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Noise influence on spike activation in a Hindmarsh–Rose small-world neural network

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Journal of Physics A: Mathematical and Theoretical
J. Phys. A: Math. Theor. 49 (2016) 285601 (16pp)

doi:10.1088/1751-8113/49/28/285601

Noise inﬂuence on spike activation in a
Hindmarsh–Rose small-world neuralnetwork
Sun Zhe1,2 and Ruggero Micheletto1,3
1

Yokohama City University, 22-2 Seto, Kanazawa Ward, Yokohama, Japan
Riken Brain Science Institute, 2-1 Hirosawa, Wako-shi, Saitama, Japan
3
Harvard University, 64 Sidney Street, Suite 170, Cambridge, MA 02139, USA
2E-mail: n145301c@yokohama-cu.ac.jp(S Zhe)
Received 23 October 2015, revised 12 April 2016
Accepted for publication 13 April 2016
Published 31 May 2016
Abstract

We studied the role of noise in neural networks, especially focusing on its
relation to the propagation of spike activity in a small sized system. We set up asource of information using a single neuron that is constantly spiking. Thiselement called initiator xo feeds spikes to the rest of the network that is initiallyquiescent and subsequently reacts with vigorous spiking after a transitional
period of time. We found that noise quickly suppresses the initiator’s inﬂuenceand favors spontaneous spike activity and, using a decibel representation ofnoise intensity, we established a linear relationship between noise amplitude andthe interval from the initiator’s ﬁrst spike and the rest of the network activation.
We studied the same process with networks of different sizes (number ofneurons) and found that the initiator xo has a measurable inﬂuence on small
networks, but as the network grows in size, spontaneous spiking emerges disrupting its effects on networks of more than about N = 100 neurons. Thissuggests that the mechanism of internal noise generation allows informationtransmission within a small neural neighborhood, but decays for bigger network
domains. We also analyzed the Fourier spectrum of the whole network membrane potential and veriﬁed that noise provokes the reduction of main θ and α
peaks before transitioning into chaotic spiking. However, network size does not
reproduce a similar phenomena; instead we recorded a reduction in peaks’
amplitude, a better sharpness and deﬁnition of Fourier peaks, but not the evident
degeneration to chaos observed with increasing external noise. This work aims
to contribute to the understanding of the fundamental mechanisms of propagation of spontaneous spiking in neural networks and gives a quantitative
assessment of how noise can be used to control and modulate this phenomenon
in Hindmarsh−Rose (H−R) neural networks.
1751-8113/16/285601+16$33.00 © 2016 IOP Publishing Ltd Printed in the UK

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J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Keywords: neural networks, noise, stochastic processes, small-world networks, discrete Fourier analysis(Some ﬁgures may appear in colour only in the online journal)
1. Introduction
The control and ﬂow of information in the brain are fundamental to the understanding of
high-level processes like attention, visual perception and other complex functions [1, 2].
Despite enormous system complexity, it has been found that information can be sourced by asingle neuron even in major sensory pathways [3]. These distinct units can trigger a chain of
events that contribute to complex perception episodes or high-level motor responses. This is
known to happen for example with feature detector neurons [4] in lower animals, and similar
phenomena are observed in mammals’ neurons which hold a higher hierarchical position in
the visual system—those that respond to complex patterns and activate elaborate responses [5].
Despite the great volume of experimental facts that point toward noise as a contributing
factor in signal transmission in brains [6–8] or generally in non-linear networks [8], we do nothave yet a universally accepted theoretical framework to quantitatively evaluate the effects ofnoise on neurons’ operations and the higher level network functions.
Here we want to contribute to this problem with a study in which we emulate a triggering
event in the brain through a single active neuron which we call initiator xo; in controlled noise
conditions, we study the ﬂow of spike activity along the network to evaluate the role of noise
amplitude on the signal propagation. In the ﬁrst part of this study, we elucidated in aquantitative manner the role of noise and we found for the ﬁrst time a linear relationshipbetween the amplitude expressed in decibels and the delay interval between the ﬁrst spike ofinitiator xo and network activation. Moreover, studying the dependence with network size, we
found that as the network grows in dimension, the spiking activity starts earlier until the roleof the initiator is completely suppressed beyond about N = 100 neurons. In the second part of
the research, we characterize the network especially focusing on its spectral characteristics,
and most particularly on its frequency dependence with noise and size. It is found that highlevels of noise introduce chaos in the network and that, again, the network size seems toreplace the function of external noise for simulations with a constant noise level and variablenetwork sizes.

