動(dòng)態(tài)突觸、神經(jīng)耦合與時(shí)間延遲對(duì)神經(jīng)元發(fā)放的影響?

于文婷 張娟 唐軍

(中國(guó)礦業(yè)大學(xué)物理學(xué)院,徐州 221116)

動(dòng)態(tài)突觸、神經(jīng)耦合與時(shí)間延遲對(duì)神經(jīng)元發(fā)放的影響?

于文婷 張娟 唐軍?

(中國(guó)礦業(yè)大學(xué)物理學(xué)院,徐州 221116)

神經(jīng)元,突觸刺激,傅里葉變換,延遲

神經(jīng)元膜電位的受激發(fā)放在神經(jīng)系統(tǒng)的信息傳遞中起著重要作用.基于一個(gè)受動(dòng)態(tài)突觸刺激的突觸后神經(jīng)元發(fā)放模型,采用數(shù)值模擬和傅里葉變換分析的方法研究了動(dòng)態(tài)突觸、神經(jīng)耦合與時(shí)間延遲對(duì)突觸后神經(jīng)元發(fā)放的影響.結(jié)果發(fā)現(xiàn):突觸前神經(jīng)元發(fā)放頻率與Hodgkin-Huxley神經(jīng)元的固有頻率發(fā)生共振決定了突觸后神經(jīng)元發(fā)放的難易,特定頻率范圍內(nèi)的電流刺激有利于神經(jīng)元激發(fā),動(dòng)態(tài)突觸輸出的隨機(jī)突觸電流中這些電流刺激所占的比率在很大程度上影響了突觸后神經(jīng)元的發(fā)放次數(shù);將突觸后神經(jīng)元換成神經(jīng)網(wǎng)絡(luò)后,網(wǎng)絡(luò)中神經(jīng)元之間的耦合可以促進(jìn)神經(jīng)元的發(fā)放,耦合中的時(shí)間延遲可以增強(qiáng)這種促進(jìn)作用,但是不會(huì)改變神經(jīng)耦合對(duì)神經(jīng)元發(fā)放的促進(jìn)模式.

1 引 言

處理和傳遞信息的神經(jīng)系統(tǒng)由大量的神經(jīng)元個(gè)體構(gòu)成,神經(jīng)信號(hào)是編碼在神經(jīng)元的動(dòng)作電位之中的[1,2].近年來(lái),人們研究了不同模式的神經(jīng)放電現(xiàn)象,Pankratova等[3]研究了Hodgkin-Huxley神經(jīng)元在不同正弦電流信號(hào)刺激下的動(dòng)作電位的出現(xiàn)情況和頻率變化,Levin和Miller[4]研究了閾下頻率電流刺激對(duì)神經(jīng)元的動(dòng)作電位的影響.與其他生物系統(tǒng)一樣[5?9],神經(jīng)系統(tǒng)中也存在不同的噪聲和延遲[10?14],它們?cè)诤艽蟪潭壬嫌绊懥松窠?jīng)元的放電現(xiàn)象.隨機(jī)電流刺激對(duì)神經(jīng)元?jiǎng)幼麟娢坏挠绊懯艿搅藦V泛的關(guān)注[15,16],文獻(xiàn)[17—22]指出神經(jīng)元的周期發(fā)放可以弱化一些隨機(jī)噪聲的影響,Tang等[23]提出延遲可以促進(jìn)神經(jīng)網(wǎng)絡(luò)的同步發(fā)放,我們以前的工作表明[24]異質(zhì)延遲能夠在很大程度上破壞神經(jīng)元發(fā)放的同步性.

在神經(jīng)系統(tǒng)中,神經(jīng)信息是通過(guò)突觸在神經(jīng)元之間傳遞的[25].在神經(jīng)信息的傳遞中,突觸具有很強(qiáng)的變異性[26],化學(xué)濃度的變化與空間異質(zhì)性都能導(dǎo)致突觸的電導(dǎo)在短時(shí)間內(nèi)波動(dòng),形成動(dòng)態(tài)突觸.在大腦中,突觸傳遞神經(jīng)信息的過(guò)程中會(huì)出現(xiàn)短時(shí)抑制(short-term depression,STD)機(jī)制[27],即如果一個(gè)突觸前神經(jīng)元的動(dòng)作電位達(dá)到高頻,那么伴隨動(dòng)作電位的神經(jīng)遞質(zhì)釋放將減少,從而使得突觸后神經(jīng)元的反應(yīng)受到抑制.這種機(jī)制影響了大腦許多功能[28]的實(shí)現(xiàn),例如皮質(zhì)的增益控制[29]、神經(jīng)網(wǎng)絡(luò)的信息存儲(chǔ)[30]、同步和選擇性注意[31,32].

