This position paper has not been peer reviewed or edited. It will be finalized, reviewed and edited
after the Royal Society meeting on ‘Integrating Hebbian and homeostatic plasticity’ (April 2016).
Cooperation across timescales between and Hebbian and
homeostatic plasticity
Friedemann Zenke1 and Wulfram Gerstner2
April 13, 2016
1) Dept. of Applied Physics, Stanford University, Stanford, CA
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2) Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne EPFL
Abstract
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We review a body of theoretical and experimental research on the interactions of homeostatic
and Hebbian plasticity, starting from a puzzling observation: While homeostasis of synapses
found in experiments is slow, homeostasis of synapses in most mathematical models is rapid,
or even instantaneous. Even worse, most existing plasticity models cannot maintain stability
in simulated networks with the slow homeostatic plasticity reported in experiments. To solve
this paradox, we suggest that there are both fast and slow forms of homeostatic plasticity with
distinct functional roles. While fast homeostatic control mechanisms interacting with Hebbian
plasticity render synaptic plasticity intrinsically stable, slower forms of homeostatic plasticity are
important for fine-tuning neural circuits. Taken together we suggest that learning and memory
relies on an intricate interplay of diverse plasticity mechanisms on different timescales which
jointly ensure stability and plasticity of neural circuits.
Introduction
Homeostasis refers to a group of physiological processes at different spatial and temporal scales that
help to maintain the body, its organs, the brain, or even individual neurons in the brain in a target
regime where they function optimally. A well-known example is the homeostatic regulation of body
temperature in mammals, maintained at about 37 degrees Celsius independent of weather condition
and air temperature. In neuroscience, homeostasis or homeostatic plasticity often refers to the
homeostatic control of neural firing rates. In a classic experiment, cultured neurons that normally
fire at, say 5Hz, change their firing rate after a modulation of the chemical conditions in the culture,
but eventually return to their target rate of 5Hz during the following 24 hours (Turrigiano and
Nelson, 2000). Thus, the experimentally best-studied form of synaptic homeostasis happens on
a slow timescale of hours or days. This slow form of synaptic homeostasis manifests itself as a
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rescaling of the strength of all synapses onto the same neuron by a fixed fraction, for instance 0.78,
a phenomenon called “synaptic scaling” (Turrigiano et al., 1998).
Mathematical models of neural networks also make use of homeostatic regulation to avoid runaway of the strength of synaptic weights or over-excitation of large parts of a neural network. In many
mathematical models, homeostasis of synaptic weights is idealized by an explicit normalization of
weights: if the weight (or efficacy) of one synaptic connection increases, weights of other connections
onto the same neuron are algorithmically decreased so as to keep the total input in the target regime
(Miller and MacKay, 1994). Algorithmic normalization of synaptic strength can be interpreted as
an extremely fast homeostatic regulation mechanism. At a first glance, the result of multiplicative
normalization (Miller and MacKay, 1994) is identical to “synaptic scaling” introduced above. Yet, it
is fundamentally different from the experimentally studied form of synaptic homeostasis mentioned
above, because of the vastly different timescales. While rescaling in the model is instantaneous, in
biology the effects of synaptic scaling manifest themselves only after hours. Interestingly, researchers
to stabilize the network activity.
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who have tested slow forms of homeostasis in mathematical models of plastic neural networks failed
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Thus, we are confronted with a dilemma: Theorists need fast synaptic homeostasis while experimentalist have found homeostasis that is much slower. In this review we try to answer the
following questions: Why do we need rapid forms of homeostatic synaptic plasticity? How fast does
a rapid homeostatic mechanism have to be, hours, minutes, seconds or less? Furthermore, what are
the functional consequences of fast homeostatic control? Can the combination of fast homeostasis
with Hebbian learning lead to stable memory formation? And finally, if rapid homeostasis is a
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requirement, what is the role of slower forms of homeostatic plasticity?
Conceptually, automatic control of a technical system is designed to keep the system in the
regime where it works well, which is called the “working point” or the “set point” of the system.
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Analogously, homeostatic regulation of synapses is important to maintain the neural networks of
the brain in a regime where they function well. However, in specific situations our brain needs to
change. For example, when we exit the subway at an unknown station, when we are confronted
with a new theoretical concept, or when we learn how to play tennis, we need to memorize the novel
environment, the new concept, or the unusual movement that we previously did not know or master.
In these situations, the neural networks in the brain must be able to develop an additional desired
“set point”, ideally without loosing the older ones.
In neuroscience it is widely accepted that learning observed in humans or animals at the behavioral level corresponds, at the level of neural networks, to changes in the synaptic connections
between neurons (Morris et al., 1986; Martin et al., 2000). Thus, learning requires synapses to
change. A strong form of synaptic homeostasis, however, keeps neurons and synapses always in the
“old” regime, i.e. the one before learning occurred. In other words, strong homeostasis prevents
learning. A weak or slow form of homeostasis, on the other hand, may not be sufficient to control
the changes induced by memory formation. Before we can solve the riddle of stability during mem2
ory formation, and before we can answer the above questions, we need to classify known forms of
synaptic plasticity.
Classification of Synaptic Plasticity
Synaptic plasticity, can be classified using different criteria. We consider three of these: Functional
relevance, synaptic specificity, and timescale of effect duration.
(i) Classification by potential function: Memory formation or homeostasis
Functionally, synaptic plasticity can be classified as useful either for homeostatic regulation or for
memory formation and action learning. Many experiments on synaptic plasticity were based on
Hebb’s postulate which suggests that synaptic changes, caused by the joint activity of pre- and
postsynaptic neurons, should be useful for the formation of memories (Hebb, 1949). Inspired by
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Hebb’s postulate, classical stimulation protocols for long-term potentiation (Bliss and Lomo, 1973;
Malenka and Nicoll, 1999; Lisman, 2003), long-term depression (Lynch et al., 1977; Levy and Steward, 1983), or spike-timing dependent plasticity (Markram et al., 1997; Bi and Poo, 1998; Sjöström
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et al., 2001), combine the activation of a presynaptic neuron, or a presynaptic pathway, with an
activation, depolarization, or chemical manipulation of the postsynaptic neurons, to induce synaptic
changes.
In parallel, theoreticians have developed synaptic plasticity rules (Willshaw and Von Der Malsburg, 1976; Bienenstock et al., 1982; Kempter et al., 1999; Song et al., 2000; Pfister and Gerstner,
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2006; Clopath and Gerstner, 2010; Brito and Gerstner, 2016; Shouval et al., 2002; Graupner and
Brunel, 2012), in part inspired by experimental data. Generically, in plasticity rules of computaas
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tional neuroscience the change of a synapse from a presynaptic neuron j to a neuron i is described
d
wij = F (wij , posti , prej )
dt
(1)
where wij is the momentary “weight” of a synaptic connection, posti describes the state of the
postsynaptic neuron (e.g. its membrane potential, calcium concentration, spike times, or firing rate)
and prej is the activity of the presynaptic neuron (Brown et al., 1991; Morrison et al., 2008; Gerstner
et al., 2014). A network of synaptic connections with weights consistent with Hebb’s postulate can
form associative memory models (Willshaw et al., 1969; Little and Shaw, 1978; Hopfield, 1982; Amit
et al., 1985; Amit and Brunel, 1997), also called attractor neural networks (Amit, 1989). However,
in all the cited theoretical studies, learning is supposed to have happened somewhere in the past,
while the retrieval of previously learned memories is studied under the assumption of fixed synaptic
weights. The reason is that a simple implementation of Hebb’s postulate for memory formation is
not sufficient to achieve stable learning (Rochester et al., 1956).
Without clever homeostatic control of synaptic plasticity, neural networks with a Hebbian rule
always move to an undesired state where all synapses go to their maximally allowed values and
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all neurons fire beyond control — or to the other extreme where all activity dies out. Therefore,
homeostasis in the sense of keeping the network in a desired regime, is a second and necessary function
of synaptic plasticity. Candidate plasticity mechanisms for the homeostatic regulation of network
function by synaptic plasticity include slow synaptic scaling (Turrigiano et al., 1998; Turrigiano and
Nelson, 2000), rapid renormalization of weights (Miller and MacKay, 1994), or other mechanisms
discussed below (Zenke et al., 2013, 2015; Chistiakova et al., 2015); but also synaptic plasticity of
inhibitory interneurons (Vogels et al., 2011, 2013) or intrinsic plasticity of neurons (Daoudal and
Debanne, 2003).
(ii) Classification by specificity of synaptic changes: Homosynaptic or heterosynaptic
In some of the classic induction protocols of long-term potentiation and depression, mentioned above,
a presynaptic pathway (or presynaptic neuron) was stimulated together with an activation of the
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postsynaptic neuron. If a change was observed in those synapses that were stimulated during the
induction protocol, the plasticity was classified as synapse-specific or “homosynaptic”. If a change
was observed in other (unstimulated) synapses onto the same postsynaptic neuron, the plasticity
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was classified as unspecific or “heterosynaptic” (Lynch et al., 1977; Abraham and Goddard, 1983;
Brown and Chattarji, 1994). Heterosynaptic plasticity has moved back in the focus recently, because
of its potential role in homeostatic regulation of plasticity (Royer and Paré, 2003; Chen et al., 2013;
Chistiakova et al., 2014, 2015; Zenke et al., 2015).
In the framework of Eq. (1), the change of a synapse may depend on the activity of the presynaptic
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neuron, the state of the postsynaptic neuron — or on both — as well as on the momentary value of
the synaptic weight. Since we do not know the function F on the right-and-side of Eq. (1), we can
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imagine that it may contain different combinations of pre- and postsynaptic variables, such as
F (wij , posti , prej ) = a0 (wij ) + f1 (wij , posti ) + f2 (wij , prej ) + a1 (wij ) H(posti , prej )
(2)
In this picture, the term a0 (wij ) would represent a drift of the synaptic strength that does not
depend on the input or the state of the postsynaptic neuron, but only on the momentary value of
the weight wij itself. More interesting is the term f1 (wij , posti ) which depends on the postsynaptic
activity, but not on the presynaptic activity. Suppose the postsynaptic neuron went through a
phase of rapid firing. Such a term will then induce changes that are not synapse-specific and would
therefore be classified as heterosynaptic. Conversely, the term f2 (wij , prej ) depends on the activity
of the presynaptic pathway, but not on the state of the postsynaptic neuron. In the following we will
call such a term transmitter-induced. If a presynaptic neuron is weakly stimulated for some time,
the change caused by the term f2 (wij , prej ) can be classified as homosynaptic, but not Hebbian. In
equation Eq.(2) only the last term is able to account for changes that depend on the joint activity of
pre- and postsynaptic neurons. This term is therefore homosynaptic and Hebbian, hence our choice
of letter H. The multiplicative factor a1 (wij ) in front of H highlights that the amount of synaptic
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plasticity may depend on the momentary state of the synaptic weight. For example, it might be
difficult or impossible to further increase a synapse that is already very strong, simply because strong
synapses require a big spine and therefore a lot of resources. The theoretical literature implements
resource limitations by keeping the weights wij below a maximum value wmax . Depending on the
implementation details, the limit is called a “hard bound” our “soft bound”.
As an aside, we note that the term “heterosynaptic plasticity” is sometimes also used for synaptic
changes that are visible at the connection from a presynaptic neuron j to a postsynaptic neuron i,
but induced by the activation of a third, typically modulatory neuron (Bailey et al., 2000). However,
in the following we do not consider this possibility.
Equations (1) and (2) that describe the change of synaptic weights as a function of neuronal
activity are called learning rules in the theoretical literature. For the rest of the paper, Eqs. (1) and
(2) provide the framework in which we analyze and compare different learning rules with each other.
