Ionatan J. Kuperwajs Howard Hughes Medical Institute Janelia Research Campus, Turaga Lab

1. Bayesian Inference of Neural Activity and Connectivity from All-Optical Interrogation of a Neural Circuit Ionatan J. Kuperwajs Howard Hughes Medical Institute Janelia Research Campus, Turaga Lab

2. Talk Outline • Problem + Dataset, VI • Model Framework • Recognition + Generative Models • Additional Considerations • Results + Conclusions • Spike Inference • Connectivity • Next Steps • Model Improvements • Applicability

3. Problem +Project Goals • All-optical techniques • 2-photon optogenetics → drive spikes in neurons • Calcium imaging → recording of fluorescence at cellular resolution • Goal: infer circuit mapping + individual neuron spiking activity • General methodology • Global Bayesian inference strategy • Jointly infer distribution over spikes and unknown connections • Why can we do this? • GPU computing • Automatic differentiation libraries (i.e. Tensorflow, Theano) • Variational autoencoders – probabilistic Bayesian inference + deep learning Simultaneous + in vivo

4. Packer Optogenetic Data • Mouse data: awake, headfixed, on treadmill • Excitatory neurons (layer 2/3 of V1) • Spatial light modulator for 2-photon excitation of C1V1 opsin in subset of neurons • Simultaneous imaging of neural activity by 2-photon calcium imaging of GCaMP6s • Twice per second, stimulated five random cells and observed activity in rest of network • Facilitates discovery of a large portion of connections within the field of view • 7200 trials for one hour • Recording of spontaneous data for 40 minutes

5. Variational Inference Bayes’ Rule posterior z latent variables prior likelihood generative model joint x data

6. Variational Inference Bayes’ Rule posterior z latent variables prior likelihood generative model joint x data Model Evidence Log likelihood function where some parameters have been marginalized • Computationally intractable • Hindered application of Bayesian techniques to large-scale data VI solves this!

7. Variational Inference Bayes’ Rule Recognition Model VI + DNN, create posterior Q(z|x) intended to approximate P(z|x) posterior z latent variables prior likelihood generative model • Optimizing this bound improves recognition model • If bound is tight, Q(z|x) = P(z|x) joint x data Model Evidence Log likelihood function where some parameters have been marginalized • Computationally intractable • Hindered application of Bayesian techniques to large-scale data VI solves this!

8. Recognition Model Key Equations Factorize the approximate posterior over the spikes

9. Recognition Model Key Equations Factorize the approximate posterior over the spikes v(t) depends on the fluorescence trace and optogenetic input Neural network that takes a window of traces from t - T to t + T and nonlinearly maps as follows: 20 features  2 standard NN layers  single value

10. Framework Overview of the Generative Model Part 1: Generate spikes at time step t Time Step t - 1 Previous spikes Inputs from the rest of the brain External photostimulation

11. Framework Overview of the Generative Model Part 1: Generate spikes at time step t Time Step t - 1 Time Step t Previous spikes Inputs from the rest of the brain Current spikes External photostimulation

12. Framework Overview of the Generative Model Part 1: Generate spikes at time step t Part 2: Generate calcium fluorescence signals from the spikes Time Step t - 1 Time Step t Previous spikes 20 Ca Calcium Inputs from the rest of the brain Current spikes Observed calcium fluorescence signal External photostimulation

13. Generative Model Key Equations Generative Model ) Fluorescence signals at time t Reconstruction with Gaussian noise Learned, diagonal covariance matrix

14. Generative Model Key Equations Generative Model ) Fluorescence signals at time t Reconstruction with Gaussian noise Learned, diagonal covariance matrix Reconstruction Additive offset Optogenetic stimulation Convolution kernel Spikes Diagonal scaling matrix Convolution Kernel (unique to each cell)

15. Generative Model Key Equations Generative Model ) Spiking Depends on the Input Dynamical System Generates Low-Dimensional Input Representing Activity in the Rest of the Brain ) Fluorescence signals at time t Reconstruction with Gaussian noise Learned, diagonal covariance matrix Low-dimensional latents Weighted matrices Reconstruction Normalized, exponentially decaying kernel Additive offset External photostimulation Learned offset Optogenetic stimulation Convolution kernel Spikes Diagonal scaling matrix Convolution Kernel (unique to each cell)