2. Models and methods
We used a Newman−Watts small-world network to simulate the neural network. A realisticneural network is neither regular nor completely random [9] and this network has both
deterministic and random properties [10]. The simulation of relatively small artiﬁcial networks can be a good metaphor for real brains, for example, in 2003 Izhikevich demonstrated
that a network of simulated spiking neurons exhibited collective waves and frequencies [11]in a range similar to the human brain.
To simulate the neurons we used the Hindmarsh−Rose (H−R) model [12, 13], changing
the intensity of noise and the network size to elucidate the role of spiking frequencies infunction of various parameters. Even though the precise arrangement of neural connections inreal brains is not known, small-world network models are widely used to simulate known2

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 1. Simulation of a H−R model for a single neuron not affected by noise orconnections. This shows the dynamical behavior of the membrane potential. The
variable x(t) represents the difference of voltage between extracellular and intracellularpotentials. Spiking variable y(t) instead describes the rate of change of the fast ion
channels (sodium ion, potassium ion, etc) and z(t) has similar meaning for the slow ionchannels (calcium).

statistical properties of the brain’s neural connections, reproducing a high clustering coefﬁcient and a low shortest path [14–17]. The network’s random connection probability of thesmall-world structure has been kept to p = 0.4 as a reasonable value with enough random
connections to induce a fast diffusion of spikes, but still small enough to represent real
biological systems. This network structure and p values are considered reasonable innumerous literature studies, for example Bassett and Bullmore’s work [15] where regions of
the human brain are found to have clustering coefﬁcients as low as 0.14 equivalent top > 0.4, or also Zheng and Lu [18] and Ozer et al’s works [19] which ﬁnd values of
0.1 < p < 0.6 in small-world neural networks. To measure the global behavior of the network we monitored the integral of the total membrane potential. This is considered to beanalogous to the electroencephalogram in a real brain. By means of Fourier analysis, wefound that this collective signal has distinct frequency peaks which are compatible withvalues observed in biological systems. This fact is not trivial since neuronal activities of acomplex network are very difﬁcult to predict analytically.
The H−R [20] system is characterized by three independent variables that represent themembrane potential and the two ion channel currents. A single neuron model is described by
the following differential equation:⎧ x˙ = y - ax 3 + bx 2 - z + I⎪⎨ y˙ = c - dx 2 - y⎪⎩ z˙ = r [s (x - c) - z ] .

(1)

In this equation, x is time dependent and represents the membrane potential, whereas yand z are often called the spiking and bursting variables, respectively [21]. These variables are
all expressed in arbitrary units and do not have a direct translation in biologically-realistic
parameters [20].
3J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 2. A sketch of the network structure. Each neuron is represented by a circle andsimulated by the H−R model. The lines represent neural connections to neighbors’
neurons. The random connections that cross the circle are those due to the p parameterof the small-world structure. The grayed neuron indicated by the symbol xo is a special
initiator that has a constant external current input Io and it is used to stimulate thewhole network and to study the diffusion of spikes in the system in the presence ofnoise. In our simulations the connections are changed randomly at every run, however
the small-world structural parameters are maintained constant. The cross-connections
parameter p is kept at p = 0.4 (for image clarity in this sketch p is lower). In someexperiments the number of neurons is also varied.

Figure 1 shows a plot of the three variable phase cycle that induces the spike. The choiceof the eight parameters in the model results in the kind of neuron simulated. Here we ﬁxed
them as a = 3, b = 3, c = 1, d = 5, s = 4, r = 0.00, c = -1.6 determined in order to produce
neuronal-bursting dynamics comparable to realistic neurons of the ‘chattering’ type. The
initiator neuron xo introduced above has a continuous input stimulus of I = 3 (a.u.) and starts
at t = 0 with the initial values x(0) = 0.3, y(0) = 0.3, z(0) = 3. Other authors [22] use similarvalues of input currents, ranging from about I = 2 to I = 10. Other neurons’ initial current
values are set to zero. The neuron is called the initiator because of its role as the single sourceof information in the network. In other terms, the initiator xo is the only neuron that provides
stimulated spikes and these are thought of as the single source of information in the network.
That is, in the absence of external noise, if other neurons are spiking it is because directly or
indirectly they receive the information to do so from xo. Our study investigates throughout therole of noise in this context. The entire network is connected as a small-world system, withthe exception of xo which receives input from the neighboring neurons and stimulates
accordingly the network structure, see ﬁgure 2 for a sketch of the conﬁguration used.
The integration of the differential equation (1) is done with an Euler method, a time
consuming but straightforward integration method. The time step is set to Dt = 0.01 ms [23]and has been veriﬁed to not introduce instabilities (the slight increase of Dt does not provoke
appreciable variation in the network response).
4

