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Neural communication systems
Péter András
School of Computing Science
University of Newcastle
Newcastle upon Tyne, NE1 7RU, UK
[email protected]
Abstract – Understanding how biological neural
systems work needs appropriate models that focus on
relevant features of neural activities and ignore
irrelevant ones. Here we propose a new way of
describing biological neural systems in terms of
abstract communication systems. Such systems use
pattern languages formulated as probabilistic
continuation
rules
that
determine
which
communications follow other communications. We
apply this framework to define neural communication
systems. The formal analysis is followed by a brief
interpretation of how the crab stomatogastric ganglion
works in the context of the proposed framework. We
also discuss further implications of the description of
neural systems as abstract communication systems.
formalized description of neural systems as activity pattern
systems and outline how to apply this interpretational context
to the case of the stomatogastric ganglion (STG) of crabs. We
also discuss further implications of the description of neural
systems as abstract communication systems.
The rest of the paper is structured as follows. Section 2
presents a conceptual framework of communication systems.
Section 3 discusses the application of this framework to the
description of neural systems and shows how to apply this
context to the interpretation and analysis of the STG. Section
4 discusses the general implications of the proposed
interpretational framework in the context of analysis and
understanding of neural systems. The paper is closed by the
conclusions section.
I. INTRODUCTION
II. ABSTRACT COMMUNICATION SYSTEMS
To understand how biological neural systems work it is
crucial to select the right models providing an appropriate
level of abstraction, which can capture the characteristic
features of the system and ignore the irrelevant variations.
Several models were proposed in the past decades. Some
of these emphasize the role of single neurons as fully
functional computational building blocks (e.g., neurons
representing basis functions in the context of
approximation tasks) [12]. Others focus on the role of
neural populations and averaging computations performed
by such populations [8] or synchronisation in such
populations [4]. Another group highlights the nonlinear
dynamics of neural activity patterns and suggest that
neural computations may be performed by switching
between attractors defined in the abstract space of activity
patterns [5,6,7]. Further others concentrate on small
networks, suggesting ingenious combinations of
connections and receptive fields to achieve certain
computational functions (e.g., perception of various
elementary visual stimulus patterns) [9]. More recently it
was suggested that pattern languages [2,15] can be used to
interpret the activity patterns recorded from neural
systems [1,3]. Furthermore, models were proposed to
show how computations with such activity patterns can be
performed [1,2].
This paper follows the track of interpretation of neural
systems in terms of pattern languages. We propose a
We introduce in this section a conceptual framework for the
description of abstract communication systems [10].
Communication units are entities which produce
communication behaviours. We ignore the internal structure
of communication units. Communication units may perceive
communication behaviours of other units and in response
may generate communication behaviours. Communication is
the process of producing a communication behaviour by a
communication unit followed by the perception of this
behaviour by another communication unit.
A communication system is made of communications
between communication units. Communication units are not
part of the communication system, and they may participate
in communications that are part of different communication
systems. Communications are produced according to
probabilistic rules depending on earlier communications;
these rules are called continuation rules. The set of earlier
communications on which depends the production of a new
communication is called the referencing set of the new
communication. We say that the new communication is
referencing to the communications of the reference set. A
communication system is characterized by a set of such
continuation rules. Those communications are part of the
communication systems, which follow the rules
characterising the communication system. A communication
system consists of a dense referencing cluster of
communications (see Figure 1).
inner structure of neurons from the pure perspective of the
neural communication system and we consider the neuronal
structure only as part of the environment of the neural
communication system. The communication behaviours that
we consider are rare single spikes, slow spiking, fast spike
burst, no spiking and other similar spike sequence patterns.
We will denote neurons as X k and spike sequence patterns
as x j . The fact that neuron X k produced the spike pattern
x j at time t − τ is denoted as ( X k , x j ) −τ , where t is the
current time .
Figure 1. A communication system. The system is made
of a dense cluster of communications referencing to other
system communications. The communication units are not
part of the communication system.
The environment of a communication system is made of
other communications that are not part of the system. The
environment poses constraints on the communication
system by imposing the likelihood of producing
communication behaviours that are components of
communications.
Communication systems compete in the environment for
communications to maintain themselves. Communication
systems that fit better the environmental constraints may
out compete other communication systems that fit less
well to the environmental constraints (i.e., the better
fitting system is more likely to be maintained by
communications following its rules than a less well fitting
system).
