Stochastic fluctuations of the synaptic function

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1 BioSystems 67 (2002) 287/294 Stochastic fluctuations of the synaptic function Francesco Ventriglia *, Vito Di Maio Istituto di Cibernetica E.Caianiello del CNR, Via Campi Flegrei 34, Pozzuoli, NA, Italy Accepted 22 August 2002 Abstract The peak amplitudes of the quantal Excitatory Post Synaptic Currents in single hippocampal synapses show a large variability. Here, we present the results of a mathematical, computational investigation on the main sources of this variability. A detailed description of the synaptic cleft, rigorously based on empirically-derived parameters, was used. By using a Brownian motion model of neurotransmitter molecule diffusion, quantal EPSCs were computed by a simple kinetic schema of AMPA receptor dynamics. Our results show that the lack of saturation of AMPA receptors obtained in these conditions, combined with stochastic variations in basic presynaptic elements, such as the vesicle volume, the vesicle docking position, and the vesicle neurotransmitter concentration can explain almost the entire range of EPSC variability experimentally observed. # 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Stochastic fluctuations; Synaptic function; Quantal EPSC variability 1. Introduction The computational ability of the brain, or its aptitude to manipulate the ongoing information, is mainly based on the neurons, which constitute the basic computational elements, and the axonic network connecting them. The transduction operated at the interface between neurons, the synapses, is of basic importance to this respect. If the synapse behaves as a fairly deterministic switch, i.e. as an element which is able to produce an output (quantal Post Synaptic Current) with a peak amplitude distributed according to a narrow Gaussian distribution and with a fairly stable * Corresponding author address: franco@biocib.cib.na.cnr.it (F. Ventriglia). shape, then the noise is introduced in the computation only by the threshold characteristics of the neurons. If this is not true, the synapse behaves as a stochastic device and the activity of the neural networks is affected by noise both at the input and at the output side of its computational elements. A clear understanding of this aspect of the synaptic activity is truly necessary for any theory about the computing ability of neural brain structures or neural coding. In a recent paper, Liu et al. (1999) collected experimental evidence on the noise characteristics of an important type of synapse, the excitatory synapse of hippocampal pyramidal neurons. By a sophisticated experimental procedure they were able to demonstrate that the unitary or quantal Excitatory Postsynaptic Currents (EPSCs) produced by stimuli arriving in time /02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved. PII: S ( 0 2 )

2 288 F. Ventriglia, V. Di Maio / BioSystems 67 (2002) 287/294 to a single synapse had peak amplitudes distributed according to a non Gaussian distribution, with a mean value of 24.6 pa, a range 5/65 pa, and a high variation coefficient*/cv /0.51. The histogram of the peak amplitudes showed a long right tail. In a previous work (Ventriglia and Di Maio, 2002b), we investigated some of the main causes of stochastic variability of the synaptic response to quantal releases of neurotransmitter. We analyzed both the presynaptic and postsynaptic sources by a mathematical model and a computational setup previously settled to study the synaptic function. The simulation was rigorously based on empirically-derived parameters and the results provided evidence that, in the considered conditions, the AMPA receptors do not saturate after a single vesicle release and the main source of stochastic variability is to be attributed to the presynaptic molecular machinery involved in the neural transmission. In the same paper we demonstrated that about two thirds of peak amplitude variability could be explained by the statistical variations of the volume and of the docking position of the releasing vesicle on the Active Zone. Here, we investigated the possible effects on EPSC stochastic fluctuations, associated with the stochastic fluctuations of the glutamate concentration in the pool of vesicles of a single synapse. The present results demonstrated that the entire range of peak amplitudes reported by Liu et al. (1999) can be explained if this last stochastic source of variability is considered. 