Anti-correlations in the degree distribution increase stimulus detection performance in noisy spiking neural networks
Anti-correlations in the degree distribution increase stimulus detection performance in noisy spiking neural networks
Marijn B. Martens 0 1
Arthur R. Houweling 0 1
Paul H. E. Tiesinga 0 1
Action Editor: Gaute T. Einevoll
0 Department of Neuroscience, Erasmus University Medical Center , Rotterdam , Netherlands
1 Department of Neuroinformatics, Donders Institute for Brain , Cognition and Behaviour , Radboud University , Nijmegen , The Netherlands
Neuronal circuits in the rodent barrel cortex are characterized by stable low firing rates. However, recent experiments show that short spike trains elicited by electrical stimulation in single neurons can induce behavioral responses. Hence, the underlying neural networks provide stability against internal fluctuations in the firing rate, while simultaneously making the circuits sensitive to small external perturbations. Here we studied whether stability and sensitivity are affected by the connectivity structure in recurrently connected spiking networks. We found that anti-correlation between the number of afferent (in-degree) and efferent (out-degree) synaptic connections of neurons increases stability against pathological bursting, relative to networks where the degrees were either positively correlated or uncorrelated. In the stable network state, stimulation of a few cells could lead to a detectable change in the firing rate. To quantify the ability of networks to detect the stimulation, we used a receiver operating characteristic (ROC) analysis. For a given level of background noise, networks with anti-correlated degrees displayed the lowest false positive rates, and consequently had the highest stimulus detection performance. We propose that anti-correlation in the degree distribution may be a computational strategy employed by sensory cortices to increase the detectability of external stimuli. We show that networks with anti-correlated degrees can in principle be formed by applying learning rules comprised of a combination of spike-timing dependent plasticity, homeostatic plasticity and pruning to networks with uncorrelated degrees. To test our prediction we suggest a novel experimental method to estimate correlations in the degree distribution.
Spiking neural networks; Stability; Sensitivity; Stimulus detection; Degree distribution; Associative plasticity
A fundamental goal of neuroscience is to elucidate how
neural circuits respond to small external inputs, while
simultaneously remaining stable against neuronal noise. This is
especially a problem for cortical networks producing sparse
activity, because weak external inputs involve a number of
spikes that is comparable to the number of spikes produced
by spontaneous activity. Neuronal noise can arise from
intrinsic and extrinsic sources and influences every level of
the nervous system (Jacobson et al. 2005; Faisal et al. 2008).
Noise has in some cases been found to limit the information
capacity of neurons (Schneidman et al. 1998; London et al.
2002), but could also enhance the computational capability
of neurons in other circumstances (Rudolph and Destexhe
2001; Stacey and Durand 2001).
With the advent of recording and imaging techniques
that are not biased to record only from neurons with a
high firing rate, experiments revealed sparse firing in the
neocortex (Houweling and Brecht 2008; Barth and Poulet
2012; Wolfe et al. 2010). For example, the barrel cortex
shows spontaneous spiking at low firing rates, ranging from
less than 1 Hz in the superficial layers to a few Hz in the
deep layers (Greenberg et al. 2008; de Kock and Sakmann
2009; Barth and Poulet 2012). According to recent
experiments, a single extra spike in one neuron in the barrel
cortex is amplified and produces approximately 28
additional spikes in its postsynaptic targets, thereby causing
a detectable increase in firing rate in the local network
(London et al. 2010). The brain thus requires strategies
to remain stable against noise in the form of spontaneous
At the same time sensory systems have to be sensitive
to relevant external input. Rodents can be trained to use
their whiskers to detect an object that predicts a reward
and respond with licking to obtain this reward (Huber et al.
2012). The neural responses in barrel cortex to whisker
stimulation are hypothesized to play an important role in
performing sensory tasks (Petersen and Crochet 2013).
Whisker stimulation results in a stimulus-locked neuronal
response that can be measured in the rat barrel cortex (Stern
et al. 2002). It is even possible to train rats to respond
when they detect a small number of spikes elicited by
electrical stimulation of a single neuron in the sensory cortex
(Houweling and Brecht 2008; Doron et al. 2014).
Thus, neuronal networks need to be stable against
intrinsic fluctuations and unrelated spiking input from other
brain areas, while the aforementioned experiments showed
that these networks are also sensitive to small
perturbations. Sensitivity and stability are connected and can in
general not be optimized simultaneously, as the increase
in one causes a decrease in the other. Increases in
sensitivity to external stimuli are mostly studied in terms of
modulation of neuronal activity, for example by attention
mechanisms (for reviews see Tiesinga et al. 2008; Fries
2009). Here we examine whether specific structures in
network connectivity can improve the sensitivity to
stability trade-off in spiking neural networks (SNNs).
Experimentally, SNNs show spontaneous spiking, which can be
amplified through recurrent connectivity into synchronous
network-wide activity, referred to as a burst (Martens et al.
2014; Chiappalone et al. 2007). Such bursts can also be
evoked in SNNs by external stimulation (Chiappalone et al.
2007). We investigated recurrent SNNs and used
simulations to determine the effects of correlation between the
number of afferent (in-degree) and efferent (out-degree)
connections in neurons on the generation of bursts as part of
spontaneous activity and in response to external stimulation.
We studied whether stimulation would lead to a detectable
change in the firing rate, which in our model would often
involve amplification into a burst response. Within the
context of our model, a large fraction of the neurons in the
network participate in the burst. When comparing to barrel
cortex, this core network should be considered embedded
in a much larger network. Hence for that case, the
network detection corresponds to a smaller fraction of the
network becoming active, which is more representative for
the experimental situation.
This computational study is the first to focus on the
trade-off between sensitivity and stability with correlations
between the in- and out-degree in SNNs, rather than in
simplified binary networks (Vasquez et al. 2013). The
previously studied network of binary neurons contained no
inhibition and was captured by a first order Markov
process, hence contained no memory of past activity past the
current state. The SNNs in this study consist of different
neuronal cell types and have connection probabilities
representative of cortical networks, and show stable low firing
rate and/or brief burst responses, whereas neural networks
with binary neurons will converge to either a high or a low
firing rate state after a single stimulation (Vasquez et al.
2013). To test network sensitivity we apply nanostimulation
(single neuron stimulation) or stimulation of a few neurons
(typically four). Our guiding hypothesis is that improved
stimulus detection can be achieved through anti-correlations
in the degree distribution.
We focus on correlations within the same neurons, rather
than degree-correlations between different neurons, which
is referred to as assortativity (Newman 2003). Most
biological networks are disassortative, such that nodes with
many edges preferentially connect to nodes with a few
edges (Newman 2003). Assortative networks appear less
stable (Brede and Sinha 2005), but at the same time
assortative neural networks perform better in detecting
subthreshold stimuli and outperform disassortative networks in the
case of memory retrieval (de Franciscis et al. 2011;
Schmeltzer et al. 2015). Multi-unit recordings in
organotypic brain slices suggest a frequency-dependent network
architecture, and showed that cortical and hippocampal
connectivity is disassortative for low frequencies and cortical
connectivity is assortative for the high frequency range
in cortex (Ito et al. 2014). These studies thus show that
whether high degree neurons preferentially connect to other
neurons with low or high degree plays a role in network
functioning, and that (dis)assortativity can be found in
neuronal networks. However, few studies have focused on
correlation in the in-degree and out-degree in the same
Neuronal network connectivity is not static, but can vary
on a timescale of hours (Minerbi et al. 2009) or days
(Trachtenberg et al. 2002; Holtmaat et al. 2005), during
which synaptic contacts can form and disappear (Yuste
and Bonhoeffer 2004). Plasticity has an important role in
neuronal circuit formation, in particular in the form of
spike-timing dependent plasticity (STDP) which induces
competitive learning (Song et al. 2009). We studied
networks that were formed randomly (without correlation in
the degree distribution) and found that STDP, in
combination with a global homeostatic rescaling of synaptic weights,
shapes the network such that after pruning the weakest
synapses a stable network with anti-correlation degrees is
When we quantified network stability in the presence
of noise, we found that the onset of the high frequency
bursting state, a state we consider pathological as noise
continuously evokes bursting, was delayed to higher
levels of background noise for networks with anti-correlated
degrees compared to networks with positive correlations
in the degree distribution. Networks with anti-correlated
degrees are thus more stable against background noise. We
also tested the sensitivity to stimulation for low noise
levels, when the networks were not spontaneously bursting,
and found that networks with positively correlated degrees
were the most sensitive as they produced a burst response
for the lowest level of recurrent excitatory connection
strength. We then tested stimulus detection, which requires
simultaneous stability and sensitivity, by applying
stimulation to a few neurons (1-6) under noise levels for which
spontaneous network bursts occurred at low rates. The
anti-correlated networks outperformed networks with
positive correlations. Taken together, these results suggest that
the correlation structure is important for the stability and
stimulus detection in neuronal networks. Furthermore, we
demonstrate that the necessary anti-correlation in the degree
distribution can emerge as the result of a simple plasticity
2 Materials and methods
In this study, we determine whether correlations in the
joint in- and out-degree distribution affect stability,
sensitivity and/or stimulus detection performance. We test this in
sparsely connected networks of spiking neurons. Here we
state the network dynamics and connectivity rules used, and
describe how the analyses were performed.