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(iii) Classification by duration of effects: Short-term versus long-term plasticity versus
consolidation
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Synaptic changes induced by a sequence of four presynaptic spikes in rapid sequence typically decay
within a few hundred milliseconds (Abbott et al., 1997; Tsodyks and Markram, 1997; Tsodyks et al.,
1998) and are called short-term plasticity. The rapid decay of plasticity outcomes implies that the
changes are not useful for memory formation, but more likely involved in gain control (Abbott et al.,
1997), which implies a homeostatic role.
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Synaptic changes induced by classic induction protocols (Bliss and Lomo, 1973; Malenka and
Nicoll, 1999; Lisman, 2003), however, last for several hours and are therefore potentially useful for
memory formation. We emphasize that the induction protocol itself often lasts less than a minute.
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In other words, induction of synaptic plasticity is fast, but the changes persist for a long time, and
this is why the phenomenon is called long-term potentiation or long-term depression, which we will
abbreviate in the following as LTP or LTD.
Under suitable conditions the changes induced by a protocol of LTP or LTD are further consolidated after about an hour (Frey and Morris, 1998; Reymann and Frey, 2007; Redondo and Morris,
2011). In the rest of the paper we mainly focus on the phase of plasticity induction and neglect
consolidation and maintenance.
To summarize this part, we may state that the classification of synaptic plasticity is possible in
many different ways. A certain aspect of plasticity (or a certain mathematical term in a model)
can be treated at the same time as Hebbian, homosynaptic, and memory-building while another
term could be heterosynaptic and homeostatic, and a third term could be transmitter-induced,
homosynaptic, but non-Hebbian. Each of the terms can lead to a change that persists for a long
time or that decays within a few seconds.
Moreover, each of the terms could be strong so that its contribution is easily verified in exper-
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iments; or it could be weak so that only a precise experiments with multiple repetitions would be
able to isolate a given term. The problem of the effect size is intimately related to the question of
the timescale for induction of plasticity, which is an important topic in itself to be discussed next.
Why do we need rapid forms of homeostatic synaptic plasticity?
With the definitions and distinctions made in the previous section, we are now ready to address the
first of the questions posed in the introduction: Why do we need a fast form of homeostatic synaptic
plasticity?
Intuitively, synaptic plasticity that is useful for memory formation must be sensitive to the
present activation pattern of the pre- and postsynaptic neuron. Following Hebb’s idea of learning
and cell assembly formation, the synaptic changes should make the same activation pattern more
likely to re-appear in the future, to allow contents from memory to be retrieved. However, the
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re-appearance of the same pattern will induce further synaptic plasticity. This leads to a positive
feedback loop that can quickly run out of control. Anybody who was sitting in the audience when
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the positive feedback loop between the speaker’s microphone and the loudspeaker resulted in an
unpleasant shriek, knows what this means. Engineers have realized that positive feedback loops can
only be stabilized by rapid control mechanisms. Similarly, the interaction between the firing activity
in a neural network with synaptic plasticity for memory formation can lose stability, unless synaptic
plasticity is controlled by rapid homeostasis.
While the plasticity rules derived from experiments look at most at one or two stimulation
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pathways at a time, neurons embedded in a network can receive inputs from thousands of different
synapses. In this section, we will outline a simple, yet powerful framework which enables us to
make quantitative predictions about the stability in large networks of spiking neurons. Synapses in
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the network will exhibit different forms of plasticity which serve specific functions for both memory
formation and homeostasis.
Before we can give a precise answer to the question of “why do we need a fast form of homeostatic
synaptic plasticity”, we have to consider what it means to be “fast” or “slow”. Hebbian synaptic
plasticity manifests itself tens of seconds or a few minutes after plasticity induction (Petersen et al.,
1998; O’Connor et al., 2005). This sets a first timescale of plasticity induction. Intuitively, we will
call a homeostatic mechanism fast if it happens on the same timescale as Hebbian plasticity induction.
It will be considered as slow, if it is much slower than induction of LTP, i.e., if it manifests itself
over hours or days.
In the previous paragraph, we used the term “timescale”. To make the intuitive notion of timescale
more precise, we need a short mathematical digression. The reader who is familiar with differential
equations or dynamical systems can skip the next subsection.
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Mathematical digression: The notion of a timescale
To formally introduce the notion of a timescale, we consider a linear system defined by the differential
equation
τ
dy
= −(y − θ)
dt
(3)
where θ and τ are two fixed positive parameters. The solution y(t) of the differential equation
relaxes exponentially to the value y = θ, on the timescale τ . Practically speaking that means that
no matter how we initialize y(t) at time t = 0, if we wait for a time 3τ , the difference y(t) − θ will
have decreased by 95%. The exponential evolution is typical for a linear system. Moreover, the
equation has a nice scaling behavior: If we know the solution for a parameter setting τ = 1, we can
get the solution for the parameter setting τ = 5 by multiplying all times by a factor of five. This is
the reason why we speak of τ as the timescale of the system.
The fact that a constant value of y(t) = θ for all t is a solution means that θ is a fixed point of
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the system. To check this, remember that the derivative of a constant is zero (so that the left-hand
side of Eq. (3) vanishes). And for y = θ the right-hand side is obviously zero, too. Moreover, this
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fixed point is stable because from any initial condition y(t) will converge towards θ.
Let us now consider the case where τ is negative. In this case, y(t) = θ is still a fixed point, but
if we start with a value y(t) > θ, then y(t) will explode exponentially fast to large positive values
while for an initial value y(t) < θ it will explode exponentially fast to large negative values. The
timescale of explosion is again given by τ .
However, our intuition for timescales breaks down for nonlinear systems. Therefore, when we
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consider a nonlinear system τ dy
dt = f (y), it does not suffice for τ to be large to be able to talk about
a slow or fast timescale. We have to be specific about the behavior of f in the regime that we are
interested in. The mathematical trick to do this is to look for a fixed point of the equation, that is
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a value of y with f (y) = 0. Suppose y = y0 is a fixed point. We then study the derivative df /dy at
y0 . Let us denote this derivative by f 0 . In the neighborhood of y0 (and only there!) the nonlinear
equation is well approximated by a linear equation τ dy/dt = (y − y0 ) f 0 . Division by f 0 brings f 0 to
the other side of the equation and enables us to identify the effective timescale τ̃ = −τ /f 0 . If you
are in doubt, compare your result with equation (3).
Another example of a linear system is a low-pass filter. Suppose we pass our variable y(t) through
a low-pass filter with time constant τd to yield
(t − t0 )
y(t0 ) dt0 .
ȳ(t) =
exp −
τd
−∞
Z
t
(4)
By taking the derivative on both sides of Eq. (4), we find that the low-pass filter ȳ is the solution
of a differential equation dȳ(t)/dt = −ȳ(t)/τd + y(t). This equation is similar to equation (3), if we
replace the constant target value θ by a time-dependent target y(t)/τd .
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a
Change in synaptic efficacy
Change in synaptic efficacy
b
1.2
0.8
LTP
0.4
0
−0.4
LTD
θ
Postsynaptic activation
1
0.5
0
θ
ȳ < θ
ȳ = θ
ȳ > θ
−0.5
−1
Postsynaptic activation
c
0.2
0
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0.4
Plasticity protocol
Time
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Synaptic efficacy
0.6
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Figure 1: Most plasticity models can reproduce the notion of a plasticity threshold
reported in experiments. (a) The change in synaptic efficacy in many plasticity models is a
function of variables related to postsynaptic activation. (b) Schematic illustration of the action of
the homeostatic moving threshold in the BCM-model (Bienenstock et al., 1982). (c) Hypothetical
evolution of synaptic efficacy as a function of time during an LTP induction experiment. The purple,
linear time course is a common assumption underlying many plasticity models. However, since the
data during plasticity induction are typically unobservable, many time courses and thus plasticity
modes are compatible with the data.
The induction timescale of Hebbian plasticity
Because both learning rules and neurons are typically nonlinear, the mathematical definition of a
timescale for Hebbian plasticity requires the existence of a fixed point. The reason is that for a
nonlinear system we can define a timescale only in the vicinity of a fixed point, as we have seen
above.
Even if a plasticity rule in isolation looks linear, the combination of plasticity with the neuronal
dynamics typically makes the network as a whole nonlinear. Consider for instance a single neuron
i that is driven by N inputs arriving at synapses wij for 1 ≤ j ≤ N . In a rate
model, the state of
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the postsynaptic neuron is characterized by its firing rate yi = g
j=1 wij xj where xj is the firing
rate of the presynaptic neuron with index j. The function g denotes the frequency-current relation
of a single neuron and we assume that it is monotonically increasing, i.e., if we increase the input,
the firing rate increases as well. In the theoretical literature g is sometimes called the gain function
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of the neuron, hence our choice of letter g.
Let us first study a simple Hebbian learning rule
dwij
= η yi xj .
dt
(5)
Note that this is a special instance within the general framework of Eqs (1), and (2). To see this,
consider a Hebbian term H(posti , prej ) = yi xj , and a1 (wij ) = η and set all other terms in Eq. (2)
to zero. This learning rule is linear in xj and linear in yi . However, if we insert yi = g
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j
wij xj
into the learning rule, the learning dynamics become nonlinear in xj . Nonlinearity implies that we
cannot define a timescale of plasticity, unless we find a fixed point where dwij /dt vanishes.
Are there fixed points of the dynamics? There is a fixed point if the postsynaptic or the presynaptic rate is zero. However, if the neuron is embedded in a network, it is reasonable to assume
that several presynaptic neurons including the presynaptic neuron j are active. Unless all weights
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wij are zero, the postsynaptic neuron is therefore also active, and the weight wij increases.
The argument can be formalized to show that wij = 0 is an unstable fixed point. If we increase
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wij by just a little, the output yi also increases which increases wij even further, which closes the
positive feedback loop. More generally, models of Hebbian plasticity that are useful for memory
formation all have an unstable fixed point. One important role of homeostatic plasticity in networks
models with Hebbian plasticity is therefore to create (additional) stable fixed points for the learning
dynamics as we will see later on in this section.
The simple example above only has a trivial fixed point at zero activity (zero weights). Moreover,
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it is lacking the notion of LTD. Plausible plasticity models have additional stationary points defined
by the plasticity threshold between LTP and LTD (Fig. 1a). Inspired by experimental data (Artola
et al., 1990; Dudek and Bear, 1992; Sjöström et al., 2001), the transition from LTP to LTD depends
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in models on the state of the postsynaptic neuron, e.g., its membrane potential, calcium level, inter
spike interval (Artola and Singer, 1993; Shouval et al., 2002; Pfister and Gerstner, 2006; Clopath
and Gerstner, 2010; Graupner and Brunel, 2012). As a paradigmatic example, which stands for a
plethora of different plasticity rules with a postsynaptic threshold, we consider the following ratebased nonlinear Hebbian rule
dwij
= η xj yi (yi − θ)
dt
(6)
with a positive constant η. Whenever the postsynaptic firing rate yi equals the value of θ, the weight
wij does not change. Thus this learning rule has a fixed point at the threshold yi = θ. For yi larger
than θ the synaptic weights increase which corresponds to the induction of LTP in the model. For
yi smaller than θ the synaptic weights decrease and LTD is induced.
Let us now embed this learning
rule
in a network. For the sake of simplicity we assume the
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gain function to be linear, g
j wij xj . Moreover, we assume that (i) all weights wij
j wij xj =
have the same value wij = w and (ii) all N neurons in the network fire with the same firing rate
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xj = 1/N , so that y = w. Inserting these assumptions into Eq. (6) yields
dw
= η w (w − θ)
dt
(7)
which characterizes the change in neural activity y as governed by synaptic plasticity. We now
linearize Expression (7) at the stationary point w = θ:
dw
≈ η θ (w − θ)
dt
This is a linear differential equation with a solution w(t) that, for w(t) > θ, explodes exponentially fast on the timescale τ =
1
ηθ
(cf. discussion of Eq. (3). Thus, in this example, w = θ is
an unstable fixed point and the linearization procedure has enabled us to identify the timescale of
plasticity induction. In practical implementations of a plasticity model, the exponential growth of
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the synaptic weights wij would stop when they obtain their maximal value wmax . However, if all
synapses onto a postsynaptic neuron, or even all synapses in a neural network sit at their upper
bound, the network cannot function as a memory.