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 3. The Fourier transformation of the membrane signal of a single neuron
inserted in a p = 0.4 network of N = 10 and with a SNR of dB = 40. This is an
example of the frequency signature of a single neuronal response. The inset shows theactual spike plot where the Fourier transformation was calculated from; the horizontal
axis represents 1000 msec of time. The bar at x = 0.8 indicates the threshold used toseparate the baseline activity from the action potential. The neuron is chosen at random
and it is not connected to the initiator neuron xo.

Figure 3 shows the Fourier transformation of the membrane signal of a single neuron
inserted in a p = 0.4 network. The network itself is intentionally chosen to be smaller(N = 10) and with a minimal noise level (signal-to-noise ratio (SNR) dB = 40) in order to be
an example of the frequency signature of the neuron response. The neuron position is chosen
at random far from the active xo. Interestingly, we can observe the emergence of peaks at
10 Hz; these frequencies in the averages membrane potential can be thought as analogous to α
waves in real brains. This shows biological plausibility in simulations as already establishedfor example by Izhikevich et al and Palva and Palva [24, 25].
To generate the network, ﬁrstly, we create a regular network with a ring over the N
vertices. Each vertex stands for a neuron. Then, every vertex in the ring is connected with its
nearest neighbors at both sides. In all our tests, k = 1; this means that each neuron isconnected with two neighbors, one on its left and the other on its right. We say that the
network is of dimension k = 1 (if k = 2 each neuron has four neighbors, two on the left andtwo on the right). Starting from this regular network structure, we create random shortcuts byadding connections pointing to other neurons chosen at random with probability p (p = 0.4 inour case). In this way there are on average Nkp shortcuts in the network. Each vertex vi(i Î {1, 2, 3 ... N}) can be connected with any other with probability p. For vertex vi, auniform distribution random number in the range zero to one is generated and compared with
p. If the number is lower than p, we selected another vertex vj (i ¹ j) randomly and add ashortcut between vi and vj. A similar test is done to the adjacent neuron until all neurons are
examined (from v1 to vN). Neurons have no connections with themselves and two neurons canbe coupled by a unique segment; a single neuron can receive more than one connection fromdifferent neurons [26].
A good representation of biologically-plausible noise is controversial, however since thecentral limit theorem guarantees that a good number of arbitrary distributed sources converge
to a Gaussian signal, we opted for this type of noise [27–29]. The intensity is modulated using
different values of SNR deﬁned in decibels as dB = 10 log10
5

As 2
An

( ) , where A is the amplitudesJ. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Michelettoof the signal and An that of the noise. We use this deﬁnition because it is standard andlogarithmic, however we have to empathize that bigger values of dB mean a less noisy
network.
The random variable is generated with this formula:⎛ -(X - m) ⎞1
e ⎝ 2s 2 ⎠ .
s 2p
2f (X, m , s ) =

⎜

⎟

(2)

Here X is a ﬂat pseudo-random value generated internally by the numerical library numpy[30, 31], μ is the mean value around which the Gaussian is centered and σ is thecorresponding standard deviation. In our experiment, μ was ﬁxed as 0 and σ as 1 [32, 33]. Weuse1
Si xi ,NS
h=,
10 dB 10e = hf (X , m , s ) .
S=(3)

