Inhibitory loop robustly induces anticipated synchronization in neuronal microcircuits

We investigate the synchronization properties between two excitatory coupled neurons in the presence of an inhibitory loop mediated by an interneuron. Dynamic inhibition together with noise independently applied to each neuron provide phase diversity in the dynamics of the neuronal motif. We show that the interplay between the coupling strengths and the external noise controls the phase relations between the neurons in a counterintuitive way. For a master-slave configuration (unidirectional coupling) we find that the slave can anticipate the master, on average, if the slave is subject to the inhibitory feedback. In this nonusual regime, called anticipated synchronization (AS), the phase of the postsynaptic neuron is advanced with respect to that of the presynaptic neuron. We also show that the AS regime survives even in the presence of unbalanced bidirectional excitatory coupling. Moreover, for the symmetric mutually coupled situation, the neuron that is subject to the inhibitory loop leads in phase.

Figure 1

Neuronal motif: master-slave-interneuron (MSI). Each node represents a neuron described by the Hodgkin-Huxley model connected to other neurons by chemical synapses. The important parameters to determine the mean spike-timing difference $\tau$ between M and S are the synaptic conductances $g_{xy}$ (where $x,y=\mathrm{M}$,S,I indicate the presynaptic and the postsynaptic neurons, respectively). Each neuron is subject to an independent Poisson spike train.

Figure 2

Membrane potential of the neurons in the presence of noise. (a) Time series of the master neuron receiving both a constant external current $I_c=170$ pA and an excitatory input, obeying the Poisson statistic shown on top, mediated by chemical synapses with neurotransmitter concentration $\left[T\right]$. (b)–(d) Time traces of the slave neuron for different values of the inhibitory conductance $g_{\text{IS}}$. As we increase the inhibition, the order between M and S spikes changes from pre-post (MS) to post-pre (SM): (b) $g_{\text{IS}}=10$ nS (DS), (c) $g_{\text{IS}}=40$ nS (AS), (d) $g_{\text{IS}}=60$ nS (AS). (e) Time series of M and S neurons with lower frequencies, modeled by the modified Hodgkin-Huxley equations described in the Appendix with $g_{\text{MS}}=50$ nS and $g_{\text{IS}}=150$ nS. Note that to represent all data in the same scale, we have added +55 mV to the membrane potential in the modified model (see the Appendix for details).

Figure 3

Characterization of the delayed synchronization regime (DS) and the anticipated synchronization regime (AS). Left panels: $g_{\text{IS}}=10$ nS (DS, $\tau >0$), middle panels: $g_{\text{IS}}=40$ nS (AS, $\tau <0$), right panels: modified Hodgkin-Huxley model (AS, $\tau <0,g_{\text{MS}}=50$ nS and $g_{\text{IS}}=150$ nS). (a)–(c) The spike-timing differences (${\tau }_{n-1}$ and ${\tau }_n$) between a spike of the master neuron and a spike of the slave neuron in two consecutive periods. (d)–(f) ${\tau }_n$ as a function of the cycle number $n$. (g)–(i) Return map ${\tau }_n$ vs ${\tau }_{n-1}$.

Figure 4

Comparison of the effects of the inhibition in two different situations: when neurons are in the excitable regime ($I_c=170$ pA) and when neurons are tonically spiking ($I_c=280$ pA). (a) The spike-timing difference $\tau$ is a smooth function of the inhibitory conductance $g_{\text{IS}}$, which controls the transition from DS to AS (with $g_{\text{ext}}=2.0$ nS and $R=63$ Hz). Points are the mean $\tau$ and error bars are the standard deviation. The standard error of the mean is comparable to the size of the symbols. (b) and (c) $\tau$ as a function of the conductance of the external synapses $g_{\text{ext}}$ for different Poissonian rates $R$ and $g_{\text{IS}}=50$ nS. In the excitable regime, if $R$ or $g_{\text{ext}}$ is too small, the master does not fire a spike. The error bars are the standard error of the mean.

Figure 5

Phase diversity for a unidirectional excitatory synapse between master and slave. Histogram of the spike-timing difference between M and S in the uncoupled situation $[g_{\text{MS}}=0$, (a) and (c)] and in the MSI motif $[g_{\text{MS}}=10$ nS, (b) and (d)] for two different inhibitory conductances $g_{\text{IS}}=10$ nS [(a) and (b)] and $g_{\text{IS}}=40$ nS [(c) and (d)]. The dashed lines in (b) (DS) and in (d) (AS) indicate that the mean of these distributions is not zero. On the contrary, the distributions have an almost zero mean in panels (a) and (c).

Figure 6

Effect of the excitatory feedback from the slave to the master, $g_{\text{SM}}>0$. The presence of $g_{\text{SM}}$ facilitates the leadership of S, which is represented by negative $\tau$. Points are the mean $\tau$, and error bars are the standard deviation. Inset: the standard deviation $\sigma$ as a function of $g_{\text{SM}}$. In both plots, $g_{\text{MS}}=10$ nS.

Figure 7

Dependence of the spike-timing difference with the synaptic conductances. $\tau$ is color-coded by the right bar in the $(g_{\text{IS}},g_{\text{MS}})$ projection of parameter space for (a) $g_{\text{SM}}=0$ and (b) $g_{\text{SM}}=4$ nS. (c) $\tau$ in the $(g_{\text{SM}},g_{\text{MS}})$ projection for $g_{\text{IS}}=10$ nS. Dashed lines at $g_{\text{MS}}=10$ nS in (a) and (c) correspond, respectively, to the curve for $I_c=170$ pA in Fig. 4 and to the black curve in Fig. 6.