2. Model The Brownian motion model of synaptic transmission used in the present work was described in Ventriglia and Di Maio (2000a,b, 2002a,b). By this model we can simulate the release of glutamate from a single docked vesicle at the arrival of a spike, the diffusion in the synaptic cleft, the binding on postsynaptic receptors, the re-uptake by presynaptic transporters, and the spill over. Our previous studies showed that the number of molecules contained in a vesicle, when small, is one of the most important synaptic parameters to be considered. Hence, this value must be carefully computed from experimental data. The main presynaptic sources of variability can be related to the concentration value of Glutamate within a vesicle, the volume and the position of the vesicle. All these sources were studied by several computational experiments. Glutamate concentration ranges 60 /210 mm in CA1 excitatory synaptic vesicles (Clements et al., 1992). Data by Shikorski and Stevens (1997) show that these vesicles have an inner radius ranging 9.9/13.3 nm. By these two ranges of values we compute that a spike arriving to a presynaptic button can produce the release in the synaptic cleft of a random number of neurotransmitter molecules in the range 147/1246. In addition, a third cause of variability is present at the presynaptic side. The geometry of a synapse with an unique release site (Stevens and Wang, 1995), the main type of synapse, is constituted by a presynaptic active zone (AZ diameter of about 220 nm), with docked vesicles juxtaposed to a Post Synaptic Density (PSD) of the same diameter containing AMPA and NMDA receptors (AMPARs and NMDARs). The centers of the two zones lie on a unique axis, but vesicles are scattered over the active zone. The different spikes induce fusion pores in single vesicles which can have different distances from the central axis. Hence, for consecutive spikes arriving to the presynaptic terminal, the receptor population in the PSD can be exposed to different neurotransmitter concentration time /courses. In our model we assumed that each vesicle is filled with a predetermined number of neurotransmitter molecules distributed uniformly in space and according to the usual Maxwell distribution of thermal equilibrium in velocity. Moreover, we hypothesized that the arrival of a presynaptic spike at a time t/0, started the activation of a fusion pore. This was considered as a water filled cylindrical space, connecting the inner part of the vesicle with the synaptic space, having predetermined length and constant opening velocity. Some appropriate values, attributed to these two last parameters, were largely discussed in Ventriglia and Di Maio (2000a,b, 2002a,b). When the fusion pore diameter reached a value about equal to the computed diameter of a glutamate molecule, the

3 F. Ventriglia, V. Di Maio / BioSystems 67 (2002) 287/ diffusion of neurotransmitter in the synaptic cleft (a flat cylinder with height of 20 nm and diameter of 400 nm) could start. AMPARs and NMDARs were randomly disposed on the PSD within tiles composing a square matrix with side equal to the diameter of the PSD. Only tiles encompassed by the PSD perimeter contained receptors (one receptor per tile). The equations describing the Brownian motion of each neurotransmitter molecule were based on the following Langevin equations: d dt r i (t)v i (t) (1) m d dt v i (t)gv i (t) p ffiffiffiffiffiffiffi 2og L(t): (2) In these equations the variables r i and v i denote the position and the velocity of glutamate molecules, m denotes the molecular mass, i stands for the ith of the N m molecules contained in a vesicle, k B is the Boltzmann constant, T is the absolute temperature in Kelvin degrees, D is the diffusion coefficient of glutamate. The friction parameter g and the Langevin term (in Eq. (2)) are due to the interaction of each neurotransmitter molecule with the molecules of water assumed to fill all the model space where neurotransmitter molecules moved. As stochastic force we supposed a white Gaussian noise [ŽL i (t) L j (t/d) /d ij d(d)] with intensity 2og. The friction term g is dependent on the absolute temperature being g/k B T/D, and o / k B T. The free Brownian motion of the glutamate molecules within the synaptic diffusion space was constricted only by the interaction with the structures limiting the model space (the surface of the synaptic vesicle, that of the fusion pore and the walls of the structures composing the synaptic cleft). In the synaptic cleft, molecules of glutamate could collide on the pre synaptic surface and on the postsynaptic one while they could freely pass through the lateral surface. Molecules crossing the lateral surface of the synaptic space were considered lost from the modeled synaptic space (spillover). Transporters contained on the presynaptic surface could absorb neurotransmitter colliding molecules (re-uptake), with a prefixed probability P R. About the PSD we assumed that AMPA and NMDA receptors were co-localized and uniformly distributed. When a