Three different classes of synapse can be defined; spiking chemical, non-spiking
chemical and electrical. Each class can have an unlimited number of types which differ in the
quantitative values of their defining parameters. Only conductance increase
chemical synapses can be simulated in this neural plug-in. The way the three classes are modeled, and
the significance of the defining parameters, is described next.
Spiking Chemical Synapses
Click here for a detailed list spiking chemical synapse properties.
Click here for an explanation on managing synaptic types.
A spiking chemical synapse is modeled as a sudden increase in post-synaptic
conductance which occurs when, and only when, the pre-synaptic neuron spikes. A
delay can be set between the pre-synaptic spike and the post-synaptic response. The post-synaptic
conductance has a defined equilibrium (reversal) potential, and this determines
whether the synapses is excitatory (equilibrium potential above spike threshold)
or inhibitory (equilibrium potential below spike threshold). The conductance
increase has an instantaneous onset to a defined amplitude, and then declines
exponentially back to zero with a defined decay rate time constant. If a second
synaptic event of this type occurs in a particular neuron before the conductance
increase initiated by the first has died away, the two events sum.
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1. Pre-synaptic spike. |
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2. Delay between pre-synaptic spike and post-synaptic response. |
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3. Initial amplitude of post-synaptic conductance increase. |
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4. Decay rate (exponential time constant) of post-synaptic conductance increase.
(approximately equals time to decay to 37% of initial value). |
Spiking chemical synapses can be defined as facilitating or anti-facilitating
(decrementing). Facilitation is modeled as follows. If a PSP occurs immediately
following a PSP of the same type from the same pre-synaptic neuron, then the
amplitude of the conductance increase at the start of the second PSP is the
value of the conductance increase at the start of the first PSP, multiplied by
the relative facilitation factor. Thus a relative facilitation value of 1 means
that no facilitation occurs, a value greater than 1 means the synaptic response
increases (facilitation), while a value less than 1 means the synaptic response
decreases (anti-facilitation). As the time interval between two synaptic events
of the same type from the same pre-synaptic neuron increases, the degree of
facilitation decays back exponentially towards a value of 1 (i.e. no
facilitation) with a defined decay time constant.
NOTE: this model produces a fairly realistic qualitative simulation, but this
does not imply that the model can be used to fit real data in a quantitative
manner.
If two pre-synaptic spikes occur, the degree of facilitation depends upon the
interval between them.
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1. Amplitude of initial (unfacilitated) post-synaptic conductance increase. |
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2. Maximally facilitated amplitude of post-synaptic conductance increase.
[Relative facilitation factor = 2/1]. |
3. Decay rate (exponential time constant) of facilitation.
(approximately equals time to decay to 37% of initial value). |
If a series of pre-synaptic spikes occur, facilitation effects from several
pre-synaptic spikes may contribute to each post-synaptic response.
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1. Amplitude of initial (unfacilitated) post-synaptic conductance increase. |
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2. Maximally facilitated amplitude of post-synaptic conductance increase. |
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3. Amplitude of facilitation contributed by immediately preceding pre-synaptic
spike. |
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4. Amplitude of facilation contributed by 2nd preceding spike. |
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5. Amplitude of facilitation contributed by 3rd preceding spike. |
Spiking chemical synapses can be defined as voltage-dependent, if desired. The
underlying concept is that the post-synaptic conductance is blocked (e.g. by
extracellular Mg++ ions in the case of NMDA-type channels) when the
post-synaptic membrane is relatively hyperpolarised, and unblocked when the
post-synaptic membrane is relatively depolarised. The mechanism is modeled as
follows. First a nominal non-voltage-dependent conductance is calculated as
described in the preceding sections, with a baseline initial unfacilitated level
equal to the fully blocked conductance. This nominal conductance is then scaled
by a factor dependent on the post-synaptic membrane potential. The scaling
factor is derived from the baseline and maximum relative conductances, which
define the initial values of the fully blocked and fully unblocked conductances
for the unfacilitated synapse, and a threshold and saturation voltage, which
define the post-synaptic voltage at which the block starts to be removed
(threshold) and at which the block is fully removed (saturation).
Typical Scaling Function for a Voltage-Dependent Synapse:
Post-Synaptic Voltage -> Post-Synaptic Conductance Increase
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1. Minimum (blocked) amplitude of initial baselin (unfacilitated) post-synaptic
conductance increase. |
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2. Maximum (unblocked) amplitude of initial (unfacilitated) post-synaptic
conductance increase. |
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3. Threshold post-synaptic voltage at which conductance starts to unblock. |
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4. Saturation post-synaptic voltage at which conductance is fully unblocked. |
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5. Example: at a post-synaptic membrane potential of -40 mV, the conductance
will be scaled to about 75% of its nominal (non-voltage-dependent) value. |
Spiking chemical synapses can be defined as having Hebbian properties, if
desired. A Hebbian synapse is one in which the synaptic strength is dependent
upon the degree of conjoint activity of the pre- and post-synaptic neurons. In
other words, if the pre- and post-synaptic neurons are both active at relatively
high frequency, the synaptic strength is augmented (“trained”). Synapses with
these properties are thought to be involved in memory-related processes such as
long-term potentiation (LTP) and classical conditioning.