16. Structured Priors on the GLM Weights Little data to support inferences on GLM weights. Must specify a prior and approximate posterior Sparse Model Dense Model

17. Structured Priors on the GLM Weights Little data to support inferences on GLM weights. Must specify a prior and approximate posterior Sparse Model Dense Model • Set p = 0.1 based on prior information • Optimized • Dramatic improvements over dense GLM • Biological connectivity is also sparse

18. Structured Priors on the GLM Weights Little data to support inferences on GLM weights. Must specify a prior and approximate posterior Sparse Model Dense Model • Simpler model • Optimized • Little benefit over a model with only low-rank activity • Set p = 0.1 based on prior information • Optimized • Dramatic improvements over dense GLM • Biological connectivity is also sparse

19. Modeling Off-Target Photostimulation Photostimulation may directly excite off-target neurons. Use a sum of five Gaussians with different scales, , to flexibly model distance-dependent stimulation.

20. Modeling Off-Target Photostimulation Photostimulation may directly excite off-target neurons. Use a sum of five Gaussians with different scales, , to flexibly model distance-dependent stimulation. This allows for stimulation to take on an elliptic pattern with a shifted center. This gives a similar spatial distribution to perturbation-triggered activity after inference.

21. Modeling Off-Target Photostimulation Photostimulation may directly excite off-target neurons. Use a sum of five Gaussians with different scales, , to flexibly model distance-dependent stimulation. This allows for stimulation to take on an elliptic pattern with a shifted center. This gives a similar spatial distribution to perturbation-triggered activity after inference.

22. Spike Inference for Spontaneous and Perturbed Data Spontaneous Data

23. Spike Inference for Spontaneous and Perturbed Data Spontaneous Data Perturbed Data • Noisy fluorescence trace • Denoised reconstructions • Inferred probability

24. Spike Inference for Spontaneous and Perturbed Data Spontaneous Data Perturbed Data • Noisy fluorescence trace • Denoised reconstructions • Inferred probability Stim-Triggered Averages

25. Spike Inference for Spontaneous and Perturbed Data Spontaneous Data Perturbed Data • Noisy fluorescence trace • Denoised reconstructions • Inferred probability • Directly stimulated cells • Large increase, slow decay driven by spiking activity • Modeled by inferred spikes Stim-Triggered Averages

26. Connectivity Predictions and Correlations Spike-and-Slab Probability Distributions Connection Strength Connection Probability

27. Connectivity Predictions and Correlations Correlation Coefficients Spike-and-Slab Probability Distributions Connection Strength Raw Calcium Traces Reconstructed Traces Connection Probability

28. Synaptic Connectivity Matrices Strength of Connection Probability of Connection

29. Synaptic Connectivity Matrices Strength of Connection Probability of Connection Combined Matrix

30. Synaptic Connectivity Matrices Strength of Connection Probability of Connection Combined Matrix Inferred Connectivity of an Individual Neuron

31. Synaptic Connectivity Matrices Strength of Connection Probability of Connection Combined Matrix Find neurons with highly correlated input profiles Inferred Connectivity of an Individual Neuron Neurons 6 + 35

32. Neurons with Similar Input Profiles Have Similar Spiking Patterns Neuron 63 Raw Traces Generated Reconstruction Optogenetic Perturbations

33. Neurons with Similar Input Profiles Have Similar Spiking Patterns Neuron 63 Raw Traces Generated Reconstruction Neuron 6 Optogenetic Perturbations Neuron 35

34. Confirming Synapse Predictions + Need for Sparser Inferences Reconstructions

35. Confirming Synapse Predictions + Need for Sparser Inferences Reconstructions

36. Confirming Synapse Predictions + Need for Sparser Inferences Reconstructions

37. Confirming Synapse Predictions +Need for Sparser Inferences Reconstructions Neuron 63 Neuron 35 Generated Reconstruction Raw Traces

38. Preliminary Conclusions • The data shows evidence of sparse, as opposed to dense, connectivity • Sparse GLM vs Dense GLM • Inferred sparse weights are consistent with known properties of neural circuits • Short-range connections are predominantly excitatory • Weights positively correlated with spontaneous correlation • Perturbed data supports stronger inferences than spontaneous data • Spike-and-slab posterior over weights • Perturbations increase the number of discovered connections • Joint inference is better than a pipeline • Allow low-rank activity, GLM connectivity, and external stimulation to influence spike inference This is the first fully Bayesian model of calcium imaging designed for perturbation data that is able to extract posteriors over a wide range of parameters with such efficiency