neurotransmitter molecule hit the postsynaptic surface it could bind, with a probability P B, the receptor enclosed in the tile containing the contact point. We assumed that each receptor had two binding sites for glutamate (Clements et al., 1998) and that the probability to bind the second site was one half that to bind the first one. The following time discretized Langevin equations for diffusion were implemented in a FOR- TRAN program by using message passing interface (MPI) paralleling routines r i (td)r i (t)v i (t)d (3) v i (td)v i (t)g v i (t) pffiffiffiffiffiffiffiffiffiffi m D 2ogD m V i (4) where V i is a random vector with three components, each having a Gaussian distribution with mean value m/0 and standard deviation s/1; D is the time step which we choose very small to have a good damping term in the above equations (D/ 40/10 15 s). This small time step is important also to get a precise description of the interactions between neurotransmitter molecules and receptors when the binding is not computed by using the traditional mass equations (that we consider not precise when the number of diffusing molecules is small) but is obtained by geometrical considerations as in our case. A parallel random number generator was used to compute, at each time step, the random vector V i, and the re-uptake and binding probabilities when required. The space position r i (t)/(x i (t), y i (t), z i (t))) within the stochastic path of each molecule was saved as a function of time every iterations (corresponding to a simulation time of 2/10 9 s). The binding times of molecules of glutamate to the postsynaptic receptors were saved on a separate matrix and were used for the computation of the quantal EPSC. The program ran on a parallel computer made of 22 processors. In our computational experiments we assumed that, as it happens in the ordinary neuron resting conditions, only AMPA receptors could contribute to the EPSC formation. For AMPARs activation, we used a very simple kinetic mechan-

4 290 F. Ventriglia, V. Di Maio / BioSystems 67 (2002) 287/294 ism based on four states: Basal (B) -closed, Active (A) -open, Inactivable (I) -closed, Desensitized (D) -closed- as defined in Edelstein et al. (1996) occurring in a linear cascade B 0 X/B 1 X/B 2 X/ A 2 X/I 2 X/D 2 where 0, 1, 2 denoted the unbound, the singly-bound and the double-bound states, respectively. This cascade does not consider the unbound and singly-bound open states since they do not play a significant role during normal receptor function. Argumentations presented in Ventriglia and Di Maio (2002b) allowed us to neglect also the transition to and from the desensitized state D. NDMA receptor kinetics was considered only for the first states B 0 X/B 1 X/ B 2, being the transition to the open state A 2 highly improbable in the simulated conditions. Due to the high affinity of these receptors for glutamate, the effect of neurotransmitter binding to NMDARs in our simulation time, was only the decreasing of free gluatamate in the synaptic cleft. At the postsynaptic side other minor causes of statistical variability can be present. One is related to the transitions from states B 2 and A 2.Whena receptor goes in the open state, it remains opened for a random period of time t o after which it passes into the closed state. The opposite occurs for the closed state, the closing time being t C. The probability of receptor opening, P O, is defined as the ratio between the total opening time and the sum of the total opening and total closing time. We supposed that the opening time t o and the closing time t C are distributed according to probability density functions (PDFs) of exponential form. In our simulation we assumed that the opening probability is not constant for all the receptors but is Gaussian distributed with mean value P O /0.71 and standard deviation 0.05, which are reported in the literature as possible values related to the opening probability (Jonas et al., 1993). We supposed also that a negative exponential distribution ruled the transitions from the open state A 2 to the inactivable state I 2, with a mean value t I :/ We investigated the changes induced on the postsynaptic response by the statistical fluctuations in the receptor opening and closing times. The computed, quantal EPSC*/I EPSC (t)*/is given by the summation of all the single currents*/ I r (t)*/produced by the double bound AMPARs, where r stands for the rth receptor. The current I r (t) was 0, for t belonging to the interval of closure and was I Mr for t in the interval of opening, while in intervals of fixed duration T just after opening and just after the closing it had the form: I(t)I Mr 1exp t (5) I(t)I Mr exp t t 1 t 1 ; (6) respectively, simulating the gating properties of the