Learning (augmentation) at Hebbian synapses is modeled as follows. Each Hebbian
synaptic type has a baseline conductance which represents its naive, unaugmented
state and to which all synaptic connections mediated by that type are set
initially. Each neuron which receives Hebbian input retains a short-term memory
(with duration set by the Hebb Time-Window parameter) in which it records the
time of occurrence of any Hebbian inputs which impinge upon it. Whenever such a
post-synaptic neuron spikes, it checks its short term memory to see whether a
Hebbian input has occurred within the time window. If such an input has
occurred, then the strength of that particular connexion is augmented. The
degree of augmentation depends upon the Increment parameter, and the time
interval between the input and the spike.
newG = G + Inc * (maxG - G) * ((window-interval)/ window)
where newG is the new conductance of the connexion, G is the current conductance
of the connexion, Inc is the fractional increment, maxG is the maximum, fully
augmented conductance, interval is the time interval between the Hebbian
pre-synaptic input and the post-synaptic spike, and window is the Hebb time
window.
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A. Spike in pre-synaptic neuron mediating Hebbian input onto post-synaptic
neuron (this is the conditioned stimulus CS of classical conditioning).
Left-hand vertical red line indicates post-synaptic conductance associated with
left-hand EPSP (blue). |
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B. Spike in post-synaptic neuron caused by a different pre-synaptic neuron
mediating powerful supra-threshold input (this is the unconditioned stimulus US
of classical conditioning). |
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C. Second spike in pre-synaptic neuron mediating Hebbian input. Right-hand
vertical red line indicates post-synaptic conductance associated with right-hand
EPSP, which is augmented due to the previous conjoint occurrence of the
pre-synaptic spike A and the post-synaptic spike (blue neuron) caused by input
from pre-synaptic spike B. |
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1. Initial (baseline) conductance amplitude of Hebb-type synaptic connexion
between pre- and post-synaptic neurons. |
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2. Maximum amplitude of Hebbian synapse when fully trained. Assume 15 units. |
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3. Time interval between Hebb-type input. from pre-synaptic spike A and
post-synaptic spike B. Assume 10 msec. |
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4. Time-window of Hebb memory. Assume 30 msec. |
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5. Conductance of Hebbian synapse at time of pre-synaptic spike A. Assume 5
units. This is already partially augmented above baseline due to previous
experience (not shown). |
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6. Further augmented conductance level of Hebbian synapse at time of
pre-synaptic spike B. Assume Hebbian increment parameter is set to 0.5, then new
conductance = 5 + (15-5) * 0.5 * (30-10)/30 units = 8.3 units |
Hebbian synapses can either be set to retain augmentation indefinitely, or to
“forget” unless learning is periodically enforced. Two factors control
forgetting; the forgetting time window, and the consolidation factor. The two
factors interact, but the former determines a baseline for how quickly synapses
forget their training, and the latter determines the degree to which well
trained (strongly augmented) synapses forget more slowly than weakly trained
synapses.
If the consolidation factor is 1, then that synaptic type forgets at the same
rate, irrespective of the degree of training. The augmented conductance simply
declines back to baseline linearly during the forgetting time window. Thus
whenever a Hebbian synaptic input occurs the post-synaptic conductance is
adjusted as follows. If the time since the most recent Hebbian augmentation
(i.e. a spike in the post-synaptic neuron which augmented that synapse according
to the criteria described above) is greater than the forgetting window, then the
synaptic conductance is set to its baseline level. If the time is within the
forgetting window, then the synaptic conductance is set to a decreased level of
augmentation:
newG = G - (G - baseG) * times since last Hebb event / ForgettingWindow
where baseG is the baseline conductance, ForgettingWindow is the forgetting time
window, and other symbols are as defined above.
Example of Hebbian Forgetting Time Window
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A. Spike in post-synaptic neuron which augments a Hebbian synapse. The
pre-synaptic spike which preceded this is not shown. This post-synaptic spike
starts the clock for the forgetting time window. |
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B. Spike in pre-synaptic neuron mediating Hebbian input occurs early in
forgetting time window and elicits a conductance change which is reduced from
the nominal augmented level but is still large. |
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C. Second spike in pre-synaptic neuron mediating Hebbian input late in
forgetting time window and elicits a conductance change which is reduced from
actual augmented level of the previous input. |
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D. Third spike in pre-synaptic neuron mediating Hebbian input occurs outside
forgetting time window and elicits a conductance change which has returned to
its baseline unaugmented level. |
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1. Initial (baseline) conductance amplitude of Hebb-type synaptic connexion
between pre- and post-synaptic neurons. |
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2. Maximum amplitude of Hebbian synapse when fully trained. |
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3. Nominal value of augmented conductance immediately following post-synaptic
spike A which augments synapse. |
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4. Actual value of augmented conductance from spike B. |
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5. Actual value of augmented conductance from spike C. |
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6. Forgetting time window. |
If the consolidation factor is greater than 1, then more strongly augmented
synapses forget more slowly. For these synapses the forgetting time window is
extended by product of the fractional degree of augmentation and the amount by
which the consolidation factor exceeds 1.