Uzuntarla等[33]建立了一個(gè)受閾值下周期電流和動(dòng)態(tài)突觸電流共同刺激的HH神經(jīng)元模型,并且研究了動(dòng)態(tài)突觸對(duì)HH神經(jīng)元首次發(fā)放潛伏期的影響.在此模型的基礎(chǔ)之上,我們建立了只受動(dòng)態(tài)突觸電流激發(fā)的HH神經(jīng)元發(fā)放模型,并將突觸后神經(jīng)元擴(kuò)展成一個(gè)包含100個(gè)神經(jīng)元的神經(jīng)網(wǎng)絡(luò),網(wǎng)絡(luò)中的神經(jīng)元均受到相同的動(dòng)態(tài)突觸刺激,還受到神經(jīng)網(wǎng)絡(luò)內(nèi)部突觸的耦合刺激[34?38].本文以此模型為基礎(chǔ)探討了動(dòng)態(tài)突觸、神經(jīng)網(wǎng)絡(luò)中的突觸耦合與時(shí)間延遲對(duì)HH神經(jīng)元發(fā)放的影響.

2 模 型

2.1 神經(jīng)元發(fā)放模型

圖1(a)描述了神經(jīng)元發(fā)放模型,模型中含有多個(gè)興奮性或抑制性的突觸前神經(jīng)元和一個(gè)突觸后神經(jīng)元,所有突觸前神經(jīng)元通過(guò)一個(gè)總和的動(dòng)態(tài)突觸作用于突觸后神經(jīng)元.在模型中,突觸后神經(jīng)膜電位的時(shí)間演化遵從HH神經(jīng)元模型[39],突觸后神經(jīng)的動(dòng)力學(xué)方程為

圖1 模型示意圖 (a)受動(dòng)態(tài)突觸刺激的神經(jīng)元模型;(b)受動(dòng)態(tài)突觸刺激和背景刺激的神經(jīng)網(wǎng)絡(luò)模型Fig.1.Schematic illustration of the neuronal model:(a)A single HH neuron model subject to an activity arriving through dynamic synapses;(b)a HH neuron network model subject to both a activity arriving through dynamic synapses and background activity.

其中突觸后神經(jīng)元的膜電位表示為Vm,單位面積的膜電容表示為Cm=1μF/cm2;鈉、鉀和滲漏通道的最高電導(dǎo)分別為GNa=120 mS/cm2,GK=36 mS/cm2,GL=0.3 mS/cm2,三者的逆轉(zhuǎn)電位分別為ENa=115 mV,EK=?12 mV,EL=10.6 mV.鈉離子通道的激活和失活與鉀離子通道的激活控制變量分別為m,h和n,它們遵從方程:

其中αγ和βγ(γ=m,n,h)為變量m,h和n的壓敏率函數(shù).文獻(xiàn)[3]中給出了αγ和βγ的具體形式,其形式由實(shí)驗(yàn)數(shù)據(jù)擬合得到.

模型中神經(jīng)的動(dòng)態(tài)突觸采取Tsodyks等[37]中提出的動(dòng)態(tài)突觸模型.最近的實(shí)驗(yàn)研究表明[40],抑制性突觸與興奮性突觸一樣,都能表現(xiàn)短時(shí)突觸的可塑性,因此動(dòng)態(tài)突觸模型也可以用于抑制性突觸[41,42].在動(dòng)態(tài)突觸模型中,動(dòng)作電位是通過(guò)神經(jīng)遞質(zhì)來(lái)傳遞的,即每個(gè)突觸前神經(jīng)元的動(dòng)作電位以特定的釋放概率(ui(t))去激活一部分神經(jīng)遞質(zhì),這種激活會(huì)以指數(shù)形式失活[37,43],經(jīng)過(guò)恢復(fù)期τrec神經(jīng)遞質(zhì)和突觸都回到初態(tài).這個(gè)過(guò)程表示為[37]

其中xi,yi,zi分別為神經(jīng)遞質(zhì)處于恢復(fù)、活躍和失活狀態(tài)的比例,神經(jīng)遞質(zhì)失活的特征時(shí)間常數(shù)τin=3 ms,突觸的恢復(fù)期τrec=100 ms.系統(tǒng)中每一個(gè)突觸前神經(jīng)元的動(dòng)作電位都表現(xiàn)為一個(gè)頻率f的泊松隨機(jī)發(fā)放序列,這些動(dòng)作電位出現(xiàn)時(shí)δ函數(shù)項(xiàng)計(jì)入方程.對(duì)于相互獨(dú)立的突觸前神經(jīng)元,它們的發(fā)放率可以相等也可以不等,本文考慮了各個(gè)突觸前神經(jīng)元的發(fā)放率相等和服從高斯分布兩種情況,以f表示相同的突觸前神經(jīng)元發(fā)放率,與CV表示突觸前神經(jīng)元發(fā)放率服從高斯分布時(shí)的平均值及變異系數(shù).另外,在大腦皮層神經(jīng)元的STD機(jī)制[37,43]下,動(dòng)作電位釋放神經(jīng)遞質(zhì)的概率為常數(shù),即ui(t)=U,本文中令U=0.67[43].

