Neurological Sciences Institute, OHSU, Portland, Oregon 97209
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ABSTRACT |
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Roberts, Patrick D.. Modeling Inhibitory Plasticity in the Electrosensory System of Mormyrid Electric Fish. J. Neurophysiol. 84: 2035-2047, 2000. Mathematical analyses and computer simulations are used to study the adaptation induced by plasticity at inhibitory synapses in a cerebellum-like structure, the electrosensory lateral line lobe (ELL) of mormyrid electric fish. Single-cell model results are compared with results obtained at the system level in vivo. The model of system level adaptation uses detailed temporal learning rules of plasticity at excitatory and inhibitory synapses onto Purkinje-like neurons. Synaptic plasticity in this system depends on the time difference between pre- and postsynaptic spikes. Adaptation is measured by the ability of the system to cancel a reafferent electrosensory signal by generating a negative image of the predicted signal. The effects of plasticity are tested for the relative temporal correlation between the inhibitory input and the sensory input, the gain of the sensory signal, and the presence of shunting inhibition. The model suggests that the presence of plasticity at inhibitory synapses improves the function of the system if the inhibitory inputs are temporally correlated with a predictable electrosensory signal. The functional improvements include an increased range of adaptability and a higher rate of system level adaptation. However, the presence of shunting inhibition has little effect on the dynamics of the model. The model quantifies the rate of system level adaptation and the accuracy of the negative image. We find that adaptation proceeds at a rate comparable to results obtained from experiments in vivo if the inhibitory input is correlated with electrosensory input. The mathematical analysis and computer simulations support the hypothesis that inhibitory synapses in the molecular layer of the ELL change their efficacy in response to the timing of pre- and postsynaptic spikes. Predictions include the rate of adaptation to sensory stimuli, the range of stimulus amplitudes for which adaptation is possible, the stability of stored negative images, and the timing relations of a temporal learning rule governing the inhibitory synapses. These results may be generalized to other adaptive systems in which plasticity at inhibitory synapses obeys similar learning rules.
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INTRODUCTION |
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The importance of plasticity at
inhibitory synapses has only recently been recognized. The
restructuring of the cortical somatosensory maps following lesions or
changes in use, for example, has been found to be dependent on the
presence of GABA activity. This observation lead to the conclusion that
the restructuring involves changes in the strength of inhibitory
synaptic input (Jacobs and Donoghue 1991; Lane et
al. 1997
; Mower et al. 1984
). Similar results
have been found in the inferior colliculus of the barn owl where the sensory maps integrating auditory and visual stimuli converge (Zheng and Knudsen 1999
). These studies suggest that the
inhibitory pathways are an important component to adaptation of
neuronal responses to changing sensory conditions.
It is not always clear in these system level studies whether the
plasticity occurs at excitatory synapses onto inhibitory interneurons
or at the inhibitory synapses themselves. Plasticity localized to
inhibitory synapses themselves has been demonstrated, however, in a few
different systems including inhibitory synapses in the visual cortex
(Komatsu and Iwakiri 1993), inhibitory neurons on
Mautner cells (Korn et al. 1992
), and inhibitory neurons
onto Purkinje cells in the cerebellum (Kano et al.
1992
). Plasticity at such inhibitory synapses is likely to be
important in central processing of sensory information as well as of
other types of information. But few published modeling studies of
inhibitory plasticity (Marshall 1990
; Nelson and
Paulin 1995
; Sirosh and Miikkulainen 1994
) have
elucidated the potential roles of plasticity at inhibitory synapses.
Furthermore, the interactions between inhibitory plasticity and
plasticity at excitatory synapses are only poorly understood.
The present study uses techniques of mathematical and computer modeling
to examine the possible roles and contributions of plasticity at
inhibitory synapses in the electrosensory lateral line lobe (ELL) of
mormyrid electric fish. This structure, and other cerebellum-like
sensory structures in electroreceptive fish, have been shown to be
adaptive sensory processors that subtract out predictable features of
the sensory inflow following a period of association between centrally
originating predictive signals and particular patterns of sensory input
(Bell et al. 1997a). Synaptic plasticity at excitatory
synapses has been demonstrated experimentally in the ELL
(Bastian 1998
; Bell et al. 1997c
;
Bodznick et al. 1999
). A previous modeling study has
shown how this cellular level plasticity can yield the system level
adaptive properties of the ELL (Roberts and Bell 2000
).
The ELL is rich in inhibitory neurons and inhibitory synapses. Although
plasticity at these inhibitory synapses has not yet been demonstrated,
an adaptive system level function for the structure as a whole, the
well demonstrated plasticity at excitatory synapses, and the presence
of extensive inhibitory interactions within the ELL suggest that this
region is a good candidate to examine the potential contributions of inhibitory synaptic plasticity.
Mormyrid electrosensory system
Mormyrid electric fish have an electric organ in their tail that
generates a brief pulse of electric current, an electric organ
discharge (EOD). The EOD generates electric field pulses in the near
vicinity of the fish in response to a centrally originating motor
command. The fish can navigate without vision by detecting distortions
caused by external objects in its self-generated electric field
(Bastian 1986).
Mormyrid fish have three classes of electroreceptors that are used for three different purposes: mormyromasts, knollenorgans, and ampullary receptors. Mormyromasts are used for active electrolocation, knollenorgans are used to sense the EODs of other electric fish in electro-communication, and ampullary receptors are used to sense the low-frequency external fields that all animals, electric and nonelectric, generate in the water.
