Conditional Flow Matching for Single-Neuron Electrophysiology: Capturing Multimodal Responses Across Stimuli

Cameron Schofield; Luca Ghafourpour; Philip H. Wong; Costas A. Anastassiou; Richard E. Turner

cs.LGOct 5, 2026

Conditional Flow Matching for Single-Neuron Electrophysiology: Capturing Multimodal Responses Across Stimuli

Authors: Cameron Schofield, Luca Ghafourpour, Philip H. Wong, Costas A. Anastassiou, Richard E. Turner

Organizations: University of Cambridge · Cedars-Sinai Medical Center · California Institute of Technology · Archimedes AI, Athena Research Center

Abstract

Neurons of the brain exhibit a rich repertoire of electrophysiology dynamics with the same repeated stimulus eliciting very different voltage responses from the same cell. One common approach in biophysically detailed models is to capture this variability through ensembles of deterministic parametrizations, at a cost of hundreds of thousands of CPU hours. Existing machine learning surrogates inherit the same limitation, where a stimulus is mapped to a single voltage response. We address this challenge by learning a conditional generative model for single-neuron electrophysiology, using flow matching with a velocity field conditioned on the input current. On biophysically detailed models of two human cortical interneuron types, the generated responses closely reproduce the electrophysiological feature distributions, spike-time structure, and excitability profiles, even matching the experimental recordings from the corresponding human cortical neurons. Near the firing threshold, firing and non-firing responses coexist at the same stimulus amplitude, and at high amplitudes, ensembles may split into low- and high-firing modes near depolarization block. We show that our model recovers both modes in each case, while a neural operator baseline suppresses spiking near threshold and blurs the gap between modes at depolarization block.

Figures & tables

Figure 1 : Performance on different neurons. From left to right, columns show: (1) voltage traces generated by our model; (2) voltage traces generated by PDE simulations; (3) spike-triggered voltage averages from our model, PDE simulations, and experimental recordings; and (4) F-I curves from our model, PDE simulations, and experimental recordings. For columns (3) and (4), the solid line represents the median, and the shaded area represents the interquartile range of the responses.

Figure 2 : Flow matching transforms noise to data. Columns show: (1) a draw

X_{0}

from the pointwise Gaussian noise source; (2) and (3) intermediate states

X_{t}

at selected flow times

t

; and (4) the generated trace

X_{1}

Panel of Figure 3 — Figure 3 : PVALB feature distributions and firing probability. A) Distribution of key electrophysiological features for a PVALB neuron from experimental recordings (grey), PDE simulations (blue), NOBLE (red), and our model (green) across three representative amplitudes. Arrows indicate the fraction of values outside of the displayed axis range. B) Explained variance (EV, %) across quantile levels of generated and PDE distributions. C) $P(\mathrm{fire}\mid I)$ , the empirical firing probability given current amplitude for PDE simulations (blue; dashed), NOBLE (red), and our model (green).

Regime	AP width ( $\mu$ s)	AHP depth (mV)	mean AP amplitude (mV)	inv ISI (Hz)	mean frequency (Hz)
All	25.3	0.595	1.14	0.533	0.334
Threshold	43.7	1.186	2.23	0.509	0.783
Low-firing	27.0	0.533	1.22	0.419	0.240
High-firing	22.9	0.592	1.03	0.605	0.361

Table 1 : Performance by dynamical regime. Wasserstein-2 distances between reference and generated feature distributions, averaged across amplitudes in different regime types. These regimes are: (1) threshold amplitudes, where at least one HoF model produces a steady state response and at least one HoF model produces a firing response, (2) low-firing amplitudes, which produce mean spike counts between 5 and 17, and (3) high-firing amplitudes, which produce mean spike counts between 18 and 37.

Figure 4 : Spike-timing variability. Spike peak timing at representative PVALB and VIP current amplitudes, comparing smoothed spike-time densities from PDE simulations (blue), Generative (green), and NOBLE (red) voltage responses.

Figure 5 : Multimodal spike counts near depolarization block. Spike count versus input current amplitude for VIP neuron. Lines plot the mean spike count in the conditional voltage responses. Histograms plot the spike count distribution for the PDE (blue), Generative (green), and NOBLE (red) responses at current amplitudes 0.1nA, 0.18nA, and 0.28nA.