方程(1)中的Isyn為突觸電流,由1000個(gè)相互獨(dú)立的突觸前神經(jīng)元共同產(chǎn)生,具體形式可以表現(xiàn)為

其中,Ne為興奮性突觸前神經(jīng)元數(shù)量,Ni為抑制性突觸前神經(jīng)元數(shù)量,Ne:Ni=4:1,這與哺乳動(dòng)物大腦皮質(zhì)中興奮性神經(jīng)和抑制性神經(jīng)的比例相當(dāng)[27].K為抑制性神經(jīng)連接和興奮性神經(jīng)連接的相對(duì)強(qiáng)度,A為興奮性連接的最大突觸電流,本文中K=4,A=250 pA[43].

2.2 神經(jīng)網(wǎng)絡(luò)發(fā)放模型

為了討論神經(jīng)耦合對(duì)突觸后神經(jīng)元發(fā)放的影響,將單個(gè)突觸后神經(jīng)元換成包含100個(gè)單體的規(guī)則神經(jīng)網(wǎng)絡(luò),該神經(jīng)網(wǎng)絡(luò)模型如圖1(b)所示.圖中的100個(gè)突觸后神經(jīng)元均受到來(lái)自于1000個(gè)突觸前神經(jīng)元的動(dòng)態(tài)突觸刺激,此神經(jīng)網(wǎng)絡(luò)中神經(jīng)元的動(dòng)力學(xué)方程為

其中i為網(wǎng)絡(luò)中神經(jīng)元的編號(hào),Vi為第i個(gè)神經(jīng)元的膜電位;Isyn項(xiàng)為來(lái)自于1000個(gè)突觸前神經(jīng)元的動(dòng)態(tài)突觸電流,此項(xiàng)與2.1節(jié)中所介紹的相同,每個(gè)突觸后神經(jīng)元受到動(dòng)態(tài)突觸電流刺激均相同;Ii項(xiàng)是HH神經(jīng)元的一個(gè)分岔參數(shù),在其小于6時(shí),HH神經(jīng)元保持在靜息狀態(tài),不會(huì)出現(xiàn)周期性發(fā)放,為了保證突觸后神經(jīng)元之間具有不同的發(fā)放特性,并且使突觸后神經(jīng)元在沒(méi)受到任何突觸刺激時(shí)保持靜息狀態(tài),設(shè)Ii服從0—6之間的平均分布,且不隨時(shí)間改變;方程右側(cè)的最后一項(xiàng)表示神經(jīng)網(wǎng)絡(luò)中突觸后神經(jīng)元之間的突觸耦合項(xiàng),g為神經(jīng)間的耦合強(qiáng)度,τ為突觸后神經(jīng)元之間信息傳輸?shù)难舆t時(shí)間.本文中所有的數(shù)值積分采用四階龍格-庫(kù)塔法,時(shí)間步長(zhǎng)為10μs.

3 結(jié)果與討論

3.1 動(dòng)態(tài)突觸對(duì)突觸后神經(jīng)元發(fā)放的影響

文獻(xiàn)[3]研究表明,對(duì)HH神經(jīng)元輸入不同的電流,HH神經(jīng)元會(huì)實(shí)現(xiàn)不同的發(fā)放狀態(tài).本節(jié)首先應(yīng)用2.1節(jié)中的模型來(lái)討論隨機(jī)突觸電流刺激對(duì)突觸后神經(jīng)元(HH神經(jīng)元)發(fā)放的影響效果.

