Shunting Inhibition Controls the Gain Modulation Mediated by Asynchronous Neurotransmitter Release in Early Development

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1 Shunting Inhibition Contols the Gain Modulation Mediated by Asynchonous Neuotansmitte Release in Ealy Development Vladislav Volman 1,2,3 *, Hebet Levine 1, Teence J. Sejnowski 1,2,3,4 1 Cente fo Theoetical Biological Physics, Univesity of Califonia at San Diego, La Jolla, Califonia, United States of Ameica, 2 Computational Neuobiology Laboatoy, The Salk Institute fo Biological Studies, La Jolla, Califonia, United States of Ameica, 3 Howad Hughes Medical Institute, The Salk Institute fo Biological Studies, La Jolla, Califonia, United States of Ameica, 4 Division of Biological Sciences, Univesity of Califonia at San Diego, La Jolla, Califonia, United States of Ameica Abstact The sensitivity of a neuon to its input can be modulated in seveal ways. Changes in the slope of the neuonal input-output cuve depend on factos such as shunting inhibition, backgound noise, fequency-dependent synaptic excitation, and balanced excitation and inhibition. Howeve, in ealy development GABAegic inteneuons ae excitatoy and othe mechanisms such as asynchonous tansmitte elease might contibute to egulating neuonal sensitivity. We modeled both phasic and asynchonous synaptic tansmission in ealy development to study the impact of activity-dependent noise and shot-tem plasticity on the synaptic gain. Asynchonous elease deceased o inceased the gain depending on the membane conductance. In the high shunt egime, excitatoy input due to asynchonous elease was divisive, wheeas in the low shunt egime it had a nealy multiplicative effect on the fiing ate. In addition, sensitivity to coelated inputs was influenced by shunting and asynchonous elease in opposite ways. Thus, asynchonous elease can egulate the infomation flow at synapses and its impact can be flexibly modulated by the membane conductance. Citation: Volman V, Levine H, Sejnowski TJ (2010) Shunting Inhibition Contols the Gain Modulation Mediated by Asynchonous Neuotansmitte Release in Ealy Development. PLoS Comput Biol 6(11): e doi: /jounal.pcbi Edito: Lyle J. Gaham, Univesité Pais Descates, Cente National de la Recheche Scientifique, Fance Received July 20, 2010; Accepted Septembe 24, 2010; Published Novembe 4, 2010 Copyight: ß 2010 Volman et al. This is an open-access aticle distibuted unde the tems of the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal autho and souce ae cedited. Funding: This eseach was suppoted by the Howad Hughes Medical Institute, the NSF Cente fo Theoetical Biological Physics (PHY ) and NIH gant MH The fundes had no ole in study design, data collection and analysis, decision to publish, o pepaation of the manuscipt. Competing Inteests: The authos have declaed that no competing inteests exist. * volman@salk.edu Intoduction Gain contol of synapses egulates the flow of infomation though neual cicuits [1]. The neuophysiological mechanisms that modulate neuonal tansfe functions have been the focus of seveal studies [2 6]. In paticula, backgound noise and shunting inhibition have a divisive effect on the gain cuve of a neuon [4,5]. In contast, pesynaptic shot-tem depession extends the dynamic ange ove which a neuon can disciminate between diffeent levels of affeent activity, and theefoe epesents anothe way to contol the neuonal output [2,7]. Anothe mechanism of gain modulation that has been extensively studied elies on balanced inputs [3,8]. When the affeent excitation and inhibition ae inceased while maintaining a constant aveage membane potential, the incease in the amplitude of the fluctuations leads to an incease in neuonal fiing ate. Howeve, contol of the neuonal fiing ate by modulating the balance between excitation and inhibition equies stict coodination between activities of upsteam excitatoy and inhibitoy neuons, a condition that may be difficult to maintain [9]. This is especially tue in the ealy developmental stages befoe the achitectue of the netwok has been established. Thus, thee may be altenative ways to achieve neuonal gain modulation. Pevious studies have assumed that shot-tem synaptic plasticity and noise act independently of one anothe to affect the input-output cuve of a neuon. This assumption holds fo the phasic elease of neuotansmitte fom a pesynaptic teminal that occus shotly afte the aival of an action potential. Although most of the fast communication between neuons is though fast phasic elease, many cental synapses have an additional, asynchonous component of tansmitte elease in esponse to stimulation [10 16]. Asynchonous elease lasts fo seveal hunded milliseconds following a synaptic stimulus, and is believed to occu due to the build-up of esidual pesynaptic calcium that enhances the pobability of vesicula elease [10,17]. The extent of asynchonous elease at a given time eflects the histoy of synaptic stimulation ove hundeds of milliseconds. Asynchonous elease is often consideed a fom of noise but should instead be seen as a fom of slow, activity-dependent synaptic signaling. Seveal studies have suggested diffeent oles fo asynchonous elease, including modulation of synaptic tansmission [11,12,18], extension of the postsynaptic spike window [15] and the contol of ecuent netwok dynamics [14,19]. Asynchonous elease may be dominant duing the ealy stages of synaptic development, thus making it a pime candidate fo gain modulation in the absence of tightly balanced excitation and inhibition. We used a computational modeling appoach to investigate the possible effects of asynchonous elease on the modulation of synaptic gain, with specific focus on the ealy developmental peiod (at post-natal days 20 30) befoe affeent inhibition is fully established [20]. We show hee that the impact of asynchonous PLoS Computational Biology 1 Novembe 2010 Volume 6 Issue 11 e

