NEURAL SCALE-INVARIANT TIME-FREQUENCY DECOMPOSITION FOR DETECTION
OF SPATIOTEMPORAL FEATURES
Zoran Tiganj, Karthik H. Shankar, Marc W. Howard
Center for Memory and Brain, Boston Universtiy
ABSTRACT
along a Weber-Fechner scale; this representation can then be
used to support a variety of behaviors including representaA growing body of evidence from neuroscience and cognition of time, space and sequences as well as episodic memory
tive science suggests that the brain maintains a scale-invariant
[12, 13]. The defining property of the Weber-Fechner repretimeline in which events are represented on a logarithmicallysentation is that the error in the time dimension grows linearly
spaced Weber-Fechner scale. This Weber-Fechner represenwith the time in the past, consistent with a logarithmicallytation is scale-invariant – meaning that the array gives the
spaced scale. Intuitively, the difference between 10 and 11
same relative error for signals at any scale – and resource
is the same as the difference between 100 and 110. A scaleconserving, with the amount of resources necessary to repreinvariant representation appears to be an adaptive response to
sent a particular timescale going up logarithmically with that
a world that contains information at many scales.
timescale. In addition to this time-domain representation, it
Let us consider the problem of spectral decomposition of
is in many cases desirable to have a frequency-domain reprea
continuous
time-varying signal f (t) from the perspective of
sentation with these properties. We show that starting from a
an
organism
facing
a natural environment. Assume that, like
logarithmically-spaced timeline, a scale-invariant band-pass
a
brain,
we
do
not
have access to the entire signal and that
decomposition with log-spaced central frequencies can be
the
resources
necessary
to maintain memory for the signal at
readily constructed. The method is neurally plausible and
any
moment
are
costly.
If we are to estimate the instantacan be computed on-line. This approach can be used in neuneous
power
of
the
signal
in a particular frequency band ωo
romorphic engineering to extract spatio-temporal features of
using
standard
methods,
we
would need to maintain on the
interests, for instance slowly varying input components.
order of ωo−1 time points of the signal. That is, the resources
Index Terms— Band-pass decomposition, Scale-invariance, necessary to estimate a particular frequency band grow linLog-spacing, Time-frequency representation
early with the period of that frequency band. In addition, for
a continuously-varying natural signal we do not a priori know
the
frequency (or frequencies) that will be important for our
1. INTRODUCTION
survival. A neurally-plausible time-local method for scaleinvariant frequency-decomposition that required logarithmic
Numerous cognitive experiments, but also our everyday exresources would enable an organism to represent a very wide
perience, demonstrate that humans can remember the time at
range of frequencies with minimal resources.
which events were experienced with gradually decreasing acThe solution we propose here builds on a neurallycuracy over a wide temporal range, up to tens or even hunplausible
method for constructing a Weber-Fechner timeline
dreds of seconds [1, 2, 3, 4]. The gradual decay without
[14].
In
this
method, a set of leaky integrators indexed by
abrupt changes in the accuracy suggests that the same mechatheir
rate
constant
s maintain at each moment the Laplace
nism might be used to store the memory over such wide temtransform
of
the
input
signal leading up to that moment. This
poral range. It is widely believed that working memory is
operation
is
time-local
meaning that the only storage of the
represented through the maintenance of action potential firsignal
is
through
the
set
of s values that are maintained. The
ing (see for example [5]).
intrinsic
properties
of
neurons
can give rise to a broad range
Natural signals, including auditory, visual and natural lanof
time
scales
[15,
16];
these
intrinsic
properties can be exguage signals show correlations over a wide range of time
ploited
to
give
rise
to
leaky
integrators
with a range of rate
scales [6, 7, 8, 9]. Perhaps as a response to this rich strucconstants
in
a
biophysically
realistic
simulation
[17].
ture, different regions of the human brain contain temporal
receptive fields that respond to features across a range of disThe biologically plausible computational method for
tinct scales [10, 11]. These considerations have led to the
spectral decomposition presented here operates on the outputs
hypothesis that at each moment the brain maintains working
of the neural leaky integrators (effectively low-pass filters that
memory as an estimate of the time at which events took place
constitute working memory) and constructs a bank of band-
pass filters. The decomposition is scale-invariant and when
the ratio of the adjacent rate constants is a constant it provides log-spacing of the central-frequencies of the band-pass
filters. The method for constructing such representation is detailed in the next section. We also reflect on the construction
of a Weber-Fechner timeline and show that this band-pass
decomposition yields a Weber-Fechner frequency-line with
analogous properties to the timeline, only in the frequency
domain. Additionally we show that the frequency-line can
also be constructed from the timeline. In the third section
we demonstrate the bandpass decomposition in the presence
of multiplicative noise and contrast its performance to slow
feature analysis (SFA) [18]. In the last section we discuss the
overall properties and potential applications of this method.