圖2(a)中顯示了在突觸前神經(jīng)元發(fā)放率f=10 Hz時(shí),突觸后神經(jīng)元在一段時(shí)間內(nèi)的發(fā)放情況.圖中的黑線表示突觸后神經(jīng)元膜電位的變化,紅線表示動(dòng)態(tài)突觸電流的變化.由圖可見(jiàn),伴隨著隨機(jī)突觸電流的輸入,突觸后神經(jīng)元的發(fā)放也表現(xiàn)出了隨機(jī)性,其發(fā)放沒(méi)有明顯的周期性.統(tǒng)計(jì)不同f下突觸后神經(jīng)元在2000 ms內(nèi)的發(fā)放次數(shù),如圖2(b)所示.為了消除系統(tǒng)中隨機(jī)性對(duì)突觸后神經(jīng)元發(fā)放所帶來(lái)的影響,圖2(b)中的結(jié)果為100次計(jì)算的系綜平均.如圖2(b)中所示,突觸后神經(jīng)元發(fā)放次數(shù)的最高峰出現(xiàn)在f=50 Hz處,f在30—100 Hz之間時(shí),突觸后神經(jīng)元發(fā)放次數(shù)較多.

圖2 (網(wǎng)刊彩色)突觸后神經(jīng)元發(fā)放 (a)突觸后神經(jīng)元隨時(shí)間的發(fā)放(f=10 Hz);(b)突觸后神經(jīng)元2000 ms內(nèi)發(fā)放次數(shù)隨f的變化Fig.2. (color online)Firing of postsynaptic neuron:(a)Firing of the postsynaptic neurons over time(f=10 Hz);(b) firing number of postsynaptic neurons in 2000 ms for different f.

為了更清楚地分析突觸電流中的頻率成分,對(duì)帶有隨機(jī)成分的動(dòng)態(tài)突觸電流的時(shí)間序列做傅里葉分析.圖3(a)—(c)分別為f=30,50與100 Hz時(shí),系統(tǒng)動(dòng)態(tài)突觸電流的傅里葉分析結(jié)果,由圖可見(jiàn)突觸電流的頻率成分主要集中在相應(yīng)突觸前神經(jīng)元發(fā)放率的整數(shù)倍上,且在突觸前神經(jīng)元發(fā)放率的一倍或兩倍時(shí)出現(xiàn)電流強(qiáng)度最大的頻率,這些最大的電流強(qiáng)度大部分都在1.6—2.0之間.文獻(xiàn)[3]中介紹了HH神經(jīng)元在受到不同振幅和頻率的正弦電流刺激下的周期發(fā)放情況,發(fā)現(xiàn)電流強(qiáng)度較小時(shí),頻率在50 Hz周?chē)碾娏鲿?huì)使HH神經(jīng)元呈現(xiàn)一定頻率的周期性發(fā)放.電流強(qiáng)度在1.6—2.0之間時(shí),能使HH神經(jīng)元呈現(xiàn)一定頻率的周期性發(fā)放的電流輸入頻率范圍在30—90 Hz.這說(shuō)明在突觸前神經(jīng)元的發(fā)放率在30—90 Hz之間時(shí),動(dòng)態(tài)突觸電流中電流強(qiáng)度最大的單頻電流成分已經(jīng)在一定程度上使得突觸后神經(jīng)元進(jìn)行持續(xù)的振蕩,所以此條件下的突觸后神經(jīng)元發(fā)放次數(shù)較多.進(jìn)而將正弦電流輸入HH神經(jīng)元發(fā)放頻率與單倍和二倍突觸前神經(jīng)元發(fā)放率進(jìn)行了對(duì)比,如圖3(d).圖中所示的是頻率在30—90 Hz之間的對(duì)比結(jié)果,過(guò)高頻或過(guò)低頻的正弦電流輸入都不能使HH神經(jīng)元發(fā)放.圖中,正弦電流輸入HH神經(jīng)元發(fā)放頻率基本可以與單倍或二倍突觸前神經(jīng)元發(fā)放率重合.同時(shí),圖中頻率范圍與圖2(b)中顯示的突觸后神經(jīng)元發(fā)放次數(shù)較多的突觸前神經(jīng)元發(fā)放率范圍相符合.由此可以得到結(jié)論:突觸后神經(jīng)元發(fā)放次數(shù)較多,是由于輸入的突觸電流中一些頻率的電流刺激本身就可以使得突觸后神經(jīng)元持續(xù)發(fā)放,這些發(fā)放不會(huì)因其他頻率刺激電流的出現(xiàn)而被抑制.因而,動(dòng)態(tài)突觸電流刺激下,突觸后神經(jīng)元多次發(fā)放是由突觸前神經(jīng)元發(fā)放頻率與HH神經(jīng)元的固有頻率發(fā)生共振所產(chǎn)生的.