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Note that the occurrence of the parameter θ in the timescale τ is a hallmark of the nonlinearity
of the full system. Just like in the well-known Hodgkin-Huxley model where the timescale of the
activation and inactivation variable depends on the voltage, the effective timescale τ of a learning
rule will depend on multiple factors such as the presynaptic activity, the slope of the gain function,
or the threshold θ between LTD and LTP.
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Even though τ in the example above is not the same as the induction timescale of long-term
plasticity, in experiments where only a single presynaptic pathway is stimulated, the two are related.
With our formalism we can also account for the strength of the recurrent feedback that is received
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by a neuron embedded into a network. Just repeat the above analysis under the assumption that
the presynaptic and postsynaptic neurons are mutually connected, of the same type, and fire at the
same rate (Zenke et al., 2013). Whatever you consider as a reasonable scenario, the timescale τ
characterizes how quickly a neuron or an entire recurrent network can generate positive feedback
and is able to “run away”.
Instead of writing down a differential equation for the weights w, as in Eq. (7), we could have
written the system in terms of y, too. A formulation in terms of the firing rates y highlights the
fact that we have to think about run-away effects of synapses as being linked to run-away effects of
neuronal activity. Note further, that in realistic neuronal network the presynaptic activity fluctuates
and is different between one neuron and the next. Fluctuations give rise to a covariance matrix
C=
1
n
Pn
t=1 xi (t)xj (t)
which may look complicated at a first glance. However, due to the symmetry
of C, there always exists a basis in which C is diagonal. When working in this basis, the plasticity
equations decouple and take the shape of Eq. (7) with different values of η.
In summary, by using the formalism introduced in this subsection, we can combine a plausible
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data-based plasticity model and a given network model to find the effective timescale τ for this
compound system. In doing so we take for instance explicitly into account that even if the change
of a single synapse within one second might be small, the combined effect of millions of synapse of
thousands of recurrently synapses could still be large. By distilling all these factors into a single
number τ we can now return to the question of homeostatic plasticity. How can homeostatic control
catch up with Hebbian plasticity that makes weights “explode” on the timescale τ ?
Stability through homeostatic plasticity
To stabilize Hebbian plasticity, theorists may select and mathematically formalize homeostatic mechanisms found in nature (Turrigiano, 2011). Two mathematical approaches to synaptic homeostasis
are widespread. Either the Hebbian plasticity term is modified by a sliding threshold, or the Hebbian plasticity term for memory formation is complemented by an independent term for rate control.
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Although other choices are possible, we will now focus on those two mechanisms. Can either of these
mechanisms guarantee stability of Hebbian learning? Are there fundamental restrictions with respect
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to how fast homeostatic plasticity has to be?
Plasticity control through a sliding LTP-LTD threshold
Several models assume that the threshold that marks the boundary between LTP and LTD is not
fixed but may slide on a slow timescale (Bienenstock et al., 1982; Cooper et al., 2004; Pfister and
Gerstner, 2006). We refer to this model class as BCM models in the following, where BCM is
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the acronym for Bienenstock-Cooper-Munro. The basic idea of that model class is that a sliding
threshold could induce a homeostatic mechanism to locally control the amount of LTP or LTD in
each synapse. The question then arises whether the sliding threshold can indeed lead to stable
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memory formation without the need for a further fast homeostatic mechanism.
To formalize this idea, we use the low-pass filter ȳi of the postsynaptic firing rate (averaged over
a time τd ) as a sensor that drives homeostatic plasticity; see (Eq. (4)). Intuitively, a homeostatic
sliding threshold θ(ȳ) increases the amount of LTD if the average postsynaptic activity ȳ exceeds
a given homeostatic set point κ for extended periods of time (Fig. 1b). Conversely, when neuronal
activity falls below this homeostatic activity target, the rate of LTP is increased at the expense
of LTD. There are several different possible forms of such BCM models (Bienenstock et al., 1982;
Cooper et al., 2004). The following is a simple, but representative example
dwij
= η xj y (yi − θ(ȳi ))
dt
with a sliding threshold θ(ȳi ). A common choice for the sliding threshold is θ(ȳi ) =
(8)
ȳi2
κ.
Let us analyze this sliding threshold model. If we insert the expression for θ into Eq. (8) and
choose a constant firing rate yi = ȳi = κ, we see that the weight wij does not change. Therefore
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yi = κ is a fixed point of the dynamics. The stability of this fixed point depends on the timescale
of the low-pass filter τd which in turn determines the timescale of metaplastic changes in Eq. (8)
(Abraham (2008); Fig. 1b). It can be shown analytically that the system has a stable fixed point
only if τd is chosen smaller than some critical value τdcrit (Cooper et al., 2004; Zenke et al., 2013).
Thus, a slow homeostasis of the sliding threshold is, on its own, not sufficient to achieve stability.
Rate control through synaptic scaling
To introduce the second control mechanism, let us consider a neuron embedded in a large network
of stochastically firing neurons. If many synaptic weights onto a given postsynaptic neuron increase,
the postsynaptic firing rate y is expected to increase. If the firing rate remains high over some
timescale τd , we may consider this as an indicator that the neuron runs out of its target firing
regime. Let us therefore use the low-pass filtered rate ȳ(t) (Eq. (4)) as a driver of homeostasis (van
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Rossum et al., 2000).
Let us focus on a postsynaptic neuron i with rate yi . We adjust the weight wij from a presynaptic
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neuron j to our postsynaptic neuron according to the rule
dwij
= η xj yi (yi − θ) + η̃(κ − ȳi )wij
|
{z
} |
{z
}
dt
(9)
Homeostasis
Hebb
The first term on the right-hand side is the Hebbian plasticity term discussed earlier, now with a
fixed threshold θ that marks the transition from LTP to LTD. The multiplicative factors in the
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second term are η̃ (a parameter) and the weight wij itself. The term κ inside the parenthesis of the
second term corresponds to a target value of the postsynaptic activity and ȳi is a low-pass filtered
version of the firing rate yi . A simple and biologically plausible way of how a neuron can sense ȳi
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on-line, is by reading out the concentration of a slow activity-related intracellular factor.
Three remarks are important. First, the homeostatic term corresponds to the heterosynaptic
plasticity term f1 (wij , posti ) in Eq. (2). Second, the fact that the heterosynaptic term is proportional
to the weight wij implies that, for ȳi > κ, larger weights are depressed more than smaller ones. Thus
this heterosynaptic term implements synaptic scaling. Third, in the language of control theory the
heterosynaptic term implements homeostasis via a proportional controller. More sophisticated forms
of heterosynaptic plasticity like a proportional-integral controller are possible (van Rossum et al.,
2000), but this generally does not change the core of our argument that we will develop in the
following.
How strong and how fast does the heterosynaptic term have to be to fulfill its homeostatic
function? It can be shown analytically, that the requirement of stability in the above system (Eq. (9))
puts some tight constraints on the parameters τd and η̃ for a given value of κ (Zenke et al., 2013).
First, η̃ cannot be significantly different from η. If η̃ is much larger than η homeostatic control
creates oscillations which are not observed in biological systems. On the other hand, if η̃ is much
12
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Figure 2: The timescales of synaptic scaling or metaplasticity are faster in models than
reported in experiments. Here we plot the homeostatic timescale of either synaptic scaling
or homeostatic metaplasticity as used in influential modeling studies (black). For comparison we
plot the typical readout time for experimental studies on synaptic scaling and metaplasticity (red).
Publications suffixed with * describe network models as opposed to the other studies which relied
on single neurons. For reference we also indicate the typical timescales of short-term plasticity and
long-term plasticity induction at the top. Note, that homeostatic timescales in models fall in the
same range as the induction timescale of Hebbian long-term plasticity, which does not seem to be
the case for experimental results on homeostatic plasticity.
smaller than η, the fixed point κ system loses stability altogether. Similarly, we can derive stability
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conditions for τd and show that stability is only guaranteed if τd is smaller than some critical value
τdcrit (which explicitly depends on η). The mathematical analysis thus shows that synaptic scaling
needs to be “fast”, or at least faster than some reference. To further answer the question of how
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fast homeostatic plasticity has to be, we therefore have to quantify the speed of “normal” Hebbian
plasticity.
Thus, our analytical approach has allowed us to answer the first question that we posed in the
introduction. We need homeostatic synaptic plasticity which is faster than some critical value in
order to achieve stability of learning dynamics. This statement is true for both the sliding threshold mechanism and the rate-control framework. However, what does “fast homeostatic plasticity”
mean quantitatively? Is this critical timescale on the order of minutes, hours or days? To answer
this question we need to quantify parameters such as η with experimental data. In the following
subsection we will address this question and translate the mathematical statements of this section
into interpretable numbers.
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How fast does a rapid homeostatic mechanism have to be?
Slow homeostatic control cannot stabilize the rapid positive feedback loop induced by Hebbian
learning. However, in this statement, the meaning of “rapid” and “slow” is relative and based on a
comparison of the timescale of homeostasis with that of induction of Hebbian plasticity. But what
is the actual timescale of Hebbian plasticity?
To connect the framework of learning rules outlined in Eqs.(1–8) to data from plasticity experiments we can rely on a wealth of data on synaptic plasticity. Experiments which used STDP
protocols to study induction of Hebbian plasticity are particularly informative (Sjöström et al., 2001;
Senn et al., 2001). Theorists have made great strides in developing models which quantitatively capture a wide range of these data (Senn et al., 2001; Pfister and Gerstner, 2006; Froemke and Dan,
2002; Clopath and Gerstner, 2010). The parameters of the resulting spike-based plasticity models
capture the timescale of plasticity induction in STDP experiments.
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STDP models such as (Pfister and Gerstner, 2006) give a particular instantiation of the Hebbian
term in Eq. (9) and make it possible to extract its parameters directly from experimental data.
To bridge the conceptual gap between the spiking plasticity models and our simplified framework
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in the firing rate domain we have to consider neural firing statistics. Because neuronal spiking
in vivo is highly irregular, we make the assumption that firing statistics can be approximated by a
Poisson process. If both pre- and postsynaptic neurons fire stochastically and Poisson-like, the STDP
models can be rewritten as firing rate models (Pfister and Gerstner, 2006). In particular, modern
STDP models (Senn et al., 2001; Pfister and Gerstner, 2006; Clopath and Gerstner, 2010) exhibit
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a transition between LTD and LTD which depends on the firing rate, consistent with experimental
data (Sjöström et al., 2001). We can exploit the connection between data, STDP models, and the
effective Hebbian rate model so as to determine a plausible timescale for the induction of Hebbian
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plasticity.
(A short aside: In the context of STDP, it might seem paradoxical to assume Poisson firing,
which explicitly gets rid of any temporal correlations between the input and output spike trains.
However, as long as temporal correlations are as small as commonly seen in cortical activity or
mimicked in simulations of balanced state networks (van Vreeswijk and Sompolinsky, 1996; Renart
et al., 2010), the Poisson approximation gives accurate results.)
By plugging our estimate of the timescale of Hebbian plasticity into the analytical expressions
for the parameters which characterize the critical limiting cases τdcrit and η̃ crit (Zenke et al., 2013), we
obtain a quantitative answer for how rapidly homeostatic mechanisms need to be. By analyzing the
stability of a low-activity background state in a recurrent network model in which synaptic weights
were evolving according to a plausible Hebbian plasticity model, we found that homeostatic control
mechanisms need to act on a timescale in the range of seconds to minutes (Zenke et al., 2013).
