Synaptic dendritic activity modulates the single synaptic event

Abstract

Synaptic transmission is the key system for the information transfer and elaboration among neurons. Nevertheless, a synapse is not a standing alone structure but it is a part of a population of synapses inputting the information from several neurons on a specific area of the dendritic tree of a single neuron. This population consists of excitatory and inhibitory synapses the inputs of which drive the postsynaptic membrane potential in the depolarizing (excitatory synapses) or depolarizing (inhibitory synapses) direction modulating in such a way the postsynaptic membrane potential. The postsynaptic response of a single synapse depends on several biophysical factors the most important of which is the value of the membrane potential at which the response occurs. The concurrence in a specific time window of inputs by several synapses located in a specific area of the dendritic tree can, consequently, modulate the membrane potential such to severely influence the single postsynaptic response. The degree of modulation operated by the synaptic population depends on the number of synapses active, on the relative proportion between excitatory and inbibitory synapses belonging to the population and on their specific mean firing frequencies. In the present paper we show results obtained by the simulation of the activity of a single Glutamatergic excitatory synapse under the influence of two different populations composed of the same proportion of excitatory and inhibitory synapses but having two different sizes (total number of synapses). The most relevant conclusion of the present simulations is that the information transferred by the single synapse is not and independent simple transition between a pre- and a postsynaptic neuron but is the result of the cooperation of all the synapses which concurrently try to transfer the information to the postsynaptic neuron in a given time window. This cooperativeness is mainly operated by a simple mechanism of modulation of the postsynaptic membrane potential which influences the amplitude of the different components forming the postsynaptic excitatory response.

Introduction

Depending on the level of investigation, information processing in the brain is attributed to the neural networks activity, to the single neuron activity, to the dendritic computation (London and Häusser 2005) or to the activity of the single synapse. Whatever it is the correct level of investigation, the main structure in charge of information transfer among neurons is the synapse. In fact, spike generation in the single neuron, neural network activity, and dendrtitic computation, all depend on the synaptic input integration.

Among the different types of synapses in the brain, the Glutamatergic (Glut) synapses represent $\sim 80\%$ of the total (Gulyás et al. 1999; Megías et al. 2001; Villa and Nedivi 2016) and each pyramidal neuron (the larger majority of neurons in the cortex and in the hippocampus) receives a number of synaptic contacts ranging $5÷30\times {10}^3$ (Gulyás et al. 1999; Megías et al. 2001). In general, $10-20\%$ of these contacts are inhibitory [GABAergic of (${\text{GABA}}_{\mathrm{A}}$ type)], and are located mainly on the dendritic shafts, on the soma and its proximity, while the rest are excitatory of Glut type (Buh and Somogyi 1994; Gulyás et al. 1999; Megías et al. 2001; Villa and Nedivi 2016). The larger part of the Glut synapses are located on spines protruding from the dendritic shafts and, in a typical pyramidal neuron, the spine density ranges $2.5÷5.25/\mathrm{\mu }m$ depending on the different dendritic branches (Bourne and Harris 2011). The large number of Glut synapses per neuron and their dendritic density suggest that the larger part of information transferred among neurons in the brain is operated by this synaptic type. Not surprisingly then, their malfunction is associated with serious brain pathologies with symptoms of memory and learning deficits, as, for example, Alzheimer (Sheng et al. 2012; Rudy et al. 2015), Parkinson (Gardoni and Di Luca 2015) and autism (Rojas 2014).

Anomalies in EEG $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ and $\mathrm{\theta }$ waves, which mainly depend on Glut synaptic activity, are often associated with impairments of other important brain performances (see for example, Bennett et al. 1973; Kupper et al. 1998; Zusho et al. 2003; Kumar et al. 2010; Averbeck et al. 2006; Rashid et al. 2011).

The response of a single Glut synapse is mediated by two types of receptors: AMPA ($\mathrm{\alpha }$-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid sensitive) and the co-localized NMDA (N-methyl-D-aspartic acid sensitive) receptors.

The excitatory post synaptic current (EPSC) produced by the Glutamatergic synapse (Glut) shows a large variability ranging $4÷100$ pA (average 25 pA and Coefficient of Variation, $C.V.=0.2÷0.7$) also in the same synapse (see for example Forti et al. 1997; Liu et al. 1999; Hanse and Gustafsson 2001). This variability is due to several factors as for example structural and functional modifications, activity dependent, which can induce Short Term Potentiation (STP), Long Term Potentiation (LTP) and Long Term Depression (LTD) (Meldolesi 1995; Larkman and Jack 1995; Martin et al. 2000; Malinow and Malenka 2002; Watt et al. 2004; Nicoll and Schmitz 2005; Rao and Finkbeiner 2007; Raymond 2007; Bliss and Collingridge 2013; Zhang et al. 2013; Baudry et al. 2015; Volianskis et al. 2015; Lisman 2017) and stochastic processes (Ventriglia and Di Maio 2002, 2003b, a; Di Maio et al. 2017; Di Maio 2019; Di Maio and Santillo 2020) the origin of which can be presynaptic, postsynaptic and extrasynaptic (for a review see Di Maio et al. 2017, 2018a, b; Di Maio 2019; Di Maio and Santillo 2020). The number of molecules in the releasing vesicle and its position with respect to the central axis of the synapse play significant roles among the stochastic factors of variability of the single EPSC (Ventriglia and Di Maio 2002, 2003a, b, 2013a, b, and for reviews Di Maio et al. (2017); Di Maio (2019); Di Maio and Santillo (2020)). Moreover, the inter synaptic source of variability depends on the location of each synapse, on the biophysical properties of the dendrites, on their location (Husser 2001; Rumsey and Abbott 2006; Harnett et al. 2012; Gulledge et al. 2012; Yuste 2013; Beierlein 2014; Weber et al. 2016) and on the activity of other neighbor synapses (Di Maio et al. 2018a, b; Di Maio and Santillo 2020).

In our recent papers, we have shown that, among the extrasynaptic factors, the response of generic Glut synapse (S) can be modulated in amplitude and time course by pools of excitatory synapses firing in a time window compatible with the event of S (Di Maio et al. 2018a, b; Di Maio and Santillo 2020). This effect is due to the depolarization produced at the dendritic level by the excitatory activity of the pool (Di Maio et al. 2018a, b; Di Maio and Santillo 2020) and can either be independent or coincident with the depolarization induced by postsynaptic spike back propagation. (Sjöström et al. 2001; Rozsa et al. 2004).

Dendritic activity is not exclusively of excitatory type. The level of dendritic excitation is limited by ${\text{GABA}}_{\mathrm{A}}$ [$\mathrm{\gamma }$-amino-butirric acid sensitive (GABA)] inhibitory synapses (Buh and Somogyi 1994; Gulyás et al. 1999; Megías et al. 2001; Treiman 2001; Klausberger et al. 2003; Palmer et al. 2012; Villa and Nedivi 2016; D’Onofrio et al. 2019). The importance of the GABAergic neuron in the information processing of neural networks is well known because, although they can fire up to 200 Hz, they usually fire in the $\theta$ rhythm domain ($4÷8$ Hz) which is associated to the learning (Klausberger et al. 2003). Moreover, GABAergic activity, repolarizing the membrane volgage (see for example Gray and Johnston 1985; Treiman 2001), counterbalance the excitation induced by excitatory inputs on the pyramidal neurons (Leranth et al. 1999; Buzsáki 2002; Guo et al. 2012) preventing seizure and epilepsy (see for example Treiman 2001; Gonzlez et al. 2015).

