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Influence of inhibitory autapses on synchronization of inhibitory network gamma oscillations

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Abstract

A recent experimental study showed that inhibitory autapses favor firing synchronization of parvalbumin interneurons in the neocortex during gamma oscillations. In the present paper, to provide a comprehensive and deep understanding to the experimental observation, the influence of inhibitory autapses on synchronization of interneuronal network gamma oscillations is theoretically investigated. Weak, middle, and strong synchronizations of a globally inhibitory coupled network composed of Wang–Buzsáki model without autapses appear at the bottom-left, middle, and top-right of the parameter plane with the conductance (gsyn) and the decay constant (τsyn) of inhibitory synapses taken as the x-axis and y-axis, respectively. After introducing inhibitory autapses, the border between the strong and middle synchronizations in the (gsyn, τsyn) plane moves to the top-right with increasing the conductance (gaut) and the decay constant (τaut) of autapses, due to that interspike interval of the single neuron becomes longer, leading to that larger τsyn is needed to ensure the strong synchronization. Then, the synchronization degree of middle and strong synchronizations around the border in the (gsyn, τsyn) plane decreases, while of strong synchronization in the remaining region remains unchanged. The synchronization degree of weak synchronization increases with increasing τaut and gaut, due to that the inhibitory autaptic current becomes strong and long to facilitate synchronization. The enhancement of weak synchronization modulated by inhibitory autapses is also simulated in the random, small-world, and scale-free networks, which may provide explanations to the experimental observation. These results present complex dynamics of synchronization modulated by inhibitory autapses, which needs future experimental demonstrations.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was sponsored by the National Natural Science Foundation of China (Grant Numbers: 11802086, 12072236, 11872276, and 11762001); the Henan Provincial Science and Technology Research Project (Grant Number: 202102310410); and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant Number: NJYT-20-A09).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471000, China

    Yanbing Jia

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Huaguang Gu

  3. College of Mathematics and Computer Science, Chifeng University, Chifeng, 024000, China

    Yuye Li

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  1. Yanbing Jia
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  2. Huaguang Gu
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H.G. conceived the experiments. Y.J., H.G., and Y.L. conducted the experiments. Y.J., H.G., and Y.L. analyzed the results. Y.J. and H.G. wrote the paper. All authors reviewed the manuscript.

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Correspondence to Huaguang Gu.

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Appendix: Influence of inhibitory autapses on synchronization in the scale-free network

Appendix: Influence of inhibitory autapses on synchronization in the scale-free network

Influence of g syn and τ syn on synchronization in the scale-free network without autapses

In the absence of autapses, the scale-free network exhibits weak synchronization in the (gsyn, τsyn) plane, as shown by the blue in Fig. 19. The synchronization degree of the scale-free network is lower than that of the networks with other three topologies, due to that the average degree of the scale-free network is smaller. The average degree of the scale-free network is 4, which is much smaller than that of other three networks.

Fig. 19
figure 19

Distribution of the synchronization index S of the scale-free network without autapses on the (gsyn, τsyn) plane

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Inhibitory autapses with τaut = 4 ms enhance weak synchronization in the scale-free network

When τaut = 4 ms, distributions of S of the scale-free network with autapses on the (gsyn, τsyn) plane for different values of gaut are displayed in Fig. 20a1–c1. Differences acquired by subtracting S values in Fig. 19 (the scale-free network without autapses) from S values in Fig. 20a1–c1, labeled as ΔS, are depicted in Fig. 20a2–c2, respectively. Similar to Fig. 9, in Fig. 20a2–c2, five degrees of change are considered as follows: − 0.8 ≤ ΔS <  − 0.1 (black), − 0.1 ≤ ΔS <  − 0.01 (blue), − 0.01 ≤ ΔS < 0.01 (green), 0.01 ≤ ΔS < 0.1 (magenta), and 0.1 ≤ ΔS < 0.8 (red). The horizontal dashed lines in Fig. 20a2–c2 denote τsyn = 4 ms. The characteristics of Fig. 20a2–c2 are as follows.

For small values of gaut (e.g., gaut = 0.01 mS/cm2), as shown in Fig. 20a1 and a2, the synchronization degree increases except for the top right region of the (gsyn, τsyn) plane. For large values of gaut (e.g., gaut = 0.1 mS/cm2 and 0.2 mS/cm2), the synchronization degree increases in the entire (gsyn, τsyn) plane, as shown in Fig. 20b1, b2, c1, and c2, and the larger gaut is, the larger the increase is. In brief, inhibitory autapses with τaut = 4 ms enhance the synchronization degree of weak synchronization in the scale-free network.

Fig. 20
figure 20

Distributions of S (top row) and ΔS (bottom row) of the scale-free network with autapses on the (gsyn, τsyn) plane for different values of gaut. a1 and a2 gaut = 0.01 mS/cm2; b1 and b2 gaut = 0.1 mS/cm2; c1 and c2 gaut = 0.2 mS/cm2. Here, τaut = 4 ms. In the bottom row, the horizontal dashed lines denote τsyn = 4 ms

Full size image

Inhibitory autapses with τaut = 4 ms enhance synchronization in the scale-free network with τsyn = 4 ms

Changes of S with respect to gsyn for gaut = 0.01 mS/cm2, 0.1 mS/cm2, and 0.2 mS/cm2 are shown by the black, red, and blue curves in Fig. 21, respectively. Figure 21 shows that the synchronization degree of weak synchronization increases with increasing gaut, and the larger gaut is, the larger the increase is.

Fig. 21
figure 21

The synchronization degree of the weak synchronization in the scale-free network increases with increasing gaut. The black, red, and blue curves represent changes of S with respect to gsyn for gaut = 0.01 mS/cm2, 0.1 mS/cm2, and 0.2 mS/cm2, respectively. Here, τaut = 4 ms and τsyn = 4 ms

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Jia, Y., Gu, H. & Li, Y. Influence of inhibitory autapses on synchronization of inhibitory network gamma oscillations. Cogn Neurodyn 17, 1131–1152 (2023). https://doi.org/10.1007/s11571-022-09856-5

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