Point process of vesicle release at AZs in a steady state

Assuming that presynaptic [Ca²⁺] follows the gating of Ca²⁺ channels instantaneously (section 4.2.2), the state of a presynaptic AZ in our model is fully characterized by the states of all the presynaptic channels and vesicular release sites. In the following, we used vectors~c and ~v to denote, respectively, the set of states of all channels and the set of states of all vesicular release sites at an AZ. Particular elements of vectors~c and ~v, denoted by c_i and v_j, correspond to a particular channel i and a particular vesicular release site j at the AZ. To denote the states of all channels but the channel i and states of all vesicular release sites but the site j,~c−i and

~v−j were used, respectively. Values assigned to the elements c_i and v_j define the states of the corresponding channel or vesicular release site. We employed a notation in which these values vary from 1 to the number of states each presynaptic channel and vesicular release site has.

The state of each vesicular release site, from which the fusion takes place, was assigned the maximum possible value. The moment of the irreversible transition from that state to the state v_i = 1 was treated as a vesicle fusion event. The open state of each channel was assigned the maximum possible value, i.e., the total number of states the channel was described by.

It follows from the formulation of the model (section 4.2), that the dynamics of the presy-naptic site is described by a Markovian jump process. This process underlies the events of vesicle fusion and postsynaptic spike generation, which are treated as points in time. As we demonstrated below, quantitative characterization of the stochastic point processes of vesicle fusion and spike generation is based on the solutions of the following two sets of differential equations. The first of them, the so-called master equation, describes the dynamics of the probability ¯P(t, ~c, ~v|~c₀, ~v₀) of finding the presynaptic site in a particular state (~c, ~v) at moment

t >0 if it was in a particular state (~c₀, ~v₀) at t= 0:

The second set of equations describes the dynamics of the probability P(t, ~c, ~v|~c₀, ~v₀) that no vesicle fusion happened before moment t and that the presynaptic AZ is in a particular state (~c, ~v) at that moment: N_V – the number of presynaptic vesicular release sites, k_v^{V j}0

j→v_j – the transition rate from state v_j⁰ to statev_j of the j-th release site, n_V(j) – the number of states of the j-th vesicular release site. ∆[Ca²⁺]^s_ij – Ca²⁺ concentration at the j-th site due to thei-th channel in the open state.

Note that the summation of ∆[Ca²⁺]^s_ij over different channels exploits the assumption of the superposition, introduced in section 4.2.2. Variable δ_v^j0

j→vj,Ca²⁺ is equal to 1 if the transition from statev_j⁰ to statev_j of vesicular release sitejis Ca²⁺ dependent, and is equal to 0 otherwise.

In all other cases here, the symbol δ stands for the Kronecker delta. If no transition between two particular states of a channel or vesicular release site exists, the corresponding transition rates (k_c^Ci0

i→ci ork^V_v0

j→vj) in Eq. (4.5) are equal to zero. In our model we considered a steady state situation where the receptor potential is constant in time (section 4.2.1). Thus, the channel gating rates in Eqs. (4.5) – (4.9) were constant in time.

The probability distribution of IEIs in a steady state can be constructed by using the solutions of Eq. (4.6). Indeed, let us use P₀^s(~c₀, ~v₀) to denote the probability that, during a particular vesicle fusion event in a steady state, the system enters state (~c₀, ~v₀). Then, the

a steady state is given by Here, C and V stand for the sets of possible realizations of vectors ~c and ~v. Accordingly, the probability density function of IEIs can be expressed in the following way:

ρ_IEI(t) =−d after summing both sides of it with respect to~cand ~v overC and V. The set of initial proba-bilities,P₀^s(~c, ~v), can be found by using the requirement that, in a steady state, the probability of entering the same state after two subsequent vesicle fusion events is equal. Mathematically, this requirement is equivalent to the following system of linear equations:

P₀^s(~c, ~v) = X where T(~c, ~v|~c∗, ~v∗) is the probability that a vesicle fusion event occurs when the system tran-sits to state (~c, ~v), if during the previous fusion event the system transited to state (~c∗, ~v∗). in a steady state given that the system leaves state (~c, ~v⁰). Eq. (4.9) has a unique solution which is normalized to one (see Supplementary Material 2.1).

