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. REFERENCES [1] P. Andras, “The Sierpinski brain”, In Proceedings of the International Joint Conference on Neural Networks 2001, pp.654659, 2001. [2] P. Andras, “Understanding Neural Computation in Terms of Pattern Languages”, In Proceedings of the International Joint Conference on Neural Networks 2003, pp. 1385-1390. [3] P. Andras, “A Model for Emergent Complex Order in Small Neural Networks”, Journal of Integrative Neuroscience, vol.2., pp. 55-70, 2003. [4] A.K. Engel and W. Singer, “Temporal binding and the neural correlates of sensory awareness”, Trends in Cognitive Sciences, vol.5, pp.16-25, 2001. [5] W.J. Freeman, “Role of chaotic dynamics in neural plasticity”, Progress in Brain Research, vol.102, pp.319-333, 1994. [6] W.J. Freeman, R. Kozma, and P.J.Werbos, “Biocomplexity: adaptive behavior in complex stochastic dynamical systems”, Biosystems, vol.59, pp.109-123., 2001. [7] A. Gelperin, “Oscillatory dynamics and information processing in olfactory systems”, The Journal of Experimental Biology, vol.202, pp.1855-1864., 1999. [8] A.P. Georgopoulos, A.B. Schwartz, and R.E.Kettber, “Neuronal population coding of movement direction”, Science, 233: 14161419., 1986. [9] S. Grossberg, “Linking laminar circuits of visual cortex to visual perception: development, grouping and attention”, Neuroscience and Biobehavioral Reviews, 25: 513-526., 2001. [10] Luhmann, N, “Social Systems”, Stanford University Press, Palo Alto, CA, 1996. [11] R.M. Harris-Warrick, E. Marder, A.I. Selverston, and M. Moulins, (Eds.), Dynamic Biological Networks. The Stomatogastric Nervous System, MIT Press, Cambridge, MA. 1992. [12] S. Haykin, Neural Networks. A Comprehensive Foundation., Englewood Cliffs, NJ: Macmillan Publishers, 1994. [13] M.P. Nusbaum and M.P.Beenhakker, “A small-systems approach to motor pattern generation”, Nature, vol. 417, pp.343-350., 2002.
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