# Social Practices for Social Driven Conversations in Serious Games

Agnese Augello
ICAR - National Research Council of Italy
Viale delle Scienze - Edificio 11
90128 Palermo, Italy
<EMAIL_ADDRESS>
Manuel Gentile
ITD - National Research Council of Italy
Via Ugo La Malfa 153
90146 Palermo, Italy
<EMAIL_ADDRESS>
Frank Dignum
Utrecht University, The Netherlands
Princetonplein 5, De Uithof
3584 CC Utrecht
<EMAIL_ADDRESS>

###### Abstract

This paper describes the model of social practice as a theoretical framework
to manage conversation with the specific goal of training physicians in
communicative skills. To this aim, the domain reasoner that manages the
conversation in the Communicate! [1] serious game is taken as a basis. Because
the choice of a specific Social Practice to follow in a situation is non-
trivial we use a probabilistic model for the selection of social practices as
a step toward the implementation of an agent architecture compliant with the
social practice model.

_K_ eywords Social practice, Serious Games, Physicians Training, Communicative
Skills

##  1 Introduction

Effective communicative skills are important in different fields, and are the
basis of every social relation. Serious games can be useful in this context,
as a valid approach to train people to properly carry out conversations by
means of simulation of dialogues with virtual characters. See e.g. [2][3][4].

The use of virtual agents for this purpose is particularly useful because it
allows us to also bring social elements of interactions into the simulations
[2]. They can be used to provide learners with a continuous feedback, showing
the effects of their conversational choices on the emotions and behavioural
changes of the interlocutors [1] [5] .

An area where conversations play an important role in everyday practice are
the medical consultations. Training physicians to communicate better has
several positive effects on patients’ well-being. For example a good
communication enhances patients’ treatment adherence [6], leads to greater
patient disclosure of sensitive psychosocial information and foster a
reduction in patients’ emotional distress [7]. It is generally agreed that
better doctor-patient interaction leads to better diagnosises, which in turn
can lead to huge (financial and human) cost savings in hospital tests and
treatments.

Several Serious Games have been designed in this context. E.g. [1] [8].
Communicate![1] is a serious game used by medical students to learn the best
communication strategies to establish a valid and trusted relationship with
the patient by means of consultations with virtual characters. SimSensei Kiosk
[8] , a dialogue system that conducts interviews related to psychological
distress conditions, shows several benefits of using virtual agents from the
perspective of the patient. In particular the patient, interacting with a
virtual agent shows more openness to the conversation, not feeling judged by
another human interlocutor.

Most agents that are used in these applications are based on scripts and
reactive rules to respond to users. Unfortunately this leads to quite
predictable and simple dialogs. Thus the challenge is to design agents that
are more pro-active and are able to autonomously take their decisions
according to their expectations and the evolution of the game. This would be
more in line with Clark [9] who claims that a dialogue is a joint activity
that must consider both individual and social processes.

The social context has a key role in the deliberative process of agents and is
particularly important in a conversational context, where the deliberative
process regards the choice of the most proper utterance. Conversation has a
social effect because it contributes to changes in people (beliefs, attitudes,
etc.), social relations, and the material world [10]. In the other hand the
social structure, the social practice and the social agents involved in the
social interaction determine the actual conversation utterances that are used
[10]. As a matter of fact different communication strategies can be used
according to the specific social context.

It is important therefore to formalize how the social context influences the
deliberation process of agents. Existing techniques often do not fully model
this type of situational deliberation of agents. The social context is often
dealt with as a set of norms that add complexity to the cognitive model of the
agent, complicate the deliberative process and that restrict agents
application to the particular application domain for which they have been
designed [11].

In order to address this issue we intend to use the concept of social practice
as modeled in [11]. Social practices refer to everyday practices and the way
these are typically and habitually performed in a society. In [11] social
practice are used to create agents that are able to sense the physical and
social aspects of the current situation, but act not in a simple reactive way,
but pro-actively choosing a plan of actions suitable to reach their both
social and functional goals.

In this paper, the model of social practice as a theoretical framework to
manage conversation with the specific goal of training physicians in
communicative skills is analyzed. To this aim, the domain reasoner that
manages the conversation in the Communicate! [1] serious game is taken as a
basis. We see how scenarios build with Communicate! can be replicated using
Social Practices in order to subsequently show how the Social Practice model
enhance the scenarios in a flexible and natural way. Because the choice of a
specific Social Practice to follow in a situation is non-trivial we use a
probabilistic model for the selection of social practices as a step toward the
implementation of an agent architecture compliant with the social practice
model [11].

##  2 Training communication skills for medics

Let us consider the following scenario. A doctor is at the desk of his office
in the hospital where he works. Someone knocks at the door. Because it is his
consultation hour he expects a patient. Thus the doctor invites the person to
come in. He habitually sees a lot of patients, following standard
conversational protocols aimed to obtain useful information from the patient
such as the existence of particular pathologies and the symptoms of the
disease.

When the doctor receives a patient he activates a plan of actions related to
the conversational protocol that should lead him to understand the patient’s
problems and to give the patient the right therapeutic treatment.

When the doctor does not know the patient who has just entered the room, he
will first introduce himself properly and ask some general background
questions to establish a trusted relationship with the patient.

Once he starts the conversation he will have some expectations about the type
of responses the patient will give. If the expectations are not met he might
change his ideas about the patient and reconsider his next moves in the
conversation. E.g. if a patient seems not to acknowledge a symptom or fact, it
might be the patient is worried or distracted and needs first to be put at
rest. However, if an emergency occurs the doctor should interrupt the
consultation and start a totally different practice (belonging to the
emergency). If he did not handle this type of emergency very often he will
have to deliberate to a greater extent (based on his medical knowledge and
other cases) to determine the right plan of action.

The scenario described above shows the complexity of autonomous agents for
this kind of games.

In order to have a concrete reference model for the medical context, we start
from the work done in the Communicate! project. First the scenario model
defined in the Communicate! serious game is briefly introduced. Then, the
“anamnesis" scenario is analized in order to understand how scenarios build
with Communicate! can be replicated using Social Practices in order to manage
the conversation in a flexible and realistic way.

##  3 Communicate! dialogue management

The scenario model defined in the Communicate! project allows the designer to
specify:

  * •

a set of parameters, that represent the dialogue state;

  * •

one or more conversation trees, in which the nodes are the player’s statements
and the computer’s statements;

  * •

a sequence of interleaves, that represent a sort of dialogue phase in which
one or more conversation trees could take place;

The Communicate! serious game is based on the structural dialogue state
approach, in which the interleaves, the trees and the statements represent the
dialogue grammar. The dialogue state is defined by the scenario parameters’
scores, by the lasts computer’s statements and by a set of emotions’ scores.
The basic set of emotions managed in Communicate! are : happiness, anger,
surprise, contempt, disgust, fear and sadness.

According to the current dialogue state, a domain reasoner selects the
possible moves (the player statements) among which the player could choose the
move to carry forward the dialogue. In fact, the player statements represent
the dialogue moves that update the elements of the dialogue state (the
parameters and the emotions of the virtual character).

###  3.1 The “anamnesis" scenario

The situation described in section 2 is managed by a specific scenario defined
in the framework of the Communicate! project [1], the “anamnesis" scenario. In
this scenario, the user plays the doctor role and holds a consultation with a
virtual patient.