The communications composing the neural communication
system are spike sequence patterns produced by neurons and
perceived by the same or other neurons. These
communications depend on earlier communications between
neurons. This dependence we formalize as rules expressing
probabilities of producing spike sequence patterns x j by a
neuron
X k depending on spike sequence patterns produced
by the same or other neurons that are connected to the neuron
X k . Formally we write
PXi k ( x j | R i ( X k , x j ))
(1)
i
where R ( X k , x j ) is the referencing set of the rule. The
same spike sequence pattern
x j may be produced by several
rules with different referencing sets. In general a referencing
set is of the form
( X u , x v ) −τ | X u is a neuron  (2)
Ri (X k , x j ) = 


connected to X k , τ > 0
III. NEURAL SYSTEMS
In this section we show how to apply the above described
communication system framework to interpret and
analyse neural systems. First, we consider the
fundamental components (e.g., behaviours, rules,
environment). Second, we provide a formalized
description of neural systems using the above defined
concept. Third, we analyse this formal description.
Finally, we show how to apply this framework to interpret
the crab STG as a neural communication system.
X k sending connections to X k have to be
listed in the referencing set of all rules applying to X k , and
only a finite selection of τ values appear in any referencing
A. Neurons and communications
The environment of neural communications systems is
composed
of
other
communications,
including
communications at the level of components of neurons. The
environment imposes constraints on the neural
communications system in terms of imposing the likelihood
of communication behaviours, i.e., of spike sequence
patterns. We write these constraints as
We consider neurons as communication units and spike
sequence patterns of neurons as communication
behaviours in neural systems. This implies that in the
context of the above described framework we ignore the
Not all neurons
set. An inter-referencing cluster of neural communications
that follow a set of continuation rules constitutes a neural
communication system. A neuron may produce
communication behaviours that are part of different neural
communication systems.
(3)
Q( x j )
the probabilities of occurrence of spike sequence patterns
x j . These probabilities may change in effect of
neuromodulators or changes within the neurons (e.g.,
changing expression of genes). A neural communication
system can be maintained in such environment if an
appropriate distribution of spike sequence patterns is
produced by application of its rules.



 p kj =





rj
∑a
i =1
kij
⋅
(9)
∏p
uv
( X u , xv )∈R i ( X k , x j )
rj
∑
∏p
uv
i =1 ( X u , xv )∈R i ( X k , x j )
N
qj =
; k = 1, N ; j = 1, M
1
N
∑p
k =1
kj
; j = 1, M
with the constraints that pkj ∈ [0,1] . The parameters of the
B. Neural communication systems
above system are a kij
Considering the above descriptions we can write the
equations determining the neural communication systems
that can exist under a set of environmental constraints.
The following notations are used:
A neural communication system is characterised by a subset
of a kij values for which the above system has at least one
a kij = PXi k ( x j | R i ( X k , x j ))
(4)
q j = Q( x j )
(5)
∈ (0,1) and q j ∈ (0,1) .
solution in terms of pkj -s, with the q j -s being given by the
environmental constraints. Not all subsets of a kij values
allow a solution for the equation system (9).
Setting the environmental constraints (i.e., q j -s) selects the
The probability of producing spike sequence pattern
x j by neuron X k is
neural communication systems that are compatible with the
constraints. If there is only one subset of continuation rules
(i.e., a kij -s) that allows a solution of the (9) system, the
pkj = P ( X k , x j ) =
neural communication system corresponding to this set of
rules is selected by the environment. The neural
communications may fluctuate between participating in the
possible neural communication systems if there are more than
one subsets of continuation rules allowing a solution. New
system behaviours are learned by selecting a new subset of
continuation rules (i.e., a kij -s) that fit to new constraints
rj
∑P
i =1
i
Xk
( x j | R i ( X k , x j )) ⋅ P i ( R i ( X k , x j ))
rj
∑ P ( R( X
i
i =1
k
(6)
, x j ))
where rj is the number of rules for neuron X k that
imposed by the environment.
