Dimensionality Reduction Techniques for Neural Population Activity

Karthik Lakshmanan

(Work done at the Robotics Institute, advised by Professor Byron Yu)

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Recording from the Brain

• Recordings using multi electrode array

Photo Credit: Stanford Neural Prosthetics

Translational Laboratory

Video: A Human Patient

Collinger et al, Lancet, 2012

Outline

• Background
• Why Dimensionality Reduction?
• Existing Dimensionality Reduction methods for Neuroscience
• Looking for delays – TD-GPFA
• Results

What do Neural Signals look like?�

Neurons communicate via action potentials

Recordings from multi-electrode array

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Recordings from multi-electrode array

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20ms

Outline

• Background
• Why Dimensionality Reduction?
• Existing Dimensionality Reduction methods for Neuroscience
• Looking for delays – TD-GPFA
• Results

Recordings from multi-electrode array

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Trajectories for two different reach targets

• Dimensionality Reduction extracts a summary view of high-d data

Yu et al., K. Oweiss ed., Academic Press, 2010.

Rationale behind Dimensionality Reduction

• The brain has ~10^11 neurons

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• Neurons form networks

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• Each neuron cannot act independently due to network constraints

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• The brain has fewer degrees of freedom at its disposal than the number of neurons at play

Overview of Dimensionality Reduction

• Each neuron is a noisy sensor of an underlying low dimensional latent process

Yu et al., J Neurophysiol, 2009.

Outline

• Background
• Why Dimensionality Reduction?
• Existing Dimensionality Reduction methods for Neuroscience
• Looking for delays – TD-GPFA
• Results

Existing Methods for Neural Data

• Principal Component Analysis (PCA)

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• Probabilistic Principal Component Analysis (PPCA)

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• Factor Analysis (FA)

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• Gaussian Process Factor Analysis (GPFA)

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Model typically fit via Expectation Maximization (EM)� �Metric of choice: Likelihood of observations

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PCA, PPCA, FA

Yu et al., NIPS, 2009.

Yu et al., J Neurophysiol, 2009.

Linear Gaussian relationship from latents to neurons

(much like a Kalman Filter)

Limitations

• Neurons typically do not show sudden changes in firing rates

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• Underlying firing rates evolve smoothly in time

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• PCA, PPCA & FA do not account for temporal smoothing

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Gaussian Process Factor Analysis (GPFA)

Latent Variables

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Observations

(Spike Counts)

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FA

GP

Yu et al., NIPS, 2009.

Yu et al., J Neurophysiol, 2009.

Outline

• Background
• Why Dimensionality Reduction?
• Existing Dimensionality Reduction methods for Neuroscience
• Looking for delays – TD-GPFA
• Results

Multiple Pathways in the brain

M1

Related Work: Anechoic Blind Source Separation

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• Assumption: No echoes (“Anechoic”)
• p < q
• Omlor, JMLR, 2011
• Morup et al, NeuroImage, 2008
• Morup et al, ICA2007, 2007

p sound sources

q microphones distributed in space

Thinking about time delays

Our method: TD-GPFA

TD-GPFA applied to simulated data

TD-GPFA applied to real data

TD-GPFA in Publication

• Extracting Low-Dimensional Latent Structure from Time Series in the Presence of Delays
• Karthik C. Lakshmanan , Patrick T. Sadtler , Elizabeth C. Tyler-Kabara , Aaron P. Batista , Byron M. Yu
• Neural Computation (2015) 27 (9): 1825–1856

Link to publication

Appendix

Principal Component Analysis (PCA)

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• Find q ordered, mutually orthogonal directions of highest variance in high-d space

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• Pick the top p directions

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Probabilistic Principal Component Analysis (PPCA)

• Probabilistic extension of PCA

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• Isotropic noise model
• Latents
• Observations

Example Neural Data

• Neurons exhibit Poisson like behavior
• High mean firing rate => high variance

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Ground truth

PCA/PPCA Fail to Recover True Relationship

• Applying PCA/PPCA

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PCA/PPCA

Ground truth

Factor Analysis (FA)

• Independent noise model for each observation dimension
• Latents
• Observations

Factor Analysis (FA)

• Applying FA

PCA/PPCA

FA

Ground truth

TD-GPFA – Inference and Learning

• Since all variables are jointly Gaussian, inference can be performed exactly.

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• Learning is stable and approximation-free using the Expectation Maximization (EM) algorithm.

TD-GPFA – Inference and Learning

• E Step:

• is Gaussian and the conditional can be computed exactly

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TD-GPFA – Inference and Learning

• M Step:
• Maximize wrt

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Mixing Matrix

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Observation Noise Covariance

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Constant offset

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Timescales

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is the delay between neuron i and latent j

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Closed form

Solutions

Solved using

Gradient

methods