The dialogue is “started" by the player who assumes the role of the doctor.
The player has to choose among a number of possible expressions. Each of these
expressions leads to the activation of a conversation tree.

Let us suppose the player, among the different possibilities, chooses the
following sentence to proceed the scenario: “I see you are a patient of Dr.
Aarts" This player statement, produces a “surprise" effect in the agent that
realizes that he will have an interview with an unknown doctor.

In terms of social practice, this element of “surprise" is the result of a
violation of an expectation of the social practices activated by the patient.
Infact, as shown to the player at the begginig of the game, the patient
expects to have an interview with his own doctor, that is the patient agent
has activated the social practice consulting my doctor. This lead the patient
agent to re-evaluate the situation and to select a new practice consulting an
unknown doctor.

The scenario manages the case in which the doctor directly introduces himself
to the patient in the initial phase of the dialogue, as well as the case in
which this greeting and presentation take place later. Of course, the two
alternatives produce different effects on the virtual patient. In the former
case, a positive effect on the “happiness" of the patient is achieved; instead
in the latter case, an effect of “displeasure" of the patient is obtained.
These two alternatives can be interpreted in the light of the social practices
selected by the patient. The practice contains a norm which states that the
doctor and patient should introduce themselves if not known yet in order to
show respect. If the doctor does not make this introduction at the start this
gets a social meaning of showing disrespect to the patient. So the
“disappointment" of the patient could be seen as a consequence of the
“violation" of the social norm.

The Communicate! scenario envisages that the doctor provides more information
to the patient about himself and his role within the hospital. In particular,
the doctor is still in training with the patient’s doctor. Knowing that the
one who is in front of him is a doctor in training leads the patient agent to
a state of “anger” if he is in an emotional state of “contempt". However, if
the patient is in a state of “happiness", he goes to a state of “surprise".

The analysis of this portion of the dialogue shows that the effect of a
communicative actions depends largely on the context in which they are
uttered. Especially, the social meaning of the utterance depends on the social
practice. Thus the ensuing emotional state of the patient also depends on this
practice.

##  4 Social practice agent architecture

The structural dialogue state approach used in Communicate! is a viable
solution to manage scripted dialogues. Although this is a good starting point,
from an educational point of view it is important to allow the player to
experience more varying and realistic dialogues. In order to overcome the lack
of flexibility of this approach, several approaches have been proposed in
literature.

For example, structural dialogue state approach has often been contrasted with
plan-based approaches[12] [13]. Anyway, dialogue plan-based approaches require
a great effort in the definition of commonsense knowledge and procedural
processing, often leading to the creation of an opaque solution.

Moreover, to improve the educational effectiveness, the game should allow the
user to play both the roles (doctor or patient), allowing the doctor to
experience the patient point of view [14]. To this aim, a design of the game
that models both the agents roles is required.

In this work, the model of social practice is introduced in order to overcome
the limitations of both structural dialogue state and plan-based approaches,
sharing the motivation with the work of Traum & Larsoon [15].

Figure 1: Architecture for social reasoning [11]

The proposed agent’s architecture (fig. 1) puts the identification of the
social context at the basis of the deliberative process of the agents. The
social context is modelled according to the theory of social practice (section
4.1). The identification of the most appropriate social practice allows for a
dynamic management of the dialogue within the serious game, allowing the
conversation to follow a specific but not rigid flow, and restricting
considerably the number of possible sentences that may be issued by the agent
and at the same time facilitating the understanding of the utterances of the
interlocutor. Starting from the observation of the context an agent updates
his beliefs and goals, and activates a set of possible social practices. He
then triggers an internal reasoning process in order to select the most
appropriate social practice (section 4.2). Finally, according to the selected
social practice he starts a deliberation process in order to manage the
conversation within the context of that practice. When the available
information is not enough to activate a single social practice, the agent will
interact with the user in order to obtain additonal information that
discriminates the context further and leads to one social practice being the
most suitable.

In this paper the focus is on the social practice modeling and selection, as
highlighetd in the figure.

###  4.1 The model of social practice

In this section the model of social practices described in [11] is analysed in
the context of a doctor-patient dialogue. According to [11], the aim of the
social practice model is to provide a representation scheme that allows the
implementation of cognitive agents able to use the social practice as a first-
class construct in the agent deliberation process. In the following, the
components of the social practice model are described by referring to a
specific social practice ("Doctor Patient Dialogue") concerning the scenario
of a doctor-patient interview. Table 1 summarizes the components of the social
practice under investigation.

Table 1: The doctor-patient dialogue as a social practice Abstract Social Practice | Doctor Patient Dialogue
---|---
Physical Context |
Resources | current time,medical instruments
Places | hospital,office
Actors | doctor1, patient1
Social Context |
Social interpretation | consulting room,consulting time,doctor has medical skills
Roles | doctor, patient, relative of the patient, nurse
Norms |  patient is cooperative, doctor is polite
Activities |  welcome, presentation, patient’s data gathering, patient symptom description constative(answer,confirm,disagree,agree) directive(ask,instruct,request)
Plan patterns |
Meaning |  support the patient, create trust, eliciting patient’s problems and concerns, treatment and solution possible, empathic opportunity, empathic response
Competences |  listening effectively, being supportive and empathic, use effective explanatory skills, adapt conversation to patient, discussing treatment options understandable

  * •

Physical Context focuses on the identification of the environment elements,
that are both the physical objects with a meaningful role in the practice
(Resources), the agents (human beings or autonomous systems) involved in the
practice (Actors) and the locations of objects and actors (Places).

In the analyzed scenario at least two people are expected to take part in the
practice, the interview takes place in a hospital, inside a room and at a
specific time. Among the various resources available in this context, we can
highlight the medical instruments visible in the room including the doctor’s
coat.

  * •

Social Context describes the social interpretation of the elements sensed in
the environment, in particular in [11] a distinction is made between the so
called Roles, describing specific behaviours that can be expected from
specific actors, and the interpretation of all the other elements, called
Social Interpretation. Moreover the Norms identify the rules that are expected
inside a social practice. As an example, in the analyzed scenario, we can
identify a physician role and a patient role, and in some cases a relative of
the patient. The doctor-patient dialogue generally takes place during
consultation hours in a special room used for the purpose. Finally, the type
of clothing, and in particular the colour of the coat worn by the doctor are
usually indicative of the "role" (doctor, assistant, nurse) that the
professional has in the hospital. Different norms can be related to the roles
of doctor and patient. As an example the doctor during the interview should
totally reserve his attention to the patient; if suddenly there is an
emergency, the norm is no more applicable. For what concerns the role of the
patient it is important that he is cooperative.

  * •

Activities are possible course of actions that agents can perform. In a fine-
grained analysis, every communicative act of the agent can be considered as an
activity. At this level we tie communicative acts to the speech act
classification [16] and not to specific utterances. Listing all the possible
language productions is not possible and certainly not efficient; moreover,
this allows us to organize the individual communicative acts in communication
protocols. Non communicative acts such as moves, gestures and so on can be
also considered at this level. In a coarse-grain analysis, a set of simple
actions can be considered as an activity; it is a sort of a scene in the plan
of actions of the agent aimed to reach a specific sub-goal. As an example a
scene can be a set of communicative and non communicative acts used by the
doctor to welcome the patient into the room.

