Stochastic resonance in feedforward-loop neuronal network motifs in astrocyte field

Journal of Theoretical Biology 335 (2013) 265–275

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Stochastic resonance in feedforward-loop neuronal network motifs in astrocyte field Ying Liu, Chunguang Li n Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, PR China

H I G H L I G H T S







The FFL network motifs are modeled in astrocyte field with mutual neuron–astrocyte interactions. The stochastic behaviors of the proposed network motifs are studied. Astrocytes improve the performance of signal processing via stochastic resonance. The functional roles of astrocytes are discussed.

art ic l e i nf o

a b s t r a c t

Article history: Received 16 April 2013 Received in revised form 2 July 2013 Accepted 7 July 2013 Available online 16 July 2013

Elucidating the underlying dynamical properties of neuronal network motifs, statistically significant patterns of interconnections, is essential to understand the dynamics of the whole networks. Besides, the brain is intrinsically noisy. Various noise-induced dynamical behaviors, in particular, the stochastic resonance (SR), have been found in both neuronal systems and neuronal network motifs. However, the effect of astrocytes, active partners in neuronal signal processing, has not yet received much attention. In this paper, we study the effect of astrocytes on the stochastic behaviors of the typical triple-neuron feedforward-loop (FFL) neuronal network motifs. The neurons are described by the Hodgkin–Huxley model, while the astrocytes are modeled by extending the Li–Rinzel model to a two-dimensional field with the effect of diffusion. The mutual neuron–astrocyte interactions are established correspondingly. Simulation results indicate that the stochastic behaviors of the FFL motifs show bell-shaped dependence on the intensities of both noise and astrocyte–neuron coupling. Moreover, in the presence of astrocytes, the performance of the FFL motifs on weak signal transmission in both noisy and noise-free environments can be significantly improved. From this point of view, the astrocytes can be regarded as a possible internal source of “noise”, which assist the neurons in signal processing. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Neuron Motif Astrocyte Calcium Noise

1. Introduction Complex networks are ubiquitous in nature. In addition to the well-known small-world effect and scale-free property, a large number of complex networks also have recurring patterns of interconnections, termed as “network motifs”, which are statistically more significant than those in randomized networks with the same degree sequence (Milo et al., 2002; Shen-Orr et al., 2002; Reigl et al., 2004; Wuchty et al., 2003; Sporns and Kotter, 2004; Li et al., 2007; Li and Li, 2008). Especially, in the study of biological neuronal networks, motifs have also been found (Milo et al., 2002; Reigl et al., 2004; Song et al., 2005). In addition, it has been discovered that, the connections among neurons in the network

n

Corresponding author. E-mail address: [email protected] (C. Li).

0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.07.007

motifs tend to be stronger than other connections, which implies that the motifs are linked to each other in a way that tends to keep the independent functions of each motif (Song et al., 2005; Alon, 2007). So, the network motifs can be seen as basic building blocks of the networks, and uncovering their dynamical properties and specific functions is essential to understand the behaviors of the whole networks. Currently, several interesting functions and dynamical properties have been found in neuronal network motifs, such as the acceleration and delay of response and longand short-term memory (Li, 2008; Ren et al., 2010; Franović and Miljković, 2010). Neurons are fundamental elements for constituting neuronal network motifs. They can generate electrical signals in response to chemical and other inputs, and then transmit them to other neurons. Besides, the brain is intrinsically noisy. Suitable noise in brain can enrich the stochastic dynamics of neuronal ensembles and induce various complex behaviors (Rolls and Deco, 2010;

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Destexhe and Rudolph, 2012; Li et al., 2006). The stochastic resonance (SR), a counterintuitive phenomenon that a proper noise can enhance the output of nonlinear dynamical systems to an optimal level, is considered as one of the most important mechanisms for neuronal information transmission. Originally, the SR was used to describe the noise-enhanced complex behaviors only when the power of a signal is weaker than that of the noise. Currently, it has been extended to characterize the occurrence of any kind of enhanced signal processing by noise (either internally or externally). In particular, it can also be used to describe a SR-like firing behavior, which is caused by the intrinsic “stochastic” effect of excitable nonlinear systems even in the absence of external noise. In this case, the internal chaotic behaviors of the excitable system can serve as an internal noise for exhibiting SR. To make a difference with the SR induced by external noise, such stochastic behavior is named as “intrinsic stochastic resonance” (ISR) (Manjarrez et al., 2002; Chik et al., 2001; Gammaitoni et al., 1998; Hu et al., 1993; Wang et al., 1998; Braun et al., 1994; Anishchenko et al., 1993). Currently, many studies on SR/ISR of different neuronal systems have been reported (Manjarrez et al., 2002; Chik et al., 2001; Collins et al., 1995; Douglass et al., 1993; Longtin, 1995; Levin and Miller, 1996; Guo and Li, 2012; Gailey et al., 1997). In addition, the SR in some neuronal network motifs has also been found (Guo and Li, 2009). On the other hand, in most parts of the brain, neurons are outnumbered by glial cells. So, a more realistic picture of brain tissue is a mass of glial cells with neurons embedded. The glial cells can be in general classified into two categories: the microglia and the macroglia cells. The former mainly functions in phagocytosis that protects the neurons of the central nervous system against infections. In addition to fulfilling the classical supportive roles as the microglia cells (Fields and Stevens-Graham, 2000, 2002; Haydon, 2001; Newman, 2003), the macroglia cells have been gradually evidenced to be active partners in neuronal signal processing in different brain areas, although they are unable to generate action potentials. The star-shaped astrocytes are the most numerous type and the best-studied macroglia cells (Haydon and Carmignoto, 2006). Experimental results have shown that astrocytes can listen to the neuronal chatter, respond to it and then talk back to the neurons. In other words, there exist bidirectional communications between neurons and astrocytes (Allegrini et al., 2009; Valenza et al., 2011; Nadkarni and Jung, 2004, 2007; Wade et al., 2011). Considering this, it is more reasonable to study the stochastic behaviors of neuronal network motifs by taking the effect of astrocytes into account. Following this motivation, the effects of astrocytes on the stochastic behaviors of neuronal network motifs are systematically studied in this paper. Similar to Guo and Li (2009), the most significant triple-neuron feedforward-loop (FFL) neuronal network motifs are studied. But different from Guo and Li (2009), the effects of astrocytes and the corresponding astrocyte–neuron interactions are also considered here. In our model, the neurons are modeled by the Hodgkin–Huxley (HH) neuron model, while the astrocytes are modeled by extending the well-known Li–Rinzel model to a twodimensional astrocyte field with the effect of diffusion among astrocytes. Then, the bidirectional neuron–astrocyte interactions are defined. Four typical FFL motifs with different combinations of excitatory and inhibitory neurons are discussed, and the impacts of astrocytes on SR in these network motifs are discussed. The rest of this paper is organized as follows. In Section 2, structural configuration of the neuronal network motifs, as well as the models of neurons, astrocytes, and mutual neuron–astrocyte interactions, is introduced. In Section 3, a series of simulations is performed and the influence of astrocytes on the stochastic behaviors of network motifs is discussed. Finally, conclusion and discussion are given in Section 4.

