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A computational account for the ontogeny of
mirror neurons via Hebbian learning
Graduation Project Bachelor Artificial Intelligence
Credits: 18 EC
Author
Lotte Weerts
10423303
University of Amsterdam
Faculty of Science
Science Park 904
1098 XH Amsterdam
Supervisors
dr. Sander Bohté
Centrum Wiskunde & Informatica
Life Sciences Group
Science Park 123
1098 XG Amsterdam
dr. Rajat Mani Thomas
Netherlands Institute for Neuroscience
Social Brain Lab
Meibergdreef 47
1105 BA Amsterdam
prof. dr. Max Welling
University of Amsterdam
Faculty of Science
Science Park 904
1098 XH Amsterdam
June 26, 2015
Abstract
It has been proposed that Hebbian learning could be responsible for the ontogeny
of predictive mirror neurons (Keysers and Gazzola, 2014). Predictive mirror neurons
fire both when an individual performs a certain action and when the action they
encode is most likely to follow a concurrently observed action. Here, we show that
a variation of Oja’s rule (an implementation of Hebbian learning) is sufficient for
the emergence of mirror neurons. An artificial neural network has been created
that simulates the interactions between the premotor cortex (PM) and the superior
temporal sulcus (STS). The PM cortex coordinates self-performed actions, whereas
the STS is a region known to respond to the sight of body movements and the
sound of actions. A quantitative method for analyzing (predictive) mirror neuron
activity in the artificial neural network is presented. A parameter space analysis
shows that a variation of Oja’s rule that uses a threshold produces mirror neuron
behavior, given that the bounds for the weights of inhibitory synapses are lower than
those of excitatory synapses. By extension, this work provides positive evidence
for the association hypothesis, which states that mirror-like behavior of neurons in
the motorcortex arises due to associations established between sensory- and motor
stimuli.
Acknowledgements
My sincere thanks goes to Sander Bohté, for sharing his expertise and
for providing me with excellent guidance and encouragement
throughout the course of this project.
Furthermore, I am extremely grateful to Rajat Thomas, for the
illuminating insight and expertise he has shared with me with great
enthusiasm.
I would also like to thank Max Welling, who took the effort to be my
university supervisor regardless of his busy schedule.
Finally, I want to thank my parents and sisters for their continuous
source of support and motivation.
Contents
1 Introduction
2 Theoretical foundation
2.1 Interpretations of associative learning .
2.1.1 Hebbian learning . . . . . . . .
2.1.2 Rescorla Wagner . . . . . . . .
2.2 Hebbian learning rules . . . . . . . . .
2.2.1 Basic Hebb Rule . . . . . . . .
2.2.2 Covariance rule . . . . . . . . .
2.2.3 BCM . . . . . . . . . . . . . . .
2.2.4 Oja’s rule . . . . . . . . . . . .
2.3 Summary . . . . . . . . . . . . . . . .
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4 Analysis procedure
4.1 A visualization of mirror neuron behavior . . . . . . . . . . . . . . . . . .
4.2 Analysis procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Computational model
3.1 A motivation for the structure of the model
3.2 Modeling a single neuron . . . . . . . . . . .
3.3 Modeling action execution and observation .
3.4 Learning rules . . . . . . . . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . .
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5 Results
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5.1 Comparison of learning rules . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Parameter space analysis of Oja’s rule . . . . . . . . . . . . . . . . . . . . 20
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Conclusion
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7 Discussion and future work
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7.1 Limitations of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1
Introduction
In the early 1990s, mirror neurons were discovered in the ventral premotor cortex of
the macaque monkey (Di Pellegrino et al., 1992). These neurons fired both when the
monkeys grabbed an object and when they watched another primate grab that same
object. Mirror neuron-like activity has been observed in monkeys, birds and humans
(Catmur, 2013). More recently, evidence has started to emerge that suggests the existence
of mirror neurons with predictive properties (Keysers and Gazzola, 2014). These neurons
fire when the action they encode is the action most likely to happen next, rather than the
action that is concurrently being observed. A critical question concerns how (predictive)
mirror neurons have developed to behave the way they do. In other words: what is the
ontogeny of mirror neurons?
Two distinct views have emerged with regards to the origin of mirror neurons: the
adaptation hypothesis and the association hypothesis (Heyes, 2010). The adaptation hypothesis refers to the idea that mirror neurons are an adaptation for action understanding
and thus a result of evolution. According to this hypothesis, humans are born with mirror
neurons and experience plays a relatively minor role in their development. Alternatively,
one can view mirror neurons as a product of associative learning, which suggests that
these neurons emerge through a correlated experience of observing and executing the
same action. The mechanism responsible for associative learning itself is a product of
evolution, but associative learning did not evolve for the ’purpose’ of producing mirror
neurons. The important difference between these two theories is the role of sensorimotor
experience, which is crucial for the associative hypothesis but has a less prevalent role
in the adaptation hypothesis. In this work, the focus will lie on further developing the
association hypothesis.
While both the association- and adaptation hypotheses can account for the ontogeny
of mirror neurons, the association hypothesis has several advantages. First, it is consistent
with evidence that indicates that mirror neurons contribute to a range of social cognitive
functions that do not play a role in action understanding, such as emotions (Heyes, 2010).
Second, research suggests that the perceptual motor system is optimized to provide the
brain with input for associative learning (Giudice et al., 2009). For example, infants have
a marked visual preference for hands. Such a preference could provide a rich set of input
stimuli for mirror neurons that are active in manual actions.
Keysers and Gazzola (2014) have proposed a mechanistic perspective on how mirror
neurons in the premotor cortex could emerge due to associative learning, or more precisely, Hebbian learning. Hebbian learning was first proposed by Donald Hebb in 1949 as
an explanation for synaptic plasticity, which refers to the ability of the connections (or
synapses) between neurons to become either weaker or stronger (Cooper et al., 2013b).
