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The Journal of Neuroscience, May 1, 2003, 23(9):3697
Learning Input Correlations through Nonlinear Temporally
Asymmetric Hebbian Plasticity
R.
Gütig1, *,
R.
Aharonov2, *,
S.
Rotter1, and
Haim
Sompolinsky2, 3
1 Institute of Biology III, University of Freiburg,
79104 Freiburg, Germany, and 2 Interdisciplinary Center for
Neural Computation and 3 Racah Institute of Physics, Hebrew
University, Jerusalem 91904, Israel
 |
ABSTRACT |
Triggered by recent experimental results, temporally asymmetric
Hebbian (TAH) plasticity is considered as a candidate model for the
biological implementation of competitive synaptic learning, a key
concept for the experience-based development of cortical circuitry.
However, because of the well known positive feedback instability of
correlation-based plasticity, the stability of the resulting learning
process has remained a central problem. Plagued by either a runaway of
the synaptic efficacies or a greatly reduced sensitivity to input
correlations, the learning performance of current models is limited.
Here we introduce a novel generalized nonlinear TAH learning rule that
allows a balance between stability and sensitivity of learning. Using
this rule, we study the capacity of the system to learn patterns of
correlations between afferent spike trains. Specifically, we address
the question of under which conditions learning induces spontaneous
symmetry breaking and leads to inhomogeneous synaptic distributions
that capture the structure of the input correlations. To study the
efficiency of learning temporal relationships between afferent spike
trains through TAH plasticity, we introduce a novel sensitivity measure that quantifies the amount of information about the correlation structure in the input, a learning rule capable of storing in the
synaptic weights. We demonstrate that by adjusting the weight dependence of the synaptic changes in TAH plasticity, it is possible to
enhance the synaptic representation of temporal input correlations while maintaining the system in a stable learning regime. Indeed, for a
given distribution of inputs, the learning efficiency can be optimized.
Key words:
Hebbian learning; spike-timing-dependent
plasticity; synaptic updating; symmetry breaking; unsupervised
learning; infomax; activity-dependent development
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Introduction |
Correlation-based plasticity has
long been proposed as a mechanism for unsupervised experience-based
development of neuronal circuitry, particularly in the cortex. However,
the specifics of a biologically plausible model of plasticity that can
also account for the observed synaptic patterns have remained elusive. Two major issues are stability and competition (Miller and
MacKay, 1994 ; Miller, 1996; Abbott and
Nelson, 2000; Song et al., 2000; van
Rossum et al., 2000; Rao and Sejnowski, 2001;
van Ooyen, 2001). If maps, such as ocular dominance
maps, emerge from initially random (but statistically homogeneous)
synaptic configurations by a Hebbian mechanism (but see Crowley
and Katz, 2000), this would imply that there is an inherent
instability in the dynamics of synaptic learning that destabilizes an
initially homogeneous synaptic pattern. However, this raises the
question as to what mechanism prevents synapses from growing to
unrealistic values when driven by unstable dynamics. The emergence of
inhomogeneous synaptic patterns also requires a competition mechanism
that makes some synapses decrease their efficacies as other synapses
grow in strength. Such competition is absent in the most naive Hebb rule, which contains only a mechanism for synaptic enhancement. Recent
experiments have led to an important refinement of correlation-based or
Hebbian learning, by showing that activity-induced synaptic changes can
be temporally asymmetric with respect to the timing of presynaptic and
postsynaptic action potentials with a precision of down to tens of
milliseconds. Causal temporal ordering of presynaptic and postsynaptic
spikes induces synaptic potentiation, whereas the reverse ordering
induces synaptic depression (Levy and Steward, 1983;
Debanne et al., 1994, 1998; Magee and Johnston, 1997;
Markram et al., 1997; Bi and Poo, 1998,
2001; Zhang et al.,
1998; Feldman, 2000; Sjöström
et al., 2001).
In this work, we address the question of whether temporally asymmetric
Hebbian (TAH) plasticity rules provide an adequate mechanism for
unsupervised learning of input correlations. Two models of TAH
plasticity have been studied recently that differ in the way that they
implement the weight dependence of the synaptic changes and the
boundaries of the allowed range of synaptic efficacies. The additive
model (Abbott and Blum, 1996; Gerstner et al.,
1996; Eurich et al., 1999; Kempter et
al., 1999, 2001;
Roberts, 1999; Song et al., 2000;
Levy et al., 2001; Câteau et al.,
2002) assumes that changes in synaptic efficacies do not scale
with synaptic strength, and the boundaries are imposed as hard
constraints. This model retains inherently unstable dynamics while
exhibiting strong competition between afferent synapses. Because this
model yields binary synaptic distributions, its ability to generate graded representations of input features is restricted. Moreover, because of the strong competition, patterns in the synaptic
distribution can emerge that do not reflect patterns of correlated
activity in the input. On the other hand, the multiplicative model
(Kistler and van Hemmen, 2000; van Rossum et al.,
2000; Rubin et al., 2001) assumes linear
attenuation of potentiating and depressing synaptic changes as the
corresponding upper or lower boundary is approached. This model results
in stable synaptic dynamics. However, because of reduced competition,
all synapses are driven to a similar equilibrium value, even at
moderately strong input correlations. Thus, neither the additive nor
the multiplicative model provides a satisfactory scenario for a robust
learning rule that implements a synaptic storage mechanism of temporal
structures in the inputs. Here, we introduce a nonlinear TAH Hebbian
(NLTAH) model, a novel generalized updating rule that allows for
continuous interpolation between the additive and multiplicative
models. We demonstrate that by appropriately scaling the weight
dependence of the updating, it is possible to learn synaptic
representations of input correlations while maintaining the system in a
stable regime. Preliminary results have been published previously in
abstract form (Aharonov et al., 2001; Gütig
et al., 2001).
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Materials and Methods |
Temporally asymmetric Hebbian plasticity. We describe
TAH plasticity as a change in the synaptic efficacy w
between a pair of cells, where the range of w is normalized
to [0, 1]. A single pair of presynaptic and postsynaptic action
potentials with time difference  t  tpost  tpre induces a
change in synaptic efficacy w given by:
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(1)
|
The temporal filter K(t) = exp(|t|/ ) (Song et al.,
2000; van Rossum et al., 2000; Rubin et
al., 2001) implements the spike-timing dependence of the
learning. The time constant of the exponential decay determines the
temporal extent of the learning window. Following experimental
measurements (Bi and Poo, 1998), we let = 20 msec throughout this paper. The learning rate  , 0 <  1, scales the magnitude of individual weight changes. The temporal
asymmetry of the learning is represented by the opposite signs of the
weight changes for positive and negative time differences. The updating
functions f+(w), f (w)  0,
which are in general weight dependent, scale the synaptic changes and
implement synaptic potentiation for causal time differences
(t > 0), and depression otherwise. Here, we
introduce a family of nonlinear updating functions in which the weight
dependence has the form of a power law with a non-negative exponent
µ:
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(2)
|
with  > 0 denoting a possible asymmetry between the
scales of potentiation and depression. Figure
1A shows the updating curves (Eq. 2) for several values of µ. For µ = 0, the
updating functions are independent of the current synaptic efficacy,
and the rule recovers the additive TAH learning model. This model requires that weights, which would have left the allowed range after an
updating step, are clipped to the appropriate boundary (0 or 1). The
case µ = 1 corresponds to the multiplicative model, in which the
updating functions linearly attenuate positive and negative synaptic
changes as a synapse approaches the upper or lower boundary of the
allowed range. Intermediate values of the updating parameter µ determine the range of the boundary effects on the changes in
w. Note that any non-zero µ, given a sufficiently small
learning rate, automatically prevents the synaptic efficacies from
leaving the allowed range [0, 1], thereby preventing the runaway
problem of synaptic efficacies and removing the necessity of
artificially clipping synaptic weights. Figure 1B
provides an illustrative example of the effects of the parameter µ on
a sequence of synaptic weight changes (see legend for details).
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Figure 1.
Non-linear TAH. A, Effect of the
parameter µ on the updating functions f+(w)
(top half) and f(w) (bottom half) for µ = 0, 0.02, 0.15, 0.5, 1 ( = 1). As µ increases, the curves
change from the constant additive updating curves (µ = 0, horizontal lines at 1 and 1) to the multiplicative updating functions
with linear weight dependence (µ = 1, straight lines with slope
1). B, Illustration of the effect of µ on synaptic
changes. At each step, one pair of presynaptic and postsynaptic spikes
induces a change in the efficacy w according to Equations 1
and 2. To elucidate the weight dependency of the updating,
the magnitude of the time difference between the spikes t
is the same in all steps, namely at step 4 and
at all other steps. For illustrative purposes, we
assume that the spike pairs are sufficiently distant so that the weight
change is effected only by the current spike pair. Furthermore, the
effect of a single pair is unrealistically magnified ( = 0.4).
For additive updating (µ = 0), all potentiating changes are
of equal magnitude, because t is constant here and the
updating is independent of the synaptic efficacy. Note, however,
that the last additive efficacy change would have resulted in an
efficacy of >1, and hence the efficacy is clipped to 1. The depression
in step 4 is larger exactly by a factor of = 1.2. In contrast,
when µ > 0, although the contribution to the synaptic change
from the time dependency is constant, the actual change in synaptic
efficacy is not. Because potentiation is scaled by (1 w)µ, as the synapse becomes stronger the same time
difference induces a smaller change. This scaling effect is more
pronounced for larger values of µ (compare the cases of µ = 0.5 and µ = 1).
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Following previous work (Kempter et al., 1999;
Song et al., 2000; Rubin et al., 2001;
but see van Rossum et al., 2000), the plasticity effects
of individual spike pairs are assumed to sum independently: given a
postsynaptic spike, each synapse is potentiated according to Equations
1 and 2 by pairing the output spike with all preceding synaptic events.
Conversely, a synapse is depressed when a presynaptic event occurs,
using all pairs that the synaptic event forms with preceding output spikes.