We reflect on its possible utility in the context of neuromorphic engineering.
a
(−1)
Lk
f (t)
f˜∗ (t)
t
Fs (t)
(−1)
sLk+1
f˜ω∗ (ω)
b
(−1)
Lk
f (t)
2. DERIVING FREQUENCY DECOMPOSITION
FROM NEURAL REPRESENTATION OF THE
RECENT PAST
Fs (t)
f˜∗ (t)
t
(−1)
sL1
f˜ω∗ (ω)
We will first describe a neural mechanism for maintaining a
scale-invariant working memory and then we will show how
this mechanism can serve as a basis for constructing a neural
representation of time and frequency. The block diagram illustrating the basic idea is show on Figure 1. The mechanism
for representing the time is analogous to the one described
in [19], see also [14, 20, 12]. The main contribution of this
paper is in constructing a scale-invariant neural frequency decomposition with log-spaced central frequencies which can
be used to identify spatiotemporal signal of interest.
2.1. A neural mechanism for maintaining working memory
Fig. 1. Construction of neural support along time and frequency dimensions. a The signal from time domain, f (t),
is transformed into Laplace domain using a set of leaky integrators Fs (t), with time constants s. The leaky integrators
are chosen such that the transform maintains a scale-invariant
memory representation of the input signal. A linear combination of this memory representation (implemented through
(−1)
(−1)
operators Lk
and sL1
is sufficient to create a neural
timeline f˜∗ (t) and neural frequency-line f˜ω∗ (ω), both scalet
invariant with respect to time and frequency. b The neural frequency-line f˜ω∗ (ω) can also be constructed from neural
timeline f˜∗ (t).
t
We use a set of leaky integrators with different time constants to maintain information about history of a continuously
changing input across multiple seconds. The crucial property
of this representation is that temporal resolution gradually decays from more recent to more distant past.
Time varying input signal f (t) is sent to N nodes constituting the input layer. Each node is a leaky integrator with a
different time constant 1/s. The activity of these nodes Fs (t)
is described by the following differential equation:
dFs (t)
= −sFs (t) + f (t).
dt
(1)
The above expression effectively constitutes a set of first ors
der low-pass filters with the cutoff frequencies 2π
and gain
1/s. This can be read from the transfer function in the Fourier
domain with variable ω:
Fs (ω)
1 1
=
.
f (ω)
s 1 + iω
s
(2)
If the number of leaky integrators would be infinite, containing all the time constants from zero to infinity, the output of
this set would constitute a Laplace transform (restricted to
purely real, positive values of theR Laplace variable) of the in0
τ
put with respect to s: Fs (t) = −∞ f (t0 )e−s(t −t) dt0 . This
would be a perfect memory representation of the input signal.
Since in practice the number of leaky integrators is limited
one can obtain a discretized approximation of the Laplace
transform by choosing a finite number of neurons (restrictsj
= C,
ing s to a discrete set of values). Here we chose sj+1
C > 1 (exponentially decaying s). Log spacing of s is optimal for long range correlated signals ([14], see also [13]). We
will see that it also results in a scale-invariant frequency line.
2.2. Creating a neural timeline
A simple linear transformation on the instantaneous values of
the outputs from the leaky integrators is sufficient to construct
a neural timeline – a neural memory representation of what
happened when. This effectively means that each different
stimulus will have a set of neurons that fire sequentially for a
circumscribed period of time following each presentation of
that stimulus. This representation has two important properties: 1) It is scale-invariant (the standard deviation of the circumscribed period of elevated firing rate is proportional to its
peak time); 2) Peak firing times of the sequentially activated
cells are log-spaced. These two properties imply that the resolution of the estimate of when a stimulus happened will decay
with time elapsed since the stimulus presentation. This is consistent with the memory decay observed in everyday life and
quantified in behavioral experiments. Additionally, this is a
resource saving property: we need log of the resources that
we would need for uniformly spaced peak firing (assuming
the same maximal time resolution).