2 Autho Summay Computation in a single neuon is egulated by gain contol the change in the sensitivity of a neuon to diffeent pattens of stimulation. Hence, it is impotant to undestand the diffeent mechanisms undelying gain modulation. Ealie wok focused on gain modulation in matue netwoks with functional connectivity; howeve, the mechanisms of gain contol in the developing bain ae still unclea. We show hee that asynchonous elease of neuotansmitte, a fom of synaptic noise that is stongly expessed in synapses of the developing bain, and shot-tem synaptic plasticity can efficiently modulate the gain of the synapse without a pioi assumptions about netwok s connectivity. Asynchonous elease depends on the activity-dependent accumulation of synaptic calcium. We show that changes in asynchonous elease can have eithe a divisive o multiplicative effect on the gain, depending on the state of the neuonal membane conductance. Thus, activity-dependent synaptic noise can egulate the infomation flow at synapses, and its impact on the neuon is flexibly modulated by neuonal membane conductance. elease citically depends on the level of shunting inhibition. In the low shunt egime, neuonal esponse is detemined by the mean level of synaptic cuent, and a change in the level of asynchonous elease leads to an appoximately multiplicative modulation of the fiing ate. In a high membane conductance egime, asynchonous elease acts to divisively modulate synaptic gain. Thus, shunting inhibition can vay the influence of asynchonous elease ove a wide ange, fom amplification to eduction. Coelated stimuli wee also diffeentially pocessed by shunting inhibition and asynchonous elease, theefoe undescoing the potential impotance of asynchonous elease in netwok dynamics. Results Synchonized neuonal fiing can explain measued postsynaptic esponse Ou goal was to undestand the ole of asynchonous elease in modulating the gain of neuons. To accomplish this, we constucted a computational model that could captue some of the most salient featues pesent in expeimental data fom hippocampal cell cultue [12]. An example of this type of data is pesented in Figue 1A. Although it is expected that at most one vesicle pe stimulus was eleased fom each hippocampal synapse in cultue [21,22], nonetheless the esponse exhibited analog behavio in that thee was both depession in the phasic elease component and a small but clealy detectable asynchonous elease. Fom this, we concluded that vesicles wee eleased synchonously fom many sites immediately following the action potential and incopoated synchonous synaptic activation in ou model, as descibed below. To show how synchonization can lead to this type of expeimental data, we fist built a computational model of a synaptic teminal which featued a small numbe of vesicles that could be stochastically eleased eithe as a phasic esponse to synaptic stimulation o in an asynchonous manne (Methods). Rhythmic and coodinated stimulation of model synapses (Figue 1B) evealed the onset of synaptic depession, in qualitative ageement with the afoementioned expeimental esults fom hippocampal cultues (Figue 1A). Cucially, the data could be explained with an extemely small (1%) level of assumed synchonization of elease at diffeent synapses. When asynchonous elease was added to the model synapse, the esponse was futhe educed (compaed to the case without asynchonous elease), because, accoding to the Equations in Methods, both evoked and asynchonous eleases wee dawn fom the same pool of esouce [23]. Thus, by vitue of its competition with phasic elease, the asynchonous elease enhanced the effects of synaptic shot-tem depession [16,24]. Since long-tem simulation studies of ou detailed synaptic model became computationally demanding fo lage (thousands) numbes of affeents, we developed a educed and computationally much moe efficient model of synaptic tansmission that still incopoated both phasic and asynchonous components of elease (Methods, also schematic in Figue 1C). This eduction is possible due to the fact that input to a specific neuon is equal to be a sum ove many (,30) jointly activated synapses. This type of model was oiginally intoduced in an ealie study of the emegence and sustenance of hythmic evebeations [19], which included the details of how this model was matched to expeimental findings. The esponse of this educed model to hythmic synaptic stimulation was qualitatively simila to the esponse of the moe detailed model (Figue 1D). The educed model was theeafte used to investigate the impact of asynchonous elease and shunting conductance on neuonal gain. We have also checked that the main conclusions obtained with the educed model (namely, the dual shunting-dependent effect of AR on inputoutput cuve) can also be epoduced with the moe detailed synaptic model (Supplementay Figue S1). Asynchonous tansmitte elease modulates the time window fo spike geneation Asynchonous elease can modulate neuonal spiking in a time window that is dependent on the pio activity of synaptic affeents. Figue 1E illustates how asynchonous elease due to ealie synaptic activity can condition neuonal sensitivity to late stimuli. A conditioning set of 30 model synapses was synchonously and hythmically stimulated by 5 pulses at 20 Hz. At time DT following the last peconditioning pulse, a test pulse was synchonously deliveed to two additional (fully elaxed) model synapses. In the pesence of asynchonous elease due to an ealie, conditioning, stimulation, the test stimulus geneated a somatic spike; blockade of the asynchonous elease eliminated the spike (Figue 1E, bottom panel). As shown in Figue 1F, the pobability of geneating a somatic spike in esponse to a weak test stimulus depended both on the time DT since the conditioning and on the ealie spike patten (chaacteized hee by the fequency and numbe of spikes). The extent of asynchonous elease at model synapses is detemined by the availability of synaptic esouce, which is educed, and by the level of esidual calcium, which builds up in the couse of polonged synaptic stimulation. Fo highe-fequency stimulation, which both depleted synaptic esouce and led to a significant build up of esidual calcium, the pobability peaked late because it took longe fo the synapse to ecove and geneate a sufficient level of asynchonous elease. In contast, when the stimulation ate appoached the ate of esidual calcium cleaance, the level of asynchonous elease at model synapses was low, in which case thee was a low pobability of geneating a spike in esponse to a test stimulus (Figue 1F, left panel). Stimulation by a lage numbe of peconditioning pulses (while keeping the ate of stimulation constant) esulted in the shift of P spike to the ight (Figue 1F, ight panel). To futhe investigate the effects of asynchonous elease on synaptic tansmission we stimulated the model synapse with Poisson PLoS Computational Biology 2 Novembe 2010 Volume 6 Issue 11 e