Mean synaptic dynamics. Because in general the spike times
of the presynaptic and postsynaptic neurons are stochastic, the dynamics of synaptic changes are also a stochastic process. However, if
the learning rate is small, the noise accumulated over an appreciable amount of time is small relative to the mean change in the
synaptic efficacies, called the synaptic drift. This drift, denoted as
 , is the mean rate of change of the synaptic
efficacy. Using Fokker-Planck mean field theory, the synaptic drifts
are described in terms of the correlations between the presynaptic and
postsynaptic activity (Kempter et al., 1999,
2001; Kistler and van
Hemmen, 2000; Rubin et al., 2001). We consider a
pair of stationary presynaptic and postsynaptic processes described by
the pulse trains  pre(t) =  k (t t ) and
post(t) = k(t t ), with mean rates
rpre =  pre and
rpost = post
and the raw cross-correlation function:
|
(3)
|
The angular brackets denote averaging over time
t while keeping the time lag t between the two
spike trains fixed. This cross-correlation is the probability density
for the occurrence of pairs of presynaptic and postsynaptic spikes with
time difference t. Using this probability density, the
synaptic drift is given by integrating the synaptic
changes w (Eq. 1) over the time differences t weighted by the probabilities
 pre,post(t):
|
(4)
|
The integral in the first term represents the synaptic
depression that stems from all input-output correlations with negative time lag (i.e., acausal correlations). These correlations are filtered
by the temporal window K(t) of the learning. This
contribution from the spike-timing dependence of the learning is
multiplied by the weight-dependent scale of depressing synaptic changes
f(w). Conversely, the second term represents
the potentiating drift originating from causal input-output
correlations, which is scaled by f+(w). Note
that the weight-dependent scales f±(w) are
evaluated outside the time integrals, because when is small, w does not change appreciably during the time scale of the
temporal filter of learning. In summary, the dynamic evolution of the
synaptic weights depends on the properties of the correlation between
the presynaptic and postsynaptic activity
pre,post(t), which in turn depend on the
details of the spike generation mechanism of the postsynaptic cell as
well as on the statistics of the afferent inputs (Kuhn et al.,
2003).
Integrate-and-fire neuron. To study the implications
of the above NLTAH plasticity model in a biologically motivated spiking neuron, we simulate a leaky integrate-and-fire neuron, with parameters similar to those of Song et al. (2000). The membrane
potential of the neuron is described by:
with membrane capacitance Cm = 200 pF, membrane resistance Rm = 100 M ,
resting potential Vrest = 70 mV, and
excitatory and inhibitory synaptic reversal potentials
Eexc = 0 mV and
Einh = 70 mV, respectively. Whenever the
membrane potential exceeds a threshold of 54 mV, an action potential
is generated and the neuron is reset to the resting potential with no
refractory period. Modeling synaptic conductance dynamics by
-shaped response functions, excitatory and inhibitory
conductances are given by:
respectively, where the tj values are the
spike times of synapse j and exc = inh = 5 msec. The values
 exc = 30 nS and inh = 50 nS were chosen such that
the total charge injected per spike (at the threshold potential) is
Qexc = 0.04 pC and
Qinh = 0.02 pC, respectively. While the
efficacies w of the N = Nexc = 1000 excitatory synapses are plastic and
governed by the TAH learning rule, all Ninh = 200 inhibitory efficacies are held fixed at 1. In the numerical
simulations, the integrate-and-fire neuron is driven by Bernoulli
(i.e., zero-one) processes defined over discrete time bins of duration
T = 0.1 msec, approximating Poisson spike trains
with a stationary rate r. For the inhibitory inputs, r = 10 Hz. All equilibrium synaptic distributions
obtained with this model neuron result from an initially uniform
synaptic state with all efficacies set to 0.5. In each case, the
learning process (learning rate, = 0.001) is simulated until
the shape of the synaptic distribution ceases to change.
Linear Poisson neuron. To investigate analytically the
properties of the TAH learning rule, we consider in addition a linear Poisson neuron (Kempter et al., 2001). The spiking
activity of this neuron post(t) is a
realization of a Poisson process with the underlying instantaneous rate
function:
|
(5)
|
where, as before, the N presynaptic input spike
trains and the output spikes are characterized by a series of pulses [i.e.,  (t) = k(t t ) and post(t) = k(t t)]. The parameter 0 <  denotes a small constant delay in the output. Because this delay is
small compared with the temporal window of learning, we approximate
exp(/)  1 throughout this work. As before, wj(t)  [0, 1] denotes the efficacy of
the jth synapse. Except for Figures 5 and 9, in which we
investigate the large N limit, we let N = 100 throughout this work.
In Figure 2A, we numerically simulate the linear
Poisson neuron receiving uncorrelated Poisson input spike trains.
Generating the spike arrival times in continuous time (down to machine
precision), the postsynaptic process defined in Equation 5 is
implemented by generating a postsynaptic spike with probability
wi/N, whenever a presynaptic spike
arrives at a synapse (i) of the neuron.
Mean synaptic dynamics for the linear Poisson neuron. For
the integrate-and-fire neuron, there is no simple exact expression relating the correlations between the presynaptic and postsynaptic spike trains to the system parameters such as the rates and input correlations. However, because of the linear summation of inputs in the
linear Poisson neuron (Eq. 5), this model permits the expression of the
input-output correlations pre,post(t) in
closed form. Considering the case that all input spike trains have a
common rate r, we obtain from Equations 3 and 5 that the
correlation of the activity at synapse i with the output
activity is:
Substituting the above in Equation 4, and rearranging the terms,
we obtain the drift of the ith synapse:
where we define the normalized cross-correlations between the
input spike trains by:
|
(6)
|
We denote the integrated normalized cross-correlations appearing
in the above drift equation by:
|
(7)
|
These matrices are the effective between-input correlations for
positive and negative time lags. If C > 0, the activity at synapse j temporally follows that at
synapse i, such that its contribution to the postsynaptic
activity results in a potentiating drift on synapse i.
Conversely, if C > 0, the activity at
synapse j precedes that at synapse i and contributes to its depression. Note that the effective correlations C are zero if the ith and
jth input spike trains are uncorrelated. Finally, the synaptic drifts can be written as:
|
(8)
|
with f = f f+. Note that the first term in Equation 8 describes
competition between the synapses when f > 0;
independently from the input correlations, the amount of induced depression on a given synapse wi is large when
other synapses wj are strong. The second term
represents the cooperative increase of the synaptic weights inherent in
TAH learning. In contrast, the last term denotes depressive synaptic
interactions stemming from negative time correlations in the input activity.
Generating correlated inputs. We consider input spike trains
with rate r and instantaneous correlations defined by:
|
(9)
|
where (t) is the Dirac- function and
cij is non-negative. In this case,
|
(10)
|
The backward effective correlations
C vanish because the argument of
 (t ) in Equation 7 is
never 0. Recall that for a Poisson process (t) with rate
r, the raw autocorrelation is (t)(t + t) = r2 + r(t). Hence, the
normalized autocorrelation is cii = 1 and the between-input correlations are cij  1, with equality only if the two spike trains i and
j are identical. In the numerical simulations, we generate
populations of correlated spike trains by conditioning the binwise
spike probabilities at time bin T on the activity of a
common reference Bernoulli spike train X0(T) with the binwise spike probability rT. To obtain a
positive pairwise correlation coefficient of 0 c = Cov(Xi(T),
Xj(T))/ between two spike trains Xi(T) and
Xj(T), the conditional probabilities  = P(Xk(T) = 1|X0(T) = 1) and  = P(Xk(T) = 1|X0(T) = 0) for k = i, j are
determined by:
|
(11)
|
This choice of and for all spike trains within
the correlated group guarantees that the spike trains have rates
r and an instantaneous pairwise correlation coefficient
c (see Appendix). For small bin sizes, this process mimics
the instantaneously correlated Poisson point processes defined above
(Eq. 9). We will also consider the case of delayed correlations of the
form (t) = r1cij(t Dij). These correlations are obtained by
shifting the instantaneously correlated input spike trains relative to
each other by a time delay Dij.
Measuring the performance of learning rules. A natural way
to measure the performance of a learning rule is to quantify its ability to imprint the statistical features of the neuronal input onto
the distribution of the learned synaptic weights. One measure of this
ability is the mutual information between the neuronal inputs and the
synaptic weights. However, direct calculation of the mutual information
in cases in which the number of synaptic weights is large is
computationally not feasible. Instead, we use here a related quantity
that measures the effect of a small change in the statistics of the
input on the learned synaptic weights. We denote the features of an
ensemble of neuronal inputs by the vector  = (1, ... , R),
where the i parameterize specific input
features (e.g., mean strength of the inputs or temporal correlations
between different inputs). Given these features, we calculate the
N × R susceptibility matrix
 ij, the elements of which are:
|
(12)
|
The ijth element measures the amount of change in the
ith synaptic efficacy that is incurred by a small change in
the jth input feature, j. A global
sensitivity measure S is constructed from this matrix by
calculating:
|
(13)
|
where det(·) denotes the determinant. The average sensitivity
Savg is defined as
S where the average · is taken over the distribution of the feature vector . The rational for calculating S is that it is closely related to the mutual
information between the input features and the weight distribution.
Specifically, if the mapping from the feature space to the synaptic
weight space induced by the learning dynamics is invertible, maximizing
Savg is equivalent to maximizing the mutual
information (Bell and Sejnowski, 1995; Shriki et
al., 2001) in the limit of a small learning rate . In this
work, we focus on the equilibrium properties of the TAH learning rule
(i.e., the weight distributions that result after the learning dynamics
have converged to a stable stationary state). Therefore,
ij is evaluated at the fixed point solution w* of the drift equations in the linear Poisson neuron, Equation 8 (see Appendix). The possibility of using the analytic expressions for ij in calculating the
sensitivity S is the main advantage of using it as a measure
of performance of various learning models.
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Results |
To understand learning phenomena in biological nervous systems in
terms of neural network function, it is crucial to bridge the gap
between the microscopic mechanisms that implement experience-based changes in neuronal signaling pathways and the macroscopic properties of the learning system composed of these pathways. In this paper, we
focus on two general goals of learning that can be defined at the
network level and also investigate the importance of the updating
parameter µ of the learning rule in these contexts. First, we
consider the question of how a network can develop a functional connectivity architecture, as for example in ocular dominance columns.
As noted in the Introduction, this type of learning task typically
requires the synaptic learning dynamics to be competitive, to allow
segregation between initially homogeneous synaptic populations. Moreover, it is important that the learning process is robust in the
sense that the learned synaptic patterns faithfully reflect meaningful
features in the neuronal input activity, rather than being dominated by
contributions from random noise. Therefore, we study here how the
interplay between competition and stability in TAH plasticity affects
the learned synaptic distributions. In the second part of Results, we
turn to the conceptually different learning task of imprinting
information about the input activity of a neuron into the respective
synaptic efficacies. In this context, the sensitivity of the learning
dynamics to features in the neuronal input becomes crucial. Thus, using
the sensitivity measure introduced in Materials and Methods, the second
part of Results concentrates on a quantitative evaluation of the
performance of different TAH learning rules.
