Amortized Inference of Neuron Parameters on Analog Neuromorphic Hardware

TL;DR

This work tackles parameter inference for seven adaptive exponential integrate-and-fire neuron parameters on analog neuromorphic hardware using amortized simulation-based inference. It compares two feature extraction strategies—handcrafted statistics and a learned summary network—within a BayesFlow framework, showing that the summary network yields more focused posteriors and posterior predictive traces that better capture membrane dynamics. By constraining the parameter space with a binary classifier, the study efficiently targets moderate spike-count regimes and demonstrates that amortized inference can generalize across observations and hardware runs. The results support the viability of amortized SBI for parameterizing complex analog neuron circuits, while acknowledging calibration and temporal-noise challenges that warrant further work and extensions to diffusion-based models and electrophysiology data.

Abstract

Our work utilized a non-sequential simulation-based inference algorithm to provide an amortized neural density estimator, which approximates the posterior distribution for seven parameters of the adaptive exponential integrate-and-fire neuron model of the analog neuromorphic BrainScaleS-2 substrate. We constrained the large parameter space by training a binary classifier to predict parameter combinations yielding observations in regimes of interest, i.e. moderate spike counts. We compared two neural density estimators: one using handcrafted summary statistics and one using a summary network trained in combination with the neural density estimator. The summary network yielded a more focused posterior and generated posterior predictive traces that accurately captured the membrane potential dynamics. When using handcrafted summary statistics, posterior predictive traces match the included features but show deviations in the exact dynamics. The posteriors showed signs of bias and miscalibration but were still able to yield posterior predictive samples that were close to the target observations on which the posteriors were constrained. Our results validate amortized simulation-based inference as a tool for parameterizing analog neuron circuits.

Figures (4)

• Figure 1: Target Observation and Dataset. (a) The top panel shows the distribution of firing rates for the uniform dataset as well as for the dataset which was constrained with a binary classifier. For the uniform dataset, only a small fraction of samples had a firing rate in the range between 140. With the help of a binary classifier, the number of samples with a firing rate in this range can be increased. The sample traces in the bottom panel show the diversity of observed firing patterns. (b) Trial-to-trial variation at the target parameterization $\bm{\theta_\text{T}}$. We display the observation $\bm{x_\text{T}}$ which is used in \ref{['sec:res:posterior']} for comparison in gray. The spike times are indicated by dots. The experiment was repeated several times with the same parameterization $\bm{\theta_\text{T}}$. The traces closely match each other at the beginning of the stimulus. As the experiments progressed, temporal noise resulted in noticeable differences in the recordings.
• Figure 2: Approximated Posterior and Posterior Predictive Samples.. (a) Two- and one-dimensional distributions of samples drawn from the approximated posteriors. The parameter set $\bm{\theta_\text{T}}$ used to generate the target observations $\bm{x_\text{T}}$ is indicated by lines. The axes span the entire range of the uniform prior distribution from 01022. The posterior trained with the handcrafted summary statistics (blue) and the one trained with a summary network (orange) both show higher sample densities near the target parameters $\bm{\theta_\text{T}}$. When using the summary network, the samples are more focused on these target parameters. A strong correlation between the exponential slope factor $\Delta_\text{T}$ and the exponential threshold $V_\text{T}$ can be observed. The values of the spike-triggered adaptation $b$ are for both approximations spread over a broad range. (b) For both posterior approximations, we drew 1000.0 random samples and plot the membrane response for the parameters with the highest posterior probability. While the exact spike timings vary, the overall spiking behavior of the posterior predictive samples is similar to the target observation: all samples show accommodation and have a similar spike count. In the case of the handcrafted summary statistics, the exact evolution of the membrane potential between spikes varies considerably from the target trace. When using a summary network, the voltage evolution more closely matches the target trace.
• Figure 3: Posterior Predictive Check. We used handcrafted summary statistics to compare the predictive posterior samples with the target observation. We drew 1000.0 parameters $\bm{\theta}$ from each posterior distribution and perform an experiment to record the observations $\bm{x}$. In the case of the target, we used the same parameterization $\bm{\theta_\text{T}}$ for all experiments. The boxes span the range from the first to the third quartile, and the whiskers span the farthest data point within the 1.5 interquartile range. The horizontal white lines mark the median value; the horizontal gray line indicates the features of the target observation $\bm{x}_\text{T}$. For most features, the posterior predictive samples show a similar behavior as the target. Some features such as the time to first spike $\Delta t_\text{spike}^\text{first}$ or the fast trough depth $V_\text{FT}$ as well slow trough depth $V_\text{ST}$ are underestimated in the posterior predictive samples. The variation in the posterior predictive samples is considerably larger than the trial to trial variations.
• Figure 4: Posterior Predictive Traces. We drew 8.0 random observations from the validation set and plotted the posterior predictive trace with the highest probability out of 10000.0 draws from the corresponding posterior distribution. The gray traces indicate the target observations, and the spikes are displayed as dots. The posterior based on handcrafted summary statistics (top/blue) can recover most features of the target traces. If no spikes are present or if the target observation includes spikes outside the stimulus region, the posterior predictive traces vary significantly from the target. When a summary network (bottom/orange) is used, these posterior predictive traces closely resemble the target traces. For the fourth observation, the posterior predictive trace of the posterior which includes a summary network does not spike, whereas the target observation spikes once.