3 Figue 1. Asynchonous elease modulates the spike geneation window and attenuates the stength of phasic synaptic tansmission. A Post-synaptic cuent ecoded unde hythmic stimulation of a cultued hippocampal neuon by anothe neuon. Note that the significant level of asynchonous elease (incease in the cuent level elative to 0) develops aleady afte the fist pulse and pesists long afte the last stimulus. B Postsynaptic cuent geneated by model synapses (vesicula model, Methods) in esponse to hythmic and synchonous stimulation at 10Hz. C Diagam of synaptic model. Tansitions fom the ecoveed (X) to active (Y) state ae possible eithe via evoked (UXd t{t sp )o asynchonous (jxdðt{t a Þ) elease. D Response of model synapse to hythmic stimulation at 10 Hz: g max ~0 (solid black); g max ~0:3 (dashed gay) (D1). D2: the stength of phasic synaptic esponse, plotted vs. the stimulus numbe, fo the model synapse with (g max ~0:3, gay cicles) and without (g max ~0, black squaes) asynchonous elease. E Modulation of spike time window by asynchonous elease in model synapses. The pobability to geneate spike in esponse to a test stimulus depends on the time DT since the end of conditioning stimulation, and on the level of asynchonous elease at model synapses. F Pobability fo spike geneation, plotted vs. the time DT fom the end of conditioning stimulation. Left panel shows the dependence of P spike on the fequency of conditioning stimulation (numbe of stimuli is the same acoss all conditions, N stim ~5): 5 Hz (gay cicles), 10 Hz (black cicles) and 20 Hz (squaes). Right panel shows the dependence of P spike on the numbe of conditioning stimuli (fequency of stimulation is the same acoss all conditions, n stim ~10 Hz): 3 stimuli (squaes), 5 stimuli (black cicles), 7 stimuli (gay cicles). Data points ae aveages ove 100 ealizations. G Diffeent components of synaptic conductance plotted vs. the ate of synaptic stimulation. The model synapse was stimulated by Poisson spike tains at n stim ~10 Hz. Top panel: conductance due to asynchonous component of synaptic tansmission. Bottom panel: conductance due to phasic component of synaptic tansmission. Open squaes: g max ~0:1. Closed cicles: g max ~0:4. doi: /jounal.pcbi g001 PLoS Computational Biology 3 Novembe 2010 Volume 6 Issue 11 e

4 inputs at diffeent ates and compaed the spike-aveaged peak conductance geneated by the phasic component of elease with the time-aveaged conductance geneated by the asynchonous component (Figue 1G). The asynchonous component of synaptic conductance was aveaged ove the window ½t i z20,t iz1 Š of only those inte-spike intevals that satisfied ISIðÞ~t i iz1 {t i w20,whee t i isthetimeofi-th spike that aives at the synapse (time is measued in milliseconds). This consevative estimate ensued that the effects of conductance due to i-th phasic elease ae negligible. In Figue 1G, synaptic conductance due to the phasic mode of elease was highe fo lowe stimulation ates, but deceased significantly fo highe stimulation ates due to shot-tem synaptic depession. Stonge asynchonous elease inceased the conductance due to the asynchonous component and futhe educed the peak phasic conductance. These esults suggest that asynchonous elease can modulate how a neuon esponds to synaptic inputs, but exactly how this occus depends on the patten of ealie synaptic activity. In what follows, we examine how this type of modulation contibutes to synaptic gain contol and how it depends on the intinsic popeties of the neuon, such as the membane conductance. Asynchonous tansmission accentuates gain modulation by shot-tem depession How does the activity-coupled noise intoduced by asynchonous tansmitte elease affect the tansfe chaacteistics of a neuon, and, moe geneally, how does shot-tem synaptic plasticity affect synaptic gain contol? Cental synapses have heteogeneous mixtues of facilitation and depession in esponses to natual stimuli [25]. We fist exploed the changes in neuonal esponse that wee caused by changes in the onset of, and ecovey fom, synaptic depession. These included changes in the ecovey time fom pesynaptic shot-tem depession, t R, and the stength of phasic synaptic tansmission, We fist studied a model neuon in the low conductance egime (g shunt ~1:2 ms cm 2 ). In Figue 2A, with no asynchonous elease (g max ~0), inceasing the value of ecovey time fom pesynaptic depession t R consistently deceased the output fiing fo high input ates and had a nealy divisive effect fo lowe input fequencies. When a elatively stong asynchonous component (g max ~0:3) was added to the model fo synaptic tansmission (as in Figue 2B), a compaison of input-output cuves fo diffeent ecovey times evealed that, in this paamete egime, the effects of asynchonous elease on the inputoutput cuve depended on the ate at which model synapses ecoveed fom depession. With an asynchonous component, the output ate fo a model neuon with quickly ecoveing affeent synapses (t R ~0:3 sec) inceased fo all input ates, wheeas the fiing ate of a neuon with moe slowly ecoveing affeents (t R ~1:2 sec) deceased following the same manipulation (Figues 2A,B). When plotted vs. the aveaged ecovey time, the output ate of a model neuon was always a monotonically deceasing function of t R (Figue 2D, fo fixed input ate of n in ~10 Hz), but the slope of the Figue 2. Gain contol by shot-tem pesynaptic depession. A Slowe ecovey fom synaptic depession leads to a eduction and faste satuation of output fiing ate: t R ~0:3 sec (ed squaes); t R ~0:6 sec (black tiangles); t R ~1:2 sec (blue cicles). B An addition of asynchonous component (g max ~0:3) to synaptic tansmission accentuates the depession-induced diffeence in neuonal gain. Symbols ae the same as in A. C Gains (defined as descibed in Text) plotted vs. the output ate, fo the diffeent scenaios shown in A,B. Top panel: model synapses without asynchonous component of elease. Bottom panel: model synapses with asynchonous elease (g max ~0:3). D Fiing ate of a model neuon, plotted vs. the ecovey time fom synaptic depession, fo Poisson input stimulation at n in ~10 Hz. Dashed line: esults obtained fo a neuon diven by model synapses with no asynchonous elease. Solid lines: g max ~0:1, 0:2, 0:3. Inset: maximal absolute value of n out ðt R Þ slope, plotted vs. the level of asynchonous elease at model synapses. E Output fiing ate vs. the input ate fo diffeent values of phasic coupling stength (as captued by U): U~0:3 (black tiangles); U~0:6 (blue cicles); U~0:9 (ed squaes). F Output fiing ate vs. the input ate fo diffeent values of phasic coupling stength (symbols ae the same as in E), but with the asynchonous elease (g max ~0:3) added to model synapses. G Gains plotted vs. the output ate, fo the diffeent cases consideed in E,F. Top panel: model synapses without asynchonous component of elease. Bottom panel: model synapses with asynchonous elease (g max ~0:3). H Output fiing ate plotted vs. the stength of phasic elease, fo Poisson input stimulation at n in ~10 Hz. Dashed line: esults obtained fo a neuon diven by model synapses with no asynchonous elease. Solid lines: g max ~0:1, 0:2, 0:3. Inset: maximal absolute value of n out ðuþ slope, plotted vs. the level of asynchonous elease at model synapses. doi: /jounal.pcbi g002 PLoS Computational Biology 4 Novembe 2010 Volume 6 Issue 11 e

