Glutamatergic synapses play a pivotal role in brain excitation. The synaptic response is mediated by the activity of two receptor types (AMPA and NMDA). In the present paper we propose a model of glutamatergic synaptic activity where the fast current generated by the AMPA conductance produces a local depolarization which activates the voltage- and $[Mg^{2+}]$-dependent NMDA conductance. This cooperative effect is dependent on the biophysical properties of the synaptic spine which can be considered a high input resistance specialized compartment. Herein we present results of simulations where different values of the spine resistance and of the $Mg^{2+}$ concentrations determine different levels of cooperativeness between AMPA and NMDA receptors in shaping the post-synaptic response.

Spiny neurons of striatum receive glutamatergic synapses on dendritic spines on the neck of which project dopaminergic synapses. Dopamine modulates, by D1 type receptors, the glutamatergic synapses by inducing the phosphorylation of AMPA and NMDA receptors which produces an increased amplitude response. Herein we present a model where, in addition to phosphorylation, the direct modulation by dopamine of the spine resistance can cooperate in producing the observed effect on some of these synapses.

Due to its extreme importance for brain activity, the function of the excitatory synapse is the object of a huge amount of researches investigating the specific contribution to the synaptic response of fairly all its structural elements. Here, we utilized amodel of hippocampal synapse to describe the random dispersion of the response evaluated from the dispersion of the amplitude peak of the miniature Excitatory Post-Synaptic Current (mEPSC). The model is based on time discretized Langevin Equations which describe the Brownian motion of Glutamate molecules released by a neurotransmitter vesicle within the synaptic cleft, their collisions with the structural elements, their binding to post-synaptic receptors and their final spillover. The value of the amplitude peak of the computed mEPSCs was put in relationship with different binding probabilities and different number of AMPA receptors. The dispersion has been used to compute an appropriate value of the binding probability of Glutamate molecules to post-synaptic receptors.

The dopamine neurotransmitter regulates important neural pathways and its action in the brain is very complex. When dopaminergic neurons make synapses on spiny neurons of the striatum nucleus, they tune the responsiveness of glutamatergic synapses by means of the dopamine D1 and D2 receptors. We studied the effect of dopamine D1 receptors on glutamatergic synapse of GABAergic spiny neurons in striatum nucleus where they are located on the neck of the same spine. The action of dopamine consists essentially in promoting the phosphorylation of AMPA and NMDA receptors thus increasing the Excitatory Post Synaptic Current peak amplitude. The consequence is a cooperative effect of glutamatergic and dopaminergic synapses for the regulation of the GABAergic neuronal code. The mechanisms by which the phosphorylation induces the increase of the EPSC amplitude still remain unclear although the lack of this regulation can be involved in several pathologies as, for example, the Parkinson's disease. We tested, by computational experiments based on our model of glutamatergic synapse, three parameters of the synaptic function that could be involved in dopamine action: (a) time binding of glutamate to receptors; (b) open probability of the receptors; and (c) single receptor conductance. For different reasons, any of the three parameters could be responsible of the increased EPSC-dopamine-dependent. Our computational results were compared and discussed with experimental results found in literature. Although for our model both the open probability and the single receptor conductance can reproduce the phosphorylation effect of dopamine, we argue that the dopamine effect consists essentially in an increase of the single receptor conductance due to a 3D rearrangement of the phosphorylated receptors.

The excitatory synaptic function is subject to a huge amount of researches and fairly all the structural elements of the synapse are investigated to determine their specific contribution to the response. A model of an excitatory (hippocampal) synapse, based on time discretized Langevin equations (time-step = 40 fs), was introduced to describe the Brownian motion of Glutamate molecules (GLUTs) within the synaptic cleft and their binding to postsynaptic receptors. The binding has been computed by the introduction of a binding probability related to the hits of GLUTs on receptor binding sites. This model has been utilized in computer simulations aimed to describe the random dispersion of the synaptic response, evaluated from the dispersion of the peak amplitude of the excitatory post-synaptic current. The results of the simulation, presented here, have been used to find a reliable numerical quantity for the unknown value of the binding probability. Moreover, the same results have shown that the coefficient of variation decreases when the number of postsynaptic receptors increases, all the other parameters of the process being unchanged. Due to its possible relationships with the learning and memory, this last finding seems to furnish an important clue for understanding the basic mechanisms of the brain activity. © 2014 Springer Science+Business Media.

