Population Decoding Based on 
an Unfaithful Model 
S. Wu, H. Nakahara, N. Murata and S. Amari 
RIKEN Brain Science Institute 
Hirosawa 2-1, Wako-shi, Saitama, Japan 
{phwusi, hiro, mura, amari} @brain.riken.go.jp 
Abstract 
We study a population decoding paradigm in which the maximum likeli- 
hood inference is based on an unfaithful decoding model (UMLI). This 
is usually the case for neural population decoding because the encoding 
process of the brain is not exactly known, or because a simplified de- 
coding model is preferred for saving computational cost. We consider 
an unfaithful decoding model which neglects the pair-wise correlation 
between neuronal activities, and prove that UMLI is asymptotically effi- 
cient when the neuronal correlation is uniform or of limited-range. The 
performance of UMLI is compared with that of the maximum likelihood 
inference based on a faithful model and that of the center of mass de- 
coding method. It turns out that UMLI has advantages of decreasing 
the computational complexity remarkablely and maintaining a high-level 
decoding accuracy at the same time. The effect of correlation on the 
decoding accuracy is also discussed. 
1 Introduction 
Population coding is a method to encode and decode stimuli in a distributed way by us- 
ing the joint activities of a number of neurons (e.g. Georgopoulos et al., 1986; Paradiso, 
1988; Seung and Sompolinsky, 1993). Recently, there has been an expanded interest in 
understanding the population decoding methods, which particularly include the maximum 
likelihood inference (MLI), the center of mass (COM), the complex estimator (CE) and the 
optimal linear estimator (OLE) [see (Pouget et al., 1998; Salinas and Abbott, 1994) and the 
references therein]. Among them, MLI has an advantage of having small decoding error 
(asymptotic efficiency), but may suffers from the expense of computational complexity. 
Let us consider a population of N neurons coding a variable c. The encoding process 
of the population code is described by a conditional probability q(rlx ) (Anderson, 1994; 
Zemel et al., 1998), where the components of the vector r = {ri) for i = 1,-.-, N are 
the firing rates of neurons. We study the following MLI estimator given by the value of 
c that maximizes the log likelihood lnp(rlc), where p(rlc ) is the decoding model which 
might be different from the encoding model q(rlc ). So far, when people study MLI in a 
population code, it normally (or implicitly) assumes that p(rlc ) is equal to the encoding 
model q(rlc ). This requires that the estimator has full knowledge of the encoding process. 
Taking account of the complexity of the information process in the brain, it is more natural 
Population Decoding Based on an Unfaithful Model 193 
to assume p(r[x)  q(rlx). Another reason for choosing this is for saving computational 
cost. Therefore, a decoding paradigm in which the assumed decoding model is different 
from the encoding one needs to be studied. In the context of statistical theory, this is called 
estimation based on an unfaithful or a misspecified model. Hereafter, we call the decoding 
paradigm of using MLI based on an unfaithful model, UMLI, to distinguish from that of 
MLI based on the faithful model, which is called FMLI. The unfaithful model studied in 
this paper is the one which neglects the pair-wise correlation between neural activities. It 
turns out that UMLI has attracting properties of decreasing the computational cost of FMLI 
remarkablely and at the same time maintaining a high-level decoding accuracy. 
2 The Population Decoding Paradigm of UMLI 
2.1 An Unfaithful Decoding Model of Neglecting the Neuronal Correlation 
Let us consider a pair-wise correlated neural response model in which the neuron activities 
are assumed to be multivariate Gaussian 
1 exp[ 1 
q(rlx) = V/(27ra2)N det(A) --2a E A l(ri - fi(x))(D - fj(x))], (1) 
i,j 
where fi(:c) is the tuning function. In the present study, we will only consider the radial 
symmetry tuning function. 
Two different correlation structures are considered. One is the uniform correlation model 
(Johnson, 1980; Abbott and Dayan, 1999), with the covariance matrix 
A 0 =60+c(1-60), (2) 
where the parameter c (with - 1 < c < 1) determines the strength of correlation. 