To construct a neural timeline, since Fs (t) is an approximation of the Laplace transform of the input signal, we need
to invert it and express Fs (t) in terms of some new variable
∗
that will correspond to internal representation of time, t. This
is achieved by setting the weights between the first and the
second layer such that the output of the second layer f˜∗ (t) is
t
approximately a k th order derivative of Fs (t) with respect to
s (see [19] for details):
(−1)k k+1 (k)
s
Fs (t).
f˜∗ (t) =
t
k!
(3)
∗
where s = −k/t. We define this linear operator as: L−1
k so
−1
˜
that f∗ ≡ Lk [Fs ]. This inversion will provide a dynamically
t
updated timeline maintained by a population of neurons, such
that the firing rate of a particular neuron in the second layer
∗
represents the input activity t time ago (Figure 2a and 2b).
Neurons with these properties have been reported in multiple
areas of the mammalian brain (these are called time-cells, see
[21] for a recent review).
To explore the frequency properties of this representation we again compute the transfer function using a Fourier
transform. Notice first that the impulse response of Fs (t) for
f (t) = δ(0) is: Fs (t) = e−st and in that case the k th or(k)
der derivative of Fs (t) reads as Fs (t) = (−t)k e−st . We
can now construct a transfer function in the Fourier space by
inserting the above expression into equation (3):
f˜∗ (ω)
t
f (ω)
=
1
(1 +
iω k+1
s )
.
(4)
The above transfer function is equivalent to a low-pass filter
of order k + 1 and unit gain. This property will be useful later
to demonstrate that we can construct a frequency-line from
this representation as well.
a
sf˜∗ (t) on lin-scale
b
sf˜∗ (t) on log-scale
t
t
1000
2000
3000
4000
5000
Time
6000
7000
8000
c
f˜ω∗ (ω) on lin-scale
5
10
15
20
25
30
Frequency [Hz]
3
4
10
9000 10000
5
10
Time
10
d
f˜ω∗ (ω) on log-scale
35
40
45
50
0
10
1
2
10
10
Frequency [Hz]
Fig. 2. Scale-invariant log-spaced neural support across time
and frequency dimensions (timeline and frequency-line). The
y-axis in all 4 plots represents the firing rate. Both representations are constructed using only instantaneous firing rate
from a set of exponentially decaying cells that received an
input at time 0. a (time in lin-scale) & b (time in log-scale):
Time-cells fire sequentially for a circumscribed period of time
following the input at time 0. Each curve represents a single cell with scaled amplitude. The firing fields (intervals
with elevated firing rate) spread in a scale-invariant fashion.
Peak firing times are log-spaced. This set of cells maintains
memory of the previous values of the stimulus such that what
happened in a more distant past is represented less accurately
than what happened in a more recent past. c (frequency in linscale) & d (frequency in log-scale): Similarly to time-cells in
the time domain, frequency responses of neurons that do the
frequency decomposition span the frequency axis in a scaleinvariant fashion with log-spaced central frequencies: slower
components are represented more precisely than faster components. Each curve again represents a single cell, but in this
case what we plot is a frequency response of the cells to a
delta-pulse. This set of cells decomposes the input into ordered frequency bands allowing an estimate of the speed of
the input signal.
2.3. Creating a neural frequency-line
Similarly to the construction of a neural timeline, a simple linear transformation of the outputs from the leaky integrators
is sufficient to construct a neural frequency-line – a scaleinvariant neural decomposition of an input signal into frequency bands with log-spaced central frequencies through a
set of temporal receptive windows.
In general, it is of a wide interest (for instance in iden-
tifying slow components) to estimate the speed of the input
signal. This can be easily done by taking a time derivative
of the signal. However this approach is prone to errors when
the scale of the target signal is not a priori known, as is often
the case. Therefore, we will use our dynamical memory representation to construct derivatives on multiple (log-spaced)
sj
= C, subtracting the acscales. It turns out that given sj+1
tivity of neighboring cells in Fs (ω), scaled by s (to have unit
gain, Figure 1a), or in f˜∗ (t) (Figure 1b) provides exactly the
t
desired log-spaced support along the frequency axis. Subtracting neighboring cells is equivalent to taking a first order
derivative with respect to the cell index, which amends to using the operator sL−1
k+1 :
f˜ω∗ (ω)
f (ω)
=
1
iω k+1
sj )
(1 +
−
1
(1 +
iω k+1
Csj )
,
(5)
where when integer k = 0 we are using Fs (t) scaled by s,
while when k > 0 we are using f˜∗ (t). Integer j goes from 1
t
∗
to N − 1. ω is given as a function of s and it represents an
∗
internal frequency-line, similarly as t represents an internal
∗
timeline. The relationship between ω and s will be explicitly
derived later. Notice that s and ω always appear together as
ratio ωs . This indicates the scale invariance since if ω changes,
scaling s by the same amount will result in a same transfer
function.