5 elation was contolled by the level of asynchonous elease at the model synapses (inset of Figue 2D). Highe levels of asynchonous elease esulted in a steepe n out ðt R Þ cuve. Thus, asynchonous elease of neuotansmitte in ou model accentuated depessionelated modulation of neuonal gain. In Figue 2C the gains ae plotted as a function of output ates (fom Figues 2A,B) to examine the effects of ecovey fom depession. These gains wee computed by taking the deivatives of the best second-ode polynomial fits to the data of fiing ates in Figues 2A,B (Chance et al., 2002). Note that in Figue 2C an additive shift in the abscissa of an input-output cuve would have no effect, an upwad shift of an input-output cuve would cause tanslation along the abscissa, and a change in gain would esult in the tanslation along the odinate. Anothe way to alte synaptic depession is by modulating U, the stength of the phasic synaptic tansmission. Figues 2E,F illustate the effect of g max and U on the input-output function. Fo g max ~0, diffeent stengths of phasic elease gave ise to distinct input-output cuves; howeve, with a elatively stong asynchonous component, the effect of phasic tansmission became less ponounced (Figue 2F). The obsevation that changing the value of U did not significantly affect the esponse of a neuon in the pesence of asynchonous elease means that, in this paametic egime, the fluctuations in the synaptic signal that aise due to the phasic elease ae not likely to be a decisive facto fo spike geneation. Rathe, as is explained below, the synaptic gain was detemined by the aveage level of synaptic cuent, which was in tun affected by asynchonous elease. In contast, with inceased membane conductance spike geneation was diven by the pesence of stong events in phasic tansmission and U was influential (esults not shown). These diffeences ae also evident in Figue 2H, which shows the dependence of output ate (fo a fixed input ate) on the value of U fo diffeent levels of asynchonous elease. Without asynchonous elease, the output ate citically depended on the stength of phasic tansmission, but the addition of an asynchonous component satuated the dependence (Figue 2H). Figue 2G shows futhe that vaying U had little effect on the gain in the pesence of asynchonous elease. Membane conductance modulates the effect of asynchonous elease on synaptic gain Membane conductance stongly affects the excitability of the cell and shapes neuonal esponses to excitatoy affeents [4,26,27]. In paticula, inceasing the membane conductance (shunting) educes neuonal excitability and shifts the neuon fom integating weak petubations at low shunt values to detecting coincidences between stong inputs at high shunt values. Thee is a simila shift in the mode of pocessing fom phasic synaptic tansmission, in which the neuon esponds with high amplitude and a fast time-scale, to asynchonous tansmission chaacteized by a low amplitude and a slow time-scale. The effect of shunting on the gain change induced by asynchonous elease is shown in Figue 3. In the high shunt egime (g shunt ~1:5 ms cm 2, Figue 3A), pogessively inceasing levels of asynchonous elease led to the eduction of both the slope (gain) and the plateau levels of the fiing ate cuve, making the neuon less sensitive to changes in input ate. Fo low input ates, the effect of activity-coupled noise due to asynchonous elease poduced divisive gain modulation. By contast, when the membane conductance was inthelowshuntegime(g shunt ~1:2 ms cm 2, Figue 3B), highe levels of asynchonous elease consistently led to an incease in the slope of input-output elation (inset to Figue 3B). A plot of gains vs. output ate, shown in Figue 3C, summaizes the effects of membane conductance and diffeent asynchonous elease ates on the synaptic gain. Figue 3D shows the neuonal fiing ate (fo fixed input ate n in ~10 Hz) as a function of g shunt, fo diffeent levels of g max. Without asynchonous elease, the output fiing ate depended almost linealy on the membane conductance. Inceasing levels of asynchonous elease shapened the distinction between high- and low- shunt states, and esulted in distinctive modulation of the neuonal gain (see also insets to Figues 3A,B). Taces of the membane potential fo these diffeent egimes ae shown in Figue 4. The effects of membane conductance and asynchonous elease on the gain cuve could be meely a consequence of modulation by noise [3,4]. Altenatively, they might eflect the competition between the two elease modes fo the available synaptic esouce. To assess the contibutions of these two influences, we eplaced the noise induced by asynchonous elease by a noisy cuent I bg geneated by an Oenstein-Uhlenbeck (OU) pocess with t noise ~10 msec and govened by ffiffiffiffiffiffiffiffiffi di bg t noise dt ~{I D bgz Nð0,1Þ ð1þ t noise whee D is the intensity of backgound noise, t noise is noise coelation time, and Nð0,1Þ is an uncoelated Gaussian pocess with zeo mean and unit vaiance. The esulting input-output cuves, shown in Figues 3E,F, suggest that inceasing the backgound noise intensity has a multiplicative effect on neuonal gain in the high shunting egime (Figue 3E), and a weak multiplicative effect fo low values of membane conductance (Figue 3F). The gains fo low and high noise levels in the lowshunt egime wee almost identical (Figue 3G). With activityindependent backgound noise, the output ate was an almost linea function of membane conductance (Figue 3H), and the dependence on noise intensity was pominent only in high shunt egime (Figue 3E, compae also to Figue 3D). Note in paticula that in the high membane conductance state, the effect of asynchonous tansmitte elease was opposite to that of activityindependent backgound noise. High shunting inceased the distance to theshold of spike geneation by a single bief synaptic stimulus, thus making the neuon able to espond only to elatively stong stimuli (coincidence detecto). The mean synaptic cuent was fa below the theshold fo spike geneation and fiing was diven by stong fluctuations in synaptic cuents. This is called a fluctuation-diven egime (FDR) [9,28]. By contast, in the meandiven egime (MDR), the ate of spike geneation was set by the mean cuent, wheeas fluctuations became elatively insignificant. Because asynchonous elease and phasic elease daw fom the same pool of esouce, inceasing the ate of asynchonous tansmitte elease deceased the amplitude of fluctuations due to phasic elease; howeve, the aveage level of synaptic cuent was highe. Thus, the effects of asynchonous elease on neuonal gain wee conditioned by the membane conductance. To bette undestand the effect of shunting conductance on the change in neuonal gain caused by asynchonous elease of neuotansmitte, we consideed a scenaio with moe positive leak evesal potential (E shunt ~{60 mv as opposed to E shunt ~ {70 mv in the baseline model). With a moe depolaized leak evesal potential, addition of asynchonous elease always esulted in the incease of output fiing ate, both fo high and low shunt egimes (Supplementay Figue S2A). This is consistent with the notion that in its moe depolaized state, the membane is aleady in the mean-diven egime whee AR acts to incease the output fiing ate. In Figue 4A, the cuent distibution became moe concentated aound its mean with inceasing levels of the asynchonous component in the fluctuation-diven egime (also Figue 4C). The PLoS Computational Biology 5 Novembe 2010 Volume 6 Issue 11 e