Regime	AP width ( $\mu$ s)	AHP depth (mV)	mean AP amplitude (mV)	inv ISI (Hz)	mean frequency (Hz)
All	27.9	0.786	1.60	0.627	0.378
Threshold	43.6	1.120	2.87	0.767	0.962
Low-firing	26.8	0.865	1.59	0.620	0.289
High-firing †	27.1	0.658	1.45	0.623	0.404

Table 2 : Performance by dynamical regime when trained on time-varying stimuli. Wasserstein-2 distances between reference and generated feature distributions, averaged across square pulse currents in different regime types. † High-firing amplitude range is cut off at 0.18nA to match the max amplitude in the time-varying stimuli dataset.

Figure 6 : Generalization to unseen time-varying stimuli. Predicted PVALB responses to unseen time-varying stimuli, comparing PDE simulations with generative models trained on time-varying and DC-step currents.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

	AP width ( $\mu$ s)	AHP depth (mV)	mean AP amplitude (mV)	inv ISI (Hz)	Spike count
DiffWave	30.1	0.99	3.08	3.53	0.48
U-Net	49.3	0.87	4.19	0.76	0.25

Appendix

Table 3 : Wasserstein-2 distances between reference and generated feature distributions, averaged over amplitudes, for the DiffWave and U-Net models.

Figure 7 : U-Net model for predicting the flow matching velocity from the interpolated voltage state

X_{t}

, flow time

t

, and input current

I

. Resolution is halved at each encoder stage and restored by interpolation in the decoder; skip connections carry full-resolution features around the bottleneck. Flow time and current conditioning enter every residual block.

	Wasserstein-2 distance, spiking amplitudes					Subthreshold noise
	AP width ( $\mu$ s)	AHP depth (mV)	mean AP amplitude (mV)	inv ISI (Hz)	Spike count	residual RMS (mV)
$\mathrm{L1}$	62.6	1.33	2.20	0.80	0.34	0.07
$\mathrm{L2}$	49.3	0.87	4.19	0.76	0.25	0.21

Appendix

Table 4 : Wasserstein-2 distances between reference and generated feature distributions, averaged over amplitudes, for U-Net models trained with

\mathrm{L1}

and

\mathrm{L2}

losses. The final column is the RMS of the high-frequency residual in non-spiking traces averaged over subthreshold amplitudes, for which the reference value is

0.003

mV (closer to this is better).

Figure 8 : Wasserstein-2 distance between reference and generated feature distributions, as a function of solver and NFE. Both axes are log-scaled.

Figure 9 : Empirical distribution of DC step amplitudes in

[-0.11,0.28]

nA considered.

Sine: $a\sin\!\left(2\pi f(t-s)/1000+\phi\right)$ for $s\leq t\leq s+d$ ; zero otherwise.
Basis	Quantity	Symbol	Distribution
	count	$N$	$\mathrm{Uniform}\{0,1,2\}$
	amplitude	$a$	$\mathrm{Uniform}(0.5,1.5)$
	frequency	$f$	$\mathrm{Uniform}(20,80)$ Hz
	phase	$\phi$	$\mathrm{Uniform}(0,2\pi)$ rad
	start time	$s$	$\mathrm{Uniform}(1,200)$ ms

Appendix

Table 5 : Construction of the custom current waveforms. For each waveform, the number of components of each basis type is sampled first from the discrete distribution shown. Conditional on a component being included, its parameters are sampled independently, each from the listed distributions. Times are relative to the activation window onset.

Figure 10 : Examples of sets of reference and generated traces for square pulse input currents to the PVALB neuron at different amplitudes. From left to right, columns show voltage traces given by (1) PDE simulations, (2) our model, and (3) NOBLE .

Figure 11 : Variance ratio (

\log_{2}\,\mathrm{Var(Generated)/Var(PDE)}

) and normalized mean bias

\Delta\mu/\sigma

for PVALB feature distributions at selected amplitudes.

Figure 12 : Wasserstein-2 distances between the reference and generated inv_ISI_values distributions for sine wave inputs of varying frequencies. Voltage responses are generated for sine waves with amplitudes 0.06nA, equilibrium position 0.05nA, and frequencies from 2 Hz to 500 Hz. Smoothed Wasserstein-2 distances are plotted for the DC-step-trained model (green) and the time-varying-trained model (orange).

Figure 13 : Additional examples of sets of reference and generated traces for unseen time-varying protocols. From left to right, columns show: (1) the input current trace, (2) the voltage traces generated by PDE simulations, (3) the voltage traces generated by our model when trained on time-varying current inputs, and (4) the voltage traces generated by our model when trained on DC step inputs.

Figure 14 : Examples of sets of reference and generated traces for square pulse input currents to the VIP neuron at different amplitudes. From left to right, columns show voltage traces given by (1) PDE simulations, (2) our model, and (3) NOBLE .