N is the total number of neurons, xi the membrane potential of neuron i, η the noise
amplitude and ε the ﬁnal Gaussian noise random variable. It results in being distributed along
a Gaussian curve, with an intensity proportional to η that contains the decibel parameter that isvaried in the tests [34]. The intensity of noise is calculated at each time loop, so several
neurons receive the same noise intensity for the same time step. This is an approximation of arealistic network where noise does not have spatial speciﬁcity.
The differential equation that describes the time evolution of a single neuron xi in thesystem is:N⎧x˙i = yi - axi3 + bxi2 - zi + ei + gi å j = 1 aij (xi - xj ) + Io⎪⎪
⎨ y˙ = c - dx 2 - y
i
i
i
⎪
⎪
z
r
s
x
=
[
(
c) - z i ] .
˙
⎩ i
i

(4)

This formula is equivalent to equation (1) but extended for a system of N neurons; variables
and parameters have the same meaning explained previously. The index i represents each of
the N neurons in the network. The parameter ei is the Gaussian noise of neuron i, updated ateach time loop. To compute the input current that each neuron receives, we calculate
gi S Nj = 1aij (xi - xj ). The sum is mediated by an adjacency matrix aij that stores the connection
between the vertices. In it, aij is 1 if there is a connection between the neurons i and j,otherwise it is 0. xi and xj are the membrane potential of neurons i and j. gi is the couplingstrength [35, 36] that is normalized to the number of connections of neuron i. For the
integration of this equation, we also used Euler method. In this way:

gi =

1

å

N
a
j = 1 ij.

(5)

In this formula, N is the number of neurons in the network. If we do not impose thisnormalization, the network will have an unbalanced inﬂuence of signals coming from theneighbors’ neurons depending on the number of connections [37, 38].
As stated above, in our simulation a neuron xo receives a speciﬁc continuous current
stimulus Io that is set to a value of 3 (a.u.) in all our tests. So, in the ﬁrst equation of system (4)if xi º xo , Io = 3; in all other cases Io = 0.
6

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 4. The spike map of a noiseless small-world network of 48 H−R neurons; asketch of it is outlined in ﬁgure 2. The neuron indicated as number 0 is a special one
that is connected to an external signal of Io = 3 (a.u.) constant input. It functions as an
initiator and after a delay of time, provokes the spiking of the whole network. The
same experiment reproduced with identical parameters but with no external input(Io = 0), produced no spiking anywhere in the network (blank result not shown).

3. Results
We veriﬁed that without the contribution of the special initiator neuron xo and neuronal noise,
the network exhibits no spiking activity and membrane signals settle to a baseline value.
However, we veriﬁed that with an increasing noise level, the network indeed starts spontaneous spiking, even without any neuronal input.
On the other hand, if neuron xo receives a constant input Io it becomes active and drives
its connected neighbors to spike, and these will induce spiking on their neighbors. The
information will propagate until the whole network will be spiking in a random but stablemanner.
In the following tests, corresponding to ﬁgures from 4 to 7, a network of N = 48 neurons
was used with connections as in table 1; i and j are the index of neurons. Value 1 in each gridindicates that neuron i is wired with neuron j (aij = 1). An empty grid means there is noconnection between neuron i and neuron j (aij = 0 ).
The noiseless response of such a system is shown in the spike map of ﬁgure 4 . This map isgenerated using an arbitrary threshold to separate the basal random signal from the spikes. Aftera few tests to evaluate the membrane potential range, the threshold was chosen as x = 0.8, see
inset of ﬁgure 3. When the membrane potential exceeds this value, we assume that the neuron isﬁring and plot the mark that is shown in ﬁgure 4 and the following spike maps. To evaluate thebehavior of the network as a whole, we integrate the membrane potential for the entire
population of neurons. We use this equation to calculate the parameter E(t):å i = 1xi (t ) .
E (t ) =
N

(6)

N

7

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Table 1. An example of network connections. Values of aij represent the connection
strength within the 48 neurons small-world network. The dimension of the network k is
1 and the shortcut probability p is 0.4. Connection strength can be only of value 1(connected) or zero (not connected). Zero values are not shown for clarity). According
to this table, there are in total 60 edges and 22 shortcuts in this network.

