Ramana Dodla and Charles J. Wilson  Biol. Cybern.107:367-383.

The stability of phase-locked states of electrically coupled type-1 phase response curve neurons is studied using piecewise linear formulations for their voltage profile and phase response curves. We find that at low frequency and/or small spike width, synchrony is stable, and antisynchrony unstable. At high frequency and/or large spike width, these phase-locked states switch their stability. Increasing the ratio of spike width to spike height causes the antisynchronous state to transition into a stable synchronous state. We compute the interaction function and the boundaries of stability of both these phase-locked states, and present analytical expressions for them.We also study the effect of phase response curve skewness on the boundaries of synchrony and antisynchrony.

Figure 1. a Profile of spike time course with finite width, and b the PRC shape considered. c Boundary curves of φ delineating the components that contribute to H(φ). Different regimes along W/T are delineated by intersections of these curves. The first regime 0 ≤ W’ < 1/9 is marked by thick curves, and the corresponding interaction function is in Eq. 17. Interaction functions up to W’ = 1/6 are computed in the Appendix. d Interaction function as the parameter W/T is increased (using formulas in Eqs. 17, 18, and 19). e The growth functions corresponding to the curves in d. Synchrony is stable and antisynchrony is unstable at small spike width or frequency. As the spike width or frequency is increased, an exchange of stability takes places between these two states. The nature of this exchange is shown numerically in Fig. 2e. Further increase of W/T produces no change in the stability of the synchronous and antisynchronous states. In this and later figures, when unspecified, the voltage shape parameters are of the HH model discussed in Sect. 2

Figure 2.  The contributions of various segments of V(t) and Z(t) toward stability of synchrony (a) and antisynchrony (b) are visually represented. The arrangement of V and Z satisfies the condition 0 ≤ W/T < 1/4. The shaded regions contribute to destabilizing the corresponding phase-locked state. In the limit of W becoming zero, λ_1a and not γ_3a becomes zero, and hence synchrony is stable, and antisynchrony unstable. When W is large such that 1/4 ≤ W’ < 2/5, the corresponding eigenvalue components are represented by λ_1b, . . ., and γ_1b, .… c, d The eigenvalue components and the total eigenvalue for synchronous (c) and antisynchronous (d) states as W’ is increased at a₃/a₂ = 0.2234, T = 14.636 (HH model parameters as discussed in Sect. 2). λ_1a is zero at W = 0, but grows to become largest component with W’, but γ_3a is the dominant component for small W’, and is overpowered by the combined effect of other components at large W’. e A numerically found bifurcation diagram for the parameters of c and d as a function of W’ confirming the prediction of stability switches of synchrony and antisynchrony as the product of spike width and frequency increases. Other stable phase-locked solutions—that are not studied here—also emerged near the transition point. f Stability diagram of synchronous and antisynchronous states computed analytically in Eqs. 11 and 12 and defined by ρ₁ and ρ₂. Numerically computed boundary using the canonical type-1 PRC shape and the PWL voltage time course is shown by the dashed curve. Note that the ratio a₃/a₂ is related to the relative spike amplitude 1 − a₃/a₂

Fig. 3 Effect of PRC skewness on phase-locking. (a) PRCs with skewness A/T = −1 (i), −0.5 (ii), 0 (iii), 0.5 (iv), and 1 (v) are illustrated. (b) The interaction functions corresponding to the five PRCs illustrated in (a) in the absence of spike width (W/T = 0). (c) The growth functions for the curves in (b) indicating stable synchrony and unstable antisynchrony for A/T between −1 and 1. (d) Effect of spike width (W/T = 0.2) is shown on the growth functions at different levels of skewness. The non-zero spike width caused the symmetric and negative skewed PRCs impart instability to synchrony, but large positive skewness countered it. (e) One-parameter bifurcation diagram as a function of skewness, showing stable antisynchrony for left leaning or nearly symmetric PRCs, and stable antisynchrony for right leaning PRCs. For the value of spike width (W/T = 0.2) used, these states are replaced by non-zero phase-locked states for a small range of positive but intermediate skewness. (f,g) Stability diagrams in the plane of normalized skewness and normalized spike width depicting stable synchrony (shaded) and stable antisynchrony (hatched) regions at two spike heights. The white space holds non-zero phase-locked states. For d-f the spike amplitude parameters are chosen from time course of HH model with an applied current of 10 μA/cm² that results in a₃/a₂ = 0.2234. However, only the slopes and not the actual values of the voltage segments affect the stability.