Fig. 1. Scheme of the network motif in astrocyte field. Each circle denotes a neuron and each square stands for an astrocyte with 30 μm  30 μm. Each square is then divided into 6  6 grids. Astrocytes are coupled by intercellular flux conditions to form a two-dimensional sheet, and on their boundaries, the flux conditions for both Ca2þ and IP3 fluxes are assumed. The boundary of the whole astrocyte field is represented by the red line. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

2. Model In this paper, we model the neuronal network motifs in a twodimensional astrocyte field with mutual neuron–astrocyte interactions. Similar to Guo and Li (2009), the triple-neuron feedforward-loop (FFL) network motif, the most significant motif in neuronal networks, is focused. Fig. 1 schematically shows the architecture of this motif, in which neuron 1 drives neuron 2, and both neurons 1 and 2 drive neuron 3. In this architecture, the neurons 1 and 3 can be regarded as the input and output neurons of the FFL motif, respectively. In addition to the neurons, six astrocytes, represented by an array of square cells with each cell located at a specific point of a regular lattice, are included. There are two reasons for formulating such a network configuration. Firstly, in the real human cortex, the ratio of astrocytes to neurons is about 1.4:1 (Nedergaard et al., 2003). So, in the simulation, it is more realistic to set the ratio of astrocytes to neurons close to the nature. Secondly, to avoid the influence of unsymmetric diffusion of the gliotransmitters from neuron 1 on neurons 2 and 3 (more details will be given in Section 3.1), neurons 2 and 3 are distributed with the same distance to neuron 1. The calcium (Ca2þ ) is an essential element for establishing bidirectional neuron–astrocyte interaction. On the one hand, some glutamate (Glu) can be released from the presynaptic neuron into the synaptic cleft, when a presynaptic neuron fires an action potential. Part of glutamate can bind to the metabotropic glutamate receptors (m-GluRs) on astrocytes, triggering the release of inositol 1,4,5-triphosphate (IP3 ) in the intracellular space of astrocytes. The production of IP3 in turn leads to the IP3 -dependent Ca2þ release from endoplasmic reticulum (ER) and Ca2þ -dependent Ca2þ release (CICR) from ER. On the other hand, if the calcium concentration inside the astrocyte exceeds a certain threshold, astrocyte can feedback to neuron by a depolarization current. In such a manner, the bidirectional signaling way between neuron and astrocyte can be established (Porter and McCarthy, 1996; Parpura and Haydon, 2000; Araque et al., 1998; Nedergaard, 1994). In the following, the models for neuron and astrocyte are illustrated in detail. 2.1. Hodgkin and Huxley neuron model The dynamics of each neuron in the network motif is described by the Hodgkin–Huxley neuron model, which is a typical paradigm for describing the spiking behavior and refractory properties of real neurons based on nonlinear conductance of ion channels.

Y. Liu, C. Li / Journal of Theoretical Biology 335 (2013) 265–275

The model equations are expressed as follows (Gerstner and Kistler, 2002): C m dV i =dt ¼ g Na m3i hi ðV i V Na Þg K n4i ðV i V K Þg L ðV i V L Þ þ I syn þ I noise þ γ i I astro ; þI ext i i i i dni =dt ¼ αni ðV i Þð1ni Þβni ðV i Þni ; dmi =dt ¼ αmi ðV i Þð1mi Þβmi ðV i Þmi ; dhi =dt ¼ αhi ðV i Þð1hi Þβhi ðV i Þhi ;