In ﬁgure 5 we show the output of the network measured as the average membranepotential calculated with equation (6) for a network of 48 neurons; p = 0.4 and k = 1. Clearlyas the noise increases, the initial transitional behavior gets shorter and shorter. This suggests
two things: that noise inﬂuences the average baseline of neuronal response and that it also
affects the spike rate of the neuronal system.
Figure 6 shows the raster plot for a neuron network with 48 neurons; k = 1 and p = 0.4
as before, for different noise levels. Again, the initiator neuron xo is spiking since thebeginning, whereas the rest of the network is quiescent and responds with a variable delaythat depends on noise level.
The SNR strongly inﬂuences the initiation of secondary spiking along the network.
Calling Ss the time interval by which secondary spikes are activated, we can interpolate ourresults with a liner model

Ss = de + a .
where d = 14 msec per decibel, ε is the noise level in dB and α is an offset, and the linear
regression coefﬁcient is better than 0.9. The inﬂuence of the initiator x0 is negligible and the8J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 5. The behavior of the neuron population membrane potential in function ofnoise level. Noise seems to help the diffusion of spikes along the network. The
horizontal axis spans over 1000 msec of simulation calculated in 100 000 steps of 0.01
ms each. The vertical axis is the voltage in arbitrary units. Each simulation is taken on a
network of N = 48 neurons and averaged over eight different experiments done withsame parameters but different random generator seeds. As the SNR grows (less external
noise) the initial transition time gets longer indicating a worse diffusion of information
across the network.

linear dependence is clear. See ﬁgure 7 for a graphical representation of this with a test of
several levels of noise. This formula cannot have a general meaning because it is bound to thisspeciﬁc network, however it shows a quantitative relation to modulate the transfer of
information with noise intensity in this case.
A network with identical parameters as above was tested in a similar way but changing
the size of the network instead of the noise (noise level was kept constant at 20 dB). As shownin ﬁgure 8 for small network sizes, the initiator neuron xo makes the spread of spikes quicker,
but it does not seem to strongly inﬂuence the transitional delay period after the network grows
more than about N = 100 neurons.
To analyze the inﬂuence of noise to the collective frequencies of the network, we studied
the Fourier spectrum of the variable E in formula (6). The Fourier spectrum peaks, if present,
are very important for recognizing rhythms and regularities in the entire network that mimic
the biological phenomena of slow and fast waves in real brains.
The Fourier transformation presents evident peaks at a strong SNR (top curve,
dB = 35), whereas the more prominent peaks at α and β seem to decay with increasing
noise [24], see ﬁgure 9. The Fourier transform data are calculated over a simulation periodof 1000 msec, cutting off the ﬁrst 300 msec to avoid including slow frequencies due to theinitial transitional phase discussed above. Interestingly the prominent β peak seems to shift
to the right with noise intensity; a very weak effect that is, however, difﬁcult to prove withmathematical means. We studied the total network power average among ﬁve different
frequency bands, θ, α, β, g1 and g2 . We calculated the Fourier spectrum of a 1000 millisecond simulation as above, then we summed up the power of those ﬁve frequency bands
with this formula9

J. Phys. A: Math. Theor. 49 (2016) 285601

Neuron Index

50

S Zhe and R Micheletto
50

(a)

40

40

30

30

20

20

10

10

0
0
50

100

200

300

400

500

600

Neuron Index

(c)

0
0
50

40

40

30

30

20

20

10

10

0

0

100

200

300

400

500

600

Time (ms)

0
0

(b)

100

200

100

200

300

400

500

600

300

400

500

600

(d)

Time (ms)Figure 6. The spike maps of a network with N = 48 neurons in different noise

conditions. The SNR in each plot is (a) dB = 35, (b) dB = 31, (c) dB = 26 and (d)dB = 18. The SNR strongly inﬂuences the initiation of regular spiking along the
network. On the vertical axis we have the neuron number; the neuron indexed as zero is
the special initiator neuron xo. Notice that a better SNR means less external noise.
Intermediate dB values were calculated but are not shown.

q=

1 7
å Y ( f ) df ,Nq f = 4a=

1 14
å Y ( f ) df ,Na f = 7b=

1 30
å Y ( f ) df ,Nb f = 14

g1 =

1 40
å Y ( f ) df ,Ng1 f = 30

g2 =

1 70
å Y ( f ) df .
Ng2 f = 50

( 7)

Here Ψ represents the vector that contains the module of the fast Fourier transformationarray. The elements corresponding to frequencies within the band are added up (θ from 4 to
7 Hz, α 7 to 14 Hz, β 14 to 30 Hz, g1 30 to 40 Hz and g2 50 to 70 Hz). Each band has adifferent number of elements, so the value is normalized to the various Nθ, Nα etc, depending
on how many elements each band has. Then we plot the results against the noise, as shown inﬁgure 10.
10

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 7. The comparative plot of the delay by which the spiking activity starts in a
network with and without an active initiator xo. Round dots represent the delay with anactive (Io = 3) initiator and the squares represent an inactive one. The inﬂuence of xo is
negligible and the linear dependence is clear. The red line represents a linear
interpolation of r > 0.9 and inclination d = 14 msec per dB.

