ModelV: A vesicle release and refill model based on terminal transmembrane voltage
According to the experiment in Fig 1, a simple release and refill model will be developed in the following based on terminal transmembrane voltage VT(t) as an input parameter. In addition to transient and sustained release, two temporal components are included: refill and recovery. Each of the terms is explained below in detail.
Transient release
Transient vesicle release refers to the number of vesicles from the pool RRP that are released rapidly upon depolarization. As noted in the introduction, an RBC voltage change from -70 mV to -55, -50, -45, -40, -35, -30, and -25 mV, respectively, leads to the release of 0, 0, 2, 5, 7, 9, and 10 transiently vesicles from a single ribbon in a time unit. Figure 2 shows a fitted step function to this data where the steps are defined as (see Equation 1 in the Supplemental Files)
where [] is the floor function, which is a function that takes a real number X as an input and produces the greatest integer less than or equal to X as an output. This is necessary to bring an integer concept to the number of released vesicles.
The relationship showing the number of transiently released vesicles during a single time step when the transmembrane voltage of the terminal changes from to is simulated as (see Equation 2 in the Supplemental Files)
Sustained release
In comparison to transient release, sustained vesicles are released in essentially larger time intervals as stochastic events with specific rates (number/second). To find the rates for one ribbon, we redrew an item (Fig 3A) based on a previous work (Oesch and Diamond 2011, Fig 3c). Integration with the EPSC-time diagrams exhibits the amount of charges coming to the AII cell that is proportional to the amount of released neurotransmitters. Rescaling the diagrams by 1/5.5 in order to find the amount of charge originating from a single ribbon and once again by 1/0.36 to make the size of the RRP equal 10 shows the cumulative vesicle release versus time from a single ribbon (Fig 3B). Figure 3B shows both the transient and sustained releases; transient releases occur rapidly after depolarization and vesicle release continues afterwards with linear rates showing sustained releases. Thus, the slope of each diagram shows the average number of sustained released vesicles per second at each transmembrane voltage. Then, by fitting a function to the steady state of the Ica(V) diagram, , the sustained vesicle release rate becomes a function of the terminal voltage. We preferred to explain the sustained release rate based on Ica(V) because vesicle release is stopped as soon as the terminal transmembrane voltage reaches the calcium Nernst potential or passes through that. In these states, the inward calcium current is replaced by outward calcium current and no calcium ions would be available in the terminal to bind to the vesicles and make them ready to be released (Werginz and Rattay 2016). According to (Fig 3C) and Eq. 24, the sustained release rate is (see Equations 3 and 4 in the Supplemental Files)
where c∞ is the steady state of the gate at voltage V, and 1.3 is a number with the dimension of the vesicle/(s.mV), which is used to obtain the sustained rates at voltage V for a standard ribbon with 10 vesicles in RRP at the initial point. αc and βc are explained in the Materials and Methods section.
To consider the stochastic release, an equally distributed random number between (0,1000]/Δt is generated at each time step Δt, Δt =1 ms, by FSustained(t, VT) compared to the related value in Sus( ). If this number is less than the related number in Sus(VT), a vesicle is released from the pool; otherwise, it is not. The number 1000 is used to change the second dimension to milliseconds. The maximum release rate of 45 vesicles/s (Fig 3C) means that the average time between two releases is 22.22 ms. Thus, a time step appears to be appropriate. The sustained release is simulated as (see Equation 5 in the Supplemental Files)
Refill
The RRP should be fed via the same rates as the sustained releases because the rate of sustained vesicle release remains constant at each transmembrane voltage even when the pool is empty. The pool is empty when the transmembrane voltage reaches -25 mV from -70 mV and stays at -25 mV for one second; see Fig 1 and Fig 5. Thus, the same strategy as sustained release is applied for the refilling process. In the refilling process, a random number between (0,1000]/Δt is generated in each time step and is then compared with the related number in Sus(VT). If the random number is less than the related number in Sus(V), one vesicle is injected to the pool; otherwise, it is not. The refill process is explained by FRefill(t,VT) as follows (see Equation 6 in the Supplemental Files)
Two series of random numbers, random1 and random2, are generated by FSustained(t, VT) and FRefill(t,VT) for the same time step to consider the independence between the refill and release terms. Although any bell-shaped curve; like Poisson or Gaussian distribution, can also be used instead of the term Sus(VT), we preferred to use the term Sus(VT) as it depends on calcium current in each voltage as well as not existing experimental data for transmembrane voltages more than -20 mV. Note that because of L-type calcium ion channels included in vesicle release that remain open as far as the membrane depolarizes, sustained vesicle release should depend on kinetics of the channels or equivalently the calcium current.