Thus, Hebbian learning explains learning on a local level between single neurons. Hebb
postulated the principle as follows: "when an axon of cell A is near enough to excite
a cell B and repeatedly or persistently takes part in firing it, some growth process or
metabolic change takes place in one or both cells such that A’s efficiency, as one of the
cells firing B, is increased".
1
To investigate whether or not Hebbian learning is sufficient to lead to the emergence
of mirror neurons, we present a computational approach that implements the mechanics
described by Keysers and Gazzola (2014). This involves the usage of an artificial neural
network (ANN) to simulate activity in the premotor cortex (PM) and the superior temporal sulcus (STS). The PM cortex coordinates self-performed actions, whereas the STS
is a region known to respond to the sight of body movements and the sound of actions.
Action execution and observation will be simulated by exciting neurons in these two areas. The aim of this simulation is to find out whether or not associative learning - in
particular Hebbian learning - is sufficient to lead to the development of mirror neurons.
Specifically, the problem investigated in this work can be defined as follows:
Can artificial neural networks that evolve via a local Hebbian-like learning rule, when
exposed to action execution and observation, be sufficient to lead to the emergence of
predictive mirror neurons?
Here, we show that Oja’s rule, an implementation of Hebbian learning, is sufficient
to impose predictive mirror-like behavior. First, the proposed research question is considered in relation to other studies on the ontogeny of mirror neurons (chapter 2). This
includes a review of four implementations of Hebbian learning: basic Hebb, the covariance learning rule, BCM, and Oja’s rule. Subsequently, the computational model that
describes the artificial neural network is introduced in chapter 3. Additionally, a variation of Oja’s rule is presented. The procedure for the quantitative analysis of mirror
neuron-like behavior, as explained in chapter 4, has been applied to the recorded activity
of ANNs that evolve via several different learning rules. The results of this analysis are
described in chapter 5. Finally, it is argued in chapter 6 that an artificial neural network
that evolves via a variation of Oja’s rule is sufficient to explain the emergence of mirror
neurons. Related problems that should be considered in future work are dealt with in
chapter 7, the final chapter.
2
2
Theoretical foundation
In this chapter, research that has preceded the current work will be discussed. First, two
computational accounts for the ontogony of mirror neurons via the association hypothesis
will be discussed. Subsequently, an elaboration of several implementations of Hebbian
learning that have been proposed in the literature will be given.
2.1
Interpretations of associative learning
The association hypothesis states that mirror neurons emerge due to associative learning.
This term refers to any learning process in which a new response becomes associated with
a particular stimulus (Terry, 2006). As a refinement of the association hypothesis, two
different implementations of associative learning have been proposed, namely Hebbian
learning (Keysers and Gazzola, 2014) and Rescorla-Wagner (Cooper et al., 2013b).
2.1.1
Hebbian learning
Keysers and Gazzola (2014) have proposed a model in which Hebbian learning could
account for the emergence of mirror neurons. Whereas there are several brain regions
that contain mirror neurons, they illustrate their idea with connections between the
premotor cortex (PM), the inferior posterior parietal cortex (area PF/PFG) and superior
temporal sulcus (STS). Both the PM and PF/PFG play a role in action coordination,
whereas the STS is a region known to respond to the sight of body movements and
the sound of actions. More recently, evidence has started to emerge that some mirror
neurons in the PM have predictive properties (Keysers and Gazzola, 2014). When an
action is observed, these neurons behave as predictors for the action that is most likely
to follow next. Consider the sequence of actions relevant to grabbing a cup of coffee,
which consists of a sequence of reaching, grabbing the cup and bringing the cup to the
mouth. If one observes another person reaching his or her arm, this induces activity in
the STS. Additionally, PM neurons that encode ’grabbing’ - the action that is likely to
be performed next - fire. Such predictive properties are of great importance because it
allows us to anticipate the actions of others. The proposed model explains the emergence
of such neurons as a result of sensorimotor input of self-performed actions. When one
performs a certain action, this has been coordinated by activity in the PM. Subsequently,
one observes him- or herself perform that particular action. This is called reafference.
Once the observation of the action has reached the STS, neurons in the PM that encode
for the next action have already become active. If synaptic plasticity caused by Hebbian
learning is assumed, the simultaneous activity in the STS of the observed action and
activity in the PM for the next action causes an association between neurons in both
regions. Keysers and Gazzola (2014) suggest that such an increase in synaptic strength
could account for the emergence of predictive mirror neurons.
3
2.1.2
Rescorla Wagner
Cooper et al. (2013b) used a computational model to investigate the development of
mirror neurons. Two implementations of associative learning were considered; RescorlaWagner - a supervised learning rule that relies on reducing a prediction error - and an
implementation of Hebbian learning and . Their computational model simulates results of
an experimental study (Cook et al., 2010) that examined automatic imitation. Automatic
imitation is a stimulus-response effect where the physical representation of features that
are task-irrelevant facilitate similar responses but interfere with dissimilar responses. In
this study participants were asked to open and close their hands in response to a color
stimulus (the ’task-relevant’ stimulus) whilst watching a video of an opening or closing
hand (the ’task-irrelevant’ stimulus). Research has shown that magnetic stimulation of
the inferior frontal gyrus (an area where mirror neurons have been found) selectively
disrupts automatic imitation. Therefore, automatic imitation is thought to be an index
for mirror neuron activity. Cook et al. (2010) showed that automatic imitation is reduced
more by contingent and signalled training than by non-contingent sensorimotor training.
The results of this study have been successfully simulated by Cooper et al. (2013b)
using a model based on the Rescorla-Wagner learning rule. The Hebbian learning rule,
however, could not explain the reported data of Cook et al. (2010). Therefore, Cooper
et al. (2013b) argue that prediction error-based learning rules, not Hebbian learning, can
account for the development of mirror neurons.
