[Paper Review] A novel method for determining the phase-response curves of neurons based on minimizing spike-time prediction error

Benjamin Torben-Nielsen, Marylka Yoe Uusisaari+1 more|arXiv (Cornell University)|Jan 4, 2010

Neural dynamics and brain function2 references3 citations

TL;DR

This paper introduces the STEP (Standardized Error Prediction) method, a novel approach to estimate neuronal phase-response curves (PRCs) by minimizing spike-time prediction error using independent perturbations across discretized phases. The method achieves robust and accurate PRC estimation from experimental data with as few as 100 spikes, even at high noise levels, outperforming existing methods in convergence and data efficiency.

ABSTRACT

Regular firing neurons can be seen as oscillators. The phase-response curve (PRC) describes how such neurons will respond to small excitatory perturbations. Knowledge of the PRC is important as it is associated to the excitability type of neurons and their capability to synchronize in networks. In this work we present a novel method to estimate the PRC from experimental data. We assume that continuous noise signal can be discretized into independent perturbations at evenly spaced phases and predict the next spike based on these independent perturbations. The difference between the predicted next spike made at every discretized phase and the actual next spike time is used as the error signal used to optimize the PRC. We test our method on model data and experimentally obtained data and find that the newly developed method is robust and reliable method for the estimation of PRCs from experimental data.

Motivation & Objective

• To develop a reliable method for estimating phase-response curves (PRCs) from experimental neuronal data, particularly under high noise and limited spike counts.
• To overcome convergence issues in existing PRC estimation methods that rely on summed spike-time errors over intervals.
• To preserve temporal phase information by treating perturbations independently at discrete phase bins, improving optimization stability.
• To validate the method on model neurons and real experimental data, demonstrating accuracy and robustness.
• To reduce the number of required spikes for reliable PRC estimation compared to existing techniques like STA, WSTA, or Galan’s method.

Proposed method

• Discretize the interspike interval into N phase bins (e.g., 100 bins) to represent the neuron's oscillatory cycle from 0 to 2π.
• For each phase bin φj, predict the next spike time as ŝi,j = si−1 + x(t(φj)) · z(φj), where x(t) is the injected fluctuating current and z(φj) is the candidate PRC.
• Compute the error signal Ei,j = √(ŝi,j − si)² for each spike i and phase bin j, preserving phase-specific prediction errors.
• Optimize the PRC parameters (Fourier coefficients a and b) using least-squares minimization of the total error across all spikes and phase bins.
• Approximate the PRC using a truncated Fourier series: z(φ) ≈ Σₙ₌₀³ [aₙ sin(nφ) + bₙ cos(nφ)].
• Use the error signal to iteratively refine the PRC until convergence, avoiding error summation over time that hinders prior methods.

Experimental results

Research questions

• RQ1 Can a PRC estimation method be developed that converges reliably with fewer spikes and higher noise levels than existing methods?
• RQ2 Does preserving phase-specific spike-time prediction errors improve the accuracy and robustness of PRC estimation compared to summed error approaches?
• RQ3 Can the STEP method accurately recover known PRC types (e.g., type-II) from model neurons under continuous fluctuating current injection?
• RQ4 How does the method perform on real experimental data from mouse cortical pyramidal neurons, and is the resulting PRC reproducible across subsampled data?
• RQ5 What is the minimum number of spikes required for reliable PRC estimation using the STEP method compared to established alternatives?

Key findings

• The STEP method successfully recovers the correct PRC type (type-II) in model neurons under continuous fluctuation, even at high noise levels.
• With only 100 spikes, the STEP method produces a stable and accurate PRC estimate, significantly reducing the data requirement compared to methods needing 480–7000 spikes.
• The PRC estimated from experimental data on mouse layer 2/3 pyramidal neurons is consistent across multiple subsamples (half-data PRCs), indicating reliability and low overfitting risk.
• The method shows robustness to high noise amplitudes (up to 100 pA), maintaining accurate PRC estimation where other methods fail to converge.
• The PRCs estimated via STEP closely match those obtained via the direct method (using brief current pulses), validating its accuracy.
• The use of phase-resolved error signals prevents loss of temporal information and enables faster convergence than previous optimization-based methods.