Figure 8. The comparative plot between networks with and without an active spike

initiator xo, in the same fashion as in ﬁgure 7, but when the network size is changed.
For small network sizes of less than about N = 100 neurons the initiator makes adifference and induces a quicker propagation of spike activity. The noise level is kept
to 20 dB in all the tests.

In this plot we see that all prominent peaks decrease with noise intensity. This happens
because a low level of noise produces the transitional phenomena shown in ﬁgures 5 and 6,
during which few neurons are spiking, then the network goes to a regime with lower and more
11

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 9. A plot of the fast Fourier transformation of the membrane potential averaged
over the whole network. The curves represent its frequency spectrum averaged over thewhole network. Each line represents a simulation with different levels of noise. The
horizontal axis is in Hz and the vertical in arbitrary units. The latter has a relative
meaning since plots are shifted vertically for clarity. These show how the network’s
characteristic frequencies are inﬂuenced by different noise levels. The simulations areproduced with a total of N = 48 neurons. The small-world network is characterized bya connection probability p = 0.4 and dimension k = 1 and a spiking initiator neuron xo
with input Io = 3 (a.u.). The plot considers only the frequencies at the regime. That is,
the simulation is for 1000 msec, the ﬁrst 300 msec of data are not considered, in order
to avoid low frequencies due to the initial baseline variations. Each plot is the averageof eight identical simulations executed with different random number seeds. The
inﬂuence of an active (Io = 3) or passive (Io = 0) neuron x _o is null since we obtainequivalent results in both cases.

distributed frequency peaks. To verify that these patterns are indeed due to the initial transitional phenomena and not to the intrinsic character of the network, we cut off the ﬁrst 300
msec of simulation and we instead observe a more intricate behavior of the bands’ power, as
shown in ﬁgure 11.
Since each symbol represents a different noise level, the fact that the plots intersect eachother indicates that noise produces non-linear effect frequency bands amplitudes in this case.
We investigate the network, changing the network size as a paradigm of biological internal
noise. The analysis is conducted with other important neuron properties and characteristics
kept constant to allow comparison and discussion. Small-world dimension k = 1, probability
p = 0.4 and every simulation is repeated eight times with different random seeds, thenaveraged as in the previous tests.
As shown in ﬁgure 12, each spectrum shows distinct peaks in the α and θ range.
Interestingly, this behavior is maintained also for bigger networks. In present conditions, oursystem is limiting the simulations to these network sizes, but it would be interesting to further
increase the number of neurons to see if the diminished lower frequencies peaks phenomenon
observed with the external noise increase would be repeated with networks of much higher
sizes.
12

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 10. The power of ﬁve frequency bands for a network of N = 48 neurons. The

level of noise is represented by each symbol according to this scheme: square, dB = 5,
plus, dB = 9, diamond, dB = 13, triangle, dB = 18, triangle (bigger), dB = 22, circle,
dB = 26, circle (bigger), dB = 31, plus (bigger), dB = 35. The peaks’ magnitude seems
to keep the same relative order with noise; this effect is caused by the predominance of
the initial transition to the regime over the noise inﬂuence. The inset shows the averageand standard deviation of E(t) in function of noise for the same network. The averagepotential drops regularly because of the initial transitional period that drives potentials
to more negative values. Noise reduces this initial transition, so the curve appears lessnegative at low dB values. Refer also to ﬁgure 5. For reasons of clarity not all thedecibel labels are indicated.