圖3 (網(wǎng)刊彩色)動(dòng)態(tài)突觸電流的傅里葉變換分析 (a)f=30 Hz,(b)50 Hz和(c)100 Hz時(shí),動(dòng)態(tài)突觸電流的傅里葉分解;(d)突觸前神經(jīng)元發(fā)放率與HH神經(jīng)元發(fā)放頻率的對(duì)比Fig.3.(color online)Flourier transform analysis of synaptic currents: flourier transform analysis of dynamic synaptic currents with f=30 Hz(a),50 Hz(b),100 Hz(c);(d)comparation between presynaptic neuron firing rate and HH neurons firing frequency.

進(jìn)一步討論突觸前神經(jīng)元發(fā)放率的異質(zhì)性對(duì)突觸后神經(jīng)元發(fā)放的影響.這里用2.1節(jié)中介紹的模型,讓突觸前神經(jīng)元發(fā)放率服從高斯分布,CV體現(xiàn)了突觸前神經(jīng)元發(fā)放率異質(zhì)性的大小.圖4(a)中顯示了=50,200 Hz時(shí)突觸后神經(jīng)元發(fā)放次數(shù)隨CV的變化情況.發(fā)現(xiàn)=50 Hz時(shí)發(fā)放次數(shù)隨CV的增加明顯減少;=200 Hz時(shí)發(fā)放次數(shù)隨CV的增加而增加.圖4(b)中顯示了在10—200 Hz間突觸后神經(jīng)元發(fā)放次數(shù)的變化情況,可見(jiàn)在峰值頻率周?chē)纳窠?jīng)元的發(fā)放次數(shù)隨著CV的增加有所減小;較大(大于150 Hz)的神經(jīng)元發(fā)放次數(shù)隨著CV的增加有所增加,這個(gè)趨勢(shì)與圖4(a)中表現(xiàn)出來(lái)的現(xiàn)象相同.這表明突觸前神經(jīng)元發(fā)放率的異質(zhì)性抑制了峰值頻率周?chē)纳窠?jīng)元的發(fā)放,促進(jìn)了突觸前神經(jīng)元發(fā)放率值較大的神經(jīng)元發(fā)放.為了解釋這一現(xiàn)象,仍以為50和200 Hz為例,將不同CV條件下系統(tǒng)的動(dòng)態(tài)突觸電流進(jìn)行了傅里葉變換分析,如圖4(c)和圖4(d).圖中動(dòng)態(tài)突觸電流的頻率成分依然主要集中在相應(yīng)的整數(shù)倍上,但隨著CV的增加其他頻率成分出現(xiàn)的幅度逐漸增大.兩幅圖中的插圖的頻率范圍為30—90 Hz,正是文獻(xiàn)[3]中給出的易使HH神經(jīng)元進(jìn)行周期性發(fā)放的頻率范圍.如插圖中所示,=50 Hz時(shí)隨著CV的增加,傅里葉分解出的電流強(qiáng)度有所減小,最終小于1.6,即可以以單個(gè)頻率電流輸入誘導(dǎo)突觸后神經(jīng)元發(fā)放的電流強(qiáng)度范圍,根據(jù)圖3中得到的結(jié)論,突觸后神經(jīng)元發(fā)放的次數(shù)應(yīng)有所減少,這與圖4(a)和圖4(b)中的現(xiàn)象相同;=200 Hz時(shí),傅里葉分解出的電流強(qiáng)度有所增加,特別是CV增加到1時(shí),各個(gè)頻率對(duì)應(yīng)的電流強(qiáng)度增加的非常明顯,圖4(a)和圖4(b)中顯示此時(shí)突觸后神經(jīng)元發(fā)放的次數(shù)有所增多,這說(shuō)明雖然沒(méi)有任何頻率的電流輸入可以獨(dú)立誘導(dǎo)突觸后神經(jīng)元的周期發(fā)放,但是該頻率范圍內(nèi)的電流強(qiáng)度增加還是能夠促進(jìn)突觸后神經(jīng)元的發(fā)放.這說(shuō)明突觸電流中電流強(qiáng)度隨頻率的分布與突觸后神經(jīng)元發(fā)放的次數(shù)有著直接的關(guān)系.對(duì)于頻率范圍在30—90 Hz的突觸電流,其電流強(qiáng)度的變化在很大程度上影響著突觸后神經(jīng)元的發(fā)放.突觸前神經(jīng)元發(fā)放率的異質(zhì)性就是通過(guò)改變這些頻率電流的振幅來(lái)影響突觸后神經(jīng)元的發(fā)放次數(shù).

綜上所述,對(duì)動(dòng)態(tài)突觸刺激下的突觸后神經(jīng)元,突觸前神經(jīng)元發(fā)放頻率與HH神經(jīng)元的固有頻率發(fā)生共振使其多次發(fā)放,動(dòng)態(tài)突觸電流中包含頻率在30—90 Hz之間的電流振幅在很大程度上決定了突觸后神經(jīng)元的發(fā)放情況.

