Connectome-constrained Latent Variable Model of Whole-Brain Neural Activity

Brain-wide measurements of neural activity combined with detailed measurements of anatomical connectivity of the C. elegans nervous system in principle allow for the development of detailed mechanistic computational models . However , there are several challenges . We often do not have direct experimental access to important modeling details such as single-neuron dynamics and the signs and strengths of the synaptic connectivity . Further , neural activity can only be measured in a subset of neurons , often indirectly via calcium imaging , and significant trial-totrial variability has been observed . To overcome these challenges , we introduce a connectome-constrained latent variable model ( CC-LVM ) of the unobserved voltage dynamics of the entire C. elegans nervous system and the observed calcium signals . We used the framework of variational autoencoders to fit parameters of the mechanistic simulation constituting the generative model of the LVM to calcium imaging observations . A variational approximate posterior distribution over latent voltage traces for all neurons is efficiently inferred using an inference network , and constrained by a prior distribution given by the biophysical simulation of neural dynamics . When applied to a recent dataset , we find that connectomic constraints enable our LVM to predict the activity of neurons whose activity were withheld . We explored models with different degrees of biophysical detail , and found that models with the most realistic conductance-based synapses provide markedly better predictions than current-based synapses for this system . 1 INTRODUCTION . The anatomical connectivity of the entire C. elegans nervous system , including both chemical and electrical synapses , has been known for several decades [ 25 ; 23 ; 26 ] . However , well-calibrated and predictive connectome-constrained mechanistic models of this nervous system have yet to be demonstrated [ 10 ; 24 ; 22 ; 7 ; 5 ] . This is because currently available experimental data are insufficient to completely constrain computational models . First , the single-neuron and synapse dynamics are generally unknown . Second , the connectome does not directly inform the signs and strengths of individual synapses . Third , the response properties of sensory neurons are incompletely known . Further , it is unclear what level of biophysical detail is necessary to reproduce the essential computational function of the C. elegans nervous system . We use recently available whole brain calcium imaging data to constrain the missing parameters in a connectome-constrained , biophysically detailed model of the C. elegans nervous system . We start with a simplified non-spiking passive point-neuron model of the voltage dynamics of individual neurons in the circuit . We model inputs to the neurons from electrical synapses , and model nonlinear chemical synapses with either current-based or conductance-based biophysics . The challenge of fitting such a model to data is two-fold . First , the voltage dynamics of the neurons are not directly observed , but rather indirectly measured via their slow calcium dynamics . Second , there is significant trial-to-trial variability in neural activity , suggesting strong initial state dependence in the neural responses even to the same sensory stimulus [ 6 ] . These issues can both be addressed by treating the collective voltage signals of all neurons in the nervous system as an unobserved latent variable whose dynamics are determined by the simplified connectome-constrained biophysical model with unknown neuronal and synaptic parameters . Our connectome-constrained latent variable model ( CC-LVM ) of voltage dynamics of the entire C. elegans nervous system is a large-scale latent variable model with a very high-dimensional latent space consisting of voltage dynamics of 300 neurons over 5 minutes of time . Many sources say that C. elegans has 302 neurons , but two of them are not connected to the rest of the nervous system , so we only model the 300 that are [ 26 ] . The generative model for these latent variables are described by stochastic differential equations modeling the nonlinear dynamics of the network activity , and a novel differential equation describing how the calcium signals signals are generated from the voltage dynamics of individual non-spiking neurons . We developed a variational autoencoder based framework for inferring the unobserved voltage dynamics from the observed calcium dynamics of only a subset of the neurons in the nervous system . An inference network enables efficient variational inference of the latent variables , and fitting the unknown neuronal and synaptic parameters of our connectome and biophysics based generative model . The compact nervous system of C. elegans makes it an ideal platform for systems neuroscience . The C. elegans nervous system consists of 300 neurons , divided into 118 distinct classes ( often bilaterally symmetric pairs ) [ 21 ; 11 ; 20 ] . These neurons can largely be categorized as sensory neurons , interneurons , and motor neurons . The majority of these neurons , about 200 , are concentrated in the head , forming the brain of the worm . The synaptic connectivity of C. elegans has been mapped with electron microscopy , providing researchers with a complete connectome [ 25 ; 26 ] . We apply the CC-LVM to whole-brain calcium imaging data which captures 170 of the 300 neurons in multiple worms as they respond to chemosensory stimuli [ 27 ] . In principle , an accurate model of the nervous system constrained by incomplete activity measurements but complete connectivity measurements can enable accurate predictions