The other correlation structure is of limited-range (Johnson, 1980; Snippe and Koenderink, 
1992; Abbott and Dayan, 1999), with the covariance matrix 
Aij = b li-jl, (3) 
where the parameter b (with 0 < b < 1) determines the range of correlation. This structure 
has translational invariance in the sense that Aij = Akt, if 
The unfaithful decoding model, treated in the present study, is the one which neglects the 
correlation in the encoding process but keeps the tuning functions unchanged, that is, 
1 exp[ 1 E(ri_ fi(z))2] ' (4) 
p(rlx) = V/(27rrr2) N -2a i 
2.2 The decoding error of UMLI and FMLI 
The decoding error of UMLI has been studied in the statistical theory (Akahira and 
Takeuchi, 1981; Murata et al., 1994). Here we generalize it to the population cod- 
ing. For convenience, some notations are introduced. Vf(r, x) denotes df(r,c)/dc. 
Eq[f(r,x)] and Vq[f(r,x)] denote, respectively, the mean value and the variance of 
f(r, x) with respect to the distribution q(rl:c ). Given an observation of the population 
activity r*, the UMLI estimate : is the value of :c that maximizes the log likelihood 
Lp(r*,x) = lnp(r*lx ). 
Denote by Xop t the value of x satisfying Eq[VLp(r, Xopt) ] = 0. For the faithful model 
where p: q, Xop t = x. Hence, (Xop t - x) is the error due to the unfaithful setting, 
whereas (: - Xopt) is the error due to sampling fluctuations. For the unfaithful model (4), 
194 S. Wu, H. Nakahara, N. Murata and S. Amari 
since Eq[VLp(r, xopt)]: O, ]i[fi(x) - fi(xopt)]f'(Xopt): O. Hence, Xop t: x and 
UMLI gives an unbiased estimator in the present cases. 
Let us consider the expansion of VLp (r*, ) at x, 
VLp(r*,) _ VLp(r*,x) + VVLp(r*,x) ( - x). 
Since VLv(r* , ) = O, 
1 
VVL(*,) (- ) _ --- 
(5) 
1 
N 7Lp (r*, x), (6) 
where N is the number of neurons. Only the large N limit is considered in the present 
study. 
Let us analyze the properties of the two random variables VVLp(r*,x) and 
VLp(r*, z). We consider first the uniform correlation model. 
For the uniform correlation structure, we can write 
ri - fi(x) + (i q- T]), (7) 
where r/and {ei}, for i = 1,..., N, are independent random variables having zero mean 
and variance c and 1 - c, respectively. r/is the common noise for all neurons, representing 
the uniform character of the correlation. 
By using the expression (7), we get 
1 1 
VLp(r*,x) = No 
1 
X77Lp(r*,x) = 
(8) 
(9) 
Without loss of generality, we assume that the distribution of the preferred stimuli is uni- 
form. For the radial symmetry tuning functions,  ]i f/(x) and  Y]i f'(x) approaches 
zero when N is large. Therefore, the correlation contributions (the terms of r/) in the above 
two equations can be neglected. UMLI performs in this case as if the neuronal signals are 
uncorrelated. 
Thus, by the weak law of large numbers, 
1 
VVLp(r*,x) 
where Qp _= Eq[VVLp(r,x)]. 
1 
i 
= N ' (10) 
According to the central limit theorem, VLp (r*, x)/N converges to a Gaussian distribution 
1--C 
VLp(r*,x)  N(O, /20. 2 E 
Gp 
= N(0, -ff), (11) 
where N(0, t 2) denoting the Gaussian distribution having zero mean and variance t, and 
Op = 
Population Decoding Based on an Unfaithful Model 195 
Combining the results of eqs.(6), (10) and (11), we obtain the decoding error of UMLI, 
(: - X)UML I ,-, N(0, Q;2Gv) , 
(1 -c)a 2 
= N(0,  f?/-5 )' (12) 
In the similar way, the decoding error of FMLI is obtained, 
(-X)FL I  V(0, 
= V(0, ( - c) 
y.i f(x)2 ) , (13) 
which has the same form as that of UMLI except that Qq and G are now defined with 
respect to the faithful decoding model, i.e., p(rlz ) = q(rlz ). To get eq.(13), the condition 
Y'i f(z) = 0 is used. Interestingly, UMLI and FMLI have the same decoding error. This 
is because the uniform correlation effect is actually neglected in both UMLI and FMLI. 