produce spike sequence pattern x j . The probability of
The dynamics of changing the active neural communication
system is driven by the difference between the actual
distribution of x j -s and the distribution of them imposed by
producing spike sequence pattern x j by the system is
P( x j ) =
1
N
(7)
N
∑ P( X k , x j )
k =1
where N is the number of neurons in the system. For
simplicity, we suppose the independence of the referenced
communication. In this case the likelihood of satisfactory
referencing sets is given by
P i ( R i ( X k , x j )) =
∏ P( X
u
, xv )
(8)
i
( X u , xv )∈R ( X k , x j )
The equations determining
communication systems are
the
possible
neural
the environmental constraints. The effect of the
environmental constraints is the alteration of the output of the
continuation rules. The difference between the actual and
environmental constraints implied distribution imposes a
prior distribution over the communication behaviours x j
qˆ j = Qˆ ( x j ) ∝ Q( x j ) − P( x j )
(10)
The effective likelihood of producing spiking sequence
pattern x j by neuron X k is given by
pˆ kj =
p kj ⋅ qˆ j
M
∑p
j =1
kj
(11)
behaviours are produced by appropriate neurons.
C. Formal analysis
The system described by equations (9) can be transformed
into a Markov chain description that allows formal
analysis and solution of it.
In the first step we introduce time lagged copies of the
d
neurons, X k being the notation for the representation of
neuron X k with time lag d . In this augmented system all
reference sets can be transformed such that they contain
only the last communication of other neurons, i.e., if
( X u , xv ) −τ ∈ R i ( X k , x j ) then
we
replace
τ
( X u , x v ) −τ in the reference set by ( X u , xv ) .
1
(12)
N'
where N ' is the number of neurons in the augmented
system, and the neurons are renumbered. We can compute
the probabilities of P (α | β ) where α , β are two such
states as follows
N'
P (α | β ) = ∏ PX k ( x jα | β )
i
j
Γ = {P (α | β )}
(15)
where the transition probabilities are considered for all state
pairs. Then the largest eigenvalue of Γ is 1 and if w is the
eigenvector corresponding to this eigenvalue than the
components of w give the stationary probabilities of the
states of the augmented neural system.
Making
the
simplification
assumption
that
the
communication behaviour of the communication units (i.e.,
neurons and their lagged copies) are independent we can get
an approximation of the distribution of spiking behaviours of
the neurons. First we calculate the probability of a state α
N'
N'
P (α ) = ∏ P( X k , x jα ) =∏ p kjα = w(α )
k =1
where
and
k
k =1
(13)
k
1
⋅
m( X k , x j , β )
if
(16)
k
w(α ) is the component of w that corresponds to α ,
k
α ( j1α ,K, j αN ' ) = [( X 1 , x jα ),K , ( X N ' , x jα )]
∑ P (x
In this way we can calculate the transition probabilities
between states α , β . Having the transition probabilities we
can write the transition matrix of the augmented system
p kjα = p k ' jα
Next we define the states
k
that the conditions of
R ( X k , x j ) are satisfied by β , i.e., the required spiking
By producing spike sequence patterns forced by the
environment, but which do not fit into the existing neural
communication system, the communications move toward
a new neural communication system. In effect a new
neural communication system emerges from the
communications between neurons, such that the
communications following the rules of the new system
produce a communications behaviour distribution that fits
the environmental constraints.
PX k ( x jα | β ) =
β ≡ R i ( X k , x j ) denotes
i
⋅ qˆ j
k =1
where
(17)
k'
X k ' is a lagged copy of X k and j kα = j kα' .
The equations (16) allow us to calculate the values of
the calculation of
p j = P( x j ) for the whole neural system
as
pj =
1
N
N
∑ pkj
(18)
k =1
Those neural systems are allowed by the environmental
constraints q j = Q ( x j ) for which the system satisfies the
| R( X k , x j ))
equations
i
β ≡R ( X k ,x j )
1
=
⋅ ∑ a kij
m( X k , x j , β ) β ≡ R i ( X k , x j )
p kj and
pj = qj
(14)
(19)
For any selection of rules we can compute the
corresponding p j -s and we can check if they satisfy the
constraints (19). The sets of rules compatible with the
constraints determine the possible neural systems. If there
are more than one such systems, the relations defined
above allow to calculate the transition probabilities
between these systems.
