Probing the neural axes for representing high-dimensional objects - PowerPoint PPT Presentation

Probing the neural axes for representing high-dimensional objects with simulated neuronal firing rates: A comparison among different strategies BENG 260 Final Project 12/8/2017 Huanqiu Zhang Bin Yu Yinan Xuan Mingyuan Chen

Problems of high-dimensional objects representations Faces Acoustic signals Chang & Tsao, Cell 2017; Kozlov & Gentner, PNAS , 2016

Neural representations Different from representations in most computer science algorithms: 1. Different biological significance 2. Further processing by downstream brain areas 3. Groups of neurons involved, with single neurons having high noise levels 4. Information encoding efficiency under metabolic constraints How can we find the axes used by neurons to represent high-dimensional objects?

Traditional Approaches Look at single neuron’s receptive fields (dimensions to which its responds) • STA (spike-triggered average) • STC (spike-triggered covariance) • MID (maximally informative dimensions) • MNE (maximum noise entropy ) • Etc. However, recent literature suggests that in order to encode high dimensional objects most efficiently, a population of neurons is needed to represent a common set of dimensions, and individual neurons in that particular population encodes several dimensions from that common set. 1,2,3,4

Outline • Simulation of neural spiking data • Linear-nonlinear Poisson (LNP) Model • Algorithms for recovering the neuronal axes • Modified Spike-Triggered Covariance (STC) Method • PCA in Dual Space • Perceptron Network • Robustness to non-Gaussian stimuli • Robustness to presence of noise

Simulation of neural spiking data • Fix the common set of dimensions used by the neuronal population to be e 1 , e 2 , …, e 20 . • For each dimension, we assign a random SD: σ 1 , σ 2 , …, σ 20 . • Generate 3000 random high-dimensional objects y 1 , y 2 , …, y 3000 . • Uses LNP model to determine the firing rates of each neuron.

Linear-nonlinear Poisson (LNP) Model 1. Each stimulus vector will be linearly filtered by . 2. Each of the output from the linear filters will be projected on a non-linear function N ( ). 3. Then, by summing over the output of N ( ), the instantaneous firing rate of a Poisson spike generator is determined. Schwartz et al ., Journal of Vision , 2006

Neurons as linear filters for different features/dimensions • Assume that we recorded the data from 200 neurons that those neurons are sharing the same 20 neural axes, and each neuron is only responsive to 1-5 of those neural axes. • Then we form a random 20 X 200 matrix E for representing how each neuron responds to different neural axes. (E ij = 1 if neuron j is responsive to variations along dimension i.)

Nonlinear function N ( ) • The distribution of the firing rate of a single neuron should match the cumulative distribution of input stimuli, for most efficient information coding. 5 Firing rate distribution • Sum the firing rates of a single neuron due to all of its encoding dimensions.

Outline • Simulation of neural spiking data • Linear-nonlinear Poisson (LNP) Model • Algorithms for recovering the neuronal axes • Modified Spike-Triggered Covariance (STC) Method • PCA in Dual Space • Perceptron Network • Robustness to non-Gaussian stimuli • Robustness to presence of noise

Spike-Triggered Covariance (STC) Method • Used on characterizing single neurons’ receptive fields Axis2 • Seeking directions in the stimulus space in which the variance of the spike-triggered ensemble differs from that of the Axis1 raw ensemble. Use PCA on the covariance matrix Schwartz et al ., Journal of Vision , 2006

Modified STC Method • View the entire neuronal population as a single neuron and try to determine its receptive fields (encoding directions)

Recovery accuracy measure • Remember: the common set of dimensions used by the neuronal population to be e 1 , e 2 , …, e 20 . • For the bases we are to recover, v 1 , v 2 , …, v 20 , take the maximum element of v i , v i and then average. Accuracy index = v i

Recovery Accuracy Result for modified STC method: Chance level

PCA in Dual Space Neural axis 3 axis 3 Neuron Neural axis 1 Object axis 2 Neural axis 2 axis 1 Object space Response space • Define a dual space (isomorphic to the object space) • For every neuron, determine its responsiveness to variations along each dimension by linear regression. • PCA in the response space

PCA in Dual Space

Perceptron Network

Perceptron Network

Outline • Simulation of neural spiking data • Linear-nonlinear Poisson (LNP) Model • Algorithms for recovering the neuronal axes • Modified Spike-Triggered Covariance (STC) Method • PCA in Dual Space • Perceptron Network • Robustness to non-Gaussian stimuli • Robustness to presence of noise

Non-Gaussian stimuli • Assume exponential distribution of data and change the nonlinear function N ( ) correspondingly. Modified STC method Dual space method

Outline • Simulation of neural spiking data • Linear-nonlinear Poisson (LNP) Model • Algorithms for recovering the neuronal axes • Modified Spike-Triggered Covariance (STC) Method • PCA in Dual Space • Perceptron Network • Robustness to non-Gaussian stimuli • Robustness to presence of noise

Robustness to presence of noise (neurons are noisy as mentioned before) Modified STC method Dual space method

Future Directions • Neuronal population encodes a smaller number of dimensions than dimension of the objects • Neural Network accuracy – find a better model to fit the data • Neural axes that are not orthogonal to each other

References 1. Bell, A.J. & Sejnowski, T.J. The “independent components” of natural scenes are edge filters. Vision Res . 37 , 3327–3338 (1997). 2. Fitzgerald, J.D. & Sharpee, T.O. Maximally informative pairwise interactions in networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80 , 031914 (2009). 3. Horwitz, G.D. & Hass, C.A. Nonlinear analysis of macaque V1 color tuning reveals cardinal directions for cortical color processing. Nat. Neurosci. 15 , 913–919 (2012). 4. Sharpee, T.O. & Bialek, W. Neural decision boundaries for maximal information transmission. PLoS ONE 2 , e646 (2007). 5. Sharpee, T.O. Toward functional classification of neuronal types. Neuron 83 , 1329–1334 (2014). 6. Schwartz, O., Pillow, J.W., Rust, N.C. & Simoncelli, E.P. Spike- triggered neural characterization. Journal of Vision 6(4) , 484– 507 (2006).