Short-term Synaptic Plasticity and Direction Selectivity
Maya Chandru
Neurotransmitter Release Probability and the Synaptic Conductance
Whether or not a presynaptic action potential (or series of presynaptic action potentials) results in a postsynaptic spike depends on whether neurotransmitter is released into the synaptic cleft and whether the amount of neurotransmitter released is sufficient to elicit a postsynaptic response.
Since neurotransmitter is released in discrete vesicles, this is a stochastic process and the synaptic conductance can be thought of as a function of the probability of release.
Binding of excitatory neurotransmitter molecules to receptors on the postsynaptic membrane results in the opening of postsynaptic ion channels. And so the postsynaptic conductance increases with neurotransmitter release.
Short-term Plasticity
The short lived changes in synaptic conductance due to the history of activity at a synapse are termed “short-term plasticity”. These changes can be modeled in terms of the probability of neurotransmitter release at the synapse.
The two principle types of short-term plasticity are facilitation and depression.
In facilitation, presynaptic activity increases the probability of neurotransmitter release. Physiologically, this can be thought of in terms of calcium dynamics in the presynaptic terminal. Calcium ions enter the terminal during a presynaptic spike and are required for neurotransmitter release. So if a series of presynaptic spikes occurs close together, calcium builds up in the presynaptic terminal, increasing the release rate.
However, between spikes, the calcium is slowly transported back into the synaptic cleft and, correspondingly, the release probability relaxes back down to its resting value. (figure adapted from Markram et al, 1998)
In depression, presynaptic activity decreases the probability of neurotransmitter release. Physiologically, this can be explained by the fact that there is a limited number of neurotransmitter vesicles present in the presynaptic terminal. Each time some are released, the probability of the remaining vesicles fusing with the membrane and releasing their contents decreases.
However, neurotransmitter is eventually recycled back from the synaptic cleft and, correspondingly, the probability of release relaxes back up to its resting value. (figure adapted from Markram and Tsodkys, 1996)
Depression as Gain Control
A single cell in the visual cortex can receive inputs from many different types of afferent neurons. These afferents can fire at a wide variety of rates. In order that the cell is not entirely dominated by those afferents with high firing rates but instead responds to percentage changes in the firing rates of its afferents, some sort of gain control is required. Because firing rates may change with time, this gain control must be dynamic. By tempering the effects of afferents with high firing rates, short-term synaptic depression increases the sensitivity of the cell to small changes in the afferent firing patterns.
The Model
As described in the Chance et. al. paper (1998), short-term depression can be modeled in terms of neurotransmitter release probability.
Each time there is a presynaptic spike
at a particular synapse, the release probability at that synapse
decreases (neurotransmitter vesicles used up). This can be
modeled by multiplying the probability variable (
)
by a number between 0 and 1 each time there is a presynaptic spike.
The value of this multiplicative factor (
)
determines the strength of the depression at that particular synapse,
1 corresponding to no depression at all.
Between presynaptic spikes at a synapse, the probability variable for that synapse relaxes (exponentially according to physiological data) back to its resting value of one (neurotransmitter recycled).
Each time there is
a spike at any of the presynaptic inputs, the postsynaptic membrane
conductance (is increased (neurotransmitter binds to postsynaptic
receptors and ion channels open). The amount that the
postsynaptic conductance (
)
is increased by is dependant on the probability of neurotransmitter
release at that particular synapse ()
and on the synaptic weight (
).
Between presynaptic spikes, the
membrane conductance ()
decays (exponentially) back down to its resting value of zero (ion
channels close).
The postsynaptic
membrane potential (
)
is modeled using the familiar equation and is dependant upon the
membrane conductance (and thus indirectly upon the presynaptic firing
patterns, depression coefficients and synaptic weights).
(Note: The time constants, synaptic weights, depression coefficients and reversal potentials used were all chosen to best fit experimental data.
The differential equations for the exponential relaxations and membrane potential were simulated using an update rule.)
Depression and Recovery
In order to ensure that my model did exhibit short-term depression, I created a stimulus of a periodic spike train, followed by a break and then by a resumption of spiking.
