Conference PaperPDF Available

Non-parametric Change Point Detection for Spike Trains

Authors:
Copyrights to IEEE – to appear in CISS 2016 @ Princeton
Non-parametric Change Point Detection
for Spike Trains
Thiago Mosqueiro
BioCircuits Institute
University of California San Diego
E-mail: thiago.mosqueiro@usp.br
Martin Strube-Bloss
Theodor-Boveri-Institute of Bioscience
Biocenter University of Wuerzburg
Rafael Tuma
Institute of Physics Of S˜
ao Carlos
University of S˜
ao Paulo
Reynaldo Pinto
Institute of Physics Of S˜
ao Carlos
University of S˜
ao Paulo
Brian H. Smith
School of Life Sciences
Arizona State University
Ramon Huerta
BioCircuits Institute
University of California San Diego
Abstract—Two techniques of non-parametric change point
detection are applied to two different neuroscience datasets. In
the first dataset, we show how the multivariate non-parametric
change point detection can precisely estimate reaction times to
input stimulation in the olfactory system using joint information
of spike trains from several neurons. In the second example, we
propose to analyze communication and sequence coding using
change point formalism as a time segmentation of homogeneous
pieces of information, revealing cues to elucidate directionality
of the communication in electric fish. We are also sharing our
software implementation Chapolins at GitHub.
Index Terms—Non-parametric, Olfaction, Electric fish, Com-
munication.
I. INTRODUCTION AND MOTIVATION
In a wide range of disciplines – finance, medicine, molecular
biology, neuroscience, geology, etc –, it is often critical
to detect inhomogeneities and changes in time series. By
changes, we may refer to a sudden drop/high in the daily
stock price, rise in neural activity due to the onset of an
external stimulus, or changes in geological core samples [1].
In biology, all important processes operate based on sensing
and signaling pathways that detect changes (both externally
or internally) and trigger response mechanisms [2], [3]. As
an example, reflex motor reactions to fast stimuli (such as
a burning sensation, or a sudden loud noise) are usually
nimble responses triggered by local interneurons that are very
sensible to certain changes in their sensory input. In the field
of statistics, determining if there is a change in time series
and estimating the most likely points of change is called the
change point problem [1], [4], [5]. Change point analysis can
be traced back to the 1950’s, first appearing in the context of
industrial quality control.
As an illustration, let us consider figures 1(a-b), where we
shift the average and increase the variance at time t= 0 of a
Wiener process. The goal of change point analysis is to find
the most probable time τat which either of these changes
take place, as in figure 1(c). More interestingly, such methods
can detect events in real time series: for instance, 1(d) shows
detection of CO presence based on signals recorded from
electronic noses [6].
In neuroscience, timing is a central topic, however advanced
change point techniques are underutilized. Even though in
many cases it is possible to come up with alternative measures
or proxies [9], it would be hard to consistently generalize
them to spike trains or other neural measures. In this paper,
we revisit some of the state-of-the-art techniques for change
point detection available [10] and how to apply them to
neuroscience data. We will use two very recent datasets to
analyze different problem types: (i) assess the reaction time
of neural populations based on their spike trains and (ii)
the construction of indicators for sequence segmentation to
elucidate animal communication.
In the first case, we use the spike trains to detect change
points in activity of a neural population based on electrophysi-
ological recordings of neurons in the first olfactory processing
stage of insects. Although in several cases firing rate accurately
represents information, this is usually only true to some extent.
Individual dynamics may also enclose important patterns and
information, as is known to be the case in the insect olfac-
tion [11]. We then use a multivariate non-parametric method
(introduced in section II [10]) to jointly examine trains of
spikes. In the second case, we examine a signal recorded
from two freely swimming electric fish (Gymnotus sp.). Since
these fish are territorialists, whenever they are placed in the
same aquarium they tend to intensely interact both physically
and using their electrical pulses, until a relation of dominance
is stated. Although electrocommunication is highly accepted,
little is known in terms of how information is conveyed
in sequences of electrical pulses. Using Inter-Pulse-Intervals,
we use change point detection as an indicator to sequence
segmentation, which can potentially aid to the detection of
important symbols in the electric fish vocabulary.
This paper is intended to give a minimal introduction to
non-parametric change point techniques for neuroscientists,
while presenting applications that might be appealing to a
wider audience. With both examples, we cover two of the
mainstream classes of analyses performed in neural data,
giving a glimpse of how powerful such techniques are and
their benefits. Up to our knowledge, such techniques have not
been used as described, and to help changing this scenario we
made available our own implementations and a few examples
[12], accessible using Python and (eventually) Matlab.