The emergence of synaptic patterns by symmetry breaking in
TAH learning
One of the basic requirements for the activity-driven formation of
cortical maps is the ability of the learning to generate spatially
inhomogeneous synaptic patterns from a population of synapses with
statistically homogeneous inputs. The emergence of such symmetry
breaking is an essential property of current cortical plasticity models
(Miller, 1996). In this section, we study the conditions
under which the TAH learning models introduced above exhibit symmetry
breaking and, hence, qualify as candidate models for the development of
functional maps. Moreover, because the learning dynamics may also lead
to symmetry breaking that overrides the correlation structure of the
afferent activity, it is important to ask what learning rules ensure a
faithful representation of the input activity within the learned
synaptic connections. We address these questions in three basic types
of homogeneous afferent activities, that differ with respect to the
correlation structure of the input spike trains: uncorrelated inputs,
uniformly correlated inputs, and uniformly correlated subpopulations
without correlations between the subpopulations ("correlated
subgroups"). Before treating these specific cases, we highlight the
general features of the synaptic learning dynamics in a population of synapses with statistically homogeneous input activities. These results
apply to all three cases of homogeneous populations of inputs.
Dynamics of a population of synapses with homogeneous inputs
To study the symmetry breaking in the synaptic patterns, we
consider the learning dynamics in cases in which the input statistics are spatially homogeneous. This means that each input obeys the same
spike statistics and has the same pattern of correlations with the
other inputs. This assumption implies that the presynaptic rates
ri (where i denotes the index of the
different afferents) are all equal. Likewise, the total sum of the
correlations that each input has with the rest of the inputs is the
same. In particular, the mean effective causal correlations,
C0:
|
(14)
|
is the same for all input channels i, and similarly for
the backward correlations (Kempter et al., 2001).
To understand the implications of spatial homogeneity in the
presynaptic inputs on the learning dynamics, it is useful to concentrate on the linear Poisson neuron model (Eqs. 5, 8). For convenience, we assume that all correlations between input spikes are
instantaneous (see Materials and Methods, Eq. 10).
The important consequence of the spatial homogeneity across the
presynaptic inputs is that the product of the effective correlation matrix C with a homogeneous vector of
synaptic efficacies wo = (wo, wo, ... ,
wo) remains homogeneous, because
C+wo = NC0wo. Hence, the homogeneous synaptic state is an eigenvector of the effective correlation matrix
C+ with eigenvalue
NC0. The existence of this homogeneous
eigenvector is important for the synaptic learning, because it means
that in a homogeneous synaptic state wo,
all synapses experience identical drifts (Eq. 8). Moreover, if there is
a wo = w* such that the synaptic drifts
become zero, the learning dynamics have a steady-state solution, with
wi = w* for all synapses. We call a
solution in which all of the learned synapses are equal a homogeneous
solution. Indeed, we show in the Appendix that for all non-zero
updating parameter µ values in our model (Eq. 2), there exists a
steady-state homogeneous solution of the learning (i = 0 in Eq. 8), with w*
being the solution of the equation:
|
(15)
|
This equation expresses the weight-dependent balance between
depression and potentiation that is controlled by the mean effective correlation C0.
Although the homogeneous synaptic steady state always exists, it may be
unstable with respect to small perturbations of the synaptic
efficacies, driving them into inhomogeneous states. Because of the
important functional consequences of this emergence of inhomogeneous
synaptic patterns at the network level, it is important to understand
the features of the learning dynamics that give rise to this phenomenon
of symmetry breaking. Therefore, we analyze the effects of small
deviations of the synaptic efficacies from the homogeneous synaptic
steady-state w*. For each synapse wi, we denote a corresponding small
deviation from the homogeneous solution by wi = wi w* and express its temporal evolution as a function of all deviations wj. As we show
in the Appendix, this temporal evolution is determined by three
separate contributions:
where go = w*f+(w*) [f(w)/f+(w)]w=w* = µw*µ/(1 w*) > 0 (Appendix, Eqs.
20, 21). The first term is a local stabilizing term. It counteracts
individual deviations from the homogeneous solution, maintaining the
synaptic efficacies at the same value w*. To understand the
origin of this stabilizing term in the learning dynamics, we consider
the effect of a single synaptic deviation wi
on the balance between depression and potentiation. If a synapse is
strengthened by a deviation wi > 0, the
resulting scale of potentiation f+(w* + wi) decreases, whereas the scale of depression f(w* + wi)
increases (Eq. 2). Conversely, a weakening deviation
wi < 0 shifts the balance between
potentiation and depression in favor of potentiation. Because this
stabilizing drift stems from the weight dependence of the ratio
f(w*)/f+(w*) (Appendix), it is not
present in the additive model (µ = 0), where the
f± values themselves are constant. The second
term is proportional to the net drift f(w*) = f+(w*) f(w*). This drift is
negative, because at the homogeneous solution,
f(w*) > f+(w*) when
depression balances the potentiating correlations (see Eq. 15, and
recall that C0 > 0). The negative drift is
multiplied by the total perturbation
jwj, which denotes the
change in the output rate attributable to the changes in the synaptic
efficacies. Thus, this term represents the competition between the
synapses. This competition results from the fact that strengthening the efficacy of any synapse increases the output rate, thereby increasing the frequency of occurrence of net negative drift in all of the synapses. It is important to note that this competition is acting between all synapses, unrelated to the correlation structure in the
afferent input.
Finally, the last term is a cooperative term. Synapses that are
positively correlated cooperate to elevate their weights. This
cooperation is driven by the potentiating component of the TAH learning
and depends on the pattern of correlations among the input channels. We
emphasize that the cooperativity in the synaptic learning in general
does not originate from a possible advantage of correlated synapses to
drive a potentially nonlinear spike generator of the postsynaptic cell,
but rather already occurs because of an inherently increased
probability of correlated synapses to precede postsynaptic spikes, even
when nonlinear cooperative effects in the spike generator are absent.
The stability of the homogeneous synaptic steady state results from the
interplay between the stabilizing, the competitive, and the cooperative
drifts in the learning dynamics. As we derive in the Appendix,
perturbations of the steady state that slightly change all weights by
the same amount w (homogeneous perturbations) decay to
zero with time and, hence, do not destabilize the learning of a
homogeneous synaptic distribution. In contrast, inhomogeneous perturbations (i.e., perturbations in which the deviations of the
synaptic efficacies from w* are not identical) can grow
exponentially through the learning dynamics and drive the system into
inhomogeneous synaptic states. In the Appendix, we specifically show
that the homogeneous synaptic state becomes unstable if the largest
real part of all inhomogeneous eigenvalues (eigenvalues corresponding to inhomogeneous eigenvectors) of the effective correlation matrix C+ is sufficiently large. Denoting this
eigenvalue by NC1, we find that when:
|
(16)
|
the homogeneous state is unstable. This inequality means that
symmetry breaking occurs whenever the cooperation between synapses C1 is strong enough to outweigh the stabilizing
term go. Note that the competition coefficient
f does not enter directly into the stability criterion.
This is because the competition term is proportional to the total
weight value, and hence is not sensitive to inhomogeneous perturbations
that do not change this value. Nevertheless, this term has a crucial
role in the stability of the learning, because it suppresses the
homogeneous growth of all synapses. As is shown in the Appendix,
Equation 16 implies that the homogeneous solution is always stable in
the multiplicative model (µ = 1).
Although this analysis was performed using the plasticity equations of
the linear Poisson neuron, it is qualitatively valid as well for other
neuron models, as we show for specific cases. Below we study how the
emergence of symmetry breaking (i.e., transitions from homogeneous to
inhomogeneous synaptic distributions) depends on the nonlinearity of
the TAH dynamics, namely the parameter µ, as well as on the asymmetry
between depression and potentiation , and on the size of the
synaptic population N.
Uncorrelated inputs: linear neuron
In this section, we investigate the synaptic distributions that
result from the TAH learning process when the postsynaptic neuron is
driven by independent Poisson spike trains of equal rate r.
For this input regime, it has been found in an integrate-and-fire neuron that additive learning (µ = 0) breaks the symmetry of the statistically identical presynaptic inputs and leads to a bimodal weight distribution (Song et al., 2000; Rubin et
al., 2001). However, it was shown by Rubin et al.
(2001) that multiplicative learning (µ = 1) leads to a
unimodal distribution of synapses. As shown in the preceding section,
these qualitatively different learning behaviors originate in the
stabilizing effect of the weight dependence of the synaptic changes on
the homogeneous synaptic state. Here, we study the generalized
nonlinear TAH rule with arbitrary µ [0, 1].
In the uncorrelated case, cij = ij, and hence C = ij/r. Both its homogeneous and
inhomogeneous eigenvalues normalized by N are:
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(17)
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Thus, the only cooperation in the learning dynamics stems from the
positive feedback induced by the correlation of each synapse with its own contribution to the postsynaptic activity. The effects of
this self-correlation on the learning dynamics decrease inversely to
the effective size of the presynaptic population rN
(i.e., the expected number of spikes that arrive within the learning time window).
By inserting the above expression for C0 into
Equation 15, we obtain the steady-state efficacy w* of the
synaptic population when the learned synaptic state is homogeneous (see
Appendix, Eq. 19). In this case, the output rate of the linear neuron
is given by this steady-state efficacy times the rate of the
presynaptic inputs r (compare Eq. 5). Figure
2A depicts the output
rate of the postsynaptic neuron as a function of the presynaptic input rate r, for = 1.05. We focus on this value of here, because we want to compare the nonlinear rules with the additive
rule. In the latter case, must be close to 1; otherwise,
practically all synapses will become zero (see Appendix). For µ = 1 (multiplicative TAH), the efficacy w* is fairly
independent of r and, hence, the output rate grows linearly
with the input rate. However, if µ is sufficiently small,
w* decreases inversely with the input rate, resulting in the
output rate being nearly constant.
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Figure 2.
Firing rate responses of a neuron driven by
uncorrelated Poisson input processes. A, Output firing rate
of the linear Poisson neuron as a function of the input rate for
selected values of µ = 0, 0.024, 0.04, 1, with N = 100 and = 1.05. Solid lines show analytical results
derived from the homogeneous synaptic state (Eq. 19) (which is stable
for all values µ > 0 and r shown here) or for
additive TAH (µ = 0) from the ratio of strong synapses given by
Equation 25 (see Results and Appendix). In both cases, the output rate
is given by the input rate multiplied with the corresponding mean
synaptic efficacy. Plot symbols depict the output rates of a
numerically simulated spiking linear Poisson neuron (Materials and
Methods) with parameters as in the analytical calculation and a
learning rate = 0.001 (except for µ = 0, where we used
= 0.003). B, C, Results from the
integrate-and-fire neuron with N = 1000. All results
refer to the neuron after convergence of the learning process.
