Multi-stable oscillations in cortical networks with two classes of inhibition

Abstract

In the classic view of cortical rhythms, the interaction between excitatory pyramidal neurons (E) and inhibitory parvalbumin neurons (I) has been shown to be sufficient to generate gamma and beta band rhythms. However, it is now clear that there are multiple inhibitory interneuron subtypes and that they play important roles in the generation of these rhythms. In this paper we develop a spiking network that consists of populations of E, I and an additional interneuron type, the somatostatin (S) internerons that receive excitation from the E cells and inhibit both the E cells and the I cells. These S cells are modulated by a third inhibitory subtype, VIP neurons that receive inputs from other cortical areas. We reduce the spiking network to a system of nine differential equations that characterize the mean voltage, firing rate, and synaptic conductance for each population and using this we find many instances of multiple rhythms within the network. Using tools from nonlinear dynamics, we explore the roles of each of the two classes of inhibition as well as the role of the VIP modulation on the properties of these rhythms.

Figures (12)

• Figure 1: A. Model circuit showing connectivity between excitatory pyramidal cells (E), inhibitory parvalbumin cells (I), inhibitory somatostatin cells (S), and VIP inhibitory cells that modulate the excitability of the S cells. B. Behavior of the network for default parameters except $\mu_e=1.25$ showing two distinct rhythms in the same network. Top rows show raster plots of the three populations for the two rhythms and bottom shows the population synaptic response (filtered firing rates). (C) Power spectrum for the two rhythms showing peaks at 15 and 16 Hz.
• Figure 2: Spike phase relationships to the surrogate local field potential, $s_e(t)$ for E, I, and S populations for large (Top) and small (Bottom) limit cycles.
• Figure 3: Behavior of the mean field model compared to the spiking model. (A) Bifurcation diagram showing the maximum and minimum of $a_e(t)$ as a function of the excitatory drive, $\mu_e$. Solid red (black) lines represent stable (unstable) equilibrium points and thick green (blue) lines are stable (unstable) periodic solutions. Thin vertical line indicates $\mu_e=1.25$ where there are two stable distinct periodic solutions. Six special points are indicated with the thin arrows: Fold of limit cycles (FL) where a stable (green) and unstable (blue) oscillation collide; Andronov-Hopf bifurcation (AH) where the equilibrium changes stability and gives birth to an oscillation. Lightly shaded region depicts parameters where there are two distinct rhythms.(B) Frequency of the oscillations as a function of $\mu_e$ (C) Comparison of the synaptic variables, $s_{e,i,s}$ for the mean-field and the spiking models. Excitatory is red, inhibitory PV is blue, and inhibitory SOM is green.
• Figure 4: (A) Stimuli to different populations of neurons allows for switching between the different oscillations. At $t=1000$ msec, the SOM cells (green) are given a 200 msec input which switches from the large oscillation to the smaller one; at $t=2050$ msec, the E cells (red) are given a stimulus for 200 msec to switch back to the large oscillation; at $t=3500$ the PV cells (blue) are given a 200 msec stimulus switching from the large to the small oscillation. (B) Stimuli of appropriate amplitude to the S population can switch from the small oscillation to the large (top), but if the stimulus is too large, no switch occurs.
• Figure 5: Dependency of dynamics and multi-rhythmicity on SOM ($\lambda$). The number of oscillations is organized around the appearance and disappearance through collisions of stable and unstable oscillations (FL, shown as black curves) and the emergence and disappearance of oscillations from equilibria (AH, shown as blue curves). As the degree of SOM inhibition ($\lambda$) changes the dependence of the mean-field system on the excitatory drive ($\mu_e$) changes qualitatively. Crossing the black and blue curves changes the number of stable equilibria and oscillations. The letters a-f in the $(\mu_e,\lambda)$ diagram are shown at their corresponding values in the $(\mu_e,a_e)$ diagrams surrounding the main diagram. SL (UL):stable (unstable)oscillation; SE (UE): stable (unstable) equilibrium. The label d shows an example of an isola of oscillations where solutions are not connected to the main branch of equilibria. Once $\lambda$ falls below about 0.45, there is no bi-rhythmicity.