(a) (b) (d)
(c) (e) (f)
Fig. 1. Examples of time series where change point analysis are required. (a-b): Artificial data where simple changes in either average or variance. (c):
Results using a simple classical non-parametric approach – no hypotheses are made regarding the underlying model. (d): Real data from an array of electronic
noses at the onset of detection of CO volatiles [6]. Each line represents one sensor in one of the noses. (e): Example of neural activity where a stimulus is
presented (at t= 0) and the Instantaneous Firing Rate (IFR) of the population suddenly changes. There recordings are from honeybees’ Projection Neurons
(PN) and Mushroom Bodies Output Neurons (MBON) [7]. (f): Recordings of freely swimming, pulse-type electric fish [8]. Inter-Pulse-Intervals (IPI) are
introduced in section IV.
II. REVIEWING NO N-PARAMETRIC EVEN T DET EC TI ON
Let X={Xt:t= 1,2, . . . , n}be a stochastic process in
discrete time. Detection of change events can be reduced to
the problem of contrasting two hypotheses. For instance, for
a single change point,
H0:XtF0,tT
H1:XtF0, t = 1,2, . . . , τ 1
F1, t =τ, τ + 1, . . . , n
(1)
are the candidate hypothesis. F0and F1are the underlying
distributions during each of the regimes of the stochastic
process. As anticipated, in neuroscience the time series X
may represent, for instance, the instantaneous firing rate of
a given neural population, while τis the time when an
external stimulus is perceived. F0then describes the stimulus-
free behavior of that population, while F1indicates the new
behavior after stimulation. In this sense, the difference between
τand the time when the stimulation started is regarded as the
perception reaction time. Next, we discuss methodologies that
requires the least prior knowledge about F0and F1as possible.
The goal is to answer the following two questions:
i How likely is to find a change point?
ii What is the best estimate of the change point τ?
Naturally, (i) is answered with a p-value that indicates how
certain we are of rejecting H0in favor of H1, whereas the
estimation of the change point will come out as the result of
some optimization – explained in the following subsections.
Figure 1(c) shows examples of estimated change points. In
general, for a stochastic process with nrandom variables, most
methods are based in solving n+ 1 statistical tests [4]. The
generalization for several change points is somewhat direct,
and can be found in the literature [1], [4].
A. Single change point using non-parametric test statistics
We first approach this problem using a very simple, but
useful, methodology first proposed by Pettitt [13]. It is based
on a chosen test statistic (not necessarily parametric), say, D,
and its associated two-sample hypothesis test. For instance,
(Mann-Whitney’s) Ustatistic can easily solve the example in
figure 1(a), while failing at even strong differences in variance.
In general, Ustatistics is a good candidate when the average
shifts, while for scaling changes as in figure 1(b) the Median
test statistic is more suitable.
In general, Kolmogorov-Smirnov or Cramer-von-Mises tend
to capture a wider range of features, although possibly without
the same precision or accuracy as more specific tests. Several
analytical results and asymptotic bounds have been calculated
in the past for most test statistics [13], [14]. Finally, In case
of high confidence on the nature of F0or F1, parametric test
statistics may be used to improve the precision of the change
point estimation.
Once Dis chosen, we draw two sample fractions from the
time series and compare them. There is more than one way
to approach this problem, and we briefly introduce two in the
following paragraph: the Sliding Window and the Moving Bar
strategies. For simplicity of notation, let [a, b]Z= [a, b]Z
the discrete interval between two whole numbers aand b.
Sliding Window p-value. Consider two time ranges R0=
[t0, t0+ ∆0]Zand R(t1) = [t1, t1+ ∆0]Z, with t1[t0+
0, n 0]Z. Set R0to be located at the beginning of the
time series (t0is not necessarily 1). Then, we perform a two-
sample statistical test using Dand evaluate the p(t1)as the
p-value of {Xt:tR0}against {Xt:tR(t1)}. We set a
threshold θto indicate that the test rejects its null hypothesis
and both windows R0and R(t1)are in different regimes. For
a single change point, the estimation becomes
ˆτ= inf argmin
k[t0+∆0,n0]Z
Ip(tk)< θ,(2)
with I(·)being the indicator function.
This algorithm is especially suitable for long time series,
given its linear complexity and low computational cost –
given that the statistical tests are only computed over two
time windows. We show simple applications on figure 2(a-
b). Whenever the R(t1)goes through a change point, p(t1)
decreases significantly. From our experience, most of the times
by three or more orders of magnitude – see example in figure
2(a-b). Finally, notice that as 0and 1both increase, Dis
more likely to capture slight changes in distribution. Yet, the
bigger 1, the lower the precision of the estimated ˆτ.