4. Conclusions
We analyzed the collective behavior of a H−R small-world network of dimension k = 1 andrandom connection probability p = 0.4. The inﬂuence of noise is studied, especially concentrating on the effect it has on the propagation of information along the network by
studying the membrane average frequencies and spike activity diffusion due to an active
neuron that initiates the spiking. To isolate other effects the network is in ideal conditions;
identical neurons, no plasticity and no delay time are considered.
The network has a single active neuron xo connected to an external stimulus of
Io = 3 mA . This neuron is spiking constantly and in the absence of noise it induces the rest of
the network to spike after an initial transitional time. Most interestingly, it seems that noise
favors the initiation of secondary spikes (the spikes that follow those of xo), and this spike
activity becomes spontaneous and independent from xo. We found a linear relationshipbetween secondary spikes’ initiation time and the SNR in decibels, with an inclination
constant of about d = 14 millisecond per decibel. This phenomenon is instead found to bemuch weaker or not existent if we increase the network size.
However, for networks of sizes of N = 100 neurons or less, the initiator xo still makes adifference, favoring the quicker spread of spike activity on the network. This suggests that
locally-active neurons play a role in the spread of information in conﬁned domains outsidewhich the effect decays with distance. For the sake of simplicity and to avoid complexity this13

J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

Figure 11. The power of ﬁve frequency bands for a network of N = 48 neurons at
different noise levels as in ﬁgure 10. Noticeably, the ﬁrst 300 msec of simulation were
cut off to emphasize the network frequencies at the regime. The various plots crosseach other, with no systematic trends. This indicates that the noise is inﬂuencing thespectrum relative intensities and seems to shift the peaks’ position (see ﬁgure 9). For
reasons of clarity not all the dB labels are shown. The initiator xo has negligible
inﬂuence on these plots, see ﬁgure 7.

Figure 12. The Fourier transformation of the membrane signal in networks of different

sizes and constant 20 dB noise level. Each line is shifted vertically for clarity, in thesame fashion as ﬁgure 9. From top to bottom, the number of neurons is decreasing (top171, bottom 10 with steps of 23 neurons). The simulation is run for 1000 msec; the ﬁrst
300 msec of data are not considered to avoid low frequencies due to the initial baselinevariations. Frequencies peaks emerge in all conditions, but are less prominent for
networks of bigger sizes. Each plot is the average of eight identical simulations
executed with different random number seeds.

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J. Phys. A: Math. Theor. 49 (2016) 285601

S Zhe and R Micheletto

result has a limited meaning bound to the speciﬁc H−R network studied here, but we intend
to verify if this function of noise is a general phenomenon in spiking networks or remains
limited to the conditions and network connection structure used here. The importance of noisein neural networks is known, and what we observed reminds us of the phenomena of stochastic resonance in biological systems [6, 39–42].
The inﬂuence of noise on frequency has been investigated by Fourier analysis. We haveshown that once the transitional delay is past, noise displaces the main frequencies peaks,
provoking a slow transition to chaos. Studying the most prominent frequency bands (θ, α, βand γ) we conﬁrmed that noise quenches those biologically-signiﬁcant peaks in the network.
Still, at regime, a less keen transition to chaos is observed: frequency peaks do reduce andtend to disappear, but in a slower and more complex trend that seems to conserve some
activity in the theta and alpha band (as shown in ﬁgures 9 and 11).
In real brains, the noise is generated internally, so we scaled up the network and used the
network size as a paradigm of the external noise. We found two trends: a reduction in theamplitude of the peaks and a better sharpness and deﬁnition of the Fourier peaks, but not theevident degeneration to chaos observed when increasing external noise. This trend must be
presumably conﬁrmed with much higher network sizes, and we intend to set up the appropriate computational tools to do that in the near future.
Our work demonstrates that in a H−R network, a single neuron can lead to the excitationof the whole system; we quantitatively studied the inﬂuence of noise and have shown that it
favors the initiation of secondary spikes in a linear manner, enhancing the propagation of
spontaneous spiking activity along the network and canceling the effect of a localized
initiator neuron that instead maintains its inﬂuence in conﬁned domains.
This study gives insights on how networks utilize noise to alter the collective behavior of
the system in their operations and brings quantitative assessment on how noise can be used asa criterion to control and modulate the spread of neural activity in H−R neural networks.
Acknowledgments
This research has been partially supported by Yokohama City University basic research grant(YCU Kisokenkyu 2014) and Donghua Education and Communication Foundation (Donghua
15-31).
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