The relationship between optimal and biologically
plausible decoding of stimulus velocity
in the retina
Edmund C. Lalor,1,* Yashar Ahmadian,2and Liam Paninski2
1Trinity Centre for Bioengineering and Institute of Neuroscience, Trinity College Dublin,
College Green, Dublin 2, Ireland
2Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, New York 10027, USA
* Corresponding author: firstname.lastname@example.org
Received January 30, 2009; revised June 14, 2009; accepted July 23, 2009;
posted August 7, 2009 (Doc. ID 106996); published September 11, 2009
A major open problem in systems neuroscience is to understand the relationship between behavior and the
detailed spiking properties of neural populations. We assess how faithfully velocity information can be decoded
from a population of spiking model retinal neurons whose spatiotemporal receptive fields and ensemble spike
train dynamics are closely matched to real data. We describe how to compute the optimal Bayesian estimate of
image velocity given the population spike train response and show that, in the case of global translation of an
image with known intensity profile, on average the spike train ensemble signals speed with a fractional stan-
dard deviation of about 2% across a specific set of stimulus conditions. We further show how to compute the
Bayesian velocity estimate in the case where we only have some a priori information about the (naturalistic)
spatial correlation structure of the image but do not know the image explicitly. As expected, the performance of
the Bayesian decoder is shown to be less accurate with decreasing prior image information. There turns out to
be a close mathematical connection between a biologically plausible “motion energy” method for decoding the
velocity and the Bayesian decoder in the case that the image is not known. Simulations using the motion en-
ergy method and the Bayesian decoder with unknown image reveal that they result in fractional standard
deviations of 10% and 6%, respectively, across the same set of stimulus conditions. Estimation performance is
rather insensitive to the details of the precise receptive field location, correlated activity between cells, and
spike timing. © 2009 Optical Society of America
OCIS codes: 330.4060, 330.4150.
The question of how different attributes of a visual stimu-
lus are represented by populations of cells in the retina
has been addressed in a number of recent studies [1–8].
This field has received a major boost with the advent of
methods for obtaining large-scale simultaneous record-
ings from multiple retinal ganglion neurons that almost
completely tile a substantial region of the visual field
[9,10]. The utility of this new method for understanding
the encoding of behaviorally relevant signals was exem-
plified by , where the authors examined the question of
how reliably visual motion was encoded in the spiking ac-
tivity of a population of macaque parasol cells. These au-
thors used a simple moving stimulus and attempted to es-
timate the velocity of that stimulus from the resulting
spike train ensemble; this analysis pointed to some im-
portant constraints on the visual system’s ability to de-
code image velocity given noisy spike train responses. We
will explore these issues in more depth in this paper.
In parallel to these advances in retinal recording tech-
nology, significant recent advances have also been made
in our ability to model the statistical properties of popu-
lations of spiking neurons. For example, a statistical
model of a complete population of primate parasol retinal
ganglion cells (RGCs) was recently described . This
model was fit using data acquired by the array recording
techniques mentioned above and includes spike-history
effects and cross-coupling between cells of the same kind
and of different kinds (i.e., ON and OFF cells). The au-
thors demonstrated that the model accurately captures
the stimulus dependence and spatiotemporal correlation
structure of RGC population responses, and allows sev-
eral insights to be made into the retinal neural code. One
such insight concerns the role of correlated activity in pre-
serving sensory information. Using pseudorandom binary
stimuli and Bayesian inference, they reported that stimu-
lus decoding based on the spiking output of the model pre-
served 20% more information when knowledge of the cor-
relation structure was used than when the responses
were considered independently .
At the psychophysical level, Bayesian inference has
been established as an effective framework for under-
standing visual perception ; some recent notable ap-
plications to understanding visual velocity processing in-
clude [12–17]. In particular,  argued that a number of
visual illusions actually arise naturally in a system that
attempts to estimate local image velocity via Bayesian
methods (though see also [18,19]).
Links between retinal coding and psychophysical be-
havior have also been recently examined using Bayesian
methods; [20,21], for example, examine the contribution
of turtle RGC responses to velocity and acceleration en-
coding. This study reported that the instantaneous firing
rates of individual turtle RGCs contain information about
Lalor et al.
Vol. 26, No. 11/November 2009/J. Opt. Soc. Am. AB25
1084-7529/09/110B25-18/$15.00© 2009 Optical Society of America
speed, direction, and acceleration of moving patterns. The
firing-rate-based Bayesian stimulus reconstruction car-
ried out in that study involved a couple of key approxima-
tions. These included the assumptions that RGCs gener-
ate spikes according to Poisson statistics and that they do
so independently of each other. The work of  empha-
sizes that these assumptions are unrealistic, but the im-
pact of detailed spike timing and correlation information
on velocity decoding remains uncertain.
