Fine structure of neural spiking and synchronization in the presence of conduction delays (hippocampusyg-rhythmsydoublets)

Hippocampal networks of excitatory and in- hibitory neurons that produce g-frequency rhythms display behavior in which the inhibitory cells produce spike doublets when there is strong stimulation at separated sites. It has been suggested that the doublets play a key role in the ability to synchronize over a distance. Here we analyze the mechanisms by which timing in the spike doublet can affect the synchro- nization process. The analysis describes two independent effects: one comes from the timing of excitation from sepa- rated local circuits to an inhibitory cell, and the other comes from the timing of inhibition from separated local circuits to an excitatory cell. We show that a network with both of these effects has different synchronization properties than a net- work with either excitatory or inhibitory type of coupling alone, and we give a rationale for the shorter space scales associated with inhibitory interactions. When neurons communicate over some distance, there are conduction delays between the firing of the presynaptic neuron and the receipt of the signal at the postsynaptic cell. It is also known that cells can synchronize over distances of at least several millimeters, over which conduction delays can be significant. This raises the question of how cells can synchro- nize in spite of the delays. Traub et al. (1) and Whittington et al. (2) suggested that the fine structure of the spiking of some of the cells may play a part in the synchronization process for the g frequency rhythm, found in hippocampal and neocortical systems during states of sensory stimulation. (For references, see ref. 2.) More specifically, for some models of cortical structure, they noted that the ability to synchronize in the presence of delays is correlated with the appearance of spike doublets in the inhibitory cells. The doublets appear in slice preparations when there is strong stimulation at separated sites (1, 2). In this paper, we analyze a mechanism for such synchronization, using a simplified version of equations of Traub and colleagues. The timing of spikes within a doublet is shown to encode information about phases of local circuits in a previous cycle; the model shows how the circuit can use this information in an automatic way to bring nonsynchronous local circuits closer to synchrony. There are two independent effects in the model. The first is the response of the inhibitory (I) cells to excitation from more than one local circuit. The I-cells may produce more than one spike, whose relative timing depends on strength of excitation and recovery properties of the cell after the firing of a first spike; the latter can include effects of after- hyperpolarization or self-inhibition in a local circuit. The second effect is the response of the excitatory (E) cells to the multiple inhibitory spikes they receive from within their local circuit or other circuits. The maximal inhibition received by an E-cell can depend on the times and sizes of the inhibitory postsynaptic potentials it receives, and this affects the time until the E-cell can spike again. We show that each of the two effects is enough to allow synchronization. Together, they give the network synchronization properties that are not intuitively clear from the properties of either alone. Previous papers have analyzed mechanisms for synchroni- zation depending on interactions among I-cells (3-6) or E-cells (5-10). In this paper, the interactions between the local circuits include E3 I and I3 E. We omit the E3 E connections, which are sparse in the CA1 region of the hippocampus (11), and consider only those I3 I connections that are sufficiently local to be considered part of a local circuit. By considering networks with a subset of these connections, we shed light on the role of each of them in the synchronization process. In particular, we show that the different kinds of coupling work together to provide synchrony over a larger range of delays than either could do alone, and that the interaction provides a significant increase in the speed of synchronization. The I3 E coupling also helps provide robustness to disruption from larger excitatory conductances, but it reduces robustness to heterogeneity. The two effects together give a rationale for the shorter space scales of the inhibitory interactions. (See Discus- sion.) Our analysis considers a pair of local circuits, each having one E-cell and one I-cell; each cell represents populations of neurons. We reduce the biophysical equations for the network to a map that takes the interspike interval of the two excitatory cells to a new interspike interval after one cycle. The map does not depend on the details of the biophysical equations. (In the motivating equations in the Appendix, each cell has basic Hodgkin-Huxley-like spiking currents.) The map is derived from two subsidiary maps that encode the times that an inhibitory cell or an excitatory cell fires after receiving inputs at two different times as a function of the time difference between the inputs. From these maps, we are able to read off information about how different kinds of coupling affect stability of the synchronized state, the period of the synchro- nized solution, the rate of synchronization, and the response of the network to heterogeneity of the cells. The importance of multiple spikes in the synchronization process distinguishes the mechanisms of this paper from other mechanisms of synchronization that deal with the envelope of bursting activity (3, 9) or single pulses (5-8). Indeed, the significance of the timing of individual spikes provides a new aspect of ''temporal coding''; the spikes encode information about the synchronization process, rather than information directly related to sensory inputs.