The study of Cooper et al. (2013b) suggests that Hebbian learning is not sufficient to
impose mirror neuron behavior. However, the used implementation of Hebbian learning
suffers from a computational problem. The weights of the connections between neurons
imposed by this learning rule are in principle unbounded, which makes the learning rule
unstable. More advanced types of Hebbian learning that adhere to such weight constraints, for example Oja’s rule (Oja, 1982), were not considered. Thus, even though
Cooper et al. (2013b) have shown that supervised learning is sufficient to explain automatic imitation data, Hebbian learning should not be dismissed fully as a possible
explanation for the emergence of mirror neurons.
2.2
Hebbian learning rules
At the time Hebbian learning was introduced by Donald Hebb (1949), the field of neuroscience, let alone the field of neuroinformatics, was still in its infancy. Hebb’s postulate
was introduced in a general, qualitative manner. It implies that learning occurs due to
changes in synaptic strength, and that this is caused by a correlation between the input
of a pre- and postsynaptic neuron. These changes have later been defined as long term
potentiation (LTP) and long term depression (LTD), which refer to an increase and decrease of synaptic efficacy respectively (Kandel et al., 2000). However, the qualitative
description of Hebb leaves much room for interpretation. Over time many computational implementations of Hebb’s postulate have been proposed. Here, four of these will
be discussed: the basic Hebb rule, the covariance rule, BCM and Oja’s rule.
4
2.2.1
Basic Hebb Rule
A simple rule that uses the presynaptic activity ai and postsynaptic activity aj to change
the synaptic strength of the connection wj,i is the basic Hebb Rule (Hebb, 1949):
∆wj,i = αhai · aj i
(1)
Here, α is the learning rate, which determines how quickly the synaptic weights
change. The brackets h and i denote that ∆wj,i is the average of all synaptic changes
over a particular time. This articulates the idea that synaptic modification is a slow
process. The rationale behind the formula is that if both neurons are relatively active,
their synaptic connection will increase substantially. However, if one of them or both are
less active, the synaptic strength will increase less.
The major disadvantage of the basic Hebb rule is that the weights can only increase.
Consequently, the basic Hebbian rule accounts for LTP but does not allow LTD. Since
LTD has been recorded in neurons, a rule that fails to account for this phenomena is
biologically implausible. Intuitively, it prevents the system from being able to ’unlearn’
what it has learned.
Moreover, the basic Hebb rule is inherently unbounded, since weights can increase
indefinitely. This is an effect that does not occur in biological systems due to the physical
constraints to the growth of synapses. Aditionally, Hebb’s rule does not take baseline
activity into account. In general, neurons always have some baseline of noisy activity.
Since the simple Hebb rule will increase the weights even if the recorded activity is simply
baseline activity, the weight updates can easily get distorted .
2.2.2
Covariance rule
The covariance rule is an alternative to the basic Hebb rule that implements both LTP
and LTD (Dayan and Abbott, 2001). It is defined as follows:
cov(x, y) =
1
n−1
∆wj,i = α · cov(ai , aj )
n
X
(xi − hxi)(yi − hyi)
(2)
(3)
i=0
As in the basic Hebb rule, α refers to the learning rate. As suggested by its name,
the covariance rule calculates the covariance of the activities of the pre- and postsynaptic
neurons and uses this to determine the weight change. The covariance of two sequences
is negative if they are uncorrelated, and positive if they are correlated. Therefore, the
covariance rule allows for negative weight updates and thus accounts for both LTP and
LTD. By substracting the average activity over a certain time period, the covariance rule
naturally takes the baseline activity into account.
However, similar as the basic Hebb rule, the covariance rule is still unbounded. In
5
theory, the weights can grow infinitely if the activation sequences keep being correlated.
In biology this is not a problem because the synapses are naturally constrained by their
physical properties to grow infinitely. In a computational model, however, such constraints have to be explicitly incorporated in the model, which is not the case for neither
the covariance rule nor the basic Hebb rule.
Another problem with the covariance rule is that ∆wj,i will always be the same as
∆wi,j , since the formula does not differentiate between the pre- and postsynaptic neuron.
Separating the development of two reciprocal connections would make the learning rule
more biologically plausible.
2.2.3
BCM
BCM is a modifcation on the covariance rule that imposes a bound on the synaptic
strengths. The rule, named after its introducers Bienenstock, Cooper and Munro (1982),
is defined as follows:
∆wj,i = hai · aj (aj − θv )i
(4)
θv acts as a threshold that determines whether synapses are strengthened or weakened. To stabilizes synaptic plasticity, BCM reduces the synaptic weights of the postsynaptic neuron if it becomes too active. This can be achieved by implementing θv as a
sliding threshold:
∆θv = ha2j − θv i
(5)
The sliding threshold induces competition between the synapses, because strengthening some synapses increases aj and thus increases θv , which in effect makes it more
difficult for other synapses to increase afterwards. However, one disadvantage of BCM
is that it does not take the baseline activity of the presynaptic neurons into account.
2.2.4
Oja’s rule
Oja’s rule, introduced by Oja (1982), is an alternative learning rule that imposes bounds
on the weights:
∆w[i,j] = hai · aj − α(a2j )w[i,j] i
(6)
Oja’s rule imposes an online re-normalization by constraining the sum of squares of
the synaptic weights. More precisely, the bound of the weights is inversely proportional
to α, that is, |w|2 will relax to α1 (Oja, 1982).
By constraining the weight vector to be of constant length, Oja’s rule imposes competition between different synapses. However, similar to BCM, Oja’s rule does not take
6
the baseline activity of the presynaptic neuron into account.