In terms of information transmission, if we can assume that a sequence of spikes emitted by a neuron in a given time window, represents a “word” for a stimulus in the neuronal language (“neuronal code”), its synaptic translation should be a sequence of Excitatory Post Synaptic Potentials (EPSPs) (“synaptic code”) (Di Maio 2019; Di Maio and Santillo 2020) and the elementary bit of information (a “letter” of the “word”) should be represented by the single spike and the single EPSP for the “neuronal code” and the “synaptic code”, respectively. However, while the single spike is a stereotyped event obeying to the “all or none” law (amplitude, shape and duraton are constant), the same does not hold for the EPSP which varies according to the variation of the EPSC (see above). Moreover, the probability that a presynaptic spike will produce an EPSP range $0.2÷0.91$ (Dobrunz and Stevens 1997; Park et al. 2012). The vesicle release probability is regulated by the interaction of the free $C{a}^{2+}$ concentration in the presynaptic button and a protein complex (SNARE complex). The arrival of a presynaptic spikes transiently modify the $C{a}^{2+}$ concentration favoring the vesicle release. Depending on the past activity of the button, the SNARE complex can be in different states each associated with a different release probability (Han and Jackson 2006). The increase of the release probability activity dependent is the base of the so called presynaptic LTP (Kokaia 2000; Guerrier and Holcman 2018). However, in terms of information transfer and synaptic code, the fact that the release probability $P_r<1$ excludes that each presynaptic spike is mirrored by a synaptic EPSP and, hence, a smaller number of EPSP (synaptic code) will be produced for a given number of spikes forming a presynaptic word (Di Maio 2019; Di Maio and Santillo 2020).

The lack of coherence between presynaptic spikes and EPSP sequences rises an important question: “How is coded the presynaptic information at the synaptic level and what is the characteristic of the EPSP which codes for the single elementary bit of synaptic information? ”. Until this question will not be answered, the study of information transmission and processing in the brain has not big chance of success. Not pretending to definitively answer this question, we consider the EPSP amplitude and its variability as the best candidate for the expression of the synaptic code and then for the transfer of the correct information among neurons. To shade more needed light on the role of the EPSP amplitude on the synaptic code formation, in our most recent papers we have studied the EPSP amplitude variability as a function of the simultaneous activity of populations (pools) of excitatory Glut synapses located at a distance such to influence the EPSP induced by a presynaptic spike in a generic synapse S (Di Maio et al. 2018a, b). We have found that the main effect of the pool activity on the amplitude of the single synaptic event is due to the degree of membrane depolarization induced by the pool. The level of depolarization modulates the driving force for the EPSC generation and, consequently, the EPSP amplitude (Di Maio et al. 2018a, b).

For the present work, ${\text{GABA}}_{\mathrm{A}}$ synapses were added to the active pool in the fixed proportion of $20\%$ (Buh and Somogyi 1994; Gulyás et al. 1999; Villa and Nedivi 2016). We used two pools of size $N_p=100$ and $N_p=200$ containing respectively 20 and 40 ${\text{GABA}}_{\mathrm{A}}$ synaptic inputs. Since we are interested solely to the excitatory and inhibitory synaptic cooperativeness in shaping the single synaptic response, we excluded the contribution of a possible spike back propagation. Ideally, we simulated electrophysiological experiments where EPSP were recorded at Post Synaptic Density (PSD) level (see below) by blocking postsynaptic spikes generation (for example by TTX).

The EPSPs were generated under the influence of excitatory and inhibitory synapses active with different mean excitatory and inhibitory firing frequencies ($\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$).

The peak level of the EPSP ($V_{\mathit{EPSP}}^{\mathit{peak}}$) and its amplitude ($V_{\mathit{EPSP}}^{\mathit{amp}}$) were considered in terms of variations of the membrane potential at the PSD level ($V_{\mathit{PSD}}$).

The obtained results show that the two parameters analyzed were modulated in a wide range by the synaptic pool activity depending on the size of the pool ($N_p$) and on the relative mean firing frequencies ($\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$).

Although we cannot say an ending word on the nature of the elementary bit of the synaptic information, we conclude that most probably it is related to the EPSP or EPSC amplitude. Moreover, the elementary bit of information forming the “synaptic code” is not simply expressed by the transmission of a single presynaptic spike but strongly depends on the state of the postsynaptic neuron (dendritic activity). Both the single bit of information and the whole synaptic word depend on the integration of all the information (inputs) the postsynaptic neuron receives in a given time window and not only on the information transmitted by the single presynaptic spike.

Methods

We modified the simulation system, already used in our previous papers (Ventriglia and Di Maio 2013a, b; Di Maio et al. 2016a, b, 2018a, b) to include the inhibitory synaptic activity. In short, our simulation system is based on two different simulation programs. A first program written in Parallel (MPI) Fortran simulates the diffusion process and is based on a fine description of the synaptic space and a time step in the order of the femtoseconds [($40\times {10}^{-15}$ s, (40 fs)] (Ventriglia and Di Maio 2013a, b). This simulation produces three matrices containing the binding times of a first and a second molecule of Glu to the postsynaptic receptors and a matrix which identifies the receptor type. These matrices are used to generate the EPSP by a second $C++$ simulation program. Since we are interested in the study of the contribution of the inhibitory synapses to the single excitatory event, we kept constant all the parameters related to synaptic diffusion (see Table 1) as well as the biophysical parameters of the synaptic spine and of the electrical circuit (see Table 2 and Fig. 1). In the following a brief description of the simulation systems and of the simulation procedures.

Table 1.

Parameters for the simulation of the synaptic geometry and for the Brownian diffusion of the molecules of Glu

Parameter	Symbol	Value
Geometrical parameters
Synaptic diameter	$d_S$	440 nm
PSD (AZ) diameter	$d_{\mathit{PSD}}$	220 nm
Vesicle diameter	$d_v$	23 nm
Height of the fusion pore	$h_p$	12 nm
Synaptic Cleft Height	$h_s$	20 nm
Fibrils diameter	$d_f$	14 nm
Receptor diameter	$d_r$	14 nm
Receptor height	$h_r$	7 nm
AMPA receptors	$N_{\mathit{AMPA}}$	55
NMDA receptors	$N_{\mathit{NMDA}}$	13
Diffusion parameters
Number of Glu molecules	$N_{\mathit{glu}}$	780
Temperature	T	$298.{16}^{o}K$
Coefficient of diffusion	D	$7.6\times {10}^{-6}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$
Molecular Mass of Glu	m	$2.4658025\times {10}^{-25}$ Kg
Pore opening velocity	$v_{\mathit{areal}}$	$31.4{\mathrm{nm}}^2{\mathrm{ms}}^{-1}$
Presynaptic absorbing probability	$P_{\gamma }$	$3\times {10}^{-6}$

Table 2.

Dendritic and synaptic simulation parameters

Parameter	Variable	Value
Resting potential	$V_r$	$-65$ mV
Synaptic reversal potential	$V_{\mathit{rev}}$	0 mV
PSD input resistance	$R_{\mathit{PSD}}$	$500M\Omega$
Neck resistance	$R_n$	$100M\Omega$
AMPA receptor Conductance	$g_{\mathit{AMPA}}$	$15\pm 10p{S}^a$
NMDA receptor Conductance	$g_{\mathit{NMDA}}$	$40\pm 15p{S}^a$
AMPA mean binding time	${\beta }_{\mathit{AMPA}}$	5 ms
NMDA mean binding time	${\beta }_{\mathit{NMDA}}$	150 ms
AMPA open probability	$P_{o_{\mathit{AMPA}}}$	0.70
NMDA open probability	$P_{o_{\mathit{NMDA}}}$	0.70
Esponential rising time constant	${\tau }_1$	U(3, 10)
Esponential decaying time constant	${\tau }_2$	U(15, 30)
Exponential amplitude constant	k	U(0, 2)

^a(Dingledine et al. 1999; Traynelis et al. 2010)

Fig. 1.