Primary afferent fibers from mormyromast and ampullary electroreceptors
terminate in separate regions of the cortex of the electrosensory
lateral line lobe. The ELL is a laminar structure, and neurons of
interest for this study, the medium ganglion (MG) cells, have their
cell bodies in the ganglion cell layer (see Fig.
1). These neurons have a large dendritic
tree of apical dendrites that reach into the molecular layer. There
they receive synaptic contact from excitatory parallel fibers and
inhibitory interneurons, the largest population of which are referred
to as stellate cells (Grant et al. 1996; Meek et
al. 1996
). This study is restricted to the region of the ELL
cortex that receives ampullary receptor input.
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Recordings from the cell bodies of MG cells reveal two types of spikes;
a narrow, presumably axonal, spike is evoked by moderate depolarization, and a large, broad spike is evoked at
stronger depolarizations. Field recordings suggest that the broad
spikes propagate into the apical dendrites of the molecular layer
(Grant et al. 1998).
The motor command that initiates an EOD originates in the command
nucleus and traverses the spinal cord to the electric organ. Simultaneously, a corollary discharge signal projects to the ELL to
intersect with the afferent electrosensory information from the
receptors (Bell 1982; Bell et al. 1983
).
These signals converge on medium ganglion cells, where basilar
dendrites receive the afferent inputs via interneurons in the granular
layer, and the apical dendrites receive corollary discharge inputs via
the parallel fibers and stellate cells in the molecular layer
(Bell et al. 1992
).
Mormyrid electric fish can sense the subtle external electric fields of interest over the background of its own electric discharge. Since the fish generates the signal, it might be advantageous to develop an adaptive filtering mechanism that eliminates the predicted electrosensory image to emphasize subtle novelties in the environment.
The MG cell responses adapt to eliminate the predicted electrosensory
image within the sensory signal (Bell 1982). Recordings near fibers known to excite granule cells that project into the molecular layer as parallel fibers suggest that their responses to the
corollary discharge are not simultaneous, but are distributed in time
following the command signal (Bell et al. 1992
).
Corollary discharge timing information of the electric discharge
arrives through parallel fibers, so a likely candidate for adaptation is the synapse between the parallel fibers and the apical dendrites.
The synaptic efficacy of the parallel fibers onto the MG cells has been
shown to change depending on the relative timing of the presynaptic
volley evoked excitatory postsynaptic potential (EPSP) and the
postsynaptic broad spike (Bell et al. 1997c). The synaptic efficacy is depressed following a pairing period in which the
postsynaptic spike follows the beginning of the EPSP within a narrow
time window of about 50 ms. This effect is referred to as
associative depression. If the postsynaptic broad spike
occurs at any other delay, then the synaptic efficacy is enhanced. This increase does not depend on the occurrence of the postsynaptic spike
and is referred to as nonassociative enhancement. Modeling studies have shown (Roberts and Bell 2000
) that the
exact form of the learning rule is critical for the system level
adaptive function of the ELL's ampullary region. If the experimentally established learning rule is used in the model, then the parallel fiber
inputs adapt to generate a negative image of the previously paired sensory input. Adding this negative predictable component to the
actual input eliminates modulation of the MG cell responses to
predictable electrosensory signals. If other learning rules with other
forms of temporal dependence on the relative timing of pre- and
postsynaptic events, are used, then the model demonstrates that the
negative image is not as faithful a copy of the original sensory input
because the learning is dynamically unstable (Roberts and Bell
2000
).
A similar adaptive filter system has been found in another
electrosensory system. In the gymnotid ELL, fibers carrying
proprioceptive information adjust their synapses in a way that can
cancel the predictable changes in electroreceptor signal intensity due
to bending of the fish's body (Bastian 1995). In
addition, experiments on synaptic plasticity (Bastian
1998
) suggest that some of the adaptation is caused by
plasticity at inhibitory synapses.
Slice experiments in the mormyrid ELL have suggested but not yet
demonstrated plasticity at inhibitory synapses (Bell et al. 1997b). The associative depression of the EPSP evoked by a
parallel fiber stimulus appeared to be accompanied by an increase in
the inhibitory postsynaptic potential (IPSP) evoked by the same
stimulus, and the nonassociative increase in EPSP size appeared to be
accompanied by a decrease in the IPSP (Fig.
2A). These changes appear to
be due to plasticity at inhibitory synapses, but could also reflect an
IPSP of unchanging size that is masked or unmasked by accompanying changes in the EPSP.
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The present study explores the hypothesis that plasticity is indeed present at inhibitory synapses in the ELL and examines the consequences of such plasticity. The learning rule that is assumed to control inhibitory plasticity follows from the hypothesis that pairing delays between pre- and postsynaptic spikes that cause the EPSP to decrease cause the IPSP to increase and pairings at other delays that cause the EPSP to increase cause the IPSP to decrease. The main feature of this form of plasticity is that it follows the same timing as the learning rules for excitatory synapses, but in the opposite direction (see Fig. 2B).
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METHODS |
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Stochastic model neuron
The model of the MG cell is constructed to represent the simplest observable dynamics in response to externally applied synaptic input. This simplified model allows for analytic results that confirm conclusions drawn from computer simulations for a wide range of parameter settings.