Note that in FMLI, (q -- (]q -- Vq[VLq(rlx)] is the Fisher information. (-2Gq is the 
Cram(r-Rao bound, which is the optimal accuracy for an unbiased estimator to achieve. 
Eq.(13) shows that FMLI is asymptotically efficient. For an unfaithful decoding model, 
Qp and Gp are usually different from the Fisher information. We call Q;2Gp the gen- 
eralized Cram(r-Rao bound, and UMLI quasi-asymptotically efficient if its decoding error 
approaches Q-2Gp asymptotically. Eq.(12) shows that UMLI is quasi-asymptotic efficient. 
In the above, we have proved the asymptotic efficiency of FMLI and UMLI when the neu- 
ronal correlation is uniform. The result relies on the radial symmetry of the tuning function 
and the uniform character of the correlation, which make it possible to cancel the corre- 
lation contributions from different neurons. For general tuning functions and correlation 
structures, the asymptotic efficiency of UMLI and FMLI may not hold. This is because the 
law of large numbers (eq.(10)) and the central limit theorem (eq.(11)) are not in general 
applicable. 
We note that for the limited-range correlation model, since the correlation is translational 
invariant and its strength decreases quickly with the dissimilarity in the neurons' preferred 
stimuli, the correlation effect in the decoding of FMLI and UMLI becomes negligible when 
N is large. This ensures that the law of large numbers and the central limit theorem hold 
in the large N limit. Therefore, UMLI and FMLI are asymptotically efficient. This is 
confirmed in the simulation in Sec.3. 
When UMLI and FMLI are asymptotic efficient, their decoding errors in the large N limit 
can be calculated according to the Cram6r-Rao bound and the generalized Cram(r-Rao 
bound, respectively, which are 
((:b - x)2)UMLi 
or2 Yij Aijfi(x)fj(x) 
0-2 
, (14) 
(15) 
3 Performance Comparison 
The performance of UMLI is compared with that of FMLI and of the center of mass de- 
coding method (COM). The neural population model we consider is a regular array of N 
neurons (Baldi and Heiligenberg, 1988; Snippe, 1996) with the preferred stimuli uniformly 
distributed in the range [-D, D], that is, ci = -D + 2iD/(N + 1), for i: 1,..., N. The 
comparison is done at the stimulus x = 0. 
196 S. Wu, H. Nakahara, N. Murata andS. Amari 
COM is a simple decoding method without using any information of the encoding process, 
whose estimate is the averaged value of the neurons' preferred stimuli weighted by the 
responses (Georgopoulos et al., 1982; Snippe, 1996), i.e., 
: ._ i rii (16) 
Ei 
The shortcoming of COM is a large decoding erron 
For the population model we consider, the decoding error of COM is calculated to be 
a2 ']4j Aijcicj 
- x) )COM " [Y'i fi(x)]  ' 
(17) 
where the condition 5'i fi(x)ci = 0 is used, due to the regularity of the distribution of the 
preferred stimuli. 
The tuning function is Gaussian, which has the form 
fi(x) = exp[- (x - ci)2], 
2a 2 
(18) 
where the parameter a is the tuning width. 
We note that the Gaussian response model does not give zero probability for negative firing 
rates. To make it more reliable, we set ri = 0 when fi(x) < 0.11 (I z - cil > 3a), which 
means that only those neurons which are active enough contribute to the decoding. It is easy 
to see that this cut-off does not effect much the results of UMLI and FMLI, due to their 
nature of decoding by using the derivative of the tuning functions. Whereas, the decoding 
error of COM will be greatly enlarged without cut-off. 
For the tuning width a, there are N = Int[6a/d - 1] neurons involved in the decoding 
process, where d is the difference in the preferred stimuli between two consecutive neurons 
and the function Int[-] denotes the integer part of the argument. 
In all experiment settings, the parameters are chosen as a = I and a = 0.1. The decoding 
errors of the three methods are compared for different values of N when the correlation 
strength is fixed (c = 0.5 for the uniform correlation case and b = 0.5 for the limited-range 
correlation case), or different values of the correlation strength when N is fixed to be 50. 