2.3
Summary
In sum, the field of neuroscience is currently evidencing a debate about the ontogeny of
mirror neurons. The associative hypothesis, which assumes that mirror neurons emerge
due to associative learning, seems to be a promising explanation. Several interpretations of associative learning have been proposed, namely Rescorla-Wagner and Hebbian
learning. Research has shown that supervised learning rules such as Rescorla-Wagner are
sufficient to explain results found in studies on automatic imitation. A different model
that explains the emergence of predictive mirror neurons by means of Hebbian learning
has been proposed, but a proof of concept has yet to be created. The chapter is concluded
by a review of several implementations of Hebbian learning, including basic Hebb, the
covariance rule, BCM and Oja’s rule.
7
3
Computational model
In this chapter we will introduce the computational model that was used to simulate the
emergence of mirror neurons. First, a motivation for the structure of the artificial neuron
network will be presented. Then, the model for a single neuron is presented. Finally, it
will be shown how this model can simulate action executions and observations.
3.1
A motivation for the structure of the model
There are several regions in the brains of monkeys where mirror neurons have been
found. Here, similar as in Keysers and Gazzola (2014), the focus lies on a set of three
interconnected regions: the premotor cortex (PM, area F5), the inferior posterior parietal
cortex (PF/PFG) and the superior temporal sulcus (STS). Both the PM and the PF/PFG
are known to contain mirror neurons. Their main function is the coordination of actions,
whereas the STS is an area in the temporal cortex that responds to the sight of body
movements and the sound of actions. Neurons in the PM and PF/PFG are reciprocally
connected, which means that the connectivity goes both from PM to PF/PFG as the
other way around. Similarly, there are reciprocal connections between the STS and
PF/PFG. Figure 1 shows the structure of the model used to simulate these areas.
In order to make a simulation of the dynamics between these different brain regions
feasible, three simplifications have been introduced. First, the intermediate step via the
PF/PFG is omitted. This results in a model that has only reciprocal connections between
the STS and PM. The artificial neural network will thus consist of two layers of neurons.
Second, it is assumed that the PM and STS have similar encodings for the same
action phases. For example, the action ‘drink a cup of coffee’ could consist of an action
phase sequence of ‘grabbing’, ‘reaching’ and ‘bringing hand to mouth’. In this example,
the PM and STS will each have three neurons that encode these three action phases. It
is not claimed that a single action phase is actually encoded by a single neuron in the
brain. In contrast, the ‘neurons’ of the artificial neural network should not be considered
simulating single neurons. As will be described in the next section, neurons are modeled
by activities that can take values between 0 and 1. In reality, however, a single neuron
can only send a binary signal - the action potential - to postsynaptic neurons. Therefore,
the artificial neurons presented here represent the average spiking rate of a group of
neurons rather than the activity of a single neuron.
Third, the connections from the PM to the STS will be restricted to inhibitory connections between neurons that encode the same action. These connections are inhibitory,
because the activity from PM to STS is thought to be net inhibitory (Hietanen and Perrett, 1993). However, it is biologically implausible to have as many inhibitory connections
as excitatory connections, since inhibitory synapses only comprise a small fraction of the
total number of synapses in the brain (Kandel et al., 2000). Computationally, too many
inhibitory connections can destabilize the network since they allow the PM to completely
suppress all STS activity. For these reasons, only inhibitory connections from the PM to
STS that encode the same action are modeled.
8
Figure 1: The setup used for the simulation. It consists of a two-layered neural network that represents
the PM (red nodes) and STS (blue nodes) cortices. Each action phase, e.g. "reach" or "grasp", is
encoded in one neuron in both the PM and STS. All STS neurons are connected to all PM neurons via
excitatory connections (blue arrows), whereas PM neurons are connected via inhibitory synapses (red
arrows) to the STS neurons that encode the same action. The weight matrices store the strengths of the
connections. In the training phase, PM neurons are sequentially activated to simulate the execution of
an action sequence, "reach, grasp, bring to mouth". The order of the excitation of the PM neurons is
determined by a transition matrix, which contains the probabilities that certain actions are followed by
others. Here, the sequence "reach (A) , grasp (B), bring to mouth (C)" is encoded, since the probabilities
to transition from A → B and B → C are both 1. With a delay of 200 ms, the STS is excited in the same
order. During the testing phase action observation is modeled. Therefore, only the STS is activated
whilst still adhering to the probabilities encoded in the transition matrix.
3.2
Modeling a single neuron
The activity of a single neuron is represented by αi , which can take a value between 0
and 1. The activity, similar as in Cooper et al. (2013a), is calculated as follows:
ai (t + 1) = ρai (t) + (1 − ρ)σ(Ii (t))
X
Ij (t) =
(wj,i ai (t − 1)) + Ej + Bj + N (0, µ2 )
(7)
(8)
i
Here, ρ is a value between 0 and 1 that determines to which extent previous activity
persists over time. Ij (t) refers to the input to the neuron, which is transformed by a value
between 0 and 1 by the sigmoid function σ. The value of Ij (t) depends among others
9
Parameter
Persistance
External excitation
Bias
Gaussian noise
ρ
Ej
Bj
µ
Value
0.990
4.0
2.0
2.0
A
B
C
D
E
F
A
0
0
0
0
0.5
0.5
B
0
0
0
0
0.5
0.5
C
1
0
0
0
0
0
D
0
1
0
0
0
0
E
0
0
1
0
0
0
F
0
0
0
1
0
0
(a) The parameters used in modeling neurons, (b) Example of a transition matrix that encodes for the actions A-C-E and B-D-F. The
drawn from Cooper et al. (2013b)
matrix contains the probability that one action
phase (row) is followed by another (column).
Figure 2: Parameters used in the model (a) and an example of a transition matrix (b)
on the weighted sum of the activities of all neurons. The weight matrix w contains
the weights of the connections. When the artificial network is trained, w is altered.
