Figure 1

The transient time to ZLS is exemplified by the mixing argument [5] for a network consists of four neurons. (a) Schematic of a homogeneous network where all delays are equal to $\tau$. (b) An example where the initial stimulation ($t=0$) is given to the neuron labeled by 1. The state of each neuron is determined by the states of its driving neurons at the previous time step. Red (dark) color indicates an evoked spike, an excited neuron. The transient time is equal to 6 in units of $\tau$. (c) An example where initial stimuli are simultaneously given to neurons 1 and 2. The transient time to ZLS is equal to 5.Reuse & Permissions

Figure 2

Illustration of the transient time, $T$, for networks leading to ZLS (second column) where $\mathcal{D}$ and $\mathcal{C}$ scale either as $O\left(1\right)$ or as $O\left(N\right)$. Rows (1)–(3) exemplify networks with $\mathcal{D}$ and $\mathcal{C}$ of $O\left(N\right)$ but different scaling for ${\mathcal{V}}_{\max }$. In cases (1) and (2a), there are $N$ neurons placed on a ring and a single additional connection between neuron 2 and neuron $N$ (indicated by the red arrow) is added to the ring. In cases (2b) and (3) there is one node with $O\left(N\right)$ outgoing connections; node 2 is connected to $O\left(N\right)$ other nodes, as indicated by the red arrows. Rows (4)–(6) exemplify networks where the transient is dominated by the diameter $\mathcal{D}$. The numbers inside the loops indicate the number of the nodes contained in the loop, and the intersection between adjacent loops contains at least one node. In cases (4) and (5) all the loops contain $O\left(1\right)$ neurons, where in case (5) the loops are connected to each other by one mutual neuron. Case (6) is a fully connected graph.Reuse & Permissions

Figure 3

Fraction of neurons which are in ZLS as a function of time for various cases and initial conditions. The (red) lightning denotes the position of the initial stimulation to the network, and numbers inside the loops denote their lengths. (a) Linear growth of $\rho \left(t\right)$ obtained in simulations with $N=101$ for the two networks shown on the right. For $T=O\left(N\right)$ there is an initial time, bounded by the diameter, where $\rho \left(t\right)=1/N$. (b) 100 loops of length 4 and the rightmost loop is of length 5. The shape of $\rho \left(t\right)$ and the transient time depend on the location of the stimulations. For case (1) it takes $O\left(\mathcal{D}\right)$ time steps for a spike to arrive at the loop of length 5, which is essential to achieve ZLS. For initial conditions (2) and (3) this time is shorter and the time to ZLS is reduced.Reuse & Permissions

Figure 4

Extension of the scaling of the transient time [Eq. (3)] to the case where $\mathcal{C}$ and ${\mathcal{V}}_{\max }^{-1}$ are both $O\left(N^{\alpha }\right),0<\alpha <1$. (a) The lengths of all loops scales as $O\left(N^{\alpha }\right)$, $\alpha =1/3$ (lower blue) and $\alpha =2/3$ (upper red). Simulation results indicates that for $\alpha =1/3$ $T\propto N^{0.98}\sim O\left(N\right)$, while for $\alpha =2/3$ $T\propto N^{1.31}\sim O\left(N^{4/3}\right)$, which are both in a good agreement with the theoretical prediction [Eq. (3)]. (b) A network where one of the loops scales differently, $\mathcal{C}$$=O\left(N^{\beta }\right),0<\alpha <\beta <1$. The transient time scales as $T=O(N^{\beta +\alpha }+N)$. For $\alpha =1/4$ (lower blue) the estimated scaling is $T\propto N^{1.01}$ while for $\alpha =1/2$ (upper red) the estimation is $T\propto N^{1.18}$, in fairly good agreement with the scaling predicted by Eq. (3).Reuse & Permissions

Figure 5

Transient time to ZLS in networks composed of Hodgkin-Huxley neurons. The transient is scaled by the delay time $\tau$ between adjacent neurons. Lower (blue) line: network topology (1) in Fig. 2. The transient time scales as $T=O\left(N\right)$, as predicted by Eq. (3). Upper (red) line: network topology (2a) in Fig. 2. The transient time scales as $T=O\left(N^{2.06}\right)$, which is in a good agreement with the predicted value, $O\left(N^2\right)$.Reuse & Permissions