of activity in neurons which were not recorded . We tested this hypothesis by using the CC-LVM to predict the activity of neurons which were measured , but whose activity was withheld during model training . We also used the model to predict the activity of entire worms , by holding out single trials during training . The CC-LVM predicted activity of withheld neurons and withheld worms significantly better than models unconstrained by the connectome , demonstrating the utility of the connectome even when little is known about the signs and strengths of individual connections . Further , we found that models with conductance-based synapses provide superior predictions to models with current-based synapses . CC-LVMs thus provide a new tool for connectome and activity constrained modeling of neural circuits and for discovering the appropriate level of detail of biophysical model for a given system . 1.1 PRIOR WORK . Previous work has been focused on creating network simulations based on anatomical connectome data , with unknown single neuron synaptic biophysics [ 13 ; 5 ; 24 ; 15 ; 10 ] . These network models provided a holistic view of the entire nervous system and were validated by comparing simulated locomotion with movements of live animals [ 8 ] . However , they were not fit against prerecorded calcium fluorescence data , so their simulated neuronal activities are not quantitatively confirmed . Recent advances in machine learning have enabled increasingly sophisticated latent variable models to uncover structure from data generated by advanced neural interfacing technologies [ 17 ] . This framework has been used in other contexts to infer neuronal voltage dynamics from high-dimensional calcium fluorescence recordings [ 16 ; 1 ; 19 ] . Despite their utility , these models do not account for connectomic data which may greatly improve their predictive power . Our work employs both LVMs and connectomic constraints to create a model that is informed by both neural population dynamics and network structure . Recently , Bayesian models have been widely applied in neuronal datasets to infer spiking activity , neural dynamics , and connectivity [ 24 ; 1 ; 19 ] . In particular , Warrington et al . [ 24 ] . use sequential Monte Carlo ( SMC ) to impute the intracellular voltage potentials of C. elegans neurons from 49 recorded calcium traces [ 9 ] . Their work combined neuronal , body and calcium observation simulators in order to model locomotion [ 3 ; 14 ] . Their simulator produces a series of exemplar neuronal voltage traces but they do not evaluate the imputed traces of unmeasured neurons against real data so the accuracy of their inferences is unknown . Furthermore , they explore how different methods for approximate Bayesian inference affect parameter estimation on simulated data and show that their method is best . Our model uses a variational auto-encoder ( VAE ) to perform inference instead of SMC which is more computationally efficient because it allows direct sampling from the approximate posterior . Additionally , the CC-LVM is trained on significantly more neurons , and we validate our model by evaluating its ability to predict the activity of neurons held out from the training data . Finally , rather than study how various inference methods can affect parameter estimation in a test autoregressive model , we search a space of generative models to optimize voltage predictions by comparing them to real calcium data . 2 CONNECTOME-CONSTRAINED LATENT VARIABLE MODEL . We constructed a connectome-constrained latent variable model ( CC-LVM ) of the C. elegans nervous system , where each node in our network represents a specific neuron in the animal . The activity of each neuron was modeled with a latent variable analogous to voltage . The dynamics of each neuron in the network is represented with a stochastic non-spiking passive leaky integrator equation with learned time constants and resting membrane potentials [ 4 ] . The neurons were coupled by both chemical and electrical synapses with learned weights [ 4 ] . We allowed these weights to be non-zero only where the connectome indicates the existence of synapses . Given a set of learned parameters ( weights , time constants , and resting membrane potentials ) , the dynamics of the network defines a prior distribution over neural activity trajectories . Importantly , the stochastic nature of the dynamics allows for deviations from perfectly deterministic dynamics . This allows the model to accommodate the observed variability in single-neuron dynamics . This variability has several potential sources , including the unmeasured initial states of the neurons and our incomplete knowledge of the sensory inputs driving the nervous system . A latent variable model of this scale with a nonlinear generative model defined by the stochastic dynamics is difficult to fit . To address this challenge , we used the probabilistic inference framework of variational auto-encoders ( VAE ) [ 18 ] to train a black-box voltage inference network to predict a posterior distribution over neural activity trajectories . We then used this inference network to train the parameters of the CC-LVM . The resulting LVM has a biologically realistic generative model of the nonlinear neural dynamics of the C. elegans nervous system , and a black-box temporal convolutional inference network which , given sensory stimulus and calcium imaging measurements , predicts a factorized Gaussian distribution over the voltages of all the neurons in the network . Several variants of the LVM were developed : we tested models in which the synaptic connections were modeled as either current-based or conductance-based [ 4 ] , and evaluated different levels of connectome constraint . The LVM was optimized with the ELBO ( evidence lower bound ) objective . 