In the presence of network activity with temporal correlations or to provide robustness against
perturbations, the timescale of homeostasis must be even shorter.
14
More specifically, and most importantly for the topic of this paper, the time constant τd of the
low-pass filter in the rate-control mechanism of Eq. (9) has to be short; otherwise the induced phase
delay destabilizes the system. Possible choices for τd include τd → 0, which means that ȳ = y. In
other words, heterosynaptic plasticity needs to be nearly instantaneous to achieve its aim to control of
firing rates during ongoing Hebbian plasticity. Thus, if we want to achieve homeostasis by synaptic
scaling, the synaptic scaling has to be extremely fast (Zenke et al., 2013).
Finally, this requirement cannot be loosened by more complicated homeostatic control mechanisms which rely on complicated molecular signalling cascades. Homeostatic control mechanisms
which require the buildup of secondary or tertiary messengers, can be formalized in the language
of control theory as integral controllers. Integral controllers can pick up small deviations from a
target value and correct them. Therefore they are good for fine-tuning. However, they have to be
“slow” (or more precisely, slower than some critical value), otherwise they become unstable even in
the absence of Hebbian plasticity (Harnack et al., 2015).
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In summary, the application of our general analytical theory to data on STDP experiments gives
an answer to the second question posed in the introduction: Rapid homeostatic control implies a
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reaction of the controller on the timescale of seconds to minutes. This answer is in good agreement with most existing simulations of plastic network models (Fig. 2) — in each of these a rapid
homeostatic control mechanisms on a timescale of seconds to minutes was implemented to maintain
stability Pfister and Gerstner (2006); Lazar et al. (2009); El Boustani et al. (2012); Clopath and
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Gerstner (2010); Gjorgjieva et al. (2011); Litwin-Kumar and Doiron (2014).
What are functional consequences of rapid homeostatic control?
If we accept the notion that homeostatic plasticity acts on a short timescale, what are the functional
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consequences for plasticity and circuit dynamics? Neurons encode information in changes of their
electrical activity levels. For instance, subsets of simple cells in the visual system fire spikes in
response to specific edge-like features in the visual field (Hubel and Wiesel, 1962); cells in higher
brain areas respond with high specificity to complex concepts and remain quiescent when the concept
they are coding for is not brought to mind (Logothetis and Sheinberg, 1996; Hung et al., 2005;
Quiroga et al., 2005); and finally certain neurons respond selectively with elevated firing rates over
extended periods during working memory tasks (Miyashita, 1988; Funahashi et al., 1989; Wilson
et al., 1993). The ability of neurons to selectively indicate through periods of strong activity the
presence of specific features in the input or specific concepts in working memory is an important
condition for computation.
Homeostatic plasticity, however, by definition tries to keep neuronal activity constant. In particular, most models of homeostatic plasticity have a single homeostatic target or set point. That
means, that homeostatic plasticity tries to achieve some notion of constancy of one specific target
variable over time. In many models this is the mean neuronal firing rate, but other target variables
15
Input
a
b
Time
Slow hom., single target
c
T
Activity
Target
d
Limits
R
Activity
Rapid hom., target range
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Activity
Rapid hom., single target
D
Figure 3: The effect of slow and rapid homeostasis on the neuronal code. (a) Fluctuating
external world stimulus over time. (b) Neural activity A = w × input over time. A slow homeostatic
mechanism adjusts w to push A towards a single homeostatic target (dashed line; Homeostasis is
modelled as τslow dw
dt = (Target − A). The homeostatic target is indicated by the dashed line). With
a slow homeostatic mechanism, a neuron can track input relatively accurately. (c) Same as b, but for
a rapid homeostatic mechanism (τslow = 50τfast ). If the homeostatic timescale becomes comparable
to the timescale of the stimulus, homeostasis starts to interfere with the neuron’s ability to track
the stimulus. (d) Rapid homeostatic mechanisms enforcing an allowed range (limits indicated by
dotted lines). Even though the neuron cannot capture all the diversity of the input, it can capture
some of it. Here we modeled homeostasis as the following nonlinear extension of the above model:
τfast dw
dt = f (Target − A) with f (x) = x for kx − Targetk > Limit and f (x) = 0 otherwise.
16
such as the bulk-synaptic conductance of all afferent synapses onto the same neuron are possible.
Is homeostatic control of activity compatible with the task of neurons to selectively respond to
stimulation? If homeostatic control of firing rates is slow, neuronal firing can deviate substantially
from the mean firing rates during short times and thus encode information (Fig. 3a,b). However,
we already know that a slow homeostatic control mechanism cannot stabilize the ravaging effects of
Hebbian plasticity. So what can we say about a rapid homeostatic mechanism? If the homeostatic
mechanism acts on a short timescales (e.g. seconds to minutes) as it would be required to stabilize
Hebbian plasticity, neural codes based on neuronal activity become problematic (Fig. 3c) because
homeostatic plasticity starts to suppress activity fluctuations which might be encoding important
information. Even more alarmingly, certain forms of homeostatic plasticity, like sliding threshold
models of homeostatic metaplasticity, not only suppress high activity periods, but also “unlearn”
previously acquired selectivity and delete previously stored memories (Fig. 4a–d). Therefore rapid
homeostatic mechanisms which enforce a single homeostatic set point are not hardly desirable from
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a functional point of view. Thus the requirement of fast homeostatic control over Hebbian plasticity
poses a problem.
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It is important to appreciate that this problem arises from the combination of a single homeostatic
target with the requirement to enforce the homeostatic constraint on a short timescale. However,
there might be a simple solution to this conundrum. Suppose there are two (or more) homeostatic
set points, or a target range (instead of a target point), implemented by multiple forms of rapid
homeostatic control. For instance, one mechanism of homeostatic plasticity could activate above a
certain activity threshold and ensure that neuronal activity does not exceed this threshold. Similarly,
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a second mechanisms could activate below a low-activity threshold. The combined action of the two
mechanisms enforces neural activity to stay within an allowed range, but still permits substantial
firing rate fluctuations inside that range (Fig. 3d).
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When such a homeostatic mechanism is paired with a form of Hebbian plasticity which has
its plasticity threshold within the limits of the allowed activity regime, the neural activity of the
compound system naturally becomes bistable. Within the allowed range no homeostatic plasticity
is active, but Hebbian plasticity is intrinsically unstable. Thus any value within the homeostatically
allowed region will lead to either LTP or LTD until the system reaches the limits at which homeostatic
plasticity rapidly intervenes by undoing any excess LTP or LTD from there on. The compound
system exhibits therefore two stable equilibrium points one at low and one at high activity. If the
high-activity fixed point corresponds to a memory retrieval state, it becomes becomes irrelevant
whether the memory is recalled every other minute or if its only recalled once a year.
We can therefore answer the third of the questions raised in the introduction: The functional
consequences of rapid homeostatic control with a single set point are that neurons loose the flexibility
that is necessary for coding. The consequences are therefore undesirable. The proposed solution is
to design rapid homeostatic control mechanisms that allow for several set points (bistability) or a
target range of activity. In the next sections we will argue that the well-orchestrated combination of
17
Hebbian and non-Hebbian plasticity mechanisms naturally gives rise to such bistable dynamics. We
will explain why these dynamics are useful for learning and memory and why, despite their stability,
they still need to be accompanied by additional slow homeostatic mechanisms.
Can the combination of Hebbian and non-Hebbian plasticity lead to
stable memory formation?
We now discuss a new class of plasticity models which combines a plausible model of Hebbian
plasticity with two additional homeostatic plasticity mechanisms (Zenke et al., 2015). For sensible
combinations these compound models do not suffer from the run-away effects of purely Hebbian
plasticity and exhibit intrinsic bistability instead (cf. Fig. 3d).
The basic logic of bistable plasticity can be summarized as follows. At high activity levels
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a rapid form of heterosynaptic plasticity limits run-away LTP and creates synaptic competition.
Similarly, at low activity levels a unspecific form of plasticity which only depends on presynaptic
activity prevents run-away LTD. The well-orchestrated interplay between these adversarial plasticity
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mechanisms dynamically creates bistability of neuronal activity and prevents pathologic run-away
effects.
Our approach is general and any Hebbian plasticity model can be stabilized through the addition
of two non-Hebbian forms of plasticity. For illustration purposes, we will now focus on the triplet
STDP model for which biologically plausible sets of model parameters exist (Pfister and Gerstner,
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2006). To prevent run-away LTP in this model, we use a form of weight-dependent, multiplicative
heterosynaptic depression (Chen et al., 2013; Chistiakova et al., 2015; Zenke et al., 2015) which
has also been interpreted as a rapid form of depotentiation (Zhou et al., 2003). To prevent run-
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away LTD, we introduce hypothetical “transmitter-induced” plasticity which depends on presynaptic
activity only and slowly increases synaptic efficacy in the absence of postsynaptic spiking (Zenke
et al., 2015). These three plasticity mechanisms work in symphony to generate two stable levels of
neuronal activity.
Let us consider the weight wij from a presynaptic neuron j to a postsynaptic neuron i. Although
the full model is an STDP model, we now express its core ideas in terms of the presynaptic firing
rate xj and the postsynaptic rate yi
dwij
=
dt
δ · xj
+η · xj yi (yi − θ) −β · (wij − w̃ij ) yi4
| {z }
|
Transmitter induced
{z
Triplet model
}|
{z
Heterosynaptic
.
(10)
}
Here, δ and β are strength parameters for the two non-Hebbian components of the plasticity
model; η is the strength parameter (learning rate) for Hebbian plasticity; and w̃j serves as a reference
weight that can be related to consolidation dynamics (Zenke et al., 2015; Ziegler et al., 2015). Note
that, because the homeostatic mechanisms are “rapid”, no low-pass filtered variable ȳ appears in the
18
a
e
Active pathway
40×Poisson at 2Hz/10Hz
wact
Control pathway
400×Poisson constant at 2Hz
act
Neuron 1
ctl
Neuron 2
wctl
b
f
act
ctl
6
0
c
Triplet STDP
75
0
40
τd = 1h
τd = 10s
20
0
0.1
0.1
Neuron 2
T
50 100 150 200 250 300 350 400
0.2
0
0
50 100 150 200 250 300 350 400
AF
0
EPSP [a.u.]
h
6
4
2
0
Neuron 1
0
0
∆EPSP [1]
40
0.3
20
d
Neuron 2
g
EPSP [a.u.]
Output [Hz]
150
Neuron 1
Output [Hz]
Input [Hz]
12
Time [s]
Time [s]
D
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Figure 4: Intrinsically stabilized firing rates, synaptic weights and selectivity through
rapid homeostatic mechanisms. (a) Schematic figure of single neuron setup with two distinct
input pathways. The “active” pathway consists of 40 Poisson neurons firing at either 2 or 10Hz. The
control pathway consists of 400 neurons all firing constantly at 2Hz. All weights are initialized at the
same values. (b) Population firing rates of the input populations averaged over 2 s bins. Firing rates
in the active pathway (solid line) are temporally changed from 2Hz to 10Hz, whereas firing rates in
the control pathway are constant at 2Hz. (c) Output firing rates of a single postsynaptic neuron
with triplet STDP (Pfister and Gerstner, 2006) at all input synapses for two different time constants
τd of the homeostatic sliding threshold (see Zenke et al. (2013); κ=3Hz). Purple: Slow homeostatic
sliding threshold, τd = 1h; Green: Fast sliding threshold, τd = 10s. Top and bottom show the
same firing rate plot for different y-axis ranges. (d) Relative weight changes for 10 randomly chosen
weights from each pathway for the slow (purple) and the fast (green) sliding threshold. Darker
colors correspond to active pathway weights and lighter colors to the control pathway. (e) Simplified
sketch of the setup with two identical postsynaptic neurons which receive the same input as described
in a. All input synapses are plastic (see Zenke et al. (2015)). The only difference between the two
neurons in this setup are the different initial conditions. For Neuron 2 the active pathway weights
are initialized at a lower value than for Neuron 1 which could be thought of being the outcome of
previous heterosynaptic depression. (f) Output firing rates of the two neurons over time. Neuron 1
(black) responds selectively to the elevated inputs in the selective pathway (cf. b). Neuron 2 (orange)
does not respond with elevated firing rates during stimulation. (g) Evolution of weights over time
for Neuron 1. Active pathway weights are plotted in black and the control pathway in gray. For
Neuron 1, the synapses in the active pathway initially undergo LTP during 10Hz stimulation of
the active pathway synapses. However, weights quickly saturate. Synapses in the control pathway
undergo heterosynaptic depression. (h) Same as g, but for Neuron 2. In contrast to that, Neuron 2
continues to fire with low firing rates and barely responds to the 10Hz input in the active pathway.