D. Real neural systems
The crab stomatogastric ganglion (STG) is a simple and
well-known neural system [11,13]. Each neuron has been
thoroughly studied individually, the connectivity structure
of the ganglion and many details about the transmitters
and neuromodulators involved in the activity of the STG
are well known. It is still not very clear how this relatively
small system (i.e., 30-40 neurons) composes several
functional networks (i.e., gastric network, pyloric
network, etc.) and produces a large variety of output
activity patterns that drive a rich repertoire of muscle
movements in the gastric system of the crab.
The above described framework for interpreting activities
in neural systems provides a new way to understand how
the STG works. In our framework we can consider the
functional networks as neural communication systems
that dominate communications between a subset of the
STG neurons under certain environmental constraints.
The environmental constraints in the STG are provided by
the neuromodulators and a limited amount of input from
higher ganglions (e.g., commissural ganglions) and
sensory neurons that switch the STG between supporting
one or another functional network.
The wide variety of activity and output patterns of the
STG can be interpreted as the result of a multitude of
continuation rules for spike sequence patterns. Following
our interpretational context inputs form higher nervous
centres (e.g., commissural ganglions and oesophagal
ganglion) and smaller variations of neuromodulators
select between continuation rules that are applied and in
effect generate the variety of observed activities in the
STG.
Testing our predictions about the applicability of the
proposed interpretational framework should be possible
by analysing simultaneous activity patterns of many
neurons of the STG. Such analysis should reveal the
active continuation rules. Measuring the amount of
neuromodulators and higher inputs should allow
establishing the effective environmental constraints that
select from the continuation rules to form the functional
networks within the STG.
IV. DISCUSSION
The analysis and interpretation framework of abstract
communication systems offers a new look on how biological
neural systems work. It emphasizes the role of inner
processes of such systems in accordance with pattern
language descriptions [2], and provides a relatively easy way
to understand and analyse multiple neural communication
systems supported by the same set of interconnected
biological neurons. The abstract communication systems
approach also makes it possible to build a formal description
of the neural system and to perform formal analysis of it in
order to determine the possible neural communication
systems and their modes of working.
The proposed framework gives an appropriate formal
framework for the analysis of how pattern languages [2,15]
may be used to describe effectively biological neural systems.
This opens up a range of possibilities to formulate activity
pattern communication rules of real neural systems on the
basis of real activity data, and to generate predictions about
such systems using pattern language descriptions of them.
As we described in the interpretation of the STG activity, the
neural communication systems approach provides a new way
of understanding the role of neuromodulation. This approach
suggests that neuromodulation essentially sets environmental
constraints on the neural communication system, and by these
constraints selects one or another mode of function of the
neural system. These modes of function may mean that the
set of interconnected neurons support one or more neural
communication system, some neurons participating possibly
in more than one such communication system.
Analysing neural systems as abstract communication systems
is likely to be possible using experimental data. By collecting
data from many interconnected neurons using high resolution
optical imaging or high resolution electrode recordings will
allow to extract typical activity patterns of neurons of the
analysed system and to construct conditional continuation
rules required for the communication system description of
the neural system. These data can be complemented by
measurements of neuromodulators and their effect on activity
patterns of individual neurons, providing information about
the environmental constraints of the neural system. The
determined rules and constraints can be used to compute
predictions about the behaviour of the system, which can be
checked experimentally by further recordings. In this way we
can build testable models of small neural systems using the
framework of abstract communication systems. Such
descriptive and predictive models of small networks can be
extended to larger networks by making high precision large
scale measurements of these latter systems.
V. CONCLUSIONS
Appropriate models and interpretational frameworks are
needed to understand how neural systems work. Here we
propose a new such framework that follows the works on
pattern languages applied in the context of neural
systems. According to our proposal neural systems
provide the substrate for several neural communication
systems characterized by a set of continuation rules which
determine which spike sequence patterns follow other
spike sequence patterns. In the proposed context the
environmental constraints select the active subset of such
rules which produce spiking activity compatible with the
constraints.
The neural communication systems framework makes
possible to build new and relatively simple interpretations
of small but complex neural systems (e.g., STG)
including also the role of neuromodulation. It is also
likely that the available high resolution experimental
neural activity data can be used to build explanatory and
predictive models of small neural systems that can lead to
testable predictions and allow validation of
communication system descriptions of such neural
systems.
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