A model of DOPA modulation of AMPA activity

Based on our more recent simulations, we studied the strange phenomenon of the dispersion of the peak amplitude of the synaptic response. This is produced when, all the other parameters of the synapse model remaining unchanged, by changing only the seed for the initialization of the random number generator, the synaptic response assumes amplitude peaks with different values in different simulations.

Mathematical models of the excitatory synapse are providing valuable information about the synaptic response. The effects of several synaptic components on EPSC variability have been tested by computer simulation. Our model, based on Brownian diffusion of glutamate in the synaptic cleft, is basically the same we have used in previous papers but parameters have been upgraded according to the new experimental findings. The presence of filaments into the synaptic cleft and the number and the ratio of AMPA and NMDA receptors have been the main parameters upgraded. A different way of computing the binding probability of glutamate molecules to receptors by means of geometrical considerations has been also used. The obtained results were more precise and they suggested that the new elements can play a significant role in the stochastic variability of the synaptic response. Nevertheless, new problems arise concerning the value of the lower limit of the binding probability.

Variable Binding Probability in AMPA Receptors and AMPARs Trafficking Ventriglia Francesco Istituto di Cibernetica "E.Caianiello" del CNR Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy franco@ulisse.cib.na.cnr.it Di Maio Vito Istituto di Cibernetica "E.Caianiello" del CNR Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy vito.dimaio@cnr.it ABSTRACT Time dependent modifications in AMPA receptors (AMPARs) post-synaptic response in excitatory synapses of brain seem to have a great importance in learning and me mory formation. Although a huge amount of experimental and theoretical researches have been carried on this argument, the basic mechanisms regulating AMPARs activity and AMPARs trafficking remain not completely clarified. In this contest, and to eliminate uncertainties in the synaptic response deriving from the difficulties of the experimental procedures, modeling and simulation studies have been applied to single excitatory synapses [1], [4], [6], [8], [13]. In previous papers we modeled and simulated a hippocampal excitatory synapse at ultra-fast time scale (simulation time step: tens of femtoseconds), to obtain more detailed information on synaptic dynamics [9], [10], [11], [12]. We demonstrated that the stochastic variability of the Excitatory Post Synaptic Current (EPSC) amplitude -reported by experimental studies- can be produced by intrinsic random variations in basic pre-synaptic elements, such as volume and docking position of neurotransmitter vesicles [9], [10], [11]. Moreover, analyzing the effects of structural elements not previously considered by modelers, such as filaments extending across the synaptic cleft at the external of the AZ/PSD volume [2] , [14], it was found that their presence induced a small, but significant, increase in the response of the synapse. For a volume reduction of the free flying space of about 50%, we observed that the increase of the response reached a value of about 20% [12]. In the meantime the experimental research furnished more exact values about the dimensions of post-synaptic receptors -much larger than previously supposed- and about their number -smaller [3], [5]. These new experimental findings were utilized to improve the parameters of the model and to deepen the results of our investigation. The basic geometry of the simulated model of excitatory synapse is constituted by a pre-synaptic active zone (AZ) -where vesicles are docked- juxtaposed to a Post Synaptic Density (PSD) -at a distance of 20 nm- containing AMPA and NMDA receptors [7]. The centers of the two zones usually lie on a common unique axis and different numbers of AMPARs and NMDARs are distributed over the PSD. The two receptor types appear to be mixed together with the AMPARs far exceeding the NMDARs. Both types have been modeled as small cylinders protruding from the PSD zone in the synaptic cleft, each having two binding sites for Glutamate neurotransmitter molecules. Trans-synaptic fibrils, crossing the cleft and disposed according to a regular spacing, have been considered at the external of the AZ/PSD space till to the boundary of the synapse. The mathematical model is based on the description of the Brownian motion of Glutamate molecules within the synaptic cleft through Langevin equations, a well known example of stochastic equations. The Langevin equations, in standard form, appear as: d r i ( t ) = v i ( t ) dt (1)172 BIOCOMP2012 - Abstracts m d v i ( t ) = - g v i ( t ) + 2 e g Ë i ( t ) dt (2) where r i ( t ) and v i ( t ) are, respectively, the position and the velocity of the generic ith Glutamate molecule and m is its molecular mass. Their discrete-time form has been used to simulate the diffusion of neurotransmitter molecules within the synaptic cleft and to obtain their binding times to postsynaptic AMPA and NMDA receptors. The binding times on AMPARs have been used to compute the synaptic response, as EPSC. A new series of simulations has been carried out to take into account the new data. In fact, the availability of more precise information about the structural elements allowed us to investigate in greater detail the dynamics of the binding of glutamate molecules to postsynaptic receptors -by investigating the probability of binding- and the effects of the so-called trafficking of AMPA receptors, which is supposed to be at the base of memory and learning phenomen