Simplified dendrite/spine electric circuit: $V_{\mathit{PSD}}$ is the membrane voltage at the level of PSD; $R_{\mathit{PSD}}$ is the PSD input resistance; $G_s$ is the total synaptic conductance; $R_n$ is the neck resistance; $C_n$ is the neck capacitance; $R_l$ is the longitudinal dendritic resistance; and $R_d$ is the dendritic input resistance

Geometry of the synaptic space

The synaptic structure can be geometrically idealized as a couple of concentric cylinders having the same height [synaptic cleft ($h=20$ nm)]. The internal cylinder is delimited on the top by the so called Active Zone (AZ) which is the area of the presynaptic button where vesicles filled of glutamate (Glu) are docked. The bottom side is limited by PSD containing the receptors. For the present simulation we positioned on the PSD 55 AMPA and 13 NMDA receptors (see Table 1). The outer cylinder delimits the synaptic space. The radii of the inner and outer cylinder were 110 nm and 220 nm, respectively. These parameters are typical of a mean size glutamatergic synapse of the hippocampus (Clements et al. 1992; Clements 1996).

A cylinder with a diameter of 14 nm, protruding 7 nm from the PSD surface, with two binding sites of circular shape (b1 and b2) positioned on the upper part, was the simulated structure of a single receptor. The coronal space between the inner and the outer cylinder was filled with fibrils, simulated like cylinders regularly spaced every 22 nm, with diameter of 14 nm, anchoring each other the pre and postsynaptic cell (Zuber et al. 2005; Ventriglia 2011; Ventriglia and Di Maio 2013a, b).

The simulation starts when a spherical vesicle, with the inner diameter of 23 nm and containing 780 molecules of Glu (centered at AZ), forms a fusion pore which opens with an areal velocity. Once the diameter of the pore exceed that a molecule of Glu, the free Brownian diffusion of the neurotransmitter into the synaptic cleft could start.

Diffusion

Brownian diffusion of the molecules of Glu was computed by Langevin diffusion equations which have the following form

$$\begin{array}{cc}\hfill \frac{d}{\mathit{dt}}{\mathbf{r}}_i(t)& ={\mathbf{v}}_i(t)\hfill \end{array}$$ 1

$$\begin{array}{cc}\hfill m\frac{d}{\mathit{dt}}{\mathbf{v}}_i(t)& =-\gamma {\mathbf{v}}_i(t)+\sqrt{2ϵ\gamma }{\mathit{\lambda }}_i(t)\hfill \end{array}$$ 2

where ${\mathbf{r}}_i(t)$ is the position vector ($x_i,y_i,z_i)$, ${\mathbf{v}}_i(t)$ is the velocity in the 3D space of the ith molecule, m is its molecular mass, ${\mathit{\lambda }}_i(t)$ is a stochastic force, $\gamma$ is a friction parameter ($\gamma =\frac{k_{B}T}{D}$; $k_B$ is the Boltzman constant, T the absolute temperature and D is the diffusion coefficient) and $ϵ=K_{B}T$ (for a detailed explanation of these parameters see Glavinovic 1999; Ventriglia and Di Maio 2000a, b, 2003a, b; Cory and Glavinovic 2006; Ventriglia and Di Maio 2013a, b, and see Table 1). The starting velocity for each each molecule (${\mathbf{v}}_{\mathbf{0}}$) was obtained by a Maxwell distribution.

$$\begin{array}{cc}\hfill {\mathbf{r}}_i(t+\Delta )& ={\mathbf{r}}_i(t)+{\mathbf{v}}_i(t)\Delta \hfill \end{array}$$ 3

$$\begin{array}{cc}\hfill {\mathbf{v}}_i(t+\Delta )& ={\mathbf{v}}_i(t)-\gamma \frac{{\mathbf{v}}_i(t)}{m}\Delta +\frac{\sqrt{2ϵ\gamma \Delta }}{m}{\mathbf{\Omega }}_i\hfill \end{array}$$ 4

with $\Delta$ being the time step and ${\mathbf{\Omega }}_i$ a vector with three Gaussian ($G(\mu =0.0,\sigma =1.0)$) components describing the space position of each molecule ($x_i,y_i,z_i$). The surfaces of all the synaptic structures were considered reflecting except:

1. the presynaptic surface because presynaptic metabotropic receptors can bind, although with a small probability ($P_{\gamma }$), molecules of Glu ;

2. the lateral limits of the synaptic space because the high density of receptors for Glu on the surrounding glial cell makes negligible the probability that an escaped molecule returns into the synaptic space (absorbing boundary);

3. the receptor binding sites.

During the Brownian diffusion, the molecules of Glu were considered volume free [i.e., identified only by their central coordinates (x, y, z)] except when they approach a receptor binding site. In this case a molecule of Glu is represented by an ovoid shape because the binding probability to the receptors has been computed by geometrical considerations. (for a detailed discussion on this topic see Ventriglia 2011; Ventriglia and Di Maio 2013a, b). In order to have numerically tractable matrices, the PSD was considered as a circle inscribed in a square (the matrix) composed of $10\times 10$ tiles. An identity matrix ($\mathbf{R}$) coded for the position (randomly chosen) of the AMPA and NMDA receptors on the PSD such that

$$\begin{array}{c}\hfill r_{i,j}\in \mathbf{R}=\left\{\begin{array}{cc}0\hfill & \text{if at pos.}(i,j)\text{there is no receptor on the tile}\hfill \\ 1\hfill & \text{if at pos.}(i,j)\text{the tile is occupied by an AMPAR}\hfill \\ 2\hfill & \text{if at pos.}(i,j)\text{the tile is occupied by an NMDAR}\hfill \end{array}\right)\end{array}$$

All the tiles located out of the circle have no receptors. Two matrices of the same size (${\mathbf{T}}_{b1}$ and ${\mathbf{T}}_{b2}$) contained the binding times $t_{b1}$ and $t_{b2}$ of a first and a second molecule of Glu to the receptors, respectively

$$\begin{array}{c}\hfill t_{b{1}_{i,j}}\in {\mathbf{T}}_{b1}\left\{\begin{array}{cc}=0\hfill & \text{if}0\text{molecule of Glu has bound the receptor}\hfill \\ =t_1\hfill & \text{if}1\text{molecule of Glu has bound the receptor at time}{t}_1\hfill \end{array}\right)\end{array}$$

and

$$\begin{array}{c}\hfill t_{b{2}_{i,j}}\in {\mathbf{T}}_{b2}\left\{\begin{array}{cc}=0\hfill & \text{if}0\text{or}1\text{molecule of Glu has bound the receptor}\hfill \\ =t_2\hfill & \text{if a}2\mathit{nd}\text{molecule of Glu have bound the receptor at the time}{t}_2\hfill \end{array}\right)\end{array}$$

Since we are only interested in the variability of the EPSP amplitude, only the receptor states which contribute to the total conductance need to be considered and, consequently, the following simplified Markov chains were used for AMPA and NMDA receptors. For the AMPA receptors we used

where the $R_0,R_1,R_2\text{and}{R}_2^{+}$ states refers to the not bound, single bound, double bound and double bound active (open) AMPA receptor. The state $R_1^{+}$ is not considered neither for the AMPA nor for the NMDA receptors because the probability that a single bound receptor opens is negligible. For the NMDA receptor, the Markov chain is complicated by the $M{g}^{2+}$ block which is membrane voltage dependent (see below)

where $\ast$ denotes the $M{g}^{2+}$ block of the receptor.