Moving Bar. Let Dkbe the two-sample statistic evaluated
for {Xt:t[1, k 1]Z}against {Xt:t[k, n]Z}. To
avoid dealing with signal of Dand standardize the method,
renormalize
˜
Dk=|Dk|
σ,(3)
with σ=Var(Dk)as its variance. Finally,
ˆτ= argmax
k[1,n]Z
˜
Dk.(4)
Under the hypothesis of single change point, ˆ
Dklike presents
a single maximum.
Kolmogorov-Smirnov can be evaluated in linear time, which
means this approach has complexity O(n2). It is worth noting
that larger data series might be treated in overlapping sections,
otherwise it may get computationally expensive to compute the
two-sample test statistics. In figure 2, we show an example of
this procedure.
Notice that both methods (Sliding Window and Moving Bar)
have skewed distributions, i.e., they are statistically biased,
estimated in less than 0.8% of the whole interval used. As
expected, the bias is often positive, except when the change
point is artificially abrupt.
B. Multiple change points using a non-parametric divergence
measure
Next, we review a recent approach proposed by James &
Matteson, namely, E-Divisive, based on a divergence measure
and a bisection search for change points [10]. No fixed number
is assumed a priori, and the only strong assumption made is
the existence of the α-th momentum. For our numerical tests,
we set α= 1.
Performing the same division as in the Moving Bar method,
let Xk={Xt:t[1, k 1]Z}and Yk={Xt:t[k, n]Z}.
James & Matteson proposed the use of the following diver-
gence measure:
Eα(X,Y)=2E|X − Y|α
E|X X 0|αE|Y − Y 0|α,(5)
(a)
(b)
(c)
(d)
Fig. 2. Examples using non-parametric test statistics. For generality, we have
used two-sample Kolmogorov-Smirnov for both techniques. For each case,
we examined the interval of confidence of the estimated change point. Right
panel represents deviation from actual change point (τ= 0) for 103trials.
(a-b): Sliding window strategy, with threshold pthr = 103.(c): Moving
bar strategy. (d): E-Divisive strategy proposed by James & Matteson [10].
where X0is an independent copy of Xand |·| denotes
euclidean norm. Estimating ˆ
Eα(X,Y)is not particularly ex-
pensive. Then, for a single point-change estimation we could
have
ˆ
t= argmax
k[1,n]Z
k(nk)
nˆ
Eα(Xk,Yk) = argmax
k[1,n]Z
ˆ
Qα(Xk,Yk),
(6)
where ˆ
Qα(Xk,Yk)is known to converge in distribution, as
mand ngrow, to a non-degenerate random variable if Xk
and Ykshare the same distribution; if not, ˆ
Qα(Xk,Yk)→ ∞
[10]. Finally, James & Matteson proposed combining this
divergence with a strategy similar to bisection root-finding
method to locate multiple change points [10]. Given the simple
form of equation 5, the extension for multivariate distributions
is direct. The overall complexity of E-Divisive algorithm is
O(Ncpn2), with Ncp being the number of (unknown a priori)
change points.
Especially for neuroscience (and biology in general), as we
point out in section III, this is especially important: this opens
the possibility of identifying change points using the joint
information of spike trains. Figure 2(d) shows the detection
of change point using the E-Divisive algorithm. Note that it
does detect more than one point, highlighting several different
regimes. All p-values are at least 0.025.
(a) (b)
(c)
(d)
(e) (f)
Fig. 3. Activity of Projection Neurons (PNs) in the Antennal Lobe of
honeybees. In all cases, stimulation starts at t= 0.(a): Raster plot and
Instantaneous Firing Rate (IFR) of PNs. (b) Each color represents a different
exposal to the same odorant. (c): Change point analysis using the population
IFR. (d): We show different reaction times and second order statistics
depending on which odor is used. (e): 3 principal components obtained by
PCA are shown, presenting hidden activity patterns in the spike train invisible
to IFRs. (f): Change point detection using jointly the five most important
components found by PCA.
III. REAC TI ON TI ME S FROM SPIKE TRAINS:INSEC T
OLFACTION RECORDINGS
Olfaction is one of the sensory modalities used to capture
important cues for survival, such as in mating and finding re-
sources. In insects, the main pathway for olfaction information
is known in some detail [15], [16], although much is yet to be
explained. Timing and time integration, for instance, is a key
to elucidate the mechanisms of decision making [11], [17].
