AI to PING transition

Abstract

The full recurrent shared-input conductance network runs through three windows: an AI target, an AI-to-PING transition, and a PING target. Only direct external excitation of inhibitory neurons is reduced; source rate, Input→E gain and all recurrent weights remain fixed from initialization. An unfitted four-state mean-field extension also predicts an oscillatory loss of equilibrium stability as Input→I strength decreases. This single-seed comparison does not establish an AI state or a PING mechanism.

Results

Median E-volley spacing in the final hold is 12.0 ms in its first 100 ms and 18.5 ms in its last 50 ms, using locally detected peaks. The full-window spectrum combines this evolution.

Four-variable mean-field prediction

The displayed mean-field evidence precedes the SNNLab 0.27 migration. The current native implementation has not regenerated this trajectory; long onset trajectories can differ in phase with numerical tolerances.

Model specialisation

We use the general mean-field model with all four recurrent pathways, shared input at $r_X=0.0025$ spikes/source/ms (2.5 Hz/source) and no private input. All synaptic decays are 5 ms. Equilibrium-compatible initial conductances permit the note’s matched-decay four-state reduction, retaining $(r_E,r_I,g_e^I,g_i^E)$ despite nonzero self-coupling.

Only the Input→I matrix is multiplied by the scheduled gain $\alpha$; Input→E and recurrence remain fixed. The model replaces changing external input by its instantaneous stationary mean, so the ramp’s filtering lag is omitted. Appendix D records the experiment-specific parameters, units, coupling provenance and numerical checks. The common conductance, current and gain derivations are in the note.

Calculations for Figure 2

1. Panel A — equilibrium rates. At 201 equally spaced multipliers from $\alpha=1$ to zero, set all four time derivatives to zero and eliminate the recurrent conductances using the stationary filter relation. A star denotes equilibrium. Bounded least squares solves the stationary equations, using the preceding equilibrium as the next initial guess.

   The earlier implementation eliminated conductances and required a maximum two-rate residual below 1e-7 Hz.

   Plot the resulting E/I rates against $\alpha$; this continues one numerically found branch without establishing its global uniqueness.

2. Panel B — local stability. Write the four equations as $\dot x=F(x;\alpha)$, where $x=(r_E,r_I,g_e^I,g_i^E)$ and $F$ is the vector of time derivatives. Evaluate the analytical flow Jacobian $J_{\mathrm{flow}}=\partial F/\partial x$ at each equilibrium and plot $\max_j\operatorname{Re}(\lambda_{J,j})$, where $\lambda_{J,j}$ are its four eigenvalues in ms$^{-1}$. Negative values indicate local stability. Brent refinement locates the zero at $\alpha=$0.482. A complex pair crosses there while the other pair remains stable. Its linear onset frequency is $f_{\mathrm{Hopf}}=1000|\operatorname{Im}(\lambda_J)|/(2\pi)=$79.4 Hz, converting the critical eigenvalue’s imaginary part from radians/ms. This tests oscillatory onset without determining Hopf criticality.

3. Panel C — protocol integration. Start from the $\alpha=1$ equilibrium, add $10^{-6}$ spikes/neuron/ms to $r_E$, and integrate the four equations through 50 ms burn-in and three 750 ms windows. Hold $\alpha=1$, decrease it linearly to zero in the middle window, then hold zero. Sample model rates every 0.5 ms; solver settings are in Appendix D. For each dashed spiking mean, count spikes within that window and divide by population size and its 0.75 s duration. These means are comparisons, not fitting targets.

Numerical settings and validation checks are recorded once in Appendix D. The closure’s oscillatory onset does not establish microscopic AI firing or the spiking network’s PING mechanism.