2.1 Network dynamics
The dynamics of the neurons in the model are described by
equations proposed by Izhikevich (2003). The Izhikevich
model constitutes a simplified version of the
HodgkinHuxley model. Other appropriate models would be ones
whose subthreshold dynamics can be integrated exactly
(Rotter and Diesmann 1999), which can be simulated with
similar computationally efficient strategies (Yamauchi et al.
2011). For Izhikevich-type neurons, membrane variables v
and u are given as:
With the following after-spike reset conditions:
if v ≥ 30, then v ← c (3)
u ← u + d
where the dimensionless variable v represents the
membrane potential in mV and the dimensionless variable u
represents the membrane recovery variable, which accounts
for the activation of the K+ currents and inactivation of Na+
currents (Izhikevich 2003). The input current I is described
in Eq. (4) below. We used the Euler method for integration
of the differential equations with smaller integration time
steps d t (representing milliseconds) than in the
aforementioned references in order to increase accuracy, specifically
0.05 ms for the membrane potential and d t = 0.1 ms for
the other slower variables. The parameters a, b, c and d
describe the neuronal type, in our model we use the settings
for regular spiking (RS), fast spiking (FS) or low-threshold
spiking (LTS) model neurons. These parameters are listed
in Table 1.
The parameter a is the rate of the recovery variable u,
smaller values result in slower recovery.
The parameter b represents the sensitivity of the
recovery variable u to the subthreshold fluctuations of the
membrane potential v, where larger values yield a
stronger coupling between u and v.
The parameter c is the reset value of the membrane
potential after a spike.
The parameter d represents the change in recovery
variable u, caused by spike-activated Na+ and K+
We model two sources of noise. The first is the variability
associated with small random events, such as ion
channel noise and stochastic synaptic release and weak synaptic
inputs due to uncorrelated spiking (Jacobson et al. 2005;
O’Donnell and van Rossum 2014). These sources of noise
contribute only a small fraction to the variability in the input
(represented by If luc in Eq. (4) below). The other form of
noise we simulate is an occasional larger event, such as
correlated spiking input events from other brain areas that are
unrelated to the sensory stimulus (London et al. 2010), and
is referred to as background noise (Ibg in Eq. (4) below).
Supplementary Figure S1 shows the flow of current within
the network. The cells receive the total input I given as:
I = If luc + Ibg + Istim + Isyn
Table 1 Parameter settings
proposed by Izhikevich to
model different neuronal
classes found in the cortex
Pyramidal neurons (Pyr) are modeled as regular spiking (RS). The inhibitory population consists of different
cell classes: we modeled parvalbumin postive neurons (PV) as fast spiking (FS) and somatostatin positive
neurons (Sst) as low-threshold spiking (LTS). ± denotes variance of the underlying normal distribution,
representing the variability of parameter values across neurons in the network
Where If luc is modeled as white noise (for mean and
variance see Table 1), and Ibg is modeled as a Poisson process
where each background spike event causes a brief current
pulse to the excitatory neurons with an amplitude of 15 and a
duration of 0.1 ms. The stimulation for our sensitivity
measurements is represented by Istim (parameter settings are
given in Section 2.5). Isyn is the conductance-based synaptic
input between the recurrently connected neurons, calculated
Isyn,j (t ) =
wij · gi (t )[Ei,rev − vj (t )]
Here wij is the synaptic strength between presynaptic
neuron i and postsynaptic neuron j , g is the conductance,
Erev the reversal potential for a particular synaptic current
(0 for excitatory and -80 for inhibitory neurons) and v is
the postsynaptic membrane potential. The conductance g is
increased with 1 for each presynaptic spike and falls off
exponentially with a time constant of 2 ms for excitatory,
and 10 ms for inhibitory neurons (Fig. 1A).
2.2 Network connectivity
The model network was composed of 600 neurons, of
which 80 % were excitatory (pyramidal cells, Pyr) and
20 % were inhibitory neurons. The cortex consists of many
functionally distinct inhibitory neuron classes that can be
identified by molecular markers (interneuron nomenclature
Group 2008; Pfeffer et al. 2013; DeFelipe et al. 2013). Here
we used two main inhibitory cell types, namely the
fastspiking parvalbumin-expressing interneurons (PV) and the
low threshold somatostatin-expressing interneurons (Sst),
(Fig. 1A). The PV cells are critical for the network as they
balance the activity of excitatory neurons and stop
network bursts from making the network epileptic. The Sst
type neurons only get activated for a high level of
network activity, and inhibit the PV neurons. These different
neuron types are included to accommodate the
hypothesis that nanostimulation of inhibitory neurons, which could
lead to disinhibition, relates to increased detection
performance (see also Buia and Tiesinga 2008). This hypothesis
was explored in pilot studies, but was not included in the
For a local network of rat neocortical neurons the
Pyr-Pyr connection probability is about 5 %, whereas
each interneuron projects to most of the local Pyr cells
(Holmgren et al. 2003; Packer and Yuste 2011; Pfeffer et al.
2013; Avermann et al. 2012; Lefort et al. 2009), (Fig. 1B).
PV neurons are modeled here to receive inhibition from both
PV and Sst neurons, whereas Sst neurons only receive
excitatory input (Pfeffer et al. 2013; Gibson et al. 1999). The
relative fraction of synaptic drive that the interneurons
provide is taken from experimental data (Pfeffer et al. 2013)
(Fig. 1C-D). This method, proposed by Pfeffer et al.,
combines a number of measurements in order the determine the
strength of the interneuron projection on pyramidal cells as
well as on other interneurons. It is important to understand
their method in order to appreciate where our parameter
settings derive from. First, using paired recordings the
probability of a connection between a pre- and postsynaptic
neuron (Pcon) was estimated based on their cell type as
well as the unitary strength (uI P SQ) of such connection
expressed as the total charge that enters the cell. This is the
time-integrated current, and thus represents the product of
amplitude and duration. The individual contribution type is
then defined as I N C = uI P SQ · Pcon; I N C thus reports
how much inhibition any interneuron of a given class
contributes, on average, to any pyramidal cell. The second step
is to determine, based on the total charge I P SQPyr entering
a pyramidal cell upon stimulation of a particular interneuron
population by optogenetic light pulses, how many
interneurons (Ninc) were activated (and how many spikes per light
pulse), i.e. Ninc = I P SQPyr /I N C. An interneuron is
recorded from simultaneously with the recording of each
pyramidal neuron. The interneuron to interneuron strength
(I N CI nt−I nt ) can then be estimated using: I N CI nt−I nt =
I P SQI nt /Ninc (Pfeffer et al. 2013). The strength so
measured can be compared and were used as relative strengths in
For many of the connectivity analysis routines, for
example to calculate the shortest path length and k-core
decomposition, we used the brain connectivitiy toolbox (BCT)
(Rubinov and Sporns 2010).