The spine circuit

The biophysical properties of the dendrites vary according to their morphology and position with respect to the soma. Functional differences between different areas of the dendrites depend on several parameters like for example the diameter and the presence of voltage gated channels (Rall 1962; Rall and Rinzel 1973; Arac et al. 2007). The most peripheral branches of the dendritic tree, usually show a high input impedance because of the very small diameter (Rall 1962; Rall and Rinzel 1973). For this property, synapses located far from the soma often produce EPSPs with higher amplitude than those produced by synapses located closer determining the so called Dendritic Democracy. This definition rises because, according to several authors (see for example Husser 2001; Rumsey and Abbott 2006), this permits to the inputs arriving in the far dendritic areas to have, more or less, the same influence on the postsynapic neuronal activity of the inputs arriving closer to the soma (Husser 2001; Rumsey and Abbott 2006). Moreover, in more recent papers, also the biophysical properties of the spines have been considered for their influence in the information transfer since they can be considered as separate electrical compartments, functionally distinct from the dendrites, with the ability to modify the EPSP properties (Nimchinsky et al. 2002; Araya et al. 2006; Nuriya et al. 2006; Arellano et al. 2007; Grunditz et al. 2008; Allam et al. 2015; Gulledge et al. 2012; Yuste 2013; Araya et al. 2014; Palmer and Stuart 2009; Harnett et al. 2012; Tønnesen et al. 2014; Di Maio et al. 2016b, 2018a; Weber et al. 2016; Kwon et al. 2017). The spines, depending on their morphology, can give different levels of modulation of the information arriving at the synapses (Palmer and Stuart 2009; Tønnesen et al. 2014; Weber et al. 2016). Many electrical compartmentalization have been proposed for the spine and some of them include several sub-circuits and components each of which, can influence the output (see for example Nimchinsky et al. 2002; Harnett et al. 2012; Yuste 2013; Tønnesen et al. 2014). However, although a detailed description of the electrical equivalent circuits of a spine can be very interesting, for our goal the key factor is the voltage difference at the two sides of the PSD ($V_{\mathit{PSD}}$) (Di Maio et al. 2018a, b; Di Maio 2019; Di Maio and Santillo 2020) and then we limited our electrical spine model to the simplified one represented in Fig. 1. The value of $V_{\mathit{PSD}}$, in fact determines both the amplitude of the EPSC/EPSP and the activation of the NMDA receptors which are voltage dependent. Being the PSD crowded of proteins (the receptors) it behaves essentially as an high impedance resistor (Fig. 1) with a very small (not significant) capacitance component. Another important electrical component of the spine is the neck. Its structure and size determines the amount of current flowing either from the PSD to the dendrite and vice versa (Nimchinsky et al. 2002; Araya et al. 2006; Grunditz et al. 2008; Yuste 2013; Tønnesen et al. 2014; Araya et al. 2014). Moreover, the presence of $C{a}^{2+}$ voltage gated channel suggests an important role of this structure in the transfer of information from the spine head to the dentrite (Nimchinsky et al. 2002; Majewska et al. 2000). In order to simplify and considering that our goal in the present paper is limited to the influence of dendritic inhibitory activity on the single synaptic event, we have neglected in our model the detailed contribution of the neck and we have adopted the following simplified equivalent electrical circuit which considers the neck nothing but a simple resistor ($R_n$ Fig. 1) (Di Maio et al. 2018a, b). The value of the spine resistance in its whole is computed and considered differently by the different authors but always in an order of magnitude ranging from hundreds of $M\Omega s$ to the $G\Omega s$ (Nimchinsky et al. 2002; Araya et al. 2006; Grunditz et al. 2008; Yuste 2013; Tønnesen et al. 2014; Araya et al. 2014). It is our opinion that its values, should be considered in the order of the $G\Omega$ but, conservatively, we adopted two in series resistances: $R_{\mathit{PSD}}=500M\Omega$ for the PSD resistance and $R_n=100M\Omega$ for the neck resistance.

Dendritic activity

The value of $V_{\mathit{PSD}}$ is influenced essentially by two components: the current produced by the dendritic activity which flows through the neck resistance ($R_n$) and the EPSC produced by the activation of the receptors conductance. For both the pools (size $N_p=100$ and $N_p=200$) simulations were conducted by using each of 7 mean excitatory firing frequencies ($\delta {\overline{\varphi }}_e:{\overline{\varphi }}_e\in 0÷6$ Hz) with each of 13 inhibitory firing frequencies ($\delta {\overline{\varphi }}_i:{\overline{\varphi }}_i\in 0÷12$ Hz). Each combination of the mean firing frequencies herein is denoted by $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ such that, for example, $\Phi (3,8)$ indicates that the excitatory synapses of the pool have a mean firing frequency of 3 Hz and the inhibitory ones of 8 Hz. The firing frequency of each single synapse of the pool, in each simulation was chosen according to a Gaussian distribution (${\varphi }_j=\Psi (\overline{\varphi },\sigma$) where $\overline{\varphi }$ is the mean firing frequency for the the type of synapse (excitatory or inhibitory) and the value of $\sigma$ ($\sigma =(\frac{\overline{\varphi }}{2})$) was chosen to have a constant coefficient of variation ($C.V.=0.5$) among the different simulations. The value of $\frac{1}{{\varphi }_j}$ was the parameter ($\gamma$) of a Poisson distribution $(P(\gamma ))$ by which we generated the time of occurrence of the single events of each synapse of the pool

$$\begin{array}{c}\hfill {\mathbf{t}}_j(m)={\mathbf{t}}_j(m-1)+P\left(\frac{1}{{\varphi }_j}\right),\forall t\in [t_0÷t_{\mathit{end}}]\end{array}$$ 5

where $t_j(m)>t_0$ is the time of occurrence of the mth event of the jth synapse belonging to the pool.

The contribution to the membrane voltage given by each synaptic event at the base of the spine was computed by the difference of two exponential

$$\begin{array}{c}\hfill V_e(t)=\left\{\begin{array}{cc}0\hfill & \text{if}\Delta t=t-t_e\le 0(\mathit{Event\; not}\mathit{yet\; started})\hfill \\ \alpha k\left(e^{-\frac{\Delta t}{{\tau }_2}},-,e^{-\frac{\Delta t}{{\tau }_1}}\right)\hfill & \text{if}\Delta t=t-t_e>0\hfill \end{array}\right)\end{array}$$ 6

where $t_e$ is the starting time of the event, ${\tau }_1$ and ${\tau }_2$ are the rising and decay time constants, k is a parameter related to the EPSP amplitude and $\alpha$ defines the polarity of the modification ($\alpha =1$ or $\alpha =-1$ respectively for excitatory and inhibitory synapses). The parameters ${\tau }_1$ and ${\tau }_2$ simulate the effect of the distance from the synapse S due to the cable properties of the dendritic path. So far, ${\tau }_1\text{and}{\tau }_2$ were chosen according to uniform distributions (${\tau }_1=U(3,10)$ and ${\tau }_2=U(15,30)$) for each event and were kept constant during each single run (1 s duration) but varied across the different runs. Randomization of ${\tau }_1$ and ${\tau }_2$ among the runs (1000 runs for each $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$) randomized the distance of the synapses of the pool with respect to the synapse S. The value of k was chosen according to a uniform distribution (U(0, 2)) for each single event to account for the amplitude variability in the same synapses among the different events (see above and for a review see Di Maio et al. (2017), and to have an idea of the different effects produced by the variation of k, ${\tau }_1$ and ${\tau }_2$ see figure 8 in Di Maio et al. (2018b)). The main parameters of the model we have used in the present paper are shown in Table 2.

