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Phys. Biol. 12 (2015) 016007 doi:10.1088/1478-3975/12/1/016007
PAPER
A topological study of repetitive co-activation networks in in vitro
cortical assemblies
Virginia Pirino
1
, Eva Riccomagno
2
, Sergio Martinoia
1
and Paolo Massobrio
1
1
Department of Informatics, Bioengineering, Robotics and System Engineering (DIBRIS),University of Genova, Genova, Italy
2
Department of Mathematics (DIMA), University of Genova, Genova, Italy
E-mail: virginia.pirino@edu.unige.it,riccomagno@dima.unige.it,sergio.martinoia@unige.it and paolo.massobrio@unige.it
Keywords: betti numbers, co-activity, micro-electrode arrays, neuronal dynamics, simplicial complex
Supplementary material for this article is available online
Abstract
To address the issue of extracting useful information from large data-set of large scale networks of
neurons, we propose an algorithm that involves both algebraic-statistical and topological tools. We
investigate the electrical behavior of in vitro cortical assemblies both during spontaneous and stimu-
lus-evoked activity coupled to Micro-Electrode Arrays (MEAs). Our goal is to identify core sub-net-
works of repetitive and synchronous patterns of activity and to characterize them. The analysis is
performed at different resolution levels using a clustering algorithm that reduces the network dimen-
sionality. To better visualize the results, we provide a graphical representation of the detected sub-
networks and characterize them with a topological invariant, i.e. the sequence of Betti numbers com-
puted on the associated simplicial complexes. The results show that the extracted sub-populations of
neurons have a more heterogeneous firing rate with respect to the entire network. Furthermore, the
comparison of spontaneous and stimulus-evoked behavior reveals similarities in the identified clus-
ters of neurons, indicating that in both conditions similar activation patterns drive the global network
activity.
1. Introduction
An important feature of modern science is the
unprecedented rate at which big data are produced
[1]. This happens in part because of new experimental
methods (e.g., multi-electrode recordings, genome
sequences, DNA microarray data, etc.) and in part
because of the increasing availability of high powered
computing technology (e.g., the computer architec-
ture used for the European Blue Brain Project [2]
utilized the Blue Gene supercomputer (IBM), made of
8192 processors capable of reaching a peak processing
speed of 22.8 trillion floating-point operations per
second (22.8 teraflops)). In addition, if data are high-
dimensional, difficulties to visualize them, extract
main features, delete redundancies, etc. can arise. This
is particularly true in biology and for high throughput
data from micro-array or other sources (see [3] for a
survey). To overcome these problems, new techniques
emerged, thanks to statistical and topological tools [4–
7]. In this work, we applied such methods (after slight
adaptations) to an in vitro experimental model,
namely dissociated cortical networks coupled to
Micro-Electrode Arrays (MEAs), with the aim to
estimate their functional properties. Topological
approaches available in the literature and applied to
the field of neuroscience, include Curto et al [4] who
processed their in silico data by three main steps: first,
they reduced the signals by binning them; second, they
removed noise by applying an ad hoc threshold; then,
they deduced a graphical representation via a topolo-
gical structure called abstract simplicial complex
[3,8]. Singh et al [7] processed in vivo multi-unit
recordings in a slightly different way: first, they
selected the most active signals (i.e., with the highest
firing rate); second, they reduced the amount of data
by binning them; then, they chose an ad hoc distance
function and used a weak witness construction with 35
landmark points for representing the data in terms of a
Rips complex [3].
RECEIVED
31 July 2014
ACCEPTED FOR PUBLICATION
9 December 2014
PUBLISHED
5 January 2015
© 2015 IOP Publishing Ltd
In this work, following Curto et al [4], we devise an
algorithm (see section 2.5) and apply it to both sponta-
neous and stimulus evoked activity of in vitro cortical
networks. In particular, we infer functional character-
istics, by creating a topological structure associated to
the neural network. Such a structure should point out
a possible significant sub-network which captures
electrophysiological activity emerging from the back-
ground spontaneous activity [4]. In details: first, we
investigate the functional connectivity of the network
in terms of its intrinsic co-activity or co-firing (i.e., a
period of synchronous firing of a group of neurons), as
defined in [5]. Then, we compare recordings of spon-
taneous and stimulus evoked activity of a pool of
experiments to investigate whether possible relation-
ships exist. Our approach aims at answering three rela-
ted questions.
1. Are there specific sub-networks containing the
electrodes (i.e., group of neurons) that fire simulta-
neously and repeatedly (i.e., core sub-networks)?
2. Are the same co-activated sub-networks recruited
during both spontaneous and stimulus-evoked
activity?
3. Are there sub-network features that allow us to
distinguish between spontaneous and stimulus-
evoked activity conditions?
Guided by the previous questions, we identify rela-
tionships among the network nodes, and quantify the
strength of such interactions.
The gold standard methods applied to electro-
physiological signals (both in vivo and in vitro) seek for
statistical dependence or causal relationships between
pairs of neurons. These methods play a relevant role in
the study of functional and effective interactions at
population level. Nowadays, the most common tech-
niques to infer functional connectivity [9,10] of neu-
ronal assemblies rely on the investigation of the
statistical properties of the spontaneous activity. Two
families of methods based on pairwise spiking activity
can be found in the literature: correlation-based algo-
rithms (e.g., cross-correlation [11]) and information
theory based algorithms (e.g., mutual information
[12]). Briefly, cross-correlation measures the fre-
quency at which one neuron fires as a function of time
relative to the firing of a spike in another neuron, while
mutual information is a measure of the statistical
dependence between two spike trains. During the last
years, by exploiting the potentiality of these algo-
rithms, new approaches have been introduced. Con-
cerning correlation-based approaches, two powerful
used methods are: partial correlation [13], and func-
tional holography [14,15]. Partial Correlation mea-
sures the time between the spiking activity of two
neurons, after the linear effect of the other neurons
of the network have been removed. Functional
holography is an elegant and mathematically rigorous
method, which allows detecting hidden motifs [16]
embedded in the inter-neuron correlation matrix.
Regarding information theory approach, in the last
years two important methods have been developed:
joint-entropy [17] based on the cross inter-spike
intervals computed across pairs of neurons, and trans-
fer entropy [18] that is an information measure which
evaluates causal relationships from time series taking
into account the past activity.
