Collective firing regularity of a scale-free Hodgkin-Huxley neuronal network in response to a subthreshold signal

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Physics Letters A

www.elsevier.com/locate/pla

Collective ﬁring regularity of a scale-free Hodgkin–Huxley neuronal

network in response to a subthreshold signal

Ergin Yilmaz

a

,

∗

, Mahmut Ozer

b

a_{Department of Biomedical Engineering, Engineering Faculty, Bülent Ecevit University, 67100 Zonguldak, Turkey}

b_{Department of Electrical and Electronics Engineering, Engineering Faculty, Bülent Ecevit University, 67100 Zonguldak, Turkey}

a r t i c l e

i n f o

a b s t r a c t

Article history:

Received 12 November 2012 Received in revised form 2 March 2013 Accepted 7 March 2013

Available online 15 March 2013 Communicated by R. Wu

Keywords:

Channel noise Scale-free network Collective ﬁring regularity Temporal coherence

We consider a scale-free network of stochastic HH neurons driven by a subthreshold periodic stimulus and investigate how the collective spiking regularity or the collective temporal coherence changes with the stimulus frequency, the intrinsic noise (or the cell size), the network average degree and the coupling strength. We show that the best temporal coherence is obtained for a certain level of the intrinsic noise when the frequencies of the external stimulus and the subthreshold oscillations of the network elements match. We also ﬁnd that the collective regularity exhibits a resonance-like behavior depending on both the coupling strength and the network average degree at the optimal values of the stimulus frequency and the cell size, indicating that the best temporal coherence also requires an optimal coupling strength and an optimal average degree of the connectivity.

1. Introduction

In the last decade, the great attention has been dedicated to research complex system dynamics by using different network topologies [1–3]. The scale-free (SF) and the small-world (SW) network topologies have been widely used due to their capabil-ity of modeling many real networks. Recently, dynamics of bio-logical neuronal networks has been examined by adapting these topologies [4–8]. In the ﬁeld of computational neuroscience, un-derstanding the weak signal detection and the coding within the nervous system of noisy components is of great importance. In this context, collective spiking regularity or temporal coherence of the neuronal networks has been extensively investigated for the SW network topology. Ozer et al. [9] examined the collective ﬁring regularity of HH neurons in an SW network and showed that an optimal number of randomly added shortcuts and a certain level of noise intensity warrant the maximal temporal coherence both in presence and absence of a subthreshold periodic current. Gong et al.[10]investigated spatial synchronization and temporal coher-ence of an SW network driven by a subthreshold periodic current and found that synchronization and coherence can be enhanced by small-world network topology. Li and Gao [11] studied regu-larity of the spiking oscillation of the Fitz-Hugh–Nagumo neurons induced by colored noise and proposed that the regularity of spike

*

Corresponding author. Tel.: +90 372 257 4010; fax: +90 372 257 2140.

E-mail address:erginyilmaz@yahoo.com(E. Yilmaz).

train in the SW networks is higher than those in a regular network at an intermediate noise level.

Although the SW network topology provides more insight into the neuronal information processing, Barabasi and Albert [12] ev-idently showed that most of the real networks exhibit neither small-world nor random topology, but they display the SF topolog-ical features. The real networks such as web pages[13], the electric distribution system[14], the pattern of citations of scientific arti-cles[15]and the movie actors taking the role at the same film[12] have been shown to be constructed through the SF topology. More-over, the findings from the voxel based resting state connectivity analyses confirmed the modeling possibility with the SF topology of intra-regional connectivity in the functional brain regions[16]. In addition, Eguiluz et al.[17] reported that a scale-free architec-ture of functionally connected brain regions has been observed at a voxel scale during the performance of a number of motor and auditory tasks.

A literature survey leaves us with the impression that the col-lective spiking regularity of neuronal networks has not been inves-tigated in any detail using the SF topology. Therefore, in this study, our aim is to examine the collective spiking regularity of the SF Hodgkin–Huxley (HH) neuronal network driven by a subthreshold periodic stimulus. In addition, we use a biologically more realistic model for the stochastic behavior of the voltage-gated ion chan-nels embedded in the membrane patch, where the channel noise intensity depends on the membrane patch area or the cell size. We examine the collective regularity depending on both the stimulus frequency and the cell size. We also investigate how the network

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1302 E. Yilmaz, M. Ozer / Physics Letters A 377 (2013) 1301–1307

average degree and coupling strength affect the collective regular-ity.

