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1. Differential thalamocortical interactions in slow and fast spindle generation: A computational model

   https://doi.org/10.1371/journal.pone.0277772  

   Mushtaq, Muhammad; Marshall, Lisa; Bazhenov, Maxim; Mölle, Matthias; Martinetz, Thomas (December 2022, PLOS ONE)

   Cymbalyuk, Gennady S. (Ed.)

   Cortical slow oscillations (SOs) and thalamocortical sleep spindles are two prominent EEG rhythms of slow wave sleep. These EEG rhythms play an essential role in memory consolidation. In humans, sleep spindles are categorized into slow spindles (8–12 Hz) and fast spindles (12–16 Hz), with different properties. Slow spindles that couple with the up-to-down phase of the SO require more experimental and computational investigation to disclose their origin, functional relevance and most importantly their relation with SOs regarding memory consolidation. To examine slow spindles, we propose a biophysical thalamocortical model with two independent thalamic networks (one for slow and the other for fast spindles). Our modeling results show that fast spindles lead to faster cortical cell firing, and subsequently increase the amplitude of the cortical local field potential (LFP) during the SO down-to-up phase. Slow spindles also facilitate cortical cell firing, but the response is slower, thereby increasing the cortical LFP amplitude later, at the SO up-to-down phase of the SO cycle. Neither the SO rhythm nor the duration of the SO down state is affected by slow spindle activity. Furthermore, at a more hyperpolarized membrane potential level of fast thalamic subnetwork cells, the activity of fast spindles decreases, while the slow spindles activity increases. Together, our model results suggest that slow spindles may facilitate the initiation of the following SO cycle, without however affecting expression of the SO Up and Down states. 

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2. Sensitivity to Control Signals in Triphasic Rhythmic Neural Systems: A Comparative Mechanistic Analysis via Infinitesimal Local Timing Response Curves

   https://doi.org/10.1162/neco_a_01586  

   Yu, Zhuojun; Rubin, Jonathan E.; Thomas, Peter J. (May 2023, Neural Computation)

   Abstract Similar activity patterns may arise from model neural networks with distinct coupling properties and individual unit dynamics. These similar patterns may, however, respond differently to parameter variations and specifically to tuning of inputs that represent control signals. In this work, we analyze the responses resulting from modulation of a localized input in each of three classes of model neural networks that have been recognized in the literature for their capacity to produce robust three-phase rhythms: coupled fast-slow oscillators, near-heteroclinic oscillators, and threshold-linear networks. Triphasic rhythms, in which each phase consists of a prolonged activation of a corresponding subgroup of neurons followed by a fast transition to another phase, represent a fundamental activity pattern observed across a range of central pattern generators underlying behaviors critical to survival, including respiration, locomotion, and feeding. To perform our analysis, we extend the recently developed local timing response curve (lTRC), which allows us to characterize the timing effects due to perturbations, and we complement our lTRC approach with model-specific dynamical systems analysis. Interestingly, we observe disparate effects of similar perturbations across distinct model classes. Thus, this work provides an analytical framework for studying control of oscillations in nonlinear dynamical systems and may help guide model selection in future efforts to study systems exhibiting triphasic rhythmic activity. 

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3. Theta activity paradoxically boosts gamma and ripple frequency sensitivity in prefrontal interneurons

   https://doi.org/10.1073/pnas.2114549118  

   Merino, Ricardo Martins; Leon-Pinzon, Carolina; Stühmer, Walter; Möck, Martin; Staiger, Jochen F.; Wolf, Fred; Neef, Andreas (December 2021, Proceedings of the National Academy of Sciences)

   Fast oscillations in cortical circuits critically depend on GABAergic interneurons. Which interneuron types and populations can drive different cortical rhythms, however, remains unresolved and may depend on brain state. Here, we measured the sensitivity of different GABAergic interneurons in prefrontal cortex under conditions mimicking distinct brain states. While fast-spiking neurons always exhibited a wide bandwidth of around 400 Hz, the response properties of spike-frequency adapting interneurons switched with the background input’s statistics. Slowly fluctuating background activity, as typical for sleep or quiet wakefulness, dramatically boosted the neurons’ sensitivity to gamma and ripple frequencies. We developed a time-resolved dynamic gain analysis and revealed rapid sensitivity modulations that enable neurons to periodically boost gamma oscillations and ripples during specific phases of ongoing low-frequency oscillations. This mechanism predicts these prefrontal interneurons to be exquisitely sensitive to high-frequency ripples, especially during brain states characterized by slow rhythms, and to contribute substantially to theta-gamma cross-frequency coupling. 

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4. On the Relation between Infinitesimal Shape Response Curves and Phase-Amplitude Reduction for Single and Coupled Limit-Cycle Oscillators

   https://doi.org/10.1137/23M1575159  

   Kreider, Max; Thomas, Peter J (June 2024, SIAM Journal on Applied Dynamical Systems)

   Phase reduction is a well-established method to study weakly driven and weakly perturbed oscillators. Traditional phase-reduction approaches characterize the perturbed system dynamics solely in terms of the timing of the oscillations. In the case of large perturbations, the introduction of amplitude (isostable) coordinates improves the accuracy of the phase description by providing a sense of distance from the underlying limit cycle. Importantly, phase-amplitude coordinates allow for the study of both the timing and shape of system oscillations. A parallel tool is the infinitesimal shape response curve (iSRC), a variational method that characterizes the shape change of a limit-cycle oscillator under sustained perturbation. Despite the importance of oscillation amplitude in a wide range of physical systems, systematic studies on the shape change of oscillations remain scarce. Both phase-amplitude coordinates and the iSRC represent methods to analyze oscillation shape change, yet a relationship between the two has not been previously explored. In this work, we establish the iSRC and phase-amplitude coordinates as complementary tools to study oscillation amplitude. We extend existing iSRC theory and specify conditions under which a general class of systems can be analyzed by the joint iSRC phase-amplitude approach. We show that the iSRC takes on a dramatically simple form in phase-amplitude coordinates, and directly relate the phase and isostable response curves to the iSRC. We apply our theory to weakly perturbed single oscillators, and to study the synchronization and entrainment of coupled oscillators. 

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5. Stochastic rectification of fast oscillations on slow manifold closures

   https://doi.org/https://doi.org/10.1073/pnas.2113650118  

   Chekroun, M. D.; Liu, H.; McWilliams, J. C. (November 2021, Proceedings of the National Academy of Sciences of the United States of America)

   The problems of identifying the slow component (e.g., for weather forecast initialization) and of characterizing slow–fast interactions are central to geophysical fluid dynamics. In this study, the related rectification problem of slow manifold closures is addressed when breakdown of slow-to-fast scales deterministic parameterizations occurs due to explosive emergence of fast oscillations on the slow, geostrophic motion. For such regimes, it is shown on the Lorenz 80 model that if 1) the underlying manifold provides a good approximation of the optimal nonlinear parameterization that averages out the fast variables and 2) the residual dynamics off this manifold is mainly orthogonal to it, then no memory terms are required in the Mori–Zwanzig full closure. Instead, the noise term is key to resolve, and is shown to be, in this case, well modeled by a state-independent noise, obtained by means of networks of stochastic nonlinear oscillators. This stochastic parameterization allows, in turn, for rectifying the momentum-balanced slow manifold, and for accurate recovery of the multiscale dynamics. The approach is promising to be further applied to the closure of other more complex slow–fast systems, in strongly coupled regimes. 

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