Additionally, the value of Ij (t) depends on three factors: Ej , Bj and µ. Ej refers to the
amount of stimulation caused by other sources than neurons within the network and can
be either on or off. External stimulation of PM neurons represents choosing the next
action that will be executed. In the case of STS neurons, Ej represents the observation
of a certain action in the real world. The input signal further consist of the activation
bias Bj , which represents baseline activity of the neuron, and Gaussian noise determined
by µ. Each time step t represents 1 ms of activity.
The parameters ρ, Ej , Bj and µ are constant in this model. The values used in our
simulations (figure 2a) were derived from previous work (Cooper et al., 2013b). All weight
matrix entries are initialized to 1 before training commences. Additionally, the network
will first be subjected to a burn-in phase. Here, the weight matrix is held constant and
the neurons are allowed to reach a certain steady-state activity.
3.3
Modeling action execution and observation
Keysers and Gazzola (2014) propose that mirror neuron activity emerges due to synaptic
changes that result from simultaneous action execution and reafference (the observation
of self-performed actions). This implies that the simulation of these mechanics should
consist of two separate phases. In the first phase, the training phase, action execution
and reafference are simulated. During this phase, the weights are being updated. The
second phase, the testing phase, consists of the mere observation of actions. If the correct
weight changes have been made during the training phase, the PM neurons should start
behaving as mirror neurons. The remainder of this section explains how the training and
testing phase have been simulated.
In the artificial neural network, PM activity represents action execution whereas STS
activity represents action observation. In essence, the action observation of self-performed
actions is a delayed version of the action execution signal. It takes approximately 100
ms for a signal in the PM to lead to an actual action execution. After another 100 ms
10
t (ms)
0
300
600
PM-A
[1,
[0,
[0,
PM-B
0,
1,
0,
PM-C
0]
0]
1]
t (ms)
200
500
700
(a) An example of an excitation sequence of
PM neurons during the action phase. If 1, Ej
is added to the input Ij (see formula 8). If
0, nothing is added. Note that the external
excitation persists until a transition is made to
a new action phase, which happens after 300
ms.
STS-A
[1,
[0,
[0,
STS-B
0,
1,
0,
STS-C
0]
0]
1]
(b) An example of an excitation sequence of the
STS neurons. Note that this is simply a delayed
version of the action sequence in c, with a delay
of 200 ms.
Figure 3: Examples of trials of the STS and PM during the training phase
the action observation reaches the STS. This gives a total delay of 200 ms.
The transitions from one action phase to the next are coordinated by a motor planner,
which chooses the next action based on a probability distribution that is encoded in a
transition matrix. A simple example of such a transition matrix is shown in figure 2b. It
encodes two actions, a → c → e and b → d → f . The first sequence can be derived from
the transition matrix since the probabilities to go from both a to c (t[0, 1] = 1) and c
to e (t[1, 2] = 1) are 1. It is assumed that the probability of each entire action sequence
is the same. Therefore, each last action phase in a sequence has a probability of n1 to
go to the first action phase of a new sequence, with n being the total number of action
sequences.
Initially, the motor planner will choose one of the first action phases as a starting
point. Once an action phase is selected, the PM neuron that encodes for this particular
action is externally stimulated (value 1 in figure 3a). The complete list of the excitations
at one particular time step is referred to as a trial. It is assumed that the length of one
action phase is 300 ms, similar as in Keysers and Gazzola (2014). After 300 ms, the next
action phase is chosen at random based on the probabilities encoded in the transition
matrix, which results in a new trial. Action observation entails the same trial, only with
a delay of 200 ms (figure 3b). During the training phase, both the PM and STS trials
are active. During the testing phase, only the STS neurons are excited.
3.4
Learning rules
During the training phase, the weight vector w will be updated according to a certain
neuron rule. Each of the Hebbian learning rules presented in chapter 2 has been implemented. However, since Oja’s rule does not naturally handle the baseline activity well,
we propose an alternative to Oja’s rule. The weight updates are the same as in the
original Oja’s rule (formula 6). However, a threshold is imposed on all the activities of
the neurons:
11
(
ak , if ak ≥θ
ak =
0, otherwise
(9)
This threshold prevents the weights to change from unsubstantial activities. Note
that the addition of the threshold cannot make Oja’s rule unbounded, since the bound
of the original rule only depends on α. Moreover, because the threshold introduces zero
correlation for certain periods of time, the weights will actually be lower compared to
weights imposed by the original learning rule.
3.5
Summary
In this chapter the computational model of the artifical network that models connections
between the premotor cortex (PM) and the superior temporal sulcus (STS) has been
presented. The general structure of connectivity in the network is a result of three
simplifying assumptions. First, the intermediate step of the PM/PFG has been omitted.
Second, each neuron only represents one action phase. Consequently, the neural network
consists of two equally sized layers. Third, the inhibitory P M → ST S connections only
exist between PM and STS neurons that encode the same action. Additionally, the
formula to calculate the activity of a single neuron has been presented, together with
an elaboration on how a transition matrix can be used to simulate action execution and
observation. Finally, a variation of Oja’s rule that uses a threshold has been introduced.
12
4
Analysis procedure
In this chapter an analysis procedure that can detect mirror neuron behavior is introduced. In contrast to the work of Cooper et al. (2013b), the simulation presented here
is not a simulation of one particular experiment. Therefore, it cannot be compared with
experimental data in a supervised manner. This requires designing new ways of investigating a quantification of mirror neuron behavior. First, it will be discussed what mirror
neuron behavior would visually look like in the simulation. Subsequently, a procedure
for calculating a quantitative measurement for the level of mirror neuron behavior in a
sequence of neuron activities is proposed.