  * •

Plan Patterns. A plan pattern is a pattern on the basis of which an agent can
construct a plan to reach a goal. A plan pattern consists of an ordered set of
scenes with a specific sub-goal that restrict the type of plans that can be
used. As an example, the doctor will start and follow a specific protocol for
the patient consultation. The protocol can be structured in the following
scenes: Welcome and Presentation, Personal Data acquisition (biographic data,
pathologies, medical treatments, allergies, …), Description of the symptoms.
The goal of the protocol is to start a doctor/patient relation and to proceed
to the consult, however the situation can evolve and lead to quit (dotted
lines in the image) the social practice. For example if the doctor acquiring
the information recognizes that another doctor could be more competent for the
patient’s issues (in this case the doctor has not the competences required by
the practice), or if the patient shows an untrusted or disrespectful behavior
(in this case a violation of a norm), the doctor can switch to another
practice.

  * •

Meaning. This part of the social practice model, is used to define social
meanings for the agent’s activities and plans. Considering communicative acts
as activities, every act can assume a different meaning depending on the
context of the dialogue, the cognitive status of the agents and so on. As an
example, communicative acts could be classified from an empathic point of view
according to the classification proposed by [17] (i.e. empathic opportunity,
empathic response, etc…). At the same time, the meaning of communicative acts
includes its intended effects [9]. Of course these examples are not exhaustive
of all the other different types of meaning that could be associated to the
communicative acts.

  * •

Competences Competences are defined as the abilities an agent should have to
perform the activities of the social practice. Amongst the medical skills
required to interact with the patient there are several communication skills
recommended to physicians [18] [19]. For example, physicians should listen
effectively, being supportive and empathic, use effective explanatory skills,
discuss treatment options in a simple way. At the same time, patients should
have health literacy skills in order to have an effective dialogue.

###  4.2 A probabilistic Reasoner for Social Practice selection

Starting from the observation of the context and according to his identity the
agent activates a set of possible social practices. Based on his beliefs and
goals it subsequently selects the most appropriate social practice. We propose
a probabilistic reasoner for this selection process, because often not all
information is available or certain. In those cases the agent will use its
experience and derive as much information as possible to make a selection.

The elements of the physical environment and their social interpretation
related to a social practice, are represented in a causal graph. An example
for the practice “Doctor Patient Dialogue" is shown in figure 2. From the root
node, representing the activation of the social practice, depart causal nodes
representing its relevant elements (i.e. place, current time, …). The states
of these nodes are the possible social interpretations of such elements; for
example the consulting time is the social interpretation of the physical
resource current time. The causal links are expressed by means of conditional
probability tables. The Bayesian networks allow the designer to adopt a top-
down approach to formalize its initial knowledge about the social practice. In
later versions learning algorithms will be implemented to allow the agent to
learn the tables of conditional probabilities related to the social practices,
but we leave that out for the moment. The algorithm of social practice
selection is shown in figure 1, and explained below according to the scenario
described in section 2. First, the observation of the context and the identity
of the agent produces an initial selection of practices. If the agent has the
role of doctor and he is at the hospital, possible social practices are for
example those related to consulting patients and the management of an
emergency. According to the available information it is possible to set
evidences in the causal graphs associated with the different social practices.
In particular if the current_time is a consulting_time and the doctor is
inside a consulting_room, the analysis of the activation probabilities of the
possible practices shows that the practice related to a patient consultation
(doctor_ patient_dialogue) has the highest probability, as shown in figure 2.
In particular even if the agent does not yet know who is his interlucotor,
according to the model of the activated social practice, he estimates that he
should be a patient with a high probability.

When the situation evolves several things might happen. The person either can
be indeed a patient, an hospital employee (is wearing some hospital stuff), or
can be not alone. Depending on the actual situation the expectations of the
social practice are confirmed and the practice is continued or the social
practice is re-evaluated with the new information. When a social practice is
chosen the agent can start a further context analysis in order to discriminate
among more concrete social practices that lead to more specific plan patterns,
roles, norms etc.

Figure 2: The decision network activation for the Doctor_Patient_Dialogue
social practice

##  5 Conclusion

In this paper we have given some initial steps showing the usability of social
practices for managing dialogues in serious games. Mainly we have shown that
social practice captures both the physical as well as the social context and
thus forms an important source of information for the interpretation and
generation of speech acts. Having this context available as source of
information allows for a more accurate interpretation of the utterances as
they can be compared to the expected type of utterances according to the
practice. Because social practices do not enforce a fixed script but rather
indicate usable plan patterns they allow for flexibility in the dialogue while
still ensuring a (joint) goal to be achieved.

Of course, there are many open questions such as the connections between
different social practices, the learning and adaptation of the agent and we
have also not shown the actual dialogue manager that is used by the agent to
generate the actual utterances. The first steps are to fully implement the
agent architecture and using a simple dialogue planner as in Communicate!
produce at least the same dialogues but on the fly. Moreover, the Information
State approach [Traum2013] will be analyzed in order to verify if a dialogue
manager defined according to this approach could be used as a valuable
deliberation engine for an agent compliant with the social practice model.

##  6 Acknowledgements

We are grateful to Prof. J. Jeuring , for his useful explanation of the
Communicate! project and for providing us the scenarios of the game.

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# Nonlinear Dendritic Coincidence Detection for Supervised Learning

Fabian Schubert
Institute for Theoretical Physics
Goethe University Frankfurt
Frankfurt am Main
<EMAIL_ADDRESS>
& Claudius Gros
Institute for Theoretical Physics
Goethe University Frankfurt
Frankfurt am Main
<EMAIL_ADDRESS>

###### Abstract

Cortical pyramidal neurons have a complex dendritic anatomy, whose function is
an active research field. In particular, the segregation between its soma and
the apical dendritic tree is believed to play an active role in processing
feed-forward sensory information and top-down or feedback signals. In this
work, we use a simple two-compartment model accounting for the nonlinear
interactions between basal and apical input streams and show that standard
unsupervised Hebbian learning rules in the basal compartment allow the neuron
to align the feed-forward basal input with the top-down target signal received
by the apical compartment. We show that this learning process, termed
coincidence detection, is robust against strong distractions in the basal
input space and demonstrate its effectiveness in a linear classification task.

_Keywords_ Dendrites $\cdot$ Pyramidal Neuron $\cdot$ Plasticity $\cdot$
Coincidence Detection $\cdot$ Supervised Learning

##  1 Introduction

In recent years, a growing body of research has addressed the functional
implications of the distinct physiology and anatomy of cortical pyramidal
neurons (Spruston, 2008; Hay et al., 2011; Ramaswamy and Markram, 2015). In
particular, on the theoretical side, we saw a paradigm shift from treating
neurons as point-like electrical structures towards embracing the entire
dendritic structure (Larkum et al., 2009; Poirazi, 2009; Shai et al., 2015).
This was mostly due to the fact that experimental work uncovered dynamical
properties of pyramidal neuronal cells that simply could not be accounted for
by point models (Spruston et al., 1995; Häusser et al., 2000).

