Collective dynamics in the presence of finite-width pulses

Abstract

The idealisation of neuronal pulses as $δ$-spikes is a convenient approach in neuroscience but can sometimes lead to erroneous conclusions. We investigate the effect of a finite pulse-width on the dynamics of balanced neuronal networks. In particular, we study two populations of identical excitatory and inhibitory neurons in a random network of phase oscillators coupled through exponential pulses with different widths. We consider three coupling functions, inspired by leaky integrate-and-fire neurons with delay and type-I phase-response curves. By exploring the role of the pulse-widths for different coupling strengths we find a robust collective irregular dynamics, which collapses onto a fully synchronous regime if the inhibitory pulses are sufficiently wider than the excitatory ones. The transition to synchrony is accompanied by hysteretic phenomena (i.e. the co-existence of collective irregular and synchronous dynamics). Our numerical results are supported by a detailed scaling and stability analysis of the fully synchronous solution. A conjectured first-order phase transition emerging for $δ$-spikes is smoothed out for finite-width pulses.

Figures (12)

• Figure 1: Example of the phase response curves (PRCs): PRC1 with $\Phi_L=-0.1$ and $\Phi_U=0.9$ (black line), PRC2 (red dashed line), PRC3 (blue dashed and dot line). The vertical dot line refers to the reset membrane potential ($\Phi_{r} = 0$).
• Figure 2: Characterization of the global network dynamics with interactions through $\delta$-pulses. Mean firing rate $\nu$, mean coefficient of variations $\langle{C}_v\rangle$, and order parameter $\chi$ are plotted vs. the coupling strength $\mu$ in panels (a), (b) and (c), respectively. Black triangles, red circles, green crosses, and blue diamonds correspond to $N=10000$, 20000, 40000, and 80000, respectively, all obtained with PRC1. Orange stars and green squares correspond to $N=10000$ and 40000 obtained with PRC3. The vertical dashed line represents the critical coupling $\mu_c = 0.537$.
• Figure 3: CID properties for PRC1, $\mu=0.95$, $\alpha = \beta=100$, and $N=10000$. Panel a): time series of the mean field $\langle \Phi \rangle$. Panel b): raster plot of spiking times $t_n$ for 100 oscillators out of $N=10000$. Panel c): instantaneous probability distribution of the phases P$(\Phi)$ at two different time points $t=8363$ (red), and $t=8374$ (green). The probability distributions are normalized such that the area underneath is 1.
• Figure 4: Global network dynamics in the presence of identical finite pulse-width and PRC1. Panel a): order parameter $\chi$ vs. $\mu$ for $N=10000$ and $\alpha=1000$ (green triangles), $\alpha=100$ (blue crosses), $\alpha=10$ (orange squares), and $\alpha=1$ (red stars). The black dashed curve corresponds to the asymptotic results obtained for $\delta$ pulses (see Fig. \ref{['fig:fig2']} (c), $N=80000$) with the critical value $\mu_c$ derived therein. Panel b): order parameter $\chi$ vs. $\alpha$ for $N=10000$ and $\mu=0.2$ (black triangles), $\mu=0.47$ (red pluses), and $\mu=0.95$ (blue diamonds). In both panels the magenta circles show the results for $N=40000$ to compare with the blue curves, respectively. The arrows highlight the parameter set for which we show in Fig. \ref{['fig:fig3']} typical CID time series.
• Figure 5: Characterization of the global network dynamics for nonidentical finite pulse-width, obtained with $\alpha=100$ and PRC1. Each column refers to different coupling strengths: $\mu=0.3$ (A), $\mu=0.47$ (B), and $\mu=0.95$ (C). Rows: mean firing rate $\nu$, mean coefficient of variations $\langle{C}_v\rangle$, and order parameter $\chi$ versus $\beta$. Colours and symbols define network sizes $N$: 10000 (black triangles), 20000 (red crosses), 40000 (orange circles), and 80000 (blue stars). Each data point is based on a time series generated over 10000 time units and sampled every 1000 steps after the transient has sorted out.