圖4 (網(wǎng)刊彩色)高斯突觸前神經(jīng)元發(fā)放率下的神經(jīng)元發(fā)放(a)不同ˉf下,突觸后神經(jīng)元2000 ms內(nèi)發(fā)放次數(shù)CV的變化;(b)不同CV下,突觸后神經(jīng)元2000 ms內(nèi)發(fā)放次數(shù)隨ˉf的變化;(c)ˉf=50 Hz與(d)200 Hz時(shí),動(dòng)態(tài)突觸電流的傅里葉分解Fig.4.(color online)The firing of postsynaptic neuron under Gaussian firing rate:(a)Firing number of postsynaptic neurons in 2000 ms for different CV under differentˉf;(b) firing number of postsynaptic neurons in 2000 ms for different firing rateˉf under different CV; flourier transform analysis of synaptic currents withˉf=50 Hz(c)and 200 Hz(d).

3.2 神經(jīng)突觸耦合與時(shí)間延遲對(duì)突觸后神經(jīng)元發(fā)放的影響

在2.2節(jié)中,我們將突觸后神經(jīng)元置換成包含100個(gè)神經(jīng)元的耦合神經(jīng)網(wǎng)絡(luò),基于該模擬,下面討論神經(jīng)網(wǎng)絡(luò)中的突觸耦合與時(shí)間延遲對(duì)突觸后神經(jīng)網(wǎng)絡(luò)發(fā)放的影響.這里,主要探究耦合強(qiáng)度g與延遲量τ對(duì)不同突觸前神經(jīng)元發(fā)放率刺激下的突觸后神經(jīng)元發(fā)放次數(shù)的影響.此處計(jì)算神經(jīng)網(wǎng)絡(luò)每個(gè)神經(jīng)元在2000 ms內(nèi)的發(fā)放次數(shù),因此發(fā)放次數(shù)實(shí)際體現(xiàn)了神經(jīng)元的發(fā)放頻率.為了減小隨機(jī)性對(duì)神經(jīng)元發(fā)放次數(shù)的影響,對(duì)每一個(gè)神經(jīng)元的發(fā)放次數(shù)取100次系綜平均,并將網(wǎng)絡(luò)中的100個(gè)突觸后神經(jīng)元的發(fā)放次數(shù)取均值,得到突觸后神經(jīng)元平均發(fā)放次數(shù).

首先探討神經(jīng)突觸耦合對(duì)突觸后神經(jīng)元發(fā)放的影響,此時(shí)系統(tǒng)無(wú)延遲.圖5(a)和圖5(b)分別為f=10,200 Hz的隨機(jī)電流刺激下神經(jīng)網(wǎng)絡(luò)的發(fā)放模式圖,其中上側(cè)的兩幅g=0,也就是突觸后神經(jīng)元之間沒(méi)有突觸耦合;下側(cè)的兩幅g=0.08,網(wǎng)絡(luò)之中的突觸后神經(jīng)元存在一定的突觸耦合.從圖中明顯可見(jiàn),下側(cè)的兩幅圖中突觸后神經(jīng)元發(fā)放的次數(shù)比上側(cè)的兩幅圖多,這可能意味著在f=10,200 Hz時(shí)的隨機(jī)電流刺激下,神經(jīng)的突觸耦合會(huì)在一定程度上促進(jìn)突觸后神經(jīng)元發(fā)放.突觸后神經(jīng)元的平均發(fā)放次數(shù)隨突觸前神經(jīng)元發(fā)放率的變化如圖5(c),這里發(fā)放率取10—200 Hz之間,基本包含了所有突觸后神經(jīng)元出現(xiàn)發(fā)放的突觸前神經(jīng)元發(fā)放率.圖中紅線和藍(lán)線(g=0.08,0.20)都在黑線(g=0)上方,且紅線(g=0.08)基本上是在藍(lán)線(g=0.20)上方,這表明神經(jīng)網(wǎng)絡(luò)中的突觸耦合會(huì)在一定程度上促進(jìn)突觸后神經(jīng)元的發(fā)放,并且g=0.08是一個(gè)促進(jìn)效果比較好的耦合強(qiáng)度.圖5(d)給出了突觸后神經(jīng)元平均發(fā)放次數(shù)隨耦合強(qiáng)度的變化情況.圖中顯示,g小于0.08時(shí),突觸后神經(jīng)元平均發(fā)放次數(shù)隨耦合強(qiáng)度的增加而明顯增加;g大于0.08后,突觸后神經(jīng)元平均發(fā)放次數(shù)略有波動(dòng),但始終保持在一個(gè)比較大的數(shù)目上.綜上所述,神經(jīng)網(wǎng)絡(luò)系統(tǒng)中的突觸耦合對(duì)系統(tǒng)中突觸后神經(jīng)元的發(fā)放具有促進(jìn)作用.