ð1Þ

where i ¼ 1; 2; 3 denote the neuron index, Vi represent the membrane potentials for each neuron i in the motif, g K , g Na , and g L stand for the maximal potassium, sodium, and leakage conductances per unit area, and V K , V Na , and V L denote their corresponding reversal potentials. Without loss of generality, the maximum conductance of the potassium, solidum, and leakage channels are set as g K ¼ 36 ms/cm2, g Na ¼ 120 ms/cm2 and g L ¼ 0:3 ms/cm2, respectively, and their corresponding reversal potentials are set as V K ¼ 12 mV, V Na ¼ 115 mV and V L ¼ 10:6 mV, respectively. The membrane capacitance is set as Cm ¼1 μ F/cm2. The parameters mi, hi, ni represent the saturation values of the gating variables with those terms in (1) governed by αni ¼ 0:01ðV i þ 50Þ=½1expð0:1ðV i þ 50ÞÞ; βni ¼ 0:125 expð0:125ðV i þ 60ÞÞ; αmi ¼ 0:1ðV i þ 35Þ=½1expð0:1ðV i þ 35ÞÞ; βmi ¼ 4:0 expð0:0556ðV i þ 60ÞÞ; αhi ¼ 0:07 expð0:05ðV i þ 60ÞÞ; βhi ¼ 1=½1 þ expð0:1ðV i þ 30ÞÞ: denotes the externally applied current. In Eq. (1), the term I ext i The term I syn denotes the synaptic coupling current among the i neurons, which is modeled by the linear sum of the synaptic current onto neuron i coming from all the pre-synaptic neurons, i.e. I syn ¼ ∑j g ij r j ðEV i Þ, where gij describes the coupling strength of i the synapse from neuron j to neuron i. For simplicity, we assume the coupling strength is identical for all the connections, denoted as g ij ¼ g for short. The reversal potential E ¼0 mV if neuron j is excitatory and E ¼ 80 mV if neuron j is inhibitory. When a neuron fires an action potential, the synaptic conductance rj of its postsynaptic target neuron is increased according to r j ¼ r j þ 0:5ð1r j Þ, and in other time dr j =dt ¼ r j =τ2 , where the refractory time τ2 is set as 15 ms. The term I noise denotes the noisy i synaptic current representing the external or internal fluctuations, which is modeled by the Ornstein–Uhlenbeck (OU) process (Destexhe and Rudolph, 2012) pffiffiffiffiffiffiffi noise τd dI i =dt ¼ I noise þ 2DηðtÞ; ð2Þ i where ηðtÞ is a Gaussian white noise with zero mean and unit variance. For simplicity, here we assume the noise intensity for each neuron is the same, denoted as D. The constant τd denotes the relaxation time of the synaptic noise, which is fixed at τd ¼ 2 ms. The synaptic inward current I astro in (1) denotes the depolari ization current feedbacked from astrocyte to neuron i with a coupling strength γ i . The value of I astro is related to the calcium i concentration in a two-dimensional field. More details are given in Section 2.2. 2.2. Astrocyte field model In literature, several models have been suggested to describe the dynamics of the intracellular Ca2þ concentration (Li and Rinzel, 1994; Atri et al., 1993; Amiria et al., 2012; Höfer et al., 2002; Sneyd et al., 1995). In the study of dressed neuron (a neuron coupled by an astrocyte), the Li–Rinzel model is typically used, which

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resembles the Hodgkin–Huxley mode for electrically excitable membranes by replacing the transmembrane potential by Ca2þ concentration. In this model, the calcium only changes in the time domain, and the signaling way from astrocyte to neuron is directly modeled by a depolarization current determined by the timedependent Ca2þ . In Sotero and Cancino (2010), a dynamical mean field was proposed to model the behavior of an ensemble of interacting neurons and astrocytes, which gives a simple and efficient description of neural-glial mass. Yet, there still exist rooms for modeling improvement. Firstly, the astrocytes are spatially extended objects. The Ca2þ can be diffused spatially from astrocyte to astrocyte through gap junctions, and thus the Ca2þ concentration in different spatial domains may be different (Allegrini et al., 2009; Liu and Li, 2013). The original Li–Rinzel model is unable to represent the Ca2þ concentration in spatial domain and thus cannot give a more accurate evaluation of the depolarization current I astro , as it is position-dependent. Secondly, i for a network motif coupled by several astrocytes, it is difficult to model the precise relations between astrocytes and neurons if only the time domain is considered, since each astrocyte may interact with several neurons in its neighborhood and meanwhile each neuron may be affected by more than one astrocyte (Newman, 2003). Considering these two aspects, it is more reasonable to model the astrocytes as a two-dimensional field, and then the depolarization current from astrocyte to neuron can be directly determined by Ca2þ in the spatial domain (Allegrini et al., 2009; Liu and Li, 2013). Following this motivation, in this paper, the Li–Rinzel model is extended to a two-dimensional field, and the diffusion effects of gliotransmitters, IP3 and Ca2þ from astrocyte to astrocyte are introduced. Mathematically, the model equations can be expressed as follows (Ullah et al., 2006): ∂½IP3 =∂t ¼ Dp ð∂2 ½IP3 =∂2 x þ ∂2 ½IP3 =∂2 yÞ þ 1=τp ð½IP3 n ½IP3 Þ þ½vp ð½Ca2þ  þ ð1αÞkp Þ=ðkp þ ½Ca2þ Þ þ r j ΘðV j 25mVÞ ð3aÞ ∂½Ca2þ =∂t ¼ Dc ð∂2 ½Ca2þ =∂2 x þ ∂2 ½Ca2þ =∂2 yÞJ flux J leak J pump 2

2

τq dq=dt ¼ k2 =ðk2 þ ½Ca2þ 2 Þq:

ð3bÞ ð3cÞ

The first terms in right hand side (RHS) of (3a) and (3b) reflect the diffusion of IP3 and Ca2þ , moving from astrocyte to astrocyte via gap junction with diffusion coefficients Dp ¼ 300ðμmÞ2 s1 and Dc ¼ 20ðμmÞ2 s1 , respectively (Allegrini et al., 2009; Atri et al., 1993). In the dynamical equation (3a) for IP3 , the second term describes the degradation of IP3 with a degradation rate of 1=τp and an equilibrium point ð½IP3 Þn , which accounts for a refractory time in calcium responses of astrocytes; the third term describes the production of IP3 caused by Ca2þ , where the constants are vp ¼ 0:13 μMs1 , kp ¼ 1:1 μM, and α ¼ 0:8; the last term reflects the glutamate-induced production of the cytosolic IP3 with the production rate rj via the Heaviside function Θ. To be specific, it is activated when the membrane potential of the neuron j, V j ≥25 mV. It is also worth pointing out that as the inhibitory neurons cannot release much excitatory neurotransmitter, glutamate, which is necessary for establishing neuron–astrocyte interaction, the signaling way from neuron to astrocyte is not considered for the inhibitory neurons, i.e. rj ¼ 0 if neuron j is inhibitory. In the dynamical equation (3b) for Ca2þ concentration, J flux describes the rate of calcium released from ER into the cytoplasm through the IP3 receptors, which designates a positive feedback controlled by the slow inactivation gate q, J pump denotes the rate of calcium flux pumping from the cytoplasm into ER and into the extracellular space; J leak denotes the leakage flux from the ER to

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the intracellular space. The dynamics of these three terms are controlled by (Nadkarni and Jung, 2004; Li and Rinzel, 1994) J flux ¼ c1 v1 m31 n31 qð½Ca2þ ½Ca2þ ER Þ J leak ¼ c1 v2 ð½Ca2þ ½Ca2þ ER Þ 2

J pump ¼ ðv3 ½Ca2þ 2 Þ=ðk3 þ ½Ca2þ 2 Þ

ð4Þ

where m1 ¼ ½IP3 =ð½IP3  þ d1 Þ;

n1 ¼ ½Ca2þ =ð½Ca2þ  þ d5 Þ;

½Ca2þ ER ¼ ðc0 ½Ca2þ Þ=c1 ; and the model parameters are c0 ¼2.0, c1 ¼0.185, v1 ¼ 6 s1 , v2 ¼ 0:11 s1 , v3 ¼ 0:9ðMsÞ1 , k3 ¼ 0:1 μM, d1 ¼ 0:13 μM and d5 ¼ 0:08234 μM (Nadkarni and Jung, 2004). The third equation (3c) denotes the fraction of IP3 receptors, q, that has not been inactivated by Ca2þ with a degradation rate 1=τq 2 2 and steady state q ¼ k2 =ðk2 þ ½Ca2þ 2 Þ. Since the astrocytes are now modeled as a two-dimensional field, the concentration of ½Ca2þ  and ½IP3  depend on their coordinates in the (x,y)-plane. It was experimentally evidenced in Parpura and Haydon (2000) that, when the cytosolic concentration of astrocytes exceeds a certain threshold, some vesicles can be released through a vesicular release mechanism (Bezzi et al., 2004), which is followed by the depolarization of the neurons with its strength governed by Ca2þ . Based on these experimental data (Parpura and Haydon, 2000), such depolarization current can be modeled by a nonlinear function of Ca2þ as follows (Nadkarni and Jung, 2004): ¼ 2:11Θðln zÞ ln z I astro i

ð5Þ

2þ

where z ¼ ½Ca ðx; yÞ196:69ðin nMÞ and Θ denotes the Heaviside function. The recorded total current is then converted to a current density measured in μA=cm2 by assuming a spherical soma. To be specified again, in the considered neuronal network motifs, the signaling way from neuron to astrocyte is governed by (3a), which further promotes the IP3 -dependent-Ca2þ elevation given in (3b). Meanwhile, the astrocyte regulates the neuronal activities by injecting a depolarization current into the HH neuron (1) with its intensity determined by Ca2þ given in (5).

Fig. 2. Connection patterns of the triple-neuron FFL network motifs with different combinations of excitatory and inhibitory neurons. For simplicity, the embedded astrocytes given in Fig. 1 are not depicted here.

to avoid the influence of the boundary conditions, an external gird is added near to each boundary gird of the whole astrocyte field, see the red line in Fig. 1.

3. Numerical simulations In this section, we perform a series of simulations to investigate the influence of astrocytes on stochastic behaviors of the tripleneuron FFL network motifs depicted in Fig. 1. Considering that the neurons can be excitatory or inhibitory, there are total eight types of configurations for the network motifs (Guo and Li, 2009). As explained in Section 2.2, the neuron–astrocyte coupling for the inhibitory neurons is not considered here. So, only the four cases that neuron 1 is excitatory are focused and their configurations are illustrated in Fig. 2. Note that for simplicity, the astrocytes A–F shown in Fig. 1 are not depicted. In the following, both the simulations with and without the noisy current I noise are peri formed, which correspond to the cases of SR and ISR, respectively. 3.1. Stochastic resonance in the presence of noise

2.3. Numerical simulation methods In the numerical simulations, the stochastic differential equation of the HH neuron (1) is solved by the standard Euler– Maruyama integration algorithm with integration step fixed at Δt ¼ 0:02 ms (Kloeden et al., 1994). The two-dimensional diffusion function in (3) for astrocytes is solved by an explicit Euler stepping for the kinetics and an alternating direction implicit (ADI) scheme for computing the diffusion terms (Morton and Mayers, 1993). Space and time steps chosen for the ADI scheme are 5 μm and 0.2 ms, respectively. The equations of astrocyte model are solved numerically by discretizing them on two-dimensional grids of square cells, each 30 μm 30 μm. The dynamics of Ca2þ and IP3 within each astrocyte is governed by (3), and at the boundaries between astrocytes, the flux-conditions are assumed (Allegrini et al., 2009; Atri et al., 1993; Höfer et al., 2002; Sneyd et al., 1995). Moreover, all the cell boundaries are oriented along one of the coordinate. For instance, considering a membrane situated at x¼ 0 in the x-spatial domain, we have Dp ∂½IP3 ð0 ; tÞ=∂x ¼ Dp ∂½IP3 ð0þ ; tÞ=∂x ¼ F p ð½IP3 ð0þ ; tÞ½IP3 ð0 ; tÞÞ Dc ∂½Ca2þ ð0 ; tÞ=∂x ¼ Dc ∂½Ca2þ ð0þ ; tÞ=∂x ¼ F c ð½Ca2þ ð0þ ; tÞ½Ca2þ ð0 ; tÞÞ;

