Inhibitory autapse mediates anticipated synchronization between coupled neurons

Two identical autonomous dynamical systems unidirectionally coupled in a sender-receiver configuration can exhibit anticipated synchronization (AS) if the receiver neuron also receives a delayed negative self-feedback. Recently, AS was shown to occur in a three-neuron motif with standard chemical synapses where the delayed inhibition was provided by an interneuron. Here, we show that a two-neuron model in the presence of an inhibitory autapse, which is a massive self-innervation present in the cortical architecture, may present AS. The GABAergic autapse regulates the internal dynamics of the receiver neuron and acts as the negative delayed self-feedback required by dynamical systems in order to exhibit AS. In this biologically plausible scenario, a smooth transition from the usual delayed synchronization (DS) to AS typically occurs when the inhibitory conductance is increased. The phenomenon is shown to be robust when model parameters are varied within a physiological range. For extremely large values of the inhibitory autapse the system undergoes to a phase-drift regime in which the receiver is faster than the sender. Furthermore, we show that the inhibitory autapse promotes a faster internal dynamics of the free-running Receiver when the two neurons are uncoupled, which could be the mechanism underlying anticipated synchronization and the DS-AS transition.

Figure 1

Neuronal motif: Sender ($S$) and Receiver ($R$) neurons unidirectionally connected by an excitatory chemical synapse. The receiver neuron presents an inhibitory chemical autapse which is a dynamical version of the negative delayed self-feedback in Eq. (1).

Figure 2

Characterizing the phase-locking and phase-drift regimes. (a) Time series of the sender and the receiver neuron after a transient time for different values of the autaptic inhibition $g_I$ and fixed external current $I=10$ pA. (b) The spike-timing difference ${\tau }_i$ between $S$ and $R$ as a function of the $i\mathrm{th}$ cycle. A pre-post firing order characterizes the usual delayed (DS) synchronization regime (upper panels, $g_I=0.15$ nS). The anticipated synchronization regime (AS) occurs when the receiver neuron fires before the sender (middle panels, $g_I=1.0$ nS). When neurons fire with different frequencies, the spike timing difference changes every cycle and the neurons are in a phase-drift (PD) regime (bottom panels, $g_I=2.0$ nS).

Figure 3

Transition from delayed to anticipated synchronization regime. Spiking time difference normalized by the period of the phase-locking $\tau /T$ as a function of inhibitory conductance $g_I$ for different values of external current $I$. Positive values of $\tau$ characterizes DS, whereas $\tau <0$ represents the AS regime. The stars mark the end of the phase-locking regime and the beginning of the phase-drift (PD). In PD the value of ${\tau }_i$ in each cycle does not converge to a fixed value $\tau$ as shown in Fig. 2.

Figure 4

Normalized spike-timing difference $\tau /T$ (right bar) in the ($g_I,g_E$) projection of parameter space for different values of external current $I$: DS (blue, mostly at the upper part in which $g_I>g_E$ and $\tau /T<0$), AS (red, middle), and PD (white, meaning that no stationary value of $\tau$ was found).

Figure 5

Characterizing the phase-drift regime induced by the inhibitory autapse: the receiver is faster than the sender. (a) The mean period of the receiver normalized by its own period when $g_I=0, \overline{T_R}/T_0$ coincides with the mean period of the sender for DS and AS regimes, but it is smaller for PD. (b) In PD, the return map of the period in each cycle of the receiver is consistent with a quasi-periodic system.

Figure 6

Inhibitory autapse promotes a faster internal dynamical of the free-running receiver, which could be the mechanism promoting AS and the transition from DS to AS. For the uncoupled situation ($g_E=0$) the period of the free-running receiver (normalized by its own period for $g_{I}0, T_R/T_0$) decreases as we increase the inhibitory conductance $g_I$. For external currents $I=5,6,7$ pA the minimum value of $T/T_o$ can also be related to the minimum value of $\tau$ in Fig. 3. For $I\ge 8$ pA the system undergoes a transition from AS to PD regime before $T_R/T_0$ reaches the minima.