Because f˜ω∗ (ω) is a result of substraction of the activity of
two low-pass filters its frequency characteristics will be band∗
pass. The central frequencies of the band-pass filters, ω, are
the peaks of |f˜ω∗ (ω)|. They can be analytically expressed in a
simple form only for k = 0:
∂|f˜ω∗ (ω)|
√
∗
= 0 ⇒ ω = s C.
∂ω
(6)
∗
Inserting ω back into equation (5) gives us the amplification
of the band-pass filters: C−1
C+1 . Notice that the amplification
does not depend on s so it will be the same for all band-pass
filters.
To identify the distribution of the central frequencies
along the frequency axis we observe the ratio of the neighboring central frequencies:
∗
ωj
∗
ω j+1
=
1
.
C
(7)
The constant ratio that is smaller than 1 (since C > 1) indicates that the central frequency grows exponentially, providing log-spaced temporal receptive fields (Figure 2c and 2d).
These ordered temporal receptive windows allow us not only
to sort the signal by the speed of the components, but also to
have an estimate of the speed.
We discussed above the case k = 0, notice that for any in∗
teger k ≥ 0 the ratio
ωj
∗
ω j+1
is constant. Increasing k increases
the order of the low-pass filters and consequently makes the
resulting band-pass filters more steep.
3. DEMONSTRATION
I this section we demonstrate how the proposed method for
frequency decomposition, combined with PCA, can be used
for identification of spatiotemporal features of interest. Given
a multidimensional input, the frequency decomposition described above should be done independently for each dimension. One can then apply PCA across all the input dimensions
separately for each band (N − 1 PCAs, where N is the number of time constants s) to quantify the energy of dominant
components. Projecting the input signal on the eigenvectors
that correspond to the largest eigenvalues (across bands) will
give us the spatiotemporal components of interest. Additionally, directions of the eigenvectors will indicate which bands
are representing the same components. Notice that since PCA
is applied in each band separately we will be able to quantify
the speed of the identified components.
We constructed a two-dimensional signal such that each
dimension was composed of a weighted sum of a slow 7 Hz
sine wave (xs ) and a fast 71 Hz cosine wave (xf ), Figure 3a.
To each input we added binary multiplicative noise designed
as a square wave with a duty cycle randomly varying between
1 and 10 ms. This was introduced to mimic occlusions that
are generally present in visual and other types of input. Total
duration of the signal was 5s. Our goal here was to isolate
the components xs and xf assuming that we do not have any
prior knowledge about the input signal.
The presence of multiplicative noise made the problem
not well suited for SFA. SFA relies on taking a time-derivative
of the input data to get an estimate of the velocity. Numerical
derivative of the given input signal resulted in a rather noisy
output that obscures the true components. The output of a
classical SFA with 0.1 ms lagged time-derivative is shown in
the first column in Figure 3b. Assuming we know the scale
of the target signal we can reduce the noise by analyzing the
signal at that scale. To illustrate this we filtered the input signal with a first order low-pass filter with a cutoff frequency of
20 Hz (different choices of the cutoff frequency led to similar
or worse results). The result of applying SFA on such signal
is shown in the second column in Figure 3b. The slow, 7 Hz
component was estimated fairly well in this case, while the
fast, 71 Hz component was still not accessible.
Since in general we do not know the target frequencies a
priori it is not possible to do the filtering at the optimal scale.
This is where the proposed method becomes useful: we can
exploit the fact that we have a memory of the recent past and
decompose the signal into multiple bands spanning all the frequencies of interest (from some minimum to some maximum
frequency). We constructed a scale-invariant memory of each
of the two input dimensions with N = 20 and k = 0. From
this memory representation we built a frequency decomposi-
a
4. DISCUSSION
Original signals
Input mixture
xs = cos(14πt)
xm (xs + 2xf )
xf = cos(142πt)
xm (2xs + xf )
xm
b
SFA with dt large
x̃f
SFA with dt small
x̃f
x̃s
x̃s
c
Bandpass decomposition
x̃f
f[Hz]
100
10
x̃s
1
0
λi,1 vi,1
0
λi,1 vi,2
Fig. 3. Comparing band-pass decomposition and SFA in identifying significant frequency components in a noisy input. a
Two-dimensional input signal consists of a mixture of a sine
and a cosine wave (7 Hz and 71 HZ respectivly) and multiplicative noise. The x-axis is time and spans 1s, and the y-axis
is unitless signal amplitude. b Outputs of SFA for derivatives
taken at two different temporal scales of the signal. c The
input signal is decomposed into 19 frequency bands (N=20).