And I did, in fact, see depression of the postsynaptic response followed by a fair amount of recovery. (In light of the time constants used and the length of the pause, it is appropriate that the recovery wasn't full.)
(note: I was unable to reproduce the transient effects due to synaptic depression as described in the paper. However, since I was able to reproduce the later results of temporal phase shifts and direction selectivity, I assume that my model is correct and that it was a question of interpretation of the plots in the first section)
Temporal Phase Shifts
Next, the model was driven with simulated visual inputs. The input used was a sinusoidal contrast grating which varied in both time and space, ie – a sine wave moving to the left as time progresses:
The afferents were divided into receptive fields using a “Difference of Gaussians” spatial filter.
This particular filter creates an on-center receptive field with a slightly inhibiting surround, centered at its peak.
The filter, when convolved with the stimulus, gives a Poisson rate distribution which can be converted into input spike trains. (each vertical column corresponds to an afferent and each white line corresponds to an input spike at a given instant in time):
Feeding this input train into the
model, I ran the simulation for 3 different values of the depression
coefficient ()
and obtained the following results for the postsynaptic membrane
potential (the curves were normalized by adjusting the values of the
synaptic weights in each simulation):
From this result, it is clear that synaptic depression affects the shape of the postsynaptic potential curve. The more synaptic depression present, the earlier the potential peaks. This temporal phase shift is the property that will be exploited in order to obtain direction selectivity.
Direction Selectivity
The simplest way in which to model direction selectivity is to have two receptive fields separated in space with a time delay incorporated into one. Thus, a stimulus reaching the field with the delay first would result in a maximal summation of the two responses at the postsynaptic membrane, while a stimulus travelling in the opposite direction result in the responses being too far separated in time to sum and, hence, no postsynaptic firing.
In order to achieve this, I created two on-center receptive fields, centered at different values. I gave one a depression coefficient of 0.4 and the other a coefficient of 1 (non-depressing), thus introducing a temporal phase shift between the two sets of afferents. Only when the postsynaptic responses to the two sets of afferents line up, will the combined effect be sufficient to elicit a postsynaptic spike.
Thus, when the non-depressed afferents are stimulated an appropriate amount of time before the depressed afferents, there will be a maximal postsynaptic response.
I created a receptive field with depression coefficient 0.4 centered at zero and a second non-depressing field centered at a variable value. I looked at the maximaum combined postsynaptic response to the two sets of afferents for different positions of the non-depressing field in order to find the optimal positioning of the receptive fields for right-moving sinusoidal stimulus similar to that described in the previous section.
The optimal response is obtained when the non-depressing field is centered at x=-2. This is consistant with the model in that a right moving contrast function would first stimulate the non-depressing afferents and then, subsequently, the depressing afferents:
I then fixed the positions of the two fields at 0 and -2 and tried different motions of the stimulus. A right moving stimulus, as expected, resulted in the peaks lining up and postsynaptic firing:
A left moving stimulus on the other hand elicited no firing from the postsynaptic membrane
showing a clear preference for rightwards motion.
So it is possible to create direction selectivity using the temporal phase shift introduced by short-term depression!
(Note: Each combination of receptive fields I tried appeared to be strongly selective for a particular velocity – depending on the relative values of depression and the spacing of the fields.
It should be possible to obtain more general direction selectivity as well. That is – a combination of receptive fields that will result in a postsynaptic response for a larger range of movements in the preferred direction. One way this could be achieved would be to increase the widths of the postsynaptic potential response peaks, perhaps by trying different combinations of filters and time constants.)
References
Abbott L.F., Varela J.A., Sen K., Nelson S.B.,
Synaptic Depression and Cortical Gain Control
Science, 1997.
Chance F.S., Nelson S.B., Abbott L.F.,
Synaptic Depression and the Temporal Response Characteristics of V1 Cells
The Journal of Neuroscience, 1998.
Dayan P., Abbott L.F.,
Theoretical Neuroscience; Computational and Mathematical Modeling of Neural Systems
MIT Press, 2001.
Tsodkys M.V., Markram H.,
The Neural Code between Neurocortical Pyrimidal Neurons Depends on Neurotransmitter release probability
PNAS, 1997.