Loss of phase-locking in non-weakly coupled inhibitory networks of type-I model neurons

Abstract

Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris–Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (J Comput Nerosci, 2008) for a network of Wang-Buzsáki model neurons. Although alternating-order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris–Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such “leap-frog” dynamics. In the Morris–Lecar model network, the alternation in the firing order arises under the condition of fast closing of K⁺ channels at hyperpolarized potentials, which leads to slow dynamics of membrane potential upon synaptic inhibition, allowing the presynaptic cell to advance past the postsynaptic cell in each cycle of the oscillation. Further, we show that non-zero synaptic decay time is crucial for the existence of leap-frog firing in networks of phase oscillators. However, we demonstrate that leap-frog spiking can also be obtained in pulse-coupled inhibitory networks of one-dimensional oscillators with a multi-branched phase domain, for instance in a network of quadratic integrate-and-fire model cells. Finally, for the case of a homogeneous network, we establish quantitative conditions on the phase resetting properties of each cell necessary for stable alternating-order spiking, complementing the analysis of Goel and Ermentrout (Physica D 163:191–216, 2002) of the order-preserving phase transition map.

Appendix

1.1 Derivation of the alternating-order phase map with second-order phase resetting

We will use the diagram in Fig. 7(a) to derive the map in the case of non-negligible second-order phase resetting, Δ₂(ϕ). Let {ϕ _n , ξ _n } denote the two phases of the postsynaptic cell at the time of arrival of each of the two spikes in n-th period of the oscillation. In the case of zero second-order resetting, Fig. 7 illustrates the relationship between these phases, ξ _n  = 1 + ϕ _n  − Δ(ϕ _n ). However, due to non-zero second-order phase resetting received by the presynaptic cell in the preceding cycle, Δ₂(ξ _n − 1) (where ξ _n − 1 is its phase at the time of arrival of the first black spike in Fig. 7(a)), the interval between two spikes of the presynaptic cell in the current cycle, denoted γ _n , will not be equal to 1:

$$ \gamma_n = 1 + \Delta_2(\xi_{n-1}) > 1 \label{eqA1} $$

(16)

Therefore, the modified relationship between ξ _n and ϕ _n reads

$$ \xi_n = \gamma_n + \phi_n - \Delta(\phi_n) \label{eqA2} $$

(17)

Note that we neglect the much smaller second-order phase-resetting due to the first spike of the presynaptic cell in each period of the 2:2 mode: Δ₂(ϕ _n ) < < Δ₂(ξ _n ). Finally, given the phase ξ _n of the postsynaptic cell right before receiving its second input, one can easily find its first passage time, ϕ _n + 1 (i.e. interval ϕ ₂ in Fig. 7(a)), using the first passage time condition

$$ \xi_n - \Delta(\xi_n) + \phi_{n+1} = 1 \label{eqA3} $$

(18)

Solving this system of equations for ξ _n yields the map

$$ \xi_{n+1} \!=\! 2 \!-\! \xi_n \!+\! \Delta(\xi_n) \!-\! \Delta(1 \!-\! \xi_n \!+\! \Delta(\xi_n)) \!+\! \Delta_2(\xi_n) \label{eqA4} $$

(19)

which can be re-written in a more compact form as

$$ \xi_{n+1} = 1 + \phi_{n+1} - \Delta(\phi_{n+1}) + \Delta_2(\xi_n) \label{eqA5} $$

(20)

If we substitute the conditions for synchronous firing, ξ _n  = 1, ϕ _n  = 0, we obtain Δ₂(1) = Δ(0), which is the correct periodicity condition relating the first- and second-order STRC curves. Therefore, the synchronous solution is always a fixed point of Eq. (19).

Differentiating Eq. (20) yields the stability condition

$$ | [1 - \Delta'(\phi) ] [\Delta'(\xi) - 1] + \Delta_2'(\xi)| < 1 \label{stacondD2} $$

(21)

which agrees with Eq. (11) when Δ₂(·)=0. Close to the bifurcation from synchrony to leap-frog spiking, ξ ≈ 1, Δ′(ξ) ≈ 0, and therefore

$$ | \Delta'(\phi) + \Delta_2'(\xi) - 1 | < 1 $$

which yields

$$ 0 < \Delta'(\phi) + \Delta_2'(\xi) < 2 \label{stacondB} $$

(22)

Recall that ϕ = 1 − ξ + Δ(ξ) (Eq. (18)). A more general stability condition for the case of non-negligible Δ₂(ϕ _n ) is given by Maran and Canavier (2008).