Fig. 1 The model network was comprised of one type of excitatory
(Pyr) neuron and two inhibitory classes (PV and Sst). A: The
majority of cells was excitatory and made fast glutamatergic synapses with a
reversal potential of 0 (representing mV). The two types of inhibitory
neurons projected fast GABAergic synapses with a reversal potential
of -80 (representing mV). The synaptic decay constant τ depended on
the presynaptic neuronal class. Table 1 contains a full description of
the neuronal model parameters. B: The pyramidal cells have a sparse
recurrent connectivity to other pyramidal cells but connect with a high
2.3 Correlations in the degree distribution
Our goal is to determine whether correlations in the in- and
out-degree distribution are beneficial in that they increase
stability and stimulus detection performance relative to
uncorrelated networks. We studied the effect of
correlations in the degree distribution for the excitatory neurons,
whereas interneurons were connected densely but without
correlations in the degree distribution (Packer and Yuste
2011). We generated networks from a truncated bivariate
Gaussian for the joint in- and out-degree distribution, this
allowed the generation of networks with large variance in
the in- and out-degree distribution (Vasquez et al. 2013). We
start from a bivariate Gaussian with a diagonal covariance
matrix given in Eq. (6).
p(x, y) = √
probability to the interneuron populations. In return, both PV and Sst
interneurons connected to all Pyr and PV cells, but not to Sst
interneurons. C: We used the relative connection strength that was found for the
inhibitory populations (Pfeffer et al. 2013). D: The voltage deflection
in response to a single presynaptic action potential when the cells are
held at resting potential. The model is conductance based, hence the
deflection caused by inhibition is relatively low compared to excitation
when the cells are at resting potential
The bivariate Gaussian can be rotated 45 degrees
clockwise or anticlockwise to obtain a distribution with positive
(PCOR) and negative (ACOR) correlations, respectively.
The mean degree (μ) depended on the network size (N ) and
the connection probability (p) as μ = N · p. The long axis
was σy = μ/3 and the short axis σx was set to 0.3 · σy . The
distributions were truncated at 1 (since a zero degree neuron
would not be considered part of the network) and at twice
the mean degree to make the distribution symmetric.
Degree distributions were obtained by sampling for each
neuron i, the in- and out-degree from the corresponding
bivariate Gaussian, diin and diout , respectively. For the
uncorrelated control network (UCOR) the list of diout values was
randomly permuted. For the networks with mixed positive
and anti-correlations (XCOR), diin and diout were sampled
for 50 % of the cells from PCOR, and for 50 % of the
cells from ACOR distributions. The simplest method for
generating a realization of the corresponding network is
the configuration method (Newman 2010). A list with diout
stubs for each neuron is made and concatenated into a
list skout . Likewise, a list with diin stubs is made and
concatenated into a list sin and randomly permuted. If the
number of out-degree stubs in diout is larger than the
number of in-degree stubs in diin, the lists are ordered and stubs
are subtracted starting with the highest out-degrees (one
stub per neuron) and added starting with the lowest
indegrees (one stub per neuron) until the lists are matching
in number of connections (vice versa for more in-degrees
than out-degrees). From these two lists, pairs are picked
from the same position, i.e., the kth stub on the out-list is
matched to the kth stub on the in-list to make the connection
sout to skin.
After the initial connectivity was made, we searched
for multiple connections between the same pair of neurons
and self connections. The overlapping and self connections
were mutually permuted using k-permutation (sampling
without replacement) using the randperm function in
Matlab (The Mathworks, Natick, MA, USA). This procedure
was repeated until no overlapping or self connections were
found. In the rare case that there was no solution possible,
other connections were included in the permutation until
we arrived at a connectivity matrix without double or self
connections. The probability of obtaining multiconnections
were not significantly different between PCOR and ACOR
networks (two-sided t-test on n = 1000 networks,
probabilities are 2.5 ± 0.2 % and 2.5 ± 0.2 %, respectively).
However, PCOR networks, which contain neurons with high
in- and out-degree, have a significantly higher probability
for self-connection than ACOR networks (p < 0.001 for
two-sided t-test on n = 1000 networks, probabilities are 0.15
± 0.04 % and 0.14 ± 0.04 %, respectively). Because
overlapping and self connections were mutually permuted, these
high in- and out-degree neurons in PCOR networks have
a minor bias to preferentially connect to each other due to
there being more self connections. However, since we study
correlations between in- and out-degree, we prefer to
maintain the distribution of the in- and out-degrees compared to,
for example, discarding double and self-connections which
would lead to a more detrimental bias because more
connections will need to be discarded in PCOR networks compared
to ACOR networks.
2.4 Network stability
Cortical neuronal networks need to be stable in the sense
that stochastic fluctuations should not lead to large increases
in the firing rate that could be detected as a stimulation. The
stability of the network is quantified in the model by the rate
at which background activity triggers synchronous
networkwide activity, also called a network burst. To perform burst
detection, we used the spike density method (Martens et al.
2014; van Pelt et al. 2004), where a spike density trace is
calculated by convolving each spike with Gaussian G(t ).
Where τ is the time at which the spike occurred, A is the
amplitude of the Gaussian (set to 1) and σ the width of the
Gaussian (2.5 ms).
The start of a burst is defined as the time at which
the spike density trace crosses a threshold (10 Hz, which
requires about 3 % of the neurons to be active within a 5
ms interval), and the end of the burst is given by the time at
which the spike density drops below this threshold.
2.5 Network sensitivity
We tested the sensitivity of cortical neuronal networks to
external stimulation. The sensitivity of the network is tested
in the model by detecting whether stimulation in a few
selected neurons for a fixed duration evokes a network
response above a fixed threshold (i.e. 10 Hz); the
stimulated neurons were excluded from the burst detection. We
selected the stimulated neurons from 10 neurons with an
out-degree closest to the average out-degree. Depending on
the computer experiment, a number of neurons (np) were
sampled from these 10 neurons. For each stimulation a new
set of np neurons were sampled. A stimulus input (Istim,
Eq. (4)) was applied to the sampled neurons by injection of
Istim = 8 for 25 ms, while the networks were not bursting
spontaneously (that is for very low background noise).
2.6 ROC analysis
To produce the receiver-operating curve (ROC), we need to
determine the true and false positive rate for a set of
detection thresholds. Stimulation was applied every 70 ms. We
used a detection window of 60 ms, where we discarded the
5 ms before the stimulation and the 5 ms at the end of the
stimulus window. This was performed to avoid the leaking
in of the spike density from another stimulus window due
to smoothing. We simulated the networks with and
without stimulation. A false positive was called when the firing
rate exceeded the specified threshold in the unstimulated
condition. A true positive was called when the firing rate
exceeded the threshold in the stimulated condition. At the
start of each stimulus window, all network variables and
random number generator seeds were restored to those
corresponding to the unstimulated trial; for a fair comparison,
the network state and noise at the start of the stimulus trial
was thus identical to the stimulus-free trial.
The ROC curve was then obtained by plotting the
fraction of false positives against the fraction of true positives
for many different thresholds. When there is no effect of
the applied stimulus, the number of true positives equals
the number of false positives, hence the ROC is the
diagonal with an area under the curve (AUC) of 0.5. We tested
this protocol by stimulating 0 neurons (i.e. the network
behaviour should be exactly the same as for a stimulus-free
trial) and found an AUC of exactly 0.5. The deviation of the
ROC curves from the diagonal, or equivalently deviation of
the AUC from 0.5, is a measure for how different the
distributions are and maps for Gaussian distributions on to the
effect size of d’, which is the difference in means of the
distributions divided by their standard deviation (Kingdom and
The number of synaptic connections increases during early
development, and subsequent associative plasticity
supervises the maturation of cortical circuits, decreasing the
number of synaptic connections (Ko et al. 2012; Martens
et al. 2015; Johnson 2001). Synaptic stabilization is
activitydependent and involves the formation of PSD-95 (De Roo
et al. 2008). PSD-95 is associated with spine stability;
weak synapses containing little PSD-95 are in general easily
pruned (Holtmaat et al. 2006; Woods et al. 2011).
