Abstract
Gamma oscillations are widely seen in the cerebral cortex in different states of the wake-sleep cycle and are thought to play a role in sensory processing and cognition. Here, we study the emergence of gamma oscillations at two levels, in networks of spiking neurons, and a mean-field model. At the network level, we consider two different mechanisms to generate gamma oscillations and show that they are best seen if one takes into account the synaptic delay between neurons. At the mean-field level, we show that, by introducing delays, the mean-field can also produce gamma oscillations. The mean-field matches the mean activity of excitatory and inhibitory populations of the spiking network, as well as their oscillation frequencies, for both mechanisms. This mean-field model of gamma oscillations should be a useful tool to investigate large-scale interactions through gamma oscillations in the brain.
Similar content being viewed by others
Data availability
No datasets were generated or analysed during the current study.
Code availability
All program codes used in this publication will be made available open-access in Zenodo (Tahvili & Destexhe, 2023).
References
Belluscio, M. A., Mizuseki, K., Schmidt, R., Kempter, R., & Buzsáki, G. (2012). Cross-frequency phase-phase coupling between theta and gamma oscillations in the hippocampus. Journal of Neuroscience, 32(2), 423–435.
Berke, J. D., Okatan, M., Skurski, J., & Eichenbaum, H. B. (2004). Oscillatory entrainment of striatal neurons in freely moving rats. Neuron, 43(6), 883–896.
Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 8, 183–208.
Brunel, N., & Wang, X.-J. (2003). What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance. Journal of Neurophysiology, 90(1), 415–430.
Buzsáki, G., & Wang, X.-J. (2012). Mechanisms of gamma oscillations. Annual Review of Neuroscience, 35, 203–225.
Chrobak, J., & Buzsáki, G. (1998). Gamma oscillations in the entorhinal cortex of the freely behaving rat. Journal of Neuroscience, 18(1), 388–398.
Destexhe, A. (2009). Self-sustained asynchronous irregular states and up-down states in thalamic, cortical and thalamocortical networks of nonlinear integrate-and-fire neurons. Journal of Computational Neuroscience, 27, 493–506.
Destexhe, A., Mainen, Z., & Sejnowski, T. J. (1998). Kinetic models of synaptic transmission. In C. Koch & I. Segev (Eds.), Methods in Neuronal Modeling: From Ions to Networks (pp. 1–25). Cambridge MA: MIT Press.
Di Volo, M., Romagnoni, A., Capone, C., & Destexhe, A. (2019). Biologically realistic mean-field models of conductance-based networks of spiking neurons with adaptation. Neural Computation, 31(4), 653–680.
Di Volo, M., & Torcini, A. (2018). Transition from asynchronous to oscillatory dynamics in balanced spiking networks with instantaneous synapses. Physical Review Letters, 121(12), 128301.
El Boustani, S., & Destexhe, A. (2009). A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation, 21, 46–100.
Engel, A. K., Fries, P., & Singer, W. (2001). Dynamic predictions: Oscillations and synchrony in top-down processing. Nature Reviews Neuroscience, 2(10), 704–716.
Geisler, C., Brunel, N., & Wang, X.-J. (2005). Contributions of intrinsic membrane dynamics to fast network oscillations with irregular neuronal discharges. Journal of Neurophysiology, 94(6), 4344–4361.
Goldman, J., Kusch, L., Aquilue, D., Yalcinkaya, B., Depannemaecker, D., Ancourt, K., Nghiem, T.-A., Jirsa, V., & Destexhe, A. (2023). A comprehensive neural simulation of slow-wave sleep and highly responsive wakefulness dynamics. Frontiers in Computational Neuroscience, 16, 1058957.
Gray, C. M., König, P., Engel, A. K., & Singer, W. (1989). Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature, 338(6213), 334–337.
Kayser, C., Ince, R. A. A., & Panzeri, S. (2012). Analysis of slow (theta) oscillations as a potential temporal reference frame for information coding in sensory cortices. e1002717.
Le Van Quyen, M., Muller, L., Telenczuk, B., Cash, S., Halgren, E., Hatsopoulos, N., Dehghani, N., & Destexhe, A. (2016). High-frequency oscillations in human and monkey neocortex during the wake-sleep cycle. Proceedings of National Academy of Sciences United States of America, 11, 9363–9368.
