(A) Minimal network for investigating synchronization with conduction delays. The long E-E connections are essential for coherence of the beta rhythm across distances, but not for the gamma rhythm. (B) The maps T_I and T_β as a function of the delay δ for g_m = 1 and all others parameters as in the Appendix. All axes have units of milliseconds. (C) The linearized scaling factor D for the gamma rhythm (dashed) and for the beta rhythm at various ratios of the effective AHP conductance and the effective inhibitory synaptic conductance. The closer D is to zero, the faster the convergence to the synchronized state and the stronger is robustness to heterogeneity.

(A) Minimal network for investigating local synchronization of gamma and beta rhythms. For the gamma rhythms, the E-E connections are absent; for the beta rhythms, they are a necessary part of the circuit. (B) Gamma-to-beta transition of local rhythms occurs as the AHP is turned on and the local E-E connections are strengthened. Parameters are as in the Appendix. For the gamma rhythm, g_ee = 0 and g_m = 0. At the first arrow, g_m is set to 1, switching the rhythm from gamma to a rhythm in which the E-cells miss beats and fires nonsynchronously. At t = 400, g_ee = 0.15, and the network quickly suppresses the nonsynchronous solution, leaving only the synchronous local state. Throughout the transitions, the I cells shown below exhibit only minor changes, slowing down slightly because of the decreased excitation (excitation every other cycle).

Gamma and beta oscillations in vitro. Intracellular recordings of gamma and beta oscillations in a CA1 hippocampal pyramidal neuron. Oscillations were induced by brief tetanic stimulation [see Whittington et al. (9) for methods]. The initial posttetanic response is a gamma oscillation with action potentials (frequency 38 Hz) separated by a period of hyperpolarization made up of both AHP and inhibitory synaptic activity. After the transition to beta activity, the underlying gamma membrane potential oscillation is still apparent (frequency 42 Hz), but spiking occurs on every second or third period (frequency 18 Hz). Action potentials are separated by the initial AHP/IPSP hyperpolarization and additional IPSPs. (Bar = 1 mV, 100 ms.)

(A) Fast gamma will not synchronize with a delay of 13 ms. If the long distance E-E coupling is turned on (c_ee = 0.5), synchrony still does not occur. However, if the AHP is then turned on (g_m = 1), the network shifts to a beta rhythm and synchronizes within two cycles. Parameters are as in the Appendix. Excitatory drive is 6 in one site and 6.5 in the other; inhibitory drive is 1.15 in both sites. The voltages of the two E-cells are represented by black and gray lines. (B) The main role played by the long-distance E-E connections is to prevent the anti-phase solution. For the first 400 ms, there is no long E-E connection. At t = 400, c_ee = 0.05, which induces synchrony within a few cycles. At t = 400, c_ee is reset to zero, but synchrony remains robust. Drives and AHP are as in A.

In network model, beta remains synchronized with longer axonal delays than does gamma. The network consists of a 96 × 32 array of pyramidal cells (1.92 mm wide) and a superimposed 96 × 4 array of interneurons, as in ref. 14. In control conditions, the maximum axon conduction delay was 3.84 ms across the array. Oscillations were evoked by tonic depolarization of both pyramidal cells (E-cells) and interneurons (I-cells), with the gamma → beta transition occurring as pyramidal cell AHP conductances and excitatory postsynaptic potential conductances simultaneously increase (10, 18). In the case shown on the right, all axonal signals crossing the midline of the array were subjected to an additional 10-ms delay. (A) Gamma, followed by beta, as plotted in simultaneous traces of local average signals (224 nearby E-cells, 28 nearby I-cells). The E-cell signals appear similar, with and without the extra 10-ms delay. During beta, both E- and I-cell traces reveals an underlying oscillation at gamma frequency. (Bars = 20 mv, 200 ms.) (B) Cross-correlations of local average E-cell signals from opposite ends of the array, for 200 ms of gamma (thin lines) and 800 ms of beta (thick lines). In the control case, both gamma and beta have cross-correlation peaks within 3 ms of 0. With the extra 10-ms conduction delay, the gamma signal is almost anticorrelated between the two sites whereas the beta signal cross-correlation peak is at −1.4 ms.