The number of synapses peaks before the critical
rewiring period, and subsequently decreases during further
development (Knudsen 2004; Johnson 2001). To mimic the
reduction in synapses we initialized UCOR type networks
with an excitatory connection probability of 10 %, twice that
of the final value of 5 %. The networks were presented with
random input in the form of spontaneous release and
background spiking (see If luc and Ibg , respectively in Eq. (4)
for details). We applied a spike-timing dependent
plasticity (STDP) rule (Song et al. 2009), while the overall level
of network activity was maintained by a network
homeostasis rule (see below). The simulations were then run for 20
s. The amplitude of STDP was increased and homeostatic
plasticity was made faster in order to reduce the length of
the simulation period. The results were comparable to those
that were obtained for simulations that were run for a longer
duration of 50 s. At the end of the simulation the weakest
synapses were removed until a connectivity of exactly 5 %
2.7.1 Spike-timing dependent plasticity For the STDP rule we used a function F( t) that determined the amount of synaptic modification arising from a single pair of pre- and postsynaptic spikes separated by a time t:
F ( t ) =
Where τ+ = τ− = 20 ms, A+ = 1A.0−5 = 0.005 (Song et al.
2009). We used a hard upper bound of synaptic strength
equal to 0.013. We found that for this synaptic strength
neurons fire at rates similar to the target firing rate (Eq. (9)), for
the supplied noise level of 0.1 Hz.
2.7.2 Network homeostasis
Applying the STDP rule (Eq. (8)) has a strong effect on
the postsynaptic firing rate (Song et al. 2009). We therefore
maintained the network mean firing rate with:
= (Rtar − R¯ ) · W
Where W is the connectivity matrix containing the
postsynaptic weights of all neurons in the network. According to
this rule all synaptic weights in the matrix W are adjusted
multiplicatively when the current mean firing rate over the
last 500 ms (R¯ ) diverges from the target mean firing rate
(Rtar = 1.5 Hz); for this process we used a (sped-up)
timescale of τh = 2 s. Experimentally homeostatic
plasticity timescales are generally in the range of hours to days
(Bateup et al. 2013; Turrigiano 2008).
2.8 Statistical analysis
To test for significant differences between ACOR and
PCOR networks we used the 2-sided t-test, implemented as
ttest2 in Matlab (The Mathworks, Natick, MA, USA).
We used two methods to test whether correlations in the
degree distributions arise when we applied the plasticity
rules described above.
For the first method (referred to as the LSR-method)
we evaluated the degree correlation using the least squares
regression on the in- and out-degree of the neurons; we
used the Matlab (The Mathworks, Natick, MA, USA)
function polyfit and we tested whether the coefficient of the
linear fit was significantly different from a horizontal line
For the second method (referred to as the
quadrantmethod) we plotted the in- and out-degree of the neurons
and divided this plot into four quadrants. For the top-right
quadrant both the in- and out-degree of the neurons are
larger than the mean in- and out-degree, respectively. For the
bottom-left quadrant, both the in- and out-degree are smaller
than their mean. The number of neurons in these two
quadrants (Pn) contribute to a positive correlation in the degree
distribution. Similarly, the number of neurons in the top-left
and bottom-right quadrants (An) are counted, which
contribute to an anti-correlation in the degree distribution. We
tested whether ( Pn - 1) was significantly different from zero
using a two-sided t-test.
3.1 Networks with anti-correlated degrees have the
lowest spread in the number of synaptic contacts
Here we examined the in- and out-degree distribution of
four network types with correlated in- and out-degrees
for the neurons: no correlation (UCOR), anti-correlation
(ACOR), positive correlation PCOR or a mix of anti- and
positive correlation (XCOR, Fig. 2A). The marginal
distribution of pre- or postsynaptic connections per neuron
is identical for these different networks (Fig. 2B).
However, the distribution for the sum of in- and out-degrees
shows that ACOR networks have a tight distribution for
the sum of pre- and postsynaptic connections per cell,
whereas PCOR networks show a wide range of values of the
summed degrees, with some cells that make few pre- and
postsynaptic contacts and others that have many synaptic
contacts (Fig. 2C, in Section 4 we relate these differences to
metabolic demands on the cell).
3.2 Networks with anti-correlated degrees have longer
path lengths between pairs of neurons and larger
Having constructed networks with unique correlations in
the degree distribution, we wanted to know whether and
in what ways the structural connectivity of these networks
was different. We used concepts from graph theory that
are described in textbooks (Newman 2010). We studied the
mean shortest path between the excitatory neurons, which
is the shortest path between two nodes, averaged across
all pairs and therefore provides a measure of the
effective connectivity in the network. Mean path length could
be a relevant quantity because it describes how activity
can spread across the network to induce a network burst.
An increase in connection probability decreased the mean
shortest path length (Fig. 3A). By maintaining a constant
connection probability and varying the network size, we
observed that the mean shortest path also decreases with
network size (Fig. 3B). We tested whether the mean shortest
path length was affected by correlations in the degree
distribution and found that for the typical networks used here
(480 excitatory neurons and connection probability 0.05),
ACOR networks had a significantly longer mean
shortest path length, with an increase of 1-2 % compared to
PCOR networks (p < 0.001, significance was tested using
a two-sided t-test, Fig. 3C). These differences are small, but
become larger for more sparsely connected networks.
Intuitively, a network structural core consists of highly
interconnected neurons. To study whether correlations in
the degree distribution affected the network structural core
size, we performed a k-core analysis (Alvarez-Hamelin
postively correlated (PCOR)
mixed correlated (XCOR)
Fig. 2 Construction of networks with a correlation between in- and
out-degree. A: Scatter plots of the in- vs. out-degree for the four
network types. The degree distributions were sampled from a truncated
bivariate Gaussian, with for each network type a different covariance
matrix. For the uncorrelated (UCOR) networks, the covariance matrix
was diagonal, with equal variance of the marginal distributions for the
in- and out-degrees. To generate correlations we start from a diagonal
covariance matrix with unequal variances and rotated it by 45 degrees
anticlockwise to obtain anti-correlated (ACOR) networks and by 45
degrees clockwise to obtain positively correlated (PCOR) networks.
We also constructed networks where half of the in- and out-degree
pairs were picked from an anti-correlated distribution and the other
half from a positively correlated distribution (XCOR). B: The networks
were constructed so that the marginal distributions for the in- and
outdegree were the same for the four network types. C: The distributions
of the sum of in- and out-degree for each neuron shows that ACOR
networks have a tight distribution for the total number of connections
per cell, whereas PCOR networks show a wider range, with some cells
that have few pre- and postsynaptic contacts and others that have many
incoming and outgoing synaptic contacts
et al. 2006). The k-core is the largest subgraph comprised
of neurons with a summed in- and out-degree of at least
k, which is determined by recursively removing neurons
that have a summed in- and out-degree lower than k.
0.05 0.15 0.25
ACOR UCOR XCOR
pcon = 0.01
pcon = 0.02
pcon = 0.05
pcon = 0.11
pcon = 0.25
1000 1500 2000
1000 1500 2000
Fig. 3 Anti-correlated degrees lead to a higher mean shortest path
length between two excitatory neurons and a larger core size. A: The
shortest path length, which is the mean distance between all pairs in
the network, decreases with increasing connection probability. B: The
shortest path length also decreased with increasing network size. This
reduction is most notable for sparsely connected networks (connection
probability 0.01). C: The four network types were compared for
varying network sizes, while the connection probability was fixed to 0.05.
