Ramana Dodla and Charles J. Wilson  PLoS One 8:e58922.

We study synchronization phenomenon of coupled neuronal oscillators using the theory of weakly coupled oscillators. The role of sudden jumps in the phase response curve profiles found in some experimental recordings and models on the ability of coupled neurons to exhibit synchronous and antisynchronous behavior is investigated, when the coupling between the neurons is electrical. The level of jumps in the phase response curve at either end, spike width and frequency of voltage time course of the coupled neurons are parameterized using piecewise linear functional forms, and the conditions for stable synchrony and stable antisynchrony in terms of those parameters are computed analytically. The role of the peak position of the phase response curve on phase-locking is also investigated.

Figure 1. Prevalence of sharp jumps in the PRCs in some neuronal models. (a) PRC of leaky integrate-and-fire neuron model: dV/dt= I_app - V where V is reset to 0 when it crosses a threshold level of 1 (in normalized units). (b) PRC of quadratic integrate-and-fire neuron model: τ dV/dt=I_app + V² where V is reset to V_r when it crosses a threshold level V_th. (c) PRC of adaptive exponential integrate-and-fire model: C_m dV/dt = I_app - G_L (V - E_L) + 2 G_Le^(V+50)/2 - w, τ_w dw/dt = a (V - E_L) - w such that V and w are reset, respectively, to V_r and w+b whenever V reaches a peak level V*. The parameters are C_m=0:1 nF, G_L=0:01 µS, V_r = -60 mV, V* = -20 mV, E_L = -70 mV, and τ_w=100 ms. The level of adaptation is controlled by a. (d) PRC of Wang-Buzsáki model that is described in the Model section. Models in (a, b, d) have no adaptation and their second and higher order PRCs are identical to that of the first order.

Figure 2. Piecewise linear models of PRC and voltage, and illustration of spike width effect. (a) Sample piecewise linear PRC profiles studied. (b) Voltage time course that consists of three piecewise linear profiles, modeled after the classic Hodgkin-Huxley model. (c) Growth function G(φ) computed for the three PRCs displayed in (a) and the voltage time course in (b). The sharp drops of the PRC at the edges altered the stability of synchrony. (d) Bifurcation diagram as a function of the normalized spike width at B=0:5 and B₂=0.25. In this and later figures, solid lines indicate stability and open circles instability of the phase-locked solutions. Synchrony is unstable at all frequencies, whereas antisynchrony is stable at high frequencies.

Figure 3. Edge effects when spike width is zero (a) Stable synchrony (shaded region) and the unstable synchrony (white region) for a₃/a₂=0:2234 in the plane of B₂/C and B/C. Boundary curves for two other levels of a₃/a₂ are also shown. (b) Same as in (a) but in the plane of B₂/C and a₃/a₂ for B/C=0. Boundary curves for two other level of B/C are also displayed. These curves are obtained by inverting equation ρ₁ for a₃/a₂. Antisynchrony is unstable in the displayed parameter ranges in (a) and (b). (c) Growth function G(φ), in the limit of zero spike width, displaying unstable synchrony but a stable phase-locked state that is very close to the synchronous state for a parameter value that is in the unstable synchrony region. (d) One-parameter bifurcation diagram as a function of B₂/C.

Figure 4. Edge effects when spike width is non-zero. (a) Stable synchrony (shaded) and stable antisynchrony (hatched) regions in B₂/C and B/C space for small spike width W/T=0:05 at a₃/a₂=0.2234. The white region holds other non-zero stable phase-locked solutions. (b) Same as in (a) but for large spike width W/T=0.3. (c) One-parameter bifurcation diagram as a function of B₂/C at small spike width and B/C=0.5, and W/T=0.05. The non-zero phase-locked states are found to be bistable with anti-synchronous state, and are not very close to the synchronous state as was the case for zero spike width. (d) Growth function at three levels of W/T when B/C=0.25 and B₂/C=0.5. When B₂ and B are chosen such that the system is in a synchronous state, increasing spike width eventually makes it unstable, but the stability is maintained until the spike width is large. (e) One-parameter bifurcation diagram as a function of spike width corresponding to the parameters in (d). The antisynchrony becomes stable here at W/T=0.13. Note that the stability boundaries and transitions are functions of the ratios W/T, a₃/a₂, and B/C. Thus, for example, the diagram in (a) is valid for any period and spike width as along as W/T=0.05.

Figure 5. Effect of skewness on the PRCs that have left side jump (a), and those that have right side jump (b). The left jump PRCs are parametrized by B/C and the right jump PRCs are parameterized by B₂/C. (a) A sample set of PRCs as the skewness A is moved from negative to positive levels is illustrated in the left column along with one-parameter bifurcation diagrams at two different levels of B/C as the skewness is increased (W/T=0.05). For positive jump synchrony is mostly unstable, and at large skewness even the antisynchrony is destabilized. But for negative jump skewness helped stabilize synchrony. On the right parameter planes depicting stability regions in the plane of skewness and jump are illustrated at different levels of W/T. (b) Same as in (a) but for right side jump PRCs (B/C=0). a₃/a₂=0.2234:

Figure 6. Effect on network behavior emerging from different PRC shapes. Modified Wang and Buzsáki model neurons are used to compute PRCs at two parameter sets (a) resulting in a PRC that is nearly symmetric, and another that has very steep rise near small phases. (b) Growth functions corresponding to the PRCs in (a). (c) Voltage time courses of two coupled model neurons corresponding to the parameter sets of the two PRCs in (a). (d) Growth function for a third parameter set resulting in a non-zero phase-locked state near synchrony. (e) Spike times (plotted as dots) of a network of 100 all-to-all coupled model neurons for the three parameter sets corresponding to the curves in (b,d). The non-zero phase-locked state near synchrony leads to prolonged transients, and a jitter in the spike times in the steady state. 1 : φ_n=2, I_app=0.17791 µA/cm². 2 : φ_n=9, I_app=0.17 µA/cm². 3 : φ_n=8.3, I_app=0.172536 µA/cm². φ_h is fixed at 5 for all the three sets.