4.1
A visualization of mirror neuron behavior
Before an evaluation metric can be designed, it must be clear what is considered to be
’mirror-like behavior’. Figure 4a shows activity that we deem to be typical for mirror
neuron behavior. The STS has been excited externally in a particular sequence and the
PM mimics the behavior of the visual neurons. Figure 4b shows predictive mirror neuron
behavior. The activity of STS-A induces activity in PM-B, which is the action that is
the most likely to follow next. Figure 4c shows a combination of the two, in which the
activity of STS-A induces activity of PM-B. In contrast to figure 4b, this activity persists
until STS-B has stopped to be active. The combined activity can explain both normal
and predictive mirror neuron behavior. Note that these kind of behaviors are expected
to emerge only during the testing phase. During the training phase, the PM neurons will
not be caused by STS activity but by external excitations.
4.2
Analysis procedure
A procedure to measure mirror neuron behavior was designed for the simulations presented in this work. All three types of mirror neuron behavior follow approximately the
same sequence of spikes (except that predictive mirror neurons surpass the first action
of a sequence). Therefore, the analysis procedure consists of the creation of a transition
matrix that describes sequences of activity in both the PM and STS cortices. By comparing the transition matrices, it can be determined to what extent the PM and STS
activity is similar, and thus, to what extent mirror neuron behavior has emerged.
The following procedure was used to calculate a transition matrix from recorded
activity. First, a low-pass filter is applied to the activity signals of all neurons (figure 4a)
to smoothen the signal (figure 4b). Then, significant peaks are substracted by setting a
threshold equal to the mean plus one standard deviation (figure 4c). Subsequently, for
each peak the neuron whose peak is the closest follow-up is determined. For a peak at
time t this is done as follows. First, the thresholded peaks of all neurons from time t to
t+3000 are collected (figure 4d). If a signal contains multiple peaks, all but the first peak
are removed from the signal. Then, a cross correlation analysis is performed between all
pairs of the original signal and the other signals. From this the delays that have the
highst cross correlation can be determined (figure 4e). If the delay of a certain neuron is
13
(a) Normal mirror neuron behavior. The activity of the PM
neurons mirrors exactly the activation of the STS.
(b) Predictive mirror neuron
behavior. The activity of PM
neurons is the same as the activity of STS neurons that will
be active next in the action sequence. For example, the green
PM neuron, which usually precedes a yellow neuron, responds
to the yellow STS neuron.
(c) Combination of normal and
predictive mirror neuron behavior. PM neurons respond
both to the same action phase
and to the action phase that
usually precedes the action
phase they encode.
Figure 4: Examples of three types of mirror neuron-like behavior
very low (−25 to 25 ms) the neuron considered to be simultaneous with the peak that is
under investigation. The neuron i which signal has the lowest positive delay di is selected
as the closest successor. In the transition matrix, tj,i is increased by one (figure 4f). If
the delay dn of another neuron n is very close to the closest delay (dn − di < 50 ms)
it is also selected as the closest successor. In this case, the total probability is evenly
distributed over the transition matrix. As a final step, the rows in the transition matrix
are normalized.
The above procedure can be applied both to STS and PM activity which results in
two matrices, tsts and tpm . In order to determine the similarity s of the two matrices,
one can calculate the Frobenius norm of their difference matrix, which gives the error :
n X
n
p X
pm 2
= (
(tsts
j,i − tj,i ) )
(10)
i=1 j=1
A low error would indicate a high level of mirror neuron-like behavior. To determine
whether or not a low retrieved error is significant, it will be compared to the distribution
14
(a) The recorded activity in the neurons
(b) Activity after application of low-pass
filter
(c) All activities lower than the threshold
are set to zero
(d) Activities within 3000 ms after the
start of the first peak of neuron B
A
A
B
C
D
B
C
D
+1
(f) The transition matrix is updated for
the transition from neuron B to the neuron
C, which has the lowest positive delay.
(e) Delays are determined by calculating
the maximal cross correlation.
Figure 5: A visualization of the procedure used to create a transition matrix from recorded activity.
Step D, E and F are repeated for all peaks detected in step A B and C. Afterwards, the rows of the
transition matrix are normalized.
15
of random errors. The distribution of errors is approximated by calculating the difference
between a set of uniformly at random generated stochastic matrices and the STS matrix.
If the retrieved error is lower than 2% of the errors determined by these random matrices,
it will be deemed significant. In the case of predictive mirror neuron behavior (figure 3b),
the transition from one action sequence to the next will always surpass the first action.
Therefore, this methodology works best for normal and combined mirror behavior (figure
3a and 3c). Note that the method presented here cannot distinguish between the different
types of mirror neuron behavior.
4.3
Summary
In this chapter a quantitative measure for the detection of mirror neuron behavior has
been presented. First, three types of mirror neuron behavior (normal, predictive and
combined) have been identified. Second, the analysis procedure to detect mirror neuron
behavior is described. This procedure consists of the generation of a transition matrix of
both the STS and PM neurons. Afterwards, these transition matrices can be compared
using the Frobenius norm of their difference matrix to obtain the error . This analysis
procedure can detect all presented types (normal, predictive and combined) of mirror
neuron behavior, though it will generally return a higher error for predictive mirror
neuron behavior.
16
5
Results
In this chapter two types of results will be discussed. First, a comparative analysis of the
different learning rules that shows how well each can account for mirror neuron behavior
is presented . Second, a parameter space analysis of the thresholded Oja’s rule is used
to show to what extent different parameter settings affect the level of mirror neuron
behavior that can be observed.
5.1
Comparison of learning rules
Figure 6 shows the mean and standard deviation of the error of the transition matrices
for 10 simulations of basic Hebb, covariance rule, BCM, Oja’s rule and the thresholded
Oja’s rule after a training phase of 250.000 ms for one action sequence consisting of four
action phases. Additionally, a hinton graph of the weight matrices is presented in figure
7. Figure 8 shows the recorded activities in the testing phase for all rules.
As visualized in figure 8a, applying the basic Hebb rule does not impose mirror
neuron behavior. The weight matrix (figure 7a) shows that the weights for all neurons
are approximately the same, which explains why the activities in figure 8b are also very
similar. This effect is caused by the baseline activity that is present for all neurons, which
influences the weight changes imposed by Hebb’s rule.