B, The output firing rate of the neuron as a function of the
input rate for = 1.05. C, The output firing rate as
a function of for an input rate of 10 Hz.
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To study the regime in which the synaptic learning dynamics break the
symmetry of the uncorrelated input population, we substitute Equation 17 into Equations 15 and 16, computing the homogeneous solution w* (Eq. 19) and the regime of its stability. Figure
3A depicts the critical
contour lines according to the stability condition (Eq. 16). Each line
traces the critical combination of the parameters µ and
rN for a fixed value of , such that  = 0. Outside
the corresponding contour ( < 0), the homogeneous synaptic state is
stable, and thus learning generally results in all synapses having the
same efficacy. In contrast, inside the contour line, the learning
dynamics induce symmetry breaking.
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Figure 3.
Symmetry breaking in the linear neuron driven by
uncorrelated Poisson input processes. A, The critical
contour lines of the stability criterion = 0 (Eq. 16) for = 1.05, 1.1, 1.25, 1.5, 2, 2.5, 3, 3.5 (from right to left, thick line
corresponds to = 1.05). The homogeneous solution is stable
outside the contour and unstable in its interior. Inset, Critical
contour for large rN and = 1.05. B,
Equilibrium synaptic weights as a function of µ for rN = 10 (squares), 20 (circles), and 200 (triangles), with = 1.05. Plot symbols mark the efficacy values obtained from simulating
the mean field learning dynamics of the linear neuron with N = 100 and the input rate r = 5, 10, 100 Hz,
adjusted to obtain the desired value of rN. As µ is
decreased, each simulation is initialized with the synaptic efficacies
of the equilibrium state obtained for the previous value of µ, plus a
perturbation vector ranging from 0.001 to 0.001. For small and large
values of rN (squares and triangles), the homogeneous
solution is stable for all µ (except the vanishing regime of very
small µ < µcrit in the large rN
case). For rN = 20 (circles), the synaptic
population splits into two groups (54 strong and 46 weak synapses) when µ crosses the corresponding critical contour in A. The
solid lines depict the analytically obtained values of the equilibrium
weights. For rN = 20 (circles), the line was
obtained by numerically solving the analytical expression for a bimodal
synaptic state with the same split ratio that was obtained in the
simulations.
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Figure 3A shows how the outcome of TAH learning depends on
the effective size of the presynaptic population. For a sufficiently small rN, the relative contribution of each input channel
to the postsynaptic activity is large and, hence, the resulting strong positive feedback drives all synapses to a stable homogeneous state
near the upper boundary (Fig. 3B, squares). In contrast, as
rN is increased, the effect of a single synapse on the
postsynaptic activity decreases. Therefore, for a sufficiently large
rN, the stabilizing force induced by the weight
dependence of the synaptic changes dominates the learning dynamics for
any non-zero µ, resulting in a stable homogeneous synaptic state
(Fig. 3B, triangles). In between the two extremes of small
and large rN, there is a regime of intermediate effective
population sizes for which symmetry breaking may occur, with the
synaptic population segregating into a strong and a weak group. Such a
case is shown in Figure 3B (circles).
Importantly, Figure 3A demonstrates that as the number of
afferents N increases, the regime of values of µ for which
the homogeneous solution is unstable shrinks to zero. The inset of
Figure 3A shows the value of µ at the border between
stability and instability of the homogeneous solution, as a function of
the effective population size. It is apparent that this µ decreases
linearly with 1/(rN) when rN is large (also
see Appendix). Hence, for any sizable degree of weight dependence and
large synaptic populations, symmetry breaking does not occur.
In the purely additive TAH model, synaptic changes do not scale at all
with the efficacy of a synapse, and the weights have to be constrained
by an additional clipping to prevent unrealistic synaptic growth. As a
result, the additive learning dynamics do not possess stationary
synaptic states in the above sense that the individual synaptic drifts
become zero. Instead, synapses with positive drifts are held at
the upper boundary, whereas synapses with negative drifts saturate at
the minimum allowed efficacy. Our treatment of the additive model in
the Appendix shows that the numbers of synapses gathering at the upper
and lower boundaries critically depend on the ratio of depression and
potentiation , as well as on the effective population size
rN. As in the nonlinear TAH learning model, small
effective synaptic populations (rN < 1/(2( 1)) will lead to all synapses saturating at the upper boundary
because of the strong positive feedback. However, as rN
increases beyond a critical value, the synaptic population breaks into
two groups, one of which remains saturated at the upper boundary while
the other, losing the competition, saturates at the lower boundary. The
ratio of synapses saturating at the top boundary is
nup = 1/2rN( 1)
(Appendix). Because this ratio is inversely proportional to the input
rate r, the output rate of the postsynaptic neuron becomes
independent of the input rate, as shown in Figure
2A.
Uncorrelated inputs: integrate-and-fire neuron
We now turn to the behavior of TAH learning in the
integrate-and-fire neuron driven by uncorrelated inputs. Figure
2B shows the output rate of this neuron model versus
the input rate for different values of µ. As the figure demonstrates,
the output-rate normalization quickly deteriorates as µ departs from
the additive model and synaptic changes become dependent on the
efficacy of the synapse. Figure 2C demonstrates that the
sensitivity of the output rate to the parameter rapidly diminishes
as µ increases. Comparing A and B of Figure 2
shows the qualitative similarity between the output rate responses of
the linear Poisson and the integrate-and-fire model neurons. Note that
we have not attempted to match the overall scale of the output rates in
the two models. The output rate of the linear neuron can be arbitrarily
changed by a gain factor without affecting any other results.
Figure 4 displays the histograms of the
equilibrium distributions of learned synaptic efficacies as a function
of the updating parameter µ. Recovering the behavior of additive
(Song et al., 2000) and multiplicative (Rubin et
al., 2001) updating models for µ = 0 and µ = 1, respectively, the plot reveals the transition between these models for
intermediate values of µ. Specifically, it shows the emergence of
symmetry breaking as µ approaches zero.
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Figure 4.
Symmetry breaking in the integrate-and-fire neuron
driven by uncorrelated Poisson input processes. A,
B, Histograms of the equilibrium synaptic distributions in
logarithmic gray-scale as a function of the updating parameter µ for
N = 1000 and = 1.05. The input rate is 10 Hz
in A and 40 Hz in B. The arrow marks the critical µ where the first bimodal distribution occurs. Insets, The synaptic
distributions at µ = 0.019. Note that whereas the histograms
shown in the main figures are single realizations of the equilibrium
synaptic distributions for each µ, the histograms shown in the insets
were obtained by averaging 30 different readouts of the converged
synaptic weights taken at consecutive intervals of 500 sec.
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As expected from the analysis of the linear neuron, we find that in the
integrate-and-fire neuron also, the critical value of µ at which the
synaptic distribution becomes bimodal decreases as the effective
population size rN increases. Increasing the rate of the
input processes from 10 Hz (Fig. 4A) to 40 Hz (Fig. 4B) lowers the first occurrence of a bimodal weight
distribution from µcrit = 0.023 to
µcrit = 0.017. The inset in each panel depicts the
equilibrium weight distribution for the intermediate value of µ = 0.019, showing a clearly bimodal distribution for the 10 Hz input
(Fig. 4A) and a clearly unimodal distribution for the 40 Hz input (Fig. 4B). Moreover, as expected from the
equations describing the homogeneous steady state in the linear neuron
(Eqs. 15 and 17), the synaptic efficacy of the homogeneous state at a given µ decreases when the input rate increases.
It is interesting to note the close similarity in the µ dependence of
the learned synaptic distributions in the linear and the
integrate-and-fire neurons. For example, in both cases, the critical µ for symmetry breaking is close to 0.023 for input rates of 10 Hz
[compare Fig. 4A with Fig. 3B
(circles)]. This is despite the fact that the two models have very
different spike generators and different sizes of synaptic populations.
The reason for this similarity is that the input-output correlations
in the integrate-and-fire neuron with 1000 synapses turn out to match
in magnitude the corresponding correlations of the linear neuron with
100 synapses (data not shown).
In summary, for uncorrelated inputs and biologically realistic sizes of
the presynaptic population, N, on the order of thousands, and for rates on the order of 10 Hz, the regime in µ and in which symmetry breaking between uncorrelated inputs as well as output
rate normalization occur is extremely narrow. Thus, the learning
behavior changes qualitatively as soon as synaptic plasticity becomes
weight dependent.
Uniformly correlated inputs
We briefly discuss here the case in which the presynaptic inputs
have positive uniform instantaneous correlations, namely that for all
i  j, cij (Eq. 9) are equal. This
situation may, for instance, occur when the entire presynaptic pool of
a neuron is driven by a common source. Treating the behavior of the
linear Poisson neuron, we show in the Appendix that positive uniform
correlation increases the value of the synaptic efficacy in the
homogeneous synaptic steady state. Moreover, the uniform correlation
does not alter the 1/(rN) dependence of the destabilizing
drifts. As a result, in nonadditive learning, when the effective
synaptic population is sufficiently large, the homogeneous steady state remains stable for any positive uniform correlation strength. In fact,
these correlations increase the stability of the homogeneous state
(Appendix) and, hence, oppose the emergence of spontaneous symmetry breaking.
Correlated subgroups
We now consider afferent input activity to a neuron that is
composed of M equally sized groups. These groups are defined
by a uniform within-group correlation coefficient
cij = c > 0 (compare Eq. 9) that is
equal within all groups. For pairs of inputs belonging to different
groups, the cross-correlation is zero. In this scenario, the
M different presynaptic groups compete for control over
firing of the postsynaptic neuron. We first treat the linear neuron
and, for simplicity, focus on the case in which the overall number of
presynaptic input channels N is large. In this limit, the
homogeneous and largest inhomogeneous eigenvalues of
C+ normalized by N are:
Comparing these expressions with their respective values in the
case of N uncorrelated inputs (Eq. 17), we note that the
learning behavior in both input scenarios is equivalent when
N is identified with M/c, the number of
correlated subgroups divided by the strength of the within-group
correlation. The stability of the homogeneous synaptic steady state in
a large network comprising M correlated synaptic subgroups,
each with within-group correlation c, behaves as in an
uncorrelated network of finite size M/c. Thus, in the limit
of a large presynaptic population, the correlation strength c scales the effective number of presynaptic inputs from
M for c = 1 to infinity for c = 0. Importantly, the largest inhomogeneous eigenvector is such that
when the homogeneous solution loses stability, the symmetry is broken
between the correlated subgroups and not within each subgroup.