The active pathway weights are slightly depressed during initial stimulation.
19
expression. Due to its rapid action and the high power, the heterosynaptic term in Eq. (10) acts
as a burst detector which dominates at high activity levels and prevents LTP run-away dynamics
(Fig. 4f).
For sensible choices of δ and β, neuronal firing rates remain in intermediate regimes (Fig. 4f) and
synaptic weights in the model converge towards stable weights ŵ whose values are dependent on the
activation history and allow the formation of long-term memories. Importantly, the model preserves
the plasticity threshold between LTD and LTP of the original triplet STDP model. The triplet model
together with the non-Hebbian plasticity mechanisms, dynamically creates two stable equilibrium
points. Overall, the synaptic weights converge rapidly towards one of two possible stable equilibrium
states (Fig. 4g–h). First, there is a “selective” equilibrium state associated with high postsynaptic
activity. In this state some weights are strong while other weights onto the same postsynaptic neuron
remain weak. Thus the neuron becomes selective to features in its input (neuron 1 in Fig. 4f and g).
Second, there is a low-activity fixed point without selectivity (Fig. 4f and h). Which fixed point a
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neuron converges to depends on initial conditions and the details of the activation pattern (Fig. 4g–
h). Once weights have converged to one of the respective stable states, weights keep fluctuating,
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but do not change on average. Moreover, since the homeostatic mechanism does not impose a single
set-point, activity patterns are not unlearned when a certain input is kept active for longer times
(compare Fig. 4d with 4g).
There are several aspects worth noting about the model. First, heterosynaptic plasticity does
not only stabilize Hebbian plasticity in the active pathway, it also introduces synaptic competition
between the active and the control pathway (Fig. 4g). Unlike BCM-like models in which heterosy-
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naptic depression of the inactive pathway depends on intermediate periods of background activity
in between stimuli (Cooper et al., 2004; Jedlicka et al., 2015), here the heterosynaptic depression
happens simultaneously to LTP induction (Fig. 4g). Second, although the learning rule effectively
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implements a rapid rescaling of the synaptic weights, it is still a fully local learning rule which only
relies on information which is locally available at the synapse (cf. Eq. (10)). Third, in general the
reference weight w̃ is not fixed, but follows its own temporal dynamics on a slower timescale (≈20min
and more). Such complex synaptic dynamics are essential to capture experiments on synaptic consolidation (Frey and Morris, 1997; Redondo and Morris, 2011; Ziegler et al., 2015). Consolidation
dynamics also play an important role in protecting memories from being overwritten by heterosynaptic plasticity. Finally, the stability properties of the learning rule Eq. (10) are not limited to
simple feed-forward circuits, but generalize to more realistic scenarios. Specifically, it enables stable
on-line learning and recall of cell assemblies in large spiking neural networks (Zenke et al., 2015).
In summary, orchestrating Hebbian and homeostatic plasticity on comparable timescales to dynamically create bistability, can reconcile the experimentally observed fast induction of synaptic
plasticity with stable synaptic dynamics and ensure stability of learning and memory at the single
neuron level. We now explore potential problems of intrinsically bistable plasticity?
20
Problems intrinsically bistable plasticity cannot solve
Consider a population of neurons with plastic synapses which follow intrinsically bistable plasticity
dynamics such as the ones described in the last section. To encode and process information efficiently,
neuronal populations need to create internal representations of the external world. Doing this
efficiently requires the response to be sparse across the population. In other words, only a subset of
neurons should respond for each stimulus. Moreover, different stimuli should evoke responses from
different subsets of neurons within the population to avoid that all stimuli look “the same” to the
neural circuit. Finally, individual neurons should respond sparsely over time. Imagine a neuron
which is active for all possible stimuli. It would be as uninformative as one which never responds to
any of the inputs. Therefore, to represent and process information in neural populations efficiently,
different neurons in the population have to develop selectivity to different features.
Intrinsically, bistable plasticity alone does not prevent neurons from responding at low firing rates
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to all stimuli (see, for example, Neuron 2 in Fig. 4f). Moreover, with similar initial conditions both
neuron 1 and 2 would have developed selectivity to the same input. Thus, in a large network where
all synapses are changed by the intrinsically bistable plasticity rule introduced above, all neurons
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could end up responding to the same feature. The question then arises, whether it is possible to
prevent such an undesired outcome.
To successfully implement network functions such as these examples, several network parameters
and properties of the learning rules themselves need to be tuned to and maintained in sensible
parameters regimes. To achieve this, additional forms of homeostatic plasticity need to be in place.
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However, due to the intrinsic stability of the learning rule, these additional homeostatic mechanisms
can now safely act on much larger timescales, similar to the ones observed in biology.
Thus we have answered the fourth question posed in the introduction. Yes, the combination
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of Hebbian and rapid homeostatic plasticity can lead to stable memory formation, if homeostasis
gives rise to intrinsic bistability. A combination of Hebbian plasticity with heterosynaptic LTD
and transmitter-induced LTP gives rise to suitable bistable dynamics. However, on its own, intrinsic
bistability of plasticity dynamics is not sufficient to induce complementary functionality across many
neurons in a network. The question then arises whether slower forms of homeostasis could come
into play in this context.
What is the role of slower homeostatic plasticity mechanisms?
Diverse homeostatic mechanisms exist in the brain at different temporal and spatial scales (Marder
and Goaillard, 2006; Turrigiano, 2012; Davis, 2013; Gjorgjieva et al., 2016). We have argued that
rapid homeostatic mechanisms are important for stability, but advantages do slow homeostatic
mechanisms have and what is their computational role?
An advantage of slow homeostatic processes is that they can integrate activity over long timescales
21
to achieve precise regulation of neural target set points (Turrigiano and Nelson, 2000; van Rossum
et al., 2000; Harnack et al., 2015). Longer integration times also allow to integrate signals from other
parts of a neural network which take time to be transmitted as diffusive factors (Turrigiano, 2012;
Sweeney et al., 2015). Slower homeostasis thus seems well suited for control problems which either
require fine-tuning or the more global homeostatic regulation of functions at the network level.
There are at least two important dynamical network properties which are not directly controllable
by the rapid homeostatic mechanisms in Eq. (10). First, temporal sparseness at the neuronal level.
A neuron that never responds to any stimulus, will never do so under bistable plasticity alone unless
the LTP threshold is lowered. Second, to ensure spatial sparseness at the network level lateral
inhibition has to decorrelate neuronal responses. As excitatory synapses change during learning, the
strength of lateral inhibition typically needs to be finely regulated.
The problem of temporal sparseness can be solved by any mechanism which ensures that a neuron
which has been completely silent for very long, eventually “gets a chance” to activate above the LTP
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threshold. This can be achieved by either lowering the threshold as in the BCM theory (Bienenstock
et al., 1982; Cooper et al., 2004; Zenke et al., 2015) or by slowly increasing the gain of either the
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neuron itself or the excitatory synapses through other forms of slow homeostatic plasticity (Daoudal
and Debanne, 2003; Turrigiano, 2011; Toyoizumi et al., 2014). Finally, similar homeostatic effects
could be achieved by decreasing inhibitory synaptic input through the action of neuron specific
inhibitory plasticity (Vogels et al., 2013).
Decorrelating neural activity and enforcing sparse activity at the population level has typically
been associated with lateral or recurrent inhibitory feedback in sparse learning paradigms (Olshausen
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and Field, 1996) and for models of associative memory (Willshaw et al., 1969; Little and Shaw, 1978;
Hopfield, 1982; Amit et al., 1985; Amit and Brunel, 1997; Litwin-Kumar and Doiron, 2014; Zenke
et al., 2015). However, to achieve sensible global network activity levels through recurrent inhibitory
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feedback requires fine-tuning (Amit and Brunel, 1997; Brunel, 2000).
Inhibitory synaptic plasticity (ISP) can achieve fine-tuning and homeostatic functions in spiking
neural networks. Specifically, Hebbian forms of ISP, for which synchronous pre- and postsynaptic
activation leads to a strengthening of inhibitory synapses, has been shown to be a powerful mechanism to regulate neural firing rates, to establish and maintain excitatory-inhibitory balance and to
decorrelate neuronal firing in neural networks (Vogels et al., 2011; Zenke, 2014; Litwin-Kumar and
Doiron, 2014). Existing Hebbian ISP models, however, increase inhibition between neurons which
are frequently active together. This is useful to create excitatory inhibitory balance and prevent
individual neurons from being synchronized. However, if neurons respond repeatedly together as
part of the same cell assembly, these neurons will start to inhibit each other. Associative memory
models, however, require the opposite situation: neurons in the same assembly should not inhibit
each other while neurons of different cell assemblies inhibit each other so as to avoid that many cell
assemblies are activated simultaneously. To achieve the latter, inhibition needs to be tuned globally.
Biological networks can achieve this type of global tuning through slow forms of homeostatic
22
plasticity (Turrigiano, 2012) which rely on diffusive signals (Sweeney et al., 2015). These signals
could either globally scale synaptic strength at inhibitory synapses directly, or indirectly influence
the levels of inhibition by modulating inhibitory learning rules through a third factor (Zenke et al.,
2015; Frémaux and Gerstner, 2016).
It is important to realize that not all homeostatic mechanisms have to be permanently active.
For instance, once the inhibitory feedback within a spiking neural network model is tuned to the
“sweet spot” at which the network can operate, moving threshold mechanisms and ISP can safely
be turned off while the network can still learn new stimuli (Zenke et al., 2015). It thus seems likely
that in nature certain forms of homeostatic plasticity could be dormant for most of the time and
spring into action only to prepare a network during the initial phase of development (Turrigiano and
Nelson, 2004) or when an extreme external manipulation changes the network dynamics (Turrigiano
et al., 1998).
Thus we are able to answer the final question as follows: Slow homeostatic mechanisms tune pa-
T
rameters of plasticity rules and neurons to enable efficient use of the available resources in networks.
For example, for the sake of efficiency, no neuron should be never active; no neuron should be always
AF
active; the number of neurons that respond to the exact same set of stimuli should stay limited.
Taken together with the results from the previous sections, these insights suggest two distinct
roles of homeostatic plasticity mechanisms on different timescales. First, homeostatic plasticity on
short timescales stabilizes Hebbian plasticity and makes synapses onto the same neuron compete with
each other. Heterosynaptic plasticity is likely to play a major role for these functionalities. Second,
homeostatic mechanisms on slower timescales achieve fine-tuning of multiple network parameters.
R
A slow shift of the threshold between LTD and LTP, the slow rescaling of all synaptic weights, or
a slow regulation of neuronal parameters are good candidates for these functionalities. Some of
these slow mechanisms could be important only in setting up the network initially or after a strong
D
external perturbation to the circuit.
Discussion
We discuss the relation of the above modeling approach to other models and to experimental data.