2. Model

In a scale-free network of HH neurons, membrane potential dy-namics is described by the following equation:

Cm dVi dt

= −

g max Na m3h(Vi

−

VNa

)

−

gmaxK n4

(V

i

−

VK

)

−

gmax_L

(V

i

−

VL

)

+

j

ε

i j

Vj

(t

)

−

Vi

(t

)

+

Iext

,

i

=

1

,

2

, . . . ,

N (1)

where gmax_Na

=

120 mS

/

cm2, g_Kmax

=

36 mS

/

cm2 and gmax_L

=

0

.

3 mS

/

cm2 _{denote the maximal sodium, the maximal potassium}

and the leakage conductance, respectively. N is the number of neurons within the network, Cm

=

1 μF

/

cm2 is the membrane ca-pacitance. VNa

=

50 mV, VK

= −

77 mV and VL

= −

54

.

4 mV are the reversal potential of sodium, potassium and leakage currents respectively. Iext

=

A sin

(

ω

t

)

is externally applied subthreshold

si-nusoidal stimulus, where A is the amplitude of the stimulus and set to 1 μA

/

cm2_.

_ε

i j denotes the coupling strength between neu-rons i and j. If neuneu-rons i and j are connected then

ε

i j

=

ε

, oth-erwise equal to zero. m and n represent the open probabilities for the sodium and potassium activation gating variables, respectively; and h represents the open probability for the sodium inactivation gating variable. The factors m3h and n4 are the mean portions of open sodium and potassium channels within the membrane patch, respectively.

In the HH model, dynamics of gating variables change over time in response to membrane potential[18]. However, if the number of ion channels is ﬁnite, the stochastic behavior of the ion channels may have remarkable effects on the neuronal dynamics. To con-sider the channel stochasticity, we use Fox’s algorithm[19]due to its widespread use and computational eﬃciency[9,20–22]. In Fox’s algorithm, the gating dynamics is described by the Langevin gen-eralization as follows[19]:

dxi

dt

=

α

xi

(

1

−

xi

)

− β

xi

+ ζ

xi

(t),

xi

=

m,n,h (2) where

α

x and

β

x are the voltage-dependent rate functions for the gating variable xi.

ζ

xi denotes the independent zero mean Gaus-sian white noise whose autocorrelation functions are given as fol-lows[19]:

ζ

m

(t

)ζ

m

t

=

2

α

m

β

m NNa

(

α

m

+ β

m

)

δ

t

−

t

,

(3a)

ζ

h

(t)ζ

h

t

=

2

α

h

β

h NNa

(

α

h

+ β

h

)

δ

t

−

t

,

(3b)

ζ

n

(t)ζ

n

t

=

2

α

n

β

n NK

(

α

n

+ β

n

)

δ

t

−

t

(3c)

where NNa and NK represent the total numbers of sodium and

potassium channels in a given membrane patch area, respectively. The total channel numbers are calculated as NNa

=

ρ

NaS and NK

=

ρ

KS, where S represents the membrane patch area or the

cell size.

ρ

Na

=

60 μm−2 and

ρ

K

=

18 μm−2 are sodium and

potassium channel densities, respectively [9,20–22]. The strength of ion channel noise in Eq.(3)is inversely proportional to the cell size S.

Following the procedure in [12], we construct the scale-free neuronal network, using N

=

200 neurons with different average degree of connectivity, kavg. To quantify the collective ﬁring

reg-ularity or the network temporal coherence, we use the inverse

of coeﬃcient of variation (CV) of the inter-spike intervals (ISIs) [9,10,22]:

λ

=

1 CV

=

T

T2

₋

_T

2 (4)

where

T

2 _and

_T2

_{denote the mean and mean-squared ISIs,}

respectively. Spike times are deﬁned by the upward crossing of the average membrane potential of the network, Vavg

(

t

)

=

1

/

N

_iN₌₁Vi

(

t

)

, past a detection threshold of 0 mV. Notably, a larger

λ

corresponds to a stronger temporal coherence. The nu-merical integration of the stochastic model is performed by using the standard Euler algorithm with a step size of 10 μs. To ensure the statistical consistency, all results are calculated over 20 differ-ent network realizations.

3. Results

In what follows, we systematically analyze how the collective firing regularity of the scale-free HH neuronal network driven by a subthreshold periodic stimulus changes with the stimulus fre-quency, the cell size, the coupling strength and the network av-erage degree. Therefore, we first investigated how the collective firing regularity of the network changes with the stimulus fre-quency. We computed

λ

during 100 second simulations for seven different cell sizes ranging from 1 to 32 μm2_{, in each case for nine}

different angular frequencies ranging between

ω

=

0

.

1–0

.

