Identifying the impact of local connectivity patterns on dynamics in excitatory-inhibitory networks

TL;DR

The paper investigates how local connectivity motifs, especially chain motifs, shape dynamics in excitatory-inhibitory networks. By decomposing the connectivity as a low-rank mean $\mathbf{J}^0$ plus a random part $\mathbf{Z}$ and analyzing eigenvalue outliers via $[\mathbf{Z}^2]$, the authors show chain motifs induce a positive outlier that can destabilize inhibition-dominated regimes and generate paradoxical responses to external inputs. They develop a dual analytical framework: a low-rank approximation based on dominant eigenmodes and an effective deterministic connectivity $\mathbf{J}^{eff}=\mathbf{J}^0+[\mathbf{Z}^2]$ that accurately predicts population-averaged responses to uniform inputs, including paradoxical effects. These results hold for both fully connected and sparse EI networks, with practical implications for interpreting optogenetic perturbation experiments and for understanding how motif structure shapes cortical dynamics.

Abstract

Networks of excitatory and inhibitory (EI) neurons form a canonical circuit in the brain. Seminal theoretical results on dynamics of such networks are based on the assumption that synaptic strengths depend on the type of neurons they connect, but are otherwise statistically independent. Recent synaptic physiology datasets however highlight the prominence of specific connectivity patterns that go well beyond what is expected from independent connections. While decades of influential research have demonstrated the strong role of the basic EI cell type structure, to which extent additional connectivity features influence dynamics remains to be fully determined. Here we examine the effects of pairwise connectivity motifs on the linear dynamics in EI networks using an analytical framework that approximates the connectivity in terms of low-rank structures. This low-rank approximation is based on a mathematical derivation of the dominant eigenvalues of the connectivity matrix and predicts the impact on responses to external inputs of connectivity motifs and their interactions with cell-type structure. Our results reveal that a particular pattern of connectivity, chain motifs, have a much stronger impact on dominant eigenmodes than other pairwise motifs. An overrepresentation of chain motifs induces a strong positive eigenvalue in inhibition-dominated networks and generates a potential instability that requires revisiting the classical excitation-inhibition balance criteria. Examining effects of external inputs, we show that chain motifs can on their own induce paradoxical responses where an increased input to inhibitory neurons leads to a decrease in their activity due to the recurrent feedback. These findings have direct implications for the interpretation of experiments in which responses to optogenetic perturbations are measured and used to infer the dynamical regime of cortical circuits.

Figures (17)

• Figure 1: Schematic of the multi-population network model. (a) The network consists of $P$ populations, each represented by a different color. This population structure defines the statistics of synaptic connectivity. (b) The corresponding connectivity matrix consists of $P^2$ blocks. Synaptic weights within each block share identical statistical properties. (c) Pairs of synapses that share a neuron can form four different types of second-order motifs. The prevalence of each motif with respect to chance is quantified by a corresponding pair-wise correlation coefficient. (d) A low-rank approximation of the connectivity matrix can integrate both the population structure and the pair-wise motif statistics.
• Figure 2: Impact of chain motifs on the eigenvalues of the connectivity matrix for fully connected excitatory-inhibitory networks. (a) Eigenspectrum in the complex plane. The spectrum consists of a circular bulk (magnification in inset), within which the eigenvalues are continuously distributed, and isolated outliers. Inhibition dominated networks with independent synapses give rise to a single negative outlier $\lambda_0$ (black circle). Progressively increasing the strength $\tau^{c}$ of chain motifs (from light to dark), this negative outlier (yellow to red) decreases from the original $\lambda_0$, while an additional positive outlier emerges and increases (light to dark green). The dots show numerically-determined outlying eigenvalues averaged over $30$ networks of $N=1000$ neurons. The inset shows the eigenvalue bulk for a single network realization with $\tau^c=0$ (purple) and $\tau^c=0.1$ (blue). (b) Dependence of outlying eigenvalues on the strength $\tau^{c}$ of chain motifs and the network size $N$. Numerically obtained eigenvalue outliers from $30$ network realizations (markers) are shown alongside the theoretical predictions (lines) calculated using Eq.\ref{['eq:twoOutliersMaintext']}. The dominant negative outlier is depicted in red, while the emergent positive outlier is denoted in green. The gray area indicates the radius of the eigenvalue bulk in networks with a size of $N=1000$, with dashed lines indicating the theoretical values dahmen2020strong. As a control, we fix $\lambda_0$ across networks with different $N$, by scaling the mean synaptic weight $J^0_{pq}$ as $1/N$. All parameter values are given in TABLE \ref{['tab:parameters']}.
• Figure 3: Impact of chain motifs on the eigenvalues of the connectivity matrix for sparsely connected excitatory-inhibitory networks. (a) Eigenspectrum in the complex plane. Inhibition dominated networks with independent synapses give rise to a single outlier $\lambda_0$ (black circle). Progressively increasing the strength $\tau^{c}$ of chain motifs (from light to dark), this negative outlier (yellow to red) decreases from the original $\lambda_0$, while an additional positive outlier emerges and increases (light to dark green). The dots show numerically-determined outlying eigenvalues averaged over $30$ networks of $N=1500$ neurons. The eigenvalue bulk for a single network realization is shown for $\tau^c=0$ (purple) and $\tau^c=0.225$ (blue). (b) Dependence of outlying eigenvalues on the strength $\tau^{c}$ of chain motifs. Numerically obtained eigenvalue outliers from $30$ network realizations (markers) are shown alongside the theoretical predictions (lines). The dominant negative outlier is depicted in red, while the emergent positive outlier is shown in green. The colored solid lines show the theoretical results calculated using Eqs. \ref{['eq:twoOutliersMaintext_sparse']}, \ref{['eq:chainperturb']}. The red and green asterisks with error bars represent the results numerically obtained from actual sparse networks (same as subplot (a)), and the gray triangles with error bars represent the results numerically obtained from the equivalent Gaussian networks. The gray area indicates the radius of the eigenvalue bulk in networks with a size of $N=1500$, with dashed lines indicating the theoretical values (dahmen2020strong for details). All parameter values are given in TABLE \ref{['tab:parameters']}.
• Figure 4: Dependence of outlying eigenvalues on the network size $N$ ranging from 1000 to 5000 for sparse excitatory-inhibitory connectivity in two different scaling limits. (a) Networks in the strongly connected regime with a constant connectivity probability of $c=0.2$ and a chain motif probability of $\tau^c=0.15$ ($\rho^c=0.064$). (b) Networks in the weakly connected regime with a fixed number of connections and chain motifs, where $C_E=240$ and $k^c_{EE}=92160$. The asterisks with error bars indicate the mean and standard deviation of numerically obtained eigenvalue outliers from 30 network instances. Other parameters: $g=6.8,~\gamma=1/4,~J=0.0129$.
• Figure 5: Impact of chain motifs on population-averaged mean of entries on connectivity vectors for Gaussian networks. (a, b) Population-averaged mean values of entries $m_i^{(r)p}$ on the right connectivity vectors. Subplots (a) and (b) respectively show the mean values for $p=E$ and $p=I$. (c, d) Population-averaged mean values of entries $n_i^{(r)p}$ on the left connectivity vectors. Subplots (c) and (d) respectively show the mean values for $p=E$ and $p=I$. The insets show the distribution of the mean values of the elements in the connectivity vectors corresponding to the eigenvalues in the bulk (red: excitatory population; blue: inhibitory population). The additional eigenvalue emerges from the bulk when $\tau^c=0.011$. Other network parameters see TABLE \ref{['tab:parameters']}.