Mathematical Models of Presynaptic Plasticity

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Mathematical Models of Presynaptic Plasticity
Richard Bertram Department of Mathematics
and Kasha Institute of Biophysics Florida State
University
Funding NSF grant DMS 0311856.
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Outline

• Examples of presynaptic enhancement
• Mathematical models of short-term enhancement
• Examples of presynaptic depression
• Mathematical models of short-term depression
• A phenomenological model of short-term plasticity

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How is Postsynaptic Response Measured?

• Postsynaptic Potential
• Excitatory postsynaptic potential (EPSP) is
  measured in
• the soma, in response to input from the
  dendrites.
• (2) End-plate potential (EPP) is measured in the
  muscle of a
• neuromuscular junction in response to synaptic
  input.

Postsynaptic Current The postsynaptic cell is
voltage clamped and the postsynaptic current
measured in response to input from dendrites
(EPSC) or in response to neuromuscular synaptic
input (EPC).
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How is Synaptic Plasticity Measured?

• Paired-pulse experiments two presynaptic
  impulses are
• induced. The ratio of the postsynaptic
  responses is the
• plasticity measure

• Impulse train experiments N presynaptic
  impulses are
• induced at a frequency of f Hz. Plasticity per
  impulse is
• then

SP gt 1 ? enhancement SP lt 1 ? depression
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Paired-Pulse Facilitation
EPSPs
In this system, facilitation decreases with
repeated testing. Aplysia sensory neuron synapse.
Jiang and Abrams (1998)
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Augmentation
Swandulla et al (1991)
The slope of the EPSP rises during a 10-sec train
of presynaptic impulses at 50 Hz. Slowly declines
after cessation of the train. Squid giant synapse.
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Forms of Presynaptic Potentiation
Facilitation Decay time constant (?) 10s to
100s of milliseconds
Augmentation ??5-10 seconds
Post-tetanic potentiation -- ?30 sec to a few
minutes
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Presynaptic Enhancement is Associated with
Accumulation of Ca2
Regehr et al (1994)
Accumulation of intracellular Ca2 during a 1 Hz
train of presynaptic impulses. Also, slow
increase of the postsynaptic response.
Hippocampal mossy fiber synapse.
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UV Flash of Ca2 Buffer Reduces Facilitation
EPPs recorded in excitor neuromuscular
junction of crayfish. Control shows decay of
facilitation induced by 10 presynaptic
impulses. UV flash greatly increases the affinity
of the Ca2 buffer diazo-2.
Kamiya and Zucker (1994)
Similar results for augmentation and post-tetanic
potentiation.
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Fogelson-Zucker Model

• First mathematical model of facilitation.
• Model consists of a PDE for 3-D Ca2 diffusion in
  the
• presynaptic terminal.
• Facilitation due to slow increase of average Ca2
  concentration
• during a train of impulses.

In simulation of squid giant synapse, average
Ca2 concentration slowly rises during 20 Hz
train of presynaptic impulses, and
falls following the train.
Fogelson and Zucker (1985)
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Problems with the Fogelson-Zucker Model

• Facilitation has been shown to be very
  temperature
• dependent. This is not consistent with a
  mechanism
• based purely on diffusion.
• The normalized release time course should look
  the
• same in facilitated and unfacilitated
  responses. This
• is not the case with the model.
• If all Ca2 binding sites for release are
  located near
• Ca2 channels, and if binding affinity is
  in the 5-20 ?M
• range, its hard to see how a small (sub
  ?M) buildup
• of Ca2 can play a large role in
  facilitation.

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A Model Based on Residual Bound Ca2
Bertram, Sherman, and Stanley model (1996)
j1,2,3,4
The Ca2 unbinding rate (k-) is large for site 4
and progressively smaller for other binding
sites. As a result, some Ca2 remains bound when
the second impulse occurs.
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Advantages of the Bertram et al. Model

• Easy to implement. Since residual free Ca2 is
  not involved,
• no need to solve the Ca2 diffusion equation.
  Instead,
• equilibrium formulas for microdomain Ca2 are
  used, assuming
• colocalization of release sites with one or
  more Ca2 channel.
• Since Ca2 acceptors would be proteins, which
  are very
• sensitive to temperature, the facilitation
  will have a strong
• temperature dependence.
• Time course of release is invariant during
  facilitation.