The left column shows eigenvectors (vi ) scaled by the largest
eigenvalue (λi ) for each frequency band. The right column
shows projections of the input signal on two eigenvectors that
correspond to peak eigenvalues.
tion into 19 frequency bands (Figure 3c). The cutoff frequencies were decaying exponentially from 142 Hz to 2.4 Hz. At
each band we applied PCA and found the largest eigenvalues
and corresponding eigenvectors. As shown in the first column in Figure 3c the direction of the eigenvectors indicates
whether a particular band dominantly represents xs or xf ,
while the eigenvalues indicate the energy in each band. Projection of the band-pass filtered input signal onto the eigenvectors that corresponded to the peak eigenvalues led to signals that resemble our original xf and xs . Notice that the
band-pass decomposition did not only identify the slow and
fast components, but also provided their frequency estimation
(on a log-spaced scale). On the other hand SFA in general
only provides the rank of the identified components with respect to their speed.
We proposed a biologically plausible neural method for constructing a scale-invariant frequency decomposition through
a set of band-pass filters with log-spaced central frequencies.
We demonstrated that this frequency decomposition has properties analogous to those of an internal timeline that can be
constructed from the same memory representation. We characterized these properties in terms of the central frequency
and gain and described how the frequency decomposition can
be constructed from neurally plausible leaky integrators.
The crucial properties of the proposed method are the
scale-invariance and log-spacing. When designing neural systems for processing natural signals it is essential to account
for processing on multiple scales that span from milliseconds
to hundreds of seconds, since the natural signals tend to be
autocorrelated on a variety of temporal scales. Maintaining the same frequency resolution for all the frequency bans
would require the number of bands to grow linearly with the
frequency. This would be rather costly in terms of resources
and computationally inefficient since we generally do not
need high resolution at high frequencies. Therefore the proposed method provides a good trade-off between resolution
and resources, in fact an optimal trade-off when the input signals are scale-invariant. Notice that memory systems based
on register-style buffers or shift-registers could not account
for the desired gradually decaying temporal resolution. The
parallel design in our approach (leaky integrators operating
in parallel, independently of each other) provides a simple
solution for gradual decay in temporal accuracy which then
allows us to construct the frequency-line with the desired
properties.
Ability to decompose the input signal into multiple frequency bands can be particularly useful for extracting features
that change slowly relative to the sensory representation. For
instance, the activity of visual receptors changes rapidly comparing to the identity or position of the object in front of the
eye. Existing unsupervised approaches such as SFA require
some prior knowledge about the temporal scale of the signal.
Additionally, band-pass decomposition allows not only identification of the components of interests, but also an estimate
of their speed on a log-scale.
The proposed method does not require convolution like
many other frequency decomposition methods. This makes
it particularly suitable for neural implementation. Since our
approach is based on biologically plausible neurons, neuromorphic platforms (see [22] for a recent example) could benefit for this approach. Designing neurons that can have long
time constants would be a step towards making these computations simple to implement in neuormorphic hardware. Even
though long time constants present a large challenge in analog electronics, the simplicity of their neural implementation
[17] could provide an inspiration. In biological systems on a
single-neuron level diffusion time constants are of the order
of 1 s, but when combined with renewal mechanisms they can
be extended to infinity (demonstrated by frequently observed
persistent firing). Having such long time constants the proposed method only requires lateral inhibition mechanisms for
taking spatial derivatives.
5. REFERENCES
[1] M. J. Hacker, “Speed and accuracy of recency judgments for events in short-term memory.,” Journal of Experimental Psychology: Human Learning and Memory,
vol. 15, pp. 846–858, 1980.
[2] P A Lewis and R C Miall, “The precision of temporal
judgement: milliseconds, many minutes, and beyond,”
Philosophical Transcripts of the Royal Society London
B: Biological Sciences, vol. 364, no. 1525, pp. 1897–
905, 2009.
[3] M. W. Howard, T. E. Youker, and V. Venkatadass, “The
persistence of memory: Contiguity effects across several minutes,” Psychonomic Bulletin & Review, vol. 15,
no. PMC2493616, pp. 58–63, 2008.
[4] B. C. Rakitin, J. Gibbon, T. B. Penny, C. Malapani, S. C.