An important finding is that the apical dendritic tree of cortical pyramidal
neurons can act as a separate nonlinear synaptic integration zone (Spruston,
2008; Branco and Häusser, 2011). Under certain conditions, a dendritic $\rm
Ca^{2+}$ spike can be elicited that propagates towards the soma, causing
rapid, bursting spiking activity. One of the cases in which dendritic spiking
can occur was termed ‘backpropagation-activated $\rm Ca^{2+}$ spike firing’
(‘BAC firing’): A single somatic spike can backpropagate towards the apical
spike initiation zone, in turn significantly facilitating the initiation of a
dendritic spike (Stuart and Häusser, 2001; Spruston, 2008; Larkum, 2013). This
reciprocal coupling is believed to act as a form of coincidence detection: If
apical and basal synaptic input co-occurs, the neuron can respond with a rapid
burst of spiking activity. The firing rate of these temporal bursts exceeds
the firing rate that is maximally achievable under basal synaptic input alone,
therefore representing a form of temporal coincidence detection between apical
and basal input.

Naturally, these mechanisms also affect plasticity, and thus learning within
the cortex (Sjöström and Häusser, 2006; Ebner et al., 2019). While the
interplay between basal and apical stimulation and its effect on synaptic
efficacies is subject to ongoing research, there is evidence that BAC-firing
tends to shift plasticity towards long-term potentiation (LTP) (Letzkus et
al., 2006). Thus, coincidence between basal and apical input appears to also
gate synaptic plasticity.

In a supervised learning scheme, where the top-down input arriving at the
apical compartment acts as the teaching signal, the most straight-forward
learning rule for the basal synaptic weights would be derived from an
appropriate loss function, such as a mean square error, based on the
difference between basal and apical input, i.e. $I_{p}-I_{d}$, where indices
$p$ and $d$ denote ‘proximal’ and ‘distal’, in equivalence to basal and
apical. Theoretical studies have investigated possible learning mechanisms
that could utilize an intracellular error signal (Urbanczik and Senn, 2014;
Schiess et al., 2016; Guerguiev et al., 2017). However, a clear experimental
evidence for a physical quantity encoding such an error is—to our
knowledge—yet to be found. On the other hand, Hebbian-type plasticity is
extensively documented in experiments (Gustafsson et al., 1987; Debanne et
al., 1994; Markram et al., 1997; Bi and Poo, 1998). Therefore, our work is
based on the question of whether the nonlinear interactions between basal and
apical synaptic input could, when combined with a Hebbian plasticity rule,
allow a neuron to learn to reproduce an apical teaching signal in its proximal
input.

We investigate coincidence learning by combining a phenomenological model that
generates the output firing rate as a function of two streams of synaptic
input (subsuming basal and apical inputs) with classical Hebbian, as well as
BCM-like plasticity rules on basal synapses. In particular, we hypothesized
that this combination of neural activation and plasticity rules would lead to
an increased correlation between basal and apical inputs. Furthermore, the
temporal alignment observed in our study could potentially facilitate apical
inputs to act as top-down teaching signals, without the need for an explicit
error-driven learning rule. Thus, we also test our model in a simple linear
supervised classification task and compare it with the performance of a simple
point neuron equipped with similar plasticity rules.

##  2 Model

###  2.1 Compartamental Neuron

The neuron model used throughout this study is a discrete-time rate encoding
model that contains two separate input variables, representing the total
synaptic input current injected arriving at the basal (proximal) and apical
(distal) dendritic structure of a pyramidal neuron, respectively. The model is
a slightly simplified version of a phenomenological model proposed by Shai et
al. (2015). Denoting the input currents $I_{p}$ (proximal) and $I_{d}$
(distal), the model is written as

$\displaystyle\begin{split}y\left(t\right)&=\alpha\sigma\left(I_{p}(t)-\theta_{p0}\right)\left[1-\sigma\left(I_{d}(t)-\theta_{d}\right)\right]\\\
&+\sigma\left(I_{d}(t)-\theta_{d}\right)\sigma\left(I_{p}(t)-\theta_{p1}\right)\end{split}$
(1) $\displaystyle\sigma(x)$ $\displaystyle\equiv\frac{1}{1+\exp(-4x)}\;.$ (2)

Here, $\theta_{p0}>\theta_{p1}$ and $\theta_{d}$ are threshold variables with
respect to proximal and distal inputs. Equation (1) defines the firing rate
$y$ as a function of $I_{p}$ and $I_{d}$. Note that the firing rate is
normalized to take values within $y\in[0,1]$. In the publication by Shai et
al. (2015), firing rates varied between $0$ and $150\,\mathrm{Hz}$. High
firing rates typically appear in the form of bursts of action potentials,
lasting on the order of $50$–$100\,\mathrm{ms}$ Larkum et al. (1999); Shai et
al. (2015). Therefore, since our model represents “instantaneous" firing rate
responses to a discrete set of static input patterns, we conservatively
estimate the time scale of our model to be on the order of tenths of seconds.

In general, the input currents $I_{p}$ and $I_{d}$ are meant to comprise both
excitatory and potential inhibitory currents. Therefore, we did not restrict
the sign of of $I_{p}$ and $I_{d}$ to positive values. Moreover, since we
chose the thresholds $\theta_{p0}$ and $\theta_{d}$ to be zero, $I_{p}$ and
$I_{d}$ should be rather seen as a total external input relative to intrinsic
firing thresholds.

Note that the original form of this phenomenological model by Shai et al.
(2015) is of the form

$y(I_{p},I_{d})=\sigma\left(I_{p}-A\sigma(I_{d})\right)\left[1+B\sigma(I_{d})\right]\;,$
(3)

where $\sigma$ denotes the same sigmoidal activation function. This equation
illustrates that $I_{d}$ has two effects: It shifts the basal activation
threshold by a certain amount (here controlled by the parameter $A$) and also
multiplicatively increases the maximal firing rate (to an extent controlled by
$B$). Our equation mimics these effects by means of the two thresholds
$\theta_{p0}$ and $\theta_{p1}$, as well as the value of $\alpha$ relative to
the maximal value of y (which is $1$ in our case).

Overall, equation (1) describes two distinct regions of neural activation in
the $(I_{p},I_{d})$-space which differ in their maximal firing rates, which
are set to $1$ and $\alpha$, where $0<\alpha<1$. A plot of (1) is shown in
Fig. 1.

When both input currents $I_{d}$ and $I_{p}$ are large, that is, larger than
the thresholds $\theta_{d}$ and $\theta_{p1}$, the second term in (1)
dominates, which leads to $y\approx 1$. An intermediate activity plateau, of
strength $\alpha$ emerges in addition when $I_{p}>\theta_{p0}$ and
$I_{d}<\theta_{d}$. As such, the compartment model (1) is able to distinguish
neurons with a normal activity level, here encoded by $\alpha=0.3$, and
strongly bursting neurons, where the maximal firing rate is unity. The
intermediate plateau allows neurons to process the proximal inputs $I_{p}$
even in the absence of distal stimulation. The distal current $I_{d}$ acts
therefore as an additional modulator.

Figure 1: Two-compartment rate model. The firing rate as a function of
proximal and distal inputs $I_{p}$ and $I_{d}$, see (1). The thresholds
$\theta_{p0}$, $\theta_{p1}$ and $\theta_{d}$ define two regions of neural
activity, with a maximal firing rate of $1$ and a plateau in the lower-left
quadrant with a value of $\alpha=0.3$. That is, the latter region can achieve
$30\%$ of the maximal firing rate.