The MG cell is modeled as a single compartment, stochastic threshold
device. All synaptic inputs are summed to yield a "noiseless membrane
potential" that represents the excitation level of the neuron at each
point in time. The total membrane potential is a combination of the
noiseless potential with a background noise term representing
uncorrelated synaptic activity. If the total membrane potential exceeds
a specified threshold, a spike is generated. This modeling approach is
similar to the "spike response" model (Gerstner and van
Hemmen 1992) that has been used to study the auditory system in
barn owls (Gerstner et al. 1996
). The MG cell model has
two thresholds: a lower threshold that generates a narrow (axonal)
spike, and a higher threshold that generates a broad (dendritic) spike.
Only the broad spike influences synaptic plasticity.
Two time scales are of importance to the model: a fast scale and a slow
scale. The fast scale characterizes the response of the MG cell to the
EOD over the course of tens of milliseconds and is limited to the
duration of each electric EOD. The slow scale represents the adaptation
of synaptic strengths due to synaptic plasticity over the course of
several minutes and lasting many EOD cycles. To represent these
processes independently, we separate these time scales into two
separate components. The x-component represents the time in
milliseconds following the EOD, and the t-component
represents the number of EOD cycles. The x-component is
discretized with xn = n(x) for n an integer. In the
simulations,
x = 1 ms, and the number of time steps
N = 150. Thus the dynamical variables in the model are
dependent on two temporal variables. For instance, the noiseless
membrane potential, denoted by
V(xn, t), is a
function of both xn and t. The
probability of a broad spike during cycle t at time
xn is a threshold (sigmoid) function of
the noiseless membrane potential. With threshold
, and noise parameter µ, the spike probability is given by the expression
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(1) |
Network architecture
The model MG cell receives three inputs (see Fig. 3): parallel fiber and stellate cell postsynaptic potentials representing inputs from the molecular layer, and deep layer inputs that represent the electrosensory image. The electrosensory image, Vel(xn), is based on recordings from the ampullary region of the ELL and is designed to duplicate the MG cell response to an EOD before adaptation takes place (Fig. 3).
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The parallel fiber inputs are modeled as a time-delayed series of
excitatory postsynaptic potentials. Each EPSP begins at a specified
delay following the beginning of each EOD cycle. The sequence of
delayed EPSPs is represented in Fig. 3 as the weighted parallel fiber
inputs each beginning at a different x-delay
(x1, x2,
x3, ...). There is one EPSP
beginning at each discretization step xn.
The waveform, E(xn), used for
all the EPSPs is shown in Fig. 4. The
EPSP waveform was obtained from recordings in vitro of MG cells while
inhibitory inputs were pharmacologically blocked (Grant et al.
1998). The contribution of each EPSP to the membrane potential
is obtained by multiplying the waveform by a synaptic weight,
w(xm, t). The slow time
scale t-dependence is due to the adaptability of the
synaptic weights resulting from the rules of synaptic plasticity. The
total contribution of the parallel fiber synapses,
Vpf(xn,
t), to the membrane potential is the sum of all individual
contributions
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(2) |
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The stellate cell inputs are modeled similarly to the parallel fiber
inputs, but as a series of IPSPs. The contribution of each IPSP is the
negative of the product of an IPSP waveform, I(xn), with a synaptic
weight,
(xm, t). The IPSP
waveform is positive
[I(xn)
0 for all
xn] and is based on the difference between a postsynaptic potential measured with and without inhibition blocking agents from experiments in vitro (Grant et al.
1998
). However, in the analyses and simulations, IPSP
initiations are not always delayed by regular series of intervals that
are correlated with the EOD. Because the timing of IPSP inputs from
stellate cells with respect to the EOD has not yet been experimentally confirmed, the model is used to determine the effects of different delay schemes. The contribution for all of the stellate cells, Vst(xn,
t), to the MG cell membrane potential is the sum of all of the
individual inputs (with N = 150 in the simulations)
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(3) |
Shunting inhibition
The linear summation of EPSPs and IPSPs given above may not
reflect the complete contribution of inhibitory synaptic inputs. Current from the excitatory synaptic inputs can be shunted through inhibitory ion channels, thereby preventing the excitatory current from
depolarizing the site of spike initiation (Koch et al.
1983; Rall 1964
; Tuckwell 1986
).
The effects of shunting inhibition are compared with calculations with
simple linear inhibition to determine what effects, if any, shunting
has on the learning dynamics.
Since the effect of shunting inhibition is to reduce the effective
injected current from the excitatory synapses, inhibitory synapses will
reduce the overall weight of EPSPs during the time course of the open
inhibitory receptors (see Fig. 4). For this purpose we use the open
time of GABAA receptors (Otis and Mody 1992) because IPSPs in the ELL are mediated by
GABAA. The time course of the IPSP differs from
the time course of the normalized conductance,
G(xn), due to the electrical
properties of the neuron. This algorithm for this representation of the
shunting has been chosen because it is in the spirit of the spike
response model, that is, the model yields the change in the response of
the postsynaptic neuron's output due to presynaptic spikes.
The shunted weight,
ws(xn,
t), is reduced by an amount that is dependent on the strength of
inhibition. Thus shunting is proportional to the inhibitory weights,
(xn, t). Since these
inhibitory synapses can change in a use-dependent manner, the amount of
shunting also changes under synaptic plasticity of inhibitory synapses.