The primary goal of this paper is to investigate the fi-
delity with which the velocity of a visual stimulus may be
estimated, given the detailed spiking responses of the pri-
mate RGC population model of , using Bayesian decod-
ers, with and without full prior knowledge of the image.
We begin by describing the mathematical construction of
the Bayesian decoders, and then compare these estimates
to those based on a biologically plausible “net motion sig-
nal” derived directly from the spike trains without any
prior image information . We derive a mathematical
connection between these decoders and investigate the
decoders’ performance through a series of simulations.
The generalized linear model (GLM) [22,23] for the spik-
ing responses of the sensory network used in this study
was described in detail in . It consists of an array of ON
and OFF retinal ganglion cells (RGC) with specific base-
line firing rates. Given the spatiotemporal image movie
sequence, the model generates a mean firing rate for each
cell, taking into account the temporal dynamics and the
center-surround spatial stimulus filtering properties of
the cells. Then, incorporating spike history effects and
cross-coupling between cells of the same type and of the
opposite type, it generates spikes for each cell as a sto-
chastic point process.
In response to the visual stimulus I, the ith cell in the
observed population emits a spike train, which we repre-
sent by a response function
??t − ti,??,
where each spike is represented by a delta function, and
ti,?is the time of the ?th spike of the ith neuron. We use
the shorthand notation riand r for the response function
of one neuron and the collective spike train responses of
all neurons, respectively. The stimulus I represents the
spatiotemporal luminance profile I?n,t? of a movie as a
function of the pixel position n and time t.
In the GLM framework, the intensity functions (instan-
taneous firing rate) of the responses riare given by
?i?t? ? f?bi+ Ji?t? +?
hij?t − tj,???,
where f?·? is a positive, strictly increasing rectifying func-
tion. As in , we adopt the choice f?·?=exp?·?. The birep-
resents the log of the baseline firing rate of the cell, the
coupling terms hijmodel the within- and between-neuron
spike history effects noted above, and the stimulus input
Ji?t? is obtained from I by linearly filtering the spatiotem-
Ji?t? =??ki?t − ?,n?I??,n?d2nd?,
where ki?t,n? is the spatiotemporal receptive field of the
cell i. The parameters for each cell were fit using 7 min of
spiking data recorded during the presentation of a nonre-
peating stimulus, with the baseline log firing rate being a
constant and the various filter parameters being fit using
a basis of raised cosine “bumps” . Given Eq. (2), we can
write down the point process log-likelihood in the stan-
dard way 
log p?r?I? ??
log ??ti,?? −?
For movies arising from images rigidly moving with
constant velocity v we have
I?t,n? = x?n − vt?,
where x?n? is the luminance profile of a fixed image. Sub-
stituting Eq. (5) into Eq. (3) and shifting the integration
variable n by v?, we obtain
where we defined
Ki,v?t;n? ??ki?t − ?,n + v??d?.
In the following we replace p?r?I? with its equivalent
p?r?x,v? [since, via Eq. (5), I is given in terms of x and v]
and use the short-hand matrix notation Ji=Ki,v·x for Eq.
(6). An important point is that in the case of a convex and
log-concave GLM nonlinearity, f?·? [conditions that are
true for our choice, f?·?=exp?·?], the GLM log-likelihood,
Eq. (4), is a concave function of x?n?.
In order to estimate the speed of the moving bar given the
simulated output spike trains r of our RGC population,
we employed three distinct methods. The first method in-
volved a Bayesian decoder with full image information,
the second method utilized a Bayesian decoder with less
than full image information, while the third method in-
volved an “energy-based” algorithm introduced by  that
used no explicit prior knowledge of the image. For reasons
that will become clear, these decoders will be hereafter
known as the optimal decoder, the marginal decoder, and
the energy method, respectively. Given a simulated out-
put spike train ensemble, we use each of these methods to
estimate the speed of the stimulus that evoked the en-
semble by maximizing some function across a range of
possible or “putative” speeds.
1. Bayesian Velocity Estimation
To compute the optimal Bayesian velocity decoder we
need to evaluate the posterior probability for the velocity
B26 J. Opt. Soc. Am. A/Vol. 26, No. 11/November 2009Lalor et al.
p?v?r? conditional on the observed spike trains r. Given a
prior distribution pv?v?, from Bayes’ rule we obtain
If the image x (e.g., a narrow bar with a luminance dis-
tinct from the background) is known to the decoder, then
we can replace p?r?v? with the likelihood function
p?r?x,v? is provided by the forward model Eq. (4), and
therefore computation of the posterior probability is
straightforward in this case.
