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

• bjects 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 e1, e2, …, e20.

• For each dimension, we assign a random SD: σ1, σ2,

…, σ20.

• Generate 3000 random high-dimensional objects

y1, y2, …, y3000.

• 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

• n a non-linear function N( ).
• 3. Then, by summing over the
• utput 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.

(Eij = 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

• Sum the firing rates of a single neuron due to all of

its encoding dimensions.

Firing rate distribution

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

Schwartz et al., Journal of Vision, 2006

• Used on characterizing single

neurons’ receptive fields

• Seeking directions in the

stimulus space in which the variance of the spike-triggered ensemble differs from that of the raw ensemble. Use PCA on the covariance matrix

Axis1 Axis2

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 e1, e2, …, e20.

• For the bases we are to recover, v1, v2, …, v20, take

the maximum element of vi , vi and then average. Accuracy index = vi

Recovery Accuracy Result for modified STC method:

Chance level

PCA in Dual 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

Object space Response space

Object Neuron Neural axis 1 Neural axis 2 Neural axis 3 axis 1 axis 2 axis 3

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”
• f 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).