圖5 (網(wǎng)刊彩色)神經(jīng)網(wǎng)絡(luò)中的神經(jīng)元發(fā)放 耦合強(qiáng)度g=0(a),(b);0.08(c),(d);f=10 Hz(a),200 Hz(b);(c)突觸后神經(jīng)元平均發(fā)放次數(shù)隨f的變化;(d)突觸后神經(jīng)元平均發(fā)放次數(shù)隨耦合強(qiáng)度的變化Fig.5.(color online)The firing pattern in neuron network with coupling strength g=0(a),(b);0.08(c),(d);f=10 Hz(a)and 200 Hz(b);(c) firing number of postsynaptic neurons for different f;(d) firing number of postsynaptic neurons for different coupling strength.

進(jìn)一步討論時(shí)間延遲對(duì)突觸后神經(jīng)元發(fā)放的影響,此時(shí)取神經(jīng)耦合強(qiáng)度為0.08,突觸后神經(jīng)元在此耦合強(qiáng)度下發(fā)放的次數(shù)較多.圖6(a)以延遲量τ為10 ms、突觸前神經(jīng)元發(fā)放率f為10,200 Hz時(shí)的神經(jīng)發(fā)放模式圖,對(duì)比圖5(a)和圖5(b)下側(cè)的兩幅圖可以清楚地發(fā)現(xiàn),τ為10 ms時(shí)突觸后神經(jīng)元的發(fā)放次數(shù)比無(wú)延遲時(shí)的發(fā)放次數(shù)多,這似乎說(shuō)明延遲可以促進(jìn)突觸后神經(jīng)元的發(fā)放.為了進(jìn)一步說(shuō)明這一現(xiàn)象,統(tǒng)計(jì)了τ在1—1000 ms之間時(shí),突觸后神經(jīng)元的發(fā)放次數(shù),如圖6(b)所示.圖中的三條線(f=10,50,200 Hz)上的點(diǎn)均大于延遲較小時(shí)突觸后神經(jīng)元發(fā)放次數(shù),這說(shuō)明在比較廣泛的延遲范圍內(nèi),延遲對(duì)突觸后神經(jīng)元發(fā)放具有促進(jìn)作用.并且,圖中三條線的峰都在τ=10 ms處出現(xiàn),說(shuō)明此處延遲對(duì)突觸后神經(jīng)元發(fā)放的促進(jìn)作用最強(qiáng).對(duì)比τ為0 ms和10 ms時(shí)突觸后神經(jīng)元發(fā)放個(gè)數(shù)隨耦合強(qiáng)度g的變化,圖6(c)和圖6(d)分別為f=10與200 Hz的情況,可見(jiàn)兩幅圖中的四條線的變化模式相同,均經(jīng)歷了g小于0.03時(shí)發(fā)放次數(shù)較少、g在0.03—0.08之間的發(fā)放次數(shù)快速增長(zhǎng)和g大于0.08時(shí)發(fā)放次數(shù)保持在較高的值.這說(shuō)明時(shí)間延遲促進(jìn)了突觸后神經(jīng)元的發(fā)放,卻不改變耦合強(qiáng)度對(duì)突觸后神經(jīng)元發(fā)放的促進(jìn)模式.

圖6 (網(wǎng)刊彩色)延遲神經(jīng)網(wǎng)絡(luò)中的發(fā)放 (a)τ=10 s時(shí),突觸后神經(jīng)發(fā)放圖[f=10 Hz(上),f=200 Hz(下)];(b)突觸后神經(jīng)平均發(fā)放次數(shù)隨延遲量τ的變化;(c)f=10 Hz,(d)200 Hz時(shí),突觸后神經(jīng)平均發(fā)放次數(shù)隨耦合強(qiáng)度的變化Fig.6.(color online)The firing in delayed neuron network:(a)Firingpattern of postsynaptic neuronswith τ=10 s[f=10 Hz(top),f=200 Hz(bottom)];(b) firing number of postsynaptic neurons for different delay τ;(c)f=10 Hz and(d)200 Hz firing number of postsynaptic neurons for different firing rate.

