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Arnold tongue of two coupled PING circuits#

Abstract#

Reciprocal excitation should synchronize two PING circuits over a wider natural-frequency mismatch as coupling increases, while preserving the rule that the intrinsically faster circuit leads in phase[1]. We test that claim in a graph-native sweep over measured detuning and coupling. The 80 E / 20 I network recovers a centred Arnold tongue that widens across 4 successive coupling steps, and all 715 primary trials are valid. One near-resolution cell violates the strict phase-sign criterion, so the 80 E / 20 I reproduction failed despite 98.7% sign agreement. A focused 800 E / 200 I confirmation resolves that exception: all 40 coupled trials lock with the expected phase sign.

Methods#

1. Fix the graph and coupling intervention. Each circuit contains 80 conductance-based excitatory neurons and 20 inhibitory neurons. A private 80-channel Poisson population drives each excitatory population. Local E-to-I AMPA and I-to-E GABA-A projections generate PING activity. Four reciprocal AMPA projections share coupling weight $𝐾$: ${𝐸}_{𝐴}$ projects to ${𝐸}_{𝐵}$ and ${𝐼}_{𝐵}$, and ${𝐸}_{𝐵}$ projects to ${𝐸}_{𝐴}$ and ${𝐼}_{𝐴}$. The graph and network seed remain fixed. Figure 1 shows the intervention; the evidence-scale result verifies the realized parameter tensors.

2. Construct and measure the natural-detuning axis before coupling. First, both circuits receive the same Poisson input rate with $𝐾=0$. Five trials at each rate measure how uncoupled gamma frequency changes with drive. The longest interval that is fully valid, monotonic, and has a within-rate frequency interquartile range (IQR) no greater than 0.8 Hz is retained.

   The retained curve maps an input rate to its median uncoupled gamma frequency. Second, this mapping is used in reverse to construct the drive for each of 13 requested signed frequency differences $𝑑$. The midpoint of the retained output-frequency range is ${𝑓}_{𝑐}=40.8$ Hz. For each $𝑑$, circuit A is assigned the desired frequency ${𝑓}_{𝐴}^{\ast }={𝑓}_{𝑐}+\frac{𝑑}{2}$, while circuit B is assigned ${𝑓}_{𝐵}^{\ast }={𝑓}_{𝑐}-\frac{𝑑}{2}$. Linear interpolation on the measured curve then gives the input rates ${𝑟}_{𝐴}$ and ${𝑟}_{𝐵}$ expected to produce those two frequencies. For example, $𝑑=+4$ Hz requests ${𝑓}_{𝐴}^{\ast }=42.8$ Hz and ${𝑓}_{𝐵}^{\ast }=38.8$ Hz, which map to ${𝑟}_{𝐴}=106.0$ and ${𝑟}_{𝐵}=86.36$ Hz per channel. Those two rates form one A/B input-rate pair.

   Third, each pair is run with $𝐾=0$ using five matched seeds. For each seed, the measured natural detuning is

   $$\mathrm{\Delta }{𝑓}_0={𝑓}_{𝐴}^0-{𝑓}_{𝐵}^0,\text{(1)}$$

   where ${𝑓}_{𝐴}^0$ and ${𝑓}_{𝐵}^0$ are the gamma peak frequencies measured under the unequal drives while the circuits remain uncoupled. This seed-specific measured value, not ${𝑟}_{𝐴}-{𝑟}_{𝐵}$ and not the requested target, is the natural- detuning coordinate assigned to the corresponding coupled trial. Figure 2 shows both the equal-drive calibration and the resulting unequal-drive detuning measurements.

3. Measure phase and concentration. Excitatory spikes are converted to population rates with a fixed 5 ms Gaussian kernel. Gamma frequency comes from the post-transient spectrum. A zero-phase 25–90 Hz band-pass filter and analytic signal define relative phase

   $$𝜑\left(𝑡\right)=\text{unwrap}\left({𝜃}_{𝐴\left(𝑡\right)}-{𝜃}_{𝐵\left(𝑡\right)}\right),\text{(2)}$$

   where $𝜑\left(𝑡\right)$ is unwrapped relative phase at time $𝑡$, and ${𝜃}_{𝐴\left(𝑡\right)}$ and ${𝜃}_{𝐵\left(𝑡\right)}$ are instantaneous phases. Phase concentration is

   $$𝑅=\left|\frac{1}{𝑇}\sum _{𝑡=1}^{𝑇}exp\left(𝑖𝜑\left(𝑡\right)\right)\right|,\text{(3)}$$

   where $𝑅$ is phase-locking value (PLV), $𝑇$ is the number of analysed time samples, and $𝑖$ is the imaginary unit. Figure 3 shows their time-domain behaviour; Figure 4 maps them across the primary grid.