Mann, E. O., Radcliffe, C. A., & Paulsen, O. (2005). Hippocampal gamma–frequency oscillations: From interneurones to pyramidal cells, and back. The Journal of Physiology, 562(1), 55–63.
Moghaddam, B., Adams, B., Verma, A., & Daly, D. (1997). Activation of glutamatergic neurotransmission by ketamine: A novel step in the pathway from NMDA receptor blockade to dopaminergic and cognitive disruptions associated with the prefrontal cortex. Journal of Neuroscience, 17(8), 2921–2927.
Pinault, D., & Deschênes, M. (1992). Voltage-dependent 40-Hz oscillations in rat reticular thalamic neurons in vivo. Neuroscience, 51, 245–258.
Popescu, A. T., Popa, D., & Paré, D. (2009). Coherent gamma oscillations couple the amygdala and striatum during learning. Nature Neuroscience, 12(6), 801–807.
Shaw, A. D., Saxena, N., Jackson, L. E., Hall, J. E., Singh, K. D., & Muthukumaraswamy, S. D. (2015). Ketamine amplifies induced gamma frequency oscillations in the human cerebral cortex. European Neuropsychopharmacology, 25(8), 1136–1146.
Sirota, A., Montgomery, S., Fujisawa, S., Isomura, Y., Zugaro, M., & Buzsáki, G. (2008). Entrainment of neocortical neurons and gamma oscillations by the hippocampal theta rhythm. Neuron, 60(4), 683–697.
Su, T., Lu, Y., Geng, Y., Lu, W., & Chen, Y. (2018). How could N-methyl-D-aspartate receptor antagonists lead to excitation instead of inhibition? Brain Science Advances, 4(2), 73–98.
Susin, E., & Destexhe, A. (2023). A network model of the modulation of gamma oscillations by NMDA receptors in cerebral cortex. eNeuro, 10, 0157–23.
Susin, E., & Destexhe, A. (2021). Integration, coincidence detection and resonance in networks of spiking neurons expressing Gamma oscillations and asynchronous states. PLOS Computational Biology, 17(9), e1009416.
Tahvili F., & Destexhe, A. (2023). Program code for mean-field models of gamma-frequency oscillations in networks of excitatory and inhibitory neurons. Zenodo.
Tian, F., Lewis, L. D., Zhou, D. W., Balanza, G. A., Paulk, A. C., Zelmann, R., Peled, N., et al. (2023). Characterizing brain dynamics during ketamine-induced dissociation and subsequent interactions with propofol using human intracranial neurophysiology. Nature Communications, 14(1), 1748.
Tort, A. B. L., Komorowski, R., Eichenbaum, H., & Kopell, N. (2010). Measuring phase-amplitude coupling between neuronal oscillations of different frequencies. Journal of Neurophysiology, 104(2), 1195–1210.
Wang, X.-J., & Buzsáki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of Neuroscience, 16(20), 6402–6413.
Zerlaut, Y., Chemla, S., Chavane, F., & Destexhe, A. (2018). Modeling mesoscopic cortical dynamics using a mean-field model of conductance-based networks of adaptive exponential integrate-and-fire neurons. Journal of Computational Neuroscience, 44, 45–61.
Zerlaut, Y., Teleńczuk, B., Deleuze, C., Bal, T., Ouanounou, G., & Destexhe, A. (2016). Heterogeneous firing rate response of mouse layer V pyramidal neurons in the fluctuation‐driven regime. The Journal of Physiology, 594, 3791–3808.
Funding
Research supported by the CNRS and the European Union (Human Brain Project, H2020-945539).
Author information
Authors and Affiliations
Contributions
A.D. conceived and supervised the study, F.T. made the analysis and prepared the figures. Both authors discussed the results and wrote the manuscript.
Corresponding author
Ethics declarations
Ethical approval
N/A.
Conflict of interest
The authors declare no conflict of interest.
Additional information
Action Editor: Albert Compte
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: Delay asymmetries
In this Appendix, we give more details on the asymmetry in delays that is needed in the mean-field model.