The nature of the synaptic pattern that emerges once the homogeneous
synaptic state loses stability depends on the number of afferent
subgroups. Here we focus on the example of two equally sized subgroups
(i.e., M = 2). A similar scenario, which is motivated by the problem of the activity-driven development of ocular dominance maps, has recently been studied by Miller and MacKay
(1994) and Song and Abbott (2001). The regimes
of symmetry breaking in which the learned synaptic efficacies segregate
according to the two correlated input groups are depicted in Figure
5A (this figure is equivalent
to Fig. 3A, with c replacing 2/N).
Thus, symmetry breaking between two correlated subgroups can occur in
nonadditive TAH learning models even when the number of presynaptic
inputs N is large. This is demonstrated in Figure
5B (solid black line), which plots the learned synaptic
efficacies as µ as varied, with c held fixed at 0.11. As
is evident from Figure 5, for this level of correlation, symmetry
breaking occurs below a fairly high value of µ 0.15. Note
that in contrast to our treatment of the uncorrelated inputs, here we
do not use close to 1 but rather set it to a generic value of
= 1.5.
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Figure 5.
Symmetry breaking in the linear neuron driven by
two equally sized correlated groups in the large N limit.
A, The critical contour lines of the stability criterion = 0 (Eq. 16) for = 1.25, 1.5, 2, 2.5, 3, 3.5 (from left to
right, thick solid line corresponds to = 1.5, thick dashed
line to = 3) plotted as a function of the within-group
correlation c. The homogeneous solution is stable outside a
contour line and unstable in its interior. The lines depicted in
B-D correspond to the equilibrium synaptic states obtained
by numerically solving the analytical expressions for the two-group
solutions with r = 10 Hz. Circles depict synaptic
efficacies obtained by simulating the mean field learning dynamics of
N = 1000 synapses. We set N = 1000 here
to obtain a better approximation of the analytical results derived for
the large N limit. Note that small discrepancies between the
analytical curves and the simulated mean field equilibrium, stemming
from the finiteness of N in the simulation, can be observed
in B. B, Equilibrium synaptic weights (black line) as a
function of µ for c = 0.11 (horizontal line in
A) with = 1.5. Gray lines show the equilibrium
weights for c = 0.05, 0.06, ... , 0.2.
C, D, Equilibrium synaptic weights (black lines)
as a function of c for µ = 0.15 (vertical line in
A) with = 1.5 in C and = 3 in
D. The gray lines in C show how the region in
which the homogeneous solution is unstable vanishes as µ is increased
from 0.15 to 0.1725 in steps of 0.0025. The gray lines in D
show the change in the equilibrium synaptic weights when is
increased from 2 (largest group separation) to 4 (smallest group
separation) in steps of 0.1. The dashed lines in C and
D depict the equilibrium synaptic weights for the
multiplicative case (µ = 1).
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Figure 5, C and D, describes the behavior of the
system as the within-group correlation is gradually turned on. As
expected from the analysis of the uncorrelated input scenario, the
substantial weight dependence of the synaptic changes induced when µ = 0.15 (solid black lines), yields a stable homogeneous
synaptic state if the within-group correlation is sufficiently weak.
However, when the correlation reaches a critical value, the homogeneous state becomes unstable and the synaptic efficacies segregate into the
two input groups, with the one winning the competition suppressing the
other. As the correlation increases still further, another transition
may occur at a higher value of c, above which the
homogeneous synaptic state becomes stable again. The presence of this
second transition (which is discontinuous) depends on the values of
r, the expected number of input spikes per synapse
arriving within its learning time window, and the ratio of depression
and potentiation (Fig. 5, compare C and D).
Importantly, for large values of µ, in particular in the
multiplicative model (µ = 1), the stabilizing force is so strong
that the homogeneous synaptic state remains stable for all positive
correlation strengths (Fig. 5C, D, dashed black
lines; also see Appendix), and no segregation is possible.
The behavior described above for the linear neuron is reproduced
qualitatively in simulations of the integrate-and-fire neuron, as shown
in Figure 6A. To
address the question of whether symmetry breaking in the
integrate-and-fire neuron can also occur at higher values of µ, we
follow the linear Poisson neuron analysis shown in Figure
5A, which suggests that increasing the value of extends the µ range of bimodal synaptic distributions. Figure
6B displays the learned synaptic distributions as a
function of µ for a within-group correlation c = 0.05, with = 1.5 and r = 10 Hz. Similar
to the linear neuron findings shown in Figure 5B, symmetry
breaking occurs here in a wide regime of µ.
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Figure 6.
Symmetry breaking in the integrate-and-fire neuron
driven by two equally sized correlated groups. Keeping N = 1000, each group consists of 500 synapses receiving Poisson
inputs. A, The learned synaptic weights of the two input
groups as a function of the within-group correlation c, for
an input rate of 40 Hz, µ = 0.019 and = 1.05. For each
correlation, the group means (circles) are computed for 30 different
realizations of the synaptic weight distribution, taken after
convergence at successive intervals of 500 sec. Error bars indicate the
corresponding SDs. The inset shows how the regime of symmetry breaking
vanishes as µ is increased through µ = 0.019, 0.021, 0.022, 0.023, 0.024. B, The learned distributions of synaptic
weights as a function of µ for an input rate of 10 Hz, c = 0.05 and = 1.5. In all bimodal weight distributions
depicted here, the splitting corresponds to the two input groups.
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To emphasize important differences between symmetry breaking in
nonlinear versus additive TAH learning, Figure
7 shows corresponding learned synaptic
efficacies for selected cases of low, intermediate, and high
within-group correlations. Figures 7A-D depicts learned weight distributions from Figure 6A for which µ = 0.019. For each correlation, synaptic efficacies resulting
from additive learning are depicted on the right (Fig.
7E-H). Except in Figure 7, A and B, where c = 0 (i.e., no input subgroups are
defined), the synaptic distribution of the subgroup with higher mean
efficacy is depicted in light gray, whereas that of the subgroup with
lower mean efficacy is displayed in dark gray.
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Figure 7.
Symmetry breaking in nonlinear and additive TAH
learning. The learned synaptic efficacies of the integrate-and-fire
neuron driven by two equally sized correlated groups are depicted for
selected within-group correlations c = 0, 0.03, 0.1, 0.3. Each input group consists of 500 synapses receiving Poisson
inputs with a rate of 40 Hz (compare Fig. 6A). The
asymmetry parameter = 1.05. For c = 0
(A, E), the histogram of the total synaptic
population is depicted. For c > 0, the light bars
describe the distribution of the presynaptic input group with the
higher mean, and the dark bars (stacked on top of the light ones)
depict the distribution of the weaker group. All histograms were
obtained by averaging 30 different readouts of the converged synaptic
weights taken at consecutive intervals of 500 sec. A-D,
Results from nonlinear TAH learning with µ = 0.019. E-H, Results from additive TAH learning, µ = 0. In
E, a total of 112 synapses are in the upper mode. In
F, the stronger group (light bars) has a bimodal
distribution with 175 synapses (35%) in the upper mode.
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Inspection of Figure 7, A and B versus
E and F, shows that in the regime of low
correlations, the learning behavior induced by the two types of
plasticity is qualitatively different. While in nonlinear TAH learning
(Fig. 7A,B), the homogeneous synaptic state is stable and
all synapses distribute around the same mean efficacy, unstable
additive learning induces symmetry breaking (Fig.
7E,F). Importantly, this symmetry breaking in general
does not reflect the correlation structure in the afferent input. As shown in Figure 7F, when the within-group correlation is
0.03, the 500 synapses of the group winning the competition (light
gray) split into two fractions of 325 versus 175 synapses, of which the
larger fraction tends to zero efficacy and mixes with the efficacies of
the losing input group. In contrast, in the nonlinear TAH model,
unfaithful splitting of the weights occurs only for extremely small
values of µ, of the order of 1/rN (Fig. 7, compare A and B with E and
F). This is because symmetry breaking within a
uniformly correlated group does not occur for µ > 1/rN, and hence the weights of each subgroup remain the same.
For intermediate strengths of the within-group correlation, both
learning rules induce symmetry breaking that faithfully reflects the
structure of the input correlation, with the synaptic distributions of
the two input groups well separated. This is shown in Figure 7,
C and G, for a correlation of c = 0.1. Note, however, that whereas in additive learning the
efficacies of both input groups reach the respective boundaries of the
allowed range (i.e., are clipped to saturation), the weights resulting
from nonlinear TAH learning do not saturate. As we show in the next
section, this property of NLTAH plasticity enhances the sensitivity of
the synaptic population to changes in the strength of the within-group
correlation. Finally, when the within-group correlation is strong, in
both types of learning all efficacies become large (Fig.
7D,H).
Clearly, the detailed quantitative properties of the learned synaptic
patterns, as well as the parameter values at which symmetry breaking
occurs, depend on the neuron model, and specifically on the spike
generating mechanism. Nevertheless, the striking qualitative similarity
in the findings from both neuron models investigated here suggests that
the symmetry breaking induced by the within-group correlations is a
general property of the nonlinear TAH rule with small but non-zero µ,
independent of the specifics of the spike generator.
Synaptic representation of input correlations
In the previous section, we studied the emergence of symmetry
breaking in homogeneous synaptic populations for different types of
instantaneously correlated input activity. In this section, we study
the more general issue of how information about the spatiotemporal structure of the afferent input is imprinted into the learned synaptic
efficacies by TAH plasticity. Specifically, we investigate how the
weight dependence of the synaptic changes affects the sensitivity of
the learning to features embedded in the input spike trains.
An example of the associated phenomena is shown in Figure
8. Here we study the effect of weight
dependence on the steady-state synaptic efficacies of the
integrate-and-fire neuron receiving 1000 Poisson inputs that comprise a
small subgroup of 50 correlated synapses (c = 0.1)
while all other input cross-correlations are zero. In this scenario,
the subgroup is statistically distinct from the rest of the synaptic
population. The coherence of spikes within the subgroup increases the
causal correlation of the member synapses with the spiking activity of
the postsynaptic neuron. Because of the ensuing cooperation between the
correlated synapses, they grow stronger than those of the uncorrelated
background. Figure 8 shows how the strength of the stabilizing drift
induced by the weight dependence of the synaptic updating modulates the degree of separation between the two subpopulations. For decreasing values of µ, learning becomes increasingly affected by the
correlation structure in the input, and the separation between the
subgroup and the background is more pronounced. However, below a
critical µ, the homogeneous state of the uncorrelated population
loses stability and splits, resulting in a bimodal distribution of the background synapses. As a consequence, the representation of the afferent correlation structure in associated groups of synaptic efficacies is confounded by the mixing of the high-efficacy mode of the
background with the subgroup of correlated synapses. This example
raises the general problem of finding an optimal learning rule that,
for a given type of input activity, compromises best between
sensitivity and stability.