In addition to these pair-wise approaches, the pre-
vious described topological and algebraic- statistics
methods have been introduced to study biological sys-
tems [3,10]. The usual approach is to choose ad hoc
thresholds to edit out background noise and to isolate
the most relevant signal data. The co-activity informa-
tion can be synthesized into the links (or edges) among
the network nodes. The connection strengths among
nodes can be translated into a weight; weak connec-
tions are simply removed from the network. In this
way, the most significant structure of the network is
extracted from the background. A family of networks
called filtration, created by varying one or more
thresholds, can be studied via persistent homology
[19–21]. The so constructed network is a weighted
abstract simplicial complex summarizing the neural
network activity over the variation of a selected thresh-
old [5]. In this work, we proceed as follows. We associ-
ate to each electrode of the MEA a meta-neuron
defined as the number of neurons coupled to the
same electrode or to the same active area. Generally,
one to three neurons are coupled into a single elec-
trode [22] and their activity is recorded by means of
that electrode. Then we aggregate the activity of
nearby electrodes, i.e. spike trains coming from
adjacent electrodes are merged to form a cumula-
tive spike train. For each aggregate a meta-neuron is
considered, whose spiking activity is the cumulative
activity recorded by those electrodes. At different
level of complexity, the meta-neurons represent a
small population within a complex large-scale net-
work [23]. At the meta-neuron level the weighted
simplicial complexes are constructed adopting the
process presented in [4] and topological properties
are obtained in form of graphs. A widely used sta-
tistics is provided by the Betti numbers of the sim-
plicial complex [3].
The simplicial complex and the related graphical
representation of the network activity are static sum-
maries of the distribution of the strongest co-activity
over the whole period of recording. The proposed
method gives a multidimensional analysis of the com-
plex network and summarizes the whole neural activ-
ity. We choose these techniques to analyze each
recording from a static point of view and to compare
recordings of spontaneous and stimulus-evoked
activity.
2
Phys. Biol. 12 (2015) 016007 V Pirino et al
2. Materials and Methods
2.1. Cell cultures
Cortical neurons are obtained from brain tissue of
Sprague Dawley rats at embryonic day 18 (E18) and
plated onto 60 planar TiN/SiN microelectrodes
(30 μm diameter, 200 μm spaced) at the density of 5-
8 × 104 cells/device, which means about 1200-
1400 cells/mm
2
. The procedure is approved by the
European Animal Care Legislation and by the guide-
lines of the University of Genova. Cerebral cortices are
dissociated by enzymatic digestion in Trypsin 0.125%
—20 min at 37
°
C—and then triturated with a fire-
polished Pasteur pipette. No antimitotic drug is added
to prevent glia proliferation. Micro-Electrode Arrays
(MEAs) are coated with adhesion promoting mole-
cules (poly-D-lysine and laminin). Neurons are main-
tained in culture dishes, each containing 1 ml of
nutrient medium (i.e., serumfree Neurobasal medium
supplemented with B27 and Glutamax-I) [24] and
placed in a humidified incubator having an atmo-
sphere of 5% CO
2
at 37
°
C.
2.2. Recording and stimulation system
The analyzed data concern the electrophysiological
activity of cortical cultures, recorded with two types of
configurations: (I) homogeneous neuronal popula-
tions coupled to MEAs and (II) interconnected
neuronal populations by means of a dual-chamber
MEA system [25]. The first configuration consists of
non-constrained cortical cultures coupled to MEAs
(Multi Channel Systems, MCS, Reutlingen, Germany)
made up by 60 TiN/SiN planar round electrodes
(30 μm diameter, 200 μm pitch-to-pitch inter-elec-
trode distance) arranged in an 8 × 8 square grid
excluding corners (figure 1(a)). The second configura-
tion is a dual-chamber system. It is given by the
integration of a MEA with a closed-compartment
microfluidic device with micro-channels connecting
the two compartments. Specifically, the two compart-
ments are 100
μ
mhigh and 1.5 mm long and the
micro-channels (
××
1
0 3 150
μ
mwidth × height ×
length) are spaced at regular intervals of 60
μ
m
(figure 1(b)) [25].
The activity of all cultures is recorded by means of
the MEA60 System (MCS). After 1200 × amplifica-
tion, signals are sampled at 10 kHz and acquired
through the data acquisition card and MC_Rack soft-
ware (MCS).
Stimulus-evoked recordings are tested only on
homogeneous cultures and electrical stimuli are deliv-
ered by using a commercial general-purpose stimulus
generator (STG 1008, MCS), which allows the applica-
tion of current and voltage pulses to selected electro-
des of the MEAs. In the presented experiments, the
stimulation protocol consists of stimuli sent sequen-
tially to a selected electrode at the frequency of 0.2 Hz,
for five minutes. Probe pulse amplitude was fixed at
1.5 Vpeak-to-peak. The stimulus pulse was biphasic
(positive phase first) and lasted for 500 μs with a 50%
duty cycle [26].
2.3. Dataset and experimental protocol
The results on homogeneous networks come from
n= 8 cortical cultures while results on interconnected
networks come from n= 9 cortical cultures; for both
datasets, the recordings were collected during the third
week in vitro.
The experimental protocol for the homogeneous
networks consists of six spontaneous five-minute long
recordings and four or six stimulus-evoked five-min-
ute long recordings. For each experiment, recordings
were acquired from four different sites. For the inter-
connected networks, each experiment consists of a
spontaneous ten-minute long recording.
2.4. Spike detection and first-order statistics
Extracellular recorded spikes are detected using the
threshold-based algorithm described in [27]. Peak-to-
peak thresholds are calculated as eight times the
standard deviation of the biological noise. They were
determined separately for each recording channel
before off-line processing of the experiments. From
the obtained peak trains, we analyzed both the spiking
and bursting activity by means of simple global
parameters like firing rate, bursting rate and some first
order statistics such as inter spike interval (ISI) and
inter burst interval distributions (IBI). Firing rate is
calculated from the spike trains by means of the
algorithm devised in [28]. Briefly, the firing rate is
computed as the sum of the recorded spikes occurred
in the window divided by the window width. The
choice of the window width is arbitrary, however there
are two limit cases: the mean firing rate (MFR) and the
instantaneous firing rate (IFR). For MFR the window
width is the entire period of registration. Whilst a
narrow time window, e.g. obtained using a Gaussian
kernel of 100 ms width, gives the IFR. The algorithm
used to detect bursts is described in [29]: defining the
ISI as the time interval between two consecutive spikes
in the peak train, the burst is a collection of consecutive
spikes. Specifically, setting the maximum ISI value at
100 ms and the minimum number of spikes at 5 for
each channel, bursts are detected as sequences of
spikes with ISI value smaller than the maximum ISI,
and containing at least 5 spikes. IBI is computed as the
time occurred between two consecutive bursts in each
site and its distribution gives the relative probability
density (see [29,30]). Finally, a network burst is a
period of synchronized firing among many recording
sites. Specifically, in this work we applied the algo-
rithm described in [29] that computes for a time bin of
25 ms the product of the total number of spikes and of
the number of active sites in each bin window. A
network burst is detected if the number of active sites
reaches the 60% of the total amount. For each network
3
Phys. Biol. 12 (2015) 016007 V Pirino et al
burst, the network burst duration (NBD) is defined as
the time occurred between the first involved spike and
the last one.