綜上所述,在具有動(dòng)態(tài)突觸的神經(jīng)網(wǎng)絡(luò)發(fā)放模型中,神經(jīng)突觸耦合可以促進(jìn)突觸后神經(jīng)元的發(fā)放,突觸耦合中的時(shí)間延遲可以加強(qiáng)這種促進(jìn)效果,但是不會(huì)改變神經(jīng)突觸耦合對(duì)突觸后神經(jīng)元發(fā)放的促進(jìn)模式.

4 結(jié) 論

基于一個(gè)動(dòng)態(tài)突觸電流刺激下的神經(jīng)模型,研究了影響突觸后神經(jīng)元(HH神經(jīng)元)發(fā)放的因素.通過(guò)傅里葉變換分析法和系綜平均法,分別分析了隨機(jī)突觸電流的頻率——振幅分布和隨機(jī)系統(tǒng)中突觸后神經(jīng)元的發(fā)放次數(shù)的變化.研究表明:突觸前神經(jīng)元發(fā)放頻率與HH神經(jīng)元的固有頻率發(fā)生共振決定了突觸后神經(jīng)元的發(fā)放率,動(dòng)態(tài)突觸電流中頻率為30—90 Hz的電流強(qiáng)度在很大程度上影響了突觸后神經(jīng)元的發(fā)放頻率;神經(jīng)之間的突觸耦合可以促進(jìn)突觸后神經(jīng)元的發(fā)放,突觸耦合中的時(shí)間延遲可以強(qiáng)化這種促進(jìn)作用,但是不會(huì)改變神經(jīng)突觸耦合對(duì)突觸后神經(jīng)元發(fā)放的促進(jìn)模式.

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E ff ects of dynamic synapses,neuronal coupling,and time delay on firing of neuron?

Yu Wen-Ting Zhang Juan Tang Jun?

(School of Physics,China University of Mining and Technology,Xuzhou 221116,China)

15 May 2017;revised manuscript

29 June 2017)

Neuronal firing plays a key role in the neuronal information transmission,and different neuronal firing patterns are reported,such as spiking,bursting.A number of neuron models are introduced to reproduce the firing patterns of single neuron or neuronal network.The key factors determining the firing pattern gain more and more attention in the study of neuron system,such as noise,network topology.Noise is able to induce sub-or super-threshold coherent neuronal firing easily,and a number of coherence resonances are reported in the noise induced firing.The network topology determines the synchronization of the firing patterns of the neuronal network,and the change of network topology may induce fruitful synchronization transitions.It is well known that synapses exhibit a high variability with a diverse origin during information transmission,such as the stochastic release of neurotransmitters,variations in chemical concentration through synapses,and spatial heterogeneity of synaptic response over dendrite tree.The collective e ff ect of all of these factors might result in the notion of dynamic synapses.In reality,the neuronal network often involves time delay due to the?nite signal propagation time in biological networks.Recently,neuronal networks with time delay have received considerable attention.Delay-sustained neuronal firing patterns may be relevant to neuronal networks for establishing a concept of collective information processing in the presence of delayed information transmission.According to the above-mentioned motivations,the firing dynamics of the single postsynapic neuron is investigated based on a simple postsynaptic neuron model by using numerical simulation and Fourier transform analysis.In this model,the postsynapic neuron receives dynamic synaptic currents from a population of presynaptic neurons.It is found that the firing rate resonance between the pre-and postsynaptic neuron determines the firing of the postsynaptic neuron.Stimulus currents in speci fi c frequency range are easy to stimulate postsynaptic neuron firing.The random currents released from dynamic synapses determine the postsynaptic firing rate.Then the single postsynaptic neuron is extended to a neuronal network,in which 100 neurons connect to its 4 nearest neighbors regularly and receive delayed synaptic currents from connected neurons.All the neurons in the network receive the same dynamic synaptic currents from the presynaptic neurons.The results show that the synaptic coupling in the network is able to promote the neuron firing in the network,and time delay in the synaptic coupling could reinforce the promotion,but the mode of the promotion is not changed.

neuron,synaptic stimulation,Flourier transform,time delay

(2017年5月15日收到;2017年6月29日收到修改稿)

10.7498/aps.66.200201

?中央高?；究蒲袠I(yè)務(wù)費(fèi)(批準(zhǔn)號(hào):2015XKMS080(TJ))資助的課題.

?通信作者.E-mail:tjuns1979@126.com

?2017中國(guó)物理學(xué)會(huì)Chinese Physical Society

http://wulixb.iphy.ac.cn

PACS:02.30.Nw,05.45.–a,05.40.–aDOI:10.7498/aps.66.200201

*Project supported by the Fundamental Research Funds for the Central Universities,China(Grant No.2015XKMS080(TJ)).

?Corresponding author.E-mail:tjuns1979@126.com