System identification of biophysical neuronal models

TL;DR

The paper addresses identifying nonlinear neuronal dynamics governed by fading-memory operators, proposing a direct identification approach that preserves the biophysical split between membrane integration and ion-channel–driven currents. It introduces a model structure based on a series of Generalized Orthonormal Basis Functions (GOBFs) followed by a static Artificial Neural Network (ANN) nonlinearity, and proves this composition can universal-accurately approximate fading-memory operators. A practical pole-selection heuristic, grounded in the system's ultra-sensitivity at a critical voltage $v^*$, guides GOBF design, and a voltage-clamp–assisted identification protocol enables small-signal characterization. The method is demonstrated on a bursting stomatogastric ganglion model, showing improved spike-timing predictions over time-delay bases and highlighting the role of long timescales in bursting dynamics. Overall, the work offers a simple, data-grounded pathway to identify complex neuronal dynamics without direct access to internal states, with implications for scalable neurophysiological modeling and analysis.

Abstract

After sixty years of quantitative biophysical modeling of neurons, the identification of neuronal dynamics from input-output data remains a challenging problem, primarily due to the inherently nonlinear nature of excitable behaviors. By reformulating the problem in terms of the identification of an operator with fading memory, we explore a simple approach based on a parametrization given by a series interconnection of Generalized Orthonormal Basis Functions (GOBFs) and static Artificial Neural Networks. We show that GOBFs are particularly well-suited to tackle the identification problem, and provide a heuristic for selecting GOBF poles which addresses the ultra-sensitivity of neuronal behaviors. The method is illustrated on the identification of a bursting model from the crab stomatogastric ganglion.

Figures (4)

• Figure 1: The general model structure of a neuronal system.
• Figure 2: Left column: Typical aggregate ionic current behavior in a biological neuron under voltage-clamp. Three different experiments are shown in which the reference $r_k$ is stepped to $-45$, $-35$ and $-25$ mV at $10$ ms. Right column: Typical behavior of a neuron (without voltage-clamp) subject to steps of different amplitudes at the input $i_k$. Voltages are in $\mathrm{[mV]}$, and currents are in $\mathrm{[\upmu A/cm^2]}$.
• Figure 3: Comparison between impulse responses (bottom) and spike responses (middle) of four GOBFs given by \ref{['eq:gobfs']} with $d=0$, $\xi_0 = 0.9802$, $\xi_1= 0.9980$, $\xi_2= 0.9998$ and $\xi_3= 0.9995$. The spike (top) used to compute the responses $g_i * v$ was normalized so that the area between $t=0$ and $t=t_1$ equals one (the same area of a Dirac impulse). The impulse responses $g_i$ are mutually orthogonal, while the responses $g_i * v_s$ are not. However, the responses $g_3*v_s$ and $g_4*v_s$ are close to $g_3$ and $g_4$, respectively, and thus are close to being orthogonal to each other. This can be explained by the fact that the timescales of $g_3$ and $g_4$ are much larger than those contained in the spike. The sampling time used is $t_s = 0.01$ ms.
• Figure 4: Top: validation $v_k$ of the true STG model (blue, solid) and output $\hat{v}_k^{\mathrm{gobf}}$ of the model defined with twelve GOBFs (red, dashed). Bottom: validation $v_k$ of the true STG model (blue, solid) and output $\hat{v}_k^{\mathrm{td}}$ of the model defined with $n_\mathrm{td} = 12$ time delays (red, dashed).

Theorems & Definitions (7)

• Definition 1: park_criteria_1992boyd_fading_1985
• Definition 2
• Lemma 1
• Definition 3
• Lemma 2
• Theorem 1
• proof