2.5. Data processing: from peak trains to a weighted
network
In this work, data are spike trains from in vitro cortical
cultures. The goal is to individuate co-activation and
related sub-networks supporting the global network
activity. The analysis was performed at different
resolution levels by aggregating input data into meta-
neurons. A meta-neuron is a small spatially confined
number of actual electrodes which perform single
macroscopic functions [23]. The aggregation is essen-
tial only in the very last part of the analysis, namely the
computing of the sequence of Betti numbers, for
computational reasons. From the spike trains to the
analysis of weighted complexes and their associated
sequence of Betti numbers, the algorithm we imple-
ment can be summarized in the following five steps.
2.5.1. From peak trains to meta-neurons: an
undersampling of the original MEA through electrodes
aggregation via instantaneous firing rate
The peak trains of the 60 electrodes are organized in a
matrix with 60 rows and as many columns as sampling
time points (here over 3 billions). Electrodes are
clustered together in meta-neurons of equal size. By
varying the size of the meta-neurons, we are able to
analyze the activity at different levels of resolution:
from a macro to a micro point of view. We present
results for three levels of resolution and different
aggregation criteria. For MEA dual-chamber we
maintain the two compartments separated also in the
meta-neuron construction to avoid destructing the
intrinsic feature of the network.
1. Meta-neurons are obtained by dividing the MEA in
14 blocks arranged in four lines. Each block has
four electrodes. In figure 2(a), the green box shows
the first meta-neuron. Note that the electrodes 12,
82, 17 and 87 are excluded. For MEA dual-
chamber, in order to keep separated the two
chambers, the first and last rows are chosen slightly
differently, e.g. the upper left green block is formed
by just the three meta-electrodes 12, 21, 22. This
choice brings to have a total of 16 meta-neurons.
2. The meta-neurons configuration is obtained by
dividing the MEA in 30 blocks, each formed by two
consecutive horizontal electrodes. In figure 2(a),
the blue box shows the first meta-neuron. As
above, for MEA dual-chamber the construction of
meta-neurons in the first and last rows is slightly
changed: the first meta-neuron is formed by only
the electrode 21, the second by the electrodes 31
and 41 and so on. This choice brings to have a total
of 32 meta-neurons.
3. Each meta-neuron coincides with the electrode. In
figure 2(a), the magenta box gives the first meta-
neuron formed by the electrode 21.
The choice of meta-neurons is arbitrary. We
observe that each electrode of the MEA records a sig-
nal coming from a small cluster of neurons lying
beneath it (generally up to three [22]). Furthermore,
Figure 1. Configuration and construction of the meta-neuron signal. Panel (a) and (b) show the classical MEA system and the dual-
chamber MEA system respectively. Panel (c) shows a raw data of spontaneous activity (300 s), panel (d) its relative peak train and
panel (e) the consequent thresholded IFR train.
4
Phys. Biol. 12 (2015) 016007 V Pirino et al
this type of data are known to be expression of rather
homogeneous activities.
From the peak trains of the collected electrodes
(figure 1(d)) the IFRs of each meta-neuron are
computed (see section 2.4): in this way, an IFR
train for the meta-neuron is defined as the sum of
the IFRs of the single electrodes at each time
point. If the meta-neuron is the single microelec-
trode then IFR train and microelectrode IFR coin-
cide. In order to keep the strongest components of
the signal, i.e. high firing, we compute the overall
mean of the non null values of the meta-neuron
IFR. Then, we choose an arbitrary threshold, said
θ
1
, set at mean
3
·(figure 1(e)). The resulting train is
a 0/1 sequence visualized as rectangles of unitary
height corresponding to the thresholded activity in
the original peak trains. For each level of resolution,
the thresholded IFRs are collected into a binary
matrix with as many rows as meta-neurons and a
variable number of columns.
Figure 2. Graphical description of the method. On panel (a) the colored boxes show the first meta-neuron at each size as in
section 2.5.1. Panel (b) shows the raster plot relative to a matrix of 14 meta-neurons and the computation of its binned matrix as in
section 2.5.2. On panel (c) and (d) the binary matrix and the reduced matrix derived from the binned matrix, as in section 2.5.2, are
shown. Panel (e) provides the graphical representation of >
M
1derived in section 2.5.2. In panel (f) the sequence of Betti numbers for
the simplicial complex associated to the matrix in panel (d) is given, as described in section 2.5.3.
5
Phys. Biol. 12 (2015) 016007 V Pirino et al
2.5.2. From IFR matrix to simplicial complex and
graphical representation
The recordings are five-minute long at a sampling
frequency of 10000 samples/s. The most evident
feature is that there is a very big amount of data and
that the IFR matrices for meta-neurons are very sparse,
with around 90% zeros. Thus, data are binned
preserving their intrinsic sparseness. Specifically, we
analyzed the main features of the NBD in order not to
break this phenomenon. The burst duration calculated
on all the experiments is 244.43 ± 142.94 ms
(mean ± standard deviation) and varies from 55 ms to
860 ms. Accordingly to that, ten lengths of the bin
window, said
θ
2
, have been selected spanning from
100 ms to 1000 ms with a 100 ms step. IFR values in
each window are summed. The percentage of zeros
decreases slowly with increasing bin size thus keeping
sparsity (see Supplementary Material Figure S1). In
contrast, the percentage of zeros decreases very rapidly
with bin windows larger than 1000 ms. The result is a
collection of ten binned matrices for each recording
and each level of resolution, where each row corre-
sponds to a meta-neuron and each column to a bin:
each entry contains the number of peaks of the meta-
neuron IFR occurred in a specific time window, as
described in figure 2(b).
The obtained binned matrices are converted into
binary matrices in order to associate them an abstract
simplicial complex. The MFR of each meta-neuron
(i.e., each row) is calculated for each binned matrix
and different thresholds have been tested, specifically
3, 5, 6, 7 × MFR (following the approach described in
[4]). For high thresholds, the sub-networks are too
sparse and for too low thresholds they are too dense.
Hence an arbitrary threshold to five times the MFR has
been set.
A binary matrix is associated to each binned
matrix: with entry 1 if the number of spikes in that spe-
cific bin window is greater than the threshold and 0
otherwise (see figure 2(c)).
