Neurocomputing 38}40 (2001) 769}774
Independent component analysis of temporal sequences
forms place cells
AndraH s Lo rincz *, GaH bor Szirtes , BaH lint TakaH cs , GyoK rgy BuzsaH ki
Department of Information Systems, Eo( tvo( s Lora& nd University, Pa& zma& ny Pe& ter se& ta& ny 1/D,
Budapest, H-1117, Hungary
Center for Molecular and Behavioral Neuroscience, Rutgers, The State University of New Jersey,
197 University Ave., Newark, NJ 07102, USA
Abstract
It has been suggested that sensory information processing makes use of a factorial code. It
has been shown that the major components of the hippocampal-entorhinal loop can be derived
by conjecturing that the task of this loop is forming and encoding independent components
(ICs), one type of factorial codes. However, continuously changing environment poses additional requirements on the coding that can be (partially) satis"ed by extending the analysis to
the temporal domain and performing IC analysis on concatenated inputs of time slices. We use
computer simulations to decide whether IC analysis on temporal sequences can produce place
"elds in labyrinths or not. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Hippocampus; Cognitive map; Generative network; Independent component
analysis; Temporal sequences
1. Introduction
It has been argued (see, e.g., [1,5], and references therein) that the processing of
sensory information makes use of a factorial code for "ltering. Such "lters would code,
e.g., the properties of natural scene ensembles that occur independently of each other.
The argument is supported by recent numerical studies [6,16,2,17]. It was found that
training "lters on natural images by using the related techniques of sparse representation formation and independent component (IC) analysis (ICA) produces receptive
* Corresponding author. Tel.: #36-1-463-3515; fax: #36-1-463-3490.
E-mail address: [email protected] (A. Lo rincz).
0925-2312/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 4 5 3 - 2
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A. Lo rincz et al. / Neurocomputing 38}40 (2001) 769}774
"elds alike to the receptive "elds of the primary visual cortex. However, if the analysis
is to be implemented by neural networks, a new condition arises, namely, inputs to the
analyzing network should be drawn independently. This is not ful"lled in a continuously changing environment. In other words, if temporal correlation between the
inputs exists then the inputs should be shu%ed before learning. Shu%ing requires the
storage of all of the inputs before learning could start. This condition means batch
learning and contradicts on-line learning. If inputs to the analyzing network are
provided in discrete time instances then ICA can be extended to temporal sequences
by concatenating a number of subsequent inputs. Concatenation will be called
embedding and the number of concatenated inputs as embedding dimension.
ICA performed in this larger dimensional space can project onto the original
(non-concatenated) number of dimensions simultaneously. This extension of ICA to
temporal sequences decreases temporal correlation between the components of the
representation within the time window of embedding. IC analysis on temporal
sequences may serve as the basis of a hierarchical IC procedure that creates factorial
code and removes temporal correlations in a systematic fashion. Interestingly,
Hateren and Ruderman found [7] that extending IC analysis to temporal sequences
improves the agreement between the emerging receptive "elds and the experimentally
found receptive "elds of the primary visual cortex.
It has been shown that the major components of the hippocampal-entorhinal loop
can be derived by conjecturing that the forming and the encoding of ICs are the tasks
of this loop [14,15]. According to the model, the hippocampal-entorhinal loop forms
a reconstructing dynamical network that makes use of orthogonal projections for
whitening and separation, and computes a top}down generative IC code, that forms
long-term memories. Here we examine one aspect of this model by asking the question
whether IC analysis on temporal sequences can produce place "elds.
2. Methods
A random model labyrinth was generated with a circular path. We have assumed
that sensory information at time t is made of local observations. At each point of the
labyrinth, the processed sensory information (PSI) consists of eight bits. The "rst four
bits indicate the presence or the absence of the walls at the north, east, south and west
side of the &animal'. The presence (absence) of a wall makes the respective input 1 (0).
The next four bits indicate the direction (north, east, south or west) of the path. The
resulting eight bit inputs are convolved with a 0.5, 1.0, 0.5 "lter along the path that
decreases the precision of local information. In the computations, the FastICA
algorithm [9] was utilized. Concatenated temporal sequences of PSIs were input to
the algorithm. Embedding depths varied from 1 (8 dimensional input) to 24
(24;8"192 dimensional input). The IC transformation matrix was trained on the
labyrinth of Fig. 1(a). Input concatenation corresponded to one (counter-clockwise,
referred to as forward) direction. After training, the generated matrix was used for
further analysis. A normalization procedure was used: The absolute values of the IC
outputs were added at time t and, in turn, each output was divided by this sum at each
A. Lo rincz et al. / Neurocomputing 38}40 (2001) 769}774
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Fig. 1. (a) The arrows show the random path used. Processed sensory information is made of two parts. The
"rst four digits show the presence or the absence of the walls to the south, east, north and west side of the
&animal'. The presence (absence) of a wall makes the respective input 1 (0) (see second inlet). The second four
digits show the direction of the path (south, east, north or west) (third inlet). The length of the path is 58
steps. (b) Histograms of IC output. Upper left: trained (original) direction, upper right: novel (reverse)
direction for embedding depths 8, 16, and 24. Lower left: exponential "ts to curves of the upper left "gure.
time instant. We assumed that such normalization is achieved, e.g., by inhibition [3].