Figure 10 shows a magnified image of a Type 1 network’s raster plot over a gamma cycle. In a gamma cycle, the sequence of spiking activity begins with FS neurons followed by RS cells as can be seen in the figure. This sequential pattern results in a slight phase advance for FS neurons compared to RS cells. Notably, this phenomenon has been experimentally observed in human and primates in vivo (Le Van Quyen et al., 2016).
The mean-field with a constant delay does not robustly generate gamma oscillations, and we needed to consider a slight asymetry of the delays between populations, as \(exc \rightarrow exc\) and \(inh \rightarrow inh\) equal to \((1+\epsilon )\tau _d\) while the delays for the other two connections (\(exc \rightarrow inh\) and \(inh \rightarrow exc\)) equal \((1-\epsilon )\tau _d\). This means that the excitatory recurrent input is received by the excitatory population a little later (\(2 \epsilon \tau _d\)) than the inhibitory input and the excitatory input is received by the inhibitory population a little sooner (\(2 \epsilon \tau _d\)) than the recurrent inhibitory input. This slight asymmetry in delays is compatible with the phenomenon that the FS cells start to spike sooner than the RS cells.
For the spiking network, however, the presence of this asymetry had little impact. The behavior of the spiking network with this asymmetry in delays, is almost similar to the one without the asymmetry. Figure 11 shows the same network as Fig. 10, but with a delay asymmetry.
Appendix 2: Calculation of the transfer function
In this Appendix, we give more details on the calculation of the transfer function in the mean-field model, following a procedure developed by Zerlaut et al. (2016) and Di Volo et al. (2019).
We suppose that the neuron’s firing rate can be expressed as a function of the statistical characteristics of its sub-threshold voltage dynamics. These statistical characteristics are the mean sub-threshold voltage (\(\mu _v\)), its standard deviation (\(\sigma _v\)), and the time correlation decay time (\(\tau _v\)).
First we calculate the mean and standard deviation of synaptic conductances which lead to \(\mu _{G e}\left( v_E, v_I\right) = v_E N_E p \tau _E Q_E\), \(\sigma _{G e}\left( v_E, v_I\right) = \sqrt{\frac{v_E N_E p \tau _E}{2}} Q_E\), \(\mu _{G i}\left( v_E, v_I\right) = v_I N_I p \tau _I Q_I\), and \(\sigma _{G i}\left( v_E, v_I\right) = \sqrt{\frac{v_I N_I p \tau _I}{2}} Q_I\) where \(N_E = 8000\) and \(N_I = 2000\) are the number of pre-synaptic excitatory and inhibitory neurons, respectively. Therefore, the input conductance and the effective membrane time constant of a neuron become \(\mu _G\left( v_E, v_I\right) = \mu _{G e}+\mu _{G i} + g_L\) and \(\tau _m^{\text{ eff } } = \frac{C}{\mu _G}\). Then assuming that the time scale of the adaptation current is much slower than the time scale of voltage fluctuations and also neglecting the exponential term in Eq. (1) we derive the statistics of the membrane sub-threshold voltage as follows.
Now one can write the neuron’s output firing rate as
which defines the neuron’s transfer function. Here, erfc is the complementary error function (\({\text {erfc}}(x)=\frac{2}{\sqrt{\pi }} \int _x^{\infty } \textrm{e}^{-t^2} \mathrm {~d} t\)) and \(V_{\text{ th }}^{\text{ eff } }\) is the effective voltage threshold which can be itself written as a function of \(\mu _V\), \(\sigma _V\), and \(\tau _V\) as shown by Zerlaut et al. (2016). Specifically, we take \(V_{\text{ th }}^{\text{ eff } }\) as a second-order polynomial as in Di Volo et al. (2019).
and we use the fitted parameters (Table 3) which were obtained by Di Volo et al. (2019).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Tahvili, F., Destexhe, A. A mean-field model of gamma-frequency oscillations in networks of excitatory and inhibitory neurons. J Comput Neurosci 52, 165–181 (2024). https://doi.org/10.1007/s10827-024-00867-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10827-024-00867-1