Supplementary material for Fuchs et al. (2001) Proc. Natl. Acad. Sci. USA 98 (6), 3571–3576. (10.1073/pnas.051631898)

To better understand the motivation for the present study, it is helpful to consider the mathematical reasons why interneuron doublets can be useful for synchrony, as analyzed in a four-cell/two-site model by Ermentrout and Kopell [Ermentrout, G.B. & Kopell, N. (1998) Proc. Natl. Acad. Sci. USA 95, 1259–1264]. The system that these authors considered had two pyramidal cells (e-cells, e₁ and e₂), and two interneurons (i-cells, i₁ and i₂), and an axonal conduction delay between e₁/i₁ and e₂/i₂. A critical factor for between-site synchrony was the timing of a spike in i₁ induced by a spike in e₂ (respectively, i₂ and e₁).

The ideas of the explanation given below of normal synchronization and disruption of synchronization are illustrated in Fig.5, which gives details from representative simulations selected from Fig. 1, one simulation from each regime. Fig. 5a is from regime I, b is from regime II, and c is from regime III. The reader should note the relatively small differences in corresponding t values.

In the mathematical analysis of the smaller network, the timing depends on (i) the fixed conduction delay, and (ii) the relation between interneuron refractoriness and the lag to spike onset following an EPSP. The extra inhibition from the doublet spike slows down both e-cells and reverses the relative order of the leading and lagging cells (see Fig. 1b, trace of e-cells). This alone would not bring the cells closer together. Synchronization depends on the following property of the interneurons: synaptic excitation that arrives very soon after a spike may lead to another spike, but with a delay that decreases as the interval between previous spike and EPSP increases (i.e., there is relative refractoriness, a natural consequence of the fast after-hyperpolarization). Suppose e₂ leads on a given cycle. Because the excitatory inputs (local and distant) to i₂ are more temporally dispersed than those to i₁, there is less of a delay in i₂—between distant input and production of the doublet spike—than there is in i₁. Combined with the reversal of spike order in the e-cells, this differential delay creates a negative feedback that can stabilize synchrony within some parameter domain (of delays and of e-i conductance). With precise average synchrony (Fig. 5b, regime II), the e-cell signals ("population spikes") are not in exact register, for every single event—Fig. 5 shows a mismatch in timing—but the system appears to correct itself.

The mechanism above is robust enough to work in large networks, with heterogeneity in cellular and connectivity parameters, but it can be disrupted if the doublets themselves are disrupted. The above analysis shows how and why. The feedback stabilization relies on having doublets form at both ends of the network.

Consider first regime III (c). If, in a cycle with e₂ leading, there is a doublet/triplet mismatch such that i₂ has the extra spike, then the extra inhibition to e₂ creates a further delay for e₂, overcoming any feedback from e₁, and pushing e₁ and e₂ further apart in time. (See c, third cycle.) Furthermore, if e₂ starts ahead, there is more time between the firing of i₂ and the next excitatory input to i₂ than for i₁, making it more likely that i₂ will have an extra (locally induced) triplet spike on that cycle if the EPSPs onto i-cells are long enough. Thus, once the cells are separated, the dynamics work to separate them more. The cycles in c having interneuron triplets display the inappropriately long intervals in the respective local e-cell signals. We see analogous behavior in the biological network, when interneuron EPSPs become slightly prolonged. In regime I, on the other hand (a), if there is a singlet/doublet mismatch, then only one e-cell has a delaying signal, and there is no feedback loop for the circuit to perform the synchronization. This is seen in a, in which the time difference between the spikes of e₁ and e₂ appears to change erratically from cycle to cycle.

The above explanation makes predictions about firing patterns, especially in regime III. It predicts that there will be periods of time with alternation of triplet spikes in i₁ and i₂, because after a triplet in i₂ the leading e-cell is e₁, and this leads to a triplet in i₁. (See c, cycles 3-5.) Furthermore, it also predicts alternation of long and short intervals in each e-cell and a long interval in one e-cell when the other has a short one (c); this follows from the alternation of triplet spikes in the two i-cells, and lengthening effect of the triplets on the e-cell interspike interval. This pattern is what gives rise to the multiple peaks in the cross-correlation.