Ephaptic Quantal eqDUO

The model eqDUO represents a further development of the model eqUNO. For the sake of user's convenience, the previous design of the interface was preserved. However, programming of the application was essentially different from that of eqUNO. A limited range of use for the latter model allowed to simplify calculations, while such simplifications could not be used in eqDUO no more. As a result, it works almost 10 times slower. Therefore it is advisable to use it only for the tasks that include paired-pulse presynaptic stimulation for which eqUNO is inapplicable.

As in the model eqUNO, neurotransmitter release from each site i,  i = 1 ... n , was taken to be random. At each instant t, the release probability is described by the following formula:

pi = a·f(t)·di(t)·exp(Vp/Vexp).
In other words, the probability depends exponentially on the potential at the presynaptic membrane, Vp . The constant a is a normalization factor defining its magnitude, and the parameter Vexp defines the slope of the exponent. Values for both of them can be set in the Presynaptic Settings window.

Fig.1. The Presynaptic Settings window.

Two time-dependent factors f(t) and di(t) in the aforecited formula reflect a facilitation and depression of transmitter release following presynaptic activation. Phenomena of paired-pulse facilitation (PPF) and depression (PPD) are well known for synapses of various types. These short-term forms of synaptic plasticity are expressed in electrophysiological experiments as changes in the amplitude of a test EPSC evoked by a second presynaptic spike that follows the first (conditioning) one in the paired-pulse paradigm. The presence of PPF or PPD is dependent on types of synapses and experimental conditions and strongly varies according to the interval between the conditioning and test pulses.

In the model, initial values for both f(t) and di(t) are taken to be unit. An arrival of the presynaptic spike temporarily increases a release probability for all releasing sites. The model imitates this post-stimulus facilitation by an additional exponential term:

f(t) = 1 + CF·exp(–(t–t' )/tF )
at t>t', where t' is the moment of termination of the presynaptic spike; factor CF reflects a magnitude of facilitation and tF is a facilitation time constant.

When transmitter release happens, release probability can temporarily decrease (post-release depression), which is imitated in the model by the factor

di(t) = 1 – CD·exp(–(t–t'i )/tD )
at t>ti', where ti' is the moment of transmitter release at the releasing site i; factor CD reflects a magnitude of depression and tD is depression time constant. If the right part of this expression is negative, the release probability considered to be zero, that correspond to absolute refractoriness of the given release site.

Explanation of Terms:
An example of the model response to presynaptic stimulation by a pair of spikes (of 1 ms duration and with an interstimulus interval of 10 ms) is shown in the figure at left. Five transmitter quanta were released during the first presynaptic spike, which evoked a wave of synaptic current shown in black. Second spike triggered already 10 release sites (due to both the post-stimulus facilitation, and the influence of a positive feedback from continued synaptic current of the first response), which evoked second wave of synaptic current shown in red.
It is supposed in the model that the subsynaptic membrane may possess some nonzero conductivity even at rest. In this case some background current will flow through the membrane. Consequently the amplitudes of the synaptic current, Is1, measured from zero level, and EPSC1 measured from the level of background current may differ. The same concerns the response evoked by the second spike. In this case it is natural to measure an amplitude of the second EPSC from a continuation of the first response.
The presence of an appreciable resistance of the synaptic cleft inevitably limits magnitude of a synaptic current. In no way the synaptic current can exceed certain saturation level defined by expression: Isat= V2 /Rg, where V2 is a membrane potential clamped in a postsynaptic neuron and Rg is a resistance of the synaptic cleft.

Results of simulation experiments performed with different models are given below in overlapping windows of Figs. 2 and 3.

Fig.2. Numbers of quanta released (left diagrams) and EPSC amplitudes (right diagrams) in response to the first (black curves) and to the second (red curves) presynaptic spike as functions of an interstimulus interval.

In Fig.2, results of simulation for two models (with and without feedback) are shown. In the first one (back window), the ephaptic feedback was removed (Rg=0), in the second model (front window) the value of the synaptic cleft resistance was set sufficiently high to influence effectively on the synaptic transmission. The results indicate that, due to the feedback, the response to the second spike considerably increases (after the period of absolute refractoriness terminate).

Fig.3. Correlation between responses (numbers of quanta released or EPSC amplitudes) to the first and to the second spike at small interstimulus interval. Calculated correlation coefficients, r, are shown.

Results of simulation for the model lacking the ephaptic feedback (Rg=0), shown in the back window of Fig.3, have revealed a weak inverse correlation between numbers of quanta released (due to the post-release depression). Results of simulation for the model with effective feedback are shown in the front window. In both models the post-stimulus facilitation was absent (CF=0). Nevertheless, in the second model, due to the feedback, the number of sites released to the second spike appreciably increases with growth of the response to the first spike.

Application eqDUO is available here as a self-extracting archive (compiled 22.08.2003):
Download the last version (~230K)

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