Bertram et al (1996)
Datyner and Gage (1980)
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Problem with a Model Based on Residual Bound Ca2
Cant explain the Kamiya-Zucker experiment,
where facilitation was reduced following
photolysis of the Ca2 buffer diazo-2.
Kamiya and Zucker (1994)
Also cant explain the reduction of facilitation
in the presence of the Ca2 buffer EGTA-AM.
Atluri and Regehr (1996)
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A Third Facilitation Model Residual Free Ca2
with a Distant Facilitation Site
If the facilitation site responds to residual
free Ca2, it must be high affinity (KD lt 5 ?M).
If this is close to a Ca2 channel it will
saturate with each impulse. Solution of Tang et
al. (2000) is to postulate a set of low-affinity
release sites (X) close to a channel (10-20 nm),
and a high-affinity facilitation site (Y)
farther (80-100 nm) from the nearest channel.
Tang et al (2000)
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Model Geometry of the Terminal
Ca2 reaction-diffusion equations are solved,
assuming a mobile endogenous buffer and a mobile
exogenous buffer (fura-2 or BAPTA).
Tang et al. (2000)
Low affinity (X) binding sites are close to a
Ca2 channel. High affinity (Y) sites could
be on the other side of the vesicle (which has
?50 nm diameter).
Filled circles represent open Ca2 channels.
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Ups and Downs of the Tang et al. Model

• The model is able to reproduce data on the
  effects of fura-2
• on facilitation in the crayfish neuromuscular
  junction. However,
• only if several assumptions are made
• Fura-2 is immobilized and the Ca2 coefficient is
• reduced fivefold in a 200 nm layer around
  the active zone.
• This is postulated to be due to a high
  degree of tortuosity.
• In the rest of the terminal, the diffusion
  coefficient of fura-2
• is reduced 100-fold (presumably because of
  binding to
• cytosolic compounds).

Experimental data, Tang et al. (2000)
Model, Matveev et al. (2002)
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A Fourth Facilitation Model Buffer Saturation
Klingauf and Neher (1997) modeled buffered Ca2
diffusion in a neuroendocrine cell (chromaffin
cell). Granules thought to be farther from Ca2
channels (200-300 nm) than in synapses. At these
distances, buffers can have a large effect on the
Ca2 time course.
Model simulations, Ca2 at different distances
from a channel.
No exogenous buffer
500 ?M Fura-2
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Buffer Saturation Increases Ca2 Signal
During a train of impulses the buffer can become
saturated. This is due to residual Ca2 binding
between pulses. Matveev et al. (2004) recently
did a numerical study of this form of
facilitation that was first suggested by Klingauf
and Neher (1997). For more on this come to
Victors talk next Friday!
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Synaptic Dpression
Many synapses exhibit depression rather than
potentiation, particularly the smaller synapses
in the central nervous system. This is thought to
be due primarily to a depletion of vesicles in
the Readily Releasable Pool (RRP).
Depression in a pyramidal neuron from the rat
cortex.
Markram et al. (1998a)
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Depression and Gain Control
Cortical neurons integrate input from about
10,000 synapses. The presynaptic afferents
produces impulses with rates from about 1 Hz to
200 Hz. Why dont the high-frequency
inputs dominate the low-frequency inputs?
Gain control synapses firing at high frequency
are depressed, so response to each impulse is
smaller.
Relative response amplitude, A(r), declines as
1/r for r greater than some limiting frequency.
Total synaptic conductance is rA(r).
Abbott et al. (1997)
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Postsynaptic Cell Responds to the RelativeChange
in the Presynaptic Input Rate
If input frequency is changed by ?r, then
?R?rA(r)??r/r. The change in the initial
response is proportional to the relative change
in input frequency. Thus, the synapse responds to
the derivative of the presynaptic input
frequency,
See Abbott et al. (1997) and Tsodyks and Markram
(1997).
Data from cortical slice.
Abbott et al. (1997)
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Depression Due to G Protein-Mediated Ca2 Channel
Inhibition

• Many chemical messengers can act as ligands for
• G-protein-coupled receptors. These include
• GABA
• Adenosine
• Glutamate
• Dopamine
• Serotonin

Binding of G?? puts Ca2 channel into a reluctant
state.
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G Protein-Mediated Autoinhibition
Transmitter molecules released by a synapse may
bind to G protein-coupled receptors on the
synapse itself, inhibiting Ca2 channels and
reducing future transmitter release.
EPSCs from dopamine synapses in the striatum. S1
is a train of 3 impulses at 100 Hz. S2 is
a single impulse 200 ms later.
Depression in wild-type (closed) vs. D2-R
knockouts (open).
Benoit-Marand et al. (2001)
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A Model for G Protein Inhibition
The Bertram and Behan model (1999) focuses on the
binding of activated G proteins to Ca2 channels.
When bound, the channel enters a reluctant state,
where opening during an impulse is unlikely. G
proteins unbind when the membrane is depolarized.
Come to my talk next Friday!
where
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A Phenomenological Model for Short-Term Plasticity
Markram and collaborators developed a
phenomeonological model that focuses on 4 main
parameters for short-term plasticity (Tsodyks
and Markram, 1997 Markram et al., 1998a).
A absolute synaptic efficacy
U fraction of efficacy used up by first impulse
?rec recovery from depression
?facil decay of facilitaion
un running value of U (facilitation
variable)
Rn remaining efficacy (depression
variable)
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Equations
From Markram et al. (1998a)
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Differential Effects of Secretion Parameters on
Frequency Response
A increased 1.7-fold
U increased 1.7-fold
Markram et al. (1998b)
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Differential Effects (cont)
?rec decreased 10-fold
?facil increased 3-fold
Markram et al. (1998b)
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Facilitation vs. Depression
In a facilitating synapse between a neocortical
pyramidal neuron and an interneuron, peak
steady-state EPSP amplitude is at ?20 Hz.
Beyond this, depression overcomes the effects of
facilitation. Beyond the limiting frequency ?,
EPSP amplitude decays as 1/f and the synapse
responds to changes in the stimulus frequency
(the derivative) rather than the frequency itself.
Markram et al. (1998a)
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Summary

