A Single-Equation Approach to Classifying Neuronal Operational Modes

TL;DR

This work addresses how neurons classify into operational modes such as coincidence detection and integration by proposing a rest-centered, nondimensionalized one-dimensional polynomial drift: $\frac{dy}{ds}=\sum_{i=1}^n \beta_i y^i + \eta(s)$. The key insight is that the sign and magnitude of the lowest-order nonlinear coefficient $\beta_m$ govern mode switching, with $\beta_m<0$ favoring coincidence detection and $\beta_m>0$ promoting integration, as shown by phase-portrait analysis and channel surrogate models. The authors demonstrate that a polynomial surrogate can faithfully reproduce Hodgkin–Huxley dynamics (up to degree seven for channel currents) and provide fast, tractable means to explore parameter regimes and predict how biophysical changes (e.g., channel mutations or neuromodulation) shift operating modes. This framework yields experimentally testable predictions and offers a compact mechanistic lens for understanding neuronal coding and potential channelopathies, while acknowledging its simplifications (e.g., absence of adaptation and dendritic structure).

Abstract

The neural coding is yet to be discovered. The neuronal operational modes that arise with fixed inputs but with varying degrees of stimulation help to elucidate their coding properties. In neurons receiving {\it in vivo} stimulation, we show that two operation modes can be described with simplified models: the coincidence detection mode and the integration mode. Our derivations include a simplified polynomial model with non-linear coefficients ($β_i$) that capture the subthreshold dynamics of these modes of operation. The resulting model can explain these transitions with the sign and size of the smallest nonlinear coefficient of the polynomial alone. Defining neuronal operational modes provides insight into the processing and transmission of information through electrical currents. Requisite operational modes for proper neuronal functioning may explain disorders involving dysfunction of electrophysiological behavior, such as channelopathies.

Figures (5)

• Figure 1: The problem addressed in this paper. (a) We simplify Hodgkin-Huxley neuron models to (b) polynomial models that accommodate different (c) integration modes under in vivo stimulation. Simulations show that the simplified model almost perfectly matches the Hodgkin-Huxley version.
• Figure 2: Firing patterns of cubic polynomial neurons with negative leak terms. (a) Detector operational mode system with stable, negative leak coefficient ($-\beta_i$) and negative $\beta_m$ ($\beta_2$). (b) Integrator operational mode system with stable, negative leak coefficient ($-\beta_i$) and positive $\beta_m$ ($\beta_2$).
• Figure 3: Accuracy of the polynomial model compared to the Hodgkin-Huxley model for neuronal dynamics of (a) potassium and (b) sodium channels.
• Figure 4: For these polynomials of order 7, the model is able to accurately predict the trajectories of the membrane potentials from the HH models for (a) potassium and (b) sodium channels..
• Figure 5: Phase portraits for polynomials representing potassium and calcium. (a) The potassium channel's stable, negative leak term allows the dynamics of the system to reflect the single-equation approach. (b) A "postive leak" (bias current injection) shifts the equilibria of calcium's equation, allowing for the channel's integrator operational mode despite the equation's $-\beta_m$ value.