9 ms−1

(Fig. 1). We found a clear optimal island of

λ

in the S–

ω

param-eter space, with an optimal frequency of

ω

=

0

.

3 ms−1 _{and an}

optimal cell size of S

=

6 μm2_{, suggesting a double-stochastic}

res-onance. We also found a second and third smaller peaks for the same cell of S

=

6 μm2 size at

ω

=

0

.

6 ms−1 and 0

.

9 ms−1, which are the second and third harmonics of

ω

=

0

.

3 ms−1_{. This is very}

clear as shown in an inset ofFig. 1. Yu et al.[23]showed that the HH neuron shows a subthreshold oscillations with an angular fre-quency of

ω

=

0

.

3 ms−1_{. Ozer et al.}_[9]_{also obtained inter-spike}

interval histograms (ISIHs) of an undriven HH neuron, and showed that the ISIH is very broad with a distinct peak near the mean of distribution (21 ms), corresponding to the period of the sub-threshold oscillations,

ω

=

0

.

3 ms−1_{. Both ﬁndings indicate that}

the optimal frequency of

ω

=

0

.

3 ms−1 for the network corre-sponds to the frequency of intrinsic subthreshold oscillations of the HH neurons [9,23]. Therefore, we suggest that the best tem-poral coherence of the network is obtained if the frequency of the subthreshold stimulus matches that of the intrinsic oscillations of the network elements.

To provide a clear picture of the dependence of the collective regularity on the cell size, we obtained the change of the regular-ity with the cell size for the network without external stimulus,

Iext

=

0, and driven by an external subthreshold stimulus with

three values of

ω

mentioned above (Fig. 2). The collective regu-larity exhibits a resonance-like behavior depending on the cell size regardless of an external stimulus, where a cell size of S

=

6 μm2

leads to the maximum coherence, suggesting that the best tem-poral coherence requires an optimal cell size or an optimal level of the noise. Our results are consistent with the results of Gong et al.[10]and Ozer et al.[9]for an SW HH neuronal network.

To gain more insight into the dependence of the best tempo-ral coherence on the cell size, we obtained ISIHs computed from 10 000 ISIs for the network spikes driven by an external stimu-lus, Iext

=

sin

(

0

.

3t

)

, for three values of the cell size, as shown

in Fig. 3. For the network with S

=

6 μm2_{, the ISIH has a}

sin-gle sharp peak at the stimulus period, indicating a high degree of phase locking (increasing the network output, i.e., the mean ﬁring frequency), whereas for the network with larger cell sizes, the ISIHs exhibit relatively wider multiple peaks at harmonics of

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Fig. 1. Dependence of collective temporal coherence of the scale-free network on subthreshold stimuli frequencyωand cell size S (kavg=10,ε=0.1, A=1 μA/cm2). Inset: how the regularity changes with subthreshold stimuli frequencyωfor a ﬁxed cell size, S=6 μm2_.

Fig. 2. Collective temporal coherence of the scale-free network for driven and undriven cases (kavg=10,ε=0.1, A=1 μA/cm2).

the stimulus period, indicating lower phase locking and cycle skip-ping (reducing the mean ﬁring frequency). Ozer et al.[9]obtained similar results for an SW HH network, where the ISIHs exhibited a similar pattern depending on the fraction of randomly added shortcuts for a given cell size.

Then, we investigated the dependence of the collective regular-ity versus cell size for different values of coupling strength at the optimal stimulus frequency of

ω

=

0

.

3 ms−1_{as shown in}_{Fig. 4}_{. For}

ε

0

.

075, the collective ﬁring regularity exhibits the SR behavior depending on the cell size. As the value of coupling strength in-creases, the maximum coherence is achieved with smaller cell size (or stronger noise intensity). In addition, the maximum coherence (

λ

max) also exhibits the SR depending on the coupling strength,

where the best temporal coherence is obtained for the coupling strength of

ε

=

0

.

1 as shown in an insert ofFig. 4.

In order to provide more insight into the relationship between the coupling strength and the collective regularity, we investigated how the collective regularity changes with the coupling strength when both the frequency and the cell size have their optimal val-ues (

ω

=

0

.

3 ms−1 _{and S}

₌

_{6 μm}2_{). Results are shown in}_{Fig. 5}_.

We found a resonance-like behavior of the regularity with respect to the coupling strength, indicating that the best coherence is ob-tained for an optimal coupling strength of

ε

=

0

.

1.

Finally, since the network average degree is another network parameter that may inﬂuence the collective ﬁring regularity of the SF network, we computed the change of the collective regularity with the cell size for different average degree values at the opti-mal conditions, i.e. an optiopti-mal stimulus frequency of

ω

=

0

.