This means that if there is a 1 in the (i,j) entry, the
ith meta-neuron fires high with respect to itself in the
jth time window.
Repeated columns are removed and an extra row is
added whose entries are the occurrences of each col-
umn. The result is a collection of binary matrices with
as many rows as meta-neurons (plus the row for the
vector of occurrences) and with a variable number of
columns that plays an essential role in our analysis (see
figure 2(d)). Each row corresponds to a meta-neuron
and each column corresponds to a link between the
meta-neurons whose entries are 1. A column whose
entries are all ones means that the electrical activity of
all meta-neurons is high. This indicates that the neu-
rons in the network are all activated at the same
time bin.
A graphical representation of a weighted binary
matrix is given by a colored network. As mentioned in
section 1, the idea of this work is to capture any
repeated patterns of co-activation, to analyze the func-
tional connectivity of the network, and possibly to find
a sub-network that supports the entire network. For
each recording, ten binary matrices have been gener-
ated. We select the smallest bin size matrix Mthat con-
tains a column of all ones, if any exists. Otherwise we
select the matrix Mwith bin size 1000 ms. Experimen-
tally, we verified that in the selected matrix only few
columns have an occurrence greater than one. Specifi-
cally, the percentage of those columns decreases with
the decrease of the meta-neuronʼs size. In the experi-
ments with 14 meta-neurons the percentage of col-
umns with an occurrence greater than one is about the
25% on average. In the case of 30 meta-neurons the
percentage drops to 9%. The columns with weight lar-
ger than one are representative of the most recursive
patterns. They are collected in the matrix >
M
1. The
simplicial complex and a graphical representation of
>
M
1are constructed as follows.
Their vertices correspond to the rows of >
M
1in
which a one appears. For each column of >
M
1, we con-
sider the vertices corresponding to rows with a one.
For example in figure 2(d) the fourth column gives the
subset of vertices σ={5,6,8}
3, the second column
gives
σ={8}
1
and so on. The simplicial complex is
generated by all these subsets (see Appendix). The gra-
phical representation connects every pair of vertices in
each subset by an edge. For a column with only two
ones the meta-neurons corresponding to the ones are
connected by an edge in the graph, for a column with
only three ones the corresponding meta-neurons are
connected by a full triangle and so on.
This mapping has automatically been performed
with the software plex [31], which receives in input
the matrix >
M
1and returns the list of all k-dimen-
sional faces for all >k0and smaller than the number
of meta-neurons. A maximal face (see Appendix) of
the simplicial complex corresponds to a column of
>
M
1while non-maximal faces do not necessarily cor-
respond to columns of >
M
1.
2.5.3. From simplicial complex to sequence of Betti
numbers
The last part of the procedure computes the sequence
of Betti numbers associated to the simplicial complex
linked to the graphical representation obtained in
section 2.5.2. The information carried out is about the
cluster connectivity of the network. The first Betti
number counts the number of connected compo-
nents: if two meta-neurons belong to different con-
nected components then they do not significantly fire
together simultaneously. The second Betti number
counts the number of 2-dimensional holes, namely
the number of loops. An example of loop is an empty
triangle: in our work, it means that there is simulta-
neous activity between all pairs of the triangle vertices
but that the three vertices do not co-work significantly.
Higher Betti numbers count higher-dimensional holes
6
Phys. Biol. 12 (2015) 016007 V Pirino et al
and can be interpreted as a measure of the activity
density in a connected component.
In conclusion, the graph gives a topological repre-
sentation of IFR matrix that in turn summarizes the
functional connectivity of the network throughout
meta-neurons at a certain resolution. The sequence of
Betti numbers gives an algebraic characterization of
the simplicial complex derived from the graph, hence
a numeric summary of the functional connectivity on
different levels/resolution by varying the meta-neu-
ron size.
2.5.4. Parameters
The data processing described from section 2.5.1 to
section 2.5.3 depends on four parameters: the meta-
neuron size (section 2.5.1), the threshold
θ
1
on the
averaged IFR signal (section 2.5.1), the sampling
window width
θ
2
(section 2.5.2) and the threshold
θ
3
to define binary matrices (section 2.5.2). A full
sensitivity analysis of the proposed methods to the
parameters is out of the scope of this paper, never-
theless it is worthwhile to give some indications.
θ
1
has
been chosen to be three times the overall mean of the
non-null values of the meta-neuronʼs IFR. The
sampling window width
θ
2
has been varied from
100 ms to 1000 ms by step of 100 ms. We noticed that
the variation of
θ
2
had little or no effect on most
experiments we analyzed, with negligible differences
in the simplices for the largest
θ
2
and for 14 meta-
neurons (data not shown). The choice of the value for
θ
3
has been discussed in section 2.5.2. Clearly, for two
different values of
θ
3
, say
θθ<
3
13
2
, the binary matrix
associated to
θ
3
1
includes the one associated to
θ
3
2
.
3. Results
Results are organized in sections in order to address
the scientific questions of the Introduction. To sup-
port and present the results, tables of the maximal
faces of each graphical representation are provided.
Furthermore, the obtained sub-networks are charac-
terized by the analysis of the IBI distributions and
NBDs. Before presenting the results for homogeneous
networks, the utility of the methods to detect clusters
of co-activity has been tested on interconnected net-
works and presented in section 3.1.
3.1. Interconnected networks
This test has been performed on n= 9 recordings
where interconnected cortical cultures are coupled to
dual-compartment MEA. The expectation for these
experiments is to find a correspondence between the
topology of the detected sub-networks and the topol-
ogy physically defined by the dual-chamber. If the
algorithm is able to locate and extract sub-populations
of neurons with a high co-activity degree, the results
should show a main sub-network in each compart-
ment and some possible interactions between the two
compartments.
The nine recordings were analyzed following the
steps described in section 2.5. Figure 3(a)–(c) illus-
trates results for the ten minutes of the spontaneous
activity of a representative experiment at a bin window
of 800 ms for three different meta-neuron sizes. Our
expectations are met: the two compartments are
detected and highlighted in all the sub-networks.
Moreover, the number of connections inside each
compartment is higher than the number of
Figure 3. Top row: filtration of the first spontaneous recording of a representative experiment: from 16 (panel (a)) to 32 (panel (b)) to
60 (panel (c)) meta-neurons. Lower row: histograms representing the cardinality of intra (white bar) and inter (gray bar) connections
of all the experiments, for each meta-neuron size and for each bin window. Data are presented as mean ±standard error and they are
significantly different (t-student test, <
p
0.01).