ACOR networks have a significantly increased (1-2 %) mean shortest
At the macroscopic level, when applying k-core
decomposition to the connectivity at the level of anatomical brain
regions, a structural core remains which is characterized
by high metabolic activity that overlaps with the activity
in the human brain during the resting state (i.e. the human
default mode network), suggesting that a structural core
is the basis for shaping brain dynamics (Hagmann et al.
2008). At the microscopic level we found that ACOR
resulted in a significantly larger structural core (Fig. 3D).
Taken together, we found graph theoretical differences
between the different network types. The number of
synaptic connections is more homogeneously distributed
in ACOR compared to PCOR networks, which led to a
larger structural core size. However, the average shortest
path length was increased in ACOR networks compared to
PCOR networks. PCOR networks have neurons with high
in- and out-degree that function as hubs that reduce the
shortest path length. The question is whether these
differences have dynamical consequences in terms of stability and
path length. D: The results of a k-core decomposition are shown for
networks with 480 pyramidal cells and connection probability 0.05.
ACOR led to a larger core of highly connected neurons compared to
PCOR. Networks in panel A and B were of type PCOR. In all panels,
the statistics were averaged across 60 networks for each correlation
type, error bars are 1 standard error of the mean (SEM) and stars
indicate significant differences between ACOR and PCOR networks
according to a two-sided t-test
3.3 Networks with anti-correlated degrees are most
stable against background noise
In vivo recordings in the rat somatosensory cortex show that
cortical neurons fire at a low frequency, ranging from less
than 1 Hz in the superficial layers to a few Hz in the deep
layers (Greenberg et al. 2008; de Kock and Sakmann 2009;
Barth and Poulet 2012). We are primarily interested in the
state of low firing rate, in which each neuron is only active a
small fraction of the time, because this state allows a
stimulation to cause a detectable difference in the network firing
rate. We therefore quantified the stability of each of the four
network types (see Fig. 2A).
In the absence of noise, no spiking activity was detected
in any of the networks. For a low level of background
noise, the networks remained stable and fired irregularly,
while increased noise results in unstable, continuous
network bursting (Fig. 4A). The excitatory synaptic strengths
were such that low frequency spiking input could evoke a
ACOR UCOR XCOR
Fig. 4 Anti-correlation in the degree distribution increases stability
against noise. A: Averaged firing rates of excitatory (black lines), PV
(red traces) and Sst (cyan traces) neurons in response to background
noise events at frequencies between 0.08 Hz and 0.1 Hz per
neuron. The low noise input evokes background spiking activity without
bursting (lower traces), whereas the high noise input rates trigger
periodic synchronized bursting activity in the network (upper traces). B:
The PCOR networks (blue) produce network burst activity for a lower
noise rate than the ACOR networks (green). UCOR (red) and XCOR
(cyan) correlated networks showed intermediate levels of stability. C:
As expected for a lower burst rate, the ACOR networks also have a
lower firing rate compared to the PCOR network for an equal amount
of random input spikes. D: The ACOR networks have on average a
longer shortest path length (also see Fig. 3). The shortest path length
We found that ACOR networks showed fewer burst
responses for the same level of noise compared to PCOR
networks (Fig. 4B). This also related to a lower firing
rate in ACOR networks (Fig. 4C). Thus PCOR networks
were less stable than ACOR networks. Stability for UCOR
and XCOR networks was in-between ACOR and PCOR
We wanted to know whether these differences in stability
could be explained by the different graph theoretical
properties found above. For a given path length the properties
such as firing rate were broadly distributed, but there was no
statistically significant trend observable between the mean
firing rate of the network and the mean shortest path length
of the associated network (Pearson correlation values were
not significantly different from zero, Fig. 4D). We also did
not find a correlation between k-core size and firing rate
(Fig. 4E). Thus, for the same number of connections in a
network, the mean pair distance and structural core size did
not influence the network stability.
for a given network was not correlated with the firing rate in that
network, as the Pearson correlation value was not significantly different
from zero (p > 0.05). Each dot represents the mean firing rate of the
excitatory network; background noise rates varied between 0.075 and
0.11 Hz. E: No correlation was found between the mean firing rate of
a network and the largest k-core in that network. Statistics and color
convention were as in panel D. F: Dots represent the firing rate of a
single neuron plotted against its in- and out-degree for one network in
the bursting state (input rate 0.11 Hz per neuron). In panels B to E the
statistics are averaged across 120 networks, error bars are 1 SEM and
stars indicate significant differences between ACOR and PCOR
networks according to a two-sided t-test. Each network consisted of 480
pyramidal, 60 PV and 60 Sst neurons
We then studied the relation between in-degree and
firing rate for individual neurons. A high in-degree led to a
high firing rate (Fig. 4F). Given the correlation structure
in the network, this means that high out-degree neurons in
a PCOR network have high firing rates, whereas the high
out-degree neurons in the ACOR network have low firing
rates. Hence, the anti-correlated degrees directly result in a
reduced synaptic output to the network in reponse to noise,
providing an intuitive understanding of the mechanism by
which the additional stability is generated.
3.4 Positively correlated networks are most sensitive
to stimulation in the absence of spontaneous bursts
Whisker stimulation results in time-locked responses that
can be measured in the rat somatosensory cortex (Stern
et al. 2002). These neuronal responses are hypothesized to
play an important role in performing and learning sensory
tasks (Huber et al. 2012; Petersen and Crochet 2013). Rats
are better at detecting external stimulation when multiple
neurons are activated compared to when a single neuron
is stimulated (Romo et al. 1998; Houweling and Brecht
2008). Here we studied the network responses upon
stimulation of 6 neurons while the networks were not bursting
spontaneously (0.07 Hz background noise). For weak
excitatory coupling strength, only a few neurons in the network
responded to stimulation in addition to the directly
stimulated cells (Fig. 5A). Neuronal recruitment increased with
excitatory coupling strength (Fig. 5B), where PCOR
networks had a higher peak firing rate than ACOR networks
(Fig. 5C). Network-wide burst responses, detected when
the firing rate crossed a predefined threshold, were
realized for weaker coupling strengths in the case of PCOR
networks (Fig. 5D), and these networks were fastest to
reach their peak activity (Fig. 5E). Taken together, these
data show that networks with positive correlations in the
degree distribution, in the absence of spontaneous
network bursting, are most sensitive to stimulation of a few
3.5 Stimulus detection is enhanced in anti-correlated
networks for higher background noise
We showed that in the absence of spontaneous bursting the
PCOR networks were most sensitive to stimulation, whereas
ACOR networks were found to be more stable against noise.
Here we investigate the sensitivity to external stimulation
for varying degrees of background noise; this provides
a direct quantification of detection performance. For our
experiments we first supply the networks with background
noise in the form of random spiking in each of the
excitatory neurons of the network (see Methods, Section 2.5).
We then applied stimulation in 1 to 6 neurons and observed
a moderate to clearly noticable increase in the firing rate.
We used these experiments to study whether stimulation
had a detectable effect on the network activity (Fig. 6B).
For low detection thresholds, detection of both true and
false positive network events is high. For intermediate
detection thresholds we observed that ACOR had lower
Fig. 5 Networks with positively correlated degrees are more sensitive
to a small perturbation for the low noise condition, for which there
are no spontaneous bursts. A: Rastergrams wherein each dot
represents a spike. The spikes of excitatory pyramidal cells are in black,
PV interneuron spikes are in red and Sst interneurons spikes are in
cyan. Depending on the strength of recurrent excitation (wee), external
stimulation in 6 neurons of an UCOR network leads to either a weak
response of varying duration that did not recruit inhibitory neurons,
or a strong, sharp response that recruited inhibitory neuron activity
that curtailed the burst. Each neuron received input from background
spikes at a rate of 0.07 Hz. Interneurons were recruited only when a
network burst occurred. The rastergrams also show the spikes of the
stimulated neurons, but these were not included for the burst
detection and post-stimulus time histograms to avoid stimulation artifacts.
