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Title: Adversarial Training of Neural Encoding Models on Population Spike Trains

Paper content:
Neural population responses to sensory stimuli can exhibit both nonlinear stimulus- dependence and richly structured shared variability.
Here, we show how adversarial training can be used to optimize neural encoding models to capture both the deterministic and stochastic components of neural population data.
To account for the discrete nature of neural spike trains, we use the REBAR method to estimate unbiased gradients for adversarial optimization of neural encoding models.
We illustrate our approach on population recordings from primary visual cortex.
We show that adding latent noise-sources to a convolutional neural network yields a model which captures both the stimulus-dependence and noise correlations of the population activity.
Neural population activity contains both nonlinear stimulus-dependence and richly structured neural variability.
An important challenge for neural encoding models is to generate spike trains that match the statistics of experimentally measured neural population spike trains.
Such synthetic spike trains can be used to explore limitations of a model, or as realistic inputs for simulation or stimulation experiments.
Most encoding models either focus on modelling the relationship between stimuli and mean-firing rates e.g. [1] [2] [3] , or on the statistics of correlated variability ('noise correlations'), e.g. [4] [5] [6] .
They are typically fit with likelihood-based approaches (e.g. maximum likelihood estimation MLE, or variational methods for latent variable models).
While this approach is very flexible and powerful, it has mostly been applied to simple models of variability (e.g. Gaussian inputs).
Furthermore, MLE-based models are not guaranteed to yield synthetic data that matches the statistics of empirical data, particularly in the presence of latent variables.
Generative adversarial networks (GANs) [7] are an alternative to fitting the parameters of probabilistic models.
In adversarial training, the objective is to find parameters which match the statistics of empirical data, using a pair of competing neural networks -a generator and discriminator.
The generator maps the distribution of some input random variable onto the empirical data distribution to try and fool the discriminator.
The discriminator attempts to classify input data as samples from the true data distribution or from the generator.
This approach has been used extensively to produce realistic images [8] and for text generation [9] .
Recently, Molano-Mazon et al. [10] trained a generative model of spike trains, and Arakaki et al. [11] , rate models of neural populations, using GANs.
However, to the best of our knowledge, adversarial training has not yet been used to train spiking models which produce discrete outputs and which aim to capture both the stimulusdependence of firing rates and shared variability.
We propose to use conditional GANs [12] for training neural encoding models, as an alternative to likelihood-based approaches.
A key difficulty in using GANs for neural population data is the discrete nature of neural spike trains: Adversarial training requires calculation of gradients through the generative model, which is not possible for models with a discrete sampling step, and hence, requires the application of gradient estimators.
While many applications of discrete GANs use biased gradient estimators based on the concrete relaxation technique [13], we find that unbiased gradient estimators REINFORCE [14] and REBAR [15] lead to better fitting performance.
We demonstrate our approach by fitting a convolutional neural network model with shared noise sources to multi-electrode recordings from V1 [16] .
We here showed how adversarial training of conditional generative models that produce discrete outputs (i.e. neural spike trains) can be used to generate data that matches the distribution of spike trains recorded in-vivo, and in particular, its firing rates and correlations.
We used unbiased gradient estimators to train conditional GANs on discrete spike trains and spectral normalisation to stabilise training.
However, training of discrete GANs remains sensitive to the architecture of the discriminator, as well as hyper-parameter settings.
We showed that we are able to successfully train adversarial models in cases where supervised and Dichotomised Gaussian models fail.
In future, adversarial training could be used to capture higher-order structure in neural data, and could be combined with discriminators that target certain statistics of the data that might be of particular interest, in a spirit similar to maximum entropy models [4] .
Similarly, this approach could also be extended to capture temporal features in neural population data [20] such as spike-history dependence or adaptation effects.
Since we condition the discriminator on the input stimulus, adversarial training could be used for transfer learning across multiple datasets.
Generative models trained this way to produce realistic spike trains to various input stimuli, may be used to probe the range of spiking behaviour in a neural population under different kinds of stimulus or noise perturbations.

assistant: text: We show how neural encoding models can be trained to capture both the signal and spiking variability of neural population data using GANs.