7
Phys. Biol. 12 (2015) 016007 V Pirino et al
connections between the two regions, as shown by the
green and light blue edges in (a) and (b). Another
important observation can be done by looking at the
meta-neurons involved in the three different levels of
resolution. From the macro to the micro scale, the
detected sub-network is refined: the meta-neurons
involved in figure 3(a) contain the ones involved in (b)
that, in turn, contain the ones involved in (c). The
finer the resolution, the more specific the result is. We
refer to the construction of meta-neurons in
section 2.5.1 and as an example we observe that at the
highest resolution meta-neurons 47, 48, 55 in
figure 3(c), cluster into meta-neuron 13 in figure 3(a),
which unites meta-neurons 25 and 29 from
figure 3(b). To quantify this result, for each bin win-
dow of each recording we computed the percentage of
inclusion of meta-neurons involved in the finer reso-
lutions in the meta-neurons of coarser levels. On aver-
age, the results show that there is a percentage of
inclusion higher than 80%. Analyzing together the
results for the three resolution levels, we can state that
all the recorded experiments on the MEA-dual-com-
partment present a similar behavior: the detected sub-
networks are substantially divided in two regions that
correspond to the physical ones (right and left), and
also some communication between the two compart-
ments/sub-networks has been detected. As expected,
the connectivity level is stronger inside the two parts
than between them. We computed and compared the
number of connections inside each (intra compart-
ment) and from one to another (inter compartment).
Figure 3(d)–(f) shows the mean and standard error for
all the experiments, for each meta-neuron size and
each bin window (from 100 ms to 1000 ms with
100 ms step). The bar plots show that the number of
intra compartment connections (white bars) is always
significantly larger than the number of inter compart-
ment connections (gray bars, <p0.01, t-student test).
3.2. Results on homogeneous networks
3.2.1. Identification of sub-networks of repeated activity
and their qualitative characterization both during the
spontaneous and the stimulus-evoked activity
Subnetworks of meta-neurons with the strongest
simultaneous firing activity are read off the simplicial
complexes in section 2.5.2. The nodes of a simplicial
complex identify the meta-neurons of the subnet-
works for which the electrical activity occurs simulta-
neously and most strongly. The degree and intensity of
co-activation are given by the actual connections
among nodes. These can be identified as elements of
the simplicial complex with cardinality larger than one
(that is Σin Appendix). Simple measures of connectiv-
ity are given by the numbers of k-dimensional faces of
the simplicial complex, for >k0. The number of
nodes is given by the number of vertices, i.e. the 0-
dimensional faces.
For a spontaneous and a stimulus-evoked repre-
sentative recording of two experiments (said Exp_1
and Exp_5), the intermediate 14 meta-neurons case is
presented in details while results for the 30 and 60
meta-neurons cases are summarized in section 3.2.2.
Figure 4shows repeated co-activation patterns
deduced from the four recordings: in each plot, a sub-
network of co-activation is evident.
In the spontaneous phase of Exp_1 (figure 4(a)), a
sub-network formed by the meta-neurons
4
,5,6,
and
8 is shown. In the corresponding stimulated phase
(figure 4(b)), the meta-neuron 3 is recruited from the
sub-network and darker edges and larger dots indicate
stronger electrical activity. On the other way round,
Exp_5 behaves in a slightly different way: during the
spontaneous activity the sub-network (figure 4(c)) is
defined by meta-neurons
4
,6,8,9,11,12,13,1
4
and in the stimulated one (figure 4(d)) by
5
,6,7,8,9,11,12,13
. Both sub-networks are
defined by eight meta-neurons, six of which are in
both sub-networks, while two are replaced.
The list of the maximal faces (table 1) of each sub-
network allows further considerations on sub-net-
work connectivity. In Exp_1 the degree of connectivity
is enhanced in the stimulated phase: in the sponta-
neous recording there are two edges (45 and 48) and
one triangle (568), which become a triangle (345) and
a tetrahedron (4568) in the stimulated phase. In Exp_5
the connectivity degree decreases as shown by the fact
that in the spontaneous case the largest faces are two
and 5-dimensional, while in the stimulated case there
are two 4-dimensional faces.
To explain the nature of the results and character-
ize the behavior of the meta-neurons involved in the
sub-networks identified above, we computed the IBI
distribution and the NBD.
Figures 5(a) and (b) shows the raster plots of the
spontaneous and stimulus-evoked activity of Exp_1.
The rows of meta-neurons not involved in the two
sub-networks are substantially those with a very low/
high activity. Rows
4
,5,6,8
in panel (a) and
3
,4,5,6,8
in panel (b) (labeled with stars) corre-
spond to meta-neurons in the detected sub-networks
and look indeed less heterogeneous than the other
rows. Figures 5(e) and (f) shows the raster plots rela-
tive to Exp_5. Here the difference between rows corre-
spondent to meta-neurons involved in the sub-
networks is much less emphasized. In Exp_1 the meta-
neurons included in the sub-network show a hetero-
geneous bursting activity, while the excluded ones
show a more stereotyped bursting activity. This fea-
ture is quantitatively represented in figures 5(c) and
(d), where the mean IBI of the Exp_1 is shown, for the
spontaneous (figure 5(c)) and stimulus-evoked activ-
ity (figure 5(d)). The different behavior of Exp_5 is
reflected in figures 5(g) and (h): the mean IBI and the
standard error show a more similar trend among the
meta-neurons included in the sub-network and those
8
Phys. Biol. 12 (2015) 016007 V Pirino et al
not included, both in the spontaneous and in the sti-
mulus-evoked activity.
Finally, from the NBD we find that the behavior of
the meta-neurons in the sub-networks is different
from the behavior of those not involved. We verify that
the network bursts of each recording have different
duration (from less than 10 ms to some hundreds of
milliseconds) and that almost always the longest ones
recruit more sites than the other ones. At first glance,
we note that the meta-neurons involved in the sub-
networks rarely appear in the network bursts and
almost only if the network bursts have long duration.
To better quantify their occurrences, for each record-
ing we select the network bursts in which the detected
sites are involved, compute the duration of the burst-
ing period and observe the time instant in which they
appear. Moreover, for each meta-neurons size and
each recording, we compare the minimum duration of
network bursts containing the sub-networks sites with
the minimum duration of network bursts in the
selected recording. The results confirm that the detec-
ted meta-neurons appear rarely, in the very last burst-
ing period and quite only in the longest network burst.
As in the previous analysis, this behavior is clear in
Exp_1 and less evident in Exp_5. Table 2shows for the
two experiments the percentage of network bursts in
which the meta-neurons involved in the sub-networks
appear, the minimum duration of those network
bursts and the minimum of all the ones occurred in
each recording.