6 Figue 3. Membane conductance defines the modulating action of asynchonous elease on neuonal tansfe function. A Tansfe popeties of a model neuon in high conductance (g shunt ~1:5 ms cm 2 ) egime: g max ~0 (dashed line); g max ~0:1 (black tiangles); g max ~0:2 (blue cicles); g max ~0:3 (ed squaes). Inset: gain (aveaged slope of input-output cuve) plotted vs. the level of asynchonous elease. B Tansfe popeties of a model neuon in low conductance (g shunt ~1:2 ms cm 2 ) egime. Symbols ae the same ones as in A. Inset: gain (aveaged slope of input-output cuve) plotted vs. the level of asynchonous elease. C Gains, plotted vs. the output ate, fo diffeent scenaios shown in A,B. Top panel: gains fo a model neuon in high conductance egime. Bottom panel: gains fo a model neuon in low conductance egime. D Output ate plotted vs. membane conductance, fo Poisson stimulation at n in ~10 Hz. Symbols ae the same as in A. Inset: maximal absolute value of n out ðg shunt Þ slope, vs. the level of asynchonous elease at model synapses. E Tansfe popeties in high conductance egime fo a model neuon diven by the activity-independent noise (as descibed in Text) of diffeent intensity: D~0 ma 2: cm {4 (black tiangles); D~1:44 : 10 {2 ma 2: cm {4 (blue cicles); D~23 : 10 {2 ma 2: cm {4 (ed squaes). In these simulations, g max ~0. Inset: gain (aveaged slope of input-output cuve) plotted vs. the intensity of noise. F Tansfe popeties in low conductance egime fo a model neuon diven by the activity-independent noise of diffeent intensity. Symbols ae the same as in E. Inset: gain (aveaged slope of input-output cuve) plotted vs. the intensity of noise. G Gains, plotted vs. the output ate, fo diffeent scenaios shown in E,F. Top panel: gains fo a model neuon in high conductance egime. Bottom panel: gains fo a model neuon in low conductance egime. H Output fiing ate, plotted vs. membane conductance, fo a model neuon diven by activity-independent noise. Inset: maximal absolute value of n out ðg shunt Þslope, vs. the intensity of noise at model neuons. doi: /jounal.pcbi g003 distibution of cuent was above the theshold in the mean-diven egime as shown in Figues 4B,C. In this egime, an incease in the asynchonous component of postsynaptic cuent (PSC) acted to incease synaptic gain. Thus, asynchonous elease has a dual influence that is jointly detemined by pe-synaptic activity and the state of the post-synaptic membane. In Figue 4D the mean and the standad deviation of postsynaptic cuents ae plotted fo diffeent combinations of asynchonous elease and membane conductance. Fo both high and low conductance states, the mean PSC always inceased fo highe levels of asynchonous elease at model synapses (Figue 4D, top panels). The standad deviation of the PSC deceased with inceasing levels of AR at model synapses, fo both high and low shunt egimes; howeve, the standad deviation attained highe values fo the model neuon in low conductance state (Figue 4D, bottom panels). Thus, depending on the state of neuonal membane conductance, asynchonous elease of synaptic neuotansmitte diffeently affected the fiing patten (Figue 4E). Competition between phasic and asynchonous eleases enables modulation of synaptic gain Is it possible to modulate the impact of asynchonous elease on synaptic gain by contolling the dynamics of synaptic esouce availability? We have aleady seen that in the high shunting egime, vaiations in the stength of phasic tansmission (changing the value of U) can contol the effect of asynchonous elease. This occus when phasic and asynchonous eleases compete fo the common pool of synaptic esouce, and is obseved in ecodings [12,16,23,24]. Howeve, if phasic and asynchonous modes of synaptic tansmission ae instead dawn fom independent pools of synaptic esouce [13], thee would follow diffeential modulation of synaptic gain by the two types of tansmitte elease. To diectly assess the effects of competitive neuotansmitte elease dynamics in ou baseline model, we consideed an altenative (non-competitive) synaptic model in which phasic and asynchonous elease wee decoupled (Equations in Methods). Specifically, sepaate X and Y vaiables wee used fo the two diffeent elease modes (Figue 5A). Analysis of neuonal tansfe popeties evealed that, in the non-competitive model, the addition of asynchonous elease always led to an upwad shift in gain cuves (Compae Figs. 5B,C,E, with Figues 3A,B,C). These esults did not qualitatively depend on the membane conductance (Compae Figue 5D with Figue 3D), einfocing the conclusion that the dual effect of g shunt and asynchonous elease elied on the competition between the two modes of tansmitte elease. An impotant diffeence between the competitive and noncompetitive foms of asynchonous elease was futhe obseved when the mean and the standad deviation of postsynaptic cuent wee plotted against the ate of asynchonous elease (Figues 5F,G). PLoS Computational Biology 6 Novembe 2010 Volume 6 Issue 11 e

7 Figue 4. Asynchonous elease modulates the statistical popeties of post-synaptic cuent. A Distibutions of postsynaptic cuent fo a model neuon in high conductance egime, and diffeent levels of asynchonous elease at model synapses: g max ~0 (top); g max ~0:2 (middle); g max ~0:4 (bottom). Dashed line maks the theshold fo spike geneation by synaptic-like cuent of the coesponding magnitude (I max ~36:74 ma=cm 2 ) and decay time t D ~5 msec. Stimulation ate is n in ~10 Hz. B Distibutions of postsynaptic cuent fo a model neuon in low conductance egime, and diffeent levels of asynchonous elease: g max ~0 (top); g max ~0:2 (middle); g max ~0:4 (bottom). Dashed line maks the theshold fo spike geneation by synaptic-like cuent of the coesponding magnitude (I max ~22:6 ma=cm 2 ) and decay time t D ~5 msec. Stimulation ate is n in ~10 Hz. C Aveaged postsynaptic cuent (top panel) plotted vs. the level of asynchonous elease at model synapses. Squaes: low conductance egime (g shunt ~1:2 ms=cm 2 ). Cicles: high conductance egime (g shunt ~1:5 ms=cm 2 ). Bottom panel: the diffeence DI between aveaged postsynaptic cuent and theshold cuent I th. D Left: mean (top) and standad deviation (bottom) of the postsynaptic cuent plotted vs. the ate of asynchonous elease fo a model neuon in high shunt egime (g shunt ~1:5 ms=cm 2 ). Right: mean m PSC (top) and standad deviation s PSC (bottom) of the postsynaptic cuent plotted vs. the ate of asynchonous elease fo a model neuon in low shunt egime (g shunt ~1:2 ms=cm 2 ). E Examples of membane potential. Top left: g shunt ~1:2 ms=cm 2, g max ~0. Top ight: g shunt ~1:2 ms=cm 2, g max ~0:2. Bottom left: g shunt ~1:5 ms=cm 2, g max ~0. Top ight: g shunt ~1:5 ms=cm 2, g max ~0:2. doi: /jounal.pcbi g004 In the non-competitive model of