The contribution of the pool activity to the membrane potential at the base of the spine ($V_d$) will be given by the summation of all the synaptic events produced by the synapses of the pool

$$\begin{array}{c}\hfill V_d(t)=\sum _{i=1}^N\sum _{j=1}^{M_n}{V}_{i,j}(t)\end{array}$$ 7

where N is the number of active synapses and $M_n$ is the number of events produced by the jth synapse during the simulation time (a single run); $V_{i,j}(t)=V_e(t)$ is the voltage contribution of the jth event of the ith synapse (see Eq. (6)).

The EPSP generation

The matrices $\mathbf{R}$ (identity matrix), ${\mathbf{T}}_{b1}$ (first Glu binding times) ${\mathbf{T}}_{b2}$ (second Glu binding times) were used to simulate the EPSP generation at the PSD level. Each AMPA and NMDA receptor was assigned with a maximal receptor conductance (${\widehat{g}}_r$) chosen by a Gaussian distribution (${\widehat{g}}_r=\Psi (\overline{g},\sigma )$) where the mean value $\overline{g}$ and $\sigma$ were computed considering the different conductances which depend on their sub-units composition (Jahr and Stevens 1990; Jonas and Sakmann 1992; Kupper et al. 1998; Dingledine et al. 1999; Smith et al. 2000; Tichelaar et al. 2004; Greger et al. 2007; Traynelis et al. 2010, and see Table 2). These values were stored in a matrix ($\widehat{\mathit{G}}$).

$$\begin{array}{c}\hfill {\widehat{g}}_{i,j}\in \widehat{\mathit{G}}=\left\{\begin{array}{cc}0\hfill & \mathit{if}\mathit{no}\mathit{receptor}\mathit{or}\mathit{if}\mathit{single}\mathit{bound}\mathit{receptor}\hfill \\ \Psi (\overline{g},\sigma )\hfill & \mathit{if}\mathit{receptor}\mathit{is}\mathit{AMPA}\mathit{or}\mathit{NMDA}\hfill \end{array}\right)\end{array}$$

The total maximal PSD conductance (G) was then

$$\begin{array}{c}\hfill G=\sum _{i=1}^{10}\sum _{j=1}^{10}{\widehat{g}}_{r_{i,j}}\end{array}$$

To define the time evolution of the single receptor conductance, we need a time interval ($\Delta t_a$) in which the receptor remains double bound (i.e., can be in the open state). This length depends on the receptor type and has an average of $\sim 5$ ms for AMPA receptors and can reach $400-500$ ms for NMDA. To compute the $\Delta t_a$ we used a Poisson distribution ($P(\beta$) having as parameter 5 ms for AMPA and, conservatively, 100 ms for NMDA receptors to produce a matrix of the unbinding time of the first (${\mathbf{T}}_{u1}$) and of the second (${\mathbf{T}}_{u2}$) molecule of Glu from each receptor

$$\begin{array}{cc}\hfill t_{u{1}_{i,j}}\in {\mathbf{T}}_{u1}=& t_{b{1}_{i,j}}+P(\beta )\hfill \\ \hfill t_{u{2}_{i,j}}\in {\mathbf{T}}_{u2}=& t_{b{2}_{i,j}}+P(\beta )\hfill \\ \hfill \end{array}$$

The time interval of activity of a receptor was computed as $\Delta t_a=min(t_{u1},t_{u2})-t_{b2}$(i.e., the smaller unbinding time minus the biding time of the second molecule).

In the time interval $\Delta t_a$, receptors undergo the transitions $R_2\rightleftharpoons R_2^{+}$ (close or open state). As shown in the description of our simplified Markov Chains, while for the AMPA receptors the binding of two molecules of Glu is a necessary and sufficient condition to be activated, the NMDA Receptors are blocked by $M{g}^{2+}$ and a second condition is necessary: $M{g}^{2+}$ unblocking. The NMDA-$M{g}^{2+}$ blocking system is membrane voltage and $[M{g}^{2+}]$ dependent (Jahr and Stevens 1990; Vargas-Caballero and Robinson 2004). Under physiological conditions ($[M{g}^{2+}]=1mM$), the block is almost complete at the membrane resting potential ($V_r\sim -65$ mV) while the total NMDA conductance is expressed at a membrane potential of $+40$ mV. The dependence of the NMDA conductance on the membrane potential follows a sigmoid function (Jahr and Stevens 1990; Vargas-Caballero and Robinson 2004). Accordingly, as in our most recent papers (Di Maio et al. 2016a, b, 2018a, b), we have used a sigmoid relationship in terms of unblocking probability ($P_u$) dependent on $V_{\mathit{PSD}}$

$$\begin{array}{c}\hfill P_u(V_{\mathit{PSD}})=\frac{1}{1+e^{-\eta V_m}}\end{array}$$ 8

where $\eta =0.65$ is a parameter adjusted to have $P_u\sim 0$ when $V_{\mathit{PSD}}=V_r=-65$ mV and $P_u\sim 1$ at $V_{\mathit{PSD}}=40$ mV. At each time step, each NMDA receptor undergoes to a Bernoulli test (B(0, 1)) by using as parameter $P_u(V_{\mathit{PSD}},t)$. For the transitions $R_2\rightleftharpoons R_2^{+}$ (i.e., between the open/close states) both AMPA and the unblocked NMDA receptors followed the time dependent probabilistic equation

$$\begin{array}{c}\hfill P(t)=1-\frac{P_{\theta }}{P_{\theta }{e}^{(\Delta t(1-P_{\theta }))}}\end{array}$$ 9

where P(t) is the probability to undergo to a transition ($R_2\to R_2^{+}$ or $R_2^{+}\to R_2$, respectively), $P_{\theta }=P_{\mathit{open}}$ or $P_{\theta }=P_{\mathit{close}}$ depending on the state at time $t-\delta t$) and $\Delta t$ is the time elapsed from the last transition. The probability to be in the open state ($P_o$) was computed by using data from the literature abut the ratio between the time spent in the open state and the total binding time of the specific receptors (see Table 2 for the used values and Dingledine et al. 1999; Zito and Scheuss 2009; Traynelis et al. 2010) while the probability to stay in the close state is $P_c=1-P_o$ and then $P_{\theta }=P_o\text{or}{P}_c$ if at the time $t-\delta t$ the receptors is open or closed respectively.

