When entering a synapse, presynaptic pulse trains are filtered according to the recent pulse history at the synapse and also with respect to their own pulse time course. Various behavioral models have tried to reproduce these complex filtering properties. In particular, the quantal model of neurotransmitter release has been shown to be highly selective for particular presynaptic pulse patterns. However, since the original, pulse-iterative quantal model does not lend itself to mathematical analysis, investigations have only been carried out via simulations. In contrast, we derive a comprehensive explicit expression for the quantal model. We show the correlation between the parameters of this explicit expression and the preferred spike train pattern of the synapse. In particular, our analysis of the transmission of modulated pulse trains across a dynamic synapse links the original parameters of the quantal model to the transmission efficacy of two major spiking regimes, that is, bursting and constant-rate ones.

The main computational function of artificial neural networks has traditionally been modeled as an adjustment of the coupling weight between neurons. In biological nets, this coupling weight is provided by the synapse, where an incoming (presynaptic) pulse causes a release of neurotransmitters, which in turn generate a postsynaptic current (PSC) that charges the postsynaptic (i.e., receiving) neuron membrane [

Mechanisms acting on the number of release sites

Dynamic synapses also interact in a complex manner with another important component of neural information transmission, modulated pulse trains [

The plasticity of

To derive this expression, we show that for regular pulse rates, the model by Markram et al. can be expressed explicitly as an exponential decay function. We use this function in Section

The model developed by Markram et al. [

The effect of this adaption can best be described as transmission of transients, that is, changes in the presynaptic pulse rate are transmitted with their full dynamic range to the postsynaptic neuron, but the response to steady-state input pulse rates diminishes. This seems to be a universal feature of biological neural nets, where novel stimuli receive increased responses compared to static ones [

For a steady-state signal, the above response can be thought of as a signal compression, so that the high dynamic range of, for example, sensory input is adapted to the limited range of the pulse response of a neuron [

Behavior of the quantal synaptic short-term adaption, protocol similar to [_{facil}=530_{rec} = 130 milliseconds, A = 1540 pA, U = 0.03. To derive a continuous PSC from the pulse-PSC of (

Using this

As expected from the model, steady-state utilization

However, this steady-state analysis does not do justice to the complex transmission characteristics across a dynamic synapse. Consequently, in the following we analyze the response of a synapse to a single transient pulse rate transition.

Figure

Relative PSC 5 Pulses after a step change in pulse rate. The synapse was in a converged state for a pulse rate of 1 Hz, 15 Hz, and 30 Hz, respectively. Then, the pulse rate was changed to the one denoted on the abscissa for 5 consecutive pulses. The mean PSC of the time window corresponding to the 5 pulses was calculated. This value was normalized to the mean PSC of a synapse being converged to the pulse rate after the step.

For decreasing pulse rate, the PSC response will continuously decrease, making the transient response bigger than the converged value. At first glance, one would expect the opposite for increasing pulse rate: if the PSC continuously increased, the transient PSC should be smaller than the converged value, and the quotient between both values should diminish for higher pulse rate differences because of

As shown in this section, the response of dynamic synapses cannot be fully characterized by the transmission characteristics for regular pulse rates. The response for most cases of transients is amplified compared to the steady-state response. This is as expected from biology, where changes in pulse rate are a source of information, while static stimuli should be attenuated in favor of these transients [

In a generalization of the analysis carried out in Figure

From top to bottom: bursty spike train, generated from a sine-modulated Poisson process and regularized approximation with rectangular modulation between high pulse rate

A modulated pulse rate can be thought of as a sequence of bursts and as such represents a generic model for various types of neural pulse signaling, where the information is encoded in the temporal fine structure of the pulse signal [

In the upper part of Figure

Figure

Time course of

For the derivation of the PSC's modulation dependency, we start with the explicit expression of (

Dependent on the sign of the term

Evaluating (

Now, the mean synaptic release quantity

Integrating over the low-rate interval, that is, the time course between points 2 and 3, in the same way yields the corresponding value

As mentioned together with Figure

When calculating an overall mean PSC, the duty cycle (i.e., the fraction each

The explicit expressions derived in Section

Left: spike trains as applied to the original quantal model, from bottom to top: high (100 Hz) and low (2 Hz) frequency modulated spike train (

The parameters were chosen to resemble the experiment of Figure

In the following, we will thus apply the derivation of Section

To resemble the optimization regime of [

Mean synaptic efficacy per spike

Of course, as we have shown in the previous section and the appendix, the preference of the quantal model depends not only on

Figure

Optimal modulation frequency

There is almost no dependence of the optimal modulation frequency and the burst preference on the low spike rate

with

Another interesting characteristic of the above plot is the decrease of the maximum

As already stated, one of the main questions behind such analyses is, for which synapse types (i.e., parameters

Relative difference of a modulated spike train from a regular spike train, grey scale coded, with respect to the parameters of the quantal model. (a) simulation, and (b) our analytical calculation. Only positive (i.e., modulation-favored) part is shown.

(a) Steady-state values of

We also used this parameter sweep to compare our analytical calculation with the original iterative formula. This is a hard test case, because already small deviations in the calculations, for example, caused by the continuous-time idealization and the approximations made with the derivation of

The principal dependencies of the favored spike mode on the synapse parameters as suggested by Figure

In the following, we try to analyze the above parameter dependencies, based on our modeling of

There is a dependency of this preference on the relation between the time constants

A second criterion based on which it can be predicted if a bursty or regular regime is preferred by the synapse, would be the relation between the convergence time constant for

From the above postulates, two criteria can be derived where the time constants derived in this paper allow to predict if a grouped/bursty or a regular regime is preferred by the synapse. The first one would be the difference between the convergence time constants for the high rates, that is,

Correlation between criteria based on the convergence time constants of (

As can be seen, there is a definite correlation in the way suggested above, that

How could these results be applied in the wider neuroscience context? One important topic of current interest is the interaction of the different forms of plasticity on the same synapse, especially with regard to the different temporal timescales of expression [

We have derived an explicit expression for the iterative quantal model [

We have shown how the filtering characteristics might be determined from the synaptic parameters. Specifically, we have provided an explanation how the filtering characteristic of a dynamic synapse depends on the effective time constants

Our noniterative expression for the behavior of the dynamic synapses of [

The convergence of iterative equations like those of the quantal model [

However, as Figure

Comparison of simulated and analytically derived time course of

In this case, an absolute time constant for this settling may be derived, which is likely to depend on the fundamental time constants of the quantal model.

In the following, an explicit expression for the convergence of

The speed of convergence is determined by the term dependent on

An explicit expression for

Now, the product term in (

With all principal dependencies being identified, the resulting time constant of

Equations (

The authors would like to gratefully acknowledge financial support by the European Union in the framework of the Information Society Technologies program, Biologically Inspired Information Systems branch, project FACETS (No. 15879).