A scaling factor,
, is used in the model to control the maximum
amount of shunting by the inhibitory synapses. Excitatory synaptic
weights are reduced by shunting over the time course of the
GABAA receptors' open time, so that the shunted
weights are computed by
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(4) |
The noiseless membrane potential is the sum of all inputs (Eqs.
2-4)
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(5) |
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Temporal learning rules
A previous study of this system characterized the consequences
of synaptic plasticity at excitatory parallel fiber synapses (Roberts and Bell 2000). The learning rule governing
synaptic plasticity was based on an experimentally determined learning rule and depends on the precise timing of the pre- and postsynaptic spikes during repetitive pairings. These experimentally determined temporal learning rules form the basis of the model's
implementation of synaptic change. During each EOD cycle, the MG cell
is activated by different parallel fibers in a series of delays indexed
by xn. The change of the excitatory
synaptic weights,
w(xn,
t) is functionally dependent on the time,
xb, of a broad spike in the postsynaptic
neuron following the beginning of each cycle. During each EOD cycle,
there is a nonassociative enhancement of each synapse that is set by
the nonassociative learning rate parameter,
w. If a postsynaptic broad spike occurs
during a narrow time window following the beginning of the EPSP, the
synaptic weight is reduced proportionally to a learning function,
Lw(xn), scaled by the associative learning rate,
w
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(6) |
The change in the inhibitory synaptic weights,
(xn, t), are similarly
treated, but with opposite sign and a different learning function,
L
(xn)
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(7) |
To make comparisons with experimental results on the rate of
adaptation, it is essential to use realistic values for the parameters of the learning rules. Realistic values for the synaptic learning rates
can be obtained from recent data that plot the time course of EPSP
enhancement and depression in slice preparations for different delays
between the pre- and postsynaptic stimulations (Han and Bell
1999). These data constrain the values of
w and
w, the
learning rates for non-associative enhancement and associative
depression.
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The dynamics of the system were investigated by calculating the average weight change in a continuum approximation of the formalism presented above (cf. APPENDIX). The analysis was used to characterize the general dynamics of the system independent of exact parameter choices.
Computer simulations were used to illustrate the dynamics of the system and to test explicit examples of parameter choices. The above formalism was implemented in a custom software package that could generate the relevant variables and display the spike output of the MG cell (the simulation software can be obtained by anonymous FTP from reed.edu/ftp/reed/users/proberts). Edge effects were handled by applying periodic boundary conditions to the xn component. In the simulations, the noiseless membrane potential was computed for each EOD cycle. The weights were randomized in a uniform distribution within 4% of their mean value. The assignment of broad during each time step following the command signal was based on the computed spike probability (Eq. 1) using a pseudo-random number generator. The synaptic weights were updated following each cycle as determined by the timing of the broad spikes and the learning rules, Eqs. 6 and 7.
A measure of the sensory image cancellation was required to compare
different conditions and their effects on the system. We used the mean
square contingency,
2(t)/N, to obtain the
difference between the membrane potential, V(xn, t), and
the time average of V(xn,
t) over the cycle length, V(t)
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(8) |
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RESULTS |
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A spike response model (Gerstner and van Hemmen
1992) of a medium ganglion cell was used to determine the
adaptive properties of its spike output due to synaptic plasticity. The
amplitude of both EPSPs and IPSPs would change depending on the
relative timing of pre- and postsynaptic spikes. The learning rules
would drive the output broad spike frequency to an equilibrium level that is a function of the synaptic learning rates.
As shown in the APPENDIX, the final broad spike frequency,
, of the model neuron after adaptation takes
place is given by the sum of the non-associative learning rates divided
by the sum of the associative rates
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(9) |
However, if the learning rates differ, then the average of the weights
continue to drift even though the broad spike probability has attained
a constant value, . The ensemble average change in
the synaptic weights is (see APPENDIX)
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(10) |
Uncorrelated inhibitory synaptic input
If the stellate cells in the molecular layer do not fire in
response to the parallel fibers that are time locked with the EOD,
their input will be uncorrelated with the electric organ cycle. The
first simulation investigates whether any measurable effects would
result from synaptic plasticity of uncorrelated stellate inputs. The
EPSPs arrive in a delayed series of adaptable inputs for up to 150 ms
following the onset of the EOD. However, each of the 150 stellate
cell-induced IPSPs arrive at a different time during each EOD cycle.
The delay, m(t), assigned to each
IPSP is randomly distributed throughout the first 150 ms of each cycle,
with one IPSP beginning at each time step. When this delay is
t-dependent, it changes with each EOD cycle. Thus the IPSPs
are here not correlated with the EOD.
In this case of randomly timed inhibitory inputs, the plasticity of inhibitory synapses adds no observable dynamics to the system other than contributing to the background noise. The rate of adaptation to changing sensory stimuli is the same, and the range of adaptability is the same.
If the plasticity is only at excitatory synapses, it can be shown that
conditions must be imposed on the excitatory learning rule to ensure
stability of the negative image. There must be a nonassociative
enhancement component to the learning rule, and associative depression
must be close to the form of the epsp waveform (Roberts
2000; Roberts and Bell 2000
). If the inhibitory
synaptic inputs arrive at random delays, then the same conditions on
the excitatory learning rule apply as without inhibitory plasticity.