• Most synapses show some form of short-term
  plasticity.
• Some facilitate, while others primarily
  exhibit depression.
• At least four different mechanisms have been
  proposed for
• facilitation. Each of these has been developed
  and explored
• through mathematical modeling.
• There are known to be at least two mechanisms
  for
• presynaptic depression. Models have been
  developed for
• each, and used to interpret the role of
  depression in synaptic
• information processing.
• Models of plasticity range from simple to
  complex, each
• providing insight into the mechanisms and
  roles of plasticity.

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References

• Abbott, L.F, J. A. Varela, K. Sen, S. B. Nelson,
  Synaptic depression and cortical
• gain control, Science, 275220-224, 1997.
• Atluri, P. P. and W. G. Regehr, Determinants of
  the time course of facilitation
• at the granule cell to Purkinje cell synapse,
  J. Neurosci., 165661-5671, 1996.
• Benoit-Marand, M., E. Borrelli, F. Gonon,
  Inhibition of dopamine release via
• presynaptic D2 receptors Time course and
  functional characteristics in vivo,
• J. Neurosci., 219134-9141, 2001.
• Bertram, R. and M. Behan, Implications of
  G-protein-mediated Ca2 channel
• inhibition for neurotransmitter release and
  facilitation, J. Comput. Neurosci.,
• 7197-211, 1999.
• Bertram, R., A. Sherman, E. F. Stanley,
  Single-domain/bound calcium hypothesis
• of transmitter release and facilitation, J.
  Neurophysiol., 751919-1931, 1996.
• Datyner, N. B. and P. W. Gage, Phasic secretion
  of acetylcholine at a mammalian
• neuromuscular junction, J. Physiol.,
  303299-314, 1980.
• Fogelson, A. L. and R. S. Zucker, Presynaptic
  calcium diffusion from various
• arrays of single channels, Biophys. J.,
  481003-1017, 1985.

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References

• Kamiya, H. and R. S. Zucker, Residual Ca2 and
  short-term synaptic plasticity,
• Nature, 317603-606, 1994.
• Klingauf, J. and E. Neher, Modeling buffered
  Ca2 diffusion near the membrane
• Implications for secretion in neuroendocrine
  cells, Biophys. J., 72674-690, 1997.
• Markram, H., Y. Wang, M. Tsodyks, Differential
  signaling via the same axon
• of neocortical pyramidal neurons, Proc. Natl.
  Acad. Sci. USA, 955325-5328, 1998a.
• Markram, H., A. Gupta, A. Uziel, Y. Wang, M.
  Tsodyks, Information processing
• with frequency-dependent synaptic connections,
  Neurobio. Learn. Mem., 70101-112,
• 1998b.
• Matveev, V., A. Sherman, R. S. Zucker, New and
  corrected simulations of synaptic
• facilitation, Biophys. J., 831368-1373, 2002.
  
• Matveev, V., R. S. Zucker, A. Sherman,
  Facilitation through buffer saturation
• Constraints on endogenous buffering
  properties, preprint.

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References

• Jiang, X.-Y. and T. W. Abrams, Use-dependent
  decline of paired-pulse facilitation
• at Aplysia sensory neuron synapses suggests a
  distinct vesicle pool or release
• mechanism, J. Neurosci., 1810310-10319, 1998.
  
• Regehr, W. G, K. R. Delaney, D. W. Tank, The
  role of presynaptic calcium in
• short-term enhancement at the hippocampal
  mossy fiber synapse, J. Neurosci.,
• 14523-537, 1994.
• Swandulla, D., M. Hans, K. Zipser, G. J.
  Augustine, Role of residual calcium
• in synaptic depression and posttetanic
  potentiation Fast and slow calcium
• signaling in nerve terminals, Neuron,
  7915-926, 1991.
• Tang, Y.-g, T. Schlumpberger, T.-s. Kim, M.
  Lueker, R.. S. Zucker, Effects of
• mobile buffers on facilitation Experimental
  and computational studies,
• Biophys. J., 782735-2751, 2000.
• Tsodyks, M. V. and H. Markram, The neural code
  between neocortical
• pyramidal neurons depends on neurotransmitter
  release probability,
• Proc. Natl. Acad. Sci. USA, 94719-723, 1997.
  
• Zucker, R. S. and W. G. Regehr, Short-term
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• Physiol., 64355-405, 2002.