where F p ¼ 5 μm s1 and F c ¼ 1:1 μm s1 are the IP3 and Ca2þ permeabilities of the membrane, respectively, 0 and 0þ stand for the limits from the left and right, respectively. Moreover, in order

We first study the performance of stochastic resonance in the FFL motifs in astrocyte field in the presence of external noise. We focus more on the response of the neuron 3, as it is viewed as the output neuron of the considered FFL motifs. In the simulation, we set the external forcing signal for neuron 1 as I ext 1 ¼ I 0 þ As sin ð2πf s tÞ, where I0 ¼3.0 is a bias, As ¼1 is the amplitude of the periodic forcing signal, and fs ¼50 Hz is its frequency. No external currents are applied to neurons 2 and 3, i.e. ext I ext 2 ¼ I 3 ¼ 0. The noise intensity for each neuron is set as D¼ 0.5, and a weak synaptic coupling g¼0.045 is adopted. We take the Type-1 FFL motif as a simulative example, in which all the three neurons are excitatory. With these parametric settings, as shown in Fig. 3(a), only a few spikes are elicited and the neuron 3 stays around the resting potential for most of time without the effect of astrocytes. So, the information cannot be successfully transmitted from neuron 1 to 3. However, by introducing the coupling from astrocytes to neurons with suitable coefficients, γ 1 ¼ 0:1, γ 2 ¼ γ 3 ¼ 0:3, the nonlinear behavior of neuron 3 becomes more coherent with that of neuron 1, see Fig. 3(b). In this case, the neuron 3 can in general follow the dynamics of neuron 1 and thus the input information can be stochastically encoded in the response of the output neuron, although a few spikes are missed in several periods of cycles. To further verify the performance of SR, the power spectra of the spike trains are also computed. The spike train is a binary time series related to the membrane potentials. For the HH neuron, if

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Fig. 3. The stochastic behaviors of the FFL motifs in the astrocyte field in the presence of noise. (a) The membrane potentials of neurons 1 (V1) and 3 (V3) for Type-1 without astrocytes. (b) The membrane potentials of neuron 1 (V1) and 3 (V3) for Type-1 with astrocytes. (c) The power spectra of the spike trains for Type-1. (d) The power spectra of the spike trains for Type-2. (e) The power spectra of the spike trains for Type-3. (f) The power spectra of the spike trains for Type-4.

the membrane potential Vi exceeds a certain threshold, about 25 mV, the neuron is assumed to elicit a spike, represented by a binary ‘1’. Otherwise, it is set as ‘0’. Following this measure, a spike train can be obtained by recording times for the generation of spikes, that is N

V i ðtÞ ¼ ∑ δðtt j Þ; j¼1

where tj is the time at which the j-th spike generates from neuron i and N is the total number of spikes initiated in the time series. The

power spectral density is then calculated from the spike train through the fast Fourier transform (Lee and Kim, 1999). The power spectra computed from the spike trains of neuron 3 with and without astrocytes for Type-1 are depicted in the top and bottom of Fig. 3(c), respectively. We find that, in the absence of astrocytes, the peak of the spike train is rather weak, less than 200, see the top of Fig. 3(c). However, by introducing the bidirectional interactions between neurons and astrocytes, the power spectrum at 50 Hz increases to 600 (see the bottom of Fig. 3(c)), which is near to three times of the former case. Pronounced peaks at the forcing frequency 50 Hz and its second harmonics at 100 Hz,

Y. Liu, C. Li / Journal of Theoretical Biology 335 (2013) 265–275

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Fig. 4. The Ca2þ waves in the central grid of astrocytes B, D and F in different network motifs.

are noticed, thus indicating that the resonance between neurons 3 and 1 is enhanced owing to the astrocytes. Similarly, we have done the simulations for the other three types of FFL motifs, and their corresponding power spectra are depicted in Fig. 3(d)–(f), respectively. The similar astrocyteenhanced SR is observed. It is remarked that without astrocytes, the power spectra of neuron 3 in both Type-2 and Type-4 (see the top of Fig. 3(d) and (f)) are slightly smaller than those in Type-1 and Type-3 (see the top of Fig. 3(c) and (e)), since neuron 2 has an inhibition effect in the former but an excitation effect in the latter on neuron 3. When the astrocytes are included, Type-1 and Type-2 (Type-3 and Type-4) show similar spectra, see the bottom of Fig. 3(c) and (d) (Fig. 3(e) and (f)). The occurrence of the similar spectra in these two FFL motifs is due to their similar patterns of depolarization from the astrocytes to the neuron 3. This phenomenon is further explained in Fig. 4. These results suggest that the astrocytes have significant impact on the stochastic behaviors of the neuronal network motifs. The Ca2þ waves in the central grids of the astrocytes B, D, and F that have direct interactions with neurons 1, 2, and 3, respectively, are depicted in Fig. 4. We first discuss the results of Type-1. As shown in Fig. 4(a), the Ca2þ wave in astrocyte B triggered by neuron 1 initiates first. Then the Ca2þ wave spatially transmits to the astrocytes in its neighborhood by diffusion. For Type-1, there are two factors affecting the Ca2þ . Firstly, concurrently with neuronal firings, the excitatory neurotransmitter, glutamate, is released. This promotes the production of IP3 (see (3a)) and further causes IP3 -dependent Ca2þ from the ER (see (3b)). Secondly, the effect of diffusion from astrocyte to astrocyte also