In our numerical experiments, we compare the compartment model with a
classical point neuron, as given by

$y(t)=\sigma\left(I_{p}(t)+I_{d}(t)-\theta\right)\;.$ (4)

The apical input $I_{d}$ is generated ‘as is’, meaning it is not dynamically
calculated as a superposition of multiple presynaptic inputs. For
concreteness, we used

$I_{d}(t)=n_{d}(t)x_{d}(t)-b_{d}(t)\;,$ (5)

where $n_{d}(t)$ is a scaling factor, $x_{d}(t)$ a discrete time sequence,
which represents the target signal to be predicted by the proximal input, and
$b_{d}(t)$ a bias. In our experiments, we chose $x_{d}$ according to the
prediction task at hand, see (17) and (19)–(20).

Note that $n_{d}$ and $b_{d}$ are time dependent since they are subject to
adaptation processes, which will be described in the next section. Similarly,
the proximal input $I_{p}(t)$ is given by

$I_{p}(t)=n_{p}(t)\sum_{i=1}^{N}x_{p,i}(t)w_{i}(t)-b_{p}(t)\;,$ (6)

where $N$ is the number of presynaptic afferents, $x_{p,i}(t)$ the
corresponding sequences, $w_{i}(t)$ the synaptic efficacies and $n_{p}(t)$ and
$b_{p}(t)$ the (time dependent) scaling and bias. Tyical values for the
parameters used throughout this study are presented in Table 1.

###  2.2 Homeostatic Parameter Regulation

The bias variables entering the definitions (5) and (6) of the distal proximal
current, $I_{d}$ and $I_{p}$, are assumed to adapt according to

$\displaystyle b_{p}(t+1)$
$\displaystyle=b_{p}(t)+\mu_{b}\left[I_{p}(t)-I_{p}^{t}\right]$ (7)
$\displaystyle b_{d}(t+1)$
$\displaystyle=b_{d}(t)+\mu_{b}\left[I_{d}(t)-I_{d}^{t}\right]\;,$ (8)

where $I_{p}^{t}=0$ and, $I_{d}^{t}=0$ are preset targets and
$1/\mu_{b}=10^{3}$ is the timescale for the adaption. Since this is a slow
process, over time, both the distal and the proximal currents, $I_{d}$ and
$I_{p}$, will approach a temporal mean equal to $I_{p}^{t}$ and $I_{d}^{t}$
respectively while still allowing the input to fluctuate. The reason for
choosing the targets to be zero lies in the fact that we expect a neuron to
operate in a dynamical regime that can reliably encode information from its
inputs. In the case of our model, this implies that neural input should be
distributed close to the threshold (which was set to zero in our case), such
that fluctuations in the can have an effect on the resulting neural activity.
See e.g. Bell and Sejnowski (1995) and Triesch (2007) for theoretical
approaches to optimizing gains and biases based on input and output
statistics. Hence, while we chose the mean targets of the input to be the same
as the thresholds, this is not a strict condition, as relevant information in
the input could also be present in parts of the input statistics that
significantly differ from its actual mean (for example in the case of a
heavily skewed distribution).

Adaptation rules for the bias entering a transfer function, such as (8) and
(7), have the task to regulate overall activity levels. The overall magnitude
of the synaptic weights, which are determined by synaptic rescaling factors,
here $n_{d}$ and $n_{p}$, as defined in (5) and (6), will regulate in contrast
the variance of the neural activity, and not the average level (Schubert and
Gros, 2021). In this spirit we consider

$\displaystyle n_{d}(t+1)$
$\displaystyle=n_{d}(t)+\mu_{n}\left[V_{d}^{t}-\left(I_{d}(t)-\tilde{I}_{d}(t)\right)^{2}\right]$
(9) $\displaystyle n_{p}(t+1)$
$\displaystyle=n_{p}(t)+\mu_{n}\left[V_{p}^{t}-\left(I_{p}(t)-\tilde{I}_{p}(t)\right)^{2}\right]$
(10) $\displaystyle\tilde{I}_{d}(t+1)$ $\displaystyle=(1-\mu_{\rm
av})\tilde{I}_{d}(t)+\mu_{\rm av}I_{d}(t)$ (11)
$\displaystyle\tilde{I}_{p}(t+1)$ $\displaystyle=(1-\mu_{\rm
av})\tilde{I}_{p}(t)+\mu_{\rm av}I_{p}(t)\;.$ (12)

Here, $V_{p}^{t}$ and $V_{p}^{t}$ define targets for the temporally averaged
variances of $I_{p}$ and $I_{d}$. The dynamic variables $\tilde{I}_{p}$ and
$\tilde{I}_{d}$ are simply low-pass filtered running averages of $I_{p}$ and
$I_{d}$. Overall, the framework specified here allows the neuron to be fully
flexible, as long as the activity level and its variance fluctuate around
preset target values (Schubert and Gros, 2021).

Mapping the control of the mean input current to the biases and the control of
variance to the gains is, in a sense, an idealized case of the more general
notion of dual homeostasis. As shown by Cannon and Miller (2017), the
conditions for a successful control of mean and variance by means of gains and
biases are relatively loose: Under certain stability conditions, a combination
of two nonlinear functions of the variable that is to be controlled can yield
a dynamic fixed point associated with a certain mean and variance. In fact, a
possible variant of dual homeostasis could potentially be achieved by coupling
the input gains to a certain firing rate (which is a non-linear function of
the input), while biases are still adjusted to a certain mean input. This, of
course, would make it harder to predict the variance of the input resulting
from such an adaptation, since it would not enter the equations as a simple
parameter that can be chosen a priori (as it is the case for equation (11) and
(12)).

A list of the parameter values used throughout this investigation is also
given in Table 1. Our choices of target means and variances are based on the
assumption that neural input should be tuned towards a certain working regime
of the neural transfer function. In the case of the presented model, this
means that both proximal and distal input cover an area where the
nonlinearities of the transfer function are reflected without oversaturation.

Table 1: Model parameters, as defined in sections 2.1 and 2.3. $\theta_{p0}$ | $0$ | $V_{d}^{t}$ | $0.25$
---|---|---|---
$\theta_{p1}$ | $-1$ | $\mu_{b}$ | $10^{-3}$
$\theta_{d}$ | $0$ | $\mu_{n}$ | $10^{-4}$
$\alpha$ | $0.3$ | $\mu_{\rm av}$ | $5\cdot 10^{-3}$
$\mu_{w}$ | $5\cdot 10^{-5}\quad$ | $I_{p}^{t}$ | $0$
$\epsilon$ | $0.1$ | $I_{d}^{t}$ | $0$
$V_{p}^{t}$ | $0.25$ |  |

###  2.3 Synaptic Plasticity

The standard Hebbian plasticity rule for the proximal synaptic weights is
given by

$\displaystyle w_{i}(t+1)$
$\displaystyle=w_{i}(t)+\mu_{w}\big{[}\left(x_{p,i}(t)-\tilde{x}_{p,i}(t)\right)\left(y(t)-\tilde{y}\right)-\epsilon
w_{i}(t)\big{]}$ (13) $\displaystyle\tilde{x}_{p,i}(t+1)$
$\displaystyle=(1-\mu_{\rm av})\tilde{x}_{p,i}(t)+\mu_{\rm av}x_{p,i}(t)$ (14)
$\displaystyle\tilde{y}(t+1)$ $\displaystyle=(1-\mu_{\rm
av})\tilde{y}(t)+\mu_{\rm av}y(t)$ (15)