asynchonous elease, the mean PSC always inceased with inceasing levels of asynchonous elease egadless of the state of neuonal membane conductance (Figue 5F fo high conductance state; low conductance state not shown). Fo a model neuon in the high-conductance state, the standad deviation of the postsynaptic cuent in the noncompetitive model ose and then deceased with highe levels of asynchonous elease (Figue 5G; low conductance state not shown); howeve, the absolute value of PSC standad deviation was significantly highe fo the non-competitive scenaio. Thus, the coupling between phasic and asynchonous modes of neuotansmitte elease in the model undelies the pefeential switch fom integation to coincidence detection fo diffeent values of membane conductance. Stuctued affeent activity and asynchonous elease diffeentially modulate synaptic gain The ate and the vaiability of neuonal fiing can be significantly affected by coelation among affeent inputs [8]. Because phasic and asynchonous elease diffe in thei elative amplitude (high amplitude fo phasic elease vs. low amplitude fo asynchonous elease) and time-scale (fast fo phasic elease vs. slow fo asynchonous elease), they define distinct tempoal windows fo signal integation and spike geneation [15], which may diffeentially influence how input coelations affect the neuonal fiing pattens. We fist consideed the esponse of ou baseline model neuon (with competitive dynamics of phasic and asynchonous eleases) to a hythmic and coodinated stimulation of 15 model synapses. The jitte between spikes on diffeent affeents was vaied (paameteized by the standad deviation of nomal distibution fom which the jitte times fo individual synaptic channels wee dawn) (Figue 6A). Figue 6B shows that in the absence of asynchonous elease, neuonal esponses to sequences of stimuli wee highly eliable with low jitte and low membane conductance. Adding asynchonous elease accentuated the effect of jitte and membane conductance by making the tansition fom eliable to uneliable esponses shape (Figue 6C). This is consistent with the pevious obsevation that in the high conductance egime, asynchonous elease acted to educe the gain (Figue 3). We geneated coelated inputs to the synapses (see Methods) to study the effects of tempoally stuctued non-hythmic input on the gain popeties of a model neuon, with paticula emphasis on the modulating action of the slow time-scale, asynchonous, elease (Figue 7). In the high shunting egime (g shunt ~1:5 ms cm 2,Figue7A,Btoppanels),inceasingthe coelation between affeents led to a geneic incease in output ate. Highe tansient synchony between affeents in a fluctuation-diven egime (measued by the coelation paamete, c) inceased the pobability that a synaptic input would coss the theshold of spike geneation. On the othe hand, highe ates of asynchonous elease counteacted the effect of coelation by weakening the impact of phasic fluctuations, thus leading to a decease in neuonal gain (Figue 7C top panel, diffeent cuves). PLoS Computational Biology 7 Novembe 2010 Volume 6 Issue 11 e

8 Figue 5. Non-competitive neuotansmitte elease dynamics eliminates asynchonous elease mediated gain modulation of synaptic stimuli. A Schematic pesentation of non-competitive scheme fo dynamics of evoked and asynchonous neuotansmitte eleases. B Tansfe popeties fo high conductance egime (g shunt ~1:5 ms=cm 2 ) of non-competitive scheme, fo diffeent levels of asynchonous elease: g max ~0 (dashed line); g max ~0:1 (black tiangles); g max ~0:2 (blue cicles); g max ~0:3 (ed squaes). C Tansfe popeties fo a neuon in low conductance egime (g shunt ~1:2 ms=cm 2 ). Symbols ae the same as in B. D Output fiing ate plotted vs. the membane conductance, fo diffeent levels of asynchonous elease. Model synapses wee stimulated by Poisson spike tains at n in ~10 Hz. Symbols ae the same as in B. E Gains, plotted vs. the output ate, fo high conductance egime (top panel) and low conductance egime (bottom panel) and diffeent levels of asynchonous elease (symbols ae the same as in B,C). F Mean postsynaptic cuent, m PSC, plotted vs. the ate of asynchonous elease, fo a model neuon in high conductance egime (g shunt ~1:5 ms=cm 2 ). Cicles: the non-competitive model. Squaes: the competitive model. Stimulation ate is n in ~10 Hz. G Standad deviation of postsynaptic cuent, s PSC, plotted vs. the ate of asynchonous elease, fo a model neuon in high conductance egime (g shunt ~1:5 ms=cm 2 ). Symbols ae the same as in F. Stimulation ate is the same as in F. doi: /jounal.pcbi g005 In the low shunting egime (g shunt ~1:2 ms cm 2, Figue 7A,B,C second panels), the esults wee evesed. The fiing in this egime was diven by the asynchonous component of synaptic cuent. Gouping inputs togethe (as a esult of coelation) effectively deceased the window fo the postsynaptic cuent, theeby making it moe difficult to geneate a spike and deceasing the neuonal gain. In contast to the high-shunt egime, the highe asynchonous elease ates inceased the fiing ate by acting on the asynchonous component of synaptic cuent. Consistent with this explanation fo the dual ole of asynchonous elease in shaping neuonal tansfe cuve, the model neuon diven by an activity-independent noise (Equation 1) always poduced highe ates fo highe noise intensities (Figue 7, thid and fouth panels). Discussion Sheman and Guilley [29] classify thalamic affeents as eithe dives, which ae stong and initiate spikes, o modulatos, which povide a backgound context. We have shown hee that the same affeent synapse can both dive and modulate depending on the dynamics of pesynaptic calcium, which itself depends on the patten of affeent activity (Figue 8). Thus, dives and modulatos should be consideed physiological athe than anatomical concepts. Neuonal gain can be modulated by a vaiety of factos. Pevious models suggested that shunting inhibition was subtactive [30], but moe ecent modeling [5] and expeimental studies [4] have demonstated that, in the pesence of synaptic excitation, shunting inputs can have a divisive effect on neuonal gain by modulating the slope of input-output cuve. We have shown that the membane conductance can detemine the modulating action of activitycoupled asynchonous elease of neuotansmitte by switching between divisive and multiplicative modes of action (Figue 8). In addition, we showed that the impact of coelated stimuli on the output fiing ate depended jointly (and oppositely) on membane conductance and asynchonous synaptic tansmission. Thus, the postsynaptic membane conductance, when consideed togethe with pesynaptic plasticity and stuctued input pattens, had an impact beyond that of shifting o scaling the gain cuve. Fom a computational pespective, the stength of asynchonous elease is a single contol paamete that contols gain modulation. This is in contast to othe mechanisms of gain modulation, such as balanced backgound input [3] o concuent action of shunting inhibition and synaptic excitation [4], which depend on seveal contol paametes. Fo backgound input to modulate the gain in divisive manne, the excitatoy and inhibitoy fiing ates have to be inceased at the same time [3]. Similaly, divisive gain contol by shunting inhibition equies coodination of the latte with synaptic excitation [4]. This poses tight constaints on the dynamics of upsteam netwoks. Asynchonous elease affects the gain without additional constaints, and shunting inhibition acts only as a switch that detemines the type of the tansfomation (division o multiplication) pefomed by the neuon. We showed that the ability to contol the gain by single paamete citically depended on the assumption that phasic and asynchonous eleases compete fo the same pool of synaptic esouce [12,13,16,23,24] (Figues 4,5). Whethe o not such competition is a univesal featue of cental synapses will be esolved by PLoS Computational Biology 8 Novembe 2010 Volume 6 Issue 11 e

