Stable Irregular Dynamics in Complex Neural Networks

Irregular dynamics in multidimensional systems is commonly associated with chaos. For infinitely large sparse networks of spiking neurons, mean field theory shows that a balanced state of highly irregular activity arises under various conditions. Here we analytically investigate the microscopic irregular dynamics in finite networks of arbitrary connectivity, keeping track of all individual spike times. For delayed, purely inhibitory interactions we demonstrate that any irregular dynamics that characterizes the balanced state is not chaotic but rather stable and convergent towards periodic orbits. These results highlight that chaotic and stable dynamics may be equally irregular.

Figure 1

Stable irregular dynamics in a random network ($N=400$, ${\gamma }_i^{-1}\equiv 1.0$, $I_i\equiv 4.0$, ${\varphi }_{\Theta ,i}\equiv 1$, ${\tau }_{i,j}\equiv 0.1$, connection probability $p=0.2$, $\sum _j{\epsilon }_{i,j}\equiv -16$). (a) Upper panel displays the spiking times (blue lines) of the first 50 neurons. Lower panel displays the membrane potential trajectory of neuron $i=1$ (spikes of height $\Delta V=2$ added at firing times). Inset shows a histogram of the coefficients of variation $C{V}_i≔{\sigma }_i/{\mu }_i$, ${\mu }_i≔⟨t_{i,k+1}^s-t_{i,k}^s⟩$, ${\sigma }_i^2≔⟨(t_{i,k+1}^s-t_{i,k}^s-{\mu }_i{)}^2⟩$ averaged over time. (b) Exponential decay of the maximal perturbation $\underset{i}{max}|{\delta }_i^{(n)}|$ (blue dots) and the minimal margin ${\kappa }^{(n)}$ (gray line) for one given microscopic dynamics. (c) Algebraic decay of the average minimal margin, $\overline{{\kappa }^{(n)}}$ (green dashed line, averaged over 250 random initial conditions), and the analytical prediction (no free fit parameter) of $\overline{{\kappa }^{(n)}}$ (black solid line). We also show the minimal margin ${\kappa }^{(n)}$ for three exemplary initial conditions (gray lines), including that of (b).

Figure 2

Convergence towards a periodic orbit in a random network ($N=40$, ${\gamma }_i^{-1}\equiv 1.0$, $I_i\equiv 3.0$, ${\varphi }_{\Theta ,i}\equiv 1.0$, ${\tau }_{i,j}\equiv 0.1$, $p=0.2$, $\sum _j{\epsilon }_{i,j}\equiv -3.3$). (a) The average minimal margin $\overline{{\kappa }^{(n)}}$ [as in Fig. 1c] decays as a power law (region $A$) and saturates after about ${10}^7$ events (region $B$) when the periodic orbit is reached. Inset: Margin ${\mu }^{(n)}$ (black) and minimal margin ${\kappa }^{(n)}$ (gray) for a trajectory started from one specific initial condition. The margin ${\mu }^{(n)}$ fluctuates strongly on the transient; after the sequence becomes periodic the minimal margin ${\kappa }^{(n)}$ does not decrease further for future events $n$. (b),(c) Snapshots of (b) irregular spike sequences after $n\approx 2\times {10}^4$ events on the transient and (c) after $n\approx {10}^8$ events on the periodic orbit.