Other intrinsically bi-stable plasticity models
The notion of intrinsic stability (or bistability) has come up recently in other studies. Toyoizumi
et al. (2014) proposed the following plasticity rule which formally fulfills the requirements of an
intrinsically stable rule:
τw
dw
= [wmax − w]+ [xy − θ]+ − [w − wmin ]+ [θ − xy]+
dt
|
{z
} |
{z
}
LTP
LTD
23
(11)
The full rule is multiplicatively coupled with a slow homeostatic process which we have omitted in
the above equation. However, even in the absence of any additional form of homeostatic plasticity,
individual synaptic weights saturate at the soft bounds wmin and wmax respectively. Similarly, to
Eq. (10) the model possesses a plasticity threshold θ. However, here, the threshold is defined via the
product of pre- and postsynaptic activity, similar to (El Boustani et al., 2012), but in contrast to
most data-driven models in which the threshold is defined on postsynaptic quantities only (Artola
and Singer, 1993; Shouval et al., 2002; Pfister and Gerstner, 2006; Clopath and Gerstner, 2010;
Graupner and Brunel, 2012).
Another example of rapid stabilization of an otherwise unstable learning rule was devised by
Lim et al. (2015). Using an analytic framework the authors infer the following rate-based plasticity model directly for experimental data on excitatory-to-excitatory connections in infero-temporal
cortex (ITC):
T
dw
∝ x (y − θ) .
dt
Because this model is unstable and leads to responses in a recurrent network model which are
much smaller than their ITC measurements, the authors added a rapid homeostatic mechanism
AF
which maintains the average synaptic strength onto a neuron. This can be formalized as:
dw
∝ (x − x̄) (y − θ) .
dt
(12)
It is interesting to note that in this model, weights grow with the rate x̄θ in the absence of
R
activity to oppose run-away LTD which is conceptionally similar to Eq. (10). However, in contrast
to Eq. (10), in the present model, run-away LTP is counterbalanced at the network level through
recurrent inhibitory feedback which pushes the activity y of most neurons below the plasticity
D
threshold θ causing them to undergo LTD. This suggests that under some circumstances the rapid
modulation of excitatory plasticity through inhibitory feedback might be enough to avoid run-away
dynamics.
While there are several different ways to stabilize Hebbian plasticity, a plethora of theoretical
models seems to conceptually agree on two key factors: First, to achieve stability against run-away
LTP a model needs to ensure that negative feedback is rapid (Pfister and Gerstner, 2006; Lazar et al.,
2009; El Boustani et al., 2012; Clopath and Gerstner, 2010; Gjorgjieva et al., 2011; Litwin-Kumar
and Doiron, 2014). And second, the feedback needs to be strong enough to be able to compensate
for the leading power of Hebbian plasticity (Oja, 1982; Bienenstock et al., 1982; Cooper et al., 2004;
Tetzlaff et al., 2012; Zenke et al., 2013; Toyoizumi et al., 2014).
Different models capture the same plasticity experiments
A plausible plasticity model has to reproduce stable learning dynamics in simulations and it has
to capture a plethora of experimental data from rate-dependent (Bliss and Lomo, 1973), voltage-
24
dependent (Artola et al., 1990) and spike-timing-dependent (Markram et al., 1997; Bi and Poo,
1998; Sjöström et al., 2001) plasticity experiments. One salient feature captured by most models
(Artola and Singer, 1993; Shouval et al., 2002; Pfister and Gerstner, 2006; Clopath and Gerstner,
2010; Graupner and Brunel, 2012) is the notion of a plasticity threshold which correlates with
postsynaptic voltage, calcium concentration or other neuronal variables related to postsynaptic
activation (Fig. 1a). However, most existing models are purely Hebbian and do not include the
notion of a rapid homeostatic mechanism. If rapid homeostatic mechanisms exist — which is what
we argue here — then how can it be that existing plasticity models without them can quantitatively
capture the data from experiments?
There are presumably three main reason for this. First, STDP experiments typically manipulate
a single pathway, either by stimulating a presynaptic neuron are a bundle of presynaptic axons.
Sometimes a designated control pathway (i.e. a second presynaptic neuron) is missing, or, if it is not
missing, the effect size in the control pathway is considered as weak. However, from a theoretical
T
perspective, we expect that heterosynaptic effects caused by stimulation of one presynaptic pathway
are weak when it is measured at one ’control’ synapses. Even if it is weak on a per-synapses bases,
AF
its net effect is strong because it spreads out over thousands of synapses. Therefore even weak
heterosynaptic plasticity at other synapses could implement a form of rapid homeostatic control
(Chen et al., 2013; Chistiakova et al., 2015; Zenke et al., 2015).
Second, in an STDP experiments with 60 repetitions of pre-post-pairs, the total activation of
the postsynaptic neuron is still in a reasonable regime. Therefore it is unclear whether the ’burstdetector’ for heterosynaptic plasticity would would be triggered (Chen et al., 2013; Chistiakova et al.,
R
2015; Zenke et al., 2015).
Third, experiments typically rely on repeated pre- and postsynaptic activation. Moreover, during
the induction protocol, synaptic efficacy changes are usually not observable. Plasticity models are
D
thus fitted to pairs of initial and final synaptic strength. However, the unobserved intermediate
synaptic dynamics could be qualitatively very different (Fig. 1c). These differences in the dynamics
contain the answers to questions such as: Is the final synaptic strength stable or would it increase
further with additional pairings? Is there a threshold of number of pairings that needs to be reached
for an all or nothing effect?
Because the detailed dynamics of synapses during induction are not known, different plasticity
models make different assumptions about the saturation of weights. Based on these assumptions,
most existing plasticity can be coarsely classified into additive and multiplicative models. Additive
plasticity models, for instance, assume a linear growth for plasticity induction (Fig. 1c; Pfister and
Gerstner (2006); Clopath et al. (2010); Song et al. (2000)). In the absence of a hard upper bound,
which is usually present in additive models, the weight would simply keep growing. Although
additive models capture plasticity experiments for a fixed number of pairings, in simulated networks
with ongoing pre- and postsynaptic activity, synaptic weights diverge and “get stuck” at the upper
and lower bounds (Billings and van Rossum, 2009).
25
The equations of multiplicative models on the other hand depend on the synaptic weight explicitly which typically makes them converge to stable intermediate weight values (Gütig et al., 2003;
Morrison et al., 2007; Gilson et al., 2010) and for ongoing spiking activity in network simulations
gives rise to unimodal weight distributions (Morrison et al., 2007).
Both multiplicative and additive models generally do not lead to stability in network models
(Morrison et al., 2007; Kunkel et al., 2011; Zenke et al., 2013) unless accompanied by additional
homeostatic mechanisms. However, these mechanisms are typically added ad-hoc and they are not
fitted to experimental data (e.g. Pfister and Gerstner (2006); Clopath et al. (2010)).
Due to the limited amount of experimental data, it is possible to construct a set of different
models which are all consistent with experimental data. The model discussed in this paper is based
on the triplet STDP model (and is therefore consistent with existing STDP data), but includes
additional forms of activity-dependent non-Hebbian plasticity which give rise to intrinsic bistability
T
(and which are harder to measure experimentally).
Can we constrain plasticity models by experiments?
AF
There are multiple ways in which synaptic plasticity models could be constrained better through
additional data. In the past, a large body of research has focused on homosynaptic associative
plasticity, also called Hebbian plasticity, using pairing experiments with various protocols such as
STDP. Here, we argue that heterosynaptic plasticity as well as transmitter-induced plasticity might
be almost as important as Hebbian plasticity due to their possibly crucial role for network stability.
R
Heterosynaptic plasticity is a promising candidate to stabilize Hebbian plasticity models against
run-away LTP is heterosynaptic plasticity (Chen et al., 2013; Zenke et al., 2013, 2015; Chistiakova
et al., 2015; Jedlicka et al., 2015). While heterosynaptic plasticity has been observed in various
D
experiments (Royer and Paré, 2003; Chistiakova et al., 2014), a conclusive picture and quantitative models are still missing. Is it possible to measure the timescale, frequency dependence, and
weight-dependence of neuron wide heterosynaptic depression by manipulating the stimulation of the
postsynaptic neuron? Another important question for the interpretation of heterosynaptic plasticity
is whether heterosynaptic plasticity induces mostly synaptic depression similar or related to LTD
or rather resets or prevents early LTP trough depotentiation at the unstimulated pathway (Zhou
et al., 2003).
Transmitter-induced plasticity is important in models and might be present in many experiments,
even though it has not been reported as such. Here transmitter-induced plasticity refers to potentially
weak long-term potentiation that is caused by presynaptic firing in the absence of postsynaptic
activity. Why is this form of plasticity potentially important? Suppose you have a network of neurons
firing at low activity, so that any given neuron can be considered a weakly active postsynaptic neuron.
Since low activity typically induces LTD, many plastic network simulations have the tendency to fall
silent. To compensate for this theorists introduce either lower bounds on synaptic weights or add
26
weak LTP triggered by presynaptic activity (Zenke et al., 2015; Lim et al., 2015). How realistic are
these assumptions? Direct experimental evidence for such terms would for instance be the growth
of synaptic efficacy during low activity “pre only” stimulation. It is possible that such a term would
look like, and could easily be mistaken for, an unstable baseline (with positive drift) before the
beginning of a plasticity induction experiment. Given the importance of such a term, even if very
weak, from the theoretical perspective, it is worth explicit studies.
Consolidation of synapses is summarized in the present model by a reference weight w̃ (Zenke
et al., 2015; Ziegler et al., 2015). Simulations predict that synaptic consolidation renders synapses
inert against heterosynaptic plasticity. Intuitively, the measured synaptic weights become sticky
and are always attracted back to their momentary stable state, i.e. weak or strong. This prediction
requires future experimental clarification.
The path towards saturation of synaptic weights during a pairing experiment (Fig. 1c) is vital
to building better plasticity models. Virtually any information which helps theorists to constrain
T
how the synaptic weight increases would be helpful. Importantly, this also includes any information
about conditions (or experimental protocols) which do not induce plasticity, despite the fact that
Conclusion
AF
either the presynaptic or the postsynaptic neuron or both have been activated.
One of the most striking differences between plasticity models and experimental data concerns the
timescale. Hebbian plasticity can be induced within seconds to minutes (Bliss and Lomo, 1973;
R
Artola et al., 1990; Markram et al., 1997; Bi and Poo, 1998). In simulated network models such
a fast Hebbian plasticity leads to run-away activity within seconds unless Hebbian plasticity is
complemented with rapid forms of homeostatic plasticity. Here, “rapid” means that homeostatic
D
changes need to take affect on the timescale of seconds or at most a few minutes (Zenke et al., 2013).
This, however, is much faster than most forms of homeostatic plasticity observed in experiments. One
of the most extensively studied forms of homeostasis in experiments is synaptic scaling (Turrigiano
et al., 1998) which proportionally scales up (down) synapses if the network activity is too low (high).
However, even the fastest known forms of scaling take hours to days to cause measurable changes
to synaptic weights (Fig. 2; Turrigiano and Nelson (2000); Ibata et al. (2008); Aoto et al. (2008)).
This apparent difference of timescales between homeostasis required for stability in models and
experimental results is a challenge for current theories (Wu and Yamaguchi, 2006; Chen et al., 2013;
Zenke et al., 2013; Toyoizumi et al., 2014). To reconcile plasticity models and stability in networks
of simulated neurons, we need to reconsider models of Hebbian plasticity and how they are fitted to
data.