channel. In the above equations I Mr is the peak of the rth receptor current, which can vary among the receptors (Jonas et al., 1993). Hence, we assumed a receptor peak current distributed according to a normal distribution with mean value of pa and standard deviation of pa. In the above equations t 1 denotes the rise (and decay) time constant of the current conveyed by a single AMPA channel. Lacking precise information in literature, we assumed this unique value for both the rise and the decay parameter. To randomize the peak current amplitude, the opening state and the opening duration of each receptor channel we made 20 runs for each computational experiment by using a C// program. We used 3.1 ms for t I ; 14.8 ns for the time constant t 1 ; and 50 ns for T. Lacking information about the microscopic AMPA receptor opening mean rate, we used t O //12.5 ms, as derived for ACh receptor (Colquhoun, 1998). The single 5 khz filtered values of the 20 time courses of the computed current I EPSC (t) were used to draw the EPSCs shown in the figures reported in the present paper. Since we were interested mainly in the early phase of the quantal EPSC, a time duration of 3000 ms is shown for each computational experiment. 3. Simulation and results Some computational experiments were carried on by changing the Glutamate concentration value, the radius of the releasing vesicle R V (and, hence, the number of neurotransmitter molecules

5 F. Ventriglia, V. Di Maio / BioSystems 67 (2002) 287/ released N m ), and the position of the fusion pore with respect to the center of the active zone X 0. Moreover, we assumed randomly variable opening probability P O and channel current peak I Mr. The other parameters were assumed with constant values as in Ventriglia and Di Maio (2002b). We considered two different receptor densities so that the side of the receptor containing tiles was 14 nm in one case and 12 nm in another case. We had 97 and 117 AMPARs, and 107 and 132 NMDARs, respectively. Affinities for glutamate had a ratio 1:30 between AMPARs and NMDARs (Xie et al., 1997). To obtain the largest value for the range of peak amplitudes we carried out two simulations for each receptor density: one produced the worst result for the peak amplitude (i.e. the smallest value), the other produced the best result (i.e. the largest peak amplitude). To obtain the worst case we considered the smallest volume vesicle (radius/9.9 nm), with the smallest Glutamate concentration (60 mm), placed at the greatest distance from the center of AZ (and, hence, of the PSD*/X 0 /90 nm). The best case was simulated by using the largest volume vesicle (radius/ 13.3 nm), with the largest Glutamate concentration (210 mm), placed at center of the AZ (X 0 /0 nm). Molecules were 147 in the first condition and 1246 in the second one. To obtain a time course for the computed quantal EPSC comparable with the rising phase of quantal EPSCs recorded in electrophysiological experiments (Forti et al., 1997; Liu et al., 1999), each experiment ran for 5/10 9 iterations (corresponding to 200 ms and requiring from 1 to 7 days of computer elaboration on our cluster of workstations). The very short computational time step of 40 fs allowed us to get a very precise description of the dynamics of neurotransmitter diffusion and of receptor binding. From this simulation we obtained that doubleligated AMPARs ranged according to the neurotransmitter molecule number and the vesicle position from 18% to 94%, for the lower receptor density, and from 26 rm to 100%, for the higher receptor density. AMPARs saturation was achieved only in the most favorable case, since a higher receptor density, corresponding to a larger density of receptors binding sites, increases the binding probability of glutamate molecules. Let us to note that the AMPAR saturation was not achieved in our previous work (Ventriglia and Di Maio, 2002a,b), where a lower Glutamate concentration value was utilized. Vice versa, bound NMDARs ranged from 31% to 100%, for the lower receptor density, and from 33% to 100%, for the higher receptor density. Fig. 1 presents results on the glutamate concentration in the higher receptor density case for the best and the worst case. Two cylinders were considered in the synaptic space to compute concentration values. The first cylinder, having the PSD and the active zone as bases, occupied the entire height of the cleft (20 nm). The second cylinder, based on the PSD and with a length of 4 nm, gives information for a volume closer to the receptors. In the same figure are reported also concentration time courses in some parcels of synaptic cleft having as a base a tile with a side of 12 nm. Also in this case the above two lengths, 20 and 4 nm, were considered. One tile was placed