Independent component analysis of correlated neuronal responses in area MT

1

Independent component analysis of correlated neuronal responses in area MT

Jiann-Ming Wu

, Chia-Yi Lu

, Cheng-Yuang Liou

Abstract – This work explores independent com- ponents of correlated firing in area MT. The pair- wise time-varying firing rate of two neighboring MT neurons in response to the same stimulus is estimated by the spline approximation to aver- aged spike trains over trials, and processed by the PottsICA algorithm for recovering independent sources. Numerical results show independent com- ponent analysis of correlated firing able to retrieve the effective source whose behaviors are highly con- sistent with variation of the stimulus.

Keywords– mutual information, population code, correlated firing, MT neurons, blind source separa- tion

I. Introduction

Correlated firing of neighboring neurons in ar- eas from retina to visual cortex have been reported in [1]-[3]. Correlated neuronal activities have been considered as crucial materials for explor- ing population encoding of the stimulus. In this work, we employ independent component analy- sis[4][5][6] to retrieve independent sources of cor- related neuronal responses and examine the con- sistency between behaviors of the extracted inde- pendent source and variation of the stimulus.

The data that we analysis was published by the authors of the work[2]. The data, filed as emu084 in the homepage[7], contains pair-wise spike trains measured from two neighboring MT neurons in re- sponse to stochastic motions. For each trial, the stimulus realized by stochastic motions of tremen- dous dots on a video screen is characterized by a coherence parameter whose sign specifies two- alternative moving directions of coherent dots and whose absolute scalar corresponds to the number of coherent dots. Every 45 milliseconds during a

†Department of Applied Mathematics, National Dong Hwa University

∗Department of Computer Science and Information engi- neering, National Taiwan University

Correspondence: Jiann-Ming Wu, Department of Ap- plied Mathematics, National Dong Hwa University, Hualien,Taiwan. Tel. 8863-8633531, FAX : 8863-8633510, email: jmwu@mail. ndhu.edu.tw

trial, according to the absolute value of the coher- ence parameter, a portion of dots on the screen are randomly selected as coherent dots, all of which are programmed to move along the direction spec- ified the sign of the coherence parameter, and the remains are considered as random dots, each of which moves along a random direction by a small displacement.

In the previous work[2], it has been shown that each MT neuron has its own preference to the co- herent direction of stochastic motions. The exper- imenter can thus select two neighboring MT neu- rons which have similar preferred directions, and measure their correlated firing in response to a va- riety of stimuli characterized by variant coherence parameters. In the experiment[2], the two possi- ble signs of the coherence parameter respectively denote the preferred and anti-preferred directions of the monitored MT neurons.

II. Materials and methods

The data contains experimental results of 420 trials. For each trial j, the pair-wise spike train can be represented by {xj[t]}^Nt=1, where N de- notes the number of total time steps during a trial, xj[t] = (xj1[t] xj2[t])^T denotes the response of the two neurons at the tth time step, and xjk[t] ∈ {0, 1} for all j, k, t. By the representa- tion, a neuron generates at most one spike at each time step.

The experiment uses 15 possible coherence pa- rameters, denoted by C = {cⁱ}¹⁵i=1, for 420 trials.

Let c(j) ∈ C denote the coherence parameter used at trial j. The averaged pair-wise spike train over all trials with coherence parameters identical to ci

is expressed as follows, ξ_i[t] = 1

n_i X

j:c(j)=ci

xj[t],

where ni denotes the size of the set {j|c(j) = cⁱ}.

2

The averaged pair-wise spike train, {ξi[t]}^t, is further processed by the spline interpolation to form the time-varying firing rate, {λⁱ[t]}^Nt=1, of the two neurons in response to the stimulus corre- spondent to ci,where λi[t] = (λi1[t], λi2[t])^T.The spline approximation is carried out by the built-in spline tools in the MATLAB package. The time- varying firing rate of the two neurons is shown in figure 1, where the plot in figure 1a or 1b contains 15 curves. A curve in figure 1a represents the fir- ing rate of one neuron, {λⁱ¹[t]}^Nt=1, and in figure 1b represents the firing rate of the other neuron, {λⁱ²[t]}^Nt=1,in response to ci.

Independent component analysis is further em- ployed to retrieve independent sources from obser- vations, {λⁱ[t]}^Nt=1,separately for each i. Since the ICA process is the same for all i, the subindex i is omitted in the following presentation. Assume that the time-varying firing rate of the two neu- rons in responding to the stimulus correspondent to c ∈ C are linear mixtures of two independent sources, represented by

λ[t] = As[t], (1)

where A denotes the unknown mixing matrix and {s[t]}^Nt=1 denotes the unknown independent sources. An ICA algorithm aims to search for an effective demixing matrix W, by which the linear transformation,

γ[t] = Wλ[t], (2)

could recover independent sources.

Let γ = (γ₁,γ₂)^T denote a random vector that characterizes estimated sources {γ[t]}^Nt=1. The ef- fectiveness of W can be quantified by the KL- divergence between the joint pdf of γ and the product of marginal pdfs of components in γ, ex- pressed by

KL (γ) = Z

R²

p (γ) ln p (γ)

p1(γ₁) p2(γ₂)dγ

where p denotes the joint pdf of γ and pk denotes the marginal pdf of γ_k. If the two components in γ are independent, KL(γ) reduces to zero. Since KL(γ)must be non-negative, it measures the mu- tual information or dependency among compo- nents in γ. The transformation in equation (2) will

make a difference between KL(λ) and KL(γ), ex- pressed by

D(W)= KL(λ)^∆ − KL(γ),

which quantifies the reduced dependency caused by the transformation (2). Following the fact that the pdf of γ is the product of the pdf of λ and

| det(W)|⁻¹,the difference can be rewritten as fol- lows,

D(W) =X

k

H(λk) + ln| det(W)| −X

k

H(γ_k), (3) where the marginal entropy of univariate λ is de- fined by

H(γ_k) =− Z

R

pk(γ_k) ln pk(γ_k) dγ_k, (4)

and det(W) denotes the determinant of W.