contributes to diffusing Ca2þ from higher concentration to lower concentration. For Type-1, since both excitatory neurons 1 and 2 drive neuron 3, the synaptic current I syn onto neuron 3 is larger i than that onto neuron 2. Thus more spikes are initiated from the excitatory neuron 3, leading to higher Ca2þ in astrocyte F than that in astrocyte D. As the Ca2þ elevated by neuronal firing dominates that by diffusion, Ca2þ wave first initiates in astrocyte F and then in astrocyte D, although they have similar quantities of Ca2þ diffused from astrocyte B. For Type-2, as both neurons 1 and 3 keep excitatory as they are in Type-1, the Ca2þ waves in astrocytes B and F are similar to their corresponding results in Type-1, see Fig. 4(a) and (b). But, as neuron 2 is inhibitory in this case, no neuronal firing-induced calcium elevation in astrocyte D occurs. So, the Ca2þ is decreased (see Fig. 4(b)) comparing to that in Type-1 (see Fig. 4(a)). For Type-3, both neurons 1 and 2 keep excitatory as they are in Type-1 so that the Ca2þ waves in astrocytes B and D are similar, see Fig. 4(a) and 4(c). As neuron 3 turns to be inhibitory in this case, the Ca2þ mainly comes from diffusion through gap junctions, and thus long time is spent for initiating Ca2þ wave in astrocyte F since the propagation of Ca2þ in astrocytes takes several seconds. For Type-4, as demonstrated in Fig. 4(d), due to the strong inhibition effect of both neurons 2 and 3, they cannot directly promote the Ca2þ in astrocytes D and F. Thus similar Ca2þ waves within these two astrocytes are observed. Comparing the Ca2þ waves in these four types of network motifs, we find that the Ca2þ waves in astrocyte B in these four cases are similar since the neuron 1 always keeps excitatory. The Ca2þ waves in astrocyte D share similar dynamics for Type-1 and Type-3 (Type-2 and Type-4), as the neuron 2 is excitatory (inhibitory) in these two cases. Similarly, Ca2þ waves in astrocyte F for Type-1 and Type-2 (Type-3 and Type-4) are

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similar, owing to the same excitability (inhibitility) of neuron 3 in these two motifs. To show the diffusion of Ca2þ in the two-dimensional field, the Ca2þ in the (x,y)-plane at several times for Type-1 in the above simulations is depicted in Fig. 5. From Fig. 5, we find that at t¼ 1 s, the Ca2þ wave initiates in astrocyte B, which is promoted by the firings of neuron 1 intermittently. The Ca2þ wave then propagates to other astrocytes by diffusion through the gap junctions, see t¼ 2.5 s. At t¼ 3 s, the Ca2þ concentration in astrocyte F is significantly improved. Because both neurons 1 and 2 drive neuron 3, more spikes are elicited from neuron 3 than that from neuron 2, leading to a higher Ca2þ concentration in astrocyte F than that in astrocyte D. At t¼ 3.5 s, the Ca2þ within the whole astrocyte field is significantly increased. After that, with the propagation of Ca2þ wave, the Ca2þ in astrocyte B decreases, see t ¼4 s, while it increases again as neuron 1 fires action potentials, see t¼9 s. The noise intensity D is an important parameter to affect the performance of SR. Here its impacts on the stochastic behaviors of the FFL network motifs in astrocyte field are analyzed. To evaluate the influence of D on SR, some coherence measures, such as the interspike interval, SNR and coherent SNR, have been suggested in the literature (Chik et al., 2001; Hu et al., 1993; Collins et al., 1995; Guo and Li, 2009; Lee et al., 1998). In this paper, in order to further evaluate the performance of ISR in the absence of external noise, the coherent SNR is adopted, as its computation does not directly relate to the external noise. According to Hu et al. (1993) and Lee et al. (1998), the coherent SNR is defined as βs ¼ Hwp =Δw, where H is the peak height of the spectrum, wp is the frequency at which the peak occurs and Δw is the width of the peak at the half maximum height. Besides, βs can also be computed from the spike train, as it contains the information of the signal. In the simulation, the same external forcing signal with amplitude As ¼1 and frequency fs ¼50Hz is applied to neuron 1. We tune the intensity of noise from 0.1 to 1.0 and compute the corresponding βs from the spike trains. The averaged results over 20 trials for these four network motifs are depicted in Fig. 6. For comparison, the results without the effect of astrocytes are also given. Note that as shown in Fig. 2, both neurons 1 and 2 are excitatory (inhibitory) in Type-1 and Type-3 (Type-2 and Type-4).

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So, the pre-synaptic coupling currents onto neuron 3 in these two motifs are similar in the absence of astrocytes, which leads to similar responses of neuron 3. Owing to this reason, only the results of Type1 and Type-2 FFL motifs in the absence of astrocytes are presented in Fig. 6 for performance comparison. It is obvious that the coherence measure βs of all the considered four types show bell-shaped dependence on the noise intensity D. The maximum βs achieves at around Dopt ¼ 0:6 in the presence of astrocytes, while it achieves maximum at around Dopt ¼ 1:25 in the absence of astrocytes. As D moves away Dopt , βs decreases for each type of motif. Based on the simulation results, it is concluded that the performance of SR in the neuronal network motifs with astrocytes is significantly better than that without astrocytes, especially when the intensity of noise D is smaller than a certain threshold, about 0.8. The simulation results suggest that the astrocytes in deed contribute to improving the performance of weak signal transmission in the FFL motifs, especially in a weak noisy environment.

