A Minimal Network of Brain Dynamics: Hierarchy of Approximations to Quasi-critical Neural Network Dynamics

TL;DR

The paper introduces the Generalized Cortical Branching Model (GCBM), a minimal neural-dynamics framework that includes excitatory and inhibitory populations to capture quasi-critical brain activity. It develops a hierarchy of mean-field (MF) approximations based on network motifs, enabling tractable dynamical maps for activity densities and phase behavior. The results show that inhibition generally stabilizes dynamics, shifts phase boundaries, reduces peak susceptibility, and broadens the quasi-critical region, while retaining directed-percolation universality at criticality; chaotic regimes emerge in the unstable phase, with links to epileptic dynamics. These findings support the quasi-criticality hypothesis and provide a structured methodology for linking microscopic motif structure to macroscopic stability and susceptibility, with potential applications in biomarkers and seizure prediction.

Abstract

We present an interacting branching model of neural network dynamics, incorporating key biological features such as inhibition with several types of inhibitory interactions. We establish a hierarchy of analytical mean-field approximations to the model, which characterizes nonequilibrium phase transitions between disorder and ordered phases, and perform a stability analysis. Generically, inhibitory neurons increase the stability of the model dynamics. The model is consistent with the quasi-criticality hypothesis in that it displays regions of maximal dynamical susceptibility and maximal mutual information predicated on the strength of the external stimuli. Directed percolation emerges as the universality class of the critical transition of the model, consistent with some previous experimental data and models. In the unstable phase, chaotic dynamics emerge, which may be linked to the occurrence of epileptic seizures.

Figures (20)

• Figure 1: Eight types of interactions defined in the GCBM. Panels illustrate: (a--b) Interactions between a neuron and an external source. (c--f) Interactions among neurons. (g) Inhibition of the excitatory-to-excitatory $\mathrm{E}\shortrightarrow\mathrm{E}$ channel. (h) Inhibition on the spontaneous activation of an excitatory neuron. Arcs are represented by arrows from the source to the target neuron. Hyperarcs are depicted as arrows with a flat line from an inhibitory neuron to another arc. Red-colored nodes and arcs denote inhibition; green nodes and arcs denote excitation. The blue box and arrow represent external spontaneous activation.
• Figure 2: Network topology of the many-body model suitable for a MF approximation. The in-degree $k_{\sf n n'}$ is uniform across all nodes. For E-E interaction, $k_{\mathsf{ee}}=2$ and bias parameter $B>0$ (incoming arcs have different weights indicated by thickness). One transmission channel is more likely to be activated than another. While $k_{\mathsf{ei}}=k_{\mathsf{ie}}=k_{\mathsf{ii}}=1$, each type has different branching parameter $\kappa_{\sf nn'}$.
• Figure 3: Neighborhood of excitatory neuron $\mathsf{e}_{4}$. Green nodes represent excitatory neurons and red nodes represent inhibitory neurons. Directed arrows indicate synaptic interactions: green for excitation and red for inhibition. The layout includes all neurons and interactions directly linked to $\mathsf{e}_{4}$, capturing its environment. The clocks in each of the nodes represent its state $z_{\sf n}$. The arcs are labeled by the probability of activation.
• Figure 4: An example network and its subgraphs in the GCBM. (a): A translationally invariant network where every neuron has the same local presynaptic environment. Each excitatory neuron receives one incoming excitatory arc, one inhibitory arc, and one external spontaneous activation arc. Each inhibitory neuron receives one excitatory arc and one external spontaneous activation arc. (b): The subgraphs used in the MF approximation. We have $\widetilde{N}_{\mathsf{ee}_{0}}=\widetilde{N}_{\mathsf{ei}_{0}}=\widetilde{N}_{\mathsf{ie}_{0}}=1$, and $\widetilde{N}_{\mathsf{ii}_{0}}=0$. Here, $\mathsf{e}_{0}$ denotes the representative excitatory neuron and $\mathsf{i}_{0}$ the representative inhibitory neuron, while $\mathsf{\bar{e}}$ and $\mathsf{\bar{i}}$ represent the mean excitatory and inhibitory neighbors, respectively. Blue arrows denote external spontaneous activation.
• Figure 5: Example motif of the network in Fig.\ref{['fig:net_to_centralSite']}. This is an example of how we form the motif from the network subgraphs. This example includes the inhibition of a directed edge of the external spontaneous activation. The motif has all the mean neighborhood connections on the left and the representative neurons on the right, showing the progression of time from left to right. A MF diagram is also shown on the right to emphasize that there are only two representative neurons in the MF approximation. We will primarily use the unfold diagram on the left to clearly show the neighborhood.We have $\widetilde{N}_{\mathsf{ee}_{0}}=\widetilde{N}_{\mathsf{ei}_{0}}=\widetilde{N}_{\mathsf{ie}_{0}}=1$, and $\widetilde{N}_{\mathsf{ii}_{0}}=0$. Here, $\mathsf{e}_{0}$ denotes the representative excitatory neuron and $\mathsf{i}_{0}$ the representative inhibitory neuron, while $\mathsf{\bar{e}}$ and $\mathsf{\bar{i}}$ represent the mean excitatory and inhibitory neighbors, respectively. Blue arrows denote external spontaneous activation.