9 Figue 6. Asynchonous elease and membane conductance define neuonal esponse sensitivity to coelated input pattens. A Examples of hythmic stimulation with contolled level of jitte between the inputs (left panel: s jitte ~2 msec; ight panel: s jitte ~10 msec, stimulation ate is 10 Hz). B The effect of asynchonous elease (g max ~0:3, second and fouth panels; g max ~0, fist and thid panels) is to decease the esponse to late stimuli when the level of jitte is high (left: s jitte ~2 msec; ight: s jitte ~10 msec, all panels). This effect is moe ponounced in high membane conductance egime (g shunt ~1:5 ms=cm 2, thid and fouth panels; g shunt ~1:2 ms=cm 2, fist and second panels). C The pobability to geneate spike in esponse to i-th stimulus (stimulation ate is 10 Hz) fo diffeent scenaios. Top left: g max ~0, g shunt ~1:5 ms=cm 2 ; Top ight: g max ~0:3, g shunt ~1:5 ms=cm 2 ; Bottom left: g max ~0, g shunt ~1:2 ms=cm 2 ; Bottom ight: g max ~0:3, g shunt ~1:2 ms=cm 2. In all panels, open cicles ae fo s jitte ~2 msec, and closed cicles ae fo s jitte ~10 msec. Data points ae aveages ove 50 ealizations. doi: /jounal.pcbi g006 studying the stuctual oganization of these synapses. Howeve, in ou model this effect was obseved ove a elatively naow ange of paametes, suggesting that othe mechanisms fo gain modulation may also be impotant. Recently, it has been shown that the elative stength of phasic and asynchonous eleases can be dynamically modulated in hippocampal synapses [31]. These studies showed that selective alteation of asynchony in tansmitte elease can be attibuted to the potein kinase C (PKC) dependent mechanisms at pesynaptic boutons [31]. Ou computational model povides an undestanding of how modulation of asynchonous elease can affect neuonal gain. Taken togethe with the expeimental esults epoted in [31], ou wok offes new insight into the pinciples of computation at the single cell level. Additional expeiments aiming to pobe changes in synaptic tansmission in esponse to calcium and PKC-modulating second messenge molecules could enhance ou undestanding of the effects that diffeent modes of neuotansmitte elease have on gain contol. What happens in the limit of a lage (,10,000) numbe of affeent synapses? A simple scaling of synapse numbe and all elevant paametes would offe a simple way to extapolate ou findings fo the simplified lumped neuon to this ealistic limit (Supplementay Figue S1). Howeve, cental neuons often possess highly amified dendites with a multitude of active conductances [32]. The dendites of a pyamidal neuon could in some cicumstances behave as independent pocessing units and the soma could be consideed as a second laye in a two-laye atificial neual netwok, evaluating the esults of many conveging local denditic computations [33]. Stong spatial sepaation of functionally simila (synchonously activated) synapses could compomise the detection of asynchonous elease. In addition, localized changes in membane conductance (fo example, due to the local action of inhibition o adaptation) could sceen the effect of tempoally stuctued activity and asynchonous elease. Both of these poblems could in pinciple be ovecome if synapses wee distibuted on denditic banches accoding to thei functional similaity, so that the pobability of synchonously activated synapses is highe when they ae located on the same banch. Although inputs to pyamidal neuons might be spatially oganized [33], the effects of synaptic distibution on gain modulation by asynchonous elease should be futhe investigated. In the point-like neuonal model that we studied hee, the membane conductance was contolled by a single paamete, and the effect of gain modulation by asynchonous elease of neuotansmitte was obseved ove a elatively naow ange of membane conductance values. Howeve, in eal pyamidal neuons, compised of many denditic compatments, membane conductances can vay acoss diffeent banches and can be PLoS Computational Biology 9 Novembe 2010 Volume 6 Issue 11 e

10 Figue 7. Effects of stuctued affeent activity and asynchonous elease on neuonal tansfe popeties. A Top panel: Tansfe cuves fo high shunt (g shunt ~1:5 ms=cm 2 ), g max ~0, and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). Second panel: Tansfe cuves fo low shunt (g shunt ~1:2 ms=cm 2 ), g max ~0, and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). Thid panel: Tansfe cuves fo high shunt (g shunt ~1:5 ms=cm 2 ), no OU noise (D~0), and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). Fouth panel: Tansfe cuves fo low shunt (g shunt ~1:2 ms=cm 2 ), no OU noise (D~0), and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). B Top panel: Tansfe cuves fo high shunt (g shunt ~1:5 ms=cm 2 ), g max ~0:3, and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). Second panel: Tansfe cuves fo low shunt (g shunt ~1:2 ms=cm 2 ), g max ~0:3, and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). Thid panel: Tansfe cuves fo high shunt (g shunt ~1:5 ms=cm 2 ), OU noise (D~0:1 ma 2: cm {4 ), and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). Fouth panel: Tansfe cuves fo low shunt (g shunt ~1:2 ms=cm 2 ), OU noise (D~0:1 ma 2: cm {4 ), and diffeent levels of affeent coelation: c~0:09 (black), c~0:9 (gay). C Top panel: Output ate in high shunt (g shunt ~1:5 ms=cm 2 ) vs. the coelation paamete and diffeent levels of asynchonous elease. Dashed line: The case of g max ~0. Solid lines: g max ~0:1, 0:2, 0:3. Second panel: Output ate in low shunt (g shunt ~1:2 ms=cm 2 ) vs. the coelation paamete and diffeent levels of asynchonous elease. Dashed line: The case of g max ~0. Solid lines: g max ~0:1, 0:2, 0:3. Thid panel: Output ate in high shunt (g shunt ~1:5 ms=cm 2 ) vs. the coelation paamete and diffeent intensities of OU noise. Dashed line: The case of D~0. Solid lines: D~2:56, 10, 23 : 10 {2 ma 2: cm {4. Fouth