3.2.2. Refining the resolution
Similar conclusions to that of the 14 meta-neurons
case can be drawn by refining the resolution level and
performing the same type of analysis for the 30 and
60 meta-neurons. All tables and figures relative to this
section are collected in Supplementary Materials. The
results show that in both the 30 and 60 meta-neurons
cases a sub-network is detected. The finer resolution
graphs have the same behavior as the ones with
Figure 4. Two examples of sub-networks in matrices with 14 meta-neurons. A spontaneous (panel (a)) and stimulus-evoked (panel
(b)) phase relative to Exp_1 are shown. A spontaneous (panel (c)) and stimulus-evoked (panel (d)) phase relative to Exp_5 are
shown.
Table 1. Maximal faces of the subnetworks of figure 4.
14 sites Maximal faces Exp_1 Maximal faces Exp_5
spontaneous 4 5, 4 8, 5 6 8 8 12 14, 4 6 8 9 11 13,
469111213, 689111213
stimulated 3 4 5, 4 5 6 8 5 9, 6 7 11 12 13, 7 8 11 12 13
9
Phys. Biol. 12 (2015) 016007 V Pirino et al
14 meta-neurons: the sub-networks of the stimulus-
evoked recordings involve almost always the same
meta-neurons of the spontaneous ones. Raster plots
and IBI distributions confirm that the not-involved
sites are those with higher/lower firing rate and that
their stereotyped firing is in contrast with the more
heterogeneous firing of the involved sites. Finally, also
the NBD analysis bring to the same consideration
raised in the 14 meta-neurons case. All these results
are noticeably in Exp_1 and less evident in Exp_5, as in
the more rough case.
3.2.3. Quantitative comparison between spontaneous
and stimulus-evoked activity
One of the focal points of this study is to figure out
whether and how the stimulated activity is linked to
the spontaneous one. Some studies like [22] justify our
Figure 5. Raster plots of IBI distribution (mean ± standard error) for 14 meta-neurons. (a) Exp_1 raster plot of the representative
spontaneous phase and (b) of the stimulus-evoked one. The marked lines in the raster plots correspond to the meta-neurons detected
in the sub-network. (c) IBI distribution (mean ± standard error) of the representative spontaneous recording and (d) of the
representative stimulus-evoked recording of Exp_1. (e) Exp_5 raster plot of the representative spontaneous phase and (f) of the
stimulus-evoked one. The marked lines in the raster plots correspond to the meta-neurons detected in the sub-network. (g) IBI
distribution (mean ± standard error) of the representative spontaneous recording and (h) of the representative stimulus-evoked
recording of Exp_5.
10
Phys. Biol. 12 (2015) 016007 V Pirino et al
working hypothesis that the stimulation does not
change the intrinsic (i.e., spontaneous) dynamics of
the network. The expectation is that the stimulus-
evoked activity enhances the spontaneous one by a
change in the firing frequency. The results presented
in section 3.2.1 seem to confirm that the evoked
activity resembles the spontaneous one as the meta-
neurons involved in the identified sub-networks are
almost always the same. Usually, the electrical stimula-
tion increases the occurrences of meta-neurons and
edges in the sub-network. The histograms of
figure 6(a)–(c) summarize this behavior: they are
relative to the three information levels considered for
Exp_1, namely 14 (panel (a)), 30 (panel (b)) and 60
(panel (c)) meta-neurons.
Similar plots for Exp_5 seem to lead to different
considerations. In figure 6(d) and (e), the histograms
relative to the first and second information level (14
and 30 meta-neurons) are shown. The finest resolu-
tion level is not informative: in both the spontaneous
and stimulus-evoked recordings only one site appears.
As in Exp_1 the stimulation site does not affect the
network behavior: the sub-network of the sponta-
neous and stimulated recording is in the same region
of the meta-neurons involved. On the contrary, here
the stimulus does not always increase the site and edge
occurrences.
To better highlight this feature, for each recording
of Exp_1 and Exp_5 and for each meta-neuron of size
m, we consider the >
M
1matrices, count in how many
recordings of the spontaneous (stimulus evoked
respectively) case a meta-neuron appear, select the lar-
gest occurrence tand keep track of the meta-neurons
appearing more than
t
2
times. The results are recor-
ded in table 3.
For Exp_1, which has 6 spontaneous recordings
and 6 stimulus-evoked recordings, the most frequent
meta-neurons in the spontaneous case are also most
frequent in the stimulus-evoked case. Whilst for
Exp_5, 6 spontaneous and 4 stimulus-evoked, the
Figure 6. Top row: bar charts relative to Exp_1 clusterized with 14 meta-neurons (panel (a)), 30 meta-neurons (panel (b)) and 60
meta-neurons (panel (c)). Bottom row: bar charts relative to Exp_5 clusterized with 14 meta-neurons (panel (d)) and 30 meta-
neurons (panel (e)). The gray bars are relative to the electrodes occurrences in the spontaneous phase, while the black bars are relative
to the electrodes occurrences in the stimulus-evoked phase.
Table 2. Summary of some relevant characteristics of the NBD (in ms) of the two representative recordings of Exp_1 and Exp_5.
meta-neuron Exp_1 Exp_5
size % min in min tot % min in min tot
14 Spont 16.82% 112.4 46.6 38.8% 137.8 112.7
Stim-evok 20.2% 116.2 8.1 99% 91.3 91.3
30 Spont 13.67% 122.9 49.3 54.76% 155.8 142
Stim-evok 19.27% 117.7 8.1 94.28% 143.2 143.2
60 Spont 22.5% 125.7 49.3 63.41% 122 113.8
Stim-evok 61.1% 111.4 8.1 65.11% 127 115.9
11
Phys. Biol. 12 (2015) 016007 V Pirino et al
situation is less homogeneous than expected. For
m= 14, one meta-neuron is lost in the stimulus-
evoked case. For m= 30, only the meta-neuron 25
appears in all recordings of the spontaneous case and
in 6 recordings of the stimulus-evoked case; in the
spontaneous all other meta-neurons appear in less
than four recordings. For m= 60, one meta-neuron is
kept in the stimulus-evoked case and one is lost.
3.2.4. Information embedded in Betti numbers sequence
The sequence of Betti numbers provides a topological
description of the simplicial complex derived from the
detected sub-network. Each Betti number refers to a
fixed space dimension. The first Betti number (named
β
0
) describes the
th
0
-dimension: it counts the number
of connected components or the number of clusters.