Upon odor stimulation the Antennal Lobes (ALs) consti-
tute the first stage of olfaction processing in the brain. The
top panel of figure 3 showcases an experimental setup for
electrophysiological experiments involving odor stimulation
and conditioning. As one of the mainstream techniques for
studying population activity, we start with the Instantaneous
Firing Rate (IFR). Figure 3(a) shows recordings of Projec-
tion Neurons (PNs) from the ALs of honeybees [7]. If not
mentioned otherwise, stimulation starts at t= 0. Due to
the intrinsic out-of-equilibrium and noisy nature of olfaction
information, responses are slightly different each repetition, as
shown in figure 3(b).
In this context, a natural question would be how to estimate
the reaction time of that population to different odorants,
exploring its sensitivity and selectivity [9], [18], [19]. PNs can
discriminate different odorants in a time scale of hundreds of
milliseconds. Behaviorally, it is known that honeybees may
learn to react fast when trained, taking less than 0.4s for a
motor response. Since these recordings come from untrained
bees, the observed reaction in this dataset reflects honeybees
response to novelty.
As a first approach, we can use E-Divisive change point
detection on the IFRs of the PNs to estimate their reaction
time. We show in figure 3(c) the detected reaction times for
a few trials. Since we have recordings of multiple odorants
(1-hexanol, 2-octanone and their mixture), we may even
assess how PNs react to each odorant individually: using 10
recordings for 3different stimuli, we show in figure 3(d) PNs
response. We can clearly see differences that are statistically
significant across odorants, and maybe the most interesting
result is that detecting mixture – which would be the hardest
task – shows larger variance with lower median of reaction
times.
In many cases, neural populations do encode changes in
their IFRs [20], nonetheless IFRs only partially represent the
actual information. Indeed, this is the case for olfaction [21]:
although the activity in 3(a) bounces back to its original firing
rate, different spatio-temporal patterns emerge. This can be
visualized using the 3most important components captured
by a PCA analysis over all spike trains, as shown in 3(e).
The black line represents the stimulus-free condition, while
the blue line is the activity of the stimulated PNs.
Thus, to include these second order features we can employ
the multivariate version of E-Divisive in the spike trains, using
each train as a different dimension. However, there are caveats:
as the dimension of the analyzed time series grow, not only
the computational effort explodes, but also precise estimation
requires much more samples. A possible solution is to use the
first few PCA components of the spike train. In figure 3(f)
we show the result of the multivariate change point detection
using the first five PCA components (the ratio between first
and fifth eigenvalues is 0.09.). We show both the detected
reaction time, with p-value of 0.002, and the profile of each
spike train over time.
Therefore, this is a general approach that can be applied
when it is necessary to go beyond firing-rate coding, including
information from individual dynamics in the spike trains.
For instance, if the firing rates remain the same, while the
subpopulation changes – which is the case of odor identity
coding the mammalian olfactory bulb [22] –, this methodology
would successfully identify it. Also, changes in the correla-
tions among neurons, such as a change in synchrony, can be
easily captured by this multivariate analysis.
IV. IND IC ATOR S FO R SEQ UE NC E SEG ME NTATION:
PU LS E-T YP E EL EC TRIC FISH
Some electric fishes generate pulses of electric field with
stereotypical waveform, similar to neurons’ action potential
[23], [24]. These pulses (or spikes) partially bounces back
and deformations are perceived by its electrosensory system,
working as echoes for electrolocation [25]. Additionally, recent
work has shown that pulse-type electric fish may also use
these pulses to communicate with its conspecifics, possibly
stating dominance [8], [26]. If that is the case, little is known
about their language: do they present grammar-structured
language? Which kind of sequence coding do these fish use?
[8], [27]. In the following, we propose the use of change point
detection to split the time series in “homogeneous” segments
of data enabling one to identify hidden regularities in the fish
behavior and communication. This time series segmentation
may reveals cues to elucidate discretization and directionality
of communication in electric fishes. Figure 4(top) summarizes
the methodology.
Notice that communication studies often use fixed time
discretization to define symbols and words, which is not
necessary here. Segments have variable lengths instead. In
[26] it was shown that abundant information is encoded in the
Inter-Spike-Interval (IPI) time series. Let τkbe the k-th spike
of a fish, the its Inter-Spike-Interval time series is defined as
∆(tk) = tk+1 tk, in close analogy to Inter Spike Interval
(ISI). In figure 4(top) we show IPI time series of two electric
fish (Gymnotus sp.) freely swimming in the same environment
(an aquarium of 1×1×1m3). To split the signals from the two
freely behaving fish, we have used a very recent suggestion
[28].