The excitatory synaptic response, the base of all the brain activity, is the object of a huge amount of researches and fairly all the structural elements of this synapse are investigated to stress out how they contribute to shape the response. To this aim mathematical models and computer simulation are providing valuable information. A previous model of excitatory synapse, based on Brownian motion of Glutamate (GLUT) molecules within the synaptic cleft and their bind- ing to post-synaptic receptors [Ventriglia and Di Maio (2000), Ventriglia and Di Maio (2003), Ventriglia (2011)], was enhanced by upgrading some parameters to new values, produced by re- cent experimental findings [Armstrong and Gouaux (2000), Mainen et al. (2002), Santos et al. (2009), Tichelaar et al. (2004), 1]. Since in a previous paper, in which the binding to receptors was stud- ied via mEPSCs linked to AMPARs trafficking, demonstrated the possibility to define an inferior limit for the binding probability between the GLUT and receptor binding site, this fact was more thoroughly investigated by a longer series of computer simulations [Ventriglia and DiMaio (2012)]. Also, the new data, which define better the 3D structure (and dimensions) of AMPA recep- tors, allowed us to investigate the effects produced on the synaptic response of the change of the heigth of the part of AMPA receptors which protrudes in the cleft from the PSD zone [Tichelaar et al. (2004)]. Moreover, the decreasing of the response with the increasing of the excentricity of the position on AZ of the releasing vesicle has been demonstrated by using also the space distribution of number of collision of GLUT molecules on receptor binding sites. Also, the dynamics of the binding has been studied by means of the description of the inter-collision times among single Glut molecules and the binding sites of the several AMPA receptors which they visit without success till the final binding.

Mathematical models of the excitatory synapse are furnishing valuable information about the synaptic response. Based on Brownian-diffusion of glutamate molecules, a synapse model was utilized to investigate the synaptic response on a femto-second time scale by the use of a parallel computer. In particular, the presence of fibrils crossing the synaptic cleft was simulated, which could have a role in shaping the brain activity. To this aim the model of synapse was modified by considering trans-synaptic filaments with diameters ranging from 7. nm to 3. nm, disposed on a grid with spacing of 14. nm or 8. nm. The simulation demonstrated that the presence of filaments induced an increase in the synaptic response, most likely linked to an increment in the probability of encounter between glutamate molecules and receptors. The increase was small - from 5 to 20%, but metabolic and functional considerations provide substantive hints about the importance of these small changes for brain activity. Moreover, it was shown that the presence of filaments made more stable the response of the synapse to random variations of pre-synaptic elements. Originated by these computational results, some inferences about the biological bases of mind diseases such as autism, mental retardation and schizophrenia, are reported in the Discussion. © 2010 Elsevier Ireland Ltd.