39. Improving the Model Upsampling Gumbel Method Expand Convolutional NN Nonlinearity

40. Improving the Model Upsampling Gumbel Method Expand Convolutional NN Nonlinearity Increase resolution of spike probabilities Calcium Traces Perturbations linear interpolation upsample factor of 2 interpolate with zeros upsample factor of 2

41. Improving the Model Upsampling Gumbel Method Expand Convolutional NN Nonlinearity Increase resolution of spike probabilities Replace analytical sampling Calcium Traces Perturbations • Can implement this with a Relaxed Bernoulli • Extends to Poisson distributions and nonlinear models relatively easily • Doesn’t extend to multi-sample objectives and VIMCO linear interpolation upsample factor of 2 interpolate with zeros upsample factor of 2 Sample from a Logistic distribution instead of a Normal/Gaussian Propagate through a sigmoid

42. Improving the Model More complex MLP Upsampling Gumbel Method decrease number of features per layer Expand Convolutional NN Nonlinearity increase number of hidden layers Increase resolution of spike probabilities Replace analytical sampling Calcium Traces Perturbations • Can implement this with a Relaxed Bernoulli • Extends to Poisson distributions and nonlinear models relatively easily • Doesn’t extend to multi-sample objectives and VIMCO linear interpolation upsample factor of 2 interpolate with zeros upsample factor of 2 Sample from a Logistic distribution instead of a Normal/Gaussian Propagate through a sigmoid

43. Improving the Model Nonlinear Generative Model More complex MLP Upsampling Gumbel Method • Biophysical properties of calcium in neurons • Linear: instant rise, exponential decay • Nonlinear: Incorporate intracellular dynamics • Possible with Gumbel • Ideally gives a better fit for data (this one might be too noisy) decrease number of features per layer Expand Convolutional NN Nonlinearity increase number of hidden layers Increase resolution of spike probabilities Replace analytical sampling Calcium Traces Perturbations • Can implement this with a Relaxed Bernoulli • Extends to Poisson distributions and nonlinear models relatively easily • Doesn’t extend to multi-sample objectives and VIMCO linear interpolation upsample factor of 2 interpolate with zeros upsample factor of 2 Sample from a Logistic distribution instead of a Normal/Gaussian Propagate through a sigmoid

44. Adaptation to Other Datasets • Refactor model to plug-and-play different spike-inference and fitting methods • Optogenetic datasets: • Synaptic blockers experiment (Packer/Hausser lab) • 10 cell stimulation experiment (mouse S1 cortex) • Dickson lab (fly fruitless circuit) • Svoboda lab (mouse ALM cortex) • Purely observational datasets: • Mrsic-Flogel lab (mouse V1 + ground truth synapses)

45. References Aitchison, L., Russell, L., Packer, A., Yan, J., Castonguay, P., Häusser, M., & Turaga, S. Model-based Bayesian inference of neural activity and connectivity from all-optical interrogation of a neural circuit.NIPS (2017, under review). Jang, E., Gu, S., & Poole, B. Categorical Reparameterization with Gumbel-Softmax. https://arxiv.org/abs/1611.01144 (2017). JJ Allaire, Dirk Eddelbuettel, Nick Golding, &Yuan Tang. tensorflow: R Interface to TensorFlow. https://github.com/rstudio/tensorflow (2016) Maddison, C. J., Mnih, A., & Teh, Y. W. The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables. https://arxiv.org/abs/1611.00712 (2017). Speiser, A., Yan, J., Archer, E., Buesing, L., Turaga, S., & Macke J.H. Fast amortized inference of neural activity from calcium imaging data with variational autoencoders. NIPS (2017, under review). Packer, A.M., Russell, L.E., Dalgleish, H.W., & Häusser, M. Simultaneous all-optical manipulation and recording of neural circuit activity with cellular resolution in vivo. Nature Methods. vol. 12, no. 2, pp. 140–146, (2015). Packer, A.M., Roska, B. & Hausser, M. Targeting neurons and photons for optogenetics. Nat. Neurosci. 16, 805– 815 (2013).

46. Acknowledgements Erik Snapp Srini Turaga HHMI, Janelia Jinyao Yan Laurence Aitchison JUS Program Adam Packer Thomas Mrsic-Flogel