Role of Delays in Shaping Spatiotemporal Dynamics of Neuronal Activity in Large Networks

We study the effect of delays on the dynamics of large networks of neurons. We show that delays give rise to a wealth of bifurcations and to a rich phase diagram, which includes oscillatory bumps, traveling waves, lurching waves, standing waves arising via a period-doubling bifurcation, aperiodic regimes, and regimes of multistability. We study the existence and the stability of the various dynamical patterns analytically and numerically in a simplified rate model as a function of the interaction parameters. The results derived in that framework allow us to understand the origin of the diversity of dynamical states observed in large networks of spiking neurons.

Figure 1

Phase diagram of the rate model, Eq. (1), for $D=0.1$. The states are: stationary uniform (SU), stationary bump (SB), oscillatory bump (OB), oscillatory uniform (OU), traveling waves (TW), standing waves (SW), lurching waves (LW), and aperiodic patterns (A). All solid lines have been determined analytically. Stability lines of other states (dotted lines) have been determined by numerical simulations. Regions of bistability are indicated by hyphens, e.g., OU-SW. Symbols refer to the patterns in Fig. 2. (For the case $D=0$ see 6, Fig. 13.7.)

Figure 2

Space-time plots of typical patterns of activity in the different regions of Fig. 1 shown over five units of time. Left-hand column from top to bottom: $J_0=-80$ and $J_1=15,5,-46,-86$ corresponds to OB, OU, SW, and A in Fig. 1. Right-hand column from top to bottom: $J_0=-10$ and $J_1=5,-38,-70,-80$ corresponding to SB, TW, SW, and A. $D=0.1$ and $I$ is varied to maintain the mean firing rate at 0.1. Dark regions indicate higher levels of activity in gray scale. Symbols refer to the location of the patterns in the phase-diagram, Fig. 1.

Figure 3

Typical firing patterns in a NSN. See text for details. Compare with Fig. 2. The network consists of two populations of 2000 neurons each. Left-hand column from top to bottom: a localized bump with oscillations, homogeneous oscillations, a period-doubled state of oscillating bumps, and a chaotic state. Parameters: $p_0^E=0.2$, 0, 0, 0, $p_0^I=0.2$, 0.5, 0.5, 0.5, $p_1^E=0.1$, 0, 0, 0, $p_1^I=0$, 0, 0.2, 0.5, $g_E=0.1$, 0, 0, 0, $g_I=0.28$, 0.1, 0.1, 0.1, ${\nu }_{\mathrm{ext}}=2000$, 15 000, 15 000, 15 000, and $g_{\mathrm{ext}}=0.01$. $\delta =0\mathrm{ms}$. Right-hand column from top to bottom: a steady and localized bump, the stationary uniform state, oscillating bumps and a chaotic state. Parameters: $p_0^E=0.2$, $p_0^I=0.2$, $p_1^E=0.2$, 0, 0, 0, (top to bottom) $p_1^I=0$, 0, 0.2, 0.2, $g_E=0.01$, 0.01, 0.01, 0.1, $g_I=0.028$, 0.028, 0.028, 0.28, ${\nu }_{\mathrm{ext}}=500$, 500, 5000, 500, and $g_{\mathrm{ext}}=0.01$, 0.01, 0.001, 0.01. $\delta =0$, 0, 0, 2.0 ms.

Figure 4

Bistability between uniform oscillations and an oscillating bump state in a purely inhibitory network. A 30 ms inhibitory pulse is applied to neurons 500 through 1500 around time 200. This induces a state in which two groups of neurons oscillate out of phase with one another; cf. voltage trace of neuron 1000 (middle) and neuron 1 (bottom). Parameters are $p_0^I=0.4$, $p_1^I=0.2$, $g_I=0.1$, ${\nu }_{\mathrm{ext}}=4500$, $g_{\mathrm{ext}}=0.033$, and $\delta =0.5\mathrm{ms}$.