Depending on the time scale considered, each segment may
be assigned to either different symbols/words in the fish’s
vocabulary, or whole sentences/messages. After the segmen-
tation, a ad hoc similarity analyses, ranging from comparing
pairs of correlations to an unsupervised learning strategy (such
as in [5], [29]), can be used to identify clusters of symbols.
In figure 4(top), we picture ykand xkas the series of
symbols identified per segment, per fish. Then, complementary
information-theoretic tools and statistical learning methods
can assess the information content and flow in such symbols
(Pearson’s correlation, mutual information, transfer entropy,
etc).
We start in a coarse time scale in figure 4(a): evaluating
the Instantaneous Firing Rate (IFR) associated with the pulse
time series, which aggregates hundreds of spikes and captures
the overall trend of the IPIs. Notice that this is done over the
course of hours, and may be easily linked to behavior time
scale. Interestingly enough, figure 4(b) shows this analysis for
two fish interacting. The number of coincidences in the de-
tected change points is somewhat remarkable, with more than
half being minutes apart. This reflects some level expected
synchrony between fishes for several reasons, including an
effect similar to “Jamming Avoidance Response” [8].
We show in figure 4(c) segmentation of only part of the
(a) (b)
(c)
(d)
Clus te r 0
Clus te r 1
Clus te r 2
Fig. 4. An indicator to sequence segmentation in both a coarser time scale
and using Inter-Spike-Intervals (IPIs). (a): Detection on a coarse-grained time
scale. (b): Comparing the detections using recordings of two fish freely
swimming. (c): Detection in a smaller time scale: change point detection likely
reveal information units in terms of communication between both fish. (d):
Small scale example of identifying communication motifs using unsupervised
learning.
IPIs of a fish in a smaller time scale, over the course of
few seconds. Probably the most interesting feature is that the
detected change points intercalate in the vast majority of cases:
for each change point of fish 2, fish 1 has a change point
closely following. Moreover, when the same fish presents two
change points in a row (predominantly observed in fish 1),
they are separated by a larger time period – possibly reflecting
silent lapses in the communication.
Finally, we have constructed a vector of statistics for each
segment (IPI variance, average slope, area under the curve
and time interval). Using PCA combined with a clustering
algorithm – K-Means or Affinity – show three main clusters.
Defining each cluster from 0to 2and labeling xkand yk
segments, we get a discretized version of the IPI series. In
figure 4(d) we show the detected cluster per fish and some of
the symbols.
Interestingly, we estimated the Mutual Information (MI)
between the discretized series as 0.16 bits/s. To test the
significance of this number, we propose the use of boot-
strap/surrogate techniques to test for data snooping. Reshuf-
fling the IPIs obliterates all patterns and no change point is de-
tected whatsoever (repeated more than 100 times). This simply
shows that the segmentation is likely not an artifact of chance,
although it does not guarantee any kind of information flow
or correlation between the two segmented IPI series. Based on
White’s Reality Check [30], we shuffle entire segments of IPIs
based on the larger scale detection. On 100 shuffles tested, the
MI dropped by a factor 4or more.
V. DISCUSSION
We revisited three nonparametric methods to solve change
point problems using two different test cases. The E-Divisive
approach reviewed in section II-B was recently proposed as
an efficient and fast method to detect multiple change points
sequences multivariate distributions [10].
In the first case, we analyzed the timing of the neural
reaction to (novel) stimuli in the presence of extremely noisy
environment and out-of-equilibrium signals. By studying firing
rates or sequences of spike trains, we determined, for instance,
that the reaction time of honeybees to different odors is
roughly of the order of 100ms, and observed that their response
to a mixture of odors has lower median with higher variance.
In our second example, we have used change point anal-
ysis as an indicator to sequence segmentation that may be
especially suitable for elucidating questions on animal com-
munication. Since change points split the series in chunks
with homogeneous statistics, these likely qualify as bits of
information being communicated. We also suggest how to
perform segmentation in different time scales and how to test
for data snooping.
Several other applications can be proposed, especially in-
volving subtle behavioral transitions controlled altogether by
several time scales [3]. Also, the literature on change point
problems is extensive and makes use of several different top-
ics (control theory, classical/Bayesian estimation, hypotheses
testing, etc). Thus, this paper is only scratching the surface of
possibilities for computational neuroscience and biology.
ACKNOWLEDGMENT
Authors thank for useful discussion with P Matias. TS
Mosqueiro acknowledges support from CNPq 234817/2014-
3. R Huerta and B Smith acknowledge partial support from
NIDCD R01DC011422 and NIH/NIGMS R01GM113967.
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