Zero-lag synchronization despite inhomogeneities in a relay system

Abstract

A novel proposal for the zero-lag synchronization of the delayed coupled neurons, is to connect them indirectly via a third relay neuron. In this study, we develop a Poincaré map to investigate the robustness of the synchrony in such a relay system against inhomogeneity in the neurons and synaptic parameters. We show that when the inhomogeneity does not violate the symmetry of the system, synchrony is maintained and in some cases inhomogeneity enhances synchrony. On the other hand if the inhomogeneity breaks the symmetry of the system, zero lag synchrony can not be preserved. In this case we give analytical results for the phase lag of the spiking of the neurons in the stable state.

Figure 1. Time evolution of coupled phase oscillators.

(a) Time evolution of two bi-directionally coupled phase oscillators for . (b) Time evolution of two bi-directionally coupled phase oscillators for . is the period of the oscillators in the phase-locked state and is the delay time from pre-synaptic neuron to post-synaptic neuron . () is the phase difference (modulo ) of two neurons at the spiking time of first (second) neuron.

Figure 2. Homogeneous and inhomogeneous system of directly coupled neurons.

(a) Synchronized states of two directly coupled identical type-II neurons. Depending on the delay time inphase or antiphase modes are stable with negative corresponding Lyapunov exponent which are shown for both the states (left panel). Phase difference () of firing of two identical Hodgkin-Huxely neurons is plotted in the right panel. (b) Synchronized states of two directly coupled neurons in presence of frequency mismatch. Numerical solutions (Cyan) affirm the validity of the analytical results obtained from linear approximation (Dark Blue). Lyapunov exponent for each of the locked states are also shown. Shaded area show the regions where no 1∶1 locking mode is seen in numerical results, and the dashed gray lines are the boundary of stability of analytic solution with negative Lyapunov exponents. In the right panel the phase difference is shown for two HH neurons with different firing rates.

Figure 3. Directly coupled neurons with unequal delays.

(a) Phase lag of two directly coupled neurons for different transmission time delays. In (b) we have shown the results for (along diagonal in (a)) and with fixed (along horizontal lines in (a)). Synchrony can only been seen for homogeneous system . Figures in right column present the results for HH neurons. Even though the patterns are not the same, the main result still holds and synchrony can not be seen with unequal delays. in the bottom-right panel is the period of the firing of HH neurons in the locked state.

Figure 4. Phase locked state of two directly coupled neurons in presence of inhomogeneity in the coupling strengths.

The results are shown for two different values of in which homogeneous network shows different properties (inphase and antiphase) in the homogenous case. The right panel show the similar result for HH neurons. Here T stands for oscillation period of HH neurons and .

Figure 5. Homogeneous relay system.

In a homogeneous relay system, synchronized state of outer neurons is not stable for all the values of the delay time. Dark blue and cyan points show the phase difference of outer neurons in stable regions resulted from analytic calculations and direct numerical integration, respectively. Dark pink (pink) points show analytical (numerical) results for the phase difference of the outer neurons with the relay neuron. Yellow and orange lines show the characteristic exponent of the map which is negative in case of stability of synchrony. Vertical dashed lines show the boundary of stability domains (characterized by negative exponents) and shaded area indicates the domain over which numerical integration shows no synchrony. In the right panel numerical results are presented for the relay system with Hodgkin-Huxley neurons. It is notable that although the same pattern of synchronization regions is seen, but the domain over which the synchrony is unstable is quite narrower for the relay system with HH neurons.

Figure 6. Symmetric relay system with inhomogeneous firing rates.

Synchronization of the outer neurons when the relay neuron has different firing rate. In the right figure the analytic result for the borders of stable synchrony for phase oscillators is shown with blue thick lines and the numerical results for the phase lag are presented by the color code. Vertical and horizontal axis show the relative difference of the firing rate of the relay neuron (with outer neurons) and the delay time, respectively. Zero lag synchrony is coded by blue. In the right panel the numeric results are shown for HH neurons. In this case inhomogeneity is applied by changing the input current (which controls the inter-spike-interval of the neuron).

Figure 7. Asymmetric relay system with inhomogeneous firing rates.

When frequency mismatch is applied to one of the outer neurons, no synchronized state is seen, but for some range of mismatch 1∶1 phase-locked states occur. (a) and (b) show the phase lag and ratio of the periods of outer neurons in the steady state, respectively. Blue lines are borders of 1∶1 modes from analytic stability test. In the right, same results are shown for HH neurons.

Figure 8. Symmetric variation of synaptic constants.

(a) Synaptic constants of incoming links to the relay neuron are changed while outgoing ones are kept constant (). Color code shows time lag of spiking of outer neurons, resulted from numerical experiments. Solid lines are drawn based on the analytic results, showing the domain of stability of synchronous state. Note that for synchrony is seen for almost all the values of delay time. For comparison the results for the homogeneous system are shown in (b) where all the synaptic constants are equal. In this case except for very small synaptic constants, the results are insensitive to the changes of synaptic strengths and can not results a synchronous state. Right panel show the similar results for HH neurons.

Figure 9. Asymmetric variation of synaptic constants.

The time lag of firing of the neurons is shown as a function of delay time and ratio of synaptic constants in an asymmetrix case. Numerical results confirm outcome of analytic calculations for the phase lag of phase oscillators of type-II (left panel). Right panel shows the results for HH neurons. Again zero-lag synchrony is seen for and .

Figure 10. Symmetric variation of the delay times.

In the left panel the phase lag of outer neurons in a relay system with canonical type-II oscillators. Horizontal and vertical axes show the incoming and outgoing delay times, respectively. Solid lines show the boundary of stable synchrony. The similar results are shown for HH neurons in the right.

Figure 11. Asymmetric variation of delays.

The time lag of firing of the neurons is shown as a function of difference in delay times. Numerical results confirm outcome of analytic calculations for the phase lag. Lyapunov exponents are also shown. They must be both negative for a stable phase-locked mode. In the right figure similar results are shown for HH neurons.