Diverse Population-Bursting Modes of Adapting Spiking Neurons

We study the dynamics of a noisy network of spiking neurons with spike-frequency adaptation (SFA), using a mean-field approach, in terms of a two-dimensional Fokker-Planck equation for the membrane potential of the neurons and the calcium concentration gating SFA. The long time scales of SFA allow us to use an adiabatic approximation and to describe the network as an effective nonlinear two-dimensional system. The phase diagram is computed for varying levels of SFA and synaptic coupling. Two different population-bursting regimes emerge, depending on the level of SFA in networks with noisy emission rate, due to the finite number of neurons.

Figure 1

Left panel: sample orbit from Eq. (2) (gray solid line) and from simulations (black solid line), and nullclines of Eq. (2) (dashed lines). Right panel: the ${\tau }_{\nu }$ characteristic time appearing in Eq. (2) is plotted vs $\nu$ for three values of $c$ (vertical sections of the phase plane on the left).

Figure 2

Phase diagram of the reduced system of Eq. (2) in the $({\Phi }^{\prime },g{\tau }_c)$ plane (central panel). Shaded regions and bifurcation lines are described in the text. Lateral panels (a)–(e) illustrate typical time courses of $\nu (t)$ from simulations of $N$ interacting VIF neurons: (a) GO; (b) LAS and GO; (c) LAS close to the LAS and GO region; (d) LAS and HAS; (e) VAS and HAS (dashed line: unstable fixed point at 5 Hz; dotted line: time of a brief external stimulation). $N={10}^3$ unless for (a) $N=5\times {10}^5$ and (b) $N=25\times {10}^3$.

Figure 3

IBI (left panels) and burst duration (right panels) distributions from simulations of ${10}^3$ interacting VIF neurons for low (bottom panels) and high (top panels) level of SFA, at different synaptic coupling strength: from thin to tick lines ${\Phi }^{\prime }=\{0.95,1.015,1.06\}$ (top); ${\Phi }^{\prime }=\{0.96,0.99,1.015\}$ (bottom). In all cases ${\tau }_c=0.25\mathrm{s}$.