Hinton, and W. H. Meck, “Scalar expectancy theory and
peak-interval timing in humans,” Journal of Experimental Psychology: Animal Behavior Processes, vol. 24, pp.
15–33, 1998.
[5] P. Goldman-Rakic, “Cellular basis of working memory,”
Neuron, vol. 14, pp. 477–85, 1995.
[11] Yulia Lerner, Christopher J Honey, Lauren J Silbert,
and Uri Hasson, “Topographic mapping of a hierarchy
of temporal receptive windows using a narrated story,”
Journal of Neuroscience, vol. 31, no. 8, pp. 2906–15,
2011.
[12] Marc W Howard, Christopher J MacDonald, Zoran
Tiganj, Karthik H Shankar, Qian Du, Michael E Hasselmo, and Howard Eichenbaum, “A unified mathematical framework for coding time, space, and sequences in
the hippocampal region,” Journal of Neuroscience, vol.
34, no. 13, pp. 4692–707, 2014.
[13] M. W. Howard, K. H. Shankar, W. Aue, and A. H. Criss,
“A distributed representation of internal time,” Psychological Review, vol. 122, no. 1, pp. 24–53, 2015.
[14] K. H. Shankar and M. W. Howard, “Optimally fuzzy
scale-free memory,” Journal of Machine Learning Research, vol. 14, pp. 3753–3780, 2013.
[15] A. V. Egorov, B. N. Hamam, E. Fransén, M. E. Hasselmo, and A. A. Alonso, “Graded persistent activity in
entorhinal cortex neurons.,” Nature, vol. 420, no. 6912,
pp. 173–8, 2002.
[16] M. E. Sheffield, T. K. Best, B. D. Mensh, W. L. Kath,
and N. Spruston, “Slow integration leads to persistent action potential firing in distal axons of coupled interneurons.,” Nature Neuroscience, vol. 14, no. 2, pp.
200–7, 2011.
[6] R. F. Voss and J. Clarke, “1/f noise in music and speech,”
Nature, vol. 258, pp. 317–318, 1975.
[17] Zoran Tiganj, Michael E. Hasselmo, and Marc W.
Howard, “A simple biophysically plausible model for
long time constants in single neurons,” Hippocampus,
vol. 25, no. 1, pp. 27–37, 2015.
[7] Dawei W Dong and Joseph J Atick, “Statistics of natural
time-varying images,” Network: Computation in Neural
Systems, vol. 6, no. 3, pp. 345–358, 1995.
[18] Laurenz Wiskott and Terrence J Sejnowski, “Slow feature analysis: unsupervised learning of invariances,”
Neural Computation, vol. 14, no. 4, pp. 715–70, 2002.
[8] E. Alvarez-Lacalle, B. Dorow, J. P. Eckmann, and
E. Moses, “Hierarchical structures induce long-range
dynamical correlations in written texts.,” Proceedings
of the National Academy of Science, USA, vol. 103, no.
21, pp. 7956–61, 2006.
[19] K. H. Shankar and M. W. Howard, “A scale-invariant
representation of time,” Neural Computation, vol. 24,
no. 1, pp. 134–193, 2012.
[9] Vishnu Sreekumar, Simon Dennis, Isidoros Doxas,
Yuwen Zhuang, and Mikhail Belkin, “The geometry
and dynamics of lifelogs: discovering the organizational
principles of human experience,” PLoS One, vol. 9, no.
5, pp. e97166, 2014.
[10] Christopher J Honey, Thomas Thesen, Tobias H Donner, Lauren J Silbert, Chad E Carlson, Orrin Devinsky,
Werner K Doyle, Nava Rubin, David J Heeger, and Uri
Hasson, “Slow cortical dynamics and the accumulation
of information over long timescales,” Neuron, vol. 76,
no. 2, pp. 423–34, 2012.
[20] Karthik H Shankar, “Generic construction of scaleinvariantly coarse grained memory,” arXiv:1406.3185
[q-bio.NC], 2014.
[21] Howard Eichenbaum, “Time cells in the hippocampus:
A new dimension for mapping memories,” Nature Reviews Neuroscience, In press.
[22] Paul A Merolla, John V Arthur, Rodrigo AlvarezIcaza, Andrew S Cassidy, Jun Sawada, Filipp Akopyan,
Bryan L Jackson, Nabil Imam, Chen Guo, Yutaka Nakamura, et al., “A million spiking-neuron integrated circuit
with a scalable communication network and interface,”
Science, vol. 345, no. 6197, pp. 668–673, 2014.