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3.2. Intrinsic stochastic resonance in the absence of noise We also investigate the performance of stochastic resonance of the triple-neuron FFL motifs in the absence of external noise, which corresponds to the case of ISR if the coherent between neuron and input signal is achieved. In the simulation, the external forcing signal for the input neuron 1 is set as I ext 1 ¼ I 0 þ As sin ð2πf s tÞ, where I0 ¼3.2, As ¼1.0 and fs ¼60Hz. Similar to Section 3.1, no external currents are applied to neurons 2 and 3, and the synaptic coupling coefficient

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As illustrated in Section 2, the coefficient γ i determines the strength of coupling from astrocytes to neurons, and thus affects the stochastic behaviors of the neuronal network motifs. Fig. 7 depicts the coherence quantity βs as a function of γ i . As we are interested in the effect of astrocytes on the transmission ability of the network motifs, the coupling coefficients of the interneuron 2 and the output neuron 3, γ 2;3 are varied, while that of the input neuron 1, γ 1 , is fixed at 0.1. For simplicity, we assume that the values of γ 2 and γ 3 are the same, denoted as γ, i.e. γ 2 ¼ γ 3 ¼ γ. The values of βs from the spike trains against different γ are depicted in Fig. 7. Similar to Fig. 6, a bell-shaped dependence of βs on the coupling coefficient γ is observed. With the increasing of γ, the value of βs increases first, and then decreases after an optimal value around γ opt ¼ 0:3 for the considered four FFL motifs. When γ 4 0:45, its corresponding βs becomes even worse than that without astrocyte (γ ¼ 0). The results suggest that there exists an optimal γ opt , at which the optimal synaptic information transmission can be achieved via SR. The existence of the optimal γ opt can be interpreted as the existence of the optimal transmission probability of the gliotransmitter according to the vesicular release mechanism (Bezzi et al., 2004). On the one hand, extremely low level of transmission probability of the gliotransmitter may have little effect on regulating neuronal activities. On the other hand, extremely high level of transmission probability may drive the neuron into limit cycle, making it unable to encode the input information into the membrane potentials. The existence of large astrocyte–neuron coupling may be due to the overexpressed effect of astrocytes on neurons in some mental diseases (Ulas et al., 2000; Aronica et al., 2000; Hauser and Seifert, 2002). From this point of view, the coupling coefficient γ can serve as a control parameter, which has great impact on the stochastic dynamics of the FFL motifs. As reported in Chik et al. (2001) and Lee and Kim (1999), the performance of SR of a single HH neuron is sensitive to the value of signal frequency. The optimal noise intensity Dopt , which corresponds to the optimal performance of SR shows an inverse bellshaped dependence on the forcing frequency. But, does the similar phenomenon exist for the coupling coefficient γ? To answer this question, the dependence of γ opt on the frequency of the forcing signal is investigated. As the results of different network motifs are similar, only the results of Type-1 are presented for illustration. The other model parameters are the same as those used for Fig. 7. From Fig. 8, it is noticed that the SR occurs when the forcing frequency (fs) ranges from 45 Hz to 75 Hz. The optimal coupling coefficient γ opt shows nonmonotonous frequency dependence on the forcing frequency fs. Firstly, with the increasing of fs, γ opt decreases initially. At fs ¼60 Hz, which corresponds to the natural frequency of the HH neuron, γ opt achieves the minimum. Further increasing the value of fs, the value of γ opt increases again. The dependence of βs on the optimal coupling coefficient γ opt is similar to that on the optimal noise intensity Dopt (Lee and Kim, 1999), implying that γ may play a similar role as D. Consequently, the depolarization current I astro caused by astrocyte–neuron coupling i may act as noise to some extent. But it is generated internally instead of externally.

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is set as g ¼0.045. The results for Type-1 are presented for illustration. Based on these parametric settings, it is shown in Fig. 9(a) that the membrane potentials of neuron 3 oscillate around the resting potential, and the information of the input signal is totally lost. However, by introducing the coupling from the astrocytes with strength γ 1 ¼ 0:04, γ 2 ¼ γ 3 ¼ 0:45, neuron 3 fires following the dynamics of neuron 1 as shown in Fig. 9(b), which indicates that the input information is encoded in the response of the output neuron. Comparing Fig. 9(b) with Fig. 9(c), it is obvious that the neuronal firing properties are modulated by the depolarization current I astro , in particular for neurons 2 and 3, since no i external currents are injected into these two neurons. With the occurrence of I astro when the Ca2þ exceeds the threshold, the i neurons 2 and 3 become more coherent with the neuron 1, see t≥3700 ms for V2 and t≥3000 ms for V3 in Fig. 9(b) so that the information can be successfully transmitted within the network motifs. These results suggest that, in addition to the excitability of neuronal systems (Chik et al., 2001; Wang et al., 1998), the mutual neuron–astrocyte interactions can also regulate the neuronal activities via the mechanism of ISR. From the power spectrum given in Fig. 9(d), pronounced peaks located at the forcing frequency and multiples of the forcing

