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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:Xt∼F0,∀t∈T

H1:Xt∼F0, 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:t∈R0}against {Xt:t∈R(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,n−∆0]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|α−E|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 = 10−3.(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(n−k)

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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