From the matrices of the conductance ($\widehat{\mathit{G}}$), of the second binding time (${\mathbf{T}}_{b}2$) and of the first (${\mathbf{T}}_{u1}$) and second (${\mathbf{T}}_{u2}$) unbinding times, we extracted smaller vectors, of size N, where only values related to N double bound receptors were contained. These vectors were used to compute the time course of the EPSP ($V_{\mathit{PSD}}(t)$) and of the total conductance (G(t)) which are the parameters evaluated for the data analysis.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_{\mathit{PSD}}(t)& =(V_{\mathit{PSD}}(t)-V_{\mathit{rev}})\sum _{i=1}^N{\widehat{g}}_{r_i}+\left[\frac{V_d(t)-I_{\mathit{PSD}}(t)}{R_n}\right]\hfill \\ \hfill V_{\mathit{PSD}}(t)& =I_{\mathit{PSD}}(t)R_{\mathit{PSD}}\hfill \\ \hfill G(t)& =\frac{I_{\mathit{PSD}}(t)}{V_{\mathit{PSD}}(t)}\hfill \end{array}\end{array}$$ 10

where G(t) is the conductance time course, $V_{\mathit{rev}}$ is the synaptic reversal potential ($V_{\mathit{rev}}=\sim 0$ mV), N the size of the vectors, ${\widehat{g}}_{r_i}=0$ if $t\notin \{t_{b{2}_i}÷min(t_{u{1}_i},t_{u{2}_i})\}$, and $I_{\mathit{PSD}}(t)$ and $V_{\mathit{PSD}}(t)$ are the EPSC and EPSP as would be recorded by an imaginary electrode positioned at the PSD level. To compute separately the AMPA and NMDA component, the same Eq. (10) was used but ${\widehat{g}}_r$ was respectively ${\widehat{g}}_{\mathit{AMPA}}$ or ${\widehat{g}}_{\mathit{NMDA}}$ and, consequently G(t) was $G_{\mathit{AMPA}}(t)$ or $G_{\mathit{NMDA}}(t)$ and N the size of the vectors containing the double bounded AMPA and NMDA receptor parameters.

Computational procedure and data analysis

The starting value of the membrane voltage at the beginning of each simulation run was always $V_{\mathit{PSD}}=V_d=V_r=-65$ mV (i.e., iso-potential in all the compartments) and the total simulation time was always 1200 ms with a time step of 0.01 ms. After the start of the simulation, the membrane voltage starts to oscillate because of the contribution of the synaptic active pool to the membrane potential. The level of oscillation increases almost linearly up to a level which depended on the parameters used for the simulations (regime level). The regime times for the parameters we used was always reached in $100-150$ ms. For this reason, conservatively, the first 200 ms of simulation were not recorded and we set the time $t_0=0$ at 200 ms of the simulation time.

The starting time ($t_s$) of the EPSP of S was fixed to occur always at 600 ms

For each of the two synaptic pools ($N_p=100$ and $N_p=200$) and for each value of $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ results were averaged over 1000 runs.

The membrane voltage in time interval $t_0÷t_s$ (i.e., $0÷599$ ms) was used to obtain the mean value of the level of the membrane potential (see Figs. 2C and 2D for example). The single run simulations (of which some examples are in Figs. 2A and 2B) show the real membrane oscillations (not mediated) at which the EPSP of S can occur.

Fig. 2.

Examples of 1 s simulation, single run (panels 2A and 2B) and mediated over 1000 runs (panels 2C and 2D) of $V_{\mathit{PSD}}$ for $N_p=100$ and $N_p=200$ respectively for the same triplets of $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$

Results

The results presented herein describe the variability of some parameters of the EPSP produced by the synapse S as a function of the synaptic dendritic activity.

According to our model (Di Maio et al. 2018b) a synaptic pool produces current flowing through the spine neck influencing the value of $V_{\mathit{PSD}}$ (Di Maio et al. 2018a, b, and see Fig. 1).

The oscillations of the membrane potential at the PSD level are visible in Fig. 2 where examples of single runs are shown (panels A and B).

By comparing panels of Fig. 2 we observe that:

1. the synaptic activity on dendrites produces a $V_{\mathit{PSD}}$ oscillation at the synapse S constrained in some limits $V_{\mathit{PSD}}^{\mathit{min}}\le V_{\mathit{PSD}}\le V_{\mathit{PSD}}^{\mathit{max}}$ forming a band of amplitude $V_{\mathit{PSD}}^{\mathit{amp}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i))=V_{\mathit{PSD}}^{\mathit{max}}-V_{\mathit{PSD}}^{\mathit{min}}$ which depends on the size of the pool ($N_p$) and on the mean firing frequencies $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ (please compare panels 2A and 2B);

2. if we exclude the contribution of the EPSP (i.e., if we exclude the time interval $t\ge t_s)$, the mean value of $V_{\mathit{PSD}}$ averaged of 1000 runs (${\overline{V}}_{\mathit{PSD}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i))$) can be a good estimator (indicator) of the level of depolarization of $V_{\mathit{PSD}}^{\mathit{amp}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ (please compare panels 2C and 2D. The value ${\overline{V}}_{\mathit{PSD}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i))$ can be considered as the maximal likelihood value for the occurrence of the event of S and will be used to compute the EPSP amplitude ($V_{\mathit{PSD}}^{\mathit{amp}}$) and the EPSP peak level ($V_{\mathit{PSD}}^{\mathit{peak}}$).

By comparing panels of Fig. 2 it seems evident that increasing the number of active synapses ($N_p$), $V_{\mathit{PSD}}^{\mathit{amp}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i))$ increases and ${\overline{V}}_{\mathit{PSD}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i))$ increases too (it becomes more depolarized). This is not surprising if we consider excitation and inhibition as two contrasting forces. By using the same proportion of inhibitory synapses, the pool of size $N_p=100$ has 80 excitatory and 20 inhibitory synapses (difference of 60 in favor of the excitatory ones). The pool of size $N_p=200$ has respectively 160 excitatory and 40 inhibitory inputs with an excess of 120 in favor of the excitatory synapses. This explains why, although we have used a larger range for the inhibitory frequencies ($\delta ={\varphi }_{i}0÷12$ Hz) than for the excitatory ones ($\delta ={\varphi }_{e}0÷6$ Hz), the pool with the larger size has always the value of ${\overline{V}}_{\mathit{PSD}}$ more depolarized. Although not related to the topic of the present paper, we noted that, interestingly, the oscillations resulting from our combinations of firing frequencies ($\Phi (\delta {\varphi }_e,\delta {\varphi }_i)$) fall in a frequency range compatible with $\theta ,\alpha$ rhythm (as better visible in panels 2A and 2B. These oscillations persist after averaging over 1000 runs suggesting that these waves do not rise randomly (see panels 2C and 2D) but are due to the used input frequencies. Notably, these waves are connected to phenomena like LTP and memory (Bennett et al. 1973; Leranth et al. 1999; Buzsáki 2002).

The values of ${\overline{V}}_{\mathit{PSD}}(N_p,\Phi (\delta {\varphi }_e,\delta {\varphi }_i))$, are shown in Figs. 3A and 3B.

Fig. 3.

