Filtered Voltage Feedback in a Piecewise-Linear Electronic Neuron: First-Switch Threshold and a Saddle-Node of Spiking Cycles

Abstract A planar piecewise-linear electronic neuron with six switching regions produces large spike-like excursions without an imposed reset but has no internal memory variable. We add one exponentially filtered voltage state w (time constant τ , feedback gain g) and leave the switching terms unchanged. Starting from rest, the current required to reach the voltage threshold has a closed form, I first = v th /s max , where s max is the peak of the linear step response. It lies well below the current I eq = (A + g)v th at which the resting equilibrium becomes inadmissible, and the first-spike latency approaches a finite limit with a square-root cusp. The onset of persistent spiking is a saddle-node bifurcation of limit cycles. On the upward section v = v th the dynamics reduce to a one-dimensional return map w n+1 = Φ(w n) whose stable and unstable fixed points merge at I fold = 0.0348814 V/ms for g = 0.05 ms −1 and τ = 50 ms. Rest and spiking coexist for I fold < I < I eq , which spans 71% to 100% of I eq ; the rate at onset is finite (about 12.6 Hz) and the cycle branch is hysteretic. Below the fold the circuit emits a finite burst of N ≃ π/ |a| b (I fold − I) spikes, with a and b measured from the return map; this agrees with simulation from N = 2 to N = 837. Over a 5×5 grid in (g, τ) the ratio I fold /I eq lies between 0.595 and 0.957. Results beyond the first-switch analysis are numerical and were checked with an independent matrix-exponential solver and a matched instantaneous-feedback control. The saddle-node scaling itself is generic; the contribution is its quantitative realization in this discontinuous circuit.