Recovery
The time required for the RRP to become full again is called the ‘recovery’ time. The recovery time was also investigated in two paired-pulse experiments (Singer and Diamond 2006, Fig 5) where the RBC was stepped from -60 mV to +90 mV to make the RRP pool empty, then stepped again to -60 mV, causing recovery without any sustained release. Then for different time intervals, the cell was stepped again to +90 mV while the AII EPSC was simultaneously measured. To find the recovery term, a function to the data from this double pulse experiment was fitted (Fig 4). The circles in Fig 4 show the replotted experimental data, and the solid line represents the modelled recovery time explained by the following equation: (see Equation 7 in the Supplementary Files)
where [] is the floor function as previously explained.
A mathematical framework for the occupancy of the RRP
The equation explaining the occupancy of the RRP versus time is composed of four terms FSustained(t,VT), FRefill(t,VT), FTransient(t,VT), FRecovery(t,VT), which respectively stand for sustained vesicle release from the RRP, vesicles coming to the pool, rapid (transient) vesicle release, and recovery. The equation is (see Equation 8 in the Supplementary Files)
The terms showing vesicle release from the pool are marked with negative signs, and the terms showing the entrance of vesicles to the pool are marked with positive signs. Pool occupancy versus time is shown in Fig 5A for the voltage protocol used in Fig 1A; here, the average of N=10000 trials was used to reduce the stochastic influence. When the RBC transmembrane voltage jumps from -70 mV to -55 mV and -50 mV, the pool remains full, jumping to -45 mV to make 20% of the pool empty; this becomes 50%, 80%, 90%, and 100% for voltage jumps to -40, -35, -30, and -25 mV, respectively. Afterwards, when the voltage reaches -20 mV, the pool again becomes empty. Fig 5B shows the recovery time of the pool. The pool becomes full in different time intervals depending on the previous occupancy state of the pool. In other words, the emptier the pool, the more time it needs to become full. For example, it takes only 48 ms to become full when the cell is stepped from -45 to -60 mV as the pool was 80% full. The other time intervals for the states of -40, -35, -30, and -25 mV to become full are 0.96, 2.9, 6.6, and 15 s, respectively.
As the refilling and sustained release terms obstruct each other’s effect for trials (which is usually more than N=1000 trials), the number of transient vesicles released from the RRP is actually the result of the RRP occupancy change for when the change is negative, while a positive RRP occupancy change shows RRP recovery. Thus, the number of released vesicles (NRV) from the RRP at time t when the terminal membrane voltage VT is governed by RRP occupancy changes and is added to the term showing sustained release: (See Equations 9 and 10 in the Supplementary Files)
where ΔRRP(t,VT) stands for RRP occupancy changes at time t that is the average of N=10000 trials, and T0 and t0 stand for the time at which the RRP occupancy starts and stops decreasing, respectively. Note that when ΔRRP(t,VT) is less than zero, some vesicles have left the pool; when it is positive, some vesicles have been injected into the pool, which is the recovery process. Thus, the condition ΔRRP(t,VT) < 0 in Eq. 10 actually shows the number of transient released vesicles from the RRP by considering RRP occupancy, but it is not necessarily equal to FTransient(ΔVT). On the other hand, because the RRP occupancy has to be also considered for number of transient release vesicles, the term ΔRRP(t,VT) shows number of transient released vesicles when the pool contains less vesicles than what FTransient(ΔVT) suggests. This phenomena is vital during periodic signals, see Fig. 13. It is obvious that ΔRRP(t,VT) is equal to FTransient(ΔVT) if there are enough vesicles in the RRP, and ΔRRP(t,VT) < FTransient(ΔVT) if not. T0 would be too small for voltage clamp experiments because the membrane voltage changes quickly in these experiments; however, T0 would be larger for the cases in which the terminal membrane voltage depolarizes via spike or non-spike stimulations. FSustained(t,VT) is the sustained vesicle release term from the RRP explained by Eq. 5. Figure 6 shows the results of Eq. 9 for the protocol of Fig 1A using a time step and T0=1ms. T0=5ms for the spike and terminal membrane depolarization presented in Fig 11 and Fig 13 because it takes 5 ms for the membrane to reach to its maximum amplitude, T0=7ms (Fig 12).