near the center of the PSD (center at (X, Y)/(0, 0)), the other occupied a more peripheral position. In Fig. 1A, we show the results for the best case. The eccentric tile subtended volume has center at (X, Y)/(48, 48) (values in nm). Fig. 1B shows results for the worst case. The releasing vesicle is here placed at a distance of 90 nm from the center of the PSD. The eccentric tile subtended volume is nearly under the releasing vesicle ((X, Y) /(84, 0)). Beside the expected time variation and the decreasing of the concentration maximum with the distance from the release point, some more interesting information are conveyed by this figure. In particular, the concentration for tile subtended volumes assumed a spiking time /course. We can observe concentration spikes of 1.5 mm over a central tile (4 nm height), whereas a maximum value of 0.25 mm is reached in the volume of the same height subtended by the PSD (best case-fig. 1A). Vice versa, for the worst case (Fig. 1B), concentration spikes of 0.5 mm are reached in the volume over the peripheral tile (4 nm height), whereas in the volume of the same height subtended by the PSD the concentration reaches a maximum value of 0.05 mm. This figure shows how erratic can be the concentration time course

6 292 F. Ventriglia, V. Di Maio / BioSystems 67 (2002) 287/294 Fig. 1. Concentration time course of glutamate in the synaptic cleft. The different graphs compare concentration time course of glutamate computed, respectively, for cylinders based on the PSD with height of 20 and 4 nm and parallelepipeds based on tiles with side length of 12 nm and heights, respectively, of 20 nm (upper graphs) and 4 nm (lower graphs). Panel A shows the effect of a vesicle centered on the AZ and releasing 1246 molecules while panel B shows the effect of a vesicle positioned at a distance of 90 nm from the center of AZ and releasing 147 molecules. Fig. 2. Number of glutamate molecule hits for each tile of the PSD grid during a complete vesicle release. In panel A the vesicle had the biggest number of glutamate molecules (1246) and was centered on AZ (i.e. X 0 /0 nm), whereas in panel B the vesicle had the lowest number of glutamate molecules (147) and an eccentric position on AZ (i.e. X 0 /90 nm).

7 F. Ventriglia, V. Di Maio / BioSystems 67 (2002) 287 /294 in a synaptic cleft when the natural releasing conditions are addressed and synaptic volumes very close to receptors are investigated. In Fig. 2, the number of hits per tiles are reported for (A) the best case and (B) the worst case for the largest receptor density. We can note that the maxima in the two cases differ for one order of magnitude. In Fig. 3, the ranges of peak amplitude variability are presented for the two different cases of receptor densities. An overall range of 5 /40 pa in the peak amplitudes is apparent for the lower receptor density and an overall range of 10 /50 pa is apparent for the higher receptor density. This second case explain quite completely the results shown by Liu et al. (1999). 4. Discussion In the present work we extended a previous investigation on the possible causes of variability of the postsynaptic response to the stimulation of a single synaptic button, measuring the variation of 293 the EPSC. We used a Brownian motion model of the neurotransmitter diffusion and a computer simulation of the neurotransmitter activity in the synaptic cleft. All the most important presynaptic sources of variability such as the stochastic variation of glutamate concentration, volume and position of neurotransmitter releasing vesicles were considered in the new computational experiments. The parameters used in our simulation were rigorously based on empirically-derived data from literature. Data on vesicle volumes and glutamate concentration permitted us to compute that a quantal release can diffuse from 147 to 1246 neurotransmitter molecules within the synaptic cleft. Also, the starting diffusion point has a random distance from the PSD axis (from 0 to more than 90 nm). Each of the volume /concentration /distance combinations produces an EPSC with a specific amplitude peak. Our worstbest case analysis was able to reproduce the entire range of the experimentally observed EPSC amplitude peak variability. This variability has great importance in the understanding of neural code Fig. 3. EPSCs produced in two extreme cases. The upper couple of branches presents results for 147 molecules in a vesicle positioned at 90 nm from the center of AZ, while the lower couple of branches shows results for 1246 molecules in a vesicle placed at the center of AZ. The superior group of traces in each couple shows data for a PSD grid with tile side of 14 nm length; the inferior one for a PSD grid with tile side of 12 nm length.