Figure 6: The average and standard deviation of the error of 10 simulations of the five learning rules. Each simulation consisted of 250.000
ms in both the training and testing phase, for a single action sequence of
4 action phases. The blue line indicates the value under which the norm
is deemed to be significant (< 2% of 10.000.000 randomly generated matrices).
The covariance rule, shown in figure 8b, naturally takes the baseline activity into
account. The weight matrix is more diverse than that of the basic Hebb rule (figure
7b). This causes the average error to be less high than that of Basic Hebb, but still not
17
(a) Basic hebb (α = 0.044),
m = 2.7
(d) Oja’s rule (θ = 0.5, αe =
10−1 ,αi = 1−1 ), m = 3.59
(b) Covariance (α = 0.044),
m = 4.8
(c) BCM (θinit = 0), m = 20.3
(e) Oja’s rule (θ = 0.0, αe =
10−1 , αi = 1−1 ), m = 3.79
Figure 7: A hinton graph that shows the relative differences of the weights of the weight matrix of
all rules. The lower left corner shows all weights from STS to PM neurons, whereas the upper right
corner shows all the PM to STS neurons. Blue and red squares denote inhibitory and excitatory neurons
respectively. They are normalized by the maximum value of the the weight matrix, which is denoted
beneath each figure by m.
significant. Moreover, as enpicted by the large variance of the covariance rule in figure
6, the results are unstable.
Figure 8c shows the activities after applying BCM, a modification on the covariance
rule which imposes a bound on the synaptic strengths. STS activity is completely suppressed during the training phase. Theoretically, BCM should impose bounds on the
weights due to competition between the synapses. However, in this case these bounds
are still too high which causes the suppression of STS activity during the training phase
due to strong inhibitory weights from PM neurons. As a consequence, BCM performs
approximately as well as basic hebb, as shown in figure 6.
An alternative to BCM is Oja’s rule. In contrast to BCM, the bounds of the weights
can be directly affected by the α parameter. To account for differences between the
bounds for inhibitory and excitatory neurons that occur in nature, different bounds were
used. Recall that α−1 is proportional to the bounds of |w|2 . Here, αinhibitory was set
to 1−1 and αexcitatory was set to 10−1 . Figure 7d shows that, if applied directly, all the
18
(a) Basic hebb, α = 0.044
(b) Covariance, α = 0.044
(c) BCM, θinit = 0
(d) Oja’s rule, θ = 0, αe = 10−1 , αi−1
Figure 8: A visualisation of the performance of all learning rules. Each
color denotes the same action phase.
The lines above the graph of STS
activity show the time each neuron
was externally excited. These are the
last 4000 ms of a simulation with a
training- and test phase of 250.000 ms
for four neurons that encode one action sequence (a-b-c-d).
19
(e) Oja’s rule, θ = 0.5, αe = 10−1 , αi−1
(a) θ = 0.3
(b) θ = 0.5.
(c) θ = 0.7
Figure 9: Results of quantitative parameter space analysis. Different images depict the results for
different thresholds θ. The x-axis shows the bounds for inhibitory neurons (αi−1 ), whereas the y-axis
shows the bounds for excitatory neurons (αe−1 ). The lower the error (denoted by a color range from dark
red to light yellow) the better the performance was for a parameter setting. The areas delineated by the
blue line are all retrieved errors that are lower than the errors of 98% of 10 million randomly generated
matrices. If the model is set to these parameter settings, mirror neuron behavior emerges.
PM neurons to behave similarly. Since no mirror behavior occurs, the average error is
high. To account for the baseline activity of the neurons, the learning rule was altered
as proposed in section 2.3, and the threshold θ was set to 0.5. As shown in figure 8e, this
results in a combination of normal and predictive mirror neuron behavior. This results
in an error that is approximately 10 times lower than the other learning rules (figure 6).
5.2
Parameter space analysis of Oja’s rule
The thresholded Oja’s rule has three parameters, namely αinhibitory , αexcitatory and
threshold θ. A parameter space analysis was performed to see to what extent different parameter settings change the outcome of applying Oja’s rule. The thresholded
Oja’s rule was applied in 500 simulations with different parameter settings. Both the
training phase and testing phase consisted of 1.000.000 ms. 19 action phases were modeled, divided over 4 action sequences of length 5, giving a total network size of 38 nodes.
Since modeling 4 action sequences of length 5 requires 20 nodes, and we only model 19,
20
two sequences partly overlap. The procedure as described in chapter 4 was applied to
the activities in the testing phase to determine the error of the difference of the PM
and STS transition matrices. Additionally, the error of 10.000.000 random matrices was
calculated. These were used to approximate the p-values at which a low error is deemed
to be significant.
The results of the parameter space analysis are shown in figure 9. The axes show
the inverse of α since this is approximately equal to the bounds of the weight matrix.
Each figure depicts the retrieved error for a different threshold θ value. The blue line
indicates the section in which the error is lower than 2% of 10.000.000 randomly obtained
errors. It shows that if the threshold is relatively low (0.3, figure 9a), Oja’s rule does
not impose mirror neuron behavior at all. Figure 9b shows that a threshold of 0.5 does
imposes mirror behavior if, in general, the bound of excitatory connections is higher than
the bound of inhibitory connections. If both parameters are higher than approximately
15, no mirror behavior emerges. If the threshold is very high (0.7, figure 9c), neuron
mirror behavior is imposed, but to lesser extent than when the threshold is 0.5. Here,
the excitatory bounds much exceed the inhibitory bounds to a larger extent for mirror
neuron behavior to emerge.
5.3
Summary
Five different learning rules have been implemented. An analysis of simulations of one
simple action sequence shows that, of the rules considered, only the thresholded Oja’s
rule is able to produce mirror neuron behavior. More precisely, it can account for both
normal and predictive mirror neuron behavior. Furthermore, a parameter space analysis
shows that if the threshold is set too low, no mirror neuron behavior is imposed. A
threshold of 0.5 shows a large field of values in which mirror neuron behavior is imposed.