3 ms−1

and an optimal coupling strength of

ε

=

0

.

1, as shown inFig. 6(a). For kavg

=

6, the channel noise (or the cell size) has no effect on

the regularity. For kavg

>

6, the collective ﬁring regularity exhibits

a resonance-like behavior depending on the cell size. The max-imum coherence is achieved with an optimal average degree of

kavg

=

10. Fig. 6(b) shows how the collective regularity changes

with the average degree when the frequency, the cell size and the coupling strength all have optimal values (

ω

=

0

.

3 ms−1,

S

=

6 μm2 _and

_ε

₌

₀

_.

_{1). We found a resonance-like behavior of}

the regularity with respect to the network average degree, indi-cating that the best coherence is obtained for an optimal average degree of kavg

=

10.

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(a)

(b)

(c)

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Fig. 4. Dependence of collective temporal coherence of the scale-free network on cell size, S, as a function of coupling strength (ω=0.3 ms−1, kavg=10, A=1 μA/cm2).

Fig. 5. Dependence ofλon coupling strength,ε(ω=0.3 ms−1, S=6 μm2_{, k}

avg=10, A=1 μA/cm2). 4. Discussions

In this work we considered a scale-free network of stochas-tic HH neurons driven by a subthreshold periodic stimulus and investigated how the collective spiking regularity or the collective temporal coherence changes with the stimulus frequency, intrinsic noise (or cell size) and the network parameters, i.e. the network average degree and the coupling strength.

We found a distinct increase in the collective regularity on the

ω

–S plane when both the frequency and the cell size have op-timal values (

ω

=

0

.

3 ms−1, S

=

6 μm2) (Fig. 1). We also found that the collective regularity exhibits smaller peaks when the fre-quency of the stimulus closes to the second and third harmonics of the stimulus (

ω

=

0

.

6 ms−1,

ω

=

0

.

9 ms−1) for the same op-timal cell size of S

=

6 μm2 (Figs. 1 and 2). Since the optimal frequency of

ω

=

0

.

3 ms−1 _{corresponds to the frequency of}

sub-threshold oscillations for the HH neurons[23], the best temporal coherence is obtained when the frequencies of the external stimu-lus and the subthreshold oscillations match[9,22]. Although Gong et al.[10]and Ozer et al.[9]considered a different network

topol-ogy, a small-world HH network, they also arrived at the similar ﬁndings. In this context, we may suggest that the best temporal coherence of a complex network is obtained at the same opti-mal values if the individual network elements are the same, i.e. stochastic HH neurons, regardless of the network topologies.

We also show that the best temporal coherence requires a cer-tain level of noise, indicating a cercer-tain cell size of S

=

6 μm2. This level of noise or the cell size enables the highest degree of phase locking, suggesting a single dominant time scale in the spike trains. When the cell size increases, the ISIHs exhibit multi-ple peaks at the harmonics of the stimulus period, indicating lower phase locking and cycle skipping (Fig. 3). The obtained results are also consistent with the computational results of different network topologies in[9,22,24,25]. On the other hand, our results demon-strate that the intrinsic noise or the cellular noise is necessary for the weak signal encoding in the context of the temporal coherence of an SF neuronal network. As indicated in[9], the noise provides an operating regime that is speciﬁc for a neuronal network, which is absent in regular or global networks. Recent studies on a single neuron or neuronal networks indicate that the noise of different

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(a)

(b)

Fig. 6. Dependence of collective temporal coherence of scale-free network: (a) on cell size, S, as a function of average degree of connectivity (ω=0.3 ms−1,ε=0.1,

A=1 μA/cm2_{), (b) on average degree of connectivity at an optimal cell size, S}₌_{6 μm}2_.

types, Gaussian or the colored noise, has great impacts on the neu-ronal dynamics, suggesting that understanding its impacts provides a deeper insight into the complex pattern of activity of the neu-ronal circuitry[26–29].

We also investigated how the collective regularity changes with the coupling strength and arrived at two ﬁndings: Firstly, the col-lective ﬁring regularity exhibits a resonance-like behavior depend-ing on the cell size and the maximal coherence is achieved with smaller cell size as the coupling strength increases (Fig. 4). Ozer et al.[9]showed that the maximal temporal coherence is achieved for fewer added shortcuts as the coupling strength increases in an SW HH neuronal network for a given cell size. Guo and Li[26] also showed that the coupling strength has an important impact on the stochastic dynamics in a feed-forward-loop (FFL) network motifs serving as a control parameter. In our study, the maximal values of the collective regularity also exhibit a resonance-like be-havior depending on the coupling strength as shown in an inset ofFig. 4, where the best temporal coherence is obtained for the coupling strength of

ε

=

0

.

