A mean-field model of gamma-frequency oscillations in networks of excitatory and inhibitory neurons

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.

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.

Fig. 11

Raster plot similar as in Fig. 10 but with delay asymmetries. $exc \rightarrow exc$ and $inh \rightarrow inh$ delays are $(1+\epsilon )\tau _d$ while the delays for the other two connections ($exc \rightarrow inh$ and $inh \rightarrow exc$) equal $(1-\epsilon )\tau _d$. The phase advance of FS cells compared to RS neurons is similar to the system without delay asymmetries (Fig. 10)

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.

$$\begin{aligned} \mu _V\left( v_E, v_I, w\right)&= \frac{\mu _{G e} E_E+\mu _{G i} E_I+g_L E_L-w}{\mu _G} \end{aligned}$$

(10)

$$\begin{aligned} \sigma _V\left( v_E, v_I\right)&=\sqrt{\sum _{s\in \{E,I\}} K_s v_s \frac{\left( \frac{Q_s}{\mu _G}\left( E_s-\mu _V\right) \cdot \tau _s\right) ^2}{2\left( \tau _{\textrm{m}}^{\text{ eff } }+\tau _s\right) }} \end{aligned}$$

(11)

$$\begin{aligned} \tau _V\left( v_E, v_I\right)&= \frac{\sum _{s\in \{E,I\}}\left( K_s v_s\left( \frac{Q_s}{\mu _G}\left( E_s-\mu _V\right) \cdot \tau _s\right) ^2\right) }{\sum _{s\in \{E,I\}}\left( K_s v_s\left( \frac{Q_s}{\mu _G}\left( E_s-\mu _V\right) \cdot \tau _s\right) ^2 /\left( \tau _{\textrm{m}}^{\text{ eff } }+\tau _s\right) \right) } \end{aligned}$$

(12)

Now one can write the neuron’s output firing rate as

$$\begin{aligned} v_{out} = F_{\{E,I\}}(v_E , v_I , w) = \frac{1}{2 \tau _V} \cdot {\text {erfc}}\left( \frac{V_{\text{ th }}^{\text{ eff } }-\mu _V}{\sqrt{2} \sigma _V}\right) \end{aligned}$$

(13)

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).

$$\begin{aligned} \begin{aligned} V_{\text{ th }}^{eff}\left( \mu _V, \sigma _V, \tau _V^N\right) =&\ P_0+\sum _{x \in \left\{ \mu _V, \sigma _V, \tau _V^N\right\} } P_x \cdot \left( \frac{x-x^0}{\delta x^0}\right) \\&+\sum _{x, y \in \left\{ \mu _V, \sigma _V, \tau _V^N\right\} ^2} P_{x y} \cdot \left( \frac{x-x^0}{\delta x^0}\right) \left( \frac{y-y^0}{\delta y^0}\right) \end{aligned} \end{aligned}$$

(14)

and we use the fitted parameters (Table 3) which were obtained by Di Volo et al. (2019).

Table 3 Fit parameters of transfer functions (in mV)