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Figure 8.
Effect of the updating parameter µ on the
learned synaptic distribution of the integrate-and-fire neuron driven
by a small correlated group and an uncorrelated background. The neuron
is driven by 950 uncorrelated and 50 weakly correlated (c = 0.1) Poisson input processes with a rate of 10 Hz ( = 1.05). Together, A and B cover the range of µ [0, 1], but note the difference in resolution.
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To address this question, we need a quantitative measure for the
performance of a learning rule in imprinting information about the
input correlations onto the synaptic efficacies. Here we apply the
sensitivity measure S (Eq. 13, Materials and Methods), which
quantifies the sensitivity of the learned synaptic state to changes in
features embedded in the input correlation structure. When S
is high, small changes in the input features are picked up by learning
and induce a large change in the learned synaptic efficacies. We
emphasize that the goal of this performance measure is to quantify and
compare general properties of different plasticity rules. It is
therefore based only on the relationship between the afferent neuronal
inputs and the learned synaptic efficacies. In particular, it avoids
direct reference to the neuronal output activity.
We first illustrate the application of the sensitivity measure by
considering a simple example in which the input feature to be
represented by the learned synaptic efficacies is only one dimensional
(i.e., a scalar quantity). Specifically, we apply S to the
scenario discussed in the previous section, of two independent input
groups with within-group correlation c. We investigate the behavior of the linear Poisson neuron and quantify how the sensitivity of the learned synaptic distribution to the strength of the
within-group correlation is affected by the weight dependence of the
synaptic changes. We consider the sensitivity of the learning as a
function of µ for a fixed correlation of c = 0.11. As
shown in Figure 5, B and C, this correlation
represents an intermediate correlation strength in the linear Poisson
neuron treatment. Using the steady-state synaptic efficacies from
Figure 5B, we compute S for values of µ between
0 and 0.5 (see Materials and Methods, Appendix). Figure 9 shows the resulting sensitivity curve.
We note that each point quantifies the sensitivity of the learned
synaptic weights to small changes in the correlation strength around
c = 0.11.
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Figure 9.
Sensitivity to the within-group correlation
strength c in the linear neuron receiving input from two
correlated subgroups. The sensitivity is plotted as a function of µ for correlation strength c = 0.11 and = 1.5 (see Fig. 5B for parameter settings, large N
limit).
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As can be seen in Figure 5B, there are two qualitatively
distinct regimes of synaptic distributions emerging from learning in
this case. For high values of µ, no symmetry breaking takes place,
and the correlation strength is represented by the common mean value of
the synaptic efficacies. In this regime (µ  0.15), S
decreases monotonically with increasing µ (Fig. 9), because the
higher weight dependence strengthens the confinement of the homogeneous
synaptic state to the center range of the synaptic efficacies. For
lower values of µ, symmetry breaking occurs, and the correlation
strength is represented by the mean efficacy values of the two
resulting groups. In this regime, S is nonmonotonous in µ.
For very low µ, the synaptic efficacies are close to saturation at
the boundaries and, hence, a change in the correlation strength cannot
induce a large change in the efficacies. However, the centralizing drift induced by a large µ reduces the sensitivity. Thus,
S has a maximum at an intermediate µ (in the present case
around µ = 0.02). Finally, at the transition between the regions
of homogeneous and bimodal synaptic distributions (µ 0.15),
sensitivity is large, because here a small change in c may
cause an abrupt and large change in the synaptic efficacies, namely a
bifurcation from a homogeneous to an inhomogeneous synaptic
distribution. Note, however, that this transition region in µ is narrow.
We now turn to a richer input scenario in which the afferent
correlation structure is inhomogeneous and the input feature space to
be represented by the learned synaptic efficacies is high-dimensional.
Specifically, we consider presynaptic activity in which each synapse
receives spike inputs with a specific relative latency with respect to
the remaining synaptic population. Such latency or delay-line scenarios
have been studied previously in the context of additive TAH learning
(Gerstner et al., 1996; Song et al.,
2000) and can, for instance, be motivated by their analogy to
certain delay-line models in auditory processing (Jeffress, 1948).
We consider the input activity to consist of N time-shifted
versions of one common Poisson spike train with rate r.
Because the synaptic learning process depends on the relative timing of the input spikes, we fix one presynaptic input as reference, and treat
the remaining N 1 delays = (1, ... , N1) as
R = N 1 dimensional vector of input features to
be represented by the learned synaptic weights. Whereas the delays fully specify the temporal correlation structure of the neuronal input
activity, S measures the sensitivity of the learned synaptic
efficacies to small independent changes in the individual delays.
Because of the temporal sensitivity of TAH plasticity, it is
intuitively clear that the learning dynamics will critically depend on
the temporal scale of the relative delays. Although it is a natural choice to set this temporal scale through the SD of a Gaussian distribution from which the delays are drawn (Song et al.,
2000; Aharonov et al., 2001; Gütig
et al., 2001), we here apply the sensitivity measure to the
simpler case in which we fix such that the delays between the
N inputs are uniformly spaced at a fixed delay
 /(N 1) [i.e., i = i/(N 1) (i = 1, 2, ... , N 1)]. We
have checked that the qualitative behavior of S in the case
of a fixed delay spacing is similar to that of
Savg (see Materials and Methods) obtained from
averaging over an ensemble of Gaussian delay vectors with SD (Aharonov et al., 2001; Gütig et al.,
2001).
We investigate here the behavior of the linear Poisson neuron. One
important difference between the delay-line input scenario considered
here and the input correlations treated above is that here non-zero
cross-correlations between input spike trains also exist at negative
time lags. Specifically, if the delays of the input activities of
synapses i and j are given by
i and j,
respectively, and the additional delay of the postsynaptic neuron is
(Eq. 5), the delay difference i (j + ) determines the temporal position of
the sharp peak in the otherwise zero effective correlation between the
two shifted Poisson inputs (Eq. 7). If this delay difference is
negative, the output activity contributed by the jth synapse
lags behind the input spikes at the ith synapse. Hence, the
jth synapse contributes to the potentiation of synapse
i, and the respective effective causal correlation
C is positive. Correspondingly, in this
case the backward effective correlation
C contributed by synapse j to
the depression of synapse i is zero. Conversely, if the
delay difference between the ith and jth input
spike trains is positive, synapse i is depressed by the
activity of synapse j, because
C becomes positive. In both cases, the
magnitude of the effective correlation is scaled by the exponentially
decaying time dependence of the learning rule (Eq. 1). The full
expressions for the effective correlation matrices
C and C
are given in the Appendix.
To calculate S for a given delay vector , we numerically
solve the drift equation of the synaptic learning (Eq. 8) for the synaptic steady state. Using the resulting learned synaptic
distributions, we compute the susceptibility matrix (Eq. 12,
Appendix), giving S (Eq. 13). Figure
10A shows the
sensitivity S as a function of µ for different values of
the temporal delay spacing . The curves clearly show an optimal
weight dependence of the synaptic changes for which the sensitivity
peaks. For larger values of µ, the performance of the learning
deteriorates because the increasing confinement of the synaptic weights
to the central range of efficacies restricts the sensitivity of the
learning to changes in the input correlation structure. Conversely, for
lower values of µ, the sensitivity is impaired because the synaptic
efficacies are beginning to saturate at the boundaries of the allowed
range as bimodal efficacy distributions emerge. The value of µ that
optimally adjusts the weight dependence of the synaptic changes depends
both on the system parameters and on the input correlations determined
by the relative time delays between the inputs. Increasing (i.e.,
increasing the relative delays) weakens the effective correlations
between the presynaptic inputs because of the exponentially decaying
temporal extent of the learning rule (Eq. 7). Hence, a lower weight
dependence of the synaptic changes (corresponding to a lower value of
µ) is needed to pick up the correlations and allow sufficient
sensitivity of the learning to the input delays. The effect of this
change in the temporal extent of synaptic interactions on the learned efficacies is shown in Figure 11, which
for each µ depicts all N synaptic efficacies for = (Fig. 11A) and = 4 (Fig.
11B). Note that because of the equidistant delays,
the relationship between the relative temporal position of a synapse
within the presynaptic population and its steady-state efficacy is
monotonic, with the leading synapse ( = 0) taking the largest
weight. In the foreground, the corresponding sensitivity curves are
shown. The plots clearly demonstrate that the saturation regime in
which most synaptic weights accumulate at the boundaries of the allowed range (black and white) begins at higher values of µ when the temporal dispersion of the inputs is small (Fig. 11A)
(i.e., the synaptic interactions are strong). The plot also reveals
that in both cases for low values of µ, only the leading synapse
remains at a high value. Finally, it can be seen that the peaks in the sensitivity curves approximately coincide with those values of µ for
which the synaptic weights smoothly cover a large range of efficacies,
as shown by the gradual change from dark to light values in the
corresponding vertical cross sections.
View larger version (30K):
[in this window]
[in a new window]
|
Figure 10.
Sensitivity in the linear neuron evaluated at
uniformly spaced input delays. The delay vector covers the range
from 0 to (see Results). The learning dynamics are simulated for
the linear neuron, with N = 101 synapses driven by the
delayed Poisson inputs with a rate of 10 Hz and = 1.5. A, The sensitivity per input feature S/R as a
function of µ for / = 0.25, 0.5, 0.75, ... , 4.75, 5 (from bottom to top). The termination of the curves at low µ is a
result of poor numerical convergence arising as the synaptic efficacies
come close to the boundaries of the allowed range. B, The
sensitivity per feature S/R as a function of the delay
interval / for µ = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1 (from
top to bottom).
|
|
View larger version (57K):
[in this window]
[in a new window]
|
Figure 11.