ModelCa: Vesicle release model based on intracellular calcium concentration
A possible step to improve the release model is to consider the calcium concentration in the terminal as the key parameter. Vesicle release depends on the degree of depolarization of the membrane of the axon terminals that control the voltage-gated calcium channels. In BCs, these are mainly L-type channels, located linearly parallel to the bottom surface of the ribbon. The intracellular calcium concentration [Ca2+] depends on the calcium current across the terminal membrane, the surface area and volume of the terminal (A, V), and the decay time constant t (Fohlmeister et al. 1990). (see Equation 11 in the Supplementary Files)
where F=96485.33 is the Faraday constant, is the initial calcium concentration set to 0.34 µM and t=10 ms (Werginz and Rattay 2016); time step dt=10 µs was used in solving this equation.
For terminal voltage <-60 mV, there is no calcium current flow, but the onset of depolarization causes an inward calcium current flow (Fig 7A and B). The inward calcium current eventually reaches a maximum if the terminal transmembrane voltage remains constant. An increment in the calcium current results in an increased intracellular calcium concentration to a steady state value (maximum value) (Fig 7C and Eq. 11). Thus, a specific intracellular calcium concentration value corresponds to each transmembrane voltage. By finding the maximum value of the calcium concentration for each transmembrane voltage, the corresponding sustained release rate is easily obtained by using Fig 3C and can be fitted (Fig 8) as (see Equation 12 in the Supplementary Files)
where 20.14 has a dimension of µM, 1 and 0.84 are dimensionless numbers, and 57.11 has the dimension of the vesicle/s.
The stochastic release process (sustained release vesicle) can be simulated as (See Equation 13 in the Supplemental Files)
where Fsustained(t,[Ca2+]) works like Fsustained(t,VT). In other words, a random integer number between (0,1000]/dt, dt=0.01 ms is generated at each time step by Fsustained(t,[Ca2+]). All of the numbers in the interval have the same probability to be chosen compared to the related value in Sus([Ca2+]); if this number is less than the related number in Sus([Ca2+]), one vesicle is released from the pool, otherwise, it is not.
The refilling process can be calculated with the same algorithm as sustained release: (see Equation 14 in the Supplemental Files)
Again, to show the independence of the randomly generated numbers in Fsustained(t,[Ca2+]) and FRefill(t,[Ca2+]), we show them as random1 and random2.
The recovery process is simulated with the same formalism as previously: (See Equation 15 in the Supplemental Files)
By calculating the time derivation of the intracellular calcium concentration (Fig 7D), a criterion for transient vesicle release appears. The local maximum points of intracellular calcium concentration changes would correspond to the number of transient vesicles released for any change in terminal voltage (Fig 7E and using Fig 2). For example, when the terminal voltage changes from -70 mV to -45 mV (see Fig 2), two transient vesicles are released. Thus, two-vesicle release is equivalent to 0.004 µM.s-1; see the red parts in Fig 7. The transient release is explained by (see Equation 16 in the Supplemental Files)
where 16.2 and 16.39 have the dimension of Vesicle.s-1, and 1 and 0.75 have no dimension. 27.02 is a normalizing number with a dimension of s. µM-1. [] is the floor function as previously discussed. To find the transient release versus time, the times in which the local maximum points take place (T1, T2, T3,...) are extracted at first; these points are the roots of the second time derivation of the intracellular calcium concentration and show the times in which the transient releases take place. Using the indicator function and other terms already explained, the RRP occupancy reads as (see Equation 17 in the Supplemental Files)
where T is a set containing the times at which the local maximum points of the time derivation of the intracellular calcium concentration take place. Time step dt=10 µs was used to solve this equation.