G% = (G - baseG) / (maxG - baseG)
Inc = (consolidation - 1) * G%
ForgettingWindow = baseForgettingWindow + baseForgettingWindow * Inc |
where G% is the current fractional augmentation in synaptic conductance due to
Hebbian learning, baseForgettingWindow is the baseline forgetting time window
and other symbols are as defined above.
Example of Hebbian Consolidation
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1. Initial (baseline) conductance amplitude of Hebb-type synaptic connexion
between pre- and post-synaptic neurons Assume 1 unit. |
| 2. Maximal (fully augmented/trained) conductance amplitude of
Hebb-type synaptic
connexion between pre- and post-synaptic neurons. Assume 4 units. |
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3. Baseline forgetting time window. Assume 100 s Assume consolidation factor 3. |
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4. Forgetting time window for conductance augmented to 2 units (33%) Augmented
forgetting time window = 166 s. |
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5. Forgetting time window for conductance augmented to 3 units (66%) Augmented
forgetting time window = 233 s. |
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6. Forgetting time window for fully trained/augmented conductance (value 4
units, 100%) Augmented forgetting time window = 300 s. |
NOTE: The quantitative model for Hebbian synapses used in this simulation is
speculative and its only justification is that it produces a behaviour which
approximates to what is thought may occur. However, the existence of Hebbian
synapses themselves is speculativeno, and thus no real synapses have been
described from which Hebbian response characteristics can be quantified.
Non-Spiking Chemical Synapses
Click here for a detailed list spiking chemical synapse properties.
A non-spiking chemical synapse is modeled as a variable increase in
post-synaptic conductance whose level depends upon the pre-synaptic membrane
potential. The post-synaptic conductance has a defined equilibrium (reversal)
potential, and this determines whether the synapses is excitatory (equilibrium
potential above spike threshold) or inhibitory (equilibrium potential below
spike threshold). The relationship (transfer function) between the pre-synaptic
voltage and the post-synaptic conductance is determined by three paramters. The
pre-synaptic voltage threshold is the minimum pre-synaptic membrane potential at
which a post-synaptic conductance increase occurs; if the pre-synaptic voltage
is below this value, there is no post-synaptic effect. The pre-synaptic
saturation voltage is the pre-synaptic membrane potential at which the
post-synaptic response reaches its maximum value; further depolarisation of the
pre-synaptic neuron above this value does not increase the post-synaptic
response. The maximum post-synaptic conductance amplitude is exactly that; the
amplitude of the post-synaptic conductance when the pre-synaptic membrane
potential reaches or exceeds the pre-synaptic saturation voltage. Between the
pre-synaptic threshold and saturation voltages, the post-synaptic conductance
varies linearly between 0 and the maximum amplitude.
Pre-Synaptic Voltage -> Post-Synaptic Conductance Increase
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1. Pre-synaptic voltage threshold. |
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2. Pre-synaptic saturation voltage. |
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3. Maximum post-synaptic conductance amplitude. |
Click here for a detailed list spiking chemical synapse properties.
An electrical synapse is modeled as a non-specific electrical conductance
linking two neurons. Current therefore flows from one neuron to the other
whenever there is a difference between the membrane potentials of the two
neurons and the junctional (electrical synapse) conductance is greater than
zero. There are two types of electrical synapse:
Non-rectifying electrical synapses have a constant junctional conductance
irrespective of the membrane potential of either neuron.
Recifying electrical synapses have a junctional conductance which varies with
the voltage difference between the membrane potentials of the two neurons (junctional
potential). These synapses are thus polarised, and by definition the
pre-synaptic neuron is the neuron which acts as a source for positive current
(i.e. depolarising potentials pass from the pre-synaptic neuron to the
post-synaptic neuron, and not vice versa). The junctional potential is defined
as the pre-synaptic membrane potential minus the post-synaptic membrane
potential, and thus junctional potential is always positive when the rectifying
synapse is in the high-conductance state.
Each electrical synapse has a defined minimum conductance, which is the standard
conductance of a non-rectifying synapse, or the "off" conductance of a
rectifying synapse, and a defined maximum conductance, which is the "on"
conducance of a rectifying synapse. For non-rectifying synapses, the maximum and
minimum conductances should be set equal. Rectifying synapses have two further
parameters, the turn-on junctional potential, which is the potential at which
the junctional conductance starts to increase above the minimum level, and the
saturation junctional potential, which is the potential at which the junctional
conductance reaches maximum. Between the turn-on and saturation junctional
potentials, the junctional conductance varies linearly between the minimum and
maximum values.
electrical synapse: Junctional potential vs junctional conductance
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1. Minimum junctional conductance. |
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2. Maximum junctional conductance. |
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3. Turn-on junctional potential. |
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4. Saturation junctional potential. |
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