In general, the bounds for inhibitory neurons have to be strictly smaller than those of
excitatory neurons to impose mirror neuron behavior. If the threshold becomes too high,
excitatory bounds much be about twice as high as inhibitory bounds.
21
6
Conclusion
In previous work it has been proposed that Hebbian learning could be responsible for the
ontogeny of predictive mirror neurons (Keysers and Gazzola, 2014). Here, we have shown
that a variation of Oja’s rule (an implementation of Hebbian learning) is sufficient to
explain the emergence of mirror neurons. An artificial neural network that simulates the
interactions between the premotor cortex (PM) and the superior temporal sulcus (STS)
has been created. Different implementations of Hebbian learning have been compared in
performance on a simple action sequence. Additionally, a parameter space analysis has
been performed on the proposed thresholded Oja’s rule to determine the sensitivity of
the parameters on its performance.
We have identified that from the learning rules considered, only the thresholded Oja’s
rule is sufficient to impose mirror neuron behavior. The other learning rules (basic Hebb,
covariance, BCM and the original Oja’s rule) are subject to at least one of the following
three limitations. First, both the covariance rule and the basic Hebb rule are unbounded,
which makes them biologically implausible. Second, all learning rules except for the
covariance rule are sensitive to baseline activity. These changes overshadow associations
between significant STS- and PM activity. This prevents mirror neuron behavior from
emerging. Third, if the inhibitory neurons are not bounded strong enough, as in BCM and
the original Oja’s rule, PM inhibition causes a complete suppression of STS activations.
This prevents any association between the PM and STS to occur. A variation of Oja’s
rule that uses a threshold and different bounds for inhibitory and excitatory synapses
has been shown to surpass each of these limitations.
A parameter space analysis has indicated that the parameters of the proposed thresholded Oja’s rule must adhere to two constraints for mirror neuron behavior to emerge.
First, the threshold value must be higher than the baseline activity, but lower than the
highest peaks measured. If the threshold is too low, small variances in baseline activity
impose weight changes that prevent the correct associations form being made. Choosing
the proper threshold value can be viewed as intrinsic homeostatic plasticity. This refers
to the capacity of neurons to regulate their own excitability relative to network activity
(Turrigiano and Nelson, 2004). Second, mirror neuron behavior can only be imposed if
the bounds for the excitatory neurons are higher than those of the inhibitory neurons.
This is necessary because otherwise inihbitory PM neurons will completely suppress STS
activity which prevents associations from being able to be learned.
In conclusion, we have shown that a thresholded Oja’s rule is sufficient to account for
the emergence of mirror neurons in an artificial neural network that simulates interactions
between the PM and STS cortices. Therefore, this work provides positive evidence for
the proposal that Hebbian learning is sufficient to account for the emergence of mirror
neurons. In the broader sense, this work promotes the idea that statistical sensory motor
contingencies suffice in the explanation for the ontogeny of mirror neurons. Therefore, it
can be regarded as positive evidence for the association hypothesis.
22
7
7.1
Discussion and future work
Limitations of this study
One of the assumptions underlying the model presented here is that the STS encodes
observations of actions of others similarly as the observations of self-performed actions.
While it has been shown that the area STS responds to the sight and sound of actions,
it has not been shown indisputable that the encoding for reafference (sensory input from
one’s own action) and others’ actions is the same. Further research should indicate
whether or not this assumption is valid.
Another limitation of the work presented here is that the input data to the ANN is
simplified compared to action execution and observation in the real world. While the time
delays that were used are comparable with what is observed in the action observation
and execution of primates Keysers and Gazzola (2014), the real world is more noisy and
less predictable than the action sequences that were used. For example, once an action
is chosen in the simulation, the entire action is always executed without interventions.
Future research should indicate whether or not the results presented here still hold when
the network is exposed to more complicated and less predictable action sequences.
7.2
Future work
It has been argued that that Rescorla-Wagner, and not Hebbian learning, account for
the emergence of mirror neurons (Cooper et al., 2013b). However, Oja’s rule was not
considered by Cooper et al. (2013b). In this work we have shown that a modified version
of Oja’s rule which is genuinely unsupervised is sufficient for the emergence of mirror
neurons. In further research, Oja’s rule should be applied to the experimental data
presented by Cooper et al. (2013b) to determine if it is sufficient to replicate the results
of the experimental study of Cook et al. (2010).
In the current implementation of the thresholded Oja’s rule, the threshold was determined manually. An extension of this work could be to model homeostatic plasticity, in
which the threshold value is dynamically set.
Keysers and Gazzola (2014) argued that the inhibitory connections from the PM
to the STS could behave as a ’prediction error’. If the PM correctly predicts the next
action, it inhibits STS activity which would indicate that no audiovisual confirmation is
required. However, as shown in this work, this can easily lead to complete surpression of
STS activity. A chain of activities in the PM cortex should then be imposed without an
exchange of information with the STS. In the model presented here, such a sequence is
not possible because there are no lateral connections modelled. In an extension of this
work one could add lateral connections to the model. Ideally, the transition matrix would
be completely omitted and replaced by interactions of neurons via lateral connections.
As a more realistic extension of this work, one would need to model the ’mirror
circuit’. If each cortical area is considered a node of a graph, then these nodes would
have to be simulated using spiking neuronal models with at least three cortical layers;
one for input, one for prediction and one for prediction error. These models will consist of
23
approximately 1000 neurons per layer at every node and will suffice to represent a much
richer repertoire of actions. Detailed work on the anatomy of the connections between
these nodes will inform the model on connection probabilities between layers within and
across nodes. A model of this nature would be instrumental in determining the origin of
mirror neurons.
24
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