In most plasticity induction experiments neither the time course of the manipulated synaptic
weight nor changes of other synapses are observed during stimulation. Quantitative models of
synaptic plasticity thus make minimal assumptions about these unobserved temporal dynamics and
27
generally ignore heterosynaptic effects entirely. In other words, missing experimental data makes it
possible to build different models which all capture the existing experimental data, but make different
assumptions about the unobserved dynamics. Importantly, some of these models become intrinsically
stable (Oja, 1982; Miller and MacKay, 1994; Chen et al., 2013) or even bistable (Toyoizumi et al.,
2014; Zenke et al., 2015). In most situations these models can be interpreted as compound models
consisting of Hebbian plasticity and forms of rapid homeostatic plasticity. Importantly, all of the
plasticity models cited in this paragraph only rely on quantities which are locally known to the
synapse; i.e. the pre- postsynaptic activity as well as its own synaptic weight. Although such local
forms of plasticity can solve the problem of stability at a neuronal level, in practice, most network
models require additional fine-tuning of parameters to achieve plausible activity levels across a
network of neurons. This role can be fulfilled slow homeostatic mechanisms which act on timescales
of hours or days, consistent with experimental data on homeostatic plasticity.
In summary, based on theoretical arguments we suggest that Hebbian plasticity is intrinsically
T
stabilized on extremely short timescales by fast homeostatic control, likely to be implemented by
heterosynaptic plasticity while slow forms of homeostatic plasticity set the stage for stable learning.
AF
However, this hypothesis will now have to stand the test of time. It will thus be an important
challenge for the next years to go beyond homosynaptic Hebbian plasticity and to gain a more
complete understanding of the diverse interactions of Hebbian and homeostatic plasticity across
timescales.
Acknowledgements. WG was supported for this work by the European Research Council
under grant agreement number 268689 (MultiRules) and by the European Community’s Seventh
R
Framework Program under grant no. 604102 (Human Brain Project). FZ was supported by the
D
SNSF (Swiss National Science Foundation).
References
Larry F Abbott, J A Varela, K Sen, and S B Nelson. Synaptic depression and cortical gain control.
Science, 275(5297):220–224, January 1997.
W. C. Abraham and G. V. Goddard. Asymmetric relationships between homosynaptic long-term
potentiation and heterosynaptic long-term depression. Nature, 305(5936):717–719, October 1983.
Wickliffe C. Abraham. Metaplasticity: Tuning synapses and networks for plasticity. Nat Rev Neurosci, 9(5):387–387, May 2008.
D. J. Amit. Modeling brain function. Cambridge University Press, Cambridge UK, 1989.
D. J. Amit and N. Brunel. A model of spontaneous activity and local delay activity during delay
periods in the cerebral cortex. Cerebral Cortex, 7:237–252, 1997.
28
Daniel J. Amit, Hanoch Gutfreund, and H. Sompolinsky. Storing Infinite Numbers of Patterns in a
Spin-Glass Model of Neural Networks. Phys Rev Lett, 55(14):1530–1533, September 1985.
Jason Aoto, Christine I. Nam, Michael M. Poon, Pamela Ting, and Lu Chen. Synaptic Signaling
by All-Trans Retinoic Acid in Homeostatic Synaptic Plasticity. Neuron, 60(2):308–320, October
2008.
A Artola and W Singer. Long-term depression of excitatory synaptic transmission and its relationship
to long-term potentiation. Trends Neurosci, 16(11):480–487, November 1993.
A. Artola, S. Bröcher, and W. Singer. Different voltage-dependent thresholds for inducing longterm depression and long-term potentiation in slices of rat visual cortex. Nature, 347(6288):
69–72, September 1990.
T
Craig H. Bailey, Maurizio Giustetto, Yan-You Huang, Robert D. Hawkins, and Eric R. Kandel.
Is Heterosynaptic modulation essential for stabilizing hebbian plasiticity and memory. Nat Rev
AF
Neurosci, 1(1):11–20, October 2000.
Guo-Qiang Bi and Mu-Ming Poo. Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type. J Neurosci, 18(24):
10464 –10472, December 1998.
EL Bienenstock, LN Cooper, and PW Munro. Theory for the development of neuron selectivity:
1982.
R
orientation specificity and binocular interaction in visual cortex. J Neurosci, 2(1):32–48, January
Guy Billings and Mark C. W. van Rossum. Memory Retention and Spike-Timing-Dependent Plas-
D
ticity. J Neurophysiol, 101(6):2775 –2788, June 2009.
T. V. P. Bliss and T. Lomo. Long-lasting potentation of synaptic transmission in the dendate area of
anaesthetized rabbit following stimulation of the perforant path. J. Physiol., 232:351–356, 1973.
Carlos S. N. Brito and Wulfram Gerstner. Nonlinear Hebbian learning as a unifying principle in
receptive field formation. arXiv:1601.00701 [cs, q-bio], January 2016.
T. H. Brown, A. M. Zador, Z. F. Mainen, and B. J. Claiborne. Hebbian modifications in hippocampal
neurons. In M. Baudry and J. L. Davis, editors, Long–term potentiation., pages 357–389. MIT
Press, Cambridge, London, 1991.
Thomas H. Brown and Sumantra Chattarji. Hebbian Synaptic Plasticity: Evolution of the Contemporary Concept. In Professor Eytan Domany, Professor Dr J. Leo van Hemmen, and Professor Klaus Schulten, editors, Models of Neural Networks, Physics of Neural Networks, pages
287–314. Springer New York, 1994.
29
Nicolas Brunel. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci, 8(3):183–208, May 2000.
Jen-Yung Chen, Peter Lonjers, Christopher Lee, Marina Chistiakova, Maxim Volgushev, and Maxim
Bazhenov. Heterosynaptic Plasticity Prevents Runaway Synaptic Dynamics. J Neurosci, 33(40):
15915–15929, October 2013.
Marina Chistiakova, Nicholas M. Bannon, Maxim Bazhenov, and Maxim Volgushev. Heterosynaptic
Plasticity Multiple Mechanisms and Multiple Roles. Neuroscientist, 20(5):483–498, October 2014.
Marina Chistiakova, Nicholas M. Bannon, Jen-Yung Chen, Maxim Bazhenov, and Maxim Volgushev.
Homeostatic role of heterosynaptic plasticity: models and experiments. Front Comput Neurosci,
9:89, 2015.
T
Brian R. Christie and Wickliffe C. Abraham. Priming of associative long-term depression in the
dentate gyrus by theta frequency synaptic activity. Neuron, 9(1):79–84, July 1992.
AF
C. Clopath, L. Busing, E. Vasilaki, and W. Gerstner. Connectivity reflects coding: A model of
voltage-based spike-timing-dependent-plasticity with homeostasis. Nature Neuroscience, 13:344–
352, 2010.
Claudia Clopath and Wulfram Gerstner. Voltage and spike timing interact in STDP – a unified
model. Front Synaptic Neurosci, 2:25, 2010.
R
Leon N Cooper, Nathan Intrator, Brian S Blais, and Harel Z Shouval. Theory of Cortical Plasticity.
World Scientific, New Jersey, April 2004.
D
Gaël Daoudal and Dominique Debanne. Long-Term Plasticity of Intrinsic Excitability: Learning
Rules and Mechanisms. Learn Mem, 10(6):456–465, November 2003.
Graeme W. Davis. Homeostatic Signaling and the Stabilization of Neural Function. Neuron, 80(3):
718–728, October 2013.
S M Dudek and M F Bear. Homosynaptic Long-Term Depression in Area CA1 of Hippocampus and
Effects of N-Methyl-D-Aspartate Receptor Blockade. PNAS, 89(10):4363–4367, May 1992.
Sami El Boustani, Pierre Yger, Yves Frégnac, and Alain Destexhe. Stable Learning in Stochastic
Network States. J Neurosci, 32(1):194 –214, January 2012.
Uwe Frey and Richard G. M. Morris. Synaptic tagging and long-term potentiation. Nature, 385
(6616):533–6, March 1997.
Uwe Frey and Richard G. M. Morris. Synaptic tagging: implications for late maintenance of hippocampal long-term potentiation. Trends in Neurosciences, 21(5):181–188, May 1998.
30
Robert C Froemke and Yang Dan. Spike-timing-dependent synaptic modification induced by natural
spike trains. Nature, 416(6879):433–8, March 2002.
Nicolas Frémaux and Wulfram Gerstner. Neuromodulated Spike-Timing-Dependent Plasticity, and
Theory of Three-Factor Learning Rules. Front. Neural Circuits, page 85, 2016.
S. Funahashi, C. J Bruce, and P. S Goldman-Rakic. Mnemonic Coding of Visual Space in the
Monkey’s Dorsolateral Prefrontal Cortex. J Neurophysiol, 61(2):331–349, February 1989.
Wulfram Gerstner, Werner M Kistler, Richard Naud, and Liam Paninski. Neuronal dynamics: from
single neurons to networks and models of cognition. Cambridge University Press, Cambridge,
2014.
Matthieu Gilson, Anthony Burkitt, David Grayden, Doreen Thomas, and J. van Hemmen. RepreE, 82(2):1–11, August 2010.
T
sentation of input structure in synaptic weights by spike-timing-dependent plasticity. Phys Rev
AF
Julijana Gjorgjieva, Claudia Clopath, Juliette Audet, and Jean-Pascal Pfister. A triplet spiketiming–dependent plasticity model generalizes the Bienenstock–Cooper–Munro rule to higherorder spatiotemporal correlations. Proc Natl Acad Sci U S A, 108(48):19383 –19388, November
2011.
Julijana Gjorgjieva, Guillaume Drion, and Eve Marder. Computational implications of biophysical
R
diversity and multiple timescales in neurons and synapses for circuit performance. Current Opinion
in Neurobiology, 37:44–52, April 2016.
Anubhuthi Goel, Bin Jiang, Linda W. Xu, Lihua Song, Alfredo Kirkwood, and Hey-Kyoung Lee.
D
Cross-modal regulation of synaptic AMPA receptors in primary sensory cortices by visual experience. Nat Neurosci, 9(8):1001–1003, August 2006.
Michael Graupner and Nicolas Brunel. Calcium-based plasticity model explains sensitivity of synaptic changes to spike pattern, rate, and dendritic location. Proc Natl Acad Sci U S A, 109(10):
3991–3996, March 2012.
R. Gütig, R. Aharonov, S. Rotter, and Haim Sompolinsky. Learning Input Correlations through
Nonlinear Temporally Asymmetric Hebbian Plasticity. J Neurosci, 23(9):3697–3714, May 2003.
Daniel Harnack, Miha Pelko, Antoine Chaillet, Yacine Chitour, and Mark C.W. van Rossum. Stability of Neuronal Networks with Homeostatic Regulation. PLoS Comput Biol, 11(7):e1004357,
July 2015.
D. O. Hebb. The Organization of Behavior. Wiley, New York, 1949.
31
John J. Hopfield. Neural networks and physical systems with emergent collective computational
abilities. Proc Natl Acad Sci U S A, 79(8):2554, 1982.
Y Y Huang, A Colino, D K Selig, and R C Malenka. The influence of prior synaptic activity on the
induction of long-term potentiation. Science, 255(5045):730–733, February 1992.
D. H. Hubel and T. N. Wiesel. Receptive fields, binocular interaction and functional architecture in
the cat’s visual cortex. J Physiol, 160(1):106–154.2, January 1962.
C. P. Hung, G. Kreiman, T. Poggio, and J. J. DiCarlo. Fast Readout of Object Identity from
Macaque Inferior Temporal Cortex. Science, 310:863 – 866, 2005.
Keiji Ibata, Qian Sun, and Gina G. Turrigiano. Rapid Synaptic Scaling Induced by Changes in
Postsynaptic Firing. Neuron, 57(6):819–826, March 2008.
T
Peter Jedlicka, Lubica Benuskova, and Wickliffe C. Abraham. A Voltage-Based STDP Rule Combined with Fast BCM-Like Metaplasticity Accounts for LTP and Concurrent “Heterosynaptic”
AF
LTD in the Dentate Gyrus In Vivo. PLoS Comput Biol, 11(11):e1004588, November 2015.