Alternatively, if the image is not fully known, we rep-
resent the decoder’s uncertain a priori knowledge regard-
ing x with an image prior distribution px?x?. In this case,
p?r?v? is obtained by marginalization over x:
p?r?v? =?p?r,x?v?dx =?p?r?x,v?px?x?dx.
Hence, we will refer to p?r?v? as the marginal likelihood.
Given the marginal likelihood, Eq. (8) allows us to calcu-
late Bayesian estimates for general velocity priors. The
prior distribution px?x? which describes the statistics of
the image ensemble, can be chosen to have a naturalistic
correlation structure. In our simulations in Section 3 we
use a Gaussian image ensemble with power spectrum
matched to observations in natural images [28,29].
In general, the calculation of the high-dimensional in-
tegral over x in Eq. (10) is a difficult task. However, when
the integrand p?r,x?v? is sharply peaked around its
maximum [which is the maximum a posteriori (MAP) es-
timate for x—as the integrand is proportional to the pos-
terior image distribution p?x?r,v? by Bayes’ rule] the so-
called “Laplace” approximation (also known as the
“saddle-point” approximation) provides an accurate esti-
mate for this integral [for applications of this approxima-
tion in the Bayesian setting, see e.g., ]. The Laplace
approximation in the context of neural decoding is further
discussed in, e.g., [31–35]. We briefly review this approxi-
Following , we consider Gaussian image priors with
zero mean and covariance Cxchosen to match the power
spectrum of natural images . Let us define the func-
L?x,r,v? ? log px?x? + log p?r?x,v? +
where d represents the number of pixels in our simulated
image, and rewrite Eq. (10) as
Using Eq. (4) and px?x?=N?0,Cx?, we obtain the expres-
L?x,r,v? = −
where ?iare given by Eqs. (2), (6), and (7), and we made
their dependence on x and r manifest. Since both terms in
Eq. (13) are concave (see the closing remarks in Subsec-
tion 2.A), the log-posterior L?x,r,v? is concave in x. To ob-
tain the Laplace approximation, for fixed r, we first find
the value of x that maximizes L (i.e., the image MAP,
xMAP). When the integrand is sharply concentrated
around its maximum, we can Taylor expand L around
xMAPto the first nonvanishing order beyond the zeroth-
order (i.e., its maximum value) and neglect the rest of the
expansion. Since at the maximum the gradient of L and
hence the first-order term vanish, we obtain
L?x,r? ? L?xMAP,r,v? −
2?x − xMAP?TH?r,v??x − xMAP?,
where the negative Hessian matrix
H?r,v? ? ? − ?x?xL?x,r,v??x=xMAP,
is positive semidefinite due to the maximum condition.
Exponentiating this yields the Gaussian approximation
(up to normalization)
eL?x,r,v?? p?x?r,v? ? N?xMAP?r,v?,Cx?r,v??,
where N??,C? denotes a Gaussian density with mean ?
and covariance C for the integrand of Eq. (12). [An impor-
tant technical point here is that this Gaussian approxi-
mation is partially justified by the fact that the log-
posterior (13) is a concave function of x [24,26,34] and
therefore has a single global optimum, like the Gaussian
(16).] Here, the posterior image covariance Cx?r,v? is
given by the inverse of the negative Hessian matrix
H?r,v?. (Note the dependence on both the observed re-
sponses r and the putative velocity v.) The elementary
Gaussian integration in Eq. (12) then yields
for the marginal likelihood, or its logarithm
log p?r?v? ? − L?xMAP?r,v?,r,v? −
The MAP itself is found from the condition ?xL=0, which
in the case of exponential GLM nonlinearity f?·?=exp?·?
yields the equation
Note that this equation is nonlinear due to the appear-
ance of xMAPinside the GLM nonlinearity on the right-
hand side. As mentioned above, the objective function Eq.
(11) is concave and can be efficiently optimized using
Lalor et al.
Vol. 26, No. 11/November 2009/J. Opt. Soc. Am. AB27
shifted ? and t1by t? to derive the second], and obtain
?2?v?dt1?d?Bx?v?t − t1??qi?− ?,v?? − t1??.
Thus, Ri?t? is a version of the response function of the
cell i offset by its baseline log firing rate biand smoothed
out on the time scale dictated by the largest of the spa-
tiotemporal scales of the receptive fields (via qi) or the cor-
relation length of typical images (via Bx)—with spatial
scales converted to time scales by dividing by v. To see
this more precisely, let us define ??1??, ??2?t1−?, and
??3?t−t1, such that t=??1+??2+??3. On the other hand,
because of the finite support of the factors of its inte-
grand, the double integral Eq. (A21) receives nonzero con-
tributions only when ???1???k, ???2??lk/v, and ???3?
