Reliability entails input-selective contraction and regulation in excitable networks

TL;DR

The paper addresses how reliable computation can emerge from the analog dynamics of excitable neurons by linking reliability to average trajectory contraction under input. It develops a contraction-based theory using the FitzHugh–Nagumo model, identifying lower and upper contraction regions separated by an expansion region, and shows that average contraction over time yields reliable responses. It then demonstrates that in an excitatory–inhibitory (EI) network, input-driven contraction enables robust regulation to a stable steady state, with an internal-model-like controller ensuring disturbance rejection for inputs generated by an exosystem. The findings suggest a principled, input-selective form of dynamical computation that is robust to perturbations and could extend to general conductance-based models, informing neuromorphic control design and analysis of regulatory computation in neural circuits.

Abstract

The animal nervous system offers a model of computation combining digital reliability and analog efficiency. Understanding how this sweet spot can be realized is a core question of neuromorphic engineering. To this aim, this paper explores the connection between reliability, contraction, and regulation in excitable systems. Using the FitzHugh-Nagumo model of excitable behavior as a proof-of-concept, it is shown that neuronal reliability can be formalized as an average trajectory contraction property induced by the input. In excitable networks, reliability is shown to enable regulation of the network to a robustly stable steady state. It is thus posited that regulation provides a notion of dynamical analog computation, and that stability makes such a computation model robust.

Figures (4)

• Figure 1: Simulations of \ref{['s.FN']} with $a=0.7$, $b=0.8$, and $\varepsilon=1/12.5$ from $9$ different initial conditions. Light-blue patches highlight the time intervals where all solutions lie within the same contraction region; (top) the solid, dark-blue lines depict the time evolution of the $v$ variable for each of the nine solutions; dashed red lines depict the boundaries of the contraction regions; (center) pointwise maximum distance between the solutions using the metric \ref{['d.d']}; (bottom) applied input $u$.
• Figure 2: (A) The EI network; (B) the exosystem.
• Figure 3: Simulations of $10$ trials of the EI network \ref{['s.E']}, \ref{['s.I']} with $a_\text{\tiny E}=0.7$, $b_\text{\tiny E}=0.8$, $\varepsilon_\text{\tiny E}=1/12.5$, $k_\text{\tiny E}=4$, $a\text{\tiny U}=0.6$, $b\text{\tiny U}=0.7$, $k\text{\tiny U}=1/2$, and $\varepsilon_\text{\tiny U}=1/30$. (A)$u$ is generated by the exosystem \ref{['s.U']}; (B), $u(t)= 2\sin(\pi t/30)$; (C)$u$ is generated by \ref{['s.U']} with $\varepsilon_\text{\tiny U}=1/5$ (while in \ref{['s.I']}, $\varepsilon_\text{\tiny U}$ remains $1/30$). In all figures: (top) the $10$ simulated trajectories of $y(t)$; (bottom) dark blue lines depict the $10$ simulated trajectories of the variable $\eta$, orange lines depict the disturbance $u$.
• Figure 4: (left) simulation of \ref{['s.LTI.y']} from $5$ random initial conditions; (right) simulation with $u$ changed to $u(t)=\sin(2\omega t)$.

Theorems & Definitions (6)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof