Structural constraints on the emergence of oscillations in multi-population neural networks

Reviewer #1 (Public Review):

Summary:

The authors study the appearance of oscillations in motifs of linear threshold systems, coupled in specific topologies. They derive analytical conditions for the appearance of oscillations, in the context of excitatory and inhibitory links. They also emphasize the higher importance of the topology, compared to the strength of the links. Finally, the results are confirmed with WC oscillators, which are also linear. The findings are to some extent confirmed with spiking neurons, though here results are less clear, and they are not even mentioned in the Discussion.

Overall, the results are sound from a theoretical perspective, but I still find it hard to believe that they are of significant relevance for biological networks, or in particular for the oscillations of BG-thalamus-cortex loop in PD. I find motifs in general to be too simplistic for multiscale and generally large networks as is the case in the brain. Moreover, the division of regions is more or less arbitrary by definition, and having such a strong dependence on an odd/even number of inhibitory links is far from reality. Another limitation is the fact that the cortex is considered a single node. Similarly, decomposing even such a coarse network in all possible (238 in this case) motifs doesn't seem of much relevance, when I assume that the emergence of pathological rhythms is more of an emergent phenomenon.

Strengths:

From the point of view of nonlinear dynamics, the results are solid, and the intuition behind the proofs of the theorems is well explained.

Weaknesses:

As stated in the summary, I find the work to be too theoretical without a real application in biological systems or the brain, where the networks are generally very large. It is not the problem in the simplicity of the model or of the topology, it is often the case that the phenomena are explained by very reduced systems, but the problem is that the applicability of the finding cannot be extended. E.g. the Kuramoto model uses all-to-all coupling, or similar with QIF neurons which also need to follow a Lorentzian distribution in order to derive a mean field. But in those cases, relaxing the strict conditions that were necessary for the derivations, still conserves the main findings of the analysis, which I don't see being the case here. The odd/even number rule is too strict, and talking about a fixed and definite number of cycles in the actual brain seems too simplistic.

Being linear is another strong assumption, and it is not clear how much of the results are preserved for spiking neurons, even though there is such an analysis, or maybe for other nonlinear types of neuronal masses.

Delays are also mentioned, and their impact on the oscillatory networks is as expected: it reduces the amplitude, but there is no link to the literature, where this is an established phenomenon during synchronization. Finally, the authors should also discuss the time-delays as a known phenomenon to cause or amplify oscillations at different frequencies in a network of coupled oscillators, e.g Petkoski & Jirsa Network Neuroscience 2022, Tewarie et al. NeuroImage 2019, Davis et al. Nat Commun 2021.

Reviewer #2 (Public Review):

Summary:

The authors present here a mathematical and computational study of the topological/graph theory requirements to obtain sustained oscillations in neural network models. A first approach mathematically demonstrates that, a given network of interconnected neural populations (understood in the sense of dynamical systems) requires an odd number of inhibitory populations to sustain oscillations. The authors extend this result via numerical simulations of (i) a simplified set of Wilson-Cowan networks, (ii) a simplified circuit of the cortico-basal ganglia network, and (iii) a more complex, spike-based neural network of basal ganglia network, which provides insight on experimental findings regarding abnormal synchrony levels in Parkinson's Disease (PD).

Strengths:

The work elegantly and effectively combines solid mathematical proof with careful numerical simulations at different levels of description, which is uncommon and provides additional layers of confidence to the study. Furthermore, the authors included detailed sections to provide intuition about the mathematical proof, which will be helpful for readers less inclined to the perusal of mathematical derivations. Its insightful and well-informed connection with a practical neuroscience problem, the presence of strong beta rhythms in PD, elevates the potential influence of the study and provides testable predictions.

Weaknesses:

In its current form, the study lacks a more careful consideration of the role of delays in the emergence of oscillations. Although they are addressed at certain points during the second part of the study, there are sections in which this could have been done more carefully, perhaps with additional simulations to solidify the authors' claims. Furthermore, there are several results reported in the main figures which are not explained in the main text. From what I can infer, these are interesting and relevant results and should be covered. Finally, the text would significantly benefit from a revision of the grammar, to improve the general readability at certain sections. I consider that all these issues are solvable and this would make the study more complete.