Regime	AP width ( $\mu$ s)	AHP depth (mV)	mean AP amplitude (mV)	inv ISI (Hz)	mean frequency (Hz)
All	37.7	0.905	3.00	2.514	2.264
Threshold	39.2	1.903	5.91	–	0.750
Low-firing	34.8	0.860	2.60	1.748	0.981
High-firing	39.3	0.907	3.16	2.939	3.009

Appendix

Table 6 : Performance by dynamical regime, VIP neuron. Wasserstein-2 distances between reference and generated feature distributions, averaged across amplitudes in different regime types. Inv ISI is omitted in the threshold regime, where firing responses contain too few spikes to yield interspike intervals.

Panel of Figure 15 — Figure 15 : A) Distribution of key electrophysiological features for a VIP neuron from experimental recordings (grey), PDE simulations (blue), NOBLE (red), and our model (green) across three representative amplitudes. Arrows indicate the fraction of values outside of the displayed axis range. B) Explained variance (EV, %) across quantile levels of generated and PDE distributions. C) $P(\mathrm{fire}\mid I)$ , the empirical firing probability given current amplitude for PDE simulations (blue; dashed), NOBLE (red), and our model (green).

Figure 16 : Variance ratio (

\log_{2}\,\mathrm{Var(Generated)/Var(PDE)}

) and normalized mean bias

\Delta\mu/\sigma

for VIP feature distributions at selected amplitudes.

Explore similar work

Learning Hybrid Biophysical Neuron Models with Neural ODEs
Jun 15, 2026Jonas Beck, Michael Deistler, Dóra Viktória Molnár +2Neural Ordinary Differential Equations Neurons
Hertie Institute for AI in Brain Health, Tübingen, Germany · Tübingen AI Center, University of Tübingen, Tübingen, Germany · Machine Learning in Science, Excellence Cluster Machine Learning, University of Tübingen, Tübingen, Germany +2
B[FM] $^2$ : Brain Foundation Model via Flow Matching with SplitUNet
Jun 18, 2026Jaedong Hwang, Kathleen Zhang, Wei Dai +5Electroencephalography Foundation Models Electrocardiogram Foundation Models
Massachusetts Institute of Technology · KU Leuven
Let EEG Models Learn EEG
May 20, 2026Yifan Wang, Yijia Ma, Wen Li +1Electroencephalography Generative Models
Stony Brook University · University of Texas Health Center at Houston.

Jun 15, 2026, q-bio.NCJ/K move · Enter open · S save

Learning Hybrid Biophysical Neuron Models with Neural ODEs

Jonas Beck, Michael Deistler, Dóra Viktória Molnár, Jakob H. Macke+1

Hertie Institute for AI in Brain Health, Tübingen, Germany · Tübingen AI Center, University of Tübingen, Tübingen, Germany · Machine Learning in Science, Excellence Cluster Machine Learning, University of Tübingen, Tübingen, Germany+2

Neural Ordinary Differential Equations Neurons Discontinuous Gating Kinetics Voltage Channel Modeling

Open paper AskPDF arXiv

Biophysical neuron models link measurements of neural activity to underlying cellular mechanisms. Yet, a central challenge is that the kinetics of many ion channels are poorly characterized, and practical simplifications -- omitting channels or reducing morphological detail -- introduce systematic gaps between model and biology. Bridging these gaps requires approaches that can flexibly discover unmodeled dynamics while preserving mechanistic interpretability. Here, we introduce a hybrid modeling framework that embeds neural ordinary differential equations into conductance-based biophysical models to capture unknown currents or mis-specified channel kinetics. By parameterizing the neural ODE in terms of voltage-dependent steady-state and time-constant functions, we recover interpretable gating dynamics directly from voltage recordings without assuming a functional form. We show that the hybrid model fits the gating kinetics of 2400 ion channel models and recovers unknown gating dynamics from single current-clamp recordings, generalizing to out-of-distribution stimulus regimes under realistic inputs and parameter misspecification. We also use our method to reduce a multicompartment model of a cortical neuron into a single-compartment hybrid model with a learned axial current, yielding up to an order of magnitude lower computational cost. Together, our results establish a plug-and-play framework for selectively replacing unknown components of conductance-based models with neural ODEs while preserving their mechanistic structure.

Abstract

Explore similar work

Learning Hybrid Biophysical Neuron Models with Neural ODEs

B[FM]2^22: Brain Foundation Model via Flow Matching with SplitUNet

Let EEG Models Learn EEG

B[FM] $^2$ : Brain Foundation Model via Flow Matching with SplitUNet