Lowet (2015): Coupled Rhythms

Results

Reciprocal coupling and phase locking

Inspired by Lowet et al. (2015), we tested when two gamma rhythms hold a steady phase difference. Across 143 settings, 141 had valid rhythms and 13 met all finite-time locking criteria; maximum valid PLV was 0.99 (Fig. 1). PLV measures consistency of relative phase, from zero to one; a value near one need not mean zero phase difference. This sampled map describes one realization and does not establish a continuous locking boundary or robustness across seeds.

Lowet et al.’s published map provides a qualitative comparison (Fig. 2). Their conductance-density coupling, Hodgkin–Huxley neurons and signal-based phase estimate differ from our conductance-LIF model and excitatory-volley phase, so the displays do not provide a quantitative replication.

Switching on reciprocal coupling

We used the map’s −4 Hz input difference and 0.09 μS coupling, keeping its saved weights and connections in both directions. At 500 ms, we switched coupling on. S/R frequencies changed from 41.6/46 Hz to 45.7/45.7 Hz. Relative phase became nearly steady (Fig. 3); PLV was 0.99, fitted drift was 0.0077 cycles/s and phase span was 0.115 cycles over 800–1,500 ms.

This diagnostic reuses the selected map matrices and input realization; its spikes match that map trajectory over the displayed interval. It illustrates the onset of finite-time locking and is not an independent evaluation or evidence of lasting locking.

Methods

1. Construct the paired circuits. Use the saved conductance-based leaky integrate-and-fire (conductance-LIF) networks in Fig. A1, with parameters in Table A1. Binary input spikes are drawn independently per channel and timestep, with probability equal to input rate times timestep in seconds. Split each original input afferent into eight channels with one-eighth weight, preserving summed input strength and recurrent matrices. No weights are trained.

2. Sweep reciprocal coupling (Fig. 1). Cross 11 input differences (−10 to +10 Hz per channel, step 2 Hz) with 13 coupling strengths (0–0.12 μS, step 0.01 μS). Mean input stays 25 Hz per channel. Copy a shared 500 ms uncoupled prefix into each coupling condition, then simulate to 5,000 ms without resetting state. Reuse inputs across coupling strengths and underlying random draws across input differences. Exclude 300 ms after coupling onset. The zero-coupling condition supplies measured frequency detuning. Batched settings are conditions, not replicates.

3. Measure relative phase (Figs. 1 and 3). Smooth excitatory population rate with a Gaussian of 1 ms standard deviation; detect peaks separated by at least 15 ms with prominence at least 10% of the maximum smoothed rate. Interpolate phase linearly between consecutive volleys, advancing by $2\pi$ radians per interval; omit samples outside observed volleys. Compute

   $$\mathrm{PLV} = \left|\frac{1}{N}\sum_{n=1}^{N} e^{i[\phi_A(n)-\phi_B(n)]}\right|. \tag{1}$$

   Here $\phi_A(n)$ and $\phi_B(n)$ are circuit phases in radians at shared valid sample $n$, $N$ is the number of those samples, $i$ is the imaginary unit, and bars denote complex magnitude. Use S/R in place of A/B for Fig. 3. Estimate frequency as the reciprocal mean inter-volley interval, drift as the least-squares slope of unwrapped relative phase in cycles against seconds, and span as its maximum minus minimum. For the map, each rhythm needs at least 20 volleys, frequency 30–80 Hz and interval coefficient of variation (standard deviation divided by mean) ≤0.2; its uncoupled calibration must also pass. Valid pairs meet the diagnostic locking criteria when absolute drift <0.25 cycles/s, PLV >0.95 and span <1 cycle. These thresholds describe finite windows.

4. Inspect coupling onset (Fig. 3). Transfer the selected map’s input, local and both cross-network matrices with A→S and B→R, retaining the same inputs. Apply native step gain, zero before 500 ms and one thereafter, for a continuous 1,500 ms trajectory. Phase metrics use 100–500 ms before coupling and 800–1,500 ms after coupling. Raster frequencies use the first and last 500 ms; each window requires at least five volleys and the same frequency and interval-regularity checks. Retained-evidence checks compare transferred matrices, eight-afferent masks, scheduled gains and unchanged parameters; exact spike agreement with the map checks transfer consistency, not independent reproducibility.