Figure 3. Continuous distributions of self-couplings.

(a–i) In a network with a Gaussian distribution of self-couplings (mean ${\displaystyle \mu =1}$ and variance ${\displaystyle {\sigma }^2=9}$), and ${\displaystyle g=2.5}$, the stable fixed point regime exhibits a distribution of fixed point values interpolating between around the zero fixed point (for units with ${\displaystyle s_i\le 1}$) and the multi-modal case (for units with ${\displaystyle {\displaystyle s_i>1}}$). The purple curve represents solutions to ${\displaystyle x=s\tanh (x)}$. (a,b–ii) A network with a lognormal distribution of self-couplings (parameters for (a,b) ${\displaystyle {\displaystyle \mu =0.2,0.5}}$ and ${\displaystyle {\displaystyle {\sigma }^2=1,0.62}}$, and ${\displaystyle g=2.5}$ ;autocorrelation timescales ${\displaystyle {\tau }_i}$ in units of ms) in the chaotic phase, span several orders of magnitude as functions of the units’ self-couplings ${\displaystyle s_i}$. (a-ii) Mean-field predictions for the autocorrelation functions and their timescales (purple curve) were generated from Equation 13 and Equation 14 via an iterative procedure, see Methods: 'Dynamic mean-field theory with multiple self-couplings' , 'An iterative solution'. (b) Populations of neurons recorded from orbitofrontal cortex of awake monkeys exhibit a lognormal distribution of intrinsic timescales (data from Cavanagh et al., 2016) (panel b-i, red), consistent with neural activity generated by a rate network with a lognormal distribution of self-couplings (panel b-i, blue; panel b-ii). ; We note that Cavanagh et al., 2016 use fitted exponential decay time constants of the autocorrelation functions as neurons’ timescales, while we use the half widths at half max of the autocorrelation functions as the timescales. To bridge these two definitions, we multiplied (Cavanagh et al., 2016) data by a factor of ln(2) before comparing it with our model and presenting it in this figure. The model membrane time constant was assumed to be 3 ms in this example.