Mean level of ${\overline{V}}_{\mathit{PSD}}$ for $N_p=100$ (A) and $N_p=200$ (B) and the relative slopes as function of $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ ((C) and (D))

The panels Figs. 3A and 3B show the dependence of ${\overline{V}}_{\mathit{PSD}}$ on the size of the active pool and on the synaptic pool input frequencies. By the comparison of panels of Fig. 3 we can stress a couple of points:

1. The dependence of ${\overline{V}}_{\mathit{PSD}}$ on the input frequency is stronger for the case with the larger number of active synapses (range $-85÷-30$ mV; $\Delta =55$ mV and range $-75÷-55$ mV; $\Delta =20$ mV, respectively);

2. the dependence of ${\overline{V}}_{\mathit{PSD}}$ on $\Phi (\delta {\varphi }_e,\delta {\varphi }_i)$ become not linear when the size of the pool increases as shown by the first partial derivative ($\Theta$) (Fig. 3D and see Eq. (11)).

$$\begin{array}{c}\hfill \Theta =f^{\prime }{({\overline{V}}_{\mathit{PSD}}({\overline{\varphi }}_e{\overline{\varphi }}_i))}_{|{\overline{\varphi }}_i}=\frac{\partial {\overline{V}}_{\mathit{PSD}}({\overline{\varphi }}_e,{\overline{\varphi }}_i)}{\partial {\overline{\varphi }}_i}\end{array}$$ 11

The absolute value of slope ($|\Theta |$) is also always larger for the pool with the larger size (compare panels 3C and 3D). The values of membrane voltage at which the EPSP start ($V_s=V_{\mathit{PSD}}(599ms)$) influences the level of the EPSP peak ($V_{\mathit{PSD}}^{\mathit{peak}}$). This influence is function of $N_p$ and of $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ as can be noted by comparing panels Figs. 4A and 4B. Also the amplitude of the EPSP computed as $V_{\mathit{PSD}}^{\mathit{amp}}=V_{\mathit{PSD}}^{\mathit{peak}}-{\overline{V}}_{\mathit{PSD}}$ follows the dependence on the pool size and on the input frequencies (see panels 4C and 4D). Differently from our previous works, where pools were formed only of excitatory synapses firing at different input frequencies (Di Maio et al. 2018a, b), the present results did not show any dependence of the time to peak neither on the pool size nor on the input frequencies (see panels 4E and 4F).

Fig. 4.

A, B: $V_{\mathit{EPSP}}^{\mathit{Peak}}$ as a function of $\Phi (\delta {\varphi }_e,\delta {\varphi }_i)$ respectively for the pool of size $N_p=100$ and $N_p=200$. C, D: $V_{\mathit{EPSP}}^{\mathit{amp}}$ for $N_p=100$ and $N_p=200$. E, F: EPSP time to peak for the peaks of A and B

For both the pools the lowest amplitude of the EPSP was achieved by the highest excitatory frequency combined with the lower inhibitory one ($\Phi (6,0)$) while the maximal amplitude was obtained by the opposite condition ($\Phi (0,12)$).

To test the specific contribution of AMPA and NMDA receptors on the total EPSP formation, we have replaced in Eq. (10) ${\widehat{g}}_{r_i}$ with ${\widehat{g}}_{AMP{A}_i}$ and ${\widehat{g}}_{NMD{A}_i}$ respectively. The results are shown in Figs. 5 and 6. The relative contribution of the AMPA and NMDA components are regulated by different mechanisms of activation and their outputs are participated by the different number of receptors on the PSD (55 AMPA and 13 NMDA Receptors). The role of active synaptic pool on the NMDA receptors activity is crucial and depends either on the size of the pool and on $\Phi ({\varphi }_e,{\varphi }_i)$. The recruitment of NMDA receptors and consequently their contribution to the total EPSP depends, in fact, on $V_s$ (i.e., on the level of $V_m$ at which the EPSP of S starts (Di Maio et al. 2018a, b)). The contribution to the total EPSP of the AMPA receptors conductance is shown in Fig. 5 The trends and the degree of non linearity for the evaluated parameters of the AMPA-EPSP components are not much different from those obtained for the total EPSP with the exception of the lower $V_{\mathit{AMPA}}^{\mathit{amp}}$ and $V_{\mathit{AMPA}}^{\mathit{peak}}$ (please compare Figs. 4 and 5). As for the total EPSP, the AMPA component shows a larger excursion of values for the case of $N_p=200$ than for the case of $N_p=100$. The peak level shows a larger non linearity with respect to the total EPSP for the results of both synaptic pools. The times to peak are coincident with those of the total EPSP.

Fig. 5.

AMPA component of the EPSP as a function of $\Phi (\delta {\overline{\varphi }}_e,\delta {\overline{\varphi }}_i)$. Peak Level (A and B), amplitude (C and D) and time to peak (E and F) for the pool of size $N=100$ (left) and $N=200$ (right) respectively

Fig. 6.

NMDA component of the EPSP as a function of $\Phi (\delta {\overline{\varphi }}_e,\delta {\overline{\varphi }}_i)$. Peak Level (A and B), amplitude and time to peak (E, F) for the pool of size $N=100$ (left) and $N=200$ (right) respectively

Much different is the case of the NMDA component the results of which are presented in Fig. 6.

Discussion

The transfer of information among neurons by chemical synapse is a very puzzling problem. It is expected that the neuronal code, intended as a sequence of spikes in a given time window, should produce, at the synaptic level, a synaptic code in the form of a sequence of EPSP. Although a sequence of presynaptic spikes really produces a sequence of EPSPs the number of EPSP in the sequence does not mirror that of the presynaptic spikes because the release probability of a vesicle is not one for each spike (Dobrunz and Stevens 1997; Park et al. 2012). Moreover, EPSP are variable in the peak level and amplitude. This variability is due to the interaction of several systems which control the flow of information at the pre- post- and extra-synaptic level (Di Maio et al. 2018a, b; Di Maio and Santillo 2020, and for a review Di Maio et al. (2017)). Variability of the synaptic code is the origin of variability of the neural code and then the full understanding of the synaptic code is a necessary condition for the full understanding of the neuronal code. Moreover, the diffusion of the synaptic activity in the dendritic branches, determining the neuron spiking activity, is important for the neuronal modeling and for the biologically plausible neural networks because it strongly influences the modification of the synaptic weights on which many of the neuronal network models are based (Wei et al. 2017; Li et al. 2020; Wu et al. 2020; Zhang et al. 2019). Although not pretending to solve these problems, the present work has been aimed to shade some more needed light on how extra synaptic (dendritic) activity can modulate the single bit of information transmitted by a single synapse. To this goal, we simulated the activity of pools of excitatory and inhibitory synapses inputting their information in a dendritic area where a generic glutamatergic synapse S is located. The pools represent all the synapses located at a distance from the synapse S such that the EPSPs they produce can be recognized at the base of the spine of S.

Our results show that the pool activity produces oscillations of the membrane potential such to induce variations of $V_m$ in the proximity of the PSD ($V_{\mathit{PSD}}$ (see Figs. 2A and 2B). These oscillations can have a significant influence on the synaptic response of S the most important of which can be summarized as follows:

1. For each number of active synapses ($N_p$) and each mena excitatory and inhibitory firing frequency ($\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$), $V_{\mathit{PSD}}$ is constrained into a specific voltage band limited by $V_{\mathit{PSD}}^{\mathit{min}}$ and $V_{\mathit{PSD}}^{\mathit{max}}$;

2. the amplitude $V_{\mathit{PSD}}^{\mathit{amp}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i))$ is larger for the larger values of $N_p$;

3. It is possible to identify a mean value (${\overline{V}}_{\mathit{PSD}}(N_p,\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i))$) identifying the level of depolarization of the band. This value is more depolarized for the larger values of $N_p$;

4. From point a,b and c, it follows that, although the membrane voltage at which the EPSP occurs ($V_s$) can be considered random inside the band (see Figs.  2A and 2B) it can be predicted to occur in the limits of the band voltage which are function of the size of the pool and of the input frequencies. This point is important because the value of $V_s$ determines the values of $V_{\mathit{PSD}}^{\mathit{peak}}$ and $V_{\mathit{EPSP}}^{\mathit{amp}}$ which directly determine the recruitment of the NMDA receptors which play a key role in LTP formation;

5. As visible in Figs. 3C and 3D, the slope of ${\overline{V}}_{\mathit{PSD}}$ distribution as function of $\Phi (\delta {\varphi }_e,\delta {\varphi }_i)$ is always bigger (more negative) for the pool with the larger size suggesting that increasing $N_p$ a smaller variation of ${\varphi }_e$ or ${\varphi }_i$ readily produces effects both on the amplitude and on the level of the PSD;

6. The distribution of ${\overline{V}}_{\mathit{PSD}}$ in each pool estimates the positioning of the voltage band as function of $\Phi (\delta {\overline{\varphi }}_e,\delta {\overline{\varphi }}_i)$ (see Figs. 3A and 3B);

7. As expected, the more depolarized ${\overline{V}}_{\mathit{PSD}}$ belong to the highest value of ${\overline{\varphi }}_e$ and the lowest of ${\overline{\varphi }}_i$ (i.e., $\Phi (6,0)$). Alternatively, the most negative one belong to the couple $\Phi (0,12)$.