Fig. 1 compares the decoding errors of the three methods for the uniform correlation model. 
It shows that UMLI has the same decoding error as that of FMLI, and a lower error than that 
of COM. The uniform correlation improves the decoding accuracies of the three methods 
(Fig. lb). 
In Fig.2, the simulation results for the decoding errors of FMLI and UMLI in the limited- 
range correlation model are compared with those obtained by using the Cramr-Rao bound 
and the generalized Cramr-Rao bound, respectively. It shows that the two results agree 
very well when the number of neurons, N, is large, which means that FMLI and UMLI 
are asymptotic efficient as we analyzed. In the simulation, the standard gradient descent 
method is used to maximize the log likelihood, and the initial guess for the stimulus is 
chosen as the preferred stimulus of the most active neuron. The CPU time of UMLI is 
around 1/5 of that of FMLI. UMLI reduces the computational cost of FMLI significantly. 
Fig.3 compares the decoding errors of the three methods for the limited-range correlation 
model. It shows that UMLI has a lower decoding error than that of COM. Interestingly, 
UMLI has a comparable performance with that of FMLI for the whole range of correlation. 
The limited-range correlation degrades the decoding accuracies of the three methods when 
the strength is small and improves the accuracies when the strength is large (Fig.3b). 
Population Decoding Based on an Unfaithful Model 197 
0015 r 0015 1 
, 
' 
-- FMLI, UMLI FULl, UMLI 
..... COM ..... COM 
 oo10 r- c 00o  '".. 
.,-  
oo , oo -... 
N C 
(a) 
(b) 
Figure 1' Comparing the decoding errors of UMLI, FMLI and COM for the uniform cor- 
relation model. 
0015 0015 r- 
' - CRB, b=0.5 -- GCRB, b=0.5 j 
, = , SMR, b=0.5  [ = , SMR, b=0.5 
--- CRB, b=0.8 --- GCRB, b=0.8 
7. .---suR b--os '  .---suR, b--o.8 , 
,'n ''.. ,'n , 
 0005 - t' 0005 
0000 0(300 J 
20 40 60 80 100 
N 
(b) 
(a) 
Figure 2: Comparing the simulation results of the decoding errors of UMLI and FMLI in 
the limited-range correlation model with those obtained by using the Cram(r-Rao bound 
and the generalized Cram(r-Rao bound, respectively. CRB denotes the Cram(r-Rao bound, 
GCRB the generalized Cramr-Rao bound, and SMR the simulation result. In the simula- 
tion, 10 sets of data is generated, each of which is averaged over 1000 trials. (a) FMLI; (b) 
UMLI. 
4 Discussions and Conclusions 
We have studied a population decoding paradigm in which MLI is based on an unfaithful 
model. This is motivated by the facts that the encoding process of the brain is not exactly 
known by the estimator. As an example, we consider an unfaithful decoding model which 
neglects the pair-wise correlation between neuronal activities. Two different correlation 
structures are considered, namely, the uniform and the limited-range correlations. The per- 
formance of UMLI is compared with that of FMLI and COM. It turns out that UMLI has a 
lower decoding error than that of COM. Compared with FMLI, UMLI has comparable per- 
formance whereas with much less computational cost. It is our future work to understand 
the biological implication of UMLI. 
As a by-product of the calculation, we also illustrate the effect of correlation on the decod- 
ing accuracies. It turns out that the correlation, depending on its form, can either improve 
or degrade the decoding accuracy. This observation agrees with the analysis of Abbott 
and Dayan (Abbott and Dayan, 1999), which is done with respect to the optimal decoding 
accuracy, i.e., the Cram6r-Rao bound. 
198 S. Wu, H. Nakahara, N. Murata andS. Amari 
0020 r 
(a) 
= = FMLI 
*-- UMLI /" 
o 03  - .... COM / "" 
 0 02 L /' , 
00 02 o4 06 08 10 
b 
(b) 
Figure 3: Comparing the decoding errors of UMLI, FMLI and COM for the limited-range 
correlation model. 
Acknowledgment 
We thank the three anonymous reviewers for their valuable comments and insight sugges- 
tion. S. Wu acknowledges helpful discussions with Danmei Chen. 
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