An interesting result of inhibitory plasticity with randomly timed
inhibitory inputs is that saturation of the weights caused by synaptic
drift (Eq. 10) distorts the negative image of the sensory pattern. The noiseless membrane potentials for two simulations are
shown in Fig. 5A, where the
solid line represents the results of a simulation at t = 600 and the learning rates of the excitatory (parallel fiber)
synapses were set equal to the learning rates of the inhibitory
(stellate cell) synapses. The input of the parallel fibers plus
stellate cells cancel the sensory input
[2(t)/N = 1]. The
dashed line in Fig. 5A shows the resulting noiseless membrane potential at t = 600, where
w
<
w. Here the
weights have saturated at their lowest value (Fig. 5C) so
that the inputs are unable to cancel the highest peak of the sensory
image [
2(t)/N = 63]. Thus synaptic plasticity at inhibitory synapses that have a
random delay can be detrimental to the fidelity of the negative image
generated by the parallel fiber inputs unless the learning rates are
finely tuned.
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Correlated inhibitory synaptic input
When inhibitory inputs are correlated with the EOD, in contrast to the uncorrelated condition considered in the previous section, plasticity at inhibitory synapses can contribute to the formation of a negative image. In particular, plasticity at inhibitory synapses allows the sum of IPSPs to complement the contribution of the EPSPs when the weights of excitatory inputs are saturated.
The results of two simulations demonstrating this phenomenon are shown
in Fig. 6. The noiseless membrane
potential is shown by the two horizontal traces in Fig. 6A.
In the first simulation (weights shown in B) the ratio of
learning rates is
w/
w <
/
.
After 400 cycles, the inhibitory synaptic weights are reduced to their
lowest values except for an interval between 60 and 85 ms following the
command signal. It is during this interval that the IPSPs contribute to
the total membrane potential during the depolarizing sensory input (the peak of the dotted trace in Fig. 6A). This interval of
increased inhibitory current subtracts the residue to form a negative
image that the excitatory current cannot effect because its weights are
saturated at their zero level. During the remainder of the EOD cycle,
the excitatory inputs adjust to cancel the sensory image. This effect
is independent of the starting conditions for the weights. On possible
advantage of this saturation effect would be to minimize he synaptic
output required to generate a negative image, thereby reducing the use
of synaptic resources, such as neurotransmitters.
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The second simulation shows the result of setting the ratio of
inhibitory learning rates less than the excitatory rates,
/
<
w/
w. In this case,
the weights saturate near their greatest values (Fig. 6C).
The inhibitory synaptic weights are reduced during the interval where
they contribute to canceling the hyperpolarizing sensory input. In this
case, the system maximizes its use of synaptic resources.
Range of adaptability
Inhibitory plasticity introduces adaptable postsynaptic potentials
that can allow the neuron to generate a negative image to cancel a much
broader range of sensory input intensity. This is seen analytically in
the added term of the summation over IPSPs (Eq. 5). The
first two terms on the right hand side must combine to level the
variations of sensory image,
Vel(xn), over
xn. Inhibitory plasticity allows the
weights of the IPSPs, (xn,
t), to adapt so that hyperpolarizing regions of
Vel(xn) can
be canceled for higher peaks in the sensory image. Since there are more
inputs to adjust through synaptic plasticity, a greater range of input intensities can be canceled. Although the adaptive range to
hyperpolarizing sensory input could be increased with the addition of
more excitatory inputs, the increased range of adaptation to
depolarizing sensory input requires plasticity at inhibitory synapses.
Two simulations depicted in Fig. 7 demonstrate the increased range of adaptability. The first simulation increased the gain of the sensory input, Vel(xn) (···, Fig. 7A). The system could not adapt to the large stimulus gain. The total membrane potential is seen to deviate from a flat line (- - -, Fig. 7A). Under these conditions, the cancellation of the sensory input by the molecular layer inputs is incomplete. The range of the system's adaptability is limited because the inhibitory weights were constant in t. Saturation of the excitatory synaptic weights can be seen in Fig. 7B.
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The second simulation is run with the larger gain, but the IPSPs are now plastic and correlated with the EOD cycle and EPSPs (Fig. 7C). Here the inhibitory inputs are able to contribute to the formation of a negative image so that the total membrane potential is nearly constant during the EOD cycle (Fig. 7A, solid trace).
Rate of adaptation
The system level rate of adaptation measures the time it takes
from an abrupt change in the predictable sensory image to be canceled
by the generation of a negative image. As derived in the
APPENDIX, the rate at which deviations in the membrane
potential flatten is a monotonic increasing function of both excitatory and inhibitory learning rates (Eq. A16). If
a is the ratio of inhibitory learning rates to excitatory
learning rates (a = /
w =
/
w),
plasticity at inhibitory synapses increases the rate of adaptation by a
factor of (1 + a)
1.
A simulation was run with the inhibitory learning rates,
and
,
set equal to the excitatory learning rates based on the physiological
values for excitatory plasticity (Han and Bell 1999
).
The value of
2(t)/N is
plotted in Fig. 8A for three
conditions of the inhibitory synapses: no plasticity
(···), plasticity according to the
learning rule in Fig. 2B, and random timing with respect to
the EOD (
), and plasticity with the timing in a series of delays
following the beginning of the EOD (- - -). The adaptation time
course for the serially delayed plastic inhibitory synapses is
considerably shorter than the other two schemes. Fitting an exponential
curve [A + B exp(
t/
), where
A is an offset parameter, B is an overall scale
factor, and
is a decay constant] to the plot, we find that the
decay constant for the simulation with excitatory plasticity only is
E = 641 cycles. For the serially delayed,
inhibitory plasticity simulation the decay constant is
E+I = 168 cycles.