Mapping between synaptic delays and equilibrium
weights. The parameters are identical to those used in Figure
10A, with / = 1 in A and
/ = 4 in B. The background depicts the learned
synaptic efficacies as a function of µ. Each row i
reflects the equilibrium weight wi in
gray-scale, with the top row corresponding to the leading synapse with
zero delay and the bottom row to the last synapse with delay . The
solid lines depict the sensitivity curves from Figure
10A.
|
|
Finally, we ask how the learning sensitivity depends on the statistics
of the input delays for a fixed value of µ. To answer this question,
Figure 10B shows S as a function of the
delay-line spacing , demonstrating that S does not vary
monotonically with , but rather has a maximum at an optimal temporal
separation of the inputs. This is because tight spacing leads to strong
effective correlations between the inputs, driving the synapses toward
saturation. On the other hand, loose spacing reduces the effective
correlations between the presynaptic inputs to the extent that the
learning behaves essentially as if driven by an uncorrelated
presynaptic population.
| |
Discussion |
The understanding of activity-dependent refinement of neural
networks has long been one of the central interests of synaptic learning studies. In this context, most investigations of unsupervised learning using correlation-based plasticity rules have been conducted in the framework of additive plasticity models, which do not
incorporate explicit weight dependence in the changes of synaptic
efficacies. These simple models suffer from stability problems: either
all synapses decay to zero or they grow without bound. An additional problem inherent to simple Hebbian models is the lack of robust competition. Indeed, it has been found that even the inclusion of
synaptic depression mechanisms does not provide a robust source of
synaptic competition, unless synaptic plasticity is fine-tuned to
approximately balance the amount of potentiation and depression (Miller, 1996).
Recent studies of experimentally observed temporally asymmetric Hebbian
learning rules have added two new ideas. One idea is that under these
plasticity mechanisms, synapses compete against each other in
controlling the time of firing of the target cell and, thus, engage in
competition in the time domain. Although TAH learning rules are indeed
inherently sensitive to temporal correlations between the afferent
inputs, we have shown here that this sensitivity alone is not
sufficient to resolve the problems associated with either stability or
competition. In the additive model of TAH plasticity, hard constraints
need to be imposed on the maximal and minimal synaptic efficacies to
prevent the runaway of synaptic strength. In addition, as was shown
here, in this model synaptic learning is competitive only when the
ratio between depression and potentiation is fine-tuned, and even then
the emergent synaptic patterns do not necessarily segregate the
synaptic population according to the correlation structure in the
neuronal input. The second idea is that TAH rules would exhibit novel
behavior because of the role of the nonlinear spike-generation
mechanism of the postsynaptic cell (Song et al., 2000).
In fact, we have shown in this work that the qualitative features of
TAH plasticity are strikingly insensitive to the nonlinear integration
of inputs in the target cell (see also Kempter et al.,
2001). For the parameter choices studied, the properties of the
synaptic steady states in the integrate-and-fire neuron are
qualitatively similar to those found in a linear input-output model
for neuronal firing. Nevertheless, we note that there are substantial
quantitative differences between the two models, particularly with
respect to the parameters rN and c, which
effectively determine the correlations between the presynaptic and
postsynaptic spike trains. Although a quantitative analysis of
these differences is beyond the scope of our work, such a study might
reveal interesting insights into the quantitative effects of the
details of the postsynaptic spike generator on the learned synaptic
distributions. In addition, it is possible that the details of the
spike-generation mechanism will affect the transient phase (i.e., the
dynamics) of the synaptic learning process.
From the present work, we conclude that some of the underlying
difficulties in correlation-based learning are alleviated by nonlinear
plasticity rules such as the NLTAH rule. The nonlinear weight
dependence of the synaptic changes provides a natural mechanism to
prevent runaway of synaptic strength. As in additive TAH learning, synaptic competition is provided by the mixture of depression and
potentiation. However, in NLTAH plasticity, the balance between depression and potentiation is maintained dynamically by adjusting the
steady-state value of the synaptic efficacies. Indeed, we have shown
that this competition is sufficient to generate symmetry breaking
between two independent groups of correlated presynaptic inputs.
However, for this to occur, the stabilizing drift induced by the weight
dependence of the synaptic changes should not be too strong. In
particular, the simple linear weight dependence (Eq. 2, with µ = 1) assumed in the original multiplicative model is incapable of
breaking the symmetry between competing input groups. In fact, we have
shown that with µ = 1, the homogeneous synaptic state is stable
for any pattern of homogeneous input correlations, provided there are
no negative correlations in the afferent activity. The present
power-law plasticity rule with 0 < µ < 1 provides a
reasonable balance between the need for a stabilizing force and a
potential for spontaneous emergence of synaptic patterns. Our study of
symmetry breaking between two competing groups of correlated synapses
is inspired by the activity-dependent development of ocular dominance
selectivity. This scenario has also been studied recently by
Song and Abbott (2001) using the additive version of TAH
plasticity. In their model, achieving a faithful splitting between the
two competing input groups with weak correlations requires relatively
tight tuning of the depression to potentiation ratio, .
One of the surprising results of our investigation is the possibility
that when the correlation within input groups is made strong, the
stability of the homogeneous synaptic state may be restored. We have
shown that this apparently counterintuitive behavior, predicted by the
analytical study of the mean synaptic dynamics of the linear Poisson
neuron, is also seen in simulations of the full learning rule in the
integrate-and-fire neuron. It would be interesting to explore possible
experimental testing of this result, perhaps in the context of the
development of ocular dominance. In this work, we have limited
ourselves to correlated subpopulations of inputs with positive
within-group correlations. However, in general, negative correlations
are an additional potential source of competition (Miller and
MacKay, 1994). Furthermore, we have not addressed the important
issue of competition between synapses that target different cells.
Lateral inhibitory connections between target neurons may provide a
source of such competition.
The last part of the present work addresses situations with
inhomogeneous input statistics. Different inputs are distinct in their
temporal relationship to the rest of the input population. Here the
issue is not whether a spatially modulated pattern of synaptic
efficacies will form through TAH learning, but rather whether this
pattern will efficiently imprint the information embedded in the input
statistics. To quantify the imprinting efficiency of the learning rule,
we introduced a new method for measuring learning rule sensitivity. In
the present context, this measure quantifies the amount of information
about the temporal structure in the inputs that a TAH rule can store.
Using this method to study the novel class of NLTAH plasticity rules
introduced here, we find that the optimal learning rule depends on the
input statistics, in the present example on the characteristic time
scale of the temporal correlations between the inputs. This finding
suggests that biological systems may have acquired mechanisms for
metaplasticity to adapt the learning rule to slow temporal changes in
the input statistics. It should be pointed out that the sensitivity
measure S focuses entirely on how the learned synaptic
distribution changes as a result of changes in the correlation pattern
among the input channels. It does not, however, address the problem of
"readout," namely how the resulting changes in the synaptic
distribution affect the firing pattern of the output cell. A measure
that takes the postsynaptic spike train into account will in general
depend on the details of the spike-generating mechanism rather than
only capture the properties of the learning rule. In general, however, any readout mechanism will depend on the information that is available in the learned synaptic state. Hence, if the learning itself is insensitive to changes in the input features, the synaptic efficacies will fail to represent these changes and no readout mechanism will be
able to extract them. The sensitivity measure S
therefore provides an upper bound on the learning performance of the
full neural system (including readout). In summary, while quantitative claims about the optimality of specific learning rules have to consider
specific readout mechanisms, our study of the general properties of the
investigated plasticity rules provide general insights into the
mechanisms that enable unsupervised synaptic learning to remain
sensitive to input features during learning.
Present experimental results (Bi and Poo, 1998) based on
the averaging of individual efficacy changes in different synapses suggest the possibility that indeed the ratio of depressing and potentiating synaptic changes increases in a stabilizing manner as
synapses grow stronger (cf. van Rossum et al., 2000).
However, available data do not provide conclusive evidence regarding
the details of the weight dependence of the efficacy changes. Our work
clearly demonstrates the importance of the weight dependence of the TAH
updating rule. Synaptic learning rules that implement a stabilizing
weight dependence of the type introduced in this work have several
advantageous properties for the learning in neural networks.
Specifically, our results predict that synaptic changes should be
neither additive nor multiplicative, but rather should feature
intermediate weight dependencies that could, for instance, result from
a gradual saturation of the potentiating and depressing mechanisms. It
will be interesting to see whether future experimental results will
confirm such a prediction. In this context, it is also important to
note that recent experiments and modeling studies reveal important
nonlinearities in the accumulation of synaptic changes induced by
different spike pairs (Castellani et al., 2001;
Senn et al., 2001; Sjöström et al.,
2001) as well as evidence for complex intrinsic synaptic
dynamics that challenges the simple notion of a scalar synaptic
efficacy (Markram and Tsodyks, 1996). The theoretical
implications of these sources of nonlinearity and intrinsic dynamics
remain to be explored.
| |
FOOTNOTES |
Received Aug. 20, 2002; revised Feb. 20, 2003; accepted Feb. 20, 2003.
*
R.G. and R.A. contributed equally to this work.
Partial funding from the Studienstiftung des deutschen Volkes, the
Institut für Grenzgebiete der Psychologie, the Large Scale Facility Program of the European Commission, the Horowitz Foundation, the German-Israeli Foundation for Scientific Research and Development, the Volkswagen Foundation, the Israel Science Foundation (Center of
Excellence 8006/00), and the USA-Israel Binational Science Foundation
is gratefully acknowledged. We thank the staff of the Methods in
Computational Neuroscience 2000 summer course at Woods Hole, Prof. E. Ruppin, and O. Shriki for useful discussions. We gratefully acknowledge
the valuable discussions with Prof. L. Abbott that inspired our present
investigation. We thank Prof. A. Aertsen and the anonymous referees for
helpful comments and suggestions.
Correspondence should be addressed to Dr. Haim Sompolinsky, Racah
Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel. E-mail: haim{at}fiz.huji.ac.il.
R. Gütig's present address: Institute for Theoretical Biology,
Humboldt University, 10115 Berlin, Germany.
| |
APPENDIX |
Generating correlated spike trains
We show here that two spike trains that are generated by
conditioning their binwise spike probabilities on the activity of a
common reference spike train X0(T) as described
in Materials and Methods, have a pairwise correlation coefficient
c. For clarity, we denote
Xi(T) simply by Xi. The
pairwise correlation coefficient is defined by
Cov(Xi,
Xj)/ , where the covariance is Cov(Xi,
Xj) = E[XiXj] E[Xi]E[Xj]. Because Xi is either 0 or 1, E[Xi] = P(Xi = 1), and therefore:
where in the final step we use Equation 11. Note that because
E[Xi] = rT, the spike train
Xi has rate r. Similarly:
Finally, because Var(Xi) = (rT)(1 rT), the binwise correlation coefficient
becomes:
Homogeneous synaptic steady state for a homogeneous
population of synapses
We derive the homogeneous synaptic steady-state solution by
setting i = 0 and
wi = w* in Equation 8 with
C = 0:
Using Equation 14, this yields:
which, discarding the trivial solution in which all synapses are
zero, implies that at the homogeneous steady state:
|
(18)
|
From this, we find that the homogeneous solution is given by
wi = w*, where w* is the
solution to Equation 15. Because C0 is positive,
this equation has a unique solution 0 < w* < 1 for any µ > 0. Note, however, that in the additive model where µ = 0, the ratio f(w)/f+(w) = and, hence, in general there is no homogeneous synaptic
steady state unless w* = 0 or all weights are clipped to the
upper boundary.