2.1 NETWORK MODEL WITH PASSIVE POINT-NEURON VOLTAGE DYNAMICS . Neurons in the C. elegans nervous system are largely non-spiking [ 2 ] , so we model the voltage dynamics for these neurons as a passive point neurons with a single electrical compartment . Let v ∈ RN denote the voltages of the N neurons . The voltage vi ( t ) for each post-synaptic neuron i at the time t was calculated using a first-order leaky integrator equation given by τiv̇i ( t ) + vi ( t ) = s c i ( t ) + s e i ( t ) + v rest i + hi ( t ) , ( 1 ) where τi is the voltage time constant , hi is the chemosensory input provided to only the sensory neurons , sci is the chemical synaptic input , s e i is the electrical synaptic input , v rest i is the resting neuron voltage . We studied two variations of the model , the current-based model and the conductance-based model , which differ in their formulations of chemical synaptic input sci . Since neurons in the C. elegans nervous system are largely non-spiking , we model the chemical synapses as having graded release of neurotransmitter , rather than the all-or-none quantal release seen in spiking neurons . In both models , we model the amount of neurotransmitter released as proportional to the pre-synaptic voltage vj , following softplus activation g ( · ) which sets a minimum voltage below which there is no synaptic release , leading to W cjig ( vj ( t ) ) . We use g ( · ) , to denote a softplus function for the rest of the paper . In our current-based model , synaptic input sci to a post-synaptic neuron i is directly proportional pre-synaptic neurotransmitter concentration : sci ( t ) = N∑ j W cjig ( vj ( t ) ) , ( 2 ) where W cji represents the chemical synaptic weight between pre-synaptic neuron j and post-synaptic neuron i. W cji can be positive or negative depending on if the synaptic connection is excitatory or inhibitory . If neurons j and i are not connected , W cji is set to zero . In the conductance-based model , we model the synaptic current entering the post-synaptic neuron with more biophysical detail as sci ( t ) = N∑ j ( Eji − vi ( t ) ) W cjig ( vj ( t ) ) . ( 3 ) Here , the pre-synaptic neurotransmitter concentration W cjig ( vj ( t ) ) is more accurately modeled as proportional to the conductance at the post-synaptic terminal . The post-synaptic current is then given by the product of the synaptic conductance and the difference between the post-synaptic voltage vi ( t ) and the synaptic reversal potential Eji . In contrast to the current-based synapse whose input is independent of the post-synaptic voltage vi ( t ) , the conductance-based synapse model more accurately also has a dependence on the post-synaptic voltage . In addition , this model decouples the sign of the synapse from the strength . The reversal potential of a synapses Eji dictates whether a synapse is excitatory or inhibitory . A large and positive Eji corresponds to an excitatory synapse causing depolarization of the post-synaptic neuron . And an inhibitory synapse will have a negative Eji causing hyperpolarization . In this model , we can now independently train the sign of a synapse and its non-negative strength W cji , which is not easily possible with current-based synapses . In both the current-based and conductance-based models , the following equation was used to represent electrical synaptic inputs : sei ( t ) = N∑ j W eji ( vj ( t ) − vi ( t ) ) , ( 4 ) where W eji is restricted to be non-negative and vj − vi is the potential difference between presynaptic and postsynaptic neurons . We restrictW eji = W e ij because the potential differences between electrical synapses are symmetric . To directly compare the outputs of the LVM to neural activity measurements , our model must generate calcium signals from the voltage traces . We model the calcium concentration [ Ca ] i of each neuron i as a first-order leaky integrator , driven by voltage-gated calcium channels with the same nonlinear current-voltage ( I-V ) function g ( vi ) : τ [ Ca ] [ Ċa ] i ( t ) + [ Ca ] i ( t ) = g ( vi ( t ) ) , ( 5 ) where g ( · ) represents softplus activation , τ [ Ca ] is a time constant shared across all neurons . We then map calcium concentration [ Ca ] into the measured calcium fluorescence signals f via an affine transform with scalar αf and bias βf , fi ( t ) = α f [ Ca ] i ( t ) + β f + σf fi ( t ) , ( 6 ) with measurement noise represented by a noise amplitude σf and a noise term f i ( t ) ∼ N ( 0 , 1 ) .

Summary: The paper develops a latent variable model with biologically meaningful parameters for the worm *C. elegans* whole brain neural traces. Given the connectivity between the neurons, the neural traces are modeled as stochastic leaky integrators with either conductance or current based synaptic input model. Given the voltage traces, the observed calcium signals are modeled as first order leaky integrators added by noise to account for the observation noise. External input is nonlinearly transformed and fed into the neurons to account for the stimulus-dependent modulation of neural activity. Multiple connectivity models (fully trainable, fully trainable with $L_2$ regularization, trainable weight magnitudes, trainable global scale) are used to investigate whether biological connectome constraints help with neural activity predictions or not. Variational inference is used to infer a posterior distribution over neural activity trajectories with factorized Gaussian distribution as the variational family. Results on neuron-hold-out and worm-hold-out experiments suggest that connectome constraints and conductance-based modeling improves model prediction over unseen neurons and worms.