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These decay constants can be converted into the rate of adaptation in
the ELL by considering that in preparations in vivo, spontaneous
electric organ discharges occur at intervals of 150-400 ms. Thus the
ranges of decay constant values predicted by our simulations are
E = 1.6-4.3 min and
E+I = 0.4-1.2 min. The
adaptation rate measured by the difference in spike rate between the
pause and burst phase of the electric organ cycle is plotted in Fig.
8B. Although this is not the same method of measuring the
deviation from a constant spike rate as our
2(t)/N analysis, the
rates are comparable because they differ only in an overall scale
factor and offset parameter. Fitting these graphs to an exponential
curve yields the decay constants,
exp1 = 0.9 min and
exp2 = 0.5 min. Only the simulation
with a series of delayed synapses and inhibitory plasticity has a range
of
values that is consistant these data.
However, there is a discrepancy between how much the decay constant is
reduced by inhibitory plasticity as predicted by the analysis presented
in the APPENDIX and the simulation. The analysis predicts
that E = 2
E+I, but the exponential fit of
the simulation yields
E = 3.8
E+I. The reason for this
difference is that the analysis linearized the equation for synaptic
change by expanding the broad spike probability near the (constant)
equilibrium level (Eq. A10). Thus we can only
expect the analysis to be accurate when the system is near the constant
spike probability. If we fit only the regions of the graph in Fig.
8A where the mean square contingency
2(t)/N
40, then
we find the relationship between the decay constants to be
E = 2.1
E+I, bringing the analysis into
close agreement with the simulation.
Another important result that follows from calculations of the decay
constant, , is an analysis of instabilities in the learning dynamics. If the associative depression learning function does not
closely resemble the postsynaptic potential, then oscillations can
develop in the spike activity that interfere with the generation of a
negative image (Roberts and Bell 2000
). We find this to
be true of the learning rules for both the excitatory synapses and the
inhibitory synapses. Analytically, instabilities appear if the real
part of the decay constant becomes negative [Re(1/
) < 0]. We
have also used the simulations to test several timing relations for
pairing of parallel fiber spikes and postsynaptic broad spikes.
Simulations were run for 4,000 EOD cycles; long enough for unstable
oscillations to develop. The window of associative depression was
shifted to different delays from the beginning of the EPSP for each
simulation. Instabilities developed for shifts outside the range from
9 to 12 ms. These simulations have confirmed that very few learning
rules are stable. Thus if there is inhibitory plasticity in this
system, the model predicts that only a narrow range of learning rules
will replicate the results of experiments in vivo.
Shunting inhibition
Since inhibitory synapses are able to shunt depolarizing currents, we used the model to determine whether any new dynamics were introduced by such nonlinear inhibition. Simulations were run for all of the above results with the excitatory weights reduced by the inhibitory shunting as described by Eq. 4. The results indicate that no new dynamics were introduced by the addition of this nonlinear form of inhibition. No instabilities developed, and the rate of adaptation was unchanged.
The main result is that the effective strength of the inhibitory inputs was increased because they not only reduced the membrane potential by subtracting the IPSPs, but also reduced the weight of the EPSPs. When the series of adaptable IPSPs were correlated with the EOD cycle, the system generated a stable negative image to cancel the sensory input (Fig. 9, A and B). Because of the increased strength of the IPSPs due to shunting, the larger depolarizing actions of the sensory image sensory image could be effectively canceled.
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If the IPSPs were uncorrelated with the EOD so that each IPSP began at a random delay following the beginning of the cycle, then the excitatory inputs were unable to cancel the sensory stimuli without saturating, as shown in Fig. 9C. Except for the shunting effects, this simulation used the same parameter settings as the run that generated the data for Fig. 5. Thus the shunting inhibition would require a greater range of the excitatory synaptic weights to cancel the same magnitude of sensory stimuli.
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DISCUSSION |
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Summary of results
The results presented here lend support to the hypothesis that the inhibitory synapses from stellate cells to the medium ganglion cells of the ELL exhibit a form of plasticity that depends on the timing of the pre- and postsynaptic spikes. These results follow only if there are inhibitory inputs that are correlated with the EOD cycle in a series of delays following the discharge. In simulations that include these inhibitory inputs along with experimentally based learning rates for synaptic plasticity, the system level adaptation to a change in sensory stimuli occurs at a rate comparable to the rate measured in experiments in vivo.
The reason that such a simple model can accurately predict the system level rate of adaptation is that the learning dynamics depend primarily on the synaptic learning rates and the timing of broad spikes during each EOD cycle. The complex internal dynamics of MG cells do not contribute prominently to the learning dynamics on the relevant time scale of 10-100 ms, except to ensure that a few broad spikes appear every cycle at a rate that increases with depolarization.