For uncorrelated input activity C = ij/(r) and hence
C0 = 1/(rN). Substituting C0 into Equation 15, we obtain the homogeneous
solution for this input scenario:
|
(19)
|
Stability of the homogeneous synaptic steady state
We analyze the stability of the homogeneous synaptic steady
state by deriving the time evolution of small perturbations
wi = wi w* of the
synaptic efficacies wi around the steady-state value w*. If these perturbations decay to zero, the
homogeneous steady state is stable. For small perturbations, the time
evolution is given by:
Using the expression for the synaptic drifts from Equation 8, we obtain:
where:
|
(20)
|
In the second step, we used Equation 15 to substitute for
1 + C0. Note that the factor
go is proportional to the derivative of the
ratio between the scales of negative and positive synaptic changes with
respect to the weights. Hence, go = 0 in
the additive model. For µ > 0:
|
(21)
|
is positive because 0 < w* < 1. In matrix
notation, the time evolution of a synaptic perturbation w
can be rewritten as:
with the matrix:
If the eigenvalues of J are negative, all perturbations
of the homogeneous state are attenuated by the learning dynamics and,
hence, this synaptic state is stable. In contrast, if any eigenvalue of
J is positive, a perturbation along the direction of the
corresponding eigenvector will grow exponentially. The matrix
J has a homogeneous eigenvector with eigenvalue
Ngo Nf(w*) + NC0f+(w*), which using Equation 18 reduces
to Ngo. Because this eigenvalue is always
negative, the homogeneous component of any perturbation w
decays to zero with rate
r2go. In contrast, the
temporal evolution of the strictly inhomogeneous component of
w (whose elements sum to zero) comprises a spectrum of
rates that are determined by the various eigenvalues of J
that correspond to inhomogeneous eigenvectors. The largest inhomogeneous eigenvalue of J is
f+(w*)NC1 Ngo, where NC1 denotes
the largest inhomogeneous eigenvalue of
C+. Hence, the homogeneous synaptic state
is stable if f+(w*)NC1 Ngo < 0, which gives the stability criterion
stated in Results (Eq. 16). Inserting Equations 15 and 21 into the
criterion 16, we obtain an upper bound for µcrit,
the largest value of µ for which the homogeneous solution is
unstable:
|
(22)
|
An important observation is that C1 C0, and hence this bound is necessarily smaller
than 1, implying that the homogeneous solution is always stable in the
multiplicative model where µ = 1. To see that
C1 C0, recall that
because NC1 is an eigenvalue of
C+, there is an eigenvector v
such that C+v = NC1v. Specifically, for
vm, the largest component of
v, this implies that
NC1vm =  C vj, and hence, NC1 = C(vj/vm) C. But because
C = NC0 (Eq. 14), this yields C1 C0.
For uncorrelated input activity, the above bound for
µcrit becomes:
|
(23)
|
where we used C0 and
C1 from Equation 17. Hence, for large
rN, the regime of µ for which symmetry breaking exists
vanishes at least with 1/(rN).
Additive TAH in the linear neuron: uncorrelated inputs
The drift of the ith synapse of a neuron receiving
uncorrelated inputs and implementing the additive model is given by
setting µ = 0 in Equation 8 with C = 1/(r) and all other effective correlations equal 0:
|
(24)
|
As explained above, this linear system has no steady
state. Imposing the boundary conditions by clipping the efficacies
results in all synapses taking the value of either 0 or 1. Thus, the
learned synaptic distribution is fully described by
nup, the ratio of the synapses that are
saturated at the upper boundary. For a ratio nup
to be consistent, the drift of a synapse with efficacy 0 must be
nonpositive, whereas the drift of a synapse with efficacy 1 must be
non-negative. From imposing these conditions in Equation 24 we get:
The first inequality implies 1 if
nup > 0. It is important to note that the
regime of < 1 would simply yield saturation of all efficacies
at the upper boundary because all synapses experience a positive drift.
The second condition yields:
However, it can be shown using methods similar to those
of Rubin et al. (2001) that:
|
(25)
|
where if this quantity is >1, nup = 1. Therefore, if 1 + 1/(2rN), all
synapses will saturate at the upper boundary, whereas if > 1 + 1/(2r), even a single synapse at the upper boundary
will experience a negative drift, and hence no synapse will saturate at 1.
Moreover, the firing rate of the linear neuron is given by:
which is independent of the input firing rate r
[except for very low rates, where all synapses become strong (i.e.,
nup = 1 and
rpost = r)]. Thus,
output rate normalization is a property of the linear neuron when the
additive model is used.
Uniformly correlated inputs in the linear neuron
When the presynaptic inputs are uniformly correlated, namely
cij = c 0 for all i j (cii = 1), the effective correlation matrix
C = [c + ij(1 c)]/(r), and hence:
|
(26)
|
Because C0 increases with
c, the correlation increases the value of the synaptic
efficacy in the homogeneous synaptic state w* (Eq. 15). To
see the effect of the correlation on the stability of the homogeneous
solution, note that because C1 decreases with c, the correlation decreases the value of (Eq. 16), and
hence increases the stability of the homogeneous solution. Moreover, because both C0 and C1
decrease with 1/(rN), for large rN the critical µ (Eq. 22) approaches zero. Thus, for any µ > 0, the
homogeneous synaptic state is stable when the effective population is
sufficiently large.
Computing the susceptibility matrix for the
linear neuron
Here we compute the susceptibility matrix (Eq. 12,
Materials and Methods) used in Results to evaluate the sensitivity
measure S for the learning process in the linear Poisson
neuron model. This matrix is obtained by the implicit function theorem.
In the synaptic steady state (w*), the synaptic drifts are
zero by definition, and hence:
where R is the dimension of the space of input
features. Using:
and denoting:
we obtain:
The matrices and Mo are
of dimensions N by R, and M is an
N by N matrix. Below we derive for the two
input scenarios studied in Results.
Two correlated input groups
For the case of two correlated subgroups, the sensitivity to the
within-group correlation is measured. Hence, the input feature is
= c with R = 1. Using Equation 8
with C = 0, C = c/(r) if i j are in the same subgroup
(C = 1/(r)), and
C = 0 otherwise, we derive the
expressions for M and Mo:
where j  i if synapse
j and i are in the same group.
Delayed Poisson inputs
For a neuron receiving time-shifted versions of a common Poisson
spike train  , the input processes are
 (t) =  (t i)
(i = 1, ... , N), where i is
the delay of the activity at synapse i. Substituting these
into Equation 6, we obtain the effective correlation matrices (Eq. 7):
|
(27)
|
where  denotes the Heaviside step function ((x) = 1 if x 0 and (x) = 0
otherwise). In this case, the input features = and
R = N 1. Based on Equations 8 and 27, we derive
the expressions for M and Mo.
For compactness, we define the matrix of relative delays
Dij = ( + j i), with N = 0 corresponding to the delay of the reference spike train.
Thus:
where denotes the Heaviside step function and all sums are
taken over the N weights. The expressions are evaluated at
the synaptic steady-state w*, which is obtained by
numerically simulating the learning equations for all N synapses.
| |
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TOP
ABSTRACT
Introduction
![E[X<SUB>i</SUB>]=P(X<SUB>i</SUB>=1)](/content/vol23/issue9/fulltext/3697/img038.gif)
![E[X<SUB>i</SUB>X<SUB>j</SUB>]=P(X<SUB>i</SUB>=1, X<SUB>j</SUB>=1)](/content/vol23/issue9/fulltext/3697/img042.gif)
![M<SUB>ii</SUB>=<FR><NU>&tgr;r<SUP>2</SUP></NU><DE>N</DE></FR>[(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>−&agr;w<SUP>&mgr;</SUP><SUB>i</SUB>]+<FR><NU>rc</NU><DE>N</DE></FR>(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>+<FR><NU>r</NU><DE>N</DE></FR>(1−c)(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>](/content/vol23/issue9/fulltext/3697/img077.gif)
![−&mgr;<FR><NU>&tgr;r<SUP>2</SUP></NU><DE>N</DE></FR><FENCE><LIM><OP>∑</OP><LL>k=1</LL><UL>N</UL></LIM> w<SUB>k</SUB></FENCE>[(1−w<SUB>i</SUB>)<SUP>&mgr;−1</SUP>+&agr;w<SUP>&mgr;−1</SUP><SUB>i</SUB>]](/content/vol23/issue9/fulltext/3697/img078.gif)
![M<SUB>ij</SUB>=<FR><NU>&tgr;r<SUP>2</SUP></NU><DE>N</DE></FR>[(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>−&agr;w<SUP>&mgr;</SUP><SUB>i</SUB>]+<FR><NU>rc</NU><DE>N</DE></FR>(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP> j∈𝒢<SUB>i</SUB>](/content/vol23/issue9/fulltext/3697/img080.gif)
![M<SUB>ij</SUB>=<FR><NU>&tgr;r<SUP>2</SUP></NU><DE>N</DE></FR>[(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>−&agr;w<SUP>&mgr;</SUP><SUB>i</SUB>] j ∉ 𝒢<SUB>i</SUB>](/content/vol23/issue9/fulltext/3697/img081.gif)
![+&tgr;r<SUP>2</SUP>[(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>−&agr;w<SUP>&mgr;</SUP><SUB>i</SUB>]+<FENCE>(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>r</FENCE>](/content/vol23/issue9/fulltext/3697/img091.gif)
![M<SUB>ij</SUB>=<FR><NU>&lgr;</NU><DE>N</DE></FR><FENCE>&tgr;r<SUP>2</SUP>[(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>−&agr;w<SUP>&mgr;</SUP><SUB>i</SUB>]−&agr;w<SUP>&mgr;</SUP><SUB>i</SUB>r&THgr;(<UP>−</UP>D<SUB>ij</SUB>)<UP>exp</UP><FENCE><FR><NU>D<SUB>ij</SUB></NU><DE>&tgr;</DE></FR></FENCE>+(1−w<SUB>i</SUB>)<SUP>&mgr;</SUP>r&THgr;(D<SUB>ij</SUB>)<UP>exp</UP><FENCE><UP>−</UP> <FR><NU>D<SUB>ij</SUB></NU><DE>&tgr;</DE></FR></FENCE><FENCE> i, j=1…N, i≠j</FENCE></FENCE>](/content/vol23/issue9/fulltext/3697/img093.gif)