B: The smoothed post-stimulus time histogram of excitatory neurons
for 25 different values of the recurrent excitatory strength, equally
spaced between 0.015 and 0.04. For smoothing see Eq. (6). C: Mean
peak firing rate in the smoothed post-stimulus time histogram
plotted against the recurrent excitatory strength. PCOR networks (blue)
showed higher peak firing rates compared to the ACOR networks
(green) for equal recurrent strength. D: Bursts were detected when the
recurrent strength exceeded 0.015, and for recurrent strength 0.023 and
higher the network consistently showed a burst response after each
stimulation. E: The peak latency, which is the time between the onset
of stimulation and the peak of the burst, decreased for stronger
recurrent strength. For recurrent strength below 0.02, variability in the peak
latency is high due to the low number of detected bursts. For each
network type the statistics are averaged across 60 networks with one
stimulation per network, error bars are 1 SEM and stars indicate
significant differences between ACOR and PCOR networks according to
a two-sided t-test
Wee = 0.020
Wee = 0.025
Wee = 0.030
* * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * *
Fig. 6 Stimulus detectability, as evaluated by ROC analysis, is higher
for networks with anti-correlations in the degree distributions than for
networks with positive degree correlation. A: Spike density for the
excitatory neurons in a stimulated ACOR network. Stimulation was
applied every 70 ms to up to 6 neurons in the network; the stimulus
duration is indicated by the block pattern (top). At the start of each
stimulation, which is indicated by dotted line, the network variables
were reset to their baseline value (no stimulation, bottom). Under the
same noise conditions as the baseline trace, the number of stimulated
neurons increased from bottom to top with a step of one. In the top
trace stimulation was applied to 6 neurons, which often initiated a
detectable response including many neurons. Background noise was
0.098 Hz per neuron. B: Stimulation of 3 (top) and 6 (bottom)
neurons was applied to ACOR (left) and PCOR networks (right), while the
background noise was identical. Stimulation in the ACOR networks
was better detectable than stimulation in the PCOR networks because
rates of false positive events compared to PCOR networks
(Fig. 6B). From the true and false positive rates we
constructed ROC curves (Fig. 6C). From these ROC curves
we extracted the area under the curve (AUC) as a
measure of stimulus detection in noisy conditions, and show
that for stimulation of a few (1 - 6) neurons stimulus
detection in ACOR networks was enhanced compared to
PCOR networks (Fig. 6D). Nanostimulation (single
neuron) had a small but significant effect on stimulus
detection. We then studied the stimulus detection under varying
background noise levels (Fig. 6E). We found that ACOR
networks were able to detect stimuli for stronger
background noise, which can be attributed to the increased
stability against background noise as shown before
the PCOR networks were less stable to noise. C: The corresponding
ROC curves for 3 (left) and 6 (right) stimulated neurons quantify the
stimulus detectability, such that curves further away from the diagonal
relate to higher detection rates. Because neuronal networks respond
non-linearly to noise, which occasionally initiated bursting in response
to spontaneous activity, additional stimulation in that case did not
increase the response amplitude further, therefore the bottom-left part
of the ROC curves remain along the diagonal. D: The
area-undercurve (AUC) of the ROC is a quantification of stimulus detectability.
Detectability increases with the number of stimulated neurons and is
highest for ACOR networks. E: We stimulated four neurons under
different background noise levels. ACOR networks showed higher
detectability in the high noise conditions compared to PCOR networks.
For each correlation type the statistics are averaged across 120
networks, error bars are 1 SEM and stars indicate significant differences
between ACOR and PCOR networks according to a two-sided t-test
3.6 Effect size depended on connection probability
and network size
We showed that ACOR networks outperform PCOR
networks in detection of external stimulation under high levels
of noise. Next we wondered what influence connection
probability and network size had on stimulus detection.
For these simulations we maintained constant synaptic
strengths. We observed that for a high connection
probability (10 %) network bursting occurred at noise levels
around 0.07 Hz, whereas networks with a low
connection probability (1 %) did not burst until noise levels
reached rates around 0.14 Hz (Supplementary Figure S2).
For connection probability 5 % and 10 % our previous
results that ACOR outperforms PCOR were confirmed.
When connection probability was lowered to 3 % and below,
the stimulation of a few neurons was difficult to detect
and the advantage of ACOR to outperform PCOR
disappeared. When connection probability was further reduced to
1 %, the ability to detect external stimulation was almost
completely abolished. We attribute these findings to the
higher mean out-degree in the densely connected networks
compared to sparsely connected networks, thereby
allowing external stimulation of a few neurons to recruit a larger
synaptic drive to the rest of the network. For our stimulation
protocol involving 600 neurons, stimulation in four neurons
and our setting of synaptic strength, the minimal connection
probability to detect external stimulation was ∼5 %.
Furthermore, we varied the network size and observed
that for larger networks the effect size by which ACOR
networks outperformed PCOR networks in terms of
sensitivity to nanostimulation was increased (Supplementary
Figure S3). Additionally, we studied the influence of the
effective time step used for the numerical integration and
found that the results were robust when we used a smaller
time step (dt=0.02 ms). Furthermore, using an identical time
step for u as the other variables (see Methods for details),
also showed consistent results (Supplementary Figure S4).
3.7 Associative plasticity forms anti-correlations
in the degree distribution
How could networks with anti-correlations in the degree
distribution emerge? Several different models exist for the
establishment of synaptic connections, but these do not
take into account correlations in the degree distribution
(Yoshihara et al. 2009; Garc´ıa-Lo´pez et al. 2010). We
studied whether correlations in the degree distribution could
emerge from associative plasticity.
Early in development the number of synaptic connections
is high, and subsequent associative plasticity reorganizes the
cortical circuits, decreasing the number of synaptic
connections (see Section 2.7 for details). We constructed networks
with 10 % connection probability to represent the more
densely connected networks early in development. These
networks were of the UCOR type to mimic the random
organization. Uncorrelated spontaneous spiking was
supplied to the network as synaptic inputs with an amplitude
of Ibg = 15, duration of 0.1 ms and at a rate of 0.1 Hz
for each neuron. The rate of 0.1 Hz was chosen because
the ACOR networks that were generated from a bivariate
Gaussian distribution then fired at ∼1.5 Hz, for which the
occasional synchronized burst emerged. When in these
networks the synaptic strength was modified by STDP and the
set point rate for the homeostatic process was set to 1.5
Hz, spiking activity still propagated throughout the network
accompanied by the occasional synchronized network burst.
We observed that associative plasticity reorganized the
synaptic weight distribution towards a bimodal
distribution (Fig. 7A). Synapses were pruned (removed), starting
with the weakest synapses, until a connectivity of 5 % was
obtained (Fig. 7A). We summed the synaptic inputs to each
of the neurons and found that the distribution was
comparable to explicitly constructed ACOR networks from a
correlated bivariate Gaussian distribution (Fig. 7B).
By plotting the in- and out-degree of the synaptic
connections that remained after associative plasticity, we observed
anti-correlation in the degree distribution (Fig. 7C). We
calculated the correlation (Section 2.8) for 60 networks
with randomly initialized dynamics and connectivity, and
found that anti-correlation in the degree distribution was
consistently formed (Fig. 7D).
Consistent with the bimodal STDP rule, we found that
applying a weight-dependent STDP rule, as described by
Morrison et al. (2007), to UCOR networks with 10 %
connectivity, resulted in networks for which the 5 % strongest
synapses have an ACOR distribution (Supplementary
In summary, for these parameter settings, dense and
uncorrelated networks were consistently reorganized into
more sparsely connected networks with anti-correlation in
the degree distribution by synaptic pruning.
The activity produced by cortical microcircuits in sensory
areas provides the opportunity to detect external stimuli,
provided that the circuits are stable against noise
generated by spontaneous firing. Such simultaneous sensitivity
and stability is difficult to achieve (Vasquez et al. 2013).
