experimental data. This includes the distribution of firing rates across the network. However, the theory in its current form applies only to networks in which there is relatively little heterogeneity in the number of incoming and outgoing connections per neuron. Here we extend this theory to include networks with arbitrary degree distributions. Furthermore, the theory takes into account correlations in the in-degree and out-degree of neurons, which would arise, e.g., in the case of networks with hublike neurons. Finally, we show that networks with broad and positively correlated degrees can generate a large-amplitude sustained response to transient stimuli which does not occur in more homogeneous networks.

On the Structure of Cortical Microcircuits Inferred from Small Sample Sizes.

The structure in cortical microcircuits deviates from what would be expected in a purely random network, which has been seen as evidence of clustering. To address this issue, we sought to reproduce the nonrandom features of cortical circuits by considering several distinct classes of network topology, including clustered networks, networks with distance-dependent connectivity, and those with broad degree distributions. To our surprise, we found that all of these qualitatively distinct topologies could account equally well for all reported nonrandom features despite being easily distinguishable from one another at the network level. This apparent paradox was a consequence of estimating network properties given only small sample sizes. In other words, networks that differ markedly in their global structure can look quite similar locally. This makes inferring network structure from small sample sizes, a necessity given the technical difficulty inherent in simultaneous intracellular recordings, problematic. We found that a network statistic called the sample degree correlation (SDC) overcomes this difficulty. The SDC depends only on parameters that can be estimated reliably given small sample sizes and is an accurate fingerprint of every topological family. We applied the SDC criterion to data from rat visual and somatosensory cortex and discovered that the connectivity was not consistent with any of these main topological classes. However, we were able to fit the experimental data with a more general network class, of which all previous topologies were special cases. The resulting network topology could be interpreted as a combination of physical spatial dependence and nonspatial, hierarchical clustering.SIGNIFICANCE STATEMENT The connectivity of cortical microcircuits exhibits features that are inconsistent with a simple random network. Here, we show that several classes of network models can account for this nonrandom structure despite qualitative differences in their global properties. This apparent paradox is a consequence of the small numbers of simultaneously recorded neurons in experiment: when inferred via small sample sizes, many networks may be indistinguishable despite being globally distinct. We develop a connectivity measure that successfully classifies networks even when estimated locally with a few neurons at a time. We show that data from rat cortex is consistent with a network in which the likelihood of a connection between neurons depends on spatial distance and on nonspatial, asymmetric clustering.

Low-dimensional dynamics of structured random networks.

Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach [J. Aljadeff, M. Stern, and T. Sharpee, Phys. Rev. Lett. 114, 088101 (2015)], we study the relationship between the network connectivity structure and its low-dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and postsynaptic neurons through a sufficiently smooth function g of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of g. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low-dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.

To achieve an earlier IFN-γ response is not sufficient to control Mycobacterium tuberculosis infection in mice.

The temporo-spatial relationship between the three organs (lung, spleen and lymph node) involved during the initial stages of Mycobacterium tuberculosis infection has been poorly studied. As such, we performed an experimental study to evaluate the bacillary load in each organ after aerosol or intravenous infection and developed a mathematical approach using the data obtained in order to extract conclusions. The results showed that higher bacillary doses result in an earlier IFN-γ response, that a certain bacillary load (BL) needs to be reached to trigger the IFN-γ response, and that control of the BL is not immediate after onset of the IFN-γ response, which might be a consequence of the spatial dimension. This study may have an important impact when it comes to designing new vaccine candidates as it suggests that triggering an earlier IFN-γ response might not guarantee good infection control, and therefore that additional properties should be considered for these candidates.