Models of the nervous system aim to achieve biologically detailed simulations of large-scale neuronal activity through the incorporation of both structural connectomes (connectivity maps) and individual neural dynamics. The nervous system of Caenorhabditis elegans is considered a framework for such a model as the connectome of its somatic nervous system for multiple types of interaction is mapped (White et al., 1986; Varshney et al., 2011; Cook et al., 2019). In addition to the connectome, advances in electrophysiological methodology allow the recording of whole-cell responses of individual neurons. These advances provide biophysically relevant details of individual neuro-dynamical properties and warrant a type of model for the C. elegans nervous system incorporating both the connectomes and individual biophysical processes of neurons. Such a model could be referred to as ElectroPhysiome, as it incorporates a layer of individual neural dynamics on top of the layer of inter-cellular interactions facilitated by the connectome.
The development of nervous system models that are further biophysically descriptive for each neuron, that is, modeling neurons using the Hodgkin–Huxley type equations (HH-model), requires fitting a large number of parameters associated with ion channels found in the system. For a typical single neuron, these parameters could be tuned via local optimizations of individual ion channel parameters estimated separately to fit their respective in vivo channel recordings such as activation/inactivation curves (Hodgkin and Huxley, 1952; Willms, 2002; Willms et al., 1999; Nicoletti et al., 2019; Liu et al., 2018; Jiang et al., 2022). Such a method requires multiple experiments to collect each channel data, and when such experiments are infeasible, the parameters are often estimated through hand-tuning. In the context of developing the ElectroPhysiome of C. elegans, the method would have to model approximately 300 neurons each including an order of hundreds of parameters associated with up to 15 to 20 ionic current terms (with some of them having unknown ion channel composition), which would require large experimental studies (Nicoletti et al., 2019). Furthermore, the fitted model may not be the unique solution as different HH-parameters can produce similar neuron activity (Marder and Goaillard, 2006; Marder and Taylor, 2011; Prinz et al., 2003; Prinz et al., 2004). As these limitations also apply for general neuron modeling tasks beyond C. elegans neurons, there has been an increasing search for alternative fitting methods requiring less experimental data and manual interventions.
A promising direction in associating model parameters with neurons has been the simultaneous estimation of all parameters of an individual neuron given only electrophysiological responses of cells, such as membrane potential responses and steady-state current profiles. Such an approach requires significantly less experimental data per neuron and offers more flexibility with respect to trainable parameters. The primary aim of this approach is to model macroscopic cell behaviors in an automated fashion. Indeed, several methods adopting the approach have been introduced. Buhry et al., 2012 and Laredo et al., 2022 utilized the differential evolution (DE) method to simultaneously estimate the parameters of a 3-channel HH-model given the whole-cell membrane potential responses recording (Buhry et al., 2012; Laredo et al., 2022). Naudin et al., 2022b further developed the DE approach and introduced the multi-objective differential evolution (DEMO) method to estimate 22 HH-parameters of three non-spiking neurons in C. elegans given their whole-cell membrane potential responses and steady-state current profiles (Naudin et al., 2022b). The study was a significant step toward modeling whole-cell behaviors of C. elegans neurons in a systematic manner. From a statistical standpoint, Wang et al., 2022 used the Markov Chain–Monte Carlo method to obtain the posterior distribution of channel parameters for HH-models featuring three and eight ion channels (two and nine parameters, respectively) given the simulated membrane potential responses data (Wang et al., 2022). From an analytic standpoint, Valle et al. 2022 suggested an iterative gradient descent-based method that directly manipulates the HH-model to infer three conductance parameters and three exponents of activation functions given the measurements of membrane potential responses (Valle and Madureira, 2022). Recent advances in machine learning gave rise to deep learning-based methods which infer steady-state activation functions and posterior distributions of three-channel HH-model parameters inferred by an artificial neural network model given the membrane potential responses data (Gonçalves et al., 2020; Estienne, 2021).
While these methods suggest that simultaneous parameter estimation from macroscopic cell data is indeed possible through a variety of techniques, it is largely unclear whether they can be extended to fit more detailed HH-models featuring a large number of channels and parameters (e.g., C. elegans neurons) (Nicoletti et al., 2019). Furthermore, for most of the above methods, the algorithms require an independent (from scratch) optimization process for fitting each individual neuron, making it difficult to scale up the task toward a large number of neurons.
Here, we propose a new machine learning approach that aims to address these aspects for the class of non-spiking neurons, which constitute the majority of neurons in C. elegans nervous system (Goodman et al., 1998). Specifically, we develop a deep generative neural network model (GAN) combined with a recurrent neural network (RNN) Encoder called ElectroPhysiomeGAN (EP-GAN), which directly maps electrophysiological recordings of a neuron, for example, membrane potential responses and steady-state current profiles, to HH-model parameters of arbitrary dimensions (Figure 1). EP-GAN can be trained with simulation data informed by a generic HH-model encompassing a large set of arbitrary ionic current terms and thus can generalize its modeling capability to multiple neurons. Unlike typical GAN architecture trained solely with adversarial losses, we propose to implement an additional regression loss for reconstructing the given membrane potential responses and current profiles from generated parameters, thus improving the accuracy of the generative model. In addition, due to the RNN component of EP-GAN, the approach supports input data with missing features such as incomplete membrane potential responses and current profiles.
Estimation of HH-model parameters from membrane potential and steady-state current profiles.
Given the membrane potential responses (V) and steady-state current profiles (IV) of a neuron, the task is to predict biophysical parameters of the Hodgkin–Huxley-type neuron model (left). We use the Encoder-Generator approach to predict the parameters (right).
We validate our method to estimate HH-model parameters of 200 simulated non-spiking neurons followed by applying it to three previously recorded non-spiking neurons of C. elegans, namely RIM, AFD, and AIY. Studies have shown that membrane potential responses of these neurons can be well modeled with typical HH-model formulations with 22 parameters (Naudin et al., 2022b; Naudin et al., 2021). We show that when trained with a more detailed HH-model consisting of 16 ionic current terms resulting in 175 trainable parameters, EP-GAN can predict parameters reproducing their membrane potential responses with higher accuracy in the reconstruction of membrane potential with significantly faster inference speed than existing algorithms such as differential evolution and genetic algorithms. Through ablation studies on input data, we show that EP-GAN retains its prediction capability when provided with incomplete membrane potential responses and steady-state current profiles. We also perform ablation studies on EP-GAN architecture components to elucidate each component’s contributions toward the accuracy of the predicted parameters. To further test EP-GAN, we estimate HH-model parameters for six newly recorded non-spiking C. elegans neurons: AWB, AWC, URX, RIS, DVC, and HSN, whose membrane potential responses were not previously modeled.
Our results suggest that EP-GAN can learn a translation from electrophysiologically recorded responses and propose projections of them to parameter space. EP-GAN method is currently limited to non-spiking neurons in C. elegans as it was designed and trained with the HH-model describing the ion channels of these neurons. EP-GAN applications can potentially be extended toward resolving neuron parameters in other organisms since non-spiking neurons are found within animals across different species (Koch et al., 2006; Roberts and Bush, 1981; Davis and Stretton, 1989a; Davis and Stretton, 1989b; Burrows et al., 1988; Laurent and Burrows, 1989a; Laurent and Burrows, 1989b; Naudin, 2022a).