The trailing time averages $\tilde{x}_{p,i}$ and $\tilde{y}$, respectively of
the presynaptic basal activities, $x_{p,i}$, and of the neural firing rate
$y$, enter the Hebbian learning rule (13) as reference levels. Pre- and post-
synaptic neurons are considered to be active/inactive when being above/below
the respective trailing averages. This is a realization of the Hebbian rule
proposed by Linsker (1986). The timescale of the averaging, $1/\mu_{\rm av}$,
is 200 time steps, see Table 1. As discussed in Section 2.1, a time step can
be considered to be on the order of $100\,\mathrm{ms}$, which equates to an
averaging time of about $20\,\mathrm{s}$. Generally, this is much faster than
the timescales on which metaplasticity, i.e. adaptation processes affecting
the dynamics of synaptic plasticity itself, are believed to take place, which
are on the order of days (Yger and Gilson, 2015). However, it should be noted
that our choice of the timescale of the averaging process used in our
plasticity model is motivated mostly by considerations regarding the overall
simulation time: Given enough update steps, the same results could be achieved
by an arbitrarily slow averaging process.

Since classical Hebbian learning does not keep weights bounded, we use an
additional proportional decay term $\epsilon w_{i}$ which prevents runaway
growth using $\epsilon=0.1$. With $1/\mu_{w}=2\cdot 10^{4}$, learning is
assumed to be considerably slower, as usual for statistical update rules. For
comparative reasons, the point neuron model (4) is equipped with the same
plasticity rule for the proximal weights as (13).

Apart from classical Hebbian learning, we also considered a BCM-like learning
rule for the basal weights (Bienenstock et al., 1982; Intrator and Cooper,
1992). The form of the BCM-rule used here reads

$w_{i}(t+1)=w_{i}(t)+\mu_{w}\big{[}y\left(y-\theta_{M}\right)x_{i}-\epsilon
w_{i}\big{]}\;,$ (16)

where $\theta_{M}$ is a threshold defining a transition from long-term
potentiation (LTP) to long-term depression (LTD) and, again, $\epsilon$ is a
decay term on the weights preventing unbounded growth. In the variant
introduced by Law and Cooper (1994), the sliding threshold is simply the
temporal average of the squared neural activity, $\theta_{M}=\langle
y^{2}\rangle$. In practice, this would be calculated as a running average,
thereby preventing the weights from growing indefinitely.

However, for our compartment model, we chose to explicitly set the threshold
to be the mean value between the high- and low-activity regime in our
compartment model, i.e. $\theta_{M}=(1+\alpha)/2$. By doing so, LTP is
preferably induced if both basal and apical input is present at the same time.
Obviously, for the point model, the reasoning behind our choice of
$\theta_{M}$ did not apply. Still, to provide some level of comparability, we
also ran simulations with a point model where the sliding threshold was
calculated as a running average of $y^{2}$.

Figure 2: Input Space for the Linear Classification Task. Two clusters of
presynaptic basal activities were generated from multivariate Gaussian
distributions. Here, $s$ denotes the standard deviation orthogonal to the
normal vector $\mathbf{a}$ of the classification hyperplane, as defined by
(17).

##  3 Results

###  3.1 Unsupervised Alignment between Basal and Apical Inputs

As a first test, we quantify the neuron’s ability to align its basal input to
the apical teaching signal. This can be done using the Pearson correlation
coefficient $\rho[I_{p},I_{d}]$ between the basal and apical input currents.
We determined $\rho[I_{p},I_{d}]$ after the simulation, which involves all
plasticity mechanisms, both for the synaptic weights and the intrinsic
parameters. The input sequences $x_{p,i}(t)$ is randomly drawn from a uniform
distribution, in $[0,1]$, which is done independently for each $i\in[1,N]$.

For the distal current $I_{d}(t)$ to be fully ‘reconstructable’ by the basal
input, $x_{d}(t)$ has to be a linear combination

$\displaystyle x_{d}(t)$ $\displaystyle=\sum_{i=1}^{N}a_{i}x_{p,i}(t)$ (17)

of the $x_{p,i}(t)$, where the $a_{i}$ are the components of a random vector
$\mathbf{a}$ of unit length.

Given that we use with (13) a Hebbian learning scheme, one can expect that the
direction and the magnitude of the principal components of the basal input may
affect the outcome of the simulation significantly: A large variance in the
basal input orthogonal to the ‘reconstruction vector’ $\mathbf{a}$ is a
distraction for the plasticity. The observed temporal alignment between
$I_{p}$ and $I_{d}$ should hence suffer when such a distraction is present.

In order to test the effects of distracting directions, we applied a
transformation to the input sequences $x_{p,i}(t)$. For the transformation,
two parameters are used, a scaling factor $s$ and the dimension $N_{\rm dist}$
of the distracting subspace within the basal input space. The $N_{\rm dist}$
randomly generated basis vectors are orthogonal to the superposition vector
$\mathbf{a}$, as defined by (17), and to each others. Within this $N_{\rm
dist}$-dimensional subspace, the input sequences $x_{p,i}(t)$ are rescaled
subsequently by the factor $s$. After the learning phase, a second set of
input sequences $x_{p,i}(t)$ and $x_{d}(t)$ is generated for testing purposes,
using the identical protocol, and the cross correlation $\rho[I_{p},I_{d}]$
evaluated. During the testing phase plasticity is turned off.

Figure 3: Unsupervised Alignment between Basal and Apical Input. Color encoded
is the Pearson correlation $\rho[I_{p},I_{d}]$ between the proximal and distal
input currents, $I_{p}$ and $I_{d}$. A–C: Classical Hebbian plasticity, as
defined by (13). D–F: BCM rule, see (16). Data for a range $N_{\rm
dist}\in[0,N-1]$ of the orthogonal distraction directions, and scaling factors
$s$, as defined in Fig. 2. The overall number of basal inputs is $N=100$. In
the bar plot on the right the sum $\Sigma_{\rm acc}$ over
$s=0,\,0.5,\,1.0\,..$ of the results is shown as a function of $N_{\rm dist}$.
Blue bars represents the compartment model, orange the point model.

The overall aim of our protocol is to evaluate the degree $\rho[I_{p},I_{d}]$
to which the proximal current $I_{p}$ aligns in the temporal domain to the
distal input $I_{d}$. We recall that this is a highly non-trivial question,
given that the proximal synaptic weights are adapted via Hebbian plasticity,
see (13). The error $(I_{p}-I_{d})^{2}$ does not enter the adaption rules
employed. Results are presented in Fig. 3 as a function of the distraction
parameters $s$ and $N_{\rm dist}\in[0,N-1]$. The total number of basal inputs
is $N=100$.

For comparison, in Fig. 3 data for both the compartment model and for a point
neuron are presented (as defined respectively by (1) and (4)), as well as
results for both classical Hebbian and BCM learning rules. A decorrelation
transition as a function of the distraction scaling parameter $s$ is observed
for both models and plasticity rules. In terms of the learning rules, only
marginal differences are present. However, the compartment model is able to
handle a significantly stronger distraction as compared to the point model.
These findings support the hypothesis examined here, namely that nonlinear
interactions between basal and apical input improve learning guided by top-
down signals.