The second Betti number (
β
1) counts the number of 2-
dimensional holes or loops and gives an indication on
the clusters’density. The third Betti number (
β
2)
counts the number of 3-dimensional holes and is a
measure of the simultaneous activity of neurons, and
so on. In this study the non-zero Betti numbers are the
first and the second ones. Each of them is closely linked
to the specific recording: both can vary several times
within the same experiment for different values of the
parameters. The first Betti number changes in a range
1 to 6, and for Exp_1 and Exp_5 for bin window size
from 100 ms to 1000 ms by step 100 ms, on average it
takes the value 1.49. Specifically, in 89.8% of cases is
one or two: the co-activation level of each simplicial
complex is completely described by one or two
clusters. The second Betti number is zero in the 96.7%
of the resulted simplicial complexes and on average it
takes the value 0.05: in a single cluster each node/site is
linked to the others.
The information carried by
β
0
and
β
1suggests that
the identified sub-networks display very dense con-
nections, both if they consist of a single cluster
(
β=i. e ., 1
0) and if they consist of separated ones
(
β>i. e ., 1
0). The small percentage of simplicial
complexes with a high first Betti number (e.g.,
β
=3, 4, 5
0
) is characterized by disconnected sub-
networks, namely non-communicating clusters. The
number of connected components and hence
β
0
can
be read off from the sub-network plots. A stimulus-
evoked recording and a spontaneous recording of two
representative experiments are shown in figure 7,
panel (a) and (b) respectively, as an example of high
first Betti number. It holds
β
=5
0and
β
=
4
0,
respectively.
The variation of the sub-network is witnessed by
the variation of
β
0
in each recording. On the contrary,
the information contained in
β
1cannot be read gra-
phically in a 2D representation. However, the percen-
tage values written above confirm that, on average,
each cluster is highly connected inside itself, i.e. the
Figure 7. Two examples of disconnected simplicial complexes. Panel (a) is relative to a stimulus-evoked recording of a representative
experiment; panel (b) is relative to a spontaneous recording of another representative experiment.
Table 3. Summary of the meta-neurons occurrences in each recording of Exp_1 and Exp_5. S gives the list of meta-neurons appearing in
most recordings; R the percentage of meta-neurons in S spontaneous also appearing in S stimulus evoked and t the maximum number of
appearances of a meta-neuron.
meta-neuron Exp_1 Exp_5
size S t R S t R
14 Spont 4–6, 8 6 6–9, 11–13 6
Stim-evok 4–6, 8 6 100% 6, 8–9, 11–13 4 85.7 %
30 Spont 11–14 6 25 6
Stim-evok 6–14 4 100% 24–25, 27, 29 3 100%
60 Spont 23–28 6 59–60 2
Stim-evok 13–28 5 100% 51, 54–55, 57, 60 2 50 %
12
Phys. Biol. 12 (2015) 016007 V Pirino et al
electrical activity is rather intense within the sub-
network.
Importantly, there are no significant differences in
the Betti numbers for spontaneous and stimulus
evoked cases with the same bin window size. The fact
that these topological invariants are often the same
supports a positive answer to Question 2 and a con-
sequential negative answer to Question 3. However,
β
0
embeds an important information that is how many
sub-networks of the MEA work significantly together
and independently from each other, while
β
1is a rough
indication of the intensity of sub-network co-activity,
the higher
β
1the lower the clusters density.
4. Conclusion
In this paper, we present a method capable of detecting
and characterizing repetitive and synchronous co-
activation patterns in in vitro cortical assemblies
pointing out to the individuation of a possible core
sub-network that sustains the global network activity.
The hypothesis that some neurons can orchestrate the
global dynamics of a network has been already
investigated in [32] where the authors detect the
presence of highly active neurons that are the pre-
cursors of the activation of network bursts. In this
work, we followed a different methodology: we took
the peak trains relative to the 60 microelectrodes of a
MEA and reduced them to a matrix encompassing
networks co-activity and to a vector of weights
expressing the multiplicity of repeated patters of
activity. Such a reduction procedure depends on four
parameters: the meta-neuron sizes, the threshold
θ
1
on
the averaged IFR signal, the sampling window width
θ
2
and the threshold
θ
3
to define the binary matrices.
Thus, MEA data are clustered in a collection of meta-
neurons of different sizes, which maintain the intrinsic
characteristics of the network without loosing too
much information. Then, we performed a double
analysis. On the one hand, we gave a bi-dimensional
graphical representation of such a matrix from which
we read off sub-networks of co-activity and the
intensity of the interactions among the involved sites.
On the other hand, we extracted a topological
characterization of those sub-networks computing the
sequence of Betti numbers of the associated simplicial
complexes. The graphical representation we provide is
two-dimensional but information about higher-order
interactions among meta-neurons are embedded into
the k-dimensional (maximal) faces of the simplicial
complexes we constructed. The novelty of our metho-
dology is two-fold. First, its multi-dimensionality sets
our approach apart from the gold standard methods
used to infer functional connectivity (e.g., cross-
correlation [33,34] and information theory based
methods [17]) that provide pairwise analysis of
relationships among the nodes of the network. Second,
it provides a summary of the network activity over the
whole recording time. Specifically, the identified sub-
networks are a static (not time dependent) summary
of the meta-neurons particularly active simulta-
neously and recurrently over the entire recording. On
the contrary, conventional methods provide a pairwise
study of the relationship between nodes giving a time
dependent correlation analysis [17]. An intermediate
approach is the one devised by Baruchi and coworkers
[14]. In that work, the functional connectivity of
in vitro neuronal networks was studied in terms of its
clusterization: the bursting activity was investigated
through connectivity matrices and 3D diagrams point-
ing out that the structural organization of the detected
bursts is simple. Although our analysis does not
involve a correlation function, our results are topolo-
gically similar: the detected sub-networks are charac-
terized by one or rarely two clusters with a very simple
structure (the sub-networks are bi-dimensional),
accordingly to the results found in [14].
By means of the graph we generated, we can
answer the three scientific questions we raised in the
Introduction. Relative to the first question, the graphs
show the sub- networks containing the electrodes that
fire simultaneously and repeatedly. Depending on the
nature of firing of each meta-neuron, the algorithm
identifies a sub-population of neurons having a more
heterogeneous behavior with respect to the entire net-
work as shown in figure 5. The more evident the differ-
ence, the more the sub-network is characterized by
high weights in its graphical representation strength-
ening the high degree of co-activity of that neuronal
cluster. This peculiar trend has been confirmed by the
analysis based on the IBI distribution (figures 5(c),
(d), (g), (h)): the profile of the IBI of the meta-neurons
belonging to the sub-network(s) has much larger var-
iance then the profile of the IBI of the other sites.
