In recurrent networks of leaky integrate-and-fire neurons, the mean-field theory has been instrumental in capturing the statistical properties of neuronal activity, like firing rate distributions. This theory has been applied to networks with either homogeneous synaptic weights and heterogeneous connections per neuron or vice versa. Our work expands mean-field models to include networks with both types of structural heterogeneity simultaneously, particularly focusing on those with synapses that undergo plastic changes. The model introduces a spike trace for each neuron, a variable that rises with neuron spikes and decays without activity, influenced by a degradation rate r p and the neuron’s firing rate ν . When the ratio α = ν / r p is significantly high, this trace effectively estimates the neuron’s firing rate, allowing synaptic weights at equilibrium to be determined by the firing rates of connected neurons. This relationship is incorporated into our mean-field formalism, providing exact solutions for firing rate and synaptic weight distributions at equilibrium in the high α regime. However, the model remains accurate within a practical range of degradation rates, as demonstrated through simulations with networks of excitatory and inhibitory neurons. This approach sheds light on how plasticity modulates both activity and structure within neuronal networks, offering insights into their complex behavior. Author Summary Networks of spiking neurons are complex systems where the structure of connections and the activity patterns generated are deeply intertwined, a relationship often studied using mathematical approaches like the mean-field theory. However, previous studies have primarily focused on networks with limited structural variability, where either the connection strength is nearly identical across the network or the number of connections varies little from one neuron to another. This work takes a step forward by combining both types of structural variability and allowing connection strengths to adapt over time, thereby providing an extended mean-field theory. We derive exact solutions for the distribution of spiking rates and connection strengths at equilibrium and demonstrate their accuracy through numerical simulations, even beyond the defining parameter ranges, offering a more comprehensive and realistic perspective on the interplay between activity and structure in neuronal networks.

Elucidating the coupling between the structure and the function of the brain and its development across maturation has attracted a lot of interest in the field of network neuroscience in the last 15 years. Mounting evidence supports the hypothesis that the onset of certain brain disorders is linked with the interplay between the structural architecture of the brain and its functional processes, often accompanied with unusual connectivity features. This paper introduces a method called the network-based statistic–simultaneous node investigation (NBS-SNI) that integrates both representations into a single framework, and identifies connectivity abnormalities in case-control studies. With this method, significance is given to the properties of the nodes, as well as to their connections. This approach builds on the well-established network-based statistic (NBS) proposed in 2010. We uncover and identify the regimes in which NBS-SNI offers a gain in statistical resolution to identify a contrast of interest using synthetic data. We also apply our method on two real case-control studies, one consisting of individuals diagnosed with autism and the other consisting of individuals diagnosed with early psychosis. Using NBS-SNI and node properties such as the closeness centrality and local information dimension, we found hypo- and hyperconnected subnetworks and show that our method can offer a 9 percentage points gain in prediction power over the standard NBS. Author Summary We propose an extension to the well-known network-based statistic (NBS) dubbed NBS-SNI, where the extension SNI stands for simultaneous node investigation. The goal of this approach is to integrate nodal properties such as centrality measures into the statistical network-based framework of NBS to probe for abnormal connectivity between important nodes in case-control studies. We expose the regimes where NBS-SNI offers greater statistical resolution for identifying a contrast of interest using synthetic data and test the approach with a real autism-healthy dataset that contains both the structural ( DTI ) and functional (fMRI) brain networks of each individual. We also tested our approach on a second dataset of individuals diagnosed with early psychosis. In the second case, our framework is supplemented by incorporating the anatomically derived measures of intrinsic curvature index and gray matter volume directly as a node property, rather than the structural networks, thereby illustrating the versatility of our approach.