4. Freeze the locking rule. Repeated uncoupled equal-drive trials set the admissible emergent frequency difference, relative-phase slope, and phase-slip count. A valid trial is locked when

   $$𝐿=\mathrm{𝟏}\left[\left|{𝑓}_{𝐴}-{𝑓}_{𝐵}\right|\le {𝜀}_{𝑓}\right]\mathrm{𝟏}\left[\left|{𝑎}_{𝜑}\right|\le {𝜀}_{𝜑}\right]\mathrm{𝟏}\left[{𝑁}_{\text{slip}}\le {𝜀}_{\text{slip}}\right],\text{(4)}$$

   where $𝐿$ is the binary locking label, $\mathrm{𝟏}[\cdot ]$ is an indicator, ${𝑓}_{𝐴}$ and ${𝑓}_{𝐵}$ are coupled gamma frequencies, ${𝜀}_{𝑓}$ is the frozen frequency-difference tolerance, ${𝑎}_{𝜑}$ is the fitted slope of Equation 2, ${𝜀}_{𝜑}$ is its tolerance, ${𝑁}_{\text{slip}}$ is the complete phase-slip count, and ${𝜀}_{\text{slip}}$ is its tolerance. PLV from Equation 3 is descriptive, not part of $𝐿$. Figure 4 reports the component estimators, and Figure 5 applies Equation 4 without post hoc threshold changes.

5. Execute the frozen primary grid. Thirteen target detunings cross eleven coupling levels with 5 paired input seeds per cell. Each trajectory lasts 3000 ms at a 0.1 ms timestep; the first 500 ms are discarded. A trial is invalid if recorded state is non-finite, any E or I firing rate is below 1 Hz, or either E population lacks a resolved 25–80 Hz peak. Figures 3–5 report this frozen grid.
6. Apply the registered verdict. Geometry passes only if the locked region is contiguous, centred near zero detuning, and widens across at least three successive nonzero coupling levels. Every grid cell must contain a valid trial. In every locked nonzero-detuning cell, the circuit with greater natural frequency in Equation 1 must lead in phase under Equation 2. The verdict subsection applies these clauses literally.
7. Run a focused finite-size confirmation. Because only one phase-sign cell fails at 80 E / 20 I, the follow-up increases each population tenfold and tests mirrored target detunings at the uncoupled, disputed, and stronger coupling levels. It retains E-population spikes and summary observables, uses ten paired seeds per cell, and lengthens the post-transient window to 5 s. Figure 6 reports the result without reopening the primary grid.

The evidence archive bit-packs exact input and population spikes for every primary and follow-up cell. The primary archive also retains population-mean voltage and projection conductance at 1 ms resolution, compiled bundles, input hashes, seed ledger, and one realized parameter set per coupling level.

Results#

The authored graph isolates reciprocal coupling#

Method 1 varies only the four cross-circuit AMPA projections. The local PING loops, private drives, graph topology, and seed are fixed, so a change across $𝐾$ is attributable to the registered coupling intervention.

The realized parameter archive confirms that all non-coupling tensors remain invariant across 11 coupling levels, as required by Method 1.

Calibration defines a stable detuning axis#

Method 2 yields a stable monotonic interval from 70 to 130 Hz per input channel. This interval supplies 13 paired A/B input-rate conditions. Measuring those conditions without coupling produces the natural-detuning coordinate used in the primary map.

The frozen thresholds used in Equation 4 are 0.8 Hz for frequency difference, 11.53 rad/s for absolute phase slope, and 4 complete slips. The grid is identified by checksum 8813df0395d0aeb00eb4613d28f776cc09cb4ff783694a7c7603e749c79f3c0f.

Representative trajectories distinguish locking from drift#

Conditions selected before inspection show the transition in time: relative phase remains bounded after locking and drifts at the immediately preceding coupling.

Component estimators localize the locking transition#

All 715 primary trials are valid. Across the grid, low coupled-frequency difference, low relative-phase drift, few phase slips, and high phase concentration emerge together as coupling increases.

Coupling produces a widening Arnold tongue#

Applying Equation 4 to the component estimators produces a contiguous locked region centred near zero detuning. Its width reaches 11.92 Hz at the largest coupling and increases across 4 successive coupling steps.

The strict 80 E / 20 I verdict fails one phase-sign cell#

Method 6 gives a split result. Geometry and validity pass, but one qualifying cell has the wrong phase sign. Agreement is 98.7%, while the registered clause requires every qualifying cell to agree. The overall 80 E / 20 I reproduction therefore failed. The exception lies near the estimator’s registered resolution, so Method 7 tests that boundary rather than repeating the full grid.

The 800 E / 200 I confirmation resolves the exception#

The focused confirmation passes: all 40 coupled trials are valid and locked under Equation 4, and all 40 have the phase sign required by Equation 1 and Method 6. At the disputed negative-detuning cell, all 10 seeds have the correct sign with mean phase −0.543 rad. The mirrored positive condition gives 0.526 rad.

The follow-up supports finite-size or estimator-resolution noise, rather than a systematic phase-ordering contradiction, as the cause of the original exception.

Evidence scale#

Dense primary retention would occupy 27.82 GB. The published exact-event and state-summary archive occupies 165.5 MB, a 168.1-fold reduction. The 143 primary cells required a median 19.03 s each locally.

References#

1. Lowet, Roberts, Hadjipapas, Peter, van der Eerden & De Weerd (2015): Input-Dependent Frequency Modulation of Cortical Gamma Oscillations Shapes Spatial Synchronization and Enables Phase Coding. PLOS Computational Biology. doi:10.1371/journal.pcbi.1004072