8 294 F. Ventriglia, V. Di Maio / BioSystems 67 (2002) 287/294 formation. If it can be extended to all cortical neurons, in fact, the debate on the transmission code among neurons would be biased towards the traditional hypothesis that the brain reacts to the average firing rates of neurons. In conclusion, we stress that the effectiveness of the presynaptic sources of variability is strictly linked to the small volumes of the neurotransmitter vesicles of the hippocampal regions. This volume allows only the packing of a small number of molecules and, hence, the saturation of the postsynaptic AMPA receptors can be achieved only when vesicles with the largest values of volume and glutamate concentration, docked near the PSD axis, are released. The lack of saturation induces different number of bound receptors and as a consequence different peak amplitudes. The large AMPA-EPSC variability experimentally observed canot be explained if the number of released molecules is much higher than we used because a much higher number of molecules would induce saturation of AMPA receptors. If saturation occurs following the release of a single vesicle, only a very small EPSC variability, mainly due to the receptor opening probability, should be observed. The observed stochastic variability of the AMPA response is then a direct consequence of the parameters we have tested in the present simulations. References Clements, J.D., Lester, R.A., Tong, J., Jahr, C.E., Westbrook, G.L., The time course of glutamate in the synaptic cleft. Science 258, 11498/ Clements, J.D., Feltz, A., Sahara, Y., Westbrook, G.L., Activation kinetics of AMPA receptor channels reveal the number of functional agonist binding sites. J. Neurosci. 18, 119/127. Colquhoun, D., Binding, gating, affinity and efficacy: the interpretation of structure-activity relationships for agonists and of the effects of mutating receptors. Br. J. Pharmacol. 125, 923/947. Edelstein, S.J., Schaad, O., Henry, E., Bertrand, D., Changeux, J.P., A kinetic mechanism for nicotinic acetylcholine receptors based on multiple allosteric transitions. Biol. Cybern. 75, 361/379. Forti, L., Bossi, M., Bergamaschi, A., Villa, A., Malgaroli, A., Loose-patch recordings of single quanta at individual hippocampal synapses. Nature 388, 874/878. Jonas, P., Major, G., Sakmann, B., Quantal components of unitary EPSCs at the mossy fibre synapse on CA3 pyramidal cells of rat hippocampus. J. Physiol. (London) 472, 615/663. Liu, G., Choi, S., Tsien, R.W., Variability of neurotransmitter concentration and nonsaturation of postsynaptic AMPA receptor at synapses in hippocampal cultures and slices. Neuron 22, 395/409. Shikorski, T., Stevens, F., Quantitative ultrastructural analysis of hippocampus excitatory synapses. J. Neurosci. 17, 5858/5867. Stevens, C.F., Wang, Y., Facilitation and depression at single central synapses. Neuron 14, 795/802. Ventriglia, F., Di Maio, V., 2000a. A Brownian simulation model of glutamate synaptic diffusion in the femtosecond time scale. Biol. Cybern. 83, 93/109. Ventriglia, F., Di Maio, V., 2000b. A Brownian model of glutamate diffusion in excitatory synapses of Hippocampus. Biosystems 58, 67/74. Ventriglia, F., Di Maio, V., 2002a. Synaptic fusion pore parameters and AMPA receptor activation investigated by Brownian simulation of glutamate diffusion. Biol. Cybern., in press. Ventriglia, F., Di Maio, V., 2002b. Stochastic fluctuations of the quantal EPSC amplitude in computer simulated excitatory synapses of hippocampus. Biol. Cybern., in press. Xie, X., Liaw, J.S., Baudry, M., Berger, T.W., Novel expression mechanism for synaptic potentiation, alignment of presynaptic release site and postsynaptic receptor. Proc. Natl. Acad. Sci. USA 94, 6983/6988.