###  3.2 Supervised Learning in a Linear Classification Task

Next, we investigated if the observed differences would also improve the
performance in an actual supervised learning task. For this purpose, we
constructed presynaptic basal input $x_{p}(t)$ as illustrated in Fig. 2.
Written in vector form, each sample from the basal input is generated from,

$\mathbf{x}_{p}(t)=\mathbf{b}+\mathbf{a}\big{[}c(t)+\sigma_{a}\zeta_{a}(t)\big{]}+s\cdot\sum_{i=1}^{N_{\rm
dist}}\zeta_{dist,i}(t)\mathbf{v}_{{\rm dist},i}\;,$ (18)

where $\mathbf{b}$ is a random vector, where each entry is drawn uniformly
from $[0,1]$, $\mathbf{a}$ is random unit vector as introduced in Section 3.1,
$c(t)$ is a binary variable drawn from $\\{-0.5,0.5\\}$ with equal probability
and $\zeta_{a}(t)$ and the $\zeta_{dist,i}(t)$ are independent Gaussian random
variables with zero mean and unit variance. Hence, $\sigma_{a}$ simply denotes
the standard deviation of each Gaussian cluster along the direction of the
normal vector $\mathbf{a}$ and was set to $\sigma_{a}=0.25$. Finally, the set
of $\mathbf{v}_{{\rm dist},i}$ forms a randomly generated orthogonal basis of
$N_{\rm dist}$ unit vectors which are—as in Section 3.1—also orthogonal to
$\mathbf{a}$. The free parameter $s$ parameterizes the standard deviation
along this subspace orthogonal to $\mathbf{a}$. As indicated by the time
dependence, the Gaussian and binary random variables are drawn for each time
step. The vectors $\mathbf{b}$, $\mathbf{a}$, and $\mathbf{v}_{{\rm dist},i}$
are generated once before the beginning of a simulation run.

For the classification task, we use two output neurons, indexed 0 and 1,
receiving the same basal presynaptic input, with the respective top-down
inputs $x_{d,0}$ and $x_{d,1}$ encoding the desired linear classification in a
one-hot scheme,

$\displaystyle x_{d,0}(t)$
$\displaystyle=1-\Theta\left(\left(\mathbf{x}_{p}(t)-\mathbf{b}\right)^{T}\mathbf{a}\right)$
(19) $\displaystyle x_{d,1}(t)$
$\displaystyle=\Theta\left(\left(\mathbf{x}_{p}(t)-\mathbf{b}\right)^{T}\mathbf{a}\right)\;,$
(20)

where $\Theta(x)$ is the Heaviside step function.

As in the previous experiment, we ran a full simulation until all dynamic
variables reached a stationary state. After this, a test run without
plasticity and with the apical input turned off was used to evaluate the
classification performance. For each sample, the index of the neuron with the
highest activity was used as the predicted class. Accuracy was then calculated
as the fraction of correctly classified samples.

Figure 4: Binary Classification Accuracy. Fraction of correctly classified
patterns as illustrated in Fig. 2, see Section 3.2. A–C: Classical Hebbian
plasticity. D–F: BCM rule. In the bar plot on the right the sum $\Sigma_{\rm
acc}$ over $s=0,\,0.5,\,1.0\,..$ of the results is given as a function of
$N_{\rm dist}$. Blue bars represents the compartment model, orange the point
model.

The resulting accuracy as a function of $N_{\rm dist}$ and $s$ is shown in
Fig. 4, again for all four combinations of neuron models and learning rules.

For classical Hebbian plasticity, the differences between compartmental and
point neuron are small. Interestingly, the compartment model performs
measurably better in the case of the BCM rule (16), in particular when the
overall accuracies for the tested parameter range are compared, see Fig. 4D.
This indicates that during learning, the compartmental neuron makes better
use, of the three distinct activity plateaus at $0$, $\alpha$ and $1$, when
the BCM rule is at work. Compare Fig. 1. We point out in this respect that the
sliding threshold $\theta_{M}$ in (16) has been set to the point halfway
between the two non-trivial activity levels, $\alpha$ and $1$.

It should be noted that the advantage of the compartment model is also
reflected in the actual correlation between proximal and distal input as a
measure of successful learning (as done in the previous section), see Fig. 5
in the appendix. Interestingly, the discrepancies are more pronounced when
measuring the correlation as compared to the accuracy. Moreover, it appears
that above-chance accuracy is still present for parameter values where
alignment is almost zero. We attribute this effect to the fact that the
classification procedure predicts the class by choosing the node that has the
higher activity, independent of the actual “confidence" of this prediction,
i.e. how strong activities differ relative to their actual activity levels.
Therefore, marginal differences can still yield the correct classification in
this isolated setup, but it would be easily disrupted by finite levels of
noise or additional external input.

##  4 Discussion

Pyramidal neurons in the brain possess distinct apical/basal
(distant/proximal) dendritic trees. It is hence likely that models with at
least two compartments are necessary for describing the functionality of
pyramidal neurons. For a proposed two-compartment transfer function (Shai et
al., 2015), we have introduced both unsupervised and supervised learning
schemes, showing that the two-compartment neuron is significantly more robust
against distracting components in the proximal input space than a
corresponding (one-compartment) point neuron.

The apical and basal dendritic compartments of pyramidal neurons are located
in different cortical layers Park et al. (2019), receiving top-down and feed-
forward signals, respectively. The combined action of these two compartments
is hence the prime candidate for the realization of backpropagation in multi-
layered networks (Bengio, 2014; Lee et al., 2015; Guerguiev et al., 2017).

###  4.1 Learning Targets by Maximizing Correlation

In the past, backpropagation algorithms for pyramidal neurons concentrated on
learning rules that are explicitly dependent on an error term, typically the
difference between top-down and bottom-up signals. In this work, we considered
an alternative approach. We postulate that the correlation between proximal
and distal input constitutes a viable objective function, which is to be
maximized in combination with homeostatic adaptation rules that keep proximal
and distal inputs within desired working regimes. Learning correlations
between distinct synaptic or compartmental inputs is as a standard task for
Hebbian-type learning, which implies that the here proposed framework is based
not on supervised, but on biologically viable unsupervised learning schemes.

The proximal input current $I_{p}$ is a linear projection of the proximal
input space. Maximizing the correlation between $I_{p}$ and $I_{d}$ (the
distal current), can therefore be regarded as a form of canonical correlation
analysis (CCA) (Härdle and Simar, 2007). The idea of using CCA as a possible
mode of synaptic learning has previously been investigated by Haga and Fukai
(2018). Interestingly, according to the authors, a BCM-learning term in the
plasticity dynamics accounts for a principal component analysis in the input
space, while CCA requires an additional multiplicative term between local
basal and apical activity. In contrast, our results indicate that such a
multiplicative term is not required to drive basal synaptic plasticity towards
a maximal alignment between basal and apical input, even in the presence of
distracting principal components. Apart from the advantage that this avoids
the necessity of giving a biophysical interpretation of such cross-terms, it
is also in line with the view that synaptic plasticity should be formulated in
terms of local membrane voltage traces (Clopath et al., 2010; Weissenberger et
al., 2018). According to this principle, distal compartments should therefore
only implicitly affect plasticity in basal synapses, e.g. by facilitating
spike initiation.