IC outputs can be either positive or negative, which has many possible neural
interpretations. Olshausen and Field [17,16] say that deviation from the average
"ring rate provides the sign of the signal. Another interpretation can be provided by
positive coding networks [4]. That is why only the absolute values of the IC outputs
are shown in our "gures. Testing was performed by employing the trained ICA matrix
on several di!erent labyrinths. The "ring rate distributions in novel labyrinths were
the same (within statistical error) then the "ring rate distribution in the original
labyrinth with concatenation in reverse (clockwise. i.e., reverse) direction. Here, inputs
to the IC matrix in reverse direction will be referred to as &novel' inputs.
3. Results
The "rst experiment was concerned with the forming of place cells in the case when
the training inputs were generated both in forward and in reverse directions. Most ICs
formed localized &place cells' in the trained direction, which means their activity was
high at around a given point along the path. Alas, there was no sign change in and
around this point that supports the idea of positive coding [4]. In reverse direction,
none of the ICs can be termed &place cell'. In turn, these &receptive "elds' were direction
sensitive. That is, if an IC was strongly responding in one direction, then it was much
less responsive in the other (see also Fig. 2).
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Fig. 2. IC outputs when 24 embedding dimensions are used. The position along the path is shown be axis y.
Di!erent IC outputs are depicted at di!erent x values. The FastICA algorithm reduces the dimension of the
IC subspace to the highest non-redundant dimension. Left sub"gure: normalized IC outputs in the training
(forward) direction, center "gure: normalized IC outputs in the novel (reverse) direction. Right sub"gure:
normalized, thresholded and centered IC outputs in the forward direction. Threshold was set to 0.4 of the
maximal value. Centering means that for easier visualization the origin of the path was set di!erently for
each IC. It corresponded to the position where the output of the IC was the highest. Place cell formation is
clearly visible in this sub"gure.
The IC output distributions were analyzed. It was found that in the training
direction the IC outputs follow exponential distribution, whereas in the reverse
direction, the distributions are peaked at non-zero values and resemble to truncated
Gaussian distributions (Fig. 1(b)).
Fig. 2 depicts the absolute values of the IC outputs normalized when using 24
embedding dimension. We found that larger embedding depth improved place cell
formation. This was not a monotone feature however. Place cell formation can fully
disappear when embedding time depth corresponds to the round trip time of the
labyrinth.
4. Discussion
Our computer simulations demonstrate that place cells can be formed by IC
analysis of temporal sequences. These cells are formed according to the particular
form of training. In a one-dimensional labyrinth (&single path') ICs represent places
and directions. IC activities are peaked in localized regions and the activities are
conditioned on the direction of motion. Indeed, this is the case for the radial arm
maze, linear track and running wheel, i.e., when the animal traverses repeated paths.
We have studied this case here. Directed place cells were formed also when both
directions were used during the training session. The formation of invariant place
"elds that can develop in other scenes, e.g., in free two-dimensional areas is highly
non-trivial if only relative directions and relative position di!erences are provided for
IC analysis. One important issue is whether invariant place cells can be formed by IC
analysis alone when both local and global information are input to the algorithm.
This point remains to be seen.
A. Lo rincz et al. / Neurocomputing 38}40 (2001) 769}774
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Hippocampal principal cells may be able to analyze temporal sequences and thus
these cells may be able to perform IC analysis on temporal sequences. Distance on the
dendritic trees of principal cells measured from the soma in the CA3 and CA1
sub"elds can be viewed as time on the 10}100 ms time scale because of propagation
delays along the dendritic trees [8]. Recent modeling e!orts of Ja!e and Carnevale
[10] indicate that dendritic signals far from the soma are not damped relative to
dendritic signals close to the soma in principal cells of CA3. Arguments that propagation delays, in e!ect, make the dendritic tree to concatenate information from di!erent
time depths have also been published [13,11,12].
We found that in trained cases the distribution of IC component amplitudes follows
exponential behavior whereas in novel situations the distribution is peaked and is
more reminiscent to a truncated Gaussian distribution. In the brain, for certain sets of
inputs and for certain neurons, "ring rate distributions resemble better to exponential
distributions, whereas in other cases these resemble better to a truncated Gaussian
distributions [18]. Our "ndings suggest that the novelty content of the information
may play an important role in the "ring rate distribution. In this respect it is
important that neither exponential nor Gaussian distribution can be "tted to most of
the experimental data [18].
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Andras Lorincz is at Eotvos Lorand University, Budapest. His background is experimental and
theoretical physics and physical chemistry. He has received his Ph.D. from Eotvos Lorand university
in physics in 1978. He has been involved in arti"cial intelligence research for about 10 years. He is
presently leading the Neural Information Processing Group at the Department of Information Systems of
Eotvos Lorand University. Research "elds of the group include neurobiological foundation of AI and AI
applications.
Gabor Szirtes has been doing his Ph.D. studies in informatics in Andras Lorincz's
group at Eotvos Lorand University, Budapest. He took good place at National
Competition in Chemistry and Biology for High School Students. He graduated
in chemistry and besides studied physics as well. His major interests are brain
modelling, ANN, Independent Component Analysis and its applications.
Balint Takacs "nished his M.Sc. studies in biophysics last year and he has been
doing his M.Sc. studies in computer science and mathematics in Andras Lorincz's
group at Eotvos Lorand University, Budapest. His major interests are neurobiology, brain modelling and ANN.