Our results show advantages to having plasticity at both excitatory and
inhibitory synapses. Advantages include an increased rate of adaptation
and an ability to adapt to a wider range of stimulus intensities. These
results of the model could be tested experimentally by blocking
inhibition in the ELL and measuring the rate and range of adaptation of
MG cells due to changing electrosensory stimuli. In addition, the
combination of excitatory plus inhibitory plasticity can provide a
means of regulating the overall synaptic current injected into the
apical dendrites by taking advantage of the "drift" of the synaptic
weights when the ratio of the excitatory learning rates
(w/
w) is less than
the ratio of the inhibitory learning rates
(
/
). Under these conditions the injected current will be reduced to the
lowest level that is still capable of sculpting a negative image to
cancel the predictable sensory input. The actual ratios of learning
rates have not been measured experimentally for inhibition, but one
would not expect the values for excitation and inhibition to match exactly.
The introduction of plasticity at inhibitory synapses increases the
number of storage sites for learning. Thus the computational capacity
is expanded by allowing temporal patterns to be encoded and stored in
the strengths of inhibitory as well as excitatory synapses (Kano
1996). In addition, plasticity at inhibitory synapses provide a
wider-range control of postsynaptic neuronal activity. We have shown
this with our model by the expanded range of adaptability acquired
using inhibitory plasticity.
The benefits of inhibitory plasticity can only be reaped if the inhibitory inputs are correlated with the EOD in a series of delays. In fact, if the inhibitory inputs are not correlated with the EOD, and the learning rate ratios are not perfectly equal, then inhibitory plasticity would reduce the effectiveness the sensory image cancellation. In addition, there is no increased rate of adaptation, and the range of adaptation is actually reduced relative to what it would be without inhibitory synaptic plasticity, particularly if shunting inhibition is present in the model.
The study of uncorrelated inhibitory input reveals a situation where there is a gradual decay of inhibitory synaptic strength that is counteracted by randomly distributed broad spikes relative to stellate cell spikes. As seen in Fig. 5, this has the effect of normalizing the inhibitory input.
The treatment of shunting inhibition in this study did not introduce
any new dynamics to the model. The shunting inhibition only increased
contributions of inhibitory input relative to that of excitatory input.
With shunting inhibition there is not only the linear contribution of
the weighted sum of IPSPs, but also a divisive effect
(Carandini and Heeger 1994) due to the reduction of the
EPSPs by a multiplicative factor. However, because the stellate cells
are distributed diffusely throughout the molecular layer, they
are excited by parallel fibers that also excite the medium ganglion
cells. Thus in this model the nonlinearity of the shunting is
proportional to the linear effects of inhibition, and no marked change
in system dynamics is observed.
Further research
The present model represents the activity and adaptation of a
single MG cell in the ELL. However, the ELL is a cortical structure with complicated interconnections between the resident neurons. Physiological and morphological studies (Grant et al.
1998; Han et al. 1999
) have suggested a basic
modular structure with excitatory (E) modules and inhibitory (I)
modules. The efferent neurons of the E-modules are excited by the
electrosensory stimuli in the center of their receptive fields, and the
efferent neurons of the I-modules are inhibited by stimuli in the
center of their receptive fields. Recent anatomical studies (Han
et al. 1999
) suggest that the MG cells of each of these modules
are synaptically interconnected, thus inhibiting each other. One
inhibition of the present model is that such mutual inhibition and
other circuitry features have not been included.
One possible explanation of the data on inhibitory plasticity in the
ELL that has not been addressed in this model is that plasticity at the
synapse from parallel fibers onto stellate cells could be responsible
for the apparent plasticity of IPSPs. This type of plasticity has been
observed in the hippocampus (Fortunato et al. 1996;
Gupta et al. 2000
). Slice experiments could be used in
the ELL to isolate the inhibitory plasticity to the synapse from
stellate cells onto MG cells. A paring paradigm similar to that used
for excitatory plasticity (Bell et al. 1997c
) could measure the change of IPSPs with glutamate blockers in the bath. The
presynaptic stimulation in the molecular layer would have to be strong
enough to elicit an IPSP in the MG cell. This type of experiment could
show if inhibitory plasticity really exists in this system. However, it
would be very difficult to eliminate the possibility that there is also
plasticity at the synapse from parallel fibers onto stellate cells in vivo.
There is a theoretical argument against the relevance of this latter type of plasticity to the effect presently investigated: the adaptation of MG cell responses to changes in predictable electrosensory stimuli. The learning rules investigated here are triggered by the timing of broad spikes in MG cells. The broad spikes are the carriers of information about the electrosensory stimuli. For the synapses from parallel fibers onto stellate cell to change in concert with the synapses onto MG cells, one would need to hypothesize another information pathway to signal the stellate synapse about the predictable aspects of electrosensory stimuli. Although it is possible that such a pathway exists, this would not lead to a parsimonious description of the system dynamics with the known anatomy.
Another possible limitation to the present model is the assumption that the stellate cells fire only once per EOD cycle. No recordings of identified stellate cells have been made in support this assumption. As we saw in the results, completely uncorrelated synapses tend to adjust to a level that contributes proportionally to the equilibrium broad spike frequency. If there were several uncorrelated stellate cell spikes per cycle, but one spike per cycle was consistently at the same delay following the EOD, then that one spike would be able to drive the synaptic input to cancel the predicted sensory pattern. The present model tested two extreme conditions: stellate cells fire perfectly correlated with the EOD or perfectly uncorrelated. The true timing of stellate cells with respect to the EOD most likely lies somewhere in between these extreme cases.