1 by a ﬁxed cell size of S

=

6 μm2_.

Sec-ondly, we found that the collective regularity exhibits a resonance-like behavior depending on the coupling strength at the optimal values of the stimulus frequency and the cell size, indicating that the best temporal coherence requires an optimal coupling strength of

ε

=

0

.

1 (Fig. 5).

Finally, we investigated the impact of the network average de-gree of the connectivity on the collective ﬁring regularity. We found that the collective regularity exhibits a resonance-like be-havior depending on the cell size as the value of the average de-gree increases, where the best temporal coherence is achieved for the average degree of kavg

=

10 by a ﬁxed cell size of S

=

6 μm2

(Fig. 6(a)). Following the above investigation, where we found that the collective ﬁring regularity exhibits a well-expressed maximum at an optimal frequency of

ω

=

0

.

3 ms−1 _{and an optimal coupling}

strength of

ε

=

0

.

1 by an optimal cell size of S

=

6 μm2, we also obtained the change of the collective regularity with the network average degree for given optimal values, and found that the best temporal coherence is obtained for an optimal average degree of

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Therefore, we conclude that appropriately adjusted parame-ter values of the stimulus frequency, the cell size, the coupling strength and the average degree of the connectivity lead to-gether to the best temporal coherence of a scale-free network of stochastic HH neurons driven by a subthreshold periodic stimu-lus.

References

[1] R. Albert, A.-L. Barabasi, Rev. Mod. Phys. 74 (2002) 47. [2] F. Qi, Z.H. Hou, H.W. Xin, Phys. Rev. Lett. 91 (2003) 046102. [3] Z.H. Hou, H.W. Xin, Phys. Rev. E 68 (2003) 055103.

[4] H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411 (2001) 41. [5] J. Camacho, R. Guimer, L.N. Amaral, Phys. Rev. Lett. 88 (2002) 228102. [6] R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86 (2001) 3200.

[7] L.F. Lago-Fernandes, R. Hureta, F. Corbacho, J.A. Siguenza, Phys. Rev. Lett. 84 (2000) 2758.

[8] O. Kwoon, H.-T. Moon, Phys. Lett. A 298 (2002) 319.

[9] M. Özer, M. Uzuntarla, T. Kayıkçıo˘glu, L.J. Graham, Phys. Lett. A 372 (2008) 6498.

[10] Y. Gong, M. Wang, Z. Hou, H. Xin, Chem. Phys. Chem. 6 (2005) 1042. [11] Q. Li, Y. Gao, Biophys. Chem. 130 (2007) 41.

[12] A.L. Barabasi, R. Albert, Science 286 (1999) 509. [13] R. Albert, H. Jeong, A.L. Barabási, Nature 401 (1999) 130. [14] A.L. Barabasi, R. Albert, H. Jeong, Physica A 272 (1999) 173. [15] S. Render, Eur. Phys. J. B 4 (1998) 131.

[16] M.P. Heuvel, C.J. Stam, M. Boersma, H.E. Hulshoff Pol, NeuroImage 43 (2008) 528.

[17] V.M. Eguiluz, D.R. Chialvo, G.A. Cecchi, M. Baliki, A.V. Apkarian, Phys. Rev. Lett. 94 (2005) 018102.

[18] A.L. Hodgkin, A.F. Huxley, J. Physiol. 117 (1952) 500. [19] R.F. Fox, Biophys. J. 72 (1997) 2068.

[20] G. Schmid, I. Goychuk, P. Hänggi, Europhys. Lett. 56 (2001) 22. [21] M. Özer, Phys. Lett. A 354 (2006) 258.

[22] M. Özer, M. Perc, M. Uzuntarla, Europhys. Lett. 86 (2009) 0008. [23] Y. Yu, W. Wang, J.F. Wang, F. Liu, Phys. Rev. E 63 (2001) 021907. [24] M. Özer, M. Perc, M. Uzuntarla, Phys. Lett. A 373 (2009) 964.

[25] M. Özer, M. Perc, M. Uzuntarla, E. Koklukaya, NeuroReport 21 (2010) 338. [26] D. Guo, C. Li, Phys. Rev. E 79 (2009) 051921.

[27] H.C. Tuckwell, J. Jost, B.S. Gutkin, Phys. Rev. E 80 (2009) 031907. [28] H.C. Tuckwell, J. Jost, PLoS Comput. Biol. 6 (2010) e10000794. [29] D. Guo, Cogn. Neurodyn. 5 (2011) 293.