Figure 2. Dynamical and fixed point properties of networks with two self-couplings.

(a) Ratio of autocorrelation timescales ${\displaystyle {\tau }_2/{\tau }_1}$ of units with self-couplings ${\displaystyle s_2}$ and ${\displaystyle s_1}$, respectively (${\displaystyle {\tau }_i}$ is estimated as the half width at half max of a unit’s autocorrelation function, see panels iii, iv), in a network with ${\displaystyle n_1=n_2=0.5}$ and ${\displaystyle g=2}$ and varying ${\displaystyle s_1,s_2}$. A central chaotic phase separates four different stable fixed point regions with or without transient activity. Black curves represent the transition from chaotic to stable fixed point regimes, which can be found by solving consistently Equation 15, Equation 16, and Equation 18 (using equal to 1 in the latter), see Methods ('Fixed points and transition to chaos' and 'Stability conditions') for details. (i, ii) Activity across time during the initial transient epoch (left) and distributions of unit values at their stable fixed points (right), for networks with ${\displaystyle N=1000}$ and (i) ${\displaystyle s_1=3.2,s_2=-1.5}$, (ii) ${\displaystyle s_1=3.2,s_2=1.2}$. (iii, iv) Activity across time (left) and normalized autocorrelation functions ${\displaystyle C(\tau )/C(0)}$, (right) of units with (iii) ${\displaystyle s_1=0.8,s_2=-1.5}$, (iv) ${\displaystyle s_1=0.8,s_2=3.2}$. (b) Timescales ${\displaystyle {\tau }_2,{\tau }_1}$ (left) and their ratio ${\displaystyle {\tau }_2/{\tau }_1}$ (right) for fixed ${\displaystyle s_1=1}$ and varying ${\displaystyle s_2}$, as a function of the relative size of the two populations ${\displaystyle n_1=N_1/N,n_2=N_2/N}$ (at ${\displaystyle g=2}$, ${\displaystyle N=2000}$; average over 20 network realizations). All points in b. were verified to be within the chaotic region using Equation 18.