panel: Output ate in low shunt (g shunt ~1:2 ms=cm 2 ) vs. the coelation paamete and diffeent intensities of OU noise. Dashed line: The case of D~0. Solid lines: D~2:56, 10, 23 : 10 {2 ma 2: cm {4. Fo all cases in C synapses wee stimulated by Poisson spike tains at n in ~10 Hz. doi: /jounal.pcbi g007 modulated locally by a vaiety of factos including locally acting shunting inhibition due to the inputs fom inteneuons, and the activation of potassium cuents, paticulaly the slow aftehypepolaization (sahp). Consideed togethe with the notion of two-laye netwok and the existence of asynchonous elease, such local modulation of denditic excitability could povide pyamidal neuons with significant computational capacity. In this view, the pyamidal neuon should be viewed as composed of many denditic computational units [33], each of which could adapt the natue of its computation (divisive o multiplicative) based on the pattens of ongoing activity. Local denditic modulation of membane conductance could also potentially widen the ange ove which asynchonous elease affects gain modulation. Moe detailed computational models should examine whethe this could indeed occu. The size of the active zone of synapses and thei elease chaacteistics ae heteogeneous [25,34]. The ability of an active zone to suppot asynchonous elease might depend on its size. Small active zones contain fewe eadily-eleasable vesicles, and theefoe tend to exhibit low elease pobabilities, which would ende asynchonous elease less fequent. Lage synapses can accommodate significantly moe vesicles that can be eleased spontaneously. Asynchonous elease has not been as well studied as phasic elease. Asynchonous elease onto a dendite could be sceened o boosted, depending on the ion channels in the denditic tee. Futue integated expeimental and modeling studies could esolve the question of how the distibution of denditic mechanisms affects the impact of synaptic plasticity and diffeent modes of neuotansmitte elease in detemining neuonal spike dischage pattens. Methods We constucted both a vesicula model of neuonal excitation as well as a simplified appoach, which makes use of the coodinated PLoS Computational Biology 10 Novembe 2010 Volume 6 Issue 11 e

11 C dv dt ~{I ionðþzi t syn ðþ t ð5þ Figue 8. Membane shunting dependent gain modulation by asynchonous elease of neuotansmitte. The action of asynchonous neuotansmitte elease on neuonal tansfe cuve depends on the state of postsynaptic membane conductance. In high shunt egime, asynchonous elease educes neuonal fiing ate. In low shut egime, asynchonous elease acts to incease neuonal fiing ate. doi: /jounal.pcbi g008 activation of 1% of the synapses impinging on a fixed neuon. The educed model used hee is based on one that was peviously matched semi-quantitatively with expeimental findings and used to study evebeatoy netwoks in hippocampal cultues [12,19]. Because we focused hee on shot-tem influences, the pesent model did not include slow synaptic depession, which is needed to teminate the evebeatoy activity in ecuent netwoks. We modified the neuonal dynamics (following [27]) by including biophysical mechanisms to account fo the popeties of spike geneation in cental neuons. All equations wee integated with custom softwae witten in C, using the second-ode Runge-Kutta method with a fixed time step Dt = msec. Neuonal dynamics We used a conductance-based model with one compatment [27,35], which is a compomise between a detailed multicompatmental model that encompass ealistic denditic mophologies and an ovesimplified integate-and-fie model. The ionic cuent though the model neuonal membane was taken to be: I ion ðþ~g t Na : m? ðv Þ : ðv{e Na zg shunt : ð V{Eshunt Þ Þzg K : wv ð Þ : ð V{EK Þ ð2þ m? ðvþ~0:5 : 1ztanh V{V 1 V 2 w? ðvþ~0:5 : 1ztanh V{V ð3þ 3 V 4 dw dt ~ : ðw? ðvþ{wþ : cosh V{V 3 2V 4 The membane potential was govened by: ð4þ whee I syn ðþis t the synaptic cuents discussed in detail in the next section. The following paamete values wee used in the model: E Na ~50 mv, E K ~{100 mv, E shunt ~{70 mv, V 1 ~ {1:2 mv, V 2 ~23 mv, V 3 ~{2 mv, V 4 ~21 mv, g Na ~ 10 ms cm 2, g K ~10 ms cm 2, C~1 mf cm 2, ~0:15. To establish the effect of membane conductance on neuonal pocessing of synaptic stimuli, the value of g shunt was vaied in the ange ½1{1:5Š ms cm 2. With this choice of paametes, the model neuon exhibited Type-2 excitability, with the tansition between quiescent and spiking states descibed by a Hopf bifucation [27]. In additional simulations (Text S1) we also investigated the changes that would ensue with Type-1 neuonal dynamics (tansition to spiking via saddlenode bifucation). Results obtained with Type-1 model wee qualitatively simila to those epoted in Text, einfocing ou assumption that the phenomenon we epot is faily geneal with espect to the detailed bifucation stuctue undelying the neual excitability. Vesicula model of synaptic dynamics The model we used hee is based on the classical quantal model of synaptic tansmission [36], extended to account fo the existence of asynchonous elease. We assume that each synaptic connection is composed of N elease sites (N = 5). Each site can accommodate at most one vesicle that is available fo elease, and the elease fom each of the N sites that constitute the synaptic connection is independent of the elease fom all othe sites. Immediately following the aival of action potential each site can elease its vesicle (if available) with pobability ~U ( ~U~0:3). In addition to this phasic elease, asynchonous elease can occu in the time inteval ½t,tzdtŠ with pobability ~g Ca 2z dt that depends on the level of pesynaptic esidual calcium - ~g Ca 2z Ca 2z 4 ~~gmax Ka 4z 4 ð6þ ½ Ca2z d½ca 2z 2 Š {b Ca2z ~ dt Kp 2z 2 zclog ½ Ca2z Š Š! Ca 2z out ½Ca 2z d t{ts j Š zip ð7þ In Equation 6, the ate of asynchonous elease depends on the pesynaptic concentation of esidual calcium, Ca 2z, which inceases due to action potentials in popotion to the electochemical gadient acoss the synaptic membane, and decays nonlinealy due to extusion by active calcium pumps. The tem I p ensues that in the absence of any pesynaptic spikes, synaptic calcium is maintained at a non-zeo steady-state level (,50 nm). Once the elease of the vesicle fom the elease site occus (eithe in phasic o asynchonous way), the site can be efilled in any time inteval ½t,tzdtŠ with pobability dt=t R. We typically investigated the neuonal esponse to the stimulation of 2000 synapses modeled as descibed above (since each synapse has 5 independent elease sites, oveall thee ae 10 4 elease sites diving the model neuon in this scenaio). Collective neuonal activity in ealy development is chaacteized by synchonized busting events, duing which most of the ecoded neuons ae engaged in highly coelated activity [37]. Thus, we PLoS Computational Biology 11 Novembe 2010 Volume 6 Issue 11 e