Stability Analysis of Asynchronous States in Neuronal Networks with Conductance-Based Inhibition

Oscillations in networks of inhibitory interneurons have been reported at various sites of the brain and are thought to play a fundamental role in neuronal processing. This Letter provides a self-contained analytical framework that allows numerically efficient calculations of the population activity of a network of conductance-based integrate-and-fire neurons that are coupled through inhibitory synapses. Based on a normalization equation this Letter introduces a novel stability criterion for a network state of asynchronous activity and discusses its perturbations. The analysis shows that, although often neglected, the reversal potential of synaptic inhibition has a strong influence on the stability as well as the frequency of network oscillations.

Figure 1

Simulation of cellular network vs pool theory. Comparison of the network activity obtained from a simulation (diamonds) of 8000 fully interconnected neurons and theory [solid lines; Eq. (3)] for an external step current of amplitude $I/(C\theta )=2.6\mathrm{k}\mathrm{H}z$ at $t=0\mathrm{m}\mathrm{s}$. For a constant value $NE\overline{g}/C=-0.4\theta \mathrm{k}\mathrm{H}z$ of the stationary inhibitory current, a reduction of the inhibitory reversal potential $E$ yields a transition from a stable oscillation (top trace) to a stable asynchronous state (traces two and three) and back again to a stable oscillation (bottom).

Figure 2

Complex roots of the linear normalization kernel $\mathcal{L}[\Lambda ]$ determine stability and oscillation frequency. Parameters are those of Fig. 1. For $E/\theta =-0.04$ there is a single root with a negative real part $\lambda$. Its imaginary part (122 Hz) roughly corresponds to the frequency of stable oscillation. In the case of $E/\theta =-0.057$ (diamonds) there is a root with a positive but very small real part at an oscillation frequency of 125 Hz. At $E/\theta =-0.067$ (circles) the real part of the root at about 125 Hz increases, whereas the real parts of the roots at lower frequencies decrease, which yields a mixture of different oscillatory components; see Fig. 1 (third trace). A further decrease to $E/\theta =-0.24$ while keeping $\overline{g}E$ constant, again yields a root with a negative real part and therefore an unstable asynchronous state. Roots with $\lambda >0.15\mathrm{k}\mathrm{H}z$ are not shown.

Figure 3

Stability and time course of the relaxation to asynchronous states. Regions of unstable asynchronous state are plotted black, stability regions are color-coded. Regions of low oscillation index $\sigma$ (see the text) are depicted through grayish colors; high oscillation indices are marked with bright colors. Insets (a)–(f) are obtained by numerical integration of Eq. (3). For high values of $|E|$ oscillations are in a “current-based” limit and show only little dependence on $E$. Note that the $E$ axis has a logarithmic scale. The vertical solid line denotes the value $I/(C\theta )=2.6\mathrm{k}\mathrm{H}z$, the horizontal line $E/\theta =-0.057$.