###  4.2 Generalizability of the Model to Neuroanatomical Variabillity

While some research on cortical circuits suggests the possibility of generic
and scalable principles that apply to different cortical regions and their
functionality (Douglas and Martin, 2007; George and Hawkins, 2009; Larkum,
2013), it is also well known that the anatomical properties of pyramidal
neurons, in particular the dendritic structure, varies significantly across
cortical regions (Fuster, 1973; Funahashi et al., 1989). More specifically,
going from lower to higher areas of the visual pathway, one can observe a
significant increase of spines in the basal dendritic tree (Elston and Rosa,
1997; Elston, 2000), which can be associated with the fact that neurons in
higher cortical areas generally encode more complex or even multi-sensory
information, requiring the integration of activity from a higher number and
potentially more distal neurons (Elston, 2003; Luebke, 2017).

With respect to a varying amount of basal synaptic inputs, it is interesting
to note that the dimensionality $N$ of the basal input patterns did not have a
large effect on the results of our model, see Fig. 3–5, as long as the
homeostatic processes provided weight normalization.

Apart from variations in the number of spines, variability can also be
observed within the dendritic structure itself (Spruston, 2008; Ramaswamy and
Markram, 2015). Such differences obviously affect the internal dynamics of the
integration of synaptic inputs. Given the phenomenological nature of our
neuron model, it is hard to predict how such differences would be reflected,
given the diverse dynamical properties that can arise from the dendritic
structure (Häusser et al., 2000). The two models tested in our study can be
regarded as two extreme cases, where the point neuron represents a completely
linear superposition of inputs and the compartment model being strongly
nonlinear with respect to proximal and distal inputs. In principle, pyramidal
structures could also exhibit properties in between, where the resulting
plasticity processes would show a mixture between the classical point neuron
behavior (e.g. if a dimensionality reduction of the input via PCA is the main
task) and a regime dominated by proximal-distal input correlations if top-down
signals should be predicted.

###  4.3 Outlook

Here we concentrated on one-dimensional distal inputs. For the case of higher-
dimensional distal input patterns, as for structured multi-layered networks,
it thus remains to be investigated how target signals are formed. However, as
previous works have indicated, random top-down weights are generically
sufficient for successful credit assignment and learning tasks (Lillicrap et
al., 2016; Guerguiev et al., 2017). Therefore, we expect that our results can
be also transferred to deep network structures, for which plasticity is
classically guided by local errors between top-down and bottom-up signals.

##  5 Appendix

###  5.1 Alignment in the Classificaction Task

Instead of measuring the model performance in the classification task
presented in Sect. 3.2 by the fraction of correctly classified patterns, as
shown in Fig. 4, one can also use the correlation between $I_{p}$ and $I_{d}$,
as done in Sect. 3.1. This is shown in Fig. 5. One observes a more pronounced
difference between the point model and the compartment model, where the latter
results in an overall better alignment for the tested parameter space.

Figure 5: Alignment between Basal and Apical Input after Binary Classification
Learning. Correlation between proximal and distal inputs after training, as
described in Sect. 3.2. A–C: Classical Hebbian plasticity. D–F: BCM rule. In
the bar plot on the right the sum $\Sigma_{\rm acc}$ over
$s=0,\,0.5,\,1.0\,..$ of the results is shown as a function of $N_{\rm dist}$.
Blue bars represents the compartment model, orange the point model.

###  5.2 Objective Function of BCM Learning in the Compartment Model

To gain a better understanding of why the BCM-type learning rule in
combination with the implemented compartment model drives the neuron towards
the temporal alignment between $I_{p}$ and $I_{d}$, we can formalize the
learning rule for the proximal weights in terms of an objective function. For
this purpose, we further simplify (1) by replacing the sigmoid functions
$\sigma(x)$ by a simple step function $\Theta(x)$. This does not change the
overall shape or topology of the activation in the $(I_{p},I_{d})$ space but
merely makes the smooth transitions sharp and instantaneous. Using $\Delta
w_{i}\propto y\left(y-\theta_{M}\right)x_{i}$, we find in this case

$\Delta w_{i}\ \propto\
\Big{[}(1-\alpha)\Theta(I_{d}-\theta_{d})\Theta(p-\theta_{p1})\\\
+\alpha(\alpha-1)\Theta(\theta_{d}-I_{d})\Theta(p-\theta_{p0})\Big{]}x_{i}\;.$
(21)

Noting that $\Theta(x)$ is the first derivative of the ReLu function
$[x]^{+}\equiv\max(0,x)$, we find that this update rule can be written as

$\displaystyle\Delta w_{i}$ $\displaystyle\propto$
$\displaystyle\frac{\partial\mathcal{L}_{p}}{\partial w_{i}}$
$\displaystyle\mathcal{L}_{p}$ $\displaystyle=$
$\displaystyle(1-\alpha)\Theta(I_{d}-\theta_{d})[p-\theta_{p1}]^{+}+\alpha(\alpha-1)\Theta(\theta_{d}-I_{d})[p-\theta_{p0}]^{+}\;.$
(22)

The objective function $\mathcal{L}_{p}$ is shown in Fig. 6. One observes that
states closer to the $I_{p}$-$I_{d}$ diagonal are preferred since they tend to
yield higher values of $\mathcal{L}_{p}$, while the opposite is the case for
off-diagonal states.

Figure 6: Objective Function for the Proximal Weight Update. The approximate
objective function for the proximal weights as given in (22) as a 3d-plot (A)
and color-coded (B). This corresponds to a combination of using (1) together
with (16). Note the ridge-like structure along the $I_{p}$-$I_{d}$ diagonal,
which supports the alignment between proximal and distal input.

It should be noted, though, that the objective function is not scale-invariant
(as would be e.g. if the squared error was used) in the sense that the prior
distributions of both proximal and distal inputs need a certain mean and
variance to cover a region of input states for which the described effects can
take place. As a counterexample, one could imagine that the input samples only
covered a flat area of $\mathcal{L}_{p}$, as for example in Fig. 6B in the
lower-left quadrant, leading to a zero average gradient. This is prevented,
however, by the homeostatic processes acting simultaneously on the gains and
biases, making sure that the marginal distributions of $I_{p}$ and $I_{d}$ are
such that higher correlations are preferred. For example, if we assume a
Gaussian marginal distribution for both $I_{p}$ and $I_{d}$ with zero means
and a standard deviation of $0.5$ (which is used as a homeostatic target in
the simulations), the expected value of $\mathcal{L}(I_{p},I_{d})$ is $-0.055$
if $I_{p}$ and $I_{d}$ are completely uncorrelated, and $0.07$ in the
perfectly correlated case.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

Both authors, F.S. and C.G., contributed equally to the writing and review of
the manuscript. F.S. provided the code, ran the simulations, and prepared the
figures.

## Acknowledgments

The authors acknowledge the financial support of the German Research
Foundation (DFG)

## Data Availability Statement

The simulation datasets for this study can be found under
https://cloud.itp.uni-frankfurt.de/s/mSRJ6BPXjwwHmfq. The simulation and
plotting code for this project can be found under
https://github.com/FabianSchubert/frontiers_dendritic_coincidence_detection.

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