Computation by Switching in Complex Networks of States

Complex networks of dynamically connected saddle states persistently emerge in a broad range of high-dimensional systems and may reliably encode inputs as specific switching trajectories. Their computational capabilities, however, are far from being understood. Here, we analyze how symmetry-breaking inhomogeneities naturally induce predictable persistent switching dynamics across such networks. We show that such systems are capable of computing arbitrary logic operations by entering into switching sequences in a controlled way. This dynamics thus offers a highly flexible new kind of computation based on switching along complex networks of states.

Figure 1

Selecting cyclic switching dynamics in complex networks of saddle states [29]. (a)–(d) Black: complex networks of states in the absence of inputs. Colors: trajectories in state space induced by specific symmetry-breaking currents, $\mathbf{\Delta }=(4,3,2,1,0)\times {10}^{-4}$ in (a),(b) and $\mathbf{\Delta }=(1,4,3,2,0)\times {10}^{-4}$ in (c),(d). (a),(c) State space representation. Whenever oscillator $i=1$ is reset, the potentials of oscillators $i\in \{2,3,4\}$ are plotted as a three-dimensional trajectory; the potential of oscillator $i=5$ is given by a color gradient from zero (light color) to one (dark color). Small black dots represent all possible switching trajectories (generated by low-noise-induced switchings). Gray spheres represent the vicinity of approached saddle states. (b),(d) Abstract representation of the same network of 30 saddles. Colored lines indicate the specific cyclic sequence of saddles approached by the trajectories shown in (a) and (b).

Figure 2

XOR binary logic gate via cyclic switching dynamics across networks of states. In (a)–(c), white background represents 0 (nonactivity) and gray 1 (activity). (a) Fixing the three inputs $R_{3,4,5}=(0,0,0)$ leaves two inputs to vary $R_{1,2}\in \{(0,0),(1,0),(0,1),(1,1)\}$. Each of the four input combinations yields (b) a specific limit cycle. The potentials of all oscillators are plotted whenever oscillator $i=1$ is reset. (c) A single output unit decodes the limit cycles by spiking or not spiking, thereby providing a binary output.

Figure 3

One 20-winner-take-all computation via switching dynamics for $N=100$; cluster symmetry $S_{21}\times S_{21}\times S_{21}\times S_{21}\times S_{16}$ [30]. The phase of all oscillators is plotted when oscillator $i=1$ is reset. The phase is a monotonic function of the potential given by ${\varphi }_i=(e^{b{V}_i}-1)/(e^b-1)$. Asymmetric external currents $\mathbf{\Delta }=({\Delta }_1,\dots ,{\Delta }_{100})$ induce specific limit cycles that serve as internal representations of the computational outcome. The initial condition is such that all 20 largest inputs are delivered to elements in the same cluster. Close to every saddle, the units in one specific, desynchronizing cluster that are subject to the five largest inputs leave that cluster. The resulting dynamics is such that four groups of five units that in total form the 20 largest inputs repeatedly leave their clusters when close to the saddle where it is unstable.