In terms of synchronization we can say that the more synchronized are the excitatory inputs, the more depolarized is the position of the band of voltage and the opposite occurs when the more synchronized are the inhibitory ones. The distribution of ${\overline{V}}_{\mathit{PSD}}$ as a function of $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ is not linear. This is clearly notable from the slopes obtained by the partial derivative (see Eq. (11)) shown in panels 3C and, especially, 3D. The larger non linearity visible in Fig. 4D is clearly dependent on the pool size and on the more depolarized level of the band voltage which induces a larger NMDAR recruitment (i.e., $V_{\mathit{EPSP}}^{\mathit{peak}}$ and $V_{\mathit{EPSP}}^{\mathit{amp}}$) depends essentially on the different contribution of the NMDA receptors). This point is relevant. If we assume that the starting time ($t_s$) of the EPSP is an independent random event (i.e., it does not follow any coincidence with one of the phases of ${\overline{V}}_m$ oscillation), it follows that $V_s$ acquires a random value inside the limits of $V_s:V_s\in V_{\mathit{PSD}}^{\mathit{min}}÷V_{\mathit{PSD}}^{\mathit{max}}$ which is characteristics of each $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$.

The peak level and amplitude in this case can have any random value inside the limits of the dependence on $N_p$ and $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ shown respectively in Fig. 4A–D. Alternatively, if $t_s$ is not a random variable but coincides with a specific phase of the membrane oscillation then the transfer of information follows a coincidence detector mechanism of information transfer (Tabone and Ramaswami 2012). In this case we should assume that exists a band for tuning the response such that the coincidence is modulated in the interval $V_s\in [V_B^{\mathit{min}}÷V_B^{\mathit{max}}]$ in order to mach a specific phase of the oscillation to transmit a specific kind of information (i.e., depending on the coincidence of $V_s$ of different values the same information can be transmitted with different characteristics).

In both cases, the starting time mostly determines the contribution of the NMDA component of the EPSC which depends strongly on the value of $V_{\mathit{PSD}}$ (Vargas-Caballero and Robinson 2004, and see below).

Peak level, amplitude and time to peak of the AMPA component of the EPSP of S, in the present results do not differ too much from the results obtained for the whole EPSP (compare Figs. 5 and 6). These results are not surprising although in our previous works, by using only excitatory synaptic pools with much larger size, we have noted some significant differences in the amplitude and time to peak (Di Maio et al. 2018a, b). The introduction of inhibitory synapses in the pool, reduced the involvement of NMDA reducing the differences between the total EPSP and its AMPA-dependent component (Di Maio et al. 2018a, b).

The NMDA component of the EPSP is critically dependent on the membrane voltage and consequently on the limits of voltage band inside which $V_s$ occurs. This component is the most interesting when considering the effect of the inhibitory synapses. As visible in Fig. 6, peak level (6A and 6B) amplitude (6C and 6D) and time to peak (6E and 6F), the NMDA component of the EPSP undergoes to different modulations depending on the couples $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$. This component is also responsible for the non linearity observed in the parameters of the total EPSP and of the AMPA-component. The variation of $V_{\mathit{PSD}}$ depending on the pool activity, in fact, produces two different and opposite effects. On one side, a greater membrane depolarization increases the probability that more NMDA receptors are recruited but, on the other side, depolarization reduces the NMDA-dependent current because reduces the driving force producing the EPSC (see Eq. (10)). In the different combinations of $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$ the different balance between these two opposite effects produces the large non linearity visible in Fig. 6 for all the analyzed parameters.

It is important to remember that phenomena like LTP, LTD and STP are connected to the $C{a}^{2+}$ influx through NMDA receptors (Lu et al. 2001; Raymond 2007; Molnár 2011; Bliss and Collingridge 2013) and this means that the recruitment of NMDA receptors dependent on the active synaptic pool directly modulates the LTP and LTD of the single synapse.

Conclusions

In the present work we have shown the dependence of the information flow by a single synapse on the activity of other neighbor synapses (active pool).

Spontaneous or induced firing of neurons in a network, in fact, produces synaptic inputs in dendrites which result in oscillations of the membrane potential.

A first important conclusion emerging from our results is that the number of active synapses and their firing frequency determine a band of voltage in the limits of which the single synaptic input can be modulated. The amplitude of this band, as well as its minimal ($V_B^{\mathit{min}}$) and maximal ($V_B^{\mathit{max}}$) limits are characteristics for any given number of active synapses ($N_p$) and for any given mean couple of firing excitatory and inhibitory frequencies ($\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$). The level of the membrane voltage at which the band takes position can be identified by its mean value. Inside the limits of the voltage band, depending on the oscillation frequency of $V_{\mathit{PSD}}$, the EPSP can start at any level of $V_s$. For a given size of the pool and for a given couple $\Phi ({\overline{\varphi }}_e,{\overline{\varphi }}_i)$, all the parameters of the EPSP depend on its time of occurrence.

Increasing the mean excitatory frequency, the voltage band moves forward the 0 mV and the same does the value of $V_s$. The effect of excitation then consists in the increase of the peak level but decrease the amplitude of the EPSP of S. On the contrary, increasing the inhibitory frequency, the band move in the hyperpolarizing direction and this decreases the peak level but increases the amplitude of the response.

In order to conclude, in the present paper we have shown a powerful modulation system of the single synaptic event of extrasynaptic origin. Independently of the possible spike back propagation, the dendritic synaptic activity produces, for each synaptic event, a modulation system which regulates the information passed by a single synapse in function of the information passed by all the other synapses active in the same time window. The modulation is essentially operated by oscillations of the membrane potential the amplitude of which depends on the number of active synapses and on their level of synchronization (inhibitory and excitatory firing frequencies).

Synaptic information transfer is then not only a process depending on the single synapse properties. It depends on the cooperativeness of all the synapses located at a distance such that, according to the cable properties of the area where the synapses are located, can influence the membrane potential in the proximity of a given synapse.

The information transfer by synaptic transmission is not likely to be of digital type. Unlikely to the neuronal code which is embedded in spike sequences, the corresponding synaptic code is modulated in amplitude by the cooperativeness of other active synapses. This observation suggests that any form of dendritic computation is essentially of analog rather than of digital type.

If the amplitude of the EPSP somehow connected to the elementary bit of information in the synaptic code this means that the information transfer by a synapse can be decreased or nullified by the activity of other synapses in the case the excitation drives the level of $V_{\mathit{PSD}}$ close to the excitatory reverse potential. Interestingly, the action of inhibitory synapses, re-polarizing the membrane voltage, favor the transfer of information by the excitatory synapses because re-polarizing the membrane voltage increase the amplitude of the EPSP.