In this work, we were quite often interested only in the first two moments ofρ_IEI(t). Calcu-lation of these moments can be reduced to solving systems of linear algebraic equations. This idea can be expressed most clearly in the matrix notation. Thus, let us first rewrite Eq. (4.6) accordingly,

where elements of the probability vector Pand the transition matrix A are indexed by a pair (~c, ~v). Then, as shown in Supplementary Material 2.2, the m-th moment of ρ_IEI(t) is given by ht^mi_IEI =m!·P^s₀^T·((−A^T)^−m·1), (4.14) where vectorP^s₀corresponds toP₀^s(~c, ~v) introduced above. 1is a vector of the same dimension as P^s₀and with all its elements equal to 1. IfP^s₀ is known, Eq. (4.14) can be evaluated analytically, at least in principle, even when ρIEI(t) cannot. In particular, the first moment can always be calculated analytically:

Time intervals between subsequent vesicle fusion events in a steady state are not uncorre-lated in general. Indeed, in the most general case, the state through which the system transits to during a particular fusion event correlates with the waiting time for that fusion event, as well as with the waiting time for the next and all the following fusion events. The serial correlation function of IEIs in a steady state, denoted by C_IEI(l), can be expressed as

C_IEI(l) = ht_it_i+li − ht_iiht_i+li pht²_ii − ht_ii²·q

ht²_i+li − ht_i+li²

= ht_it_i+li_IEI − hti²_IEI

ht²iIEI− hti²_IEI , (4.16) where t_i and t_i+l are the i-th and the (i+l)-th IEIs of vesicle fusion. Writing (4.16) we used the fact that, in a steady state, the statistical properties of the system are invariant to a shift in time, thus, ht^m_i i =ht^m_i+li = ht^mi_IEI. In a steady state, ht_it_i+li_IEI can be expressed through the solutions of Eq. (4.6) and their derivatives in the following way:

htiti+liIEI =1^T·(J·T^l−1·G·P^s₀), (4.17) where elements of matrices G, J, andT are

G_{(~c,~v)(~c∗,~v∗)}=

and during the next fusion event transits into state (~c, ~v). T_(~⁽ⁿ⁻¹⁾_{c,~v)(~c∗,~v∗)} is the probability that the system transits into state (~c, ~v) just after (n−1) subsequent vesicle fusion events, given that it started in state (~c∗, ~v∗). J_{(~c,~v)(~c∗,~v∗)} is the mean time to the next vesicle fusion event, given that the system is initially in state (~c, ~v) = (~c∗, ~v∗). Writing Eq. (4.17) we used the fact that, once the initial conditions for the (i+n)-th fusion event are fixed, waiting times for the i-th and (i+n)-th events are statistically independent. This follows from the fact that the dynamics of the system underlying the vesicle fusion process is Markovian.

Eqs. (4.4) and (4.6) are systems of homogeneous linear differential equations with constant coefficients. In general, the solutions to these equations can be expressed as linear combinations of functions from a set {t^αi · e^λi·t, t^αi ·cos (λ_i·t), t^αi ·sin (λ_i·t)}. Solving these equations then reduces to diagonalization of the corresponding transition matrices. This can be done analytically in a number of relevant cases, as is demonstrated throughout this work.

In the case of the Ca²⁺-nanodomain, Ca²⁺-microdomain, and intermediate Ca²⁺-domain coupling scenarios (Fig. 4.1C), vesicular release sites within the AZ are statistically independent.

Then, solutions of Eqs. (4.4) and (4.6) can be factorized:

P¯(t, ~c, ~v|~c0, ~v0) = for an AZ with a single vesicular release site i. ~cⁱ denotes the state of the Ca²⁺ channels corresponding to the site i while v_i denotes the state of the vesicular release site i. The initial condition vectorP₀(~c, ~v) for calculatingρ_IEI(t) can also be factorized for AZs with independent vesicular release sites: a single vesicular release site. P_i^r is the probability that a vesicle fusion at the AZ in a steady state is due to sitei. P_i^r is equal to the fraction of the overall release rate at the AZ contributed by sitei,

In particular, expressions of ρ_IEI(t) and hti_IEI are now considerably simpler compared to the

Writing Eq. (4.23) we used the fact that a vesicular release site which results in a vesicle fusion transitions to a state with v_i = 1 during the fusion event. The term in the square brackets in Eq. (4.24) is the vesicle fusion rate due to one of the vesicular release sites at the AZ. In other words, the vesicle fusion rate at the AZ is a sum of the vesicle fusion rates at separate vesicular release sites.

Expressions (4.19) – (4.24) can be further simplified when considering the Ca²⁺-nanodomain, Ca²⁺-microdomain or intermediate Ca²⁺-domain scenarios individually. For the shared Ca²⁺ -domain scenario, equivalent expressions to (4.19) – (4.24) apply for the groups of vesicular release sites sharing the same channels (see Supplementary Material 2.3). The fact that so-lutions of the original Eqs. (4.4) and (4.6) can be factorized is, in practice, crucial to making the analytical approach applicable to systems with a realistic number of vesicular release sites.

Indeed, the number of possible states of the system increases exponentially with the number of vesicular release sites. For example, let us assume an AZ with fourteen Ca²⁺-nanodomain driven release sites and that each unit, consisting of a channel and a vesicular release site, is described by 2×5 = 10 states. Then, the number of possible states of the system is equal to 10¹⁴. However, when the factorization is exploited, we have to instead deal with 14 independent systems each characterized by only 10 states.