Mean-Field Analysis of PING Bifurcations

Abstract

We analysed the four-state E→I→E specialisation of the mean-field model. Analytical-Jacobian continuation identified a Hopf bifurcation at 0.593905 nA with onset frequency 27.57 Hz. Ascending and descending near-onset integrations produced coincident sampled branches and continuous amplitude growth consistent with a supercritical transition. This numerical criterion does not exclude a narrower bistable interval or unstable cycle. Hopf frequency decreased as inhibitory decay lengthened.

Results

A reversible Hopf onset slows

At $I_{\mathrm{ext},E} =$ 0.593905 nA, a complex-conjugate eigenvalue pair crossed the imaginary axis at $f_{\text{Hopf}} =$ 27.57 Hz (Fig. 1A). The sampled amplitude branches differed by at most 0.00687 Hz and had $R_{\text{fit}}^{2} =$ 0.9994, consistent with a supercritical transition (Fig. 1B). Across the inhibitory-decay sweep, $f_{\text{Hopf}}$ decreased from 30.2 to 17.94 Hz (Fig. 1C). Finite spacing cannot exclude narrower bistability or an unstable periodic orbit.

Methods

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1. Model. The mean-field note supplies the shared derivation; Appendix A records this experiment’s specialisation.

2. Hopf bifurcation and onset frequency. At the reference inhibitory-pathway decay $\tau_{I\to E}=6$ ms, we evaluated the equilibrium over 401 equally spaced external drives from 0 to 4 nA.

   2.1 — Solve for the equilibrium.

   Set the two recurrent conductance derivatives to zero in the note’s equations, apply the experimental specialisation, and substitute the inhibitory equilibrium rate into the excitatory equation. The scalar residual is the excitatory rate minus its predicted gain.

   Brent’s bracketed method searched $0 \le r_{E} \le \frac{1}{\tau_{\mathrm{ref},E}}$ with absolute tolerance $10^{- 13}$ ${\text{ms}}^{- 1}$ and relative tolerance $10^{- 12}$. We recovered $r_{I}$, $g_{e}^{I}$ and $g_{i}^{E}$ by substitution and required an absolute residual below $10^{- 11}$ ${\text{ms}}^{- 1}$.

   2.2 — Calculate local stability. At every equilibrium, we evaluated the analytical reduced Jacobian and numerically calculated its four eigenvalues. Eigenvalues with positive imaginary part greater than $10^{- 8}$ ${\text{ms}}^{- 1}$ were eligible for the leading complex pair. We plotted all four eigenvalues in the complex plane and coloured them by external drive; negative and positive real parts denoted local decay and growth, respectively (Fig. 1A).

   2.3 — Identify and verify the crossing.

   The first change of the leading complex pair from negative to nonnegative real part supplied a coarse drive bracket. Brent’s method refined the zero to absolute drive tolerance $10^{- 12}$ nA and relative tolerance $10^{- 12}$, recomputing the equilibrium and analytical Jacobian at every trial drive. We confirmed nonzero imaginary part, negative real parts for the remaining modes, and positive transversality by a centred difference with $10^{- 5}$ nA half-step. The recorded crossing slope was 0.29 ${\text{ms}}^{- 1} {\text{nA}}^{- 1}$ and the largest remaining real part was -0.458 ${\text{ms}}^{- 1}$.

   2.4 — Calculate the onset frequency. At the refined crossing, $\omega_{\text{Hopf}}$ was the magnitude of the critical pair’s imaginary part in radians per millisecond. We converted it to hertz using $f_{\text{Hopf}} = 1000 \frac{\omega_{\text{Hopf}}}{2 \pi}$ (Fig. 1A).

3. Numerical criticality test. We integrated 25 equally spaced drives from 0.1 nA below to 0.55 nA above the refined Hopf point, first in ascending and then in descending order. The initial state was the lowest-drive equilibrium with an added $10^{- 3}$ ${\text{ms}}^{- 1}$ excitatory-rate perturbation; each subsequent condition began from the preceding endpoint. LSODA integrated each condition for 2,000 ms with relative tolerance $10^{- 7}$, absolute tolerance $10^{- 10}$ and maximum step 1 ms.

   For each drive, peak-to-peak excitatory-rate amplitude $A_{\text{pp}}$ was measured over the final 500 ms. We classified the sampled dynamics as consistent with a supercritical transition when the maximum difference between ascending and descending amplitudes was below $10^{- 4}$ ${\text{ms}}^{- 1}$, and ordinary least-squares regression of $A_{\text{pp}}^{2}$ on $I_{\mathrm{ext},E} - I^{\ast}$ above onset had positive slope and $R_{\text{fit}}^{2} > 0.9$. This finite numerical criterion tests reversible, continuous onset; it does not detect an unstable cycle or prove the absence of narrower bistability.[1]

4. Inhibitory-decay sweep. We repeated the equilibrium continuation and refined complex-pair crossing at $\tau_{I\to E} = 4.5 , 6 , 9 , 12 , 18$ and $27$ ms while holding all other model parameters fixed. At every decay constant, we applied the same nonzero-frequency, remaining-mode stability and transversality checks used at the 6-ms reference condition, then converted the critical angular frequency to $f_{\text{Hopf}}$ in hertz.