Previously, in a simple recurrent network of stochastic
binary neurons, it was numerically shown that stability was
increased for ACOR relative to PCOR networks.
Nevertheless, these ACOR networks consisting of binary neurons had
the same level of sensitivity compared to PCOR (Vasquez
et al. 2013).
Here we studied the effects of correlation between
inand out-degree on stimulus detection in recurrent spiking
neuronal networks. We found that ACOR networks had
increased network stability, whereas in our simulations of
the low noise state, without the spontaneous bursting
activity, sensitivity was highest for PCOR networks. The rat
somatosensory cortex shows spontaneous spiking at
firing rates of up to a few Hz (Greenberg et al. 2008; de
Kock and Sakmann 2009; Barth and Poulet 2012). When
we performed stimulation in the more realistic setting of
spontaneous background spiking, representative of these
experimentally observed network states, we found that
0 .004 .008 .012 .016
-1 -.8 -.6 -.4
Fig. 7 Associative plasticity forms networks with anti-correlation in
the degree distribution. A: Top: an UCOR network with a connection
probability of 10 % with an upper bound on synaptic strength of 0.013
was run for 20 seconds with spike-timing dependent plasticity. The
amplitude of STDP was increased and timescale of homeostatic
plasticity was decreased compared to their values in the literature in order
to reduce the duration of the simulation. At the end of the simulation
period the synaptic distribution was bimodal. The weak synapses (red)
were pruned and removed from the distribution until a connectivity
of exactly 5 % was obtained. The synaptic strength of the remaining
synapses (green bars) was comparable to an ACOR network
explicitly constructed from a bivariate Gaussian distribution (bottom). B:
The summed excitatory synaptic strength for the STDP-generated
network (top) and the explicitly constructed ACOR network (bottom).
detection performance was highest for the ACOR networks.
High noise levels bring the recurrent networks to a
pathological bursting regime, with high frequency spontaneous
bursting which results in a high false positive rate.
Anticorrelations in the degree distribution provide stability to
the network, and as a consequence a lower false positive
rate. At the same time, these ACOR networks remain
sensitive to external stimulation, thus simultaneously improving
stability and stimulus detection compared to PCOR
networks. Our hypothesis is that stimulation detection
corresponds to a nonlinear increase in neural activity in sensory
areas. In our model, we use bursts as a proxy for such
an event. As our model networks represent only a small
part of the entire barrel cortex network, the bursts
correspond to a more modest increase in the barrel cortex activity.
Specifically, they should be experimentally observable as
a modestly increased rate coupled to a strongly increased
level of synchronization in sparsely active networks
(Houweling and Brecht 2008; Barth and Poulet 2012; Wolfe
et al. 2010). We further speculate that downstream neurons
in areas that plan actions (i.e. the initiation of licking) have
become more sensitive to these synchronously active
neurons, for instance, through a Hebbian mechanism during
C: In- and out-degrees of the STDP-generated network (top) and the
explicitly constructed ACOR network (bottom). Black line is fitted
using the LSR-method. D: 60 STDP-generated networks (top) and
60 explicitly constructed ACOR networks (bottom) were tested for
correlations in the degree distribution using the LSR-method and the
quadrant-method (see Materials and Methods). All STDP-generated
networks showed anti-correlations in the degree distribution that were
comparable to the generated ACOR networks. The LSR-method shows
that the angle of the slope is similar for the explicitly generated and
STDP-generated networks, whereas the lower values found for the
STDP-generated networks using the quadrant-method are due to the
increased variance across independent realizations
By dissecting the firing rate based on in-degree, we found
that in the ACOR networks the neurons with high
outdegrees had on average a lower firing rate; this effectively
reduces the excitatory input to the network during
spontaneous activity. Concurrently, the high in-degree neurons
collect inputs from many neurons in the network, and have
a higher than average firing rate, but project their output to
a relatively small portion of the neurons so as to not
destabilize the network. As a consequence, stimulating the average
out-degree neurons in the ACOR networks results in a burst
response even though the networks remained more stable
against the noise-induced bursts compared to the PCOR
networks. These findings provide an intuitive
understanding of the mechanism by which the ACOR networks were
more stable to noise than PCOR networks, which improved
the stimulus detection performance in ACOR networks. It
was recently shown in vivo that network firing patterns are
largely dictated by basic circuit variables (Okun et al. 2015;
Harris and Mrsic-Flogel 2013). We suggest correlations in
the degree distribution contribute as a basic network
property to the maintenance of stable spiking activity in neuronal
We investigated the stability and sensitivity in spiking
neurons, where, due to the presence of inhibitory neurons,
the network can be in a regime with no or a few bursts
during spontaneous activity, and brief bursts terminated by
recruited inhibition can be induced by electrical stimulation.
Key relevant features of the neuron dynamics are integration
of multiple synaptic inputs into an output spike, a refractory
period as well as an effective inhibitory feedback. These
features are also present in other spiking neuronal
models, i.e. LIF neurons and multicompartmental models with
Hodgkin-Huxley currents, and when properly
parameterized we expect similar results. The results will of course
be different, when, in the latter, the model neurons can
switch between spiking and (single neuron) bursting states,
as this will make induction of a network burst easier and
less dependent on network structure, and when the
integration properties are different, i.e. higher sensitivity for inputs
with a certain values for inter-input intervals, for
example for the resonate-and-fire neuron proposed by Izhikevich
For the neuronal networks in this study we found that
sensitivity to stimulation of a few neurons requires a
minimal connection probability. The effect size by which
ACOR networks outperformed PCOR networks increased
with connection probability and network size. Although
many synaptic connectivity features are ubiquitous among
cortical system, experimentally observed connectivities
differ between species and sensory modality (for review see
(Chapeton et al. 2012)). It is interesting whether the ability
to detect nanostimulation is, for example, different between
rats and mice, and whether visual, auditory and
somatosensory regions show a difference in detection performance. We
predict that densely connected regions show better
performance compared to sparsely connected regions. Inhibitory
neurons are less abundant in cortical circuits than excitatory
neurons, but are more densely connected to the
excitatory population (Pfeffer et al. 2013). Nanostimulation of
inhibitory neurons might therefore have an increased
detection performance compared to nanostimulation of excitatory
Synaptic communication places a disproportionally high
demand on energy consumption (Mink et al. 1981; Harris
et al. 2012). Pre- and postsynaptic parts of the neuron
consume a comparable amount of energy (Attwell and Laughlin
2001; Harris et al. 2012). Cells that are stressed by excessive
ATP consumption can produce damaging levels of reactive
oxygen and nitrogen species (ROS/RNS) in the cell, leading
to protein dysfunction and potential cell death (for review
see (Wang and Michaelis 2010)). Cell death by oxidative
stress is linked to neurodegenerative diseases (Wang and
Michaelis 2010). When networks have anti-correlations in
the degree distribution, the energy demand is more
homogeneously distributed over the neurons (Fig. 8A). Thus,
by making cellular demands on energy consumption more
homogeneous, the anti-correlation in the degree distribution
provides another level of robustness to brain networks.
For standard growth models the number of pre- and
postsynaptic connections for each neuron is set to be
independent, and there is no correlation in the in- and
outdegrees. These networks will be of type UCOR. Previously,
STDP was shown to lead to non-random structures (Kato
and Ikeguchi 2010; Masuda and Kori 2007) and
disassortivity in network connectivity (Kato et al. 2009). These
authors also studied the distribution of in-degree, out-degree
and sum of in- and out-degree. They observed a general
reduction in the out-degree, particularly for neurons with
high in-degrees, but did not quantify the correlations in the
degree distribution (Kato et al. 2009). We demonstrated that
STDP can reorganize UCOR networks into networks with
anti-correlation in the degree distribution.