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frequency are observed, indicating the occurrence of ISR. Moreover, several small peaks at f ′ ¼ f s =nðn ¼ 2; 3; 4; 5Þ and the integer multiples of these frequencies are noticed, due to the deterministic features of the neuron in a chaotic firing motion. These small peaks can be reduced if an external noise is applied to the neurons, consistent with the results obtained in Chik et al. (2001) and Wang et al. (1998). The power spectrum reflects the multimodal aperiodic firing pattern of the neuron, which is enhanced by the astrocytes, and thus a reliable transmission of weak signals can be ensured. In order to further elucidate the functional role of astrocytes, we replace the I astro by some externally applied currents, and compare i their power spectra. For a fair comparison, we keep some statistics of I astro induced by astrocytes and those of the external currents the i same. Two cases are considered here. In the first case, we only add a bias current into each neuron i with an intensity equivalent to the mean of I astro given in Fig. 9(c). In the second case, the currents i distributed in Gaussian with the same means and variances as their corresponding I astro are applied to the neurons. The other parametric i settings are the same as those for Fig. 9. The power spectra of neuron 3 computed from the spike trains of the above two cases are depicted in Fig. 10. We find that the power spectrum of neuron 3 with external bias currents can be significantly increased comparing to the case without astrocytes given in Fig. 9(a) (In this case, neuron 3 stays at the resting potential, and thus its power spectrum based on the spike train is nil.). Pronounced peaks at the forcing frequency and its harmonics

are observed. The power spectrum can be further increased when the current sharing the same mean and variance with the I astro is i acted on each neuron i, see Fig. 10(b). But it is still worse than the case that the astrocytes are incorporated (Fig. 9(c)). This implies that the depolarization current I astro from the astrocytes to i neurons may behave more like a noise current with both mean and variance, but not that only. These results may also give a possible explanation for the experimental results obtained in Braun et al. (1994) that the neuron shows some SR-like firing behaviors even with a steady thermal stimulation, which is modeled by a constant stimulus plus a weak periodic signal. Similar to the study of SR in Section 3.1, the effect of the coupling coefficient γ 2 ¼ γ 3 ¼ γ on the performance of ISR is investigated, in which γ 1 is fixed at 0.04. The corresponding values of βs are computed and the results are given in Fig. 11. Similar to the case of SR, the bellshaped βs curves against γ are observed for our considered four types of network motifs. As reflected in Fig. 11, the maximal βs for Types 1–3 achieves at around γ opt ¼ 0:3, while it achieves at a larger value of γ opt (γ opt ¼ 0:45) for Type-4. The larger value of γ opt for Type-4 may be due to the strong inhibition of neurons 2 and 3.

4. Conclusion and discussion Network motifs, significantly occurring patterns in biological neuronal networks, are considered as the basic building blocks of neuronal networks. So, elucidating the dynamics of neuronal network

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motifs and their functions would shed new light on understanding the behaviors of the whole brain. Considering that the astrocytes also participate in neuronal signal processing, the presented work studied the performance of stochastic resonance in the FFL motifs in astrocyte field with bidirectional neuron–astrocyte interactions. The HH model is used to describe the dynamics of the neurons in the FFL motifs, while an extension of Li–Rinzel model to a two-dimensional field is proposed for modeling the dynamics of astrocytes in spatial domain. The mutual neuron–astrocyte interactions are defined correspondingly. Four possible structural configurations of the triple-neuron FFL network motifs are considered. Comparing to the results obtained in Guo and Li (2009), the response of output neuron to a weak periodic signal can be enhanced by a suitable nonzero coupling from astrocyte to neuron via the mechanism of SR/ISR in the presence/absence of external noise, respectively. In this way, the performance of neuronal network motifs for weak signal transmission, in particular in a weak noisy environment, is significantly improved. Although the phenomena of SR in neuronal systems have been extensively studied in the literature, the exact sources of internal noise in the neuronal systems still remain as an open question. Current evidence for neuron exploiting SR is mostly obtained indirectly with a non-natural setting as both signal and noise were applied externally to sensory receptors and neurons. In fact, it is more natural that noise generated internally instead of externally in the real nervous systems (Levin and Miller, 1996). Based on our simulation results, the internally mutual neuron–astrocyte coupling plays a similar role as noise in neuronal signal processing of network motifs. Therefore, it can be considered as a possible source of internal noise to some extent. Some possible future works are suggested here. Firstly, in this paper, the neurons and astrocytes are modeled by the HH model and Li–Rinzel model, respectively. It may be a future work to perform similar study on different neuron and astrocyte models. Secondly, it is noted that when the neuron fires an action potential, another neurotransmitter, adenosine triphosphate (ATP) can also be released. Recent studies have shown that ATP can regulate neuronal activities but with a very complex mechanism, which may cause depolarization or hyperpolarization of nearby neurons (Smith, 2010; Halassaa et al., 2009). So, a further improvement can be performed by taking the dynamics of ATP and the corresponding ATP-mediated neuron–astrocyte interaction into account, provided that the more precise knowledge of ATP releasing mechanism and its functional roles is available. Thirdly, a possible extension of the current work is to study the stochastic behaviors of the network motifs with different configurations. In real neuronal networks, there are several different motifs with

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specific functions in signal processing. For example, the mixedfeedforward-feedback-loop (MFFL) network motifs, where twoneuron feedbacks that regulate or are regulated by a third neuron, have been shown to take active role in long-term and short-term memories (Li, 2008). In order to make a better understanding of the whole network dynamics, it is necessary to uncover the dynamic behaviors of these complex network motifs in a twodimensional astrocyte field.

Acknowledgment This work is supposed by National Natural Science Foundation of China (Grant nos. 61171153, 61101045), the Foundation for the Author of National Excellent Doctoral Dissertation of China under (Grant no. 2007B42), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, and the Zhejiang Provincial Natural Science Foundation of China (Grant no. LR12F01001).

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