Richard Kempter, Wulfram Gerstner, and J. van Hemmen. Hebbian learning and spiking neurons.
Phys Rev E, 59(4):4498–4514, April 1999.
Susanne Kunkel, Markus Diesmann, and Abigail Morrison. Limits to the development of feed-forward
R
structures in large recurrent neuronal networks. Front Comput Neurosci, 4:160, 2011.
Andreea Lazar, Gordon Pipa, and Jochen Triesch. SORN: A Self-Organizing Recurrent Neural
Network. Front Comput Neurosci, 3, October 2009.
D
W. B. Levy and O. Steward. Temporal contiguity requirements for long-term associative potentiation/depression in the hippocampus. Neuroscience, 8(4):791–797, April 1983.
Sukbin Lim, Jillian L. McKee, Luke Woloszyn, Yali Amit, David J. Freedman, David L. Sheinberg,
and Nicolas Brunel. Inferring learning rules from distributions of firing rates in cortical neurons.
Nat Neurosci, advance online publication, November 2015.
John Lisman. Long-term potentiation: outstanding questions and attempted synthesis. Phil Trans
R Soc Lond B, 358(1432):829–842, April 2003.
W. A. Little and G. L. Shaw. Analytical study of the memory storage capacity of a neural network.
Math. Biosc., 39:281–290, 1978.
Ashok Litwin-Kumar and Brent Doiron. Formation and maintenance of neuronal assemblies through
synaptic plasticity. Nat Commun, 5, November 2014.
32
Nikos K Logothetis and David L Sheinberg. Visual Object Recognition. Annual Review of Neuroscience, 19(1):577–621, 1996.
G. S. Lynch, T. Dunwiddie, and V. Gribkoff. Heterosynaptic depression: a postsynaptic correlate
of long-term potentiation. Nature, 266(5604):737–739, April 1977.
Robert C. Malenka and and Roger A. Nicoll. Long-Term Potentiation–A Decade of Progress?
Science, 285(5435):1870–1874, September 1999.
Eve Marder and Jean-Marc Goaillard. Variability, compensation and homeostasis in neuron and
network function. Nat Rev Neurosci, 7(7):563–574, July 2006.
H. Markram, J. L\protectübke, M. Frotscher, and B. Sakmann. Regulation of synaptic efficacy by
coincidence of postysnaptic AP and EPSP. Science, 275:213–215, 1997.
T
S. J. Martin, P. D. Grimwood, and R. G. M. Morris. Synaptic Plasticity and Memory: An Evaluation
of the Hypothesis. Annu Rev Neurosci, 23(1):649–711, 2000.
AF
K. D. Miller and D. J. C. MacKay. The role of constraints in \protect Hebbian learning. Neural
Computation, 6:100–126, 1994.
Yasushi Miyashita. Neural correlate of visual associative long-term memory in the primate temporal
cortex. Nature, 335(27):817–820, October 1988.
R
R. G. M. Morris, E. Anderson, G. S. Lynch, and M. Baudry. Selective impairment of learning and
blockade of long-term potentiation by an N-methyl-D-aspartate receptor antagonist, AP5. Nature,
319(6056):774–776, February 1986.
D
Abigail Morrison, Ad Aertsen, and Markus Diesmann. Spike-timing-dependent plasticity in balanced
random networks. Neural Comput, 19(6):1437–67, June 2007.
Abigail Morrison, Markus Diesmann, and Wulfram Gerstner. Phenomenological models of synaptic
plasticity based on spike timing. Biol Cybern, 98(6):459–478, June 2008.
Daniel H. O’Connor, Gayle M. Wittenberg, and Samuel S.-H. Wang. Graded bidirectional synaptic
plasticity is composed of switch-like unitary events. PNAS, 102(27):9679–9684, July 2005.
Erkki Oja. Simplified neuron model as a principal component analyzer. J Math Biol, 15(3):267–273,
1982.
Bruno A. Olshausen and David J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. , Published online: 13 June 1996; | doi:10.1038/381607a0,
381(6583):607–609, June 1996.
33
Carl C. H. Petersen, Robert C. Malenka, Roger A. Nicoll, and John J. Hopfield. All-or-none potentiation at CA3-CA1 synapses. PNAS, 95(8):4732–4737, April 1998.
Jean-Pascal Pfister and Wulfram Gerstner. Triplets of Spikes in a Model of Spike Timing-Dependent
Plasticity. J Neurosci, 26(38):9673–9682, September 2006.
R. Quian Quiroga, L. Reddy, G. Kreiman, C. Koch, and I. Fried. Invariant visual representation by
single neurons in the human brain. Nature, 435(7045):1102–1107, June 2005.
Roger L. Redondo and Richard G. M. Morris. Making memories last: the synaptic tagging and
capture hypothesis. Nat Rev Neurosci, 12(1):17–30, January 2011.
Alfonso Renart, Jaime de la Rocha, Peter Bartho, Liad Hollender, Néstor Parga, Alex Reyes, and
Kenneth D. Harris. The Asynchronous State in Cortical Circuits. Science, 327(5965):587 –590,
T
January 2010.
Klaus G. Reymann and Julietta U. Frey. The late maintenance of hippocampal LTP: Requirements,
AF
phases, ‘synaptic tagging’, ‘late-associativity’ and implications. Neuropharmacology, 52(1):24–40,
January 2007.
N. Rochester, J. Holland, L. Haibt, and W. Duda. Tests on a cell assembly theory of the action of
the brain, using a large digital computer. IEEE Trans Inf Theory, 2(3):80–93, September 1956.
Sébastien Royer and Denis Paré. Conservation of total synaptic weight through balanced synaptic
R
depression and potentiation. Nature, 422(6931):518–522, April 2003.
Walter Senn, Henry Markram, and Misha Tsodyks. An Algorithm for Modifying Neurotransmitter
D
Release Probability Based on Pre- and Postsynaptic Spike Timing. Neural Comput, 13(1):35–67,
January 2001.
Harel Z. Shouval, Mark F. Bear, and Leon N. Cooper. A Unified Model of NMDA ReceptorDependent Bidirectional Synaptic Plasticity. Proc Natl Acad Sci U S A, 99(16):10831–10836,
August 2002.
Per Jesper Sjöström, Gina G Turrigiano, and Sacha B Nelson. Rate, Timing, and Cooperativity
Jointly Determine Cortical Synaptic Plasticity. Neuron, 32(6):1149–1164, December 2001.
Sen Song, Kenneth D. Miller, and L. F. Abbott. Competitive Hebbian learning through spiketiming-dependent synaptic plasticity. Nat Neurosci, 3(9):919–926, September 2000.
Yann Sweeney, Jeanette Hellgren Kotaleski, and Matthias H. Hennig. A Diffusive Homeostatic Signal
Maintains Neural Heterogeneity and Responsiveness in Cortical Networks. PLoS Comput Biol, 11
(7):e1004389, July 2015.
34
Christian Tetzlaff, Christoph Kolodziejski, Marc Timme, and Florentin Wörgötter. Analysis of
synaptic scaling in combination with Hebbian plasticity in several simple networks. Front Comput
Neurosci, 6:36, 2012.
Taro Toyoizumi, Megumi Kaneko, Michael P. Stryker, and Kenneth D. Miller. Modeling the Dynamic
Interaction of Hebbian and Homeostatic Plasticity. Neuron, 84(2):497–510, October 2014.
Misha Tsodyks, Klaus Pawelzik, and Henry Markram. Neural Networks with Dynamic Synapses.
Neural Comput, 10(4):821–835, 1998.
Misha V. Tsodyks and Henry Markram. The neural code between neocortical pyramidal neurons
depends on neurotransmitter release probability. PNAS, 94(2):719–723, January 1997.
Gina Turrigiano. Too Many Cooks? Intrinsic and Synaptic Homeostatic Mechanisms in Cortical
T
Circuit Refinement. Annu Rev Neurosci, 34(1):89–103, 2011.
Gina G. Turrigiano. Homeostatic Synaptic Plasticity: Local and Global Mechanisms for Stabilizing
AF
Neuronal Function. Cold Spring Harb Perspect Biol, 4(1):a005736, January 2012.
Gina G Turrigiano and Sacha B Nelson. Hebb and homeostasis in neuronal plasticity. Current
Opinion in Neurobiology, 10(3):358–364, June 2000.
Gina G. Turrigiano and Sacha B. Nelson. Homeostatic plasticity in the developing nervous system.
R
Nat Rev Neurosci, 5(2):97–107, February 2004.
Gina G. Turrigiano, Kenneth R. Leslie, Niraj S. Desai, Lana C. Rutherford, and Sacha B. Nelson.
Activity-dependent scaling of quantal amplitude in neocortical neurons. Nature, 391(6670):892–
D
896, February 1998.
M. C. W. van Rossum, G. Q. Bi, and G. G. Turrigiano. Stable Hebbian Learning from Spike
Timing-Dependent Plasticity. J. Neurosci., 20(23):8812–8821, December 2000.
C. van Vreeswijk and H. Sompolinsky. Chaos in Neuronal Networks with Balanced Excitatory and
Inhibitory Activity. Science, 274(5293):1724 –1726, December 1996.
Tim P. Vogels, Henning Sprekeler, Friedemann Zenke, Claudia Clopath, and Wulfram Gerstner.
Inhibitory Plasticity Balances Excitation and Inhibition in Sensory Pathways and Memory Networks. Science, 334(6062):1569 –1573, December 2011.
Tim P Vogels, Robert C Froemke, Nicolas Doyon, Matthieu Gilson, Julie S Haas, Robert Liu, Arianna Maffei, Paul Miller, Corette Wierenga, Melanie A Woodin, Friedemann Zenke, and Henning
Sprekeler. Inhibitory Synaptic Plasticity - Spike timing dependence and putative network function.
Front Neural Circuits, 7(119), 2013.
35
Christoph von der Malsburg. Self-organization of orientation sensitive cells in the striate cortex.
Kybernetik, 14(2):85–100, December 1973.
D. J. Willshaw, O. P. Bunemann, and H. C. Longuet-Higgins. Non-holographic associative memory.
Nature, 222:960–962, 1969.
David J Willshaw and Christoph Von Der Malsburg. How patterned neural connections can be set
up by self-organization. Proceedings of the Royal Society of London B: Biological Sciences, 194
(1117):431–445, 1976.
Fraser A. W. Wilson, Séamas P. Ó Scalaidhe, and Patricia S. Goldman-Rakic. Dissociation of Object
and Spatial Processing Domains in Primate Prefrontal Cortex. Science, 260(5116):1955–1958, June
1993.
T
Zhihua Wu and Yoko Yamaguchi. Conserving total synaptic weight ensures one-trial sequence
learning of place fields in the hippocampus. Neural Networks, 19(5):547–563, June 2006.
AF
Friedemann Zenke. Memory formation and recall in recurrent spiking neural networks. PhD thesis,
École polytechnique fédérale de Lausanne EPFL, Lausanne, Switzerland, 2014.
Friedemann Zenke, Guillaume Hennequin, and Wulfram Gerstner. Synaptic Plasticity in Neural
Networks Needs Homeostasis with a Fast Rate Detector. PLoS Comput Biol, 9(11):e1003330,
November 2013.
R
Friedemann Zenke, Everton J. Agnes, and Wulfram Gerstner. Diverse synaptic plasticity mechanisms
orchestrated to form and retrieve memories in spiking neural networks. Nat Commun, 6, April
D
2015.
Qiang Zhou, Huizhong W. Tao, and Mu-ming Poo. Reversal and Stabilization of Synaptic Modifications in a Developing Visual System. Science, 300(5627):1953–1957, June 2003.
Lorric Ziegler, Friedemann Zenke, David B. Kastner, and Wulfram Gerstner. Synaptic Consolidation:
From Synapses to Behavioral Modeling. J Neurosci, 35(3):1319–1334, January 2015.
36