Sequential Desynchronization in Networks of Spiking Neurons with Partial Reset

The response of a neuron to synaptic input strongly depends on whether or not the neuron has just emitted a spike. We propose a neuron model that after spike emission exhibits a partial response to residual input charges and study its collective network dynamics analytically. We uncover a desynchronization mechanism that causes a sequential desynchronization transition: In globally coupled neurons an increase in the strength of the partial response induces a sequence of bifurcations from states with large clusters of synchronously firing neurons, through states with smaller clusters to completely asynchronous spiking. We briefly discuss key consequences of this mechanism for more general networks of biophysical neurons.

Figure 1

Sequential desynchronization transition in a network of $N=50$ neurons ($U=U_b$, $b=-3$, $\tilde{\epsilon }=0.0175$). The phases ${\varphi }_i$ of all neurons are plotted against the $s$th spike of a reference neuron. Starting from a synchronous state and perturbing at $s=5$, the phase dynamics are shown for (a) $c_1=0.025<c_{\mathrm{cr}}^{(N)}$ (inset: magnification), (b) $c_2=0.5\in (c_{\mathrm{cr}}^{(N)},c_{\mathrm{cr}}^{(2)})$, and (c) $c_3=0.7>c_{\mathrm{cr}}^{(2)}$. (d) Probability $\mathbb{P}(a)$ of observed cluster sizes $a$ in the asymptotic dynamics of 1500 simulations for each $c\in \{0,0.0125,\dots ,1\}$ starting from random phases uniformly distributed in $[0,1)$. Solid red line: Exact theoretical prediction (10) above which clusters are unstable.

Figure 2

Coacting synchronizing and desynchronizing mechanisms underly the desynchronization transition. The phase-potential relations are shown as solid black curves. The phase differences before (${\Delta }^{-}$) and after spike reception (${\Delta }^{+}$) are indicated. (a) Synchronization due to subthreshold inputs coacting with either (b) synchronization for strongly refractory partial resets ($c\approx 0$) or (c) desynchronization for sufficiently conservative partial resets ($c\approx 1$) determine the invariance of clusters under the return map (6).