Asynchronous Response of Coupled Pacemaker Neurons

We study a network model of two conductance-based pacemaker neurons of differing natural frequency, coupled with either mutual excitation or inhibition, and receiving shared random inhibitory synaptic input. The networks may phase lock spike to spike for strong mutual coupling. But the shared input can desynchronize the locked spike pairs by selectively eliminating the lagging spike or modulating its timing with respect to the leading spike depending on their separation time window. Such loss of synchrony is also found in a large network of sparsely coupled heterogeneous spiking neurons receiving shared input.

Figure 1

Emergence of asynchrony for shared inhibition. (a) Coupled STN network. ${\omega }_1=2.7\mathrm{Hz}$ ($I_1=0$) and ${\omega }_2=9.3\mathrm{Hz}$ ($I_2=8\mathrm{pA}/\mu {\mathrm{m}}^2$). $G_{\mathrm{inh}}=1\mathrm{nS}/\mu {\mathrm{m}}^2$. $g_{\mathrm{exc}}^{*}=0.4\mathrm{nS}/\mu {\mathrm{m}}^2$. The top inset shows brief time courses at $g_{\mathrm{exc}}=0.5\mathrm{nS}/\mu {\mathrm{m}}^2$. For the same conductance parameters, the effect of frequency heterogeneity is shown in the bottom inset. (b) Frequency dependence of $A$ for the three networks: STN ($G_{\mathrm{inh}}=1\mathrm{nS}/\mu {\mathrm{m}}^2$, and three different values of $g_{\mathrm{exc}}$), $N+K$ ($I_1=0$, $I_2=8\mathrm{pA}/\mu {\mathrm{m}}^2$, $g_{\mathrm{exc}}=0.62\mathrm{nS}/\mu {\mathrm{m}}^2$, $G_{\mathrm{inh}}=1\mathrm{nS}/\mu {\mathrm{m}}^2$), and HH ($I_1=0$, $I_2=8\mu \mathrm{A}/{\mathrm{cm}}^2$, $g_{\mathrm{exc}}=0.2\mathrm{mS}/{\mathrm{cm}}^2$, $G_{\mathrm{inh}}=1\mathrm{mS}/{\mathrm{cm}}^2$). $p_1,\dots ,p_4$ mark input frequencies at which $A$ changed its character in the STN network due to $T$-current activation, and are described in the text (${\tau }_{\mathrm{inh}}=1\mathrm{ms}$).

Figure 2

(a) Time window of spike separation in $1:1$ phase-locked state in the absence of extrinsic input for the coupled STN, $N+K$, and HH excitatory neuronal pairs as a function of frequency heterogeneity ($I_1$ is fixed at $4\mathrm{pA}/\mu {\mathrm{m}}^2$ for STN and $N+K$, and at $4\mu \mathrm{A}/{\mathrm{cm}}^2$ for the HH). The inset shows scenarios of input events with respect to spike pair separation discussed in the text. (b) Excitatory STN network. Predicted and computed time windows (see text). The top inset shows the mechanism by which oncoming inhibition suppressed a lagging spike. The bottom inset shows the resultant interspike interval histogram densities. (${\tau }_{\mathrm{inh}}=1\mathrm{ms}$; $g_{\mathrm{exc}}=0.5\mathrm{nS}/\mu {\mathrm{m}}^2$.)

Figure 3

(a) Dependence of $A$ on ${\tau }_{\mathrm{inh}}$. For STN network ${\lambda }_{\mathrm{inh}}=100\mathrm{Hz}$, $g_{\mathrm{exc}}=0.5\mathrm{nS}/\mu {\mathrm{m}}^2$. For HH network $G_{\mathrm{inh}}=0.5\mathrm{mS}/{\mathrm{cm}}^2$ and $g_{\mathrm{exc}}=0.1\mathrm{mS}/{\mathrm{cm}}^2$. $I_1$ and $I_2$ are as in Fig. 1b. $1/\beta =3\mathrm{ms}$ for both the networks. (b) Mutually inhibitory network of two coupled HH neurons ($I_1=0$, $I_2=8\mu \mathrm{A}/{\mathrm{cm}}^2$) displaying spike time asynchrony for inhibitory input. At ${\lambda }_{\mathrm{inh}}=0$, $t_w=7.8$ for #1 and 9.6 ms for #2. Voltage traces transitioning from phase-locked (${\lambda }_{\mathrm{inh}}=0$) to asynchronous state for finite input rate are illustrated in the top inset. The phase response curves corresponding to the two phase-locked neurons are illustrated in the left inset, bottom. The right inset shows the densities of spike pair separation in the asynchronous state.

Figure 4

Emergence of asynchrony from coherent states in networks of $N=10$ excitatory all-to-all coupled STN (a) and sparsely coupled inhibitory HH (b) neurons measured by average spike rate as a function of mutual coupling without and with shared inhibition. (a) $I_j=(j-1)10/9\mathrm{pA}/\mu {\mathrm{m}}^2$, $j=1,\dots ,N$, and $G_{\mathrm{inh}}=0.5\mathrm{nS}/\mu {\mathrm{m}}^2$. (b) Inset shows $M_{ij}$ (with $n=4$); a filled square represents a synaptic connection. $I_j=5+(j-1)8/9\mu \mathrm{A}/{\mathrm{cm}}^2$, $j=1,\dots ,N$, $G_{\mathrm{inh}}=0.1\mathrm{mS}/{\mathrm{cm}}^2$.