Fig 1.
Visualization of the control parameters density , balance
, and symmetry
for eight example networks (A-H).
Neurons are represented by gray circles, non-zero connections between neurons by arrows. One-headed arrows stand for uni-directional, two-headed arrows for bi-directional connections. Blue/magenta connections are excitatory (wij > 0), red/orange connections inhibitory (wij < 0).
Fig 2.
Dynamical phases in recurrent neural networks and characteristic output signals of individual neurons.
(A) Two-dimensional phase diagram, showing the fraction of positive Lyapunov exponents as a function of the control parameters balance and density, for a constant symmetry parameter
(Note that in part G, we show the average Lyapunov exponent along a a one-dimensional cut through the 2D phase space of balance and density, for constant density 0.2). In the heat map, dark blue colors indicate fλ>0 ≈ 0, dark red colors fλ>0 ≈ 1. The red region in the center of the phase diagram is the chaotic regime, consistent with the irregular outputs of selected neurons (F). The ‘left’ blue region at negative balance values is the regime of cyclic attractors, often with small period lengths T ≈ 2, as demonstrated with the neuron output (B). The ‘right’ blue region at positive balance values is the regime of fixed points, as exemplified with the constant neuron output (D). Note that, in both cases (B, D) the fraction of positive Lyapunov exponents is zero since the dynamics is non-chaotic, and hence the color coding is identical in both cases. The most interesting dynamics is found at the edges of the chaotic regime (C, E), where one finds cases of periodic behavior with large period length T > 2, periodic behavior with intermittent bursts, decaying oscillatory behavior, and ‘beating’ oscillatory behavior. Note that, the sampled time traces depicted in the figure are from selected neurons, not necessarily from within the same network.
Fig 3.
Comparing different dynamical measures, and the effect of system size.
The columns correspond to the quantities fλ>0 (left), Tav (middle) and ρrms (right), as defined in the methods section. The rows from top to bottom correspond to increasing system sizes, characterized by the number of neurons N in the neural networks. For each of the 12 cases, a two-dimensional phase diagram is shown as a function of balance and density, keeping a constant symmetry parameter of . The three dynamic phases become apparent only for systems with a minimum size of N ≥ 100. The three different dynamical measures are mutually consistent. In particular, the chaotic regime is characterized by a fλ>0 close to one, by a diverging Tav, and by a vanishing ρrms. For large systems with N ≥ 10000, the density parameter has no more effect on the system dynamics, which is then controlled by the balance only.
Fig 4.
Effect of symmetry on system dynamics.
(A): Standard plot of fλ>0 as a function of balance and density, for constant symmetry . (B): Plot of fλ>0 as a function of balance and symmetry, for constant density
(see orange box in (A)). (C): Plot of fλ>0 as a function of symmetry and density, for constant balance
(see green box in (A)). For too large symmetry
, the system ends up in fixed point attractors, irrespective of balance and density. Note that the phase diagram shown in (A) is the same as shown in Fig 3D.