Moreover, the presence of sub-network(s) is con-
firmed by the network bursts analysis which shows
that the meta-neurons in the sub-network identified
by our method appear rarely in each single network
burst and, if they do, only in the last part of longest
network bursts (see table 2). This feature confirms that
our algorithm selects regions of autonomous meta-
neurons that only sometimes contribute to the global
behavior.
To address the second question in the Introduc-
tion, we compared the behavior of spontaneous and
stimulus-evoked recordings. The stimulation enhan-
ces the natural firing of the network without bringing
significant changes in the qualitative features of the
detected sub-networks (see figure 6). Specifically, the
assembly of meta-neurons identified in the evoked
recordings corresponds to the same group identified
in the spontaneous recordings: on average, the sites in
the sub-network for the evoked cases have larger
weights, meaning that they occur more often, and the
maximal faces have higher dimension indicating a
greater co-activity level (see table 1).
13
Phys. Biol. 12 (2015) 016007 V Pirino et al
Our answer to the third question is provided by
the sequence of Betti numbers, used in topological
data analysis [3–5,7]. In all cases, we observed that
only the first two Betti numbers were non zero. Thus,
the sequence gives the number of connected compo-
nents of the sub-network and their level of con-
nectivity. In particular, there is not a specific Betti
numbers sequence for the spontaneous and for the
evoked recordings: the stimulation excites the net-
work maintaining the main features of the sponta-
neous phase.
In the future, we could also consider a dynamic
version, time dependent, of the simplicial complexes:
in that case, the analysis would likely give rise to a fil-
tration depending on a parameter variation (as done in
[4,5,7]). This different approach would produces a
more varying Betti number sequences enhancing the
information about network connectivity. Other future
work could be related to improve the algorithm mini-
mizing the loss of information, analyze the parameters
sensitivity, vary the extraction procedure of sub-net-
works and improve their graphical representation. To
get closer to some classical methods (e.g., cross-corre-
lation), we could consider not only the synchronous
firing but also the propagation delay using a time shift-
ing-window instead of the single static time bin. Fur-
thermore, it would be interesting to further analyze
cases in which neuronal populations are partly con-
fined-segregated or to analyze the activity of 3D neu-
ronal cultures in which the level of co-activation and
the spontaneous formation of sub-networks are more
likely to take place [35].
Acknowledgments
The research leading to these results has received
funding from the European Unionʼs Seventh Frame-
work Programme (ICT-FET FP7/2007–2013, FET
Young Explorers scheme) under grant agreement n
284772 (BrainBow). VP is grateful to Professor
Vladimir Itskov for useful discussion and email
exchange. The authors also wish to thanks Dr
Mariateresa Tedesco for the neuronal cell cultures and
Dr Bruno Benedetti for the thorough revision of the
manuscript and his valuable suggestions.
Appendix: simplicial complexes and Betti
numbers
Afinite set of high dimensional data somehow
connected to each other through unknown relation-
ships can be more easily interpreted through a
graphical representation. The more intuitive idea is to
tie together these data by building upon them a
skeleton that allows organizing their interactions. One
of the most common structure is that of abstract
simplicial complex. For recent references on topics
related to those in this paper see [3,8] and references
therein.
An abstract simplicial complex is a pair
Σ=
X
V(, )
where Vis a finite non-empty set and Σis
a non-empty family of subsets of Vsuch that if σΣ∈
and
τσ
⊆then
τ
Σ∈. The simplicial complex Σσ
generated by a non-empty set σis the set of all subsets
of σand the simplicial complex generated by σand τis
the union of Σσand
Στ
. In essence Vis in one-to-one
correspondence with data points and Σwith relation-
ships among data points. The simplex condition
imposes that a relationship holding true for a set of
data is transmitted to all its subsets.
More specifically, a k-dimensional face aggregates
+k1
data points and the all k-dimensional faces are
collected into a k-simplex defined as
ΣσΣσ=∈∣♯=+k{1}
k, where
σ
♯
is the number of
elements of Vin σ[8]. A face is maximal if it is not
contained in any other face of the simplex. The faces of
dimension 0, 1 and 2 are called vertices, edges, trian-
gles, respectively.
Ak-dimensional face can be represented in a k-
dimensional space.
A total ordering on the points in Vis chosen and
two fundamental families of linear operators are
defined. For i=1,…,kand
+
s
i1
the +i(1)
th element
of
σΣ∈
k
, the first type of operators is
ΣΣ σσ⟶⟼−
−+
ds:, {}(A.1)
ik k i11
and for <
i
0or
>
i
k
,d
i
is the null operator. To define
the second type of operators, or board operators,
consider the so-called k-chains: linear combinations
of k-dimensional faces over a suitable coefficient field
⎪
⎪
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
∑
Σσ== ∈
σΣ
σσ
∈
()
CX a a() Span : (A.2)
kk
k
For k= 0 to the number of vertices minus one, the k-
board operator is
∑
∂⟶ ∂=−
−
=
CX C X d:() (), (1)(A.3)
kk k k
i
k
ii1
0
and for <k0or >−kn1it is equal to the zero map.
The chain resolution of the simplicial complex
Σ=
X
V(, )
on
is defined as the sequence of maps:
⟶…⟶ ⟶ ⟶
⟶⟶…⟶
+∂∂
−∂−
+
−
CC
CC
0
0(A.4)
kk
kk
1
12
kk
k
1
1
From the definitions of d
i
and ∂
k
, note that: (1) the
board operator is a homomorphism of vector spaces;
(2) ∂◦∂ ≡
+
0
kk1
; (3) ∂⊆ ∂
+
Imm Ker() ()
kk1.
Definition 1. The k-th homology group of the
simplicial complex Xis =∂HX Ker(, ) ( )
kk
∂+
Imm ()
k1. The dimension of HX(, )
kas a quotient
space of homology is called the k-th Betti number, and
it is written as
14
Phys. Biol. 12 (2015) 016007 V Pirino et al
β=∂∂
+
()
() ( )
dim Ker Imm .(A.5)
kkk1
An intuitive interpretation of Betti numbers for the
analysis presented in this work is that the kth- Betti
number counts the number of k-dimensional
holes. Thus:
-
β
0
counts the connected components;
-
β
1counts the missing polygons (also known as 2-
dimensional polytopes), e.g. the number of missing
triangles, missing squares, missing circles and so on;
-
β
2counts the 3-dimensional holes or missing 3-
polytopes (e.g., the missing tetrahedrons, missing
spheres);
-
β
kcounts the number of +k(1)
-dimensional holes,
i.e. missing +k(1)
-polytopes.
Since Betti numbers are a topological invariant, sim-
plicial complexes arising from different data set can be
compared and classified. More details about the homol-
ogy groups are reported in [36] and references therein.
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