?lcorr/v [where ?kand lkare the typical temporal and spa-
tial size of the receptive field filters ki?t,n?, respectively,
and lcorris the correlation length of typical images in the
naturalistic prior ensemble]. Thus if ?t?=??1+??2+??3? is
much larger than the sum of the three scales ?k, lk/v, and
lcorr/v, the filter w?t? is bound to vanish. This leads to the
discussion of Subsection 2.B.3 following Eq. (25).
APPENDIX B: O„D… DECODING
Here, we discuss how to implement the Newton–Raphson
optimization algorithm such that it finds the maximum a
posteriori estimate for the image xMAP[satisfying Eq.
(19)] in cases where the image depends only on one spa-
tial dimension and in a computational time that scales
only linearly with the spatial dimensionality of the image
vector d. The Newton–Raphson algorithm for minimizing
the function L?x? works as follows [we have in mind the
objective function defined in Eq. (13), but for simplicity we
drop r and v from its arguments in this appendix]. At
each iteration of this algorithm, starting from the vector
x, we change this vector by an amount ?x which is found
by solving the set of linear equations H?x??x=?xL?x?.
Here, the right-hand side is the gradient of L?x?, and H?x?
is its negative Hessian matrix [as in Eq. (15)], both evalu-
ated at x. In general, the solution of a set of d linear equa-
tions can be calculated in O?d3? elementary operations
. This would make the decoding of images with even
moderate angular extension forbidding. Fortunately, as
we will now explain, the quasi-locality of the GLM model
allows us to overcome this limitation. The negative Hes-
sian of L?x?, Eq. (13), is given by
H?x? = Cx
where the matrices Ji,t?x? have the elements
Ji,t?n1,n2;x? = Ki,v?t;n1?Ki,v?t;n2??i?t;x?dt,
and Ki,v?t;n? was defined in Eq. (7) in terms of the recep-
tive field filter of the cell i, ki??,n? [as we are considering
the one-dimensional image case, ki??,n? is understood to
be the full receptive field integrated along the transverse
spatial dimension]. Here, we turned the integral over t in
Eq. (13) into a discrete sum in Eq. (B1), as is done in the
numerical implementation, and for simplicity we wrote
Eq. (B2) for the case of exponential GLM nonlinearity.
Generalization of this equation and the rest of the argu-
ment to general nonlinearities is straightforward. Be-
cause of the finite spatial size of the receptive field and
the finite duration of the temporal filter, ki??,n? is nonzero
only when ???0,Tk? and n??nmin
are upper bounds on the cells’ temporal
integration windows and the size of the cells’ receptive
field surrounds, respectively. It follows then from Eq. (7)
+vTk(we assumed v?0, but generalization to v?0 is
straightforward). Thus Ji,t?n1,n2;x? vanishes if ?n1−n2?
??n+vTk, regardless of i and t. In other words, for all
?i,t?, Ji,t?x? are banded matrices with a band width of
?n+vTk, and so is their sum. If we further use a prior co-
variance Cxwith a banded inverse, then the full Hessian
Eq. (B1) will be banded [e.g., the naturalistic prior cova-
riance introduced after Eq. (A15) can be defined as the in-
verse of a tridiagonal matrix in the numerical implemen-
Unlike in the general case, the solution of a set of d lin-
ear equations with a banded equation matrix of band
width B can be found in a computational time ?B2d—i.e.,
in our case, in a computational time scaling only linearly
(as opposed to cubically) with the image size d. On the
other hand, we have observed empirically that the num-
ber of necessary Newton–Raphson iterations is more or
less constant and does not scale with d. Hence the overall
optimization procedure for finding xMAPcan be performed
in O?d? computational time. This allows us to decode the
velocity of large moving images. Similar methods with
O?d? computational cost have been used in inference and
[48,49], e.g., in applications to neural data analysis .
?, where Tkand
Thanks to E. P. Simoncelli for very detailed comments on
the manuscript, to J. Pillow for providing us with the pa-
rameters for the network model introduced in , and to
D. Pfau and E. J. Chichilnisky for many useful comments.
YA and LP are partially supported by NEI grant R01
EY018003 and by a McKnight Scholar award to LP. YA is
additionally supported by a Patterson Trust Fellowship in
Brain Circuitry. EL is supported by an Irish Research
Council for Science, Engineering and Technology (IRC-
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