Theta nested Gamma

Question

Does a theta-frequency envelope on otherwise independent afferent excitation organise fast population bursting? We extend the circuit in Susin and Destexhe (2021) with periodic input and a constant-input comparison. The circuit is a modified conductance-based LIF version of the published PING network.[1]

Both populations receive excitatory afferents. From the start of the simulation, with no burn-in, the theta condition follows

Here $t$ is time in seconds from initialization, $\lambda(t)$ is the rate of each afferent in spikes/s, $\lambda_0=2$ spikes/s is the mean, $A=1.8$ spikes/s is the modulation amplitude, and $f_\theta=5$ Hz is the envelope frequency. The rate ranges from 0.2 to 3.8 Hz and starts at its rising mean crossing. The comparison remains at 2 Hz. The two conditions therefore have the same prescribed mean over complete cycles; their realised spike counts and neuronal firing rates need not match.

Results

Recorded activity

Paired comparison

We measured activity from 0.2 to 0.8 seconds, spanning three complete theta cycles and excluding 200 ms at each end from measurement. The full simulation, including the initial transient, is shown; no burn-in was simulated. Mean E firing rates were 16.76 Hz with theta input and 15.82 Hz with constant input; mean I rates were 19.67 and 18.29 Hz, respectively.

These observations characterise this circuit under the specified input envelopes. They do not establish a separate gamma oscillator: sharp theta-locked bursts can contribute harmonics to the filtered band, and a maximum at a search boundary is not evidence of an isolated spectral peak. Repeated seeds, spectral-background checks and feedback interventions would be needed for a stronger mechanistic claim.

Methods

1. Pair the input conditions. Reuse connectivity, weights and the initial state; generate target-specific Poisson counts with the same random streams. Only the prescribed source-rate envelope differs. Parameters are listed in the appendix.
2. Measure population rates and spectra. Count all spikes in 1 ms bins and divide by population size and bin duration. Use 200–800 ms for rate and spectrum summaries. Welch spectra use a Hann window, 500 ms segments and 50% overlap; report the raw maximum within 30–90 Hz, with 2 Hz bin spacing.
3. Measure phase concentration. Apply a fourth-order Butterworth 30–90 Hz bandpass in second-order sections forward and backward over the full recording. Take the Hilbert-transform magnitude as the amplitude envelope, then crop to the measurement interval. Concentration is the magnitude of the amplitude-weighted mean unit phase vector at the prescribed 5 Hz phase. The constant condition uses the same clock. This is descriptive and has no surrogate correction.