STDP rules need to lead to competition and prevent
divergent weights. There are a number of strategies to
achieve that, such as additive STDP, which leads to strong
competition, but needs a hard weight cut off. The weight
cut off can be avoided by using multiplicative STDP, which
generally leads to weaker competition. A recent
alternative, the so called log-STDP rule (Gilson and Fukai 2011)
has competition but does not need a weight cut off because
it leads to a long-tailed weight distribution. We did not
determine explicitly whether log-STDP would lead to
anticorrelated networks, but as this is a consequence of the
competitive nature of the STDP rule we used, we expect that
logSTDP works in the same way, and in addition leads to more
realistic log-normal distributions of connection strength.
What could trigger this structural organization in a
developing brain? One possibility is that the network
restructuring towards anti-correlations occurs after the
transition from immature to mature STDP (Itami and Kimura
2012). For the rat somatosensory cortex, this
developmental switch coincides with the critical learning period and
a period of rapid reorganization of the cortical circuitry
(Martens et al. 2015).
Alternatively, in the mature brain plasticity rules could
form anti-correlated degrees to obtain cortical circuits
sensitive to (nano-)perturbation. After a training period rats
respond significantly more to stimulation of a single neuron
in the somatosensory cortex than to catch trials, consistent
with a sparse cortical code for sensation (Houweling and
Brecht 2008; Doron et al. 2014). Thalamic activity that is
triggered by whisker stimulation could project preferentially
to neurons with high out-degrees. Here, an anti-correlated
network configuration could provide simultaneous stability
to noise, and sensitivity to (nano)stimulation.
In the networks studied here, synaptic strength was held
constant. Consequently, the variability in in-degree results
in variable firing rates. For our plasticity experiments, we
XCOR 20 40 60 80 energie consumption (a.u.)
single cell RT-qPCR
Fig. 8 Metabolic consequences of, and a method to experimentally
confirm, anti-correlated degrees in the cortex. A: The pre- and
postsynaptic parts of the neuron have a comparable energy consumption
((Harris et al. 2012; Attwell and Laughlin 2001), estimates in the
diagram are from (Harris et al. 2012)). By assigning equal levels of energy
consumption to the pre- and postsynaptic part of each synapse, we find
that energy demands are more homogeneously distributed over cells
in ACOR networks compared to PCOR networks. B: To estimate the
in- and out-degree of single neurons, we propose to use single cell
RTqPCR and compare the relative expression of pre- and postsynaptic
markers. This can for example be performed for RNA encoding
presynaptic proteins Neurolexin (NRXN), Vesicular glutamate transporter
1 (vGlut1) and Synaptotagmin-1 (SYT1) (Sudhof 2008; Beaudoin
et al. 2012; Tang et al. 2006); and postsynaptic proteins Neuroligin
(NRGN), postsynaptic density-95 (PSD-95) and Glutamate receptor 2
(GluR2) (Sudhof 2008; Beaudoin et al. 2012; Dingledine et al. 1999)
applied homeostatic scaling such that the network scales
towards a specific target firing rate. However,
homeostatic scaling could also be applied to individual neurons
(Turrigiano 2008). Firing rates of individual neurons
converging to a target firing rate could lead to variability in
the synaptic strength, thereby reducing the stability of the
ACOR network and abolishing the competitive advantage of
ACOR compared to PCOR networks.
The recurrent networks were organized without any
laminar structure. However, the cortex is organized as a layered
structure, generally thought to be comprised of functional
cortical columns (Mountcastle 1998; Douglas and Martin
2004), which can improve computational efficiency beyond
the capabilities of recurrent networks without such spatial
organization (Treves 2003; Raizada and Grossberg 2003;
Haeusler and Maass 2007). As we showed here,
correlations in the degree distribution can also provide additional
capabilities for stimulus detection. Neurons in cortical
networks with an anti-correlation in the degree distribution can
perform unique roles in the network. The neurons with low
in-degree and high out-degree could amplify a signal by
projecting to many neurons within a layer or across layers. The
neurons with high in-degree and low out-degree could
provide improved detection of a network burst by integrating
many inputs, and send the detection signal to specific target
Due to experimental limitations, correlations in the
degree distribution have not been directly quantified
experimentally (Vasquez et al. 2013). Classical tracing
techniques are not appropriate for single neuron studies because
they involve connections to or from multiple nearby
neurons (Lanciego and Wouterlood 2011). Electron microscopy
(EM) based reconstruction of cortical circuitry could
provide the complete connectivity structure of a local network.
Analyses in a recent review paper (Helmstaedter 2013)
show that it could be experimentally feasible to image an
appropriately sized block of cortical tissue. However, the
main bottleneck is analysis: the detection of synapses and
properly identifying the pre- and postsynaptic neuron. The
combination of new technologies, such as crowdsourcing
(Arganda-Carreras et al. 2015), interactive machine
learning (Sommer et al. 2011) and molecular biology (Hirokawa
2011) will make the EM more feasible within a decade.
Alternatively, viral based techniques allow crossing of
exactly one single synaptic connection which could help
to visualize neurons (Wickersham et al. 2007; Osakada
et al. 2015). However, to obtain the pre- and
postsynaptic connections, single cells should be infected with both
anterograde and retrograde crossing viruses, making this
a challenging approach. Another method is to
simultaneously record from multiple cells and assess connections
by inducing action potentials in one neuron at a time and
recording the postsynaptic responses in other nearby cells
(Song et al. 2005; Perin et al. 2011). Such recordings
can be used to estimate the degree distribution indirectly
by subsampling. Alternatively, correlations in the degree
distribution can be estimated by studying motifs, for
example from triplets of neurons (Vasquez et al. 2013). Taken
together, we feel that these techniques do not provide a
feasible strategy for experimentally confirming our hypothesis
on connectivity. We have therefore formulated an alternative
The diversity of interneuron subtypes, generally defined
by particular molecular markers such as parvalbumin
and somatostatin, have been elegantly interrogated by
simultaneous use of molecular, anatomical and
electrophysiological techniques on single neurons (Tricoire et al.
2011; Toledo-Rodriguez and Markram 2014). For the
excitatory cells here we propose a similar approach: by
patch clamping single neurons (1) the electrophysiological
profile can be tested, (2) the cell can be colored by dye or
virus injection such that the anatomical structure can be
reconstructed and (3) by single-cell Reverse
Transcriptasequantative Polymerase Chain Reaction (RT-qPCR)
(Freeman et al. 1999), or RNA sequencing (RNA seq)
(Zeisel et al. 2015), the mRNA content of the cell can be
quantified. The mRNA quantity is an indirect measure of
protein expression in the cell. By quantifying the mRNA
that code for proteins that are typically found in the
presynaptic terminal (such as Neurolexin, Vesicular glutamate
transporter 1 and Synaptotagmin-1 Sudhof 2008; Beaudoin
et al. 2012; Tang et al. 2006) and proteins that are
typically found in the postsynaptic spines (such as Neuroligin,
PSD-95, and GluR2 Sudhof 2008; Beaudoin et al. 2012;
Dingledine et al. 1999), the in- and out-degree of single
neurons can be estimated (Fig. 8B). Thus, by
combining the molecular, anatomical and electrophysiological
blueprint of the cell’s degree distribution, a (layer-specific)
subclassification could be made for single excitatory
In this study, we showed that correlations in the degree
distribution can add computational capabilities for
neuronal networks. While intuitively networks that have
neurons with high in- and out-degree seem ideal for
stimulus detection, we showed that when taking network
stability into consideration the detectability was enhanced
for networks with anti-correlated degrees. We propose
experimental methods to investigate the correlation of
in- and out-degree in individual neurons. Furthermore,
we have shown how a simple plasticity rule can
organize cortical networks to obtain anti-correlations in the
degree distribution. Our results suggest that anti-correlation
in the degree distribution could be an important
strategy to increase stimulus detectability in recurrent cortical
Acknowledgments This research was funded by the Netherlands
Organisation for Scientific Research (NWO, grant numbers 62001113
Compliance with Ethical Standards
The authors declare that they have no conflict
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted
use, distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
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