On the Contraction of Excitable Systems

TL;DR

The paper analyzes contraction in conductance-based excitable systems to explain reliable spike timing. It proves incremental exponential stability of the unforced Hodgkin–Huxley flow on a compact forward-invariant set and shows contraction persists under impulsive synaptic inputs subject to an average dwell-time or periodicity condition; at high input rates, a tonic, noncontractive regime emerges due to near-constant conductance. By pairing the continuous flow with a hysteretic event detector, the authors show spike times become robust across trials when contraction holds, while dense driving can erase this reliability. Numerical experiments illustrate both regimes, linking spike-timing reliability to contraction and highlighting when rate coding becomes the appropriate description. The work provides a testable framework for when precise spike timing constitutes a faithful information carrier versus when it is subsumed by firing rate, with implications for neuromorphic control and interpretation of neural codes.

Abstract

We analyze contraction in conductance-based excitable systems and link it to reliable spike timing. For a Hodgkin-Huxley neuron with synaptic input, we prove the unforced dynamics is incrementally exponentially stable on a compact physiological set. With impulsive inputs, contraction persists under an average dwell-time condition, and for periodic spike trains we derive an explicit lower bound on the inter-impulse period. In the high-rate limit the synapse is effectively held open, the conductance is nearly constant, and self-sustained limit-cycle oscillations can arise, so contraction no longer holds. This continuous-time view explains when spike times extracted by an event detector remain robust across trials, and simulations confirm both regimes.

Figures (4)

• Figure 1: Circuit schematic of the conductance-based model \ref{['eq:exc_model']} with the external synapse \ref{['eq.synapse']}.
• Figure 2: Block diagram of the overall architecture. A spike is converted into a timing through an event detector.
• Figure 3: Hodgkin–Huxley neuron driven by periodic impulse trains. Top: membrane voltage for multiple initial conditions under the same input. Bottom: synaptic state $s(t)$ and conductance $g_{\mathrm s}(t)$. Left column (sparse, $T=15$ ms): contraction is preserved and all trajectories converge to a unique steady-state response. Right column (dense, $T=0.5$ ms): the synapse is effectively always open (constant input), a limit cycle emerges, and phase offsets persist across initial conditions. ($\alpha=0.8$, $\tau_s=5$ ms, $\bar{g}_s=0.3$ mS/cm$^2$, $E_{\text{s}}=65$ mV).
• Figure 4: Hodgkin–Huxley neuron with a conductance synapse driven by two impulse trains. Top: membrane voltage from $10$ trials under the same input, with random initial condition and parameters perturbed by $\pm 20\%$ of baseline values. Bottom: spike raster plots. Left column (sparse): a random train with dead-time yields tightly aligned spike times across trials. Right column (dense): a uniform train with period $T=0.01$ ms keeps the synapse effectively open, producing an almost constant conductance, with trajectories showing tonic-like activity with trial-dependent phase. (Baseline parameters: $\alpha=1$, $\tau_s=4$ ms, $\bar{g}_{\mathrm s}=0.425$ mS/cm$^2$, $E_{\mathrm s}=65$ mV).

Theorems & Definitions (18)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1: Incremental Exponential Stability
• Theorem 1
• proof
• Theorem 2: IES under average dwell time
• proof
• Corollary 1: Periodic impulse trains