The relevance of the spike timing of stellate cells becomes more
apparent when one considers the responsiveness of stellate cells to
parallel fiber spikes compared with that of MG cells. Although data are
not available for the mormyrid ELL, some indication appears in the
gymnotiform ELL (Berman and Maler 1998) and the mammalian cerebellum (Barbour 1989
) that stellate cells
in the molecular layer are much more sensitive to parallel fiber spikes than the principal neurons they inhibit (pyramidal cells in the gymnotiform ELL and Purkinje cells in the cerebellum). The difference in responsiveness could be a result of stellate cells being more electrotonically compact that the principal cells, a condition that
would generalize to the mormyrid ELL. Granule cells that give rise to
parallel fibers receive input from many sources besides corollary
discharge signals, so it is likely parallel fibers are active, and
therefore so are stellate cells, even when there is no EOD. The present
model restricts the activity of stellate cells to the first 150 ms
following the EOD. In the absence of MG cell broad spikes, the stellate
cell synapses onto MG cells would be depressed to their lowest possible
levels due to these asynchronous stellate spikes. However, the MG cells
that are driven by ampullary electroreceptor afferents would respond
with randomly timed broad spikes. In analogy with the uncorrelated
stellate cell spikes (Fig. 5), the learning rule acts to normalize the
inhibitory inputs to a constant broad spike output.
An extension of the model that was studied for excitatory plasticity
(Roberts and Bell 2000) is the effects of different
temporal learning rules on the dynamics of sensory adaptation. In
contrast to the excitatory learning rule, the choice of learning rule
used for the inhibitory synapses was not measured experimentally, but hypothesized to be the inverse of the temporal learning rule governing excitatory synapses. Similar results apply here as in a previous modeling study for excitatory synaptic plasticity alone (Roberts and Bell 2000
); only a near match between the postsynaptic
potential and the learning function can lead to a stable negative
image. That is, if the postsynaptic potential is excitatory, then the associative component of the learning rule must depress the synapse and
have a time course that closely matches the EPSP. This is the
theoretical reason for our choice of an associative component for the
learning rule that enhances the inhibitory synapses by an amount that
is proportional to the IPSP.
In simulations where other temporal learning rules were used at either the excitatory or inhibitory synapses, oscillations developed that prevented the cancellation of the predictable sensory signal. In addition, to generate a negative image, the non-associative component must have the same sign as the contribution of the postsynaptic potential: enhancement for the excitatory synapses and depression for the inhibitory synapses. Thus the present modeling study suggests not only the existence of synaptic plasticity at inhibitory synapses from stellate cells onto MG cells, but also predicts the temporal form of the learning rule that changes the synaptic efficacy depending on the exact timing between the pre- and postsynaptic spike.
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APPENDIX |
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In this APPENDIX we derive the analytic results
reported in RESULTS. The equilibrium spike probability can
be calculated by considering the situation when the noiseless membrane
potential is constant in t so that
V(xn, t) = 0. From the definition of the membrane potential (Eq. 5), the
only variable that changes as a function of t are the
synaptic weights. Thus the membrane potential is stationary when
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(A1) |
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(A2) |
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(A3) |
As has been shown previously (Roberts and Bell 2000),
the stationary membrane potential implies that the broad spike
probability is a constant of both x and t,
f(x, t) =
.
Substituting into Eq. A1 the expressions for the
average change in synaptic weights per cycle (Eqs.
A2 and A3), we arrive at the condition
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(A4) |
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(A5) |
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(A6) |
If these ratios are not equal, then by substituting
into the expressions for the average weight changes (Eqs.
A2 and A3), we find
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(A7) |
The rate of adaptation for the system is measured by the time it takes
to approach an equilibrium broad spike frequency. The time constant,
, associated to this rate can be calculated using the change in the
membrane potential per cycle
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(A8) |
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(A9) |
The rate of change of the membrane potential can be calculated by
expanding the broad spike probability function,
f(xn, t), about the
equilibrium value, . At the equilibrium value, the noiseless membrane potential is defined to be
so
that
= {1 + exp[
µ(
)]}
1. The first two terms of the Taylor
expansion of the broad spike probability function near equilibrium are
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(A10) |
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(A11) |
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(A12) |
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(A13) |
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(A14) |
If the real part of the decay constant () is positive, then the
oscillatory modes will not grow and the membrane potential will reach
an equilibrium value that is constant in x. On evaluating the integrals in Eq. A14, we find that the real
part of the decay constant is
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(A15) |
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(A16) |
To calculate how much the rate of adaptation is increased by plasticity
at inhibitory synapses, let =
w =
/a and
=
=
w =
/a, where a
is a constant real number. Using these values, we simplify the
expression for the decay rate (Eq. A16). If
E is the decay parameter for the EPSP waveform, and
k is the frequency mode of the decaying disturbance, we find
that
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(A17) |
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ACKNOWLEDGMENTS |
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The author thanks G. McCollum, G. Magnus, V. Han, and C. Bell for discussions and helpful suggestions on the manuscript.
This research was supported in part by National Science Foundation Grant IBN 98-08887.
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FOOTNOTES |
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Address for reprint requests: Neurological Sciences Institute, OHSU, 1120 N.W. 20th Ave., Portland, OR 97209 (E-mail: proberts{at}reed.edu).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 13 March 2000; accepted in final form 20 July 2000.
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REFERENCES |
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