The number of released vesicles at time t will be governed by solving (see Equations 18 and 19 in the Supplemental Files)
Although we used special kinetics for calcium ion channels, any type of kinetics can be used without affecting the generality of the problem. Fig 9 shows the experimental data versus the data calculated by these equations.
One of the differences between modelCa, Eq. 18, and modelV, Eq. 9, is the signal delay, i.e., the time interval between stimulation of the bipolar cell and the release of the first vesicle. The modelV has no signal delay, while the modelCa has a maximum signal delay of 0.43 ms, similar to other works that suggest millisecond delay (Baden et al. 2011, Singer and Diamond 2003; see discussion) (Fig 10). Figure 10 shows the first transiently released vesicles versus time when the terminal membrane changes from -70 to other potentials with a step voltage of 5 mV in a voltage clamped experiment. Transiently released vesicles have a time delay of 0.27, 0.34, 0.40, 0.43, and 0.43 ms when the transmembrane voltage changes from -70 mV to -45, -40, -35, -30, and -25 mV, respectively.
To compare vesicle release in spiking and non-spiking BCs, two cells with the same geometry are considered in extracellular stimulation because the geometry of the cells defines the shape and amplitude of the potential exerted to the compartments (Rattay et al. 2018). Thus, an imaginary spiking BC with the same shape was made by adding sodium and potassium channels to the axon. When the two spiking and non-spiking BCs are stimulated intracellularly with a 100 ms rectangular pulse of 500 pA, the terminal transmembrane voltage of the non-spiking BCs reaches -48 mV, leading to no transient vesicle release; however, the voltage membrane of the terminal in spiking BCs passes through -10 mV, which means that the terminal depolarizes for 50 mV, leading to the release of all 10 vesicles existing in the pool (Fig 11). Both models in Eq. 9 and 18 produced the same results. Thus, spikes make the RRP empty. Here, sustained release is not of interest because the pulse duration is not long. In average three sustained vesicles are released during a 100 ms pulse stimulation in spiking BCs, while this number is less than one in the cell with a passive membrane.
When both spiking and non-spiking bipolar cells are stimulated extracellularly with small pulse amplitudes, between 0.9-3 µA and a long pulse duration of 10 ms (Fig 12), the number of transiently released vesicles increases linearly versus the pulse amplitude for the non-spiking cell while it increases exponentially in the spiking cell because the spike has sufficient time to reach the terminal and make the terminal more depolarized. Both cells with active and passive membranes have the same behavior for short pulse durations, e.g. 3 ms, since the terminal in both cells senses the same voltage originating from the microelectrode and the spike does not have sufficient time to reach to the terminal. The major difference between the two models appears when the pulse amplitudes pass through 4 µA; the number of transiently released vesicles starts decreasing in modelCa but remains constant in modelV (Fig 12). The reduction in the number of transiently released vesicles for large pulse amplitudes is because of the outward calcium currents that take place when the terminal transmembrane voltage depolarizes more than Eca=-20mV.
Figure 13 shows the responses of a spiking BC to pulse trains. The spiking BC is stimulated with a pulse amplitude of 1500 pA and 3 µA in intra- and extracellular stimulations, respectively, with a pulse duration of 1 ms followed by a 200 ms interval (5 Hz stimulation). Spike amplitudes are 52, 45 mV in intra- and extracellular stimulations, respectively. In both cases all of the 10 vesicles existing in the pool are released by the first pulse and3 vesicles by the following pulses in the recovery time of 200 ms (Fig 13).