Inhibition causes ceaseless dynamics in networks of excitable nodes

Abstract

The collective dynamics of a network of excitable nodes changes dramatically when inhibitory nodes are introduced. We consider inhibitory nodes which may be activated just like excitatory nodes but, upon activating, decrease the probability of activation of network neighbors. We show that, although the direct effect of inhibitory nodes is to decrease activity, the collective dynamics becomes self-sustaining. We explain this counterintuitive result by defining and analyzing a "branching function" which may be thought of as an activity-dependent branching ratio. The shape of the branching function implies that, for a range of global coupling parameters, dynamics are self-sustaining. Within the self-sustaining region of parameter space lies a critical line along which dynamics take the form of avalanches with universal scaling of size and duration, embedded in a ceaseless time series of activity. Our analyses, confirmed by numerical simulation, suggest that inhibition may play a counterintuitive role in excitable networks.

FIG. 1

(Color online.) A) Time series of S(t) show typical behavior of this system: α > 0 causes the S = 0 state to become repelling, so that dynamics are self-sustaining. B) Empirical distributions of network activity show that states of critical systems are much more uniformly distributed while sub- and supercritical states fluctuate within tight bands. C) Predictions of branching function Λ [Eq. (4)] agree well with empirical measurements of S(t + 1)/S(t) for various λ and α. Three regimes corresponding to Λ > 1, Λ = 1 and Λ < 1 are visible, explaining dynamics from panels A and B. The Λ > 1 regime causes self-sustained behavior. Sub- and supercritical networks achieve Λ = 1 at a single S (arrows), around which dynamics fluctuates tightly; critical networks achieve Λ ≈ 1 over a wide range in S, allowing broad fluctuations. Λ < 1 for large values of S preventing activity from completely saturating. N = 10⁴, 〈k〉 = 200 for all panels.

FIG. 2

(Color online.) A) Empirical measurements of Λ₀ (symbols) agree well with predictions, Eq. (S15), showing that as α increases, the S = 0 state becomes more repulsive. B) Lifetime of network activity increases with inhibitory fraction α for various N and 〈k〉. Simulations began with 100 active nodes, with lifetime calculated from the fraction of simulations that ceased prior to T = 10⁴ timesteps. (C) Lifetime scales correctly with q, as shown in Eq. (6), indicated by collapse of curves.

FIG. 3

(Color online.) A) We define avalanches as excursions above a threshold S^*, with duration d the length of the excursion and size a the integral under the curve over the duration of the excursion. B) Distributions of avalanche size are power law for all α, P(a) ~ a^−3/2. The dashed line corresponds to sizes from a critical Galton-Watson branching process with S^* = 128. C) Durations are not power-law distributed but have the same distribution as durations from a critical Galton-Watson process. Durations do not show the familiar universal power-law exponent of −2 due to the conversion of ceaseless time series into avalanches (see text and [23]). Data shown: N = 10⁴ nodes over 3 × 10⁶ timesteps, 〈k〉 = 200.