Here we use the PottsICA algorithm[6] to es- timate W and D(W). The PottsICA algorithm uses the normalized histogram to represent the marginal pdf of each γ_k.By the marginal pdf rep- resentation, minimization of KL(γ) with respect to W turns tractable and can be realized by neural relaxation based on a hybrid of mean field anneal- ing and gradient descent methods. The marginal entropies in D(W) can be also estimated based on the representation of normalized histograms for marginal pdfs. So we can estimate D(W) for the demixing matrix obtained by the PottsICA algo- rithm.

III. Numerical results and Discussions For each coherence parameter ci in C, the time- varying firing rate of the monitored MT neurons, represented by {λⁱ[t]}^Nt=1, is processed by the PottsICA algorithm, and is then transformed to the independent firing rate, {γi(t)}^Nt=1,by the ob- tained demixing matrix Wi. The quantity D(Wi) that measures the reduced dependency by the demixing transformation is shown in figure 2 for all i, where the horizontal axis measures the co- herence parameter.

It is observed that the scale of the reduced de- pendency appears graded when the coherence pa- rameter is respectively set high negative, low nega- tive, low positive and high positive. The evidence

3

for dependent firing of the two MT neurons be- comes stronger as the the coherence parameter increases. For high negative coherence, the ob- tained D value is zero; the two MT neurons tend to have independent firing. This is because the co- herent direction specified by negative coherence is the anti-preferred direction of the monitored MT neurons.

For low negative coherence, the evidence for de- pendent firing is still weak. However when the co- herence parameter is set low positive, the evidence for dependent firing becomes stronger relative to the case with negative coherence. With positive coherence, the coherent direction of stochastic mo- tions is close to the preferred direction of the two MT neurons. In the occasion, both the individual firing rates of the two MT neurons are encoded with informations about the positive stimulus. An ICA algorithm helps to achieve two independent sources, one of which is expected to carry with most informations encoded within the two indi- vidual firing rates.

Two independent sources extracted from the time-varying firing rate of the two MT neurons are shown in figure 3. Since the PottsICA algorithm possesses the order-preserving property[6], we can relate the first component of γ_i to one source and the second component of γ_i to the other source for all i. The response of two independent sources to each ci is shown in figure 3. A curve in figure 3a displays the sequence of {λⁱ¹[t]}^Nt=1 for approx- imating the response of one source, and in figure 3b draws the sequence of {λⁱ²[t]}^Nt=1 for approxi- mating the response of the other source to ci.

From figure 3a, it is observed that the 15 curves within the time interval at about [200, 350] form four clusters, respectively corresponding to high negative, low negative, low positive and high posi- tive coherence. The response of this source to low positive coherence becomes distinguishable from that to low negative coherence. The response of the source in figure 3a is significantly consistent with variation of the stimulus. The same consis- tency can not be found in figure 1a and 1b, where observed firing rate of the two MT neurons is dis- played. The extracted source in figure 3a have been shown to carry with most informations en- coded within the two individual firing rates.

By independent component analysis of corre-

lated firing of the two MT neurons, we have es- timated an effective source that encodes most in- formations within the firing rate of pair-wise neu- rons for distinguishing variation of the stimulus.

The signals of neuronal activities that encode the stimulus are partially contained by the measured individual spike train. The ICA algorithm plays a role of extracting significant signals from corre- lated firing of neighboring neurons following the assumption of linear mixtures. In the near future, we will explore independent component analysis for more paired MT neurons for further investiga- tion to population codes of neuronal activities.

References

[1] Britten, K.H., Shadlen, M.N., Newsom, W.T. & Movshon, J.A., The analysis of visual motion: A comparison of neu- ronal and psychophysical performance, Journal of Neuro- science 12: 4745-4765, 1992.

[2] Bair, W., Zohary, E., Newsome, W.T., Correlated firing in macaque visual area MT: time scales and relationship to be- havior, The journal of Neuroscience, 21(5):1676-1697, 2001.

[3] Nirenberg, S., Carcieri, S.M., Jacobs, A.L., Latham, P.E., Retinal ganglion cells act largely as independent encoders, Nature, 411:698-701, 2001.

[4] Cardoso, J.,Blind signal separation:statistical principles, Proceedings of the IEEE, Vol. 9, No. 10, PP. 2009-2025, OCT.,1998.

[5] Hyvarinen, A.,Fast and robust fixed-point algorithms for in- dependent component analysis, IEEE Transactions on Neu- ral Networks 10(3):626-634, 1999.

[6] Wu, J.M. and Chiu,S.J., Independent component analysis using Potts models, Transactions on neural networks, vol.12 No.2 march 2001

[7] http://www.cns.nyu.edu/~wyeth/index.html.

Figure list

Fig. 1 The pair-wise time-varying firing rate of two neighboring MT neurons estimated by the spline approximation to the averaged spike trains over trials.

Fig. 2 The dependency reduced by the demixing matrix.

Fig. 3 Independent components of the pair-wise time-varying firing rate of two MT neurons.

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0 -0 . 0 2

0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 2 0 . 1 4

m illis e c o n d s

firing firing

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0

0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8

m i l l i s e c o n d s

firing firing

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0

- 0 . 0 1 0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8

m i l l i s e c o n d s

firing firing

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0

0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6

m i l l